Theory and Application of the Z-Transform Method

THEORY AND APPLICATION OF THE z-TRANSFORM METHOD I~. I. J URY,

DEPARTMENT OF ELECTRICAL ENGINEERING,

IINIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA

I{(>BERT E. KRIEGER PUBLISHING CO. IIUNTINGTON, NEW YORK

PREFACE

I luring the curly Ilfties. considerable interest and activity arose umong C'lIl!illeCrS und systems theorists in the relatively new area of discrete ,y,ICIlI theory, This activity and aroused interest were mainly motivated h)' Ihe advivcly ~Ieveloped and applied to many areas of discrete system IIII'III'Y, I hI' ,uhjec\ matter is mainly addressed to engineers and systems Ilwllli,I,; however. some of the material developed would also be of IJIII'I~",1 In applied mathematicians, Whenever it is found necessary. most III IIU' IhC'llrellls alld lemmus are proved 011 a somewhat rigorous mathe111011 h.d ha,i!>, Jlowever, to limit the text and to place morc cmphasis on IIII' .ll'l'lh'alioll!>, Ihe ri~lIr is sllllletimcs sacrificed, ('onsclluently, certain 1111'1111'11" an' tOlldled nil hricl1y without prnnl's, Cnnver~encc prnhlcms ,II" lI11t '"'I'nroll!>ly p"rs"ed, hut lIrc left til Ihe reader ror rurther study, "III 11111',1 .. I' Ih~' III1PI'll\'~'1I ll1alC'rial till' pC'rtiJlent rdi:rell~'c, arc ~'ited SIl 111.11 II ... 11';11"'1 ~'aJll'a,ily lilld Ihe ri~ol'Utis dcrivlllilllls or Ihl' thl'ory, C'h,ll'lI'l I .11"'11' ..1'" ill dl:lail thl' .:,lrallsl'lIrll1 Ihl'ury alld it-. 1IJ1l1lilka"till', ,11,,1 IIII' IIIII,," ...d ,'tl;\II .. l'ollll, I',xll'n .. ivl' Ii .. " III tlll'lIlI'lII', 01 rilles ,II" "llh"1 011'11\'1'" III 1"1"1'1111'" III ;11'l1"ailll 1IIl' n'a"I'1 With till' ,Il'taill'" II'," .. I 1111',1111'1111111 \'11

viii

PREFACE

In Chapter 2, the z-transform method is applied to the solution of linear difference or difference-differential equations. Several forms of difference equations are given with examples from network. circuit, and control theory. The problem of stability of linear discrete systems and the root distribution within the unit circle are discussed in detail in Chapter 3. This stability discussion is not yet available in any other text; it also represents some of my original work that is available only in papers. Thus the contents of this chapter could be very useful to applied mathematicians as well as to engineers and systems theorists. Further development of the z-transform theory is considered in Chapter 4, with the derivation of the convolution of the z-transform and the modified z-transform. The application of the convolution theorem. and of the two-sided z-transform to statistical study and design of discrete systems are demonstrated. The application of the convolution z-transform to the solution of certain types of nonlinear discrete systems or nonlinear difference equations is emphasized in Chapter 5. Examples of the application of this method are also described. (n Chapter 6 the periodic modes of oscillation of certain nonlinear discrete systems are emphasized and applied. The stability study of limit cycles and their identification is also included. The use of the z-transform method and its modification as applied to the approximate solution of differential equations or continuous systems is examined in Chapter 7. The methods developed are well suited for digital computer analysis. Finally, in Chapter 8 are found various examples of areas in discrete system theory. These areas include nonlinear sampled-data feedback system, discrete antenna array theory, information and filtering theory, economic systems, sequential circuits. and discrete Markov processes. The main emphasis is the application of the z-transform theory. developed in the preceding chapters, to various disciplines of discrete system theory. Extensive tables of z-transform, modified z-transforms. total square integrals, and summation of special series are in the Appendix. Furthermore, there are many problems that illustrate further the application of the z-transform theory and indicate the method of the proofs of cert:lin theorems omitted from the text. Some of the chapters have heen 11I\I~ht :It :1 ~r:ulU:ltc seminar fur a olle-lIllit. one-scmcstcr l'uursc at the Universily III' Califurnia. 'Thc pren"luisitc is all ullder,~radllatl' "(1111's,' ill lim'al' sysll'lI1 analysis. If used as a lI'xt.1 hl'lil'w thalnlllh"lI1all'rilll":1II h,'a,h"llIalc'IYl'llv,'n'" ill a tWII"Ullit, lilli' .,,'IIII,.,Il', rnlll'il' lit IIII' ,~, III I UIIIl' Ie'wl,

PREFACE

ix

The writing of this book was aided considerably by my students whose contributions and efforts are very much emphasized. To mention but a few, I would like to thank Professors M. A. Pai and S. C. Gupta, Doctors C. A. Galtieri and T. Nishimura. and Messrs. A. Chang. T. Pavlidis. and A. G. Dewey. The suggestions and valuable comments of Professor W. Kaplan have led to much improvement in the mathematics and they are greatly appreciated. Since the major part of this material grew both from my research activities and those of my students, I would like to thank the Air Force Office of Scientific Research for their generous support and interest in our group at the University of California. The patience and efforts of Mrs. L. Gilmore in typing this manuscript are gratefully appreciated and acknowledged. I wish to acknowledge the very helpful private discussions by correspondence, over many years, with my colleague Professor Va. Z. Tsypkin of the Institute of Automatics and Telemechanics in the U.S.S.R. t hope that the contents will serve the useful purpose of providing a unified and integrated transform method which can be applied extensively lind fruitfully by systems engineers and scientists for many years. /oi·hr/lory 1964

E. I.

JURY

CONTENTS

z-TRANSFORM DEfiNITION AND THEOREMS

1.1 1.2 1.3 1.4 1.5 1.6 1.7

I.K

1.'1

2

.'.1 .'.) .'. , .'..1 .'.... .',(1

.', I

Discrete Time Function and z-Transform Definitions Properties of z-i'ransforms Inverse z-Transform and Branch Points The Modified z-Transform Relationship between Laplace and z-Transforms Application to Sampled-Data Systems Mean Square Value Theorem Equivalence between Inverse Laplace and Modified z-Transforms Other Transform Methods Appendix. A Method of Determining the Coefficients of the .:-Tnlllsform Expansion .:-TRANSrORM METHOD OF SOLUTION OF LINEAR DIffERENCE EQUATIONS

LillcClr Dilference Equations with Constant Coefficients SlIlution of Difference Equations Whose Coefficients Are Pcrillllic Functions l.incClr l>illcrcncc-DiITerential Equations I )illi.'I'l'lIl·C ':'Iuatinns with Periodic Coefficients Tillll'·Var'yin~ Dillcrcncc E(llIutions '1 inl!" Varying .:-Translill'ln lind thc Systcm FUllction 1)01111'" .:-"('rallsliH·lIIaliun lind SCllutilln IIf Parlinl Dil1'erencc

2 3 9 15 20 28 29 31

36 41

45 45

48 51 57 S9 66 73

xii

CONTENTS

3

STABILITY CONSIDERATION FOR LINEAR DISCRETE SYSTEMS

79

3.1 Definition of Stability 80 3.2 "Stability Condition for Linear Time-Varying Discrete Systems 81 3.3 Tests for Stability 82 3.4 Stability Test Directly Applied in the z-Plane 84 3.5 Determinant Method 8S 3.6 Critical Stability Constraints for System Design 94 3.7 Number of Roots of a Real Polynomial Inside the Unit Circle 9S 3.8 Relationship between the Determinant Method and Hurwitz Criterion 97 3.9 Table Form 97 3.10 Division Method 109 3.1 I Aperiodicity Criterion for Linear Discrete Systems 112 3.12 Theorems Related to Stability and Number of Roots 116 Appendices l. Derivation of the Table Form of Stability 121 2. Singular Cases in Determinant and Table Forms 128 3. Summary of the Stability Criteria 136 4

4.1 4.2 4.3

5

5.1 5.2 5.3 5.4 (,

tt.! td

CONVOLUTION z-TRANSFORM

142

Complex Convolution Theorem Complex Convolution Theorem for the Modified z-Transform Applications of the Convolution Modified or z-Transform Method Appendices I. Proof of Complex Convol ution Formula 2. Derivation of Total Square Integrals Formula

142 145 147 167 168

CONVOLUTION z-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMS

174

Assumptions Convolution z-Transforms of Certain Functions Method of Solution for Secon
17S 176 177 179

I'FRI()()I(' MC)()I:S OF OS(,(I.I.ATION IN NONI.INI"AI( I>IS( 'IUTI: SYSTIMS

IKI)

I illlil ('yd,' A'lIIly .. i, of NOllli'"'''1" J)i",n'l(' Sy,II'III' Appli,"aliflllllfIJII'I'IIIIIIIIIIII'lIlnl 1'11'111111111 loSp"'"11I1' I"~ .. ,ilpl,'"

II)tl

I KI)

CONTENTS

6.3 6.4

6.S 6.6 6.7

xiii

Limitation on the Period of Limit Cycles of Relay Mode Oscillations 203 Stability Study of Limit Cycles 207 208 Forced Oscillations in Nonlinear Discrete Systems Direct z-Transform for Determining True Oscillation 213 Periodic Solution of Certain Nonlinear Difference Equations 21S

7 z-TRANSFORM METHOD IN 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

APPROXIMATION TECHNIQUES

219

A pproximation Methods Initial Conditions Nonzero Integrating Operators z-Forms and Modified z-Forms The Choice of the Sampling Period Analysis of the Error Low-Pass Transformation for z-Transforms Applications to Time-Varying Differential Equations Application to Nonlinear Differential Eq"uations Other Numerical Techniques

219 223 229 231 238 239 240 241 243 246

H APPLICATIONS TO VARIOUS H.I 1<'2 IU

X.4 X.5 X.(I

AREAS OF SYSTEM THEORY

248

Nonlinear Sampled-Data Feedback Systems Analysis of Discrete Antenna Array by z-Transform Method Application to Information and filtering Theory z-Transform Method Applied to Problems of Economics Linear Sequential Circuits Application to Discrete Markov Processes

248 254 258 263 267 270

APPENDIX Tahle I z-Transform Pairs Tahle" Pairs of Modified z-Transforms TlIhle III Total Square Integrals
x<1

278 289 297 300

0

1'1« IIII.I:MS

301

INUI·X

321

1 z-TRANSFORM DEFINITION AND THEOREMS

I he techniques of the z-transform method are not new, for they can be arlllally traced back as early as 1730 when DeMoivrc l introduced the "lIncept of the "generating function" (which is actually identical to the Iransform) to probability theory. The concept of the generating function \\;1" latcr extensively used in 1812 by Laplace 2 and others in probability Ih,'ory. In <\ much later article by H. L. Se<\I,3 <\ historical survey of the \1'." of the generating function in probability theory was presented. 1("n'llIly, the development and extensive applications of the :-transform3I ,35 ,.1\' IIIl1ch enhanced as a result of the use of digital computers in systems. I h,'''" ~ystems are referred to as discrete, because of the discrete nature of Ilw \igllals (lr information flowing in them. Thus a new discipline of ',\",1\'111 Ihc(lry is being developed, to be known as discrete system theory. Ilw matl'rial here is devoted for the most part to discussing the various 1"n·l .. III' this discrete theory. I hi' -I ransfnrm method constitutes one of the transform methods that • ,III J", applicd to thc solution of linear difference equations. It reduces II ... ',"ll1lilln~ IIl'such c{luations into those of algcbraic equations. The I "I'I,u'" Iran!ool'llrm method, which is well developed fur thc solution of .hlll-lI'nllal "'Illations and ('xtcnsively lI~l'd in Ihe Ii ICl'lIt 111'1', can he 1II,,,I,h,'dlo ,'\ll'lId il~ applkahilily tn Ilisl'I'cll' S),Sll'IIlS. SlIch IIImlilil'alions 1t,lIo' 1I",lIlh'" ill inll'milldll/'. Ih,' vadou), a~)'IIl'iakd Il'an"l'lIrlll 1l'l'ill1i111lCS \\ It II h ,IIC' h, idly di""II)'M',1 ill Ih,' lasl sl'l'lion (II' Ihi)' dlap"'1'. I h." dmph" i.. mainly .II'vlIlI'cl III IIIl' ,kwlol"III'UI III' Ilw Ilwol')' of 11,111,,1111111 ancllh., 1I11I1It1ic'cl IIall .. 1'01'111 , Many "",I'ul 1111'0,,'111 .. 1I'lah'" III Iho",,' 11;111',1',,"11', IIIC' "11111'1 cl"llv,',1 III' !oolah',1. III ;11""111'", IIlh,'1' 1111'1111'111',1111' 1I.1'""III'C'clll' Ih,' !,rllhle'lII ""l'lill" 11'11111'" III Ih, .. ,hap"'I,

2 THEORY AND APPLICATION OF THE z-TRANSFORM

1.1 Discrete time function and z-transform definitions14-16.24-26 In many discrete systems, the signals ftowing are considered at discrete values of I, usually at nT, n = 0, 1,2, ... , where T is a fixed positive number usually referred to as the sampling period. (n Fig. 1.1, a continuous function of time /(1) is shown where its values at 1 = nT are indicated. The study of such discrete systems may be carried through by using the z-transform method. This method will be extensively developed in this and other chapters with its modifications, extensions, and applications. Definition

Let T be a fixed pOSitive number (it could be taken as unity). Let/(/) be defined for this discussion for t ~ O. This case will be extended in Chapter 4 to cover values of I which are also negative. The z-transform of/(I) is the function 00

,1[f]

= ~(z) = I

f(nT)z-n,

for Izl

n=O

> R =-1 p

(1.1)

p = radius of convergence of the series

of the complex variable z. We use the symbol,1 to denote the z-transform

off Since only the values/n = /(nT) of/at nTare used, the z-transform is actually defined for the sequence {In}. (1.2)

i:

e....

n

' ...... _-

J

51::::: ...., i:~I I ;...1

~I

'I'

I HilllU 1.1

"'I'

1'1'

i: ~

.~.

4'"

i: \!!.. .~

',',.

~

I

nl~'IC'IC' 111111 "ClIlIiIl"CI"~ hllwIlClII\.

Z-TRANSFORM DEFINITION AND THEOREMS

3

The series in equation (1.1) can always be considered as a formal series to be'manipulated in certain ways and not necessarily to be summed. If/(r) has a jump discontinuity at a value nT. we shall always interpret lenT) as the limit of/(t) as 1-+ nT+, and we shall assume the existence of this limit. for n == 0, 1,2, ... for all/(/) considered. EXAMPLE

To obtain the z-transform Of/(l)

==

t, we use equation (1.1) as follows:

...,

§(z)

== 1[f) == I

f(nT)z-n

== Tz-1 + 2Tz-1 + 3Tz- 3 + ...

11-0

== Tz-J [1 + 2z-1 + 3z- 2 + ... ] ==

Tz 2' (z - 1)

for Izl

>1

(1.3)

1.2 Properties of z-transforms5 ,14l,25,26 In the following we shall show a few properties and theorems related to the z-transform. Some theorems witt be presented whose proofs could be easily obtained as an exercise in the problem section. Their use will enable us to develop the z-transform method and indicate its applications in the following chapters.

Linearity of the z-transJorm For all constants

CI

and

C2.

the following

prop~rty

holds:

...,

1(cdl

+ cd2) == ~ [cdl(nT) + cs.fa(nT)]z-1I limO 00

CO

flUO

fl=O

== C1 ~ /1(nT)z-fl + C2 I

.fa(nT)z-n (1.4)

Thus 1 is it linear operatur on the Iineur !lpace of ull ::-transformable functions /(t), (I :?: 0).

If" III 0' :4' (:), .)I{(/' '1')1·-:1:1'(:)

/(UI)I

4 THEORY AND APPLICATION OF THE z-TRANSFORM Proof'

By definition 1[f(t

+

GO

T)] =

I

f[(n

+

l)T)z-n

n-O GO

00

= zft-O I I[(n + l)T]c'"+ll = z I l(kT)z-k k-I

(1.6)

where k = n + 1. By adding and substracting /(0+) term under the summation sign of equation (1.6), we can write the summation over the range from k = 0 to k = 00. Thus 1[/(1

+ T)]" = zl~/(kT)Z-k - I(O+)J

= z[§"(z) - 1(0+)] (1.7)

Extending these procedures, we can readily obtain for any positive integer m the following results: 1[/(1

+ mT)] =

zm[§"(z) -

],1/(kT)Z-"]

(1.8)

COROLLARY: If 1[/(1)] = .F(z), then 1[/(t - nT)u(t - nT)] = Z-II§"(Z) where U(/)

= unit step function

for n = 0, 1,2, . .. (1.9)

Proof' By definition 00

.1[/(1 - nT)u(1 - nT)]

=I

[f(m - n)Tu(m - n)T]z-,n

",.-0 00

= z-n I {t[(m - n)T]u[(m - n)T]}z-''''-"' (1.10) m-O

Letting m - n = k, we obtain GO

.1[1(1 - nT)u(1 - nT)]

= Z-II I

f(kT)II(kT)z-"

It--II

= z-n I'" l(kT)z-k = z-"§"(z)

(I. J I)

"=0

The use of the shifting theorem is important in the solution of difference equations as indicated in Chapter 2. Following a similar procedure. we can easily obtain the z-transform of the forward difference as well as the backward difference as follows: .1[£\~f("T»)

I = (z - 1)":'(::) -:: '"l(: -

I)'"

I

1£\1(0'1")

(1.12)

I "

where

£\

I(" 'I') -

f( PI

+ 1)"1' - le".,.,

( I.ll)

Z-TRANSFORM DEFINITION AND THEOREMS

5

Furthermore,

I[V'r(nT)] = (I -

Z-l)k~(Z)

(1.14)

where

V/(nT)

== l(nT) - I(n -

(1.15)

I)T

Complex scale change If the %-transform of1(/) is ~(z), 1[e- 0 'l'(/)] = .~(e,.r%).

(1.16)

Proof' From the definition of the z-transform we have

=

1[e-"Y(t)]

co

I

co

e-tJnTf(nT)z-fI

== 1: f(nT)(~Tz)-n

'1-0

(1.17)

ftleO

Finite summalion1,.26,28 Tu obtain the %-transform of (1.18)

linit we define

1:" f(kT)

~ g[(nT)],

ft-l

or

g[(n - 1)T]

=I

f(kT)

(1.19)

k-O

k~O

We can write a relation between successive values of the sum by noting Ihis definition and equation (1.19),

K(nT) = g(n - I)Tu(n - I)T + l(nT) Al'plying the z-transform to this ~1(z)

SlIlving I'ur

= zl(9(z)

~q(::).

~1(::) =

eq~ation,

(1.20)

we have

+ F(z)

(1.21)

we finally oblain

J[i/(kT)] = _%_,:F(z), k

D

:: -

1

for Izl

>1

( 1.22)

It,ililll ,,,,tI./ilwll'tll,,,·.fl u l 1111111

Iht' tft'liniliun til' Ihc~-ll'Iln!ll'nrm, .#(,.) _

i 11·11

f('''/'): " -

,nOn +/(1..1 'I') +1(2~1') ... , . .

(1.2,\'

6

THEORY AND APPLICATION OF THE z-TRANSFORM

We readily notice that the initial value is obtained as: 1(0)

=. lim !F(z)

(1.24)

Iff(O) = 0, we can obtainf(T) as the lim %?(z). For the final value, let us write z-co

1[/(t

+ T) -

I(t)] = lim

..

I

+ l)T -

[/(k

l(kT)]z-t

(1.25)

""CO k-O

The transform of the left-hand side is obtained from equation (1.4) and (1.5) lli~ .. %?(z) - zf(OT) - !F(z) = lim ! [/(k I)T - l(kT)]z-1:

+

""'00 t-O

(1.26)

We now let z - I for both sides of equation (1.26), assuming the order of taking the limits may be interchanged.

lim (z - l)!F(z) - I(OT) 1~1

..

= lim ! [/(k + l)T - l(kT)] n-'cc 1::=0 = lim ([/(1 T) - I(OT)) "-CO + [/(2T) - 1(1 T)] + ... + [/(nD - I(n - l)T] + [/(n + l)T - l(nT)]} = lim [-/(OT) + fen + l)T] .. -co (1.27) = -f(OT) + f(oo)

so that we finally obtain

lim l(nT)

= lim (z -

""'00

(1.28)

l)!F(z)

1-1

,

if the limit exists.

Complex multiplication (real convolution) Iff1 andft have the z-transform !FI(Z) and .F.Jz), then !F1(z)!Fa(z) =

1~/I(kT)fl(n -

k)T]

(1.29)

Proof: By definition co

!F1(z)!F.(z)

=I

Il(kT)z-t!F1(z)

(1.30)

1:-0

but from equation (1.11) Z

.t,~.(z) -

ICI.(1 - k'l'))

11/, en

Ie, "'- O. fnf n· Ie

(Ul)

Z-TRANSFORM DEFINITION AND THEOREMS

7

Hence eo

'1(Z):F2(Z)

=I

fl(kT)I[ia(t - kT)J

It~O

eo

=

I

k~O

eo

/l(kT) I Ia(n - k)T]z-1I n~O

=! {i /l(kT)h[(n n-O

but 12 (n - k) T

= 0, for n <

(1,32)

k)T]}z-n

k~O

~erefore

k and

~l(z)j~'iz) = 1[~/1(kT)/2(n -

(1.33)

k)TJ

Complex differentiation or multiplication by I If ~(z) is the z-transform off, then

Proof'

l[tfJ = - Tz !!.. ~(z) dz By definition

(1,34)

ao

J[if]

=I

(nT)f(nT)z-n

=

eo

-TzIf(nT)[-nz-n-l]

II~

(1.35)

n~

The term in the brackets is a derivative of z-n with respect to eo

d

d

II~O

dz

dz

ao

,[if] = - Tz I f(nT) - z-n = - Tz -

I

l.

f(nT)z-fi

n-O

d = -Tz -:F(z) dz

(1.36)

Similarly. we cun write (1.37) Wllt'l'l'

k

>0

and integer

(1.38)

As II l'llrnllary In Ihis Ihcnrclll we can deduce

I "')'/(11) I )'" '.

( 1,39)

,/(: I)k

wlll"l' 11Il" rUIlction "fA I _

,=;: It (/~'Jf(:) .. _....-

nUl

1/(1/

iN lIivon by

• 1)(11"- 2) •. ,(II .,

k

-j' I)

( 1.4U)

8

THEORY AND APPLICATION OF THE z..TRANSFORM

As special cases of the preceding. 1[( -1)"n 1lt1(n - k

+

1)]

= zdltF(z) -dzlt

(1.41)

and l[n(n

+ 1)(n + 2) ... (n + k -

dltF(z)

1)/(n») = (-t)ltzlt-Itdz

(1.42)

Special summation theorem 00

If F(z) =

I

/(n1)z-", then

71-0

F(zlt) = I[V!t(n1)]

(1.43)

where [nlltl

/1(nT) =

I

/(mT),

m-O

and [n/k] denotes the largest integer in n/k. Proof' From the definition of the z..transform we can write [",it]

&

]

[nllt)

00

1 m~o /(mT) = 1:.0'1.-" ",~o /(mT) [

00

[nllt]

= ~o ~o z-,(mT)

(1.45)

Since n varies in integer values from zero to infinity and m also varies in integer values, we can write for the right-hand side of this equation,

IL!l/(mT)] == io ,,~,/(mT)z-" == l/(mT) «:

00

", .. 0

11_0

7I!ltz-

== ! /(mT)z-7Jlk 1: z-n

n

(1.46)

However, 'I.

00

2'1.-" =--,

.. ~o

for

1

'I. -

1'1.1>

1

Therefore equation (1.46) can be written

J[

[nllt)

!

f(mT)

]

== -

".-0

Z -

z

2 f(mT)z-mk = 00

1 ",-0

From equation (1.14), we have for k

:J[Vf(nT)] =

'I. -

=

z

-F(zlt) 'I. -

1

(1.47)

1

1

-.~(=)

.::

( 1.4K)

lIsing this cqulltinn in (1.47) we tinnily nhillin :J'(:') -

.JIVf.(,"nl

( 1.49)

%-TRANSFORM DEFINITION AND THEOREMS

9

This theorem is very important in obtaining the inverse z-transform of special functions of ~ (z) containing essential singularities. These functions are discussed in detail in the next section. We introduce the following additional theorems whose proofs are left as an exercise to the reader.

Differentiation with respect to second independent ('ariable

1[.! oa f(t, a)] = .! oa 9'"(%, a)

(1.50)

Second independent ('ariable limit t'alue .llim f(t. a)] = lim .'F(z. a)

(LS1)

4~tI'O

L;. ... oo

Integration wilh respect to second independent t'ariable

J[1:1f

(t, a) daJ =

f~1 :F(z, a) da

(1.52)

if the integral is finite.

1.3 Inverse ~-transform and branch points 14 •a7 The discrete functionf(t) at t = nT or f(nT) can be obtained from F(z) hy " process called the inverse z-transform. This process is symbolically delluted as (1.53)

or

where :~(z) is the ::-trnnsform f(t)t or f(nT). In the following, we discuss the several methods from which we can IIhlain /(tr1') from :~(::) or the inverse z-trunsform, 'J111' pm"c'r ,vc'r;c',I' mc·tlmt!

WIIl'II .... ( ,) is

I~iv"n

:as a funl'lillll an:llylic I'm

1:1> R (lind

lit ;: ,...

'I').

Iii,' v:III", 111'/(11 /') ,'an h,' j'l'mlily Ilhl:ainl'd as the l'IICnki,'nt III' .. " ill Ihe I"'WI'I "'I i," "'Pilll,illll (T:aylm"s ",· .. il·s) nl' :~ (~) IlS " l'ullctiull "I' .: I.

10

THEORY AND APPLICATION OF THE 'I.-TRANSFORM

From equation (I.l) it is observed that

§l(z)

= f(OT) + f(T)z-1 + .. .f(knz-k + .. .f(nnz-n + ... (1.54)

Thus it is noticed that f(nT) can be read off as the coefficient ofz- n and so can values off at other instants of time. If §l(z) is given as a ratio of two polynomials in 'I.-I, the coefficients f(On, . •. ,f(nT) are obtained as follows:

§l(z)

= Po + P1Z-1 + p"z-I + ... + P..z- n

+ qlZ- 1 + qaz-a + ... + qnz-n = f(OT) + f(1 T)Z-1 + f(2T)z-2 + ... qo

(1.55)

where

Po \

PI

= f(OT)qo

= f(l T)qo + f(OT)ql (1.56)

Pn

= f(nT)qo + f(n

- I)Tql

+ fen -

2)Tqa

+ ... + f(OT)q ..

It is also observed thatf(nT) can be obtained by a synthetic division of the numerator by the denominator.

Partial fraction expansion If SI"(z) is a rational function of 'I., analytic at a partial fraction expansion,

00,

it can be expressed by (1.57)

The inverse of this equation f(nT) can be obtained as the sum of the individual inverses obtained from the expansion, that is,

We can easily identify the inverse of a typical !F/t;(z). from tablest or power series and thus obtainf(nT).t

t Sec Appendix, Tnblc I. t A 11c:lerlllilllllll 1I11:lhllll fur IIhllllllllll: I("n fl'lllll('IIIIIIlillll (I. "1"1) i~ inlhc: Appendix !If Ihllll'llllpter.

11

Z-TRANSFORM DEFINITION AND THEOREMS

Complex integral formula We can also represent the coefficient f(nT) as a complex integral. Since $&"(z) can be regarded as a Laurent series, we can multiply ,~(z) in equation (1.54) by Z"-I and integrate around a circle r on which 1zl = Ro' Ro > R, or any simple closed path on or outside of which ~ (z) is analytic. This integral yields 2'If i times the residue of the integrand, which is, in this case, I( the coefficient of z- I. Hence

nn,

i$&"(Z)Z,.-1 dz

= f(nT)'

2'1Tj

(1.59)

or

f(nT)

= ~.1 $&"(Z)Z"-I dz, 21T} h'

(n

= 0, 1, 2, ... )

(1.60)

The contour r encloses all singularities of .~(z) as shown in Fig. 1.2. The contour integral in equation (1.60) can be evaluated when .~(z) has only isolated singularities by using Cauchy's integral formula,

f(nT)

= _1_. .1 $&"(z)zn-I dz = 2'1T)

J..

sum of the residues of §(Z)Z,.-I (1.61)

The following cases are discussed for the inversion formula. TIfE POLES OF .~(z) ARE SIMPLE.

.~(z)

Assume that

= .;f'(z)

(1.62)

~(z)

When ~ (z) has simple zeros only, the residue at a simple singularity given by lim (z - a).:F(z)z,,-1

= lim [(Z -

a) ,;fez)

:-a

: 'n

~(Z)

zn-1J

Imaginary

I'/)"~!;

of

:!i ( .. )

I It ;IIIU' U

('1111111'" 11110',:"1111111 III IIII' . plll\lll

lJ

is

(1.63)

12

THEORY AND APPLICATION OF THE Z-TRANSFORM POLES OF ~(Z) ARE NOT GIVEN IN A FACTORED FORM, BUT ARE SIMPLE.

The residue at the singularity am is

[~(z)

~I(Z) Z

7I-IJ

(1.64) :QIJ..

where

~'(z) = d~(z)

(1.65)

dz.

/F(Z) HAS MULTIPLE POLES. The residue at a kth-order pole of !F(z) is given by the following expression: residue at kth-order pole at a,

b l = _1_ d"-l k - I! d?!,-1 .~(Z)

[.~(z)(z _

Q)"Z7l-IJI

(1.66)

'_11

has essential singularitier7

In some cases when ~(z) has an essential singularity at z other than infinity, we can utilize the preceding theorems of the z-transform to obtain the inverse without power series expansion. This is best illustrated by the following examples. If !F(z) is given as ~(z) = e-Iz/zI-1/2.'

(1.67)

to obtain its inverse, that is,/(nT) = rl[~(z)J. We let ~(z) = e-z /, and """(z) = e- 1/ 2• 2 ; then by using the real convolution theorem on p. 6, n

lenT) = ,-l[~(Z)~(z)J

=rI h(kT)g[(n -

k)TJ

(1.68)

1<-0

The inverse of e-z /, can be readily obtained from the series expansion (1.69)

Furthermore, from the theorem on p. 8, we have for k I

,-I[e-l/t •

-=--

(n/21 (

J = h(nT) = 'i/ I

m"O

= 2,

1)m

2'"m!

(1.70)

Using equations (1.69) and (1.70) in equation (1.68). we obtuin (rcplucing k by s) (1.71)

Z-TRANSFORM DEFINITION AND THEOREMS

13

The expression in the bracketed term of equation (1.71) is different from zero only for s even, and then it is equal to (-I )"/2

(1.72)

2,/2(S/2)!

Putting s = 2k in equation (1.71) and using (1.72), we finally obtain for the inverse z-transform ["/21 (l)'a-J,: X,,-21t H ( ) l(nT) rl[e-Z/Ze-1/2Z'] ! = ~ k=O k! (II - 2k)! 21n! (1.73) where H" is the Hermite polynomial of degree II.

=

=

ff(:;) is irrational/unction 0/"Z"37

The inverse z-transform can be obtained either by power series expansion or the integral formula. If we use the latter form, we must be specially careful in the integration because of branch points in the integrand. As in the previous case, we shall illustrate the procedure by using the following example. Let ji"(z) be given as $(z) =

e~

b)"

(1.74)

where Ot can be any real value and assumed to be noninteger. To obtain/(nT) we use the two procedures of contour integration and power series expansion. To simplify the integration process we introduce ~l change of variable to normalize the constant b to be unity. Let ::=b.'l (1.75) Then .:F(::) becomes .'I/(b/I) =

(11; I)'"

(1.76)

The function .f:(h!l) hm; :1 hmllch cut in the !I-phme thut extends from Icrn h) minus unily liS lIhUWIl in Fig. 1.3. By lIsing c'Iulltiun (J.lIO), the inwl"sc is ~ivcll

1~1I'1') = .J

.

II;F(lII/)] ,

= 2fT.i .I,~.

I. (It:f.-..!)'',,II 1,/11 I'

/1

•

•

( 1.77)

wh,'rc Ih,' dusCo',J ,'unlnm' I' iN Ill! shllwn in Fi~. 1.3. W,' c.·un "I\\ily shllw Ihlll III Ihe Ii mil !I • n the inlc~1'II1 II 1"11 II III I Ihc "nlllll'·II·dc.' lie' /) ih /l'III, Fmlhc.·I"II1IlI"c.', Ihc intcl-Il"IIllllnnl-lll'A III 111'111 1(· .....

14 THEORY AND APPLICATION Of THE Z-TRANSFORM Imaginary y-plane

A

-1.0

Real

c o

)

z;; e-jlr

FIGURE 1.3 Contour integration enclosing branch point or S:(by).

Therefore f(nT) can be obtained by integrating around the barrier.

f(nT) = b ll . [ fO(xe- S"_ + 1"\x ..- 1e- SIIII dx 2.,,) Jl xe i'tt J

+ fl(xe Sr :

Jo

1\- x.. -1eSwn

J

xe 'lr

dX]

(1.78)

This equation can also be written as

f(nT) = b".[ fl X"- 1- 2 (1 _ x)Glelr(n-GlI dx 2.,,) Jo

-il

x,,-l-_{l - x)-e-I,,(n-Gl) dxJ

= b" sin (n - <x}." fl X"-l-«(t _ x)- dx

."

Jo

(1.79)

By utilizing the following identity, B(m. k) = r(m)r(k) _ fl xm-1(1 _ X)k-l dx rem + k) we obtain for fCnT)

Jo

lenT) = b1l sin (n - <x)." r(n - <x)r(ot + 1) ." r(n + I)

(1.80)

(1.81)

By using the identity

r(m)r(1 - m) =

-?!sm ."m

(1.82)

we finally obtain

l(nT) = b" and for

at

rCn

not + l) = + l)r(<x - n + 1)

(<x) b"

n

(1.83)

negative integer

(ex) _ "

~'('.I.. - ~) 1'("

+ 1)1'( ·"11)

( .... 1)"

(1.84)

Z-TRANSFORM DEFINITION AND THEOREMS

J5

We can also obtain J(nT) from the Taylor's series expansion of F(z) as follows:

(z ~ bJ = (1 + bz-

~(z) =

1 )-

«: 1 d"(1 + bz-I)'·I _ =Iz II~O n! (dz- 1)" .-1_

(1.85)

II

0

Equation (1.85) yields ~(z)

ao

J

11=0

n!

I -

=

ex(ex - I)(ex - 2) ... (0: - n

+I)b"z-

II

(1.86)

We know that

r(O(

+ I) = at(~ -

1')(7. - 2) ••• (ex - n

+ J). r(cx -

n+

1)

(1.87) and r(n

+ I) =

n!

(1.88)

By using equations (1.87) and (1.88) in equation (1.86).

~(z) =

(z + b, = i r(at + l)b" z J r(n + l)r(at - n + J) = I (at) b"z-" n

Z-II

""0 II~O

(1.89)

From the definition of the z-transform we readily ascertain

J(nT) = (:) b"

(1.90)

The entries of Table I in the Appendix readily yield the inverse ztransforms ofJ(nT). They have been obtained by using the theorems and properties of the z-transforms discussed in the preceding sections. 1t should be noted from the inverse theorem thatJ(nT) can be obtained from the power series expansion without having to evaluate the poles of .'F(z). This feature offers a decisive advantage over the continuous case using the Laplace transform. Thus the z-transform is used for approximating a continuous function, which will be discussed in Chapter 7.

1.4 The modified z-lmnsform lol ,2H,31,32

In nlllny applicalions of discretc systcms and pnrticularly ill Ihe IISl~ tlf in cClntrul systems, Ihe uutput hl'lwccn I hl' slllllpJinl~ instants is vl~ry impnrtant. In SlUdyill~ hyhrid systems (miKl'd dil~itllJ :lIId 111111111", sy!ih'lIls) thl' oUlpll1 iN II \·CllltiIlIlClIlS flllll·tiun IIf lin,,", 11111111111" thl' .:-trllnsfllnu IIll'lhud is IIClI 'Iuih' IIdl'llllll'" fur II nitklll "Iudy ClI IlIId, ,Ii~itlll l·nrnJllIll~rs

16

THEORY AND APPLICATION OF THE z-TRANSFORM I(t)

FIGURE 1.4 A fictitiously delayed output to scan values of {(I) other than at

I

= nT.

systems. However, the z-transform can be easily modified to cover the system behavior at all instants of time; The extension of this method is called the modified z-transform method. The modified z-transform is also important in the study of linear systems for periodic inputs, in sampled-data systems with pure delay, in limit cycle analysis of discrete systems, in the solution of difference equations whose coefficients are periodic functions (with period equal to unity), in the solution of mixed difference-differential equations, in approximation techniques for continuous systems, and in summing up of infinite convergent series. The introduction of the modified z-transform as well as its applications will be discussed both in the remainder of this chapter and in the following chapters.

Modified z-trans/orm definition l 4.?JJ To obtain the values of /(/) other than at 1 = nT, (n = 0, I, 2•... ) we can delay the function /(1) by a fictitious negative delay !l T as shown in Fig. 1.4. By letting !l vary between zero and unity we obtain all the points of /(1) for 1 = (n - A)T, n = 0, I, 2, ... and 0 ~ !l ~ I. As will be shown in Section 1.5 in order to avoid any convergence problem in the integral evaluation of tile modified z-transform from the Laplace transform of /(1), and also in order to utilize the existing extensive tables of the modified z-transform, we make the following change of variable:

A = I - m.

O~m~ I

With Ihis chllngc of varillhle. ( I -

en ."

I ., m)'I',

hl'l~lIml'S

n - n, 1.2 •... ,

n <~

,'t < I

e1. 1)1 )

z-TRANSFORM DEFINITION AND THEOREMS

17

We can also scan the values of the continuous function between the sampling instants by fictitiously advancing the functionf(t) by the amount 'YfT. such that 1

= (n + 77)T,

n = 0, 1.2, ...

( 1.92)

The preceding description of the time is utilized in some other works, thus avoiding any convergence difficulties in the integration process as will be explained in Section 1.5. The modified z-transform off is defined as follows:

.~(z, m) ~ Jm
i [f(n -

I

+ m)T]z-lI,

°

,,-0

~ m ~ I

(1.93) This definition also relates to the modified z-transform of the function .I [(1/ - I m)T). By using equation (1.9), the preceding equation can be written as

+

':F(z, m)

= Z-1 ~"" [fen + m)T)z-n.

( 1.94)

n=O

I-'rom this equation by letting m

= 0, we readily deduce

ex:

z.~(z, m)lon'o = ~ f(t1T)z-II

= .~(z)

( 1.95)

n~O

Therefore the z-transform is obtained as a special case from the modified transform. Furthermore, if f(t) has no discontinuity at the sampling illstants (or jumps), the :-transformation can also be obtained as follows: .?="(z)

= .:F(z. m)l ...

(1.96)

1

Whenf(l) has a discontinuity at the sampling instants. the lime function Il'Iatl'd to this equation yields the value at the left side of the discontinuity, Ihal i~. at 1= nT , for /I = 1,2, ' .. and the value zero at 1/ = O. This '0111 Ill' readily ascertained by noting equation (1.7). SlInilar to the .::-transform we can derive several properties of the IIIlIdilil'd ::-transform, These properties and important theorems only arc ',1;111'" in thl' ftlllowin~ disl'ussions and the proofs clIn be e~,sily derived by IIH' n'adl''', Thl' steps for the derivation follows exactly as for the Ilall~fllrlll, rlII'O/',·II/.\' ,."/,/{",, 10 1/11' I~H

IIAI

VAIlII

'11

m 'u

,>I""II.~j;l/'Il/ll.~';

IIIIIIHI M

11111./111 I 111),1 II

//Ior/i/i,'"

IIIII',J

I. ",

•

'II

I~. III'

II

I) II

]8

THEORY AND APPLICATION OF THE Z-TRANSFORM

Special case, the response over the first interval, n

lim fen ,.. ... 0

+ m)T =

O~m:sa

= 0, 0 s m s 1, is

lim z.F(z, m)

(1.98)

Z:~Q)

O:S:m:S:l

FINAL VALUE

limf(n, m)T = lim (z - l).F(z, m)

(1.99)

I~l

ft-Q)

if the limit exists. REAL TRANSLATION

1 ... [f(1 - kT)] = z-Ic.F(z, m),

1",[f(1 - aT)] = z-IJF[Z, m = .F(z, m -

where

+ I - a), a],

(1.100)

k - 0, 1,2, ...

0

~ m

a$ m~

I

$

a (1.101)

o < a < I, and zero initial conditions off

LINEARITY

where a l are constants independent of t (1.102) COMPLEX SCALE CHANGE

1 ... [e HTf(t))

= e:!:bT(m-l).F(ze±bT, m)

DIFfERENTIATION WITH RESPECT TO

1m

(1.103)

m

[1.om fen, m)TlJ = 1.om .F(z, m)

(1.J04)

MULTIPLICATION BY IT<

1 ... [11(1)]

= r[(m - 1).F1(z, m) -

z:z !Fl(Z, m)J

(1.105)

where .F1(z, m) = 1",[tk-Y(I)] and k integer larger than zero.

(1.106)

DIVISION BY 1

1 [-f(t)J m

t

= -T1 z"'-11zco z-m=( or z m ) d z +)'1m -f(t) ' 1-0 t ' 0$ m $ 1 (1.107)

INTEGRATION WITH RESPECT TO I

1m

[I ' 1

f(l) dl ..,. . T .. o . z .- I

il

o'lt(z, til) ,I",

C)

+.,.

i"'

:i>(z, m) (I",

41

( 1.1(8)

Z-TRANSFORM DEFINITION AND THEOREMS

19

SUMMATION OF SERIES

.... I/(n »-0

+ m)T == lim ~(z, m),

(l.l09)

%-1

if the sum exists. INVERSE

MODIFIED

Z-TRANSFORM.

The continuous time function

1(1)1/=
process called inverse modified z-transformation, that is, 1(')II~(n-1+"')T

== I(n, m)T == J';;,I[~(Z, m)1

(1.110)

Methods similar to those for the inverse z-transform exist for the inverse modified z-transform, namely the integral formula and the power series. The integral formula yields the time function in a closed form, 1(1)11-
~ ~(z, m)z ..

,( 2'ITJk

-l

dz, t

== (n

0 ::; m -

I

~I

+ m)T

(1.111)

where the closed contour r encloses in a counterclockwise direction the pules of .~(z, m). The power series method yields the continuous function in a piecewise form. z/F(z, m) == !oem) + II(m)z-l + h(m)z-Z + ...

+ In(m)z-" +, .. , Izl >

R

0

~ m ~

1 (1.1 12)t

where lo(m) with 0 ~ m ~ 1 represent the continuous time function in the first sampling interval, II(m) represents the same function in the second interval. and so forth. Actually. the time function is related to ./,,(m) as follows: f(t) == I(n

+ m)T == f~(m)

(1.113)

II should be noted that since m can be considered as a constant in the contour integration of equation (1.111), the same tables for the inverse ,:-transform are readily applicable to the inverse modified z-transform.

== I ..(m). To obtain the maxima minima poinls we clin differentiate the modified ::-transform with H'sp~~cl In m tn ohtuin. using the power series form. MAXIMUM OR MINIMUM POINTS OF I(t)

III'

o;;,·J'"(::. m)

Om

=.('( m) +.,"() +I'(z In;:) II

I '1""' ,:"III'III'lclIlII "'/.(111) "1'IM'III'h, I" Ihi" dllll"c'"

1m z

1

a

+ ... -+ l:,("'):: " -I-

' ., (1.114)

I~ IIhlilllll'" IIHill,: Ihl' IIc'll'llIlIlI;l1I1 funll IIiN"II""" III Ihr

20

THEORY AND APPLICATION OF THE z-TRANSFORM

If we let J~(m)

= 0, J;(m) = 0, ... , J~(m) = 0

(1.115)

the solution of this equation for 0 < m < 1 yields the maxima or minima points. The sign of the second derivative determines which points are maxima and which are minima. The preceding theorem is very important in determining the quality of response in discrete systems. MODIFIED z-TRANSFORM OF A

J",[flk)(t)]

kth

DERIVATIVE

= Tt10" amt§"(z, m),

provided that IimJ'nl(t) = 0 for

(1.116)

1-0

and

X

{z't-l[z§"(z, m) -

~>JI(m)z-"J}

(1.117)

where h is.a positive integer and lim pnl(t) = 0 for 0 ~ n ~ k - 1. 1-0

1.5 Relationship between Laplace and z-transforms 14 ,29,34 The one-sided Laplace transform of a function

F(s)

~ ~[f) =ii(t)e-"

dt,

"J"

Re [s]

is defined as follows:

> an

(1.1 18)

where aa is the abscissa of ahsolute convergence associated withJ(t). The inverse Laplace transform is represented by the following Bromwitch integral, that is,

J(t) = ~-l[F(s»)

1 i<+/
c-J
c > am for t

>0

(1.119)

We can obtain both .F(z, m) and .F(z) directly from F(s) by an integral transformation symbolically denoted as follows:

:F(:::. m) /~:

,Z",I/'"(·v)I

( 1.12()

lind (1.121)

z-TRANSFORM DEFINITION AND THEOREMS,

21

The purpose of this section is to show the preceding relationship and develop the required transformation to enable us to obtain the z-transform directly from the Laplace transform, The function f
(1.122)

a scale factor I/h is needed for such a conversion. Therefore we can II'place the sampled functionf(nT) by the impulse functionf*(t) provided Ih., s('alc factor (or pulse width) is accounted for. The definition of such all impulse function is given (assuming a full impulse occurring at t = 0+).
00

= f(t)! O(t -

f*(t)

nT)

=!

( 1.123)

l(nT) O(t - nT)

n~O

n=O

I aking the Laplace transform of this equation and noting that (~(t)

= 0,

t ¢ 0,

and

fO(t) dt

= 1, for a

111111

.'/'I~(t

.

-

kT)] =

<0
1 00

O(t - kT)e-" dt = e'- ks7 '

(1.124)

( 1.125)

IIblain

\\1'

,...(.~) ~ ,Y'[!*(t)] =

ae

2 !(nT)e- nT•

( 1.126)

n-O

II"" 1''111al ion is readily recognizcd as the ::-transform ofJ. if we replace bv IIt'IWI' we eSlablish

• I.

.'~(:) - /'.( , .\' >1 •

( I. ) 27)

'/,' III :

",

II

\\T

III~"',~

dl'I\1I1!-

~ "(1

II/')·-'~ '/'( I), WI'

I'l'adily

t'~lahli .. h Ih(~

,',,"m'd ion

I I)', I "I.' I :I1lt1 1,1]7) Iwlw('rll IIII' .'·Irall .. rmm IIlId Iht, 1111'1111'1'

11,,"',1111111

-'I )

I, ",. -

""I,~).!l '1'1(,·(111 - .'I'I/It) It"lnl

(I J.lH)

22

THEORY AND APPLICATION OF THE z-TRANSFORM

By using the convolution theorem for the Laplace transform and assuming that f( t) contains no impulses and is initiaUy zero, we obtain (1.129)

where

=

F(P)

F(s)

1 T

1-

e- •

1._" = 9'[f(t)] I.~"

(1.130)

= 1 + e-T• + e- 2T• + ' , , e- nT• + ' , ,

= ~[bT(t)],

for le-T'I

<1

(1.131)

and

For the case F(s) has only one degree higher denominator than numerator, Eq, 1.129 should be modified,40 oo

F·(s)

1 f.c+J = ~(z) 1._e = -, F(p) 21T} c-loo 1T'

1

-TC.- ) l>

e

dp

+ 1f(0+) (1.132)

The addition of !/(O+) is required in view of our definition that a full impulse occurs at t = 0+, Both equations (1.129) and (1.132) yield a relationship between ~(z) and F(s), hence we define the following transformation :40

~(z) ~ .('[F(s)]

= [~ 21T}

r+ C

/oo

JC-lae

F(p)

1-

~TC.-l>1 dp + V(O+)] eT'_~

e

(1.133)

El'O/uotion of .('[F(s)] To evaluate §(z), we assume first that F(s) has two degrees in s higher denominator than numerator; thus we can use equation (1.129), The path of integration in equation (1.129) should lie in an analytic strip which does not enclose or pass through the poles of the integrand, This is assured in view of the restriction imposed on c, In effecting the line integral, we can readily enclose in a negative sense (clockwise direction) the poles of 1/1 - (' .Tc. pI in the right half of the p-plane, or alternatively we may enclose in a positive sense the left half of t Where ,r - Rr 1.\'1. rI.,. - IIh ....·I\'1I or IIh."llIlr l'''"Y!',!:!'I!"!' til' /'In, ,,1"\,111 II!'II' "., - II, " - It I' 1/'1 .

CI""h~lllulr l'llllyr'W'lIl'"

If.. , -

IIhwi~~u

%-TRANSFORM DEFINITION AND THEOREMS

23

the plane. Because of the assumed form of F(s) the integrals on both the infinite semicircles are zero. If we integrate along the left half p-plane as shown in Fig. 1.S equation (1.129) becomes I ·~(%)I.~,7'O = -. 2fT)

t

F(p)

I- T ( ' dp 1 - e "'J)

(1.134)

If f~s) has only simple poles, the integral using Cauchy's formula yields sum of the residue of the function in the closed path, that is,

I he

~(z) = rontAor ~ residue of A(p) ~ B(p) 1 - e m",

where

F(p) = F(s)/.=» =

T

IJ

I

e- , .~e7'o

~~:~

(1.135)

(1.136)

When B(s)/.~P = 0 has simple roots only, this equation becomes (1.137) wh~'rc S •• Sz, S3, ••• , s"

B'(s,,) = dB

ds

are the simple roots of B(s) = 0, and

I

(1.138)

sa ••

WIII:re /'ls) has branch points in addition to regular singularities, the can also be obtained using equation (1.129); however. here

llallsful'ln

Imaginary .f ....... ~

II

/

jt"'''Poles 01

x X

I

....... Poles 0 1 - - -

x;'-

)(

I

p-plane

x

!\.·(II)

,, ,,'MY. \

c+jfJO x

1- ,.-T/o-p)

0,0

,.'

x

"

q

'~ .......

--- ...•.... ... /

I Ie .IIIU I \

",.Ih IIllIIh'/:..IIII ... III 1111' "'1'1111111111 Ih"" ""11\1'

24

THEORY AND APPLICATION OF THE z-TRANSFORM

,

p-plane

\

.,

\

Poles of

\

F(pJ

I I I

---.l'2 .. X

x

J-

xt T x

I

x }$.k ....

c -joo

'"

I

Real

'f

JI

FIGURE 1.6 Path of integration in the right half of the p-plane.

special care is required to evaluate the line integral. An example will be presented later. The integral in equation (1.129) can also be evaluated by the contour integration which encloses only the poles of 1/1 - e-'l'II-PI in the right half of the p-plane as shown in Fig. 1.6. Thus equation (1.129) can be represented by

F*(s)

1 , = ..2'[J*(I)] = -. 217)

F(p)

] -

I - T ( ) dp e I-P

(1.139)

The integral of equation (1.139) is equivalent to the negative sum of the residue according to Cauchy's theorem,

F*(s)

=

-(sum of the residue of the integrand at the poles enclosed) (1.140)

By evaluating the residues at the infinite poles of 1/1 - r finally obtain 1 k~oo F"'(s)=- I F(s+jkw r )

T

7'CS-PI,

we

(1.141)

k--co

where (I. 142)

If we replace in this equation ,,'1'. = :, we readily ohtain the ;:-transli,rm = Z[F(.\,)J, The'two fOI'IIlS III' ( I. I.Vi an" 1.141) arl' l'lluivalent clln he relldily verified till' hpcciflc ('""mple" III' 1-'(.\,),

,~(.;:)

Z-TRANSFORM DEFINITION AND THEOREMS

25

For F(s) has one degree in s higher denominator than numerator. the integral along the infinite semicircle in the left-half plane is no longer zero. whereas in the right-half plane it is still zero. There equation (1.132) should be used. which yields on the left-half plane the same as equation (1.135). whereas for integration in the right-half plane we obtain 1·1.·10 I

k~'"

pes) = - ~ F(s T k:-..,

+ jkwr) + U(O+)

(1.143)

It should be noted that here the infinite summation is not absolutely convergent. However. if the sum is evaluated by taking pairs of terms ~'vrresponding to equal positive and negative values of the index k. the ~llIll converges to a definite value, EXAMPLE

o.

(iiven F(s) = 1'( I - /3)SII 1",2R, ~ > f/ > O. and noninteger, we obtain ..... (.:) ~ Z[F(s)] as follows: \Ising the contour integration of equation (1.129). we can write for

,. . I .F(z)

= Z[F(s)] =

1 iC+iOC 'rJI 1I-lr(1 R) -. e p -.1:- I' dp 27TJ c-;-,; I - Z e

The integrand has a branch cut (for

~

(1.144)

not integer) in the left half of

lIu' plane. Using Cauchy's formula the integral of equation (1.144) is "lllIivalcnt 10 integration around the branch cut as shown in Fig. 1.7. lit-lin' (1.145)

c +joo

fl-plane

J' u -om

Rf'1I1

\

\ f'" 1,'·-'· \

,, ,

111,1 'itl I I

"-

..... ,

I',"h III

IIIh'I~IIII""1 ,111111111111

hllllll h 1111

26

THEORY AND APPLICATION OF THE z-TRANSFORM

Setting p = xe/'ff on r I' and p around r 0 vanishes, we have

g&"(z)

r(1 - ~}[iO e-«zxfJ-1e;flrr dx 2'TTj "" 1 - z-le- z

==

= xe -/.. on r 2' and

r(l - {J) sin {J'TTi"" e-«"xfJ - 1

0 1 - z-le- Z

'TT

= 1-

r(fJ)

i"" 0

noting that the integral

e-«Zx/J-1e-1fJrr' ] +1"" dx 0 1 - z-le- z

dx

e-a"xfJ - 1 dx

(1.146)

1 - z-l e-.,

i"

We note from integral tables that

-If'" fJ-l e Xl dx = r(p)lI>(z-l, o 1 - z- e-'"

p, (X),

{J

> 0, (X > 0

(l.I47)

This equation is also defined for {J integer. Therefore g&"(z)

= lI>(z-\ p, (X),

for (X

>0

and

P> 0

(and also integer) (1.148) It is known that the definition of lI>(z-1, {J. (X) is lI>(z-x,

p, (X) =

i

n=O (n

1

+ (X)fJ

z-n

(1.149)

which is the definition of the z-transform oif(t) = 1/(1 a > 0, and T = 1. This result is expected because

+

f(t) = g.>-l[F(s)] = g.>-l[r(l - P)S,s-le,,·] = 1/(t

(X)II, for

P> 0,

+ (X),s

for all p's different from I, 2. 3. Althougp (z - I, ~. Ct) can be represented only in a summation form, however. if Z-1 = I, the function <1>(1, P. (X) = '(P, (X) is tabulated for certain (X and po This function is referred to as generalized Riemann-zeta function. Therefore the z-transform does not exist in a closed form; 0

0

0

0

00

however. the infinite summation I I/(n ,,=0 ically for certain -x's and p's.

+ (Xt

can be calculated numer-

£L'aluation of ~(z. m) ~ Zm[F(s)]

The relationship between the Laplace transform and the modified ztransform can be readily obtained similarly to the relationship of the z-transform. From equations (I. 93) and (1.123), we can write o~(z. m)11

fT.

A P(s. m) = .:1'1/*(/. m>l '-j

:/'1/( t

'f"

III /') ,~ -,.(1)

I. 0

.-~

m ..:: (I.I 'til)

27

Z-TRANSFORM DEFINITION AND THEOREMS

This equation can also be written as S;(z, m)I"ae7" = !l'[f(t - T =e- Ts

+ mT) ~T(t -

.2"[/(t+mn8 T (t)].

T)] Osm
(1.151)

From the convolution theorem of Laplace transform this equation is clluivalent to 1 lei-loO F(p)e mpT t :!F(z, m)I •• = -. Z-l -TIs- I dp,

,.T.

27T}

1- e

0-1oe

P

o~ m < 1

(1.152)

It is noticed from the preceding that to evaluate this integral we require the change of variable parameter from m through A = I - m, so as to ICCI the term e mpT such that the integral vanishes on the left haIf infinite \cmi-circle. Furthermore, equation (1.152) constitutes the relationship between :~ (2:, m) and F( s)j thus we define -

A

~(z, m) = Zm[F(s))

t lC+i F(p)e mll'l = -. Z-l 27T} c-Joe o ",

I

,

-Tls- ,)

1- e

o~ m < t

dp,

I

(1.153)

Inlcgrating in the left-half plane and assuming F(s) has regular singularcq uation (1.153) becomes _ . F( )emllT .'1" (z m) = Z-l ~ resIdue of ---,,,-P,,-:::---: ~ m<1 , pnlraof l_e llTz- 1 '

1111''',

°

Jo'(p)

Wllt'n 1'\ Jlll'~sed

I'~s)

= A(s)/B(s)

(1.154)

has simple poles, equation (1.154) can be

as :~(z, m)

= Z-l [

N

~.

A(s)

e",snT

--" ---..,..n ·1 B'(s,,) I - e-TIS-'nl

]

z~.T"

o~ m < 1 (I.I 55)

\\ 111"11' ·\'s, .f:!,

•••

,s-" arc thc simple roots of R(s) = 0, and B'(s,,) =

~: I

,. "'n

I.dlh' II of Ihe Appcndix lists extensive forms of the modified zII,tll',l'ollll for variolls forms of (i(s) or F(.f). This lahle can also he lIsed I" "blalll :(/(.') ,:-I(i(s)llIsin~ the relalionship in clilialion (1.157). ''1110111011\ (I.I ~4) :Inti (1.155) arc :llsCl valid for", . , I, pl'llVilkti /-'(.1') h,I·. 1"01 dITI'I'I'~ In .\' highl'" tll'lIolllillalnr Ihall 11111111'1'01101'. Il0wl'wl'. if II I'

Jril·.

olliv lillI'

tll'/~I't'I'

in

,I'

high,'!, tll'lIolllillalor Ihall 1I1111H'1':t11l1'.

''III,SIl'"l'' (I 1'1·1) :11111 (1,1 "'I) yidd rill' Ilwi!' inwl~I's, if tli.,nlllll1l1lilil·S I \1',1. IIII' "lilli,,,, III 1111' "'1'1 .. illl· "I' 1111' di~I·ClIllillllilil·'. Thr vllltll' al I 0 'lllllllid Ill' III~I'II a', 1111 -. \I .111111 .... /1·111

28

THEORY AND APPLICATION OF THE Z-TRANSfORM

Evaluating the integral of equation (1.152) in the right half of the p-plane, we (with due care for convergence) obtain similar to equation (1.141) the infinite series form F*(s, m) ~ ,!R[/*(r, m)]

=! k~OO F(S + jk 21T)e-l'+I.lZw,nHI-mIT, Tk.-oo

T

O~

m

< It

(1.156)

If F(s) is of two degrees higher denominator than numerator, this equation can be extended for m = I. The z-transform can be obtained as a special case from the preceding by noting .~(z) = z!F(z, m)lm_o

(I.157)

and for /(0+) = 0 .~(z) = ~(z, m)lm_l

is,

(1.1 58)

The conditions for obtaining F(s) knowing either :F(z, m) or !F(z),that Z.;;1, Z-I,are discussed in Chapter 4.

].6 Application to sampled-data systems14 - 16 ,ZO One of the basic engineering applications of the z-transform theory is in the field of sampled-data or digital control systems. Such systems are used more and more often in modern technology. We shall briefly obtain the z-transforms of certain systems configurations. I. Let the sampled data systems be presented as shown in Fig. 1.8. The z-transform of the output is readily obtained by applying the z-transform to C(s) as follows: C(s) = E*(s)G(s)

(1.159)

Tv

oE~)

,... .........,

X~ FIGURE 1.8

I

A

I

0

'8(z)

Cis). '8 (z. m)

A sampled-data system.

t It should be nOled Ihlll ir we lei", this equation should he IlIl1llilird us 1"·f.I, -

~

G(s}

---0

I

,"

==

, ~ I' I. ".

I inside the SIInlJ11illilln lind before slimming, (

.f

J.,,)

I II.,.

I 6,(11".

Z-TRANSFORM DEFINITION AND TflEOREMS

29

T .,..../~ - - - < ) 'If(l)

R(s)

I

+

(.'(s). ~ (z. m)

I

FIGURE 1.9

A sampled-data feedback control system,

or 1"(,(:;) ,;1 Z[C(s)]

= 1,"(=Y1(z),

where t(::)

=

£*(s)l. r- I hI!

( 1.160)

lhe modified z-transform of equation (I.J 59) is 'fi(.:, 11/) ~ Z",[C(s)]

= t(z)~(z, m)

(1.161)

2, l.et the sampled-data feedback system be presented as shown in 1,9, The z-transform of the output transform ('(s) is given as

li~,

1"6'(.:) ~ Z[CCs)] 0(::)

=

Z[£(s»)

= Z[£*(s)G(s)) = =

l,""(zY-1(z)

= Z[R(s) x G(s)H(s)] = ,'1I(z) - t

Z[R(s) - C(s)H(s»)

(1.162) £*(s) (zY1l ..*' (.:)

(1.163)

SlIhstituting the preceding in equation (1.162), we obtain f(,(Z)

= ;7f(z)

:-1(z)

1

(1.164)

+ ,;r.'~(z)

Similarly, for the modified z-transform of the output we obtain ft.'( , :,'" )

=

"Y [r») ,,"'n \. (s

=

';/J()

,'71 Z

~(z"

1+

m)

,*,'~(z)

( 1.165)

W,' I'an alslI ohtain Ihe syslem transfer function of any configuration III ·.al1lpl~d-data or di~ital control system, In later chapters some of these W·.II·I ....

will

h,l' .. I udicd

in nHlI'I.' detail.

1.7 Mcan stillarc valuc Ihcllrcm I4 .:17 .:lH thl'lIn'lll with it, l·\tl'n~ill'" is wry tlwflll in til\' ,tlldy uf III parti"lIlar. till' 1\I('ail "Ilian' valtll' III' tla,' "lIl1lillllllllS "11111 III ',nlllpkd 1IIIIn \'111111111 ,y,h'IlI' 1'1111 hI' n':ulily IIhlailll'll. IIII'

1lIlIlIwilll~

tll'.III'II' W',II'II'"

· 30

THEORY AND APPLICATION OF THE z-TRANSFORM

THEOREM:

(1.166)

Proof:

From the infinite series form of the z-transform, we can write the following relations: 1 "'QOO Z[G(s}] = - 2 G(s + jkw,} + !g(O+} (t.t67) T",--oo

1 Z[G(s}G( -s)] =....:.

T

"'.I -00 G(s + jkw,}G( -s -

"'--00

jkw,)

(J.168)

Letting s - jw in this expression and noting that G(jw) is the conjugate of (I( -jw), we find that 1 Z[G(s}G( -S»)._I", = - 2 IG(jw + jkw,W (1.169) T "'--00

'=00

From the infinite series form of the modified z-transform, the following can be written: . G*(jw, m)

=.!.

"ioo G(jw + jnw,)e-m-m)(",+"""IT,

Tn--oo

o~ m < 1

Let

(1.170) (1.171)

Then G*(jw, m}

=! "'ioo G{jw + jkw,}eifl-m)(",tk""IT, T",--oo

o~ m < 1

(1.172)

Multiply equation (1.170) by (1.172) and note that since G(jw) is the transform of a real valued function, ~(jw)

= G( -

(1.173)

jw)

Thus

1 ["_00 IG*{jw, m)1 2 ="2 2 IGUw T "--00 n=eQ)

1c-co

+ 2 k-2 tl·

'f!

+ jnw,W1

IIlk

G{jw

+ jnflJ,)

fl.

X (;( -jill - jklfl,k

Ilk

,,1,.. ,11

"'1'1'1.

()<"'~I

(1.174)

Z-TRANSFORM DEFINITION AND THEOREMS

31

The mean value of the second summation term, evaluated from m = 0 to m = I, is zero since for n ~ k, e-ilrw,(l-m)T is orthogonal to e- jnw,(I-III)T with respect to the interval (0, 1) in m. Therefore 1 R=CO IG*(j(l), m)1 2 = I~(z, m)/.=e/wr I2 = I i L IG(j(l) + jnw r)1 2 T 11=-00 (1.175) By noting equations (1.169) and (1.175), the following theorem is shown: Z[G(s)G( -s)]

= T f'~(z, m)I.=,I""1 2 dm =

T

I~(z, m)I.~./Q)TI2,

o~ m ~

1 (1.176)

The following two corollaries can be shown similar to the preceding proof: Z[F(s)F( -s))

= T f.5F(Z, m).5F(z-I, m) dm,

0

~m~1 (1.177)

:llId

o~ m ~ 1

Z[F(s)H( -s») = T f.5F(Z, m)Jt'(z-l, m) dm,

(1.178)

Thc use of the extension of this theorem will be demonstrated in ('haplcr 4.

1.8 Equivalence between inverse Laplace and modified z-transforms 14 ,39 1111111 clllHition (1.111), it has been indicated that we can obtain the nllllillUoUS time function from the modified z-transform as follows:

lell, m)T ~ 1;;.l~(z, m)] = ~

r !F(z, m)z"-l dz,

27rJ )"

t

= (n

-

o~ m ~

I

+ m)T.

t,

II

(1.179)

= integer

\\ 111'11'

:1- (;;, m) 11\1

:1 J ",II(l) I :!.\:

~",[/'ls)]

(1.180)

I III Ihl'II1111 I'l', Ihe same funclion can l)l' ohtained from e1lualitlll (1.118) II ',III f'. IIIl' iIIV1'I'SC 1.11 plncl' I!'ailSI'llI'll!. lIenee we hll VI: the 1'1111 owing

hlt'l1lll y

1(1l1,

I" '1

",I',. .'" ..'/' 'II"(.~)I ....' J",'I:~(;:, IIIlI.

n':III
" - ill\t')I,I'" O.II! I)

32 THEORY AND APPLICATION OF THE z-TRANSFORM

In some practical situations the form of F(s) contains combinations of both eT • and s (mixed form), and thus the inverse Laplace transform yields the time function as an infinite series. However, by using the inverse modified z-transform, we obtain a closed form for the time function. This procedure is advantageously utilized to obtain (I) a closed form for a convergent infinite Fourier series, (2) a closed form solution for the response of a linear time invariant circuit to general periodic inputs, and (3) the steady-state response (in a closed form) of such circuits by applying the final value theorem for the modified z-transform. To explain in detail these applications, we shall discuss the following three cases. Summation of a Fourier series

Assume F(5) as follows: 1 1 (1.182) F(s) -- 1 -e-T. s + a The inverse Laplace transform is expressed as 1 iC+iGO e" for t > 0 (1.183) c(t) = ds 21Tj c-Ico (1 - e-T·)(s + a) The integrand in this equation has simple poles at $ = -a and s = ± j(27rklT), k = 0, I, 2, . . .. Evaluating the integral by the sum of the residues, we obtain e-at 1 2 GO c(t) = ++ Tk-I -I I - ~T aT X a cos (21TkIT)t + (27rk/T) sin (27rk/T)t (1.184)

+ (27rk/T)2

a2

The steady-state part of this equation is c(t)•• = ,

..!... + ~ i aT

a cos (27rk/T)t

TIt=l

a2

+ (2'1rk/T) sin (27rkIT)r + (27rk/T)2

(1.185) If the substitution t = (n - I + m)T is inserted in this equation. the expression is 1 2 ~ a cos 27rkm + (27rk/T) sin 27rkm () ( 1.186) ct.,=-+-k aT

TIt=l

a2

+ (27rklT)2

On the other hand, the modified z-trunsform of F( ...) is readily obtained as .'F(z, m)

= ":",(F(.~)J

'"

I

I::

I

~",(' ,~

II

/I

) '"

:: ,.,'

t -"",.

I

-- ',,.' l , .. t ..

..

lI<m<1

'

(1.IH7)

33

Z-TRANSFORM DEFINITION AND THEOREMS

The steady-state response is obtained by applying the final value theorem

.-1

= lim (z -

C(/)$"

I)SF(z, m) =

e

-amT -a1' ,

1- e

0<

<1

m

( 1.188)

Multiplying the numerator and denominator respectively by

e4 T ,

we get

e"TU-mJ

c( I)

."

= ---=-e"T _ I

( 1.189)

By equating the steady-state part of this equation to the steady-state part of the inverse Laplace transform, we obtain

i

a cos 27Tkm 02

k-l

+ (27Tk/T) sin 27Tkm = I eClTU-mJ _ ..!.. , + (27Tk/T)2 2 eaT - I 2a 0< m < I (1.190)

I hw, a dosed form of a convergent infinite series is obtained by the final

""I

Ill'

theorem of the modified z-transform,

Response of a linear circuit for periodic inpul 39 111'1 iVl' Ihe general equation for i(I). the current in the RL circuit shown in I 'I'. 1.1 () when e(1), the applied voltage, is the periodic saw tooth voltage "I,"" 11 in Fig. 1.11. Assume zero initial conditions, The Laplace transor the voltage £(s) is given as

""111

. _1_,_,

II \I

-

[~( I

- e·-I,,) _

(' ~. 52

S

I - (I ('

IIII'

1IIIIl'Il1

II \ I

i (t)

khe-''']

R

+ 0.4S)e-O.4.

(1.191)

',. ..

eW

Iransrorm /(s) is

n.s

I';h)

I< I 1..\

I

-"

I.

+ 0.4.~)e-O.4. 2 5 (5 + 0.5)

1 - (I .,..

FIGURE 1.10

(1.192)

w,·

;apply IllI' Illodilkd .::-lransl'ol"lll to Ihis "'I"alioll 10 () ..'i.~

_,

I " I"

I· - (I

~CI

+- 0.45),'

5 2(S

n ..\.

I 0.5)

." ( I I. Il .1\ ),' n 4.,.) ~ '" ~~( \ I 0 'I)

I

II 1')\)

34 THEORY AND APPLICATION OF THE %-TRANSFORM R= 1 ohm L- 2 henries h .. 0.4 second T= 1.0 second

.(t)

k-l

/11 ~----!_ _~--:~ __

T+h

/ InT+h ..£o_...1: __ _+_

nT

Time

FIOURE 1.11

In view of the convergence required for the modified %-transform of the last term in this equation for m < 0.4, we have to divide the response into the following two regions:

1.

o :s; m :s; 0.4

(1.194)

In this region the modified z-transform .f(%, m) is readily obtained by utilizing equation (I.IOI) as follows:

J(%, m) == _1_[(2% - 1.185) e- ml2 + m - 2J z - 1 % - 0.607 1 == - - "0(%' m)

(1.195)

z-1

The steady-state current for region 1 is

i..

== lim J o(z, m) == 2.07e- mI2 + m -

(1.J96)

2

The total response for region 1 is obtained by using the inverse modified %-transform as follows: .(n, m )T ==

I

==

i

1 -Z"-l- (2z -.

t == n - 1 + m, n == integer

2.

0.4

2)

t.185 e-mIl + m d% 2.".) r z - 1 z - 0.607 2.071e- mll + m - 2 - 0.07(0.607)n-le-mll,

0

:s; m :s; 0.4 (1.197)

:s; m :s; t

(1.198)

In this region we obtain i ..

== 0.114 e

(1.199)

m,l

;(n, m)7'- ().1I4,· 111,111 -- «(U07}").

I •• (II -

1 + /PI).

0.4 "'" ". "'" I

(1.2(JU)

Z-TRANSFORM DEFINITION AND THEOREMS

35

Therefore, by using the modified z-transform and dividing the inverse into two regions, we obtain the solution in a closed form. However, if we had applied the inverse Laplace transform to ./(s), the response would be obtained as infinite series which is difficult to plot and calculate exactly.

Steady-slale response of a linear time-invariant circuil" Consider the steady-state voltage of vo(t) across the condenser in Fig. 1. I2a for an input of the form of rectified sine wave shown in Fig. 1.12h. Vo(s) in this example equals V.o(s)

=

1 + e-'I'· _1_ e-'I's s + a

k

S2

+ (.".IT)2 1 -

=

1

J _ e-'I'.

V. (s)

(1.201)

01

where (1.202)

The steady-state voltage is given by vo•• = lim

,-I

<...

[V01(S»)

= lim

,-1

2kZm[

(s

2

+ (.".IT)J 2)(s + a)J .

(1.203) By using partial fraction expansion of + (""IT)2J(s + a) or availahle tables of modified z-transforms, Zm[l/(s2 + (.".IT)2)(s + a»), can be readily obtained. Thus the steady-state voltage is obtained as follows

1/[52

Vo••

=

[2 + Q

k

(""IT)

2J [

2e-(J~'I''I' -

I - e

(J

cos.".m

+

QT

sin .".mJ,

'IT

o~ m ~

J (1.204)

III slllllmurizing the application of the final value theorem of the IIIl11lilictl :-transform to obtain the steady-state response, the following t':llil'S arc discussed,

IWII

, - II

.x

HI

,/

'j

( ) ",ft)

110(0

uilt}

!

I

(M

(Ill

IlCilllU·

1.1'-

36

THEORY AND APPLICATION OF THE %-TRANSFORM

CASE

1 (1.205)

where F 1(s) is a rational function of s and has a final value /ls' that can be obtained from the lim sF(s). Hence the steady-state response

.-0

(J .206a) CASE

2 (1.206b)

The sum of all T" is less than T. In general, the steady-state f.s of this equation is composed of several regions depending on (T,,); thus in obtaining the modified %-transform of F3(s) the several regions should be observed. Mathematically. this indicates that care should be taken in observing the convergence of the integral along the infinite semicircle in evaluating the modified z-transform. The example on p. 33 illustrates such a case where equation (1.101) can be readily applied in calculating the modified z-transform for the various regions.

1.9 Other transform methods In this section various transform methods or operators other than the %-transform which have been used in the analysis of discrete systems are briefly discussed and enumerated. Pertinent references are given for each of these methods and the reader can pursue further study as needed. The close relationship between some of these methods and the z.tnlllsform can be readily ascertained after digesting the material of this chapter. Finally. this section will serve as an historical background to the z-transform and other associated transform methods.

Laplace-Slieltjes inlegral4.5 This transform integral is defined as

L,[f(t)] = F(e,T)

£ Loo/(t)e-,T drx(t)

( 1.2(7)

where rx(/) is the staircase function shown in Fig. 1.13. The integral in equation (1.207) cun be rewritten as

I·.r/WI =

f.· ..f(/k .. II

., ,1,,(1) •

i "

I
"

I jIll

(I ;"IIIK)

Z-TRANSfORM DEfiNITION AND THEOREMS

37

OI(t)

,..-I 3T

2T T -3T I

-TI

I

I I

I I

I

__ ...JI

FIGURE 1.13 Staircase function cr.(,).

III Ihi!; method. absolute convergence of the series is assumed, that is, coo

I

Ij(nT)z- 7I lz_ t f" ~ M

(1.209)

71-0

wh,'re M is finite. This condition is satisfied, for example, if j(t) is a 1'"lynomial, but not if j(t) = e,l. The magnitude of Iz-"I"js e-" 7'o, and II (1.2(9) holds for some a, say ao, it would hold for any a > ao. !lased on this definition. the use of Dirac (delta) function in the defini111111 the =-lransform could be bypassed and thus retain a certain rigor III Ihl' mathematics of this transform. It may be noted that the LaplaceSllc·lc.ics integral reduces to the one-sided Laplace transform when ~(I) = I.

nr

71,(, xeneraliud Laplace transformS IIII'. IIIdhnd, which has been introduced by W. M. Stone for the opera111111,11 ,,"lution of difference equutions. is defi!led as follows:

. \ or: 1-'(11) /.I/·WIJo: =f(·~) = l . "

/I

\\llh III,· l'IIIHlilinn limf{.\')

C'"

"

(1.210)

S"II

n.

The symhol H dennle" a shiflin~

.'1

"111"1,11111 1lt-lilll'lI hy IIIl' rl'ialiclll HI. I II FI·"(I.), lIy 1I"IIII~ IIII'. 11:I1I .. 1'11I1tI. Stiliit' ha" llt-wIIlP"cl all c'Xll'lI,iw 1i .. 1 III' 11'\11',1111111 l'ail .. (I"(/,) 1(,\,)1. whidl ,1It'l'I""dy u'lah'clll' IIIC' . 110111'01'''1111 111111 ..... ",n'pl 1111 1\ lill'lIlI ' - c,"'··,

38

THEORY AND APPLICATION OF THE %-TRANSFORM

;y=Ix r--

4

I

3

2

o FIGURE 1.14 A jump function.

The jump function method? This method was introduced by Gardner and Barnes and is similar to the Laplace-Stieltjes integral discussed earlier. The transform in this case is deli ned as yes)

6

(00

== 9"[Iy(x)] == Jo

(1.211)

Iy(x)e-·:e dx

where Iy(x) denotes the jump function of y(x). If y(x) function of y(:c) is as shown in Fig. 1.14 and gives Iy(:c)

==

== :c,

the jump

00

I:c

== Irp(:c -

(1.212)

r)

r-l

where p(:c) denotes a unit pulse of duration one, that is, p(:c) = u(:c) - u(:c - I)

(1.213)

where u(:c) is the unit step function. In general, the jump function Iy(:c) can be written as

00 Iy(:c) = !y(r)p(:I: - r). r-O

where p(:I: - r) £, u(:c - r) - u(:c - r - J) (1.214)

Dirichlet series transform8 This transform has been proposed by T. Fort and has the following definition: f(.~)

.£

'r,

/)[,,(1)) ... ~ m "tI(I),

,·-0

fur

III

>

I,!I

> I,

I -

intcoller

(1.21~)

z-TRANSFORM DEFINITION AND THEOREMS

39

It is of interest to note that this transform reduces to the z-transform when m = e, 1 = nT, that is,

..,

/(s)

=I

a(nT)z-II £ ~(z)I,_.'"

.. -0

= J[a(nT)]

(1.216)

This change of notation could also be performed without influencing the application of the method.

Discrele Laplace Iransjorm 9 •41 The discrete Laplace transformation, which is denoted by the operator symbol ~, is a transformation applied to the sequence (fll}' that is, 00

(1.217) ·~{fn} ~ ~ i"e-"' == F*(s) ,,=0 This transformation can also be described as a Laplace transformation applied to a certain pseudofunctionl*(t) described as 01)

1*(t)

== /(1) 2 ~(t - nT)

( 1.218)

n=9

I"hcrcfore the!!) transformation may be considered as a special case of the I apllicc transformation. The discrete Laplace transformation has been applied extensively to discrete system theory by Tsypkin and his colleagues in the U.S.S.R. More recently. Doctsch U has presented an illuminating exposition of this 1II1'Ihud and its relationship with both Laplace transform and the z1IIIIlsfunn methods.

1.ollren/-Cauchy·s transjorm lO II .... lransfurm is again related to the z-transform or the generating 'ullrlinn method and is defined as (1.219)

S"!/~ "P,II',lIm· II •44

:\ 11,.","1

III",ralll" Oil sl"llIl'lIl'l'S is 111l~ shift opcralor H, dclillCll hy Ihe FII,. Thl'lI II". 1':11" I /'711,. 111 gl'lIl'l'I1l,

,,'1,1111'11" II" I

I,. "", . - II""

(I.n!))

40 THEORY AND APPLICATION OF THE Z-TRANSFORM If u(t) is a function of the continuous variable t and if T is a given constant, we define the shift operator E here by the formula Eu(t)

= U(I + T)

(1.221)

or, more generally, ERu(t} = u(t

+ nT)

(I.222)

In applications to difference equations, the shift operator plays the same role as the differentiation operator D in differential equations. B. M. Brown has applied this method extensively to the analysis and design of discrete systems.

,- Transform method!3 This transform, which is applicable to sampled-data systems and to certain difference equations, is defined as

ya, e) = ~~Y ..+a}

'--I. , =e •-

co

=

I

y,,+&(l

+ ,)-R,

n-O

t,

0

~

e~ 1

n

= 0, I, 2, . . .

(1.223)

The inverse integral transform is given by y,,+&

=~ .1 Ya, e)(1 + ,)"-1 d" 2'1TJ Jpole8 or Ule Intesrand

o s: e s:

I,

n

= 0, I, i, ...

(1.224)

The advantage of this transform is that the Laplace transform is readily obtainable as a limiting process, that is,

Yes)

= .9'[y(t)] = r"" y(l)e-

Jo

sf

dt = lim TY(', 0)

(1.225)

7''''0

Mikusinski's operational calculus42.43 Mikusinski's operational calculus is the most general method that encompasses both the continuous and the discrete theory. The z-transform method can be obtained as a special case of a series of the translation operator defined by Mikusinski. In this operator the numerical cocmcients are the values of the function at the sampling instants and the exponents are 0, T, 2T, ... , nT. . . .. We can define for the functiun [j(t)) the operator

,"

f· = ll<'''I')/.,,'I' .. II

( 1.22f,)

41

Z-TRANSFORM DEFINITION AND THEOREMS

For IIltT we can write e-" 7\ and thus the identity between the operator f* and the z-transform can be established. The modified z-transform can be also obtained from Mikusinski's calculus by introducing a parametric operator with respect to the parameter l:!. 1 - m. This operator is defined in the whole domain 0 ~ l:!. < I, 0 ~ t < co. In concluding this section of the survey of the various transform methods, we may mention that any of the methods could be effectively used in the treatment of discrete systems. They represent one and the same thing with different form or notation. Which method is preferred depends on the ease and familiarity to the reader.

=

Appendix. A method of determining the coefficients of the z-transform expansion 9 •47 From equations (1.55) and (1.56), (letting qo = I). we obtain the time l'ocfficients f(11 T) as follows:

f(OT) = Po f(l T) = Pl - f(OT)ql f(2T) = P2 - '1./(1 T) - qd(OT)

f(nT)

=

L" f[(n

Pn -

(I)

- i)T]q;

;~1

hjuation (I) can be written in a determinant form as follows:

o ql

=

f(nT)

,llId

110 n

0

q2

o

ql

q" q" ~-=

0

for

11 ~

I

(2)

1

Pn.

Iqllalillll (2) is readily used ror f,,(m) by suhslitliling for I'll' Jl,,(m). 11I1I11I'11II1l1l',llll'l'lIl'l1kit'III'/;"A(k>- I)t'all hcoblaillcd as

III

"

I

~

A

,

I

I;, ,A

,If,

(.1 )

42

THEORY AND APPLICATION OF THE %-TRANSFORM REFERENCES

1. DeMoivre, Miscellanes Analytica de Seriebus et Quatratoris, London, 1730. 2. Laplace, P. S., Theorie Analytique de Probabilities, Part I: Du Calcul des Functions Generatrices, Paris, 1812. 3. Seal, H. L., "The Historical Development of the Use of Generating Functions in Probability Theory," Mitteilungen der Jlereinigung Schweizerincher Verischerungs Mathematiker, Bern, Switzerland, Vol. 49, 1949, pp. 209-228. 4. Widder, D. V., The Laplace Transform, Princeton University Press, Princeton, New Jersey, 1946. S. Helm, H. A., "The z-Transformation," B.S.T. Journal, Vol. 38, No.1, 19S6, pp. 177-196. 6. Stone, W. M., "A List of Generalized Laplace Transforms," Iowa State College Journal of Science, Vol. 22,1948, p. 21S. 7. Gardner, M. F., and J. L. Barnes, Transients in Linear Systems, John Wiley and Sons, ~ew York, 1942, Chapter 9. 8. Fort, T., "Linear Difference Equations and the Dirichlet Series Transform," Am. Math. Monthly, Vol. 62, No.9, 1955, p. 241. 9. Tsypkin, Y. Z., Theory of Pulse Systems, State Press for Physics and Mathematical Literature, Moscow, 19S8 (in Russian). 10. Ku, Y. H., and A. A. Wolf, "Laurent-Cauchy Transform for Analysis of Linear Systems Described by Differential-Difference and Sum Equations," Proc. IRE, Vol. 48, May 1960, pp. 923-931. II. Brown, B. M., "Application of Operational Methods to Sampling and Interpolating Systems," IFAC Proc., Vol. 3, pp. 1272-1276. 12. Van Der Pol, Balth, and H. Bremmer, Operational Calculus Based on Two-Sided Laplace Integral, Cambridge University Press, London, 1950. 13. TsChauner, J., Einfiihrung in die Theorie der Abtastsysteme, Verlag R. Oldenbourg, Munich,1960. 14. Jury, E. r.. Sampled-Data Control Systems. John Wiley and Sons, New York, 19S8. IS. Ragauini, J. R., and G. F. Franklin, Sampled-Data Control Systems, McGrawHill Book Co., New York, 1959. 16. Tou, J. T., Digital and Sampled-Data Control Systems, McGraw-Hili Book Co., New York, 1959. 17. Jury, E. I., and F. J. Mullin, "A Note on the Operational Solution of Linear Difference Equations," J.F.I., Vol. 266, No.3, September 1958, pp. 189-205. 18. Neufeld, J., "On the Operational Solution of Linear Mixed Difference Differential Equations," Cambridge Phil. Soc., Vol. 30, 1934, p. 289. 19. Bridgland, T. F., Jr., "A Linear Algebraic Formulation of the Theory of SampledData Control," J. Soc. Indust. Appl. Math., Vol. 7, No.4, December 1959, p. 431. 20. James H., N. Nichols, and R. Phillips, Theory of SerIH)meC'hani.fnrs, M.I.T. Rad. Lab. Series No. 2S, McGraw-Hili Book Co., Ncw York, 1947, Ch. S. 21. Meschkowski, H., Di,ffc',,'"z,'ng',';c'I""!I{I'", (illttcngcn Vllndenhucsch 111111 Ruprecht, Basclund Stuttgllrt, JlJ~'). 22. M."l1tel, I'., 1.1'e"III.r .mr 11'.1' 1'lIri~, 1·'~7.

H"I'/IIUII,'I','

rt 1.l'lIf.r AI'I,III·lItf",I.f, (illUlhicl' Villllrll,

%-TRANSFORM DEFINITION AND THEOREMS

43

23. Miller, K. S., An Introduction to the Calculus of Finite DiJference and Differential Equations. Henry Holt and Co., New York, 1960. 24. Kaplan, W., Operational Methods for Linear Systems, Addison-Wesley Publishing Co., Reading. Massachusetts, 1962. 25. Aseltine, John A., Transform Method in Linear Systems, McGraw-Hili Book Co., New York, 1958. 26. Cheng, D. K., Analysis of Linear Systems, Addison-Wesley Publishing Co., Reading, Massachusetts. 1959. 27. Jury. E. r.. "Contribution to the Modified z-Transform Theory," J. Frankli" Itlsl .• Vol. 270, No.2, 1960, pp. 114-124. 28. Tsypkin, Y. Z., DiJferenzenleichungen der Impuls-und Regeltchnik, Veb Verlag Technik. Berlin, 1956. 29. Doetsch. G .• Handbuch dl.'r Laplace Transformation, Birkhauser Verlag, Basel and Stuttgart, 1956. JO. Friedland. B., "Theory of Time-Varying Sampled-Data Systems." Dept. of Electric;11 Engineering Technical Report T-19/B. Columbia University, Ncw York. 1956. .11. Barker, R. H., "The Pulse Transfer Function and its Application to Sampling Scrvo Systems." PrOf. 1.£,£,. (London), Vol. 99. Part IV. 1952. pp. 202-317. 12. Lago. G. V.• "Additions to z-Transformation Theory for Sampled-Data Systems." "'rtll/.~. A.I.E.E., Vol. 74, Part II. 1955. pp. 403-408. 1.1. Truxal, J. G., Automatic Feedback System Synthesi.s, McGraw-Hill Book Co., New York, 1955. 1.\. I.awden. D. F.• "A General Theory of Sampling Servo Systems." PrOf. I.E.E. II "mlon). Vol. 98. Part IV, 1951, pp. 31-36. 1\. Ragazzini. J. R., and L. A. Zadeh. "Analysis of Sampled-Data Systems." Trans. ·I.I.!:'./:·.• Vol. 71. ParI II. 1952. pp. 225-232. i lit. l'l'ccman, H.• and O. Lowenschuss. "Bibli6graphy of Sampled-Data Control S~'~ICIllS ami :-Transform Applications," I.R.E. Trans. IAulomalic Control (PGAC). . Mardi 1958. pp. 28-30. t I JIII'Y, F. 1.. and C. A. Gaitieri, "A Note on thefnvenk z-Transform." I.R.E. Trans. l1l1'rJI:r (Ctlrrl'sp(mc/{'I/('e), Vol. CT-9, Septe r 1961, PP', 371-374. tK. Mllri. M.• "Statistical Treatment of Sampled- ala Control Systems for Actual 1(.111<111111 Inputs," Trun.~. ASME, Vol. 80. No.2, February 1958. N . .Iury. E. 1., "A Notc on the Steady-State Respon~ of Linear Time-Invariant Sys"·111·.... I.R.I,·. Prllc. «·tll'rI'.rptlllt/I'l/fl.'), Vol. 48, No.5, May 1960. I 'II'/lil

·111 Wi1t~, (', 11.. 1'l'il/rip"·.I· tlf FC'l'db(lck etlllimi. Addison-Wesley Publishing Co. 1I.·.ulilig. M;I~sachllsc\ls. 11)(,0 (Appcmlix n. p. 261). ·11 t 01

I )OIcI,,:h. (;uslav, (;lIid.· 111111.· Al'l'li('(llitllu ..rl.(ll'lm·,· Tr(l/I.rji'rm, D. Viln Nostrand 01'1(.1. (·h. 4.

I,' l\'ltl.u,ill,J..i •.Iall. 01",,,,,'i,,,,,,1 (,,,1"11111,,. i>cl'J:alllnn

I'rcs~,

Ncw Yllrk. V.. 1. K, 1')5').

II 1'.1'."'111. Iialln·hm·. "lh'lallll/l,hip 1ll'IWC\'1I MiJ..u~ill\J..i·s 0p"rauIIlIal ('"I,'ullls alld lilt' 1\1."hll.·.1 11.",,1 .."11." Ma''''1 IIfS.-i,·tIl·,· I'I"i",·I. ''''1'1. .. I' I .I .• tllllVl't',ily Clf t ,lhlllllll;l. "'·It.,·I,·y ..hllll· 1'1,.. (111111"1 ',up"lvhlllll .. llh,· :lIl1hlll). +I

I It

lit "" II. II 1\1. /I,.. 1111111"'/11,,1,...,1 1"" I. t h I I

\\ \ til".

n"·,,, I' ,./1 ",,',1/ ,\"·,It,·"". I.. hll Wlh-y .111.1 S....•••

44

THEORY AND APPLICATION OF THE %-TRANSFORM

4S. Volgin, L. N., Elements of the Theory of Control Machines, Soviet Series, Moscow, 1962. 46. Kuo, B. C., Analysis and Synthesis of Sampled-Dala Control Systems, PrenticeHall, Englewood Cliffs, New Jersey, 1963. 47. Tsypkin, Y. Z., Theory of Linear Sampled-DQtQ Systems, State Press for Physics and Mathematical Literature, Moscow, 1963 (in Russian). 48. Tamburelli, 0., Alternate;:..Transform, Alta Frequenza, Vol. 30. No. lQ, October 1961, pp.74S-7S0.

2 z-TRANSFORM METHOD OF SOLUTrON OF LINEAR DIFFERENCE EQUATIONS

The field of linear difference equations has been studied extensively by many mathematicians in the last century or more. Several excellent textbooks! 8 have been written on this subject that can be studied by the interested reader. Several methods exist for solution of difference equations such as the classical method, the matrix method, the recurrence method, and the transform methods. In this chapter we shall apply the :-transform method (or widely known among mathematicians as the !tenerating function method) for the solution of only certain types of linear difference equations. The formulation of the difference equation can be expressed in several forms such as the backward and forward dilTerence method or the translational form. The theorems developed in the preceding chapter will be readily applied to any of the several forms used in the mathematical formulation of these equations. Several examples l'nlln network and system theory will be presented and solved, using the ,lilTerence equations as the basis of description. Although this chapter is necessarily a brief study of difference equations, ",'vera I types of linear difference or difference-differential equations, usually encountered in physical. economic. and physiological systems, will he studied. This study will be based primarily (although other IIll'thmlli exist) Oil the ::-trunsl'orm or the modified :-lransform theory.

2.1 I.incur dilTcrcllcc c«.Juuti()ns with (·I1I1!.icll'"

Ilw

,,'OI1Slillll

rC1l1nwill~

,." ",.1'" ~

,

II

c()cllicicnts2H .:1I

,IiITcrcncc Cllllillil1l1.

,., v

I' ...... ,."

'.11" t.

(1.1 )

46

nn·:ORY AND APPLICATION OF THE Z-TRANSFORM

lind h, are constants, Y .. is the forcing function and is known for all villues of n ~ O. It is desired to find z". z" and Y,. are both functions of thr pnmmeter 1= nT. Assuming T = I in all the discussions of this dUlpter. I = n. where n is an integer equal to or greater than zero. Tilking the z-transform of equation (2.1). we obtain

tI,

(2.2) wIH~I'l' .'1'(:) ~ j[znJ. qv(z) ~ 1[Y,.], and zl' Ys are the initial conditions. SCllvill~

equation (2.2) for

:"(z) =

'W(z)

i 1-0

~(z),

we obtain

biZ; + f QiZi(I1Z/Z-/) ;-0

1-0

f b•.zi(!yr- /) ;=0

1-0

p

IQ;Zi

i-O

(2.3)

The inverse z-transformation zn A 1-1[.~(z)J can be accomplished by Cauchy's integral formula, the power series expansion, the partial II IIl·tiCln expnnsion, or the available table of inverse z-transforms. It is "l'plln~III frum the foregoing that initial conditions are readily inserted IIl1hlllllllicnlly ut the start, thus avoiding evaluating the integration conIIIIIlIls liS in the clnssical solution of difference equations. This fact with Iht' Wit' III' lahles of z-transform pairs indicates that this method of solution I... " ,'(,l'llIin ndv~\Otages over the other methods. Thl" l"xtellsinn of the z-transform to solution of simultaneous linear ,hllrlrlll't' "'JlUltions with constant coefficients can be performed in a 11111111111 fllshilln ns for the preceding case. ""illf~

II.IIISTR.ATIVE EXAMPLE·'

Irll"il'ed tn find the current in the nth loop i" for the ladder network IIhuWII ill H~. 2,1. Assume all the resistances except the load R£ have Ihr slime villue It II

III

lie II Ilet I I

Iluhl....... IWill It.

Z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

47

The equation for the loop with current in+ \ (2.4)

Equation 2.4 is true for any n except -I and k - I (the beginning and end loops). Equation (2.4) with end conditions is sufficient to describe the network. Applying the z-transformation to equation (2.4) and dividing by R, we obtain ,F(z) - 3:f(z) + 3z;0 + z2.1(Z} - Z2;0 - zi\ = 0 (2.5) Solving for ..f(z), we have ...f(z) = z(zio - 3;0 + i 1) Z2 3z + I

(2.6)

We can eliminate one of the initial conditions by writing the equation for the first loop of Fig. 2.1.

2Rio - Ri\ = V

(2.7)

'I. = 2''0 --RV

(2.8)

or

I'Iws .1'(.:::) becomes

fez) = z{z - [J + (V/Rio»}io z2-3z+1 h'ol11

(2.9)*

tables of inverse z-transform (Table I) we readily obtain ; .. as

1"lIows: i" = ,:y-l[..I'(Z)] = io[COSh won

I

l'US 1 /"11

nlll hI'

J (2.10)

3 2

(2.1 J)

J"5.,

(2.12)

= -

!oillh /"n :: • IIII'. 1'11",1111111

+ I - ~Rio) sinh won

2

III", Wlilh'n inn

1'111'111 ~lIilllhll'

II II

I

fill IlIhll'

,'16, 11'/ Rill) I \.

I I

rnllil'~ ;IS

48 THEORY AND APPLICATION OF THE Z-TRANSFORM

The value of io can be found by substituting equation (2.1O) into the equation for the end loop and solving for i o• The current in can also be obtained from the power series expansion of (2.9) in powers of Z-I.

2.2 Solution of difference equations whose coefficients are periodic functions ll8 ,31 In the solution oftQis type of difference equation the modified z-transform iii readily applied. Consider the following type of difference equation: p

(/

= i-O ! h(t)xnH + i-O ~ gi(t)Yn+i (n+p) s: t s: (n+P+ll

(2.13)

x{t)

I'ere /;(1) and gi(l) are arbitrary periodic functions of the independent vllriahle I with periods equal to unity. UcclIuse of the periodicity of/;(I) and gi(t), we can replace these functions hy 1;(111) and gi(m) , respectively, where m is a parameter which varies frolll zero to unity. With this change of variable equation (2.13) becomes p

x(n

(/

+ p + m) = i-O ~ };(m)xn+i + ~ gi(m)Yn+i i-O

(2.14)

I'wm Ihe definition of the modified z-transform, we have

1 + m)) ~ ,q-(z, m),

","[x(" -

.1mfx(n

+ k + m)] £

(2.15)

Zk+l[,q-(Z, m) - Ix(j + m - 1)z-/J, i-O

o~ m ~

1 (2.16)

'I'hl' lIIotlificd z-tran!.form of the difference equation in (2.14) gives

Z"'I[.,y·(Z, m) - Ix(j .

+m -

I)Z-IJ

I' 0

= i~.r.(m)Zfr(Z) - XXSZ-1J I

i R.(m):tr·:q(Z) - }::lIIZ '1,

nllih

,n.,

():5:m~1

(2.17)

I-~

• "

;n:,

IIl1tl III) lilt' Ullknown in Ihis ('tllllilion. lIt1wc"vl'r, :1'(z) ht' ,11'11'11111111',1 hy noling Ihlll :V'(:) - :n·,III)I", ,. Thlls,sllhslillllin~~ ", - I III rllllllllOIl p.1 II gavt''I Il'lIllily :n'). Tht' n"lIlliu/', ""pn" .. itlll

1'1111

1.111

Ill'

1111111,1.11111'11 III ()

Ill, whit-II

ellll

Ihl'lI ht' '1IIlvI'cI 1'111

.n . , m). Thill

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

49

would finally give ,q"(z, m) =

C?!I(Z)[ 11+1

f g1(1)zi f h(m)z' + I z,,+1 _ f h(l)z'

Q

.-0

z

]

gi(m)zi

.-0

i-O

t~o

x

[~oh(m)zi

%

ziz-i

+ i~glm)zi ~ y;Z-i]

(2.18)

The time function is obtained by the inverse modified z-transform t

= (n - J + m),

0!5:

m

!5: J (2.19)

ILLUSTRATIVE EXAMPLE

hnd ;r(t) which is a solution of the following equation. x(t)

=

:1:"+1(2.582 - 1.582e-1 - t) - x,,(J - e-I )

+ Yl/+l(e- I + t -

1)

+ 0.418Yn(1

- e-') (2.20)

Wc CCln construct the coefficients of X/t and ?Ik to be periodic functions pcriod one by replacing t in these functions by m. This gives the I .. llowing clJmuion: Wllh

x( n + J + m) = xn+ 1(2.582 - 1.582e- m

-

m)

+ ?I" I(£' + III - 1) + 0.418!1,,(1 - (' ./11), I

1\

+-

xn(l - e- m )

III

0 !5: m !5: 1

(2.2L)

111'11' III" ~'lldlkil'lIl!1 ur ."" II' ."". /1" .1_ !/" Clrc pcriodic. IIII' IIlllllilkd :-lransJ'1lI'1II ul'CtluHtion 12.21) givcs l.}I:f'(1..m)

Ix(rn 'l)+x(mh Il}

(2.~K2

1.5.'2c'

1·"'( l) xIII . Ie,. "' I '" I

1I.oIIH II

(I

1) 1.1

III

m)l,

c' "').'1"( l) .y( 1.)

,. "''''''11.1

)'11

I

c.'.J ,,)

50 THEORY AND APPLICATION OF THE Z-TRANSFORM 1.6 1.4

t

1.2 1.0

0.8

-=wO. 6 0.4

ra_

0.2

°0~~--~--~~~~--~--~~~~9

FIGURE 2.2 Particular solution for 1/.. = a unit step.

To obtain ~(%). we replace m = 1 in equation (2.22) to obtain

~(%)

= (0.368% + 0.264)tzV(z) + 222:0 + %(2:1 Zl -

21

+ 0.632

Zl -

21

2:0 - 0.368yo)

+ 0.632 (2.23)

If we assume y,. = unit step (Yo = I) and 2:0 = 0, we readily obtain from equation (2.21), by letting n = -1, m I, the value 2:1 = 0.368. Therefore ~(z) becomes

=

o'!'(%) =

(0.36821

+ 0.264)% Z + 0.632)

(2.24)

(z - 1)(21· -

Substituting (2.24) in equation (2.22) and noting that 2:(m - I) = O. we obtain %2[~(Z, m) - x(m)z-I] = (2.582 - l.S82e- m - m)~(z) - (1 - e-m)~'(z)

+ (1

- e- m)-%%-

%2 + (e- m + m - 1 ) - 21 - 1 - (e- m + m - 1)%

1

(2.25)

The initial value 2:(m) can be obtained from equation (2.21) by letting n = -I to give x(m)losms 1

= yo(e- m + m -

1) = e-'"

+m

- I

(2.26)

Substituting equation (2.24) and (2.26) in equation (2.25) an~l. expandinl in powers of Z·". we ohtain the particulnr Rulution .t·pC z. m) "~ f()lIuw!l: z:"p(:' III) .... (I'

III

·1 III"

I):

II

-I- (O..\fIH I O.t,]2m)..:

-f (!.f"2

I

O.M1,. Ill).. • -I· . ..

(7..27)

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

51

The coefficient of Z-k represents x,,(t) for k ~ t < k + I. The final value of X,,(I) which is known to exist is obtained from applying the final value theorem to !r(z, m) to give lim x(n - t n"'''X)

+ m) =

lim (z - l)~,,(z, m) = 1

(2.270)

z-I

The particular solution is shown in Fig. 2.2.

2.3 Linear difference-differential equations5 ,36 In many applications of control systems a pure delay (or transportation lag) often appears in the controlled process. The mathematical description of such systems is carried through the use of mixed difference-differential equations. In the following discussion only linear constant coefficient equations involving one-time delay are considered. Extension to other cases can be performed using the same or other techniques. The modified z-transform of the continuous functions C(/) can be written in the following form:

z
+ c1(m)z-1 + ... co + c ..(m)z-" +... = 11-0 I c,,(m}z-n t = (n

+ m)T,

0

~ m ~

1 (2.28)

From this series expansion of H(z, m), we may note that

co(l)

= Cl(O), Cl(1) =

cz(O), e..(I) = c ..+1(O), for n = 0, I, 2, ...

for continuous outputs as assumed in this case. Using the preceding form of the modified z-transform of C(I), we can also write the transforms of the derivatives of c(l), c(l + n) with T = I, from equations (1.116) and (1.117) as

a

(2.29)

.1m[c'(t)] = am [
a

+ n)] = 'om Im[C(1 + n)] =

~ {ZII

om

l[ZW(Z. m) -'i\'I,(m)::: I'

II

pl}

(2.30)

Similm rclatitlnships could he writtcn for highcl'-urdcr derivatives. In Ihe soluliun III' dilTcrcnce-diiTcrential c'llIalillns. it is t~llnvenitml to l'IloUSt' Iht' sampling p,~rill(l as t~'1'1lI1 ICI Ih,' lin,,! tlt'lay itsdf. Thi~ will yicltillu' J"L'sJl"nM' ill II dClSt~tl UI" l'UI,,:isl~ I'urm wilh Ih,' minimulII III' lahul'. Wt' !.hall 11111,11 all' lilt, pml'I',lmt' hy sulYinl~ lilt, l'ulluwlIIl~ Iwu t'lIlIlIIplt's III ""11111.

52

THEORY AND APPLICATION OF THE z-TRANSFORM

EXAMPLE I: FIRST-oRDER EQUATION Consider the difference-differential equation

c'(t)

+ ke(t -

I)

== 0

(2.31)

The boundary condition is specified for a sampling interval; here let c(t) == Pfor 0 ~ I ~ I, where both k and Pare constants. We must also assume that there is no discontinuity at t == O. By taking the modified z-transform of equation (2.31),

that is,

~ c;f(z, m) + kz-l~(Z, m) == 0 am a z - tff(z, m) + kfif(z, m) == 0 am

(2.32)

(2.33)

This can be solved as a first-order differential equation in m, treating z as constant. The solution of equation (2.33) is fif(z, m) =

A(z)e-'k/~)m

(2.34)

where A(z) is to be determined. Now from the definition of the modified z-transform in equation (2.28) fif(z, 1)

+ coCO) =

that is, A(z)e-k/.

+ (J =

~(z,

0)

(2.35)

zA(z)

Therefore zA(z) =

p.

(2.36)

1 - z-le-kl •

Substituting in (2.34) t"A( ) ZlI>'Z,m

...:.P--:-...,....,.... e-U:/dm = __ 1 - ,,-Ie-Ill.

Expansion of.<2.37) in powers of

Z-I ,-le/&

(2.37)

gives

COl

zfif(z, m)

= (Je-Ikldm 1: z-fle-Ikls)n ft.O

=

PI z-fte-Ikls)'n+ml COl

(2.38)

n-O

The coefficient of z-n will give the respon5c (',,(m) and if we cVlllulltC this for all n we get the cnmplclc rcspnnllc. UI(.. z·.,

M.

" .. t" ) - I'11-0 ... M

","~

v~n

I-(Ie/:)(II f·",W

III

(2.11)

Z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

53

This can be simplified as f'L7(

Z'l)'

z, m)

~ Z-/~ = (J £.. £..

[-k(/

".0

1-0

+m-

p)]p

(2.40)

p!

The coefficient of Z-I is ez(m) = {J

±

(-k)"(/

'P~O

+m-

p)'P

(2.41)

p!

Equation (2.41) is useful if one is interested in the response for large intervals of time. Generally, the process of obtaining a general time function such as (2.41) in a closed forin is very difficult if not impossible. In many engineering problems, we are interested only in the transient response. This can be obtained by actually expanding equation (2.38) and identifying the coefficients of Z-I, Z-2, •.• , etc. This approach is illustrated in the following example. The coefficients of Z-I, Z-2, •.• will give the responses for the 1st, 2nd, ... sampling intervals in terms of m which ranges from 0 to 1. EXAMPLE

2:

FEEDBACK CONTINUOUS SYSTEM WITH TIME DELAy31

Consider the feedback system shown in Fig. 2.3. The forward transfer function is G(s)e-~~ = [(s + 0.3)/S2] e-'. The overall transfer function can he written as c(.~)

R(s)

+ 0.3)e-· + (s + 0.3)e-·

(s

-= S2

(2.42)

Alllwugh we can obtain the modified z-transform of this equation directly, however. for sake of illustration we formulate first the difference equation. The difference equation is. for r( t) = tu( t). c"(t}

+ e'(1 -

1)

+ 0.3c(l -

1)

=

+ 0.3r(1 J) + 0.3r(1 -

r'(t - 1)

= U(I -

J) J) (2.43)

I 'IlIlltion (2.43) is also equivalent to e"H(t

wllh

Ih~·

+

I)

+ r'(I) + 0.3c(r) =

U(I)

+ 0.3r(t)

(2.44)

inlli"r conditions specified as ('(I) = 0, (·'(t)

=0

for 0 ~ r ~ 1

(2.45) e(t)

I I( IIIIU ], I

('lIl1liIlIlIlU~ It-I"'dhll~'k

s'yslem wilh p"re ddllY.

54 Tlllln"V AND A.'LIC'ATION 01' 'H' ..'UNIPOI.. Taking the mmliflcd ,·trlUllIl'nrm ul' (1.44) z

a' ~(z, m) + ama 'l(z, m) + 0.3'.-(a, ttl) amI = _1_ + 0.3[ mT + __ T_ ..] %-

1

%-

1

(% -

1)1

(2.46)

Treating z as constant, (2.46) is solved as a differential equation in m, land noting that T the sampling period = I, we obtain

~(z , m)

= A1e'lm + AI e..... + (z _1 1)1 + (%_m_ _ 1)

('2.47)

where Al and AI are functions of % and $1> Sa are roots of the characteristic equation . -1 ± I - 1.2% ZSI + s + 0.3 = 0, that IS, S1,I = 2% (2.48)

J

Again from the definition of the modified z-transform we have tCf(z, m)I"'=1

= [ztCf(z, m) -

(2.49)

co(m)]I",_o

and

~ am <;f(z, m)1

== ..!. [ztCf(z, m) -

am

m-l

co(m)]1

(2.50) m-O

provided that C'(I) is continuous at t = 1,2, .... Substituting (2.47) in (2.49) and solving for Al and As yields

1

ZAI = - - - - - , - - - (SI - S.)(Z-l e'l - 1)

(2.51)

-1 ---~--­ (SI - S.)(%-l e,· - 1)

(2.52)

zAa =

The complete solution for H(z, m) is z~(z

,

m) ==

1 1 e'I'" - - - - - - , - - - (SI - SI)(%-le'l - 1) (SI - SI)(%-l e,· - 1) X

e,.m +

Z

(z - 1)1

+~

(z - 1)

,/

(2.53)

I Expanding equation (2.53) in negative powers of z and identifying the coefficients of z-l, %-2•••• gives the system response for all time. Although this looks formidable, considerable simplification will result if we are

..1'IAN.'UIM ICILllTION

1111..',.1,&1 lit Ih...."nll'lO nilly :f.'C:,O) -

=

.:

-

(z -

1)1

= 1,'l'Il1n

In.umt., lilt Ihlll

I

--, I)

_1_[ I z-ne,·n -l: z-ne,tn] SI -

S8 n-O

- --

(z - 1)1

"

(sa - ,Y,)(: ',"\ -

1

Z

=

I.INIIAI 1'"'I'II.IINe'• ..,LlATIUNIl

I" •.unpltll",

'I'

(: -, I)·

CJ.

SI -

z ~. (z - 1)1 ".1

S8

"z

-ft

I

CICo

n-O

(z-nens, - z-nen'l )

n~1

e"" - en•• SI - sa

(2.54)

tables53 (2.55)

where SI,I

=

-1 ± ../1 - 1.2z 2z

(2.56)

This reduces (2.54) to

Although (2.57) is in a sense "closed" form, it is not useful in evaluating Ihe response since one has to expand the infinite product to identify coefficients of Z-I, Z-I, etc. It is easier to proceed directly from (2.54) and simplify it by actual expansion.

en•• - e'lIl ---- = n SI -

Sa

nl

+-

2!

(S,

n3

+ Sa) + -

3!

2

(SI

a

+ SIS8 + Sa)

• + 4! ~ M + s~sa + SIS~ + S~) •.• . + q! -nil (SIII-I + s,11-2Sa + ... S211-1 ) + . . .

(2.58)

56 THEORY AND APPLICATION OF THE til-TRANSFORM

The polynomials in SIS. can all be calculated in terms of til and it is noted that z appears only in powers of Z-I. %~(z, 0) can then be simplified as

7

+ !!... (Z-8 7!

1.5z-5

+ 0.54z-4 -

0.027z-3)

8

- ~ (z-' - 1.8z-8 + 0.9z-5

8!

+ !!...•(Z-R _ 9!

2.1z-7

-

+ 1.35z-8 -

0.108z-4) 0.27z-&

+ 0.OO81z-4) + ...J

(2.59)

By identifying the coefficients of z-k in this expression, the response up to the fifth sampling in~tant can be obtained as c(O) = 0

c(3)

= 2.08

c(l) = 0 c(4)

c(2)

== 4.145

== 0.55 c(S)

== 5.4135

(2.60)

This result, as is to be expected, agrees with those obtained by other methods. 5O More terms in the expansion in (2.59) have to be taken to evaluate responses for longer times. In this brief discussion the usefulness of modified z-transform has been demonstrated in solving linear constant coefficient difference-differential equations. It is particularly useful in calculating transient responses of continuous systems having time delays, and this method avoids having to evaluate roots of transcendental equations, but requires manipulation of series. Although in Example 2 only the response at sampling instants has been calculated, it is, however, necessary to use the modified z-trami· form theory to arrive at the constants AI and AI' This method can he easily extended to higher-order systems huving more tlmn one but otherwise commensurahle timl' deluYII.61 • 1\ ~llCl\lld he 111111'11111111 "10(',0) I. 11111 tll'1I1I1I11I1I .,I'lh"

I

11'1111\1'111'111, 111111 i'l, '(.(~).

z-TRANSFORM SOLlJTlON OF LINEAR DIFFERENCE EQUATIONS

57

2.4 Difference equations with periodic coefficients25 ,28 Consider the following equation: Zn+l = a,.z" + Yn (2.61) is a periodic coefficient whose period is N; that is, 0t = a.\,+i for i = 0, I, 2, 3, ... , N - 1. To solve this equation we shall first form a constant coefficient difference l'(luution that describes the solution for n = kN, k = 0, 1,2, ... , and then use this result to find the solution .for all other values of n. This pmccdure, in essence, removes the periodicity of the coefficients from the analysis. I.et ;[YkNJ = Yo + YNZ-N + Y2NZ- 2N + Ya.vz- aN + ... II"

l[YkN+lJ - Yl

,)'[]/(klll.v-d

+

YN+l Z- N

+

Y2N+l Z- 2N

+ YlcNz - kN + ... + YaN+1 Z- aN + ...

= YN-l + Y2.v_1z-N + Ya,v_ 1z-2N + Y4.\,_lZ-3N + , , ,

(2.62) With this pattern for z .. we solve this periodic coefficient difference t·tlwltiun. Let n - kN so that an = 00 (that is, an = ak .,· and letting at Ihe' <;1:lrl k = 0), We can then write for equation (2.61)

+ Y"'N ,1:"',\'12 = a1z"',Hl + Ykl\'+1 = ;1'''',\'13 = a2x k.V·.,. + YkNI.2 = a"z",.\'

X",."II -

"'U't II,\'

=

a,\,

al0 rk.'i

+ aIYk.\' + Y",.\'+l + a2a'Yk.V + a2Y"',V+! + Yk.'·,,2

a2a ,aOxk.\,

+ Yllc+lJN-l = (0.\,_l all'_2' , , aO)zk," I" • a.)ll""v + (a,\·._1 .• , 0a)Y",,\, ... I ' " aa)lllt;v 12 + ' . , + 0,\, 1111A·.II," 2 + ,'lIlt Ill.\' I (2.63)

'IXI'" 111,\'·-'

+ (aN 'I-

(ll,,'

lilt' 1;I';f ('cJllatinll is aC'flmlly :a cnnstClnt eClcnicient difference cqmltion :tilt! fhe v:arinus furcing terms, The ::-tr:ansfnrlll method ,III III' u';"IIlv applic·tl. '

1I1\'1I1\,11I1~ ,"" I

'1.1', ,f·') I

(II,

I'"

·"01 . ('"

I'"

(10):7'4",'(::)

".)'.'11" 1'(") " •. ' I

II\,

I ('"

,.,. "I)!IIA.\('~)

1';'1/" III ,. ~( .• ) I ·'l/IA'II" ,(.~)

a.M)

S8

THEORY AND APPLICATION OF THE Z-TRANSFORM

or

~"N(Z) ==

N

z -

1 (aN-I' ••

ao)

[zN:co + (aN-I'

•• )q;'tNCz)

+ ...

+ QN-ll{!J(lc+lIN-a(Z) + I{!JIt+lIN-l(z»).

(2.65)

We l"eadily obtain :C"N by inverse z-transformation of this equation. Knowing :CItN we can find :CIt N+l from the first of the series of equation (2.63). Similarly, :CItN+1 is obtained from the second, and so forth. We finally obtain :CU:+lI,v.

Solution of simultaneous difference equations with periodic coefficients Consider the two following difference equations where a". b", e", and d" are periodic coefficients of period N, and P" and Q" are known for all n(n ~ 0). :C"+1 == a":C,, + b"y" + P" (2.66) (2.67) A solution for these two equations can be obtained following the procedure for a single equation; that is, the 2N equations which correspond to a complete period of the coefficients are written and each equation is substituted into the following equation, as in (2.63). This results in two constant coefficient difference equations,iWhich involve :Ck .\·, YItN and various forcing function terms. These twp equations can be simultaneously solved for the two unknowns using the!. indicated procedure of the z-transform method. \ EXAMPLE

Find :c" which is a solution of the following equation: (2.68) with at == 0.5 for i == 0, 1,4,5,8,9, ... ,and a, == -0.5 for i 10, II, ... , and y" == (..... l)". Here N == 4

==

2,3,6,7,

4

I[Y,.;v) '1 (.

== I + z-4 + z-8 + ... == ~

r'I,,~· )== -I .... :; . -

z

I

~ ..

---

I , I f · "

IlttV,_ - IIIt,v.

(2.69)

z - I

!I. v I.

-

!I•.v, I

"'"

-~

z"- 1.

(2.70) (2.71)

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

Using this procedure. we readily obtain for !l'kN(Z)

=,

% -

59

~Ii:.v(z)

1 , 0.5

X [ Z4zo

+ 0.125z' + (0.25)( -z') z'-1

z4-1

0.5z' - ~J z4-1 z4-1 (2.72)

with

Zo

= 0 fr () Ii:." Z =

(%4 _

-1.625z4 0.0625)(z' - 1)

The inverse of this equation yields 26.0 41i: Z411: = - [l -
Z4k+1

=--

%'k+2

=- -

%411:+8

= -

IS

[1 -

(2.73)

(2.74)

(2.75)

m ]+ I 4k

(2.76)

Similarly. 13.0

30

13.0

60

4k

[1 - (!) ] 4k

[l - (i) ]

!

+ 4~

(2.77) (2,78)

2.5 Time-varying difference equations7,10,23,25 111 the following section, we shall first apply the ordinary z-transform to linear difference equations with rational coefficients and then, in the next ...,·"tinn. the time-varying z-transform will be discussed and applied. The ordinary ::-trunsform reduces the solution of the difference equation to Ihl' solution of a differential equation in the variable z. By obtaining :1 (:), we then apply the inverse z-transform (or use of tables) to obtain IIII' I ime solution. A general solution of such equations exists only for IIIl' lirst order eClse, whereas higher-order equations can be solved only I". pa!'lieular cases. We shall illustrate the procedure by considering the 1.. IIC1will~ Nth-order difference equation with rational coefficients in the \ .11 ia hi,' ". 11'1 Ih,' "int'n'lIce 1:(IIHllioll oe represented as

,

t

~ (II," I "~A )/'·~l( II) •

'X(II)

(2.79)

60

THEORY AND APPLICATION OF THE z-TRANSFORM

+

where the shift operator E denotes Ef(ll) = f(n I), and g(n) is a known forcing function. Applying the z-transform to equation (2.79), we get, using equation (1.34)

(2.80) where ~(z) is a polynomial in z containing all the initial values of equation (2.79) and '~l(Z) = 1[[(n + k)] = Zk§(Z) - (initial values). Equation (2.80) can also be conveniently written as

d.F(z) dz

+ B(z) §(z) =

D(z} A(z)

A(z)

(2.81)

where .\'

Ia~+]

A(z) = -

(2.82)

k~l

.V

I

B(z) =

(ka rr

+ bk)Zk

(2.83)

k=l

and

D(z) = fI(z)

+ <8(z)

(2.84)

It may be noted that equation (2.81) is a first-order differential equation in the variable z. The solution of this equation is generally known and is given by the following:

.~(z) =

exp[-J

exp[f

B(z) dzJf D(z) A(z) A(z;) -

B(z) dzJdZ A(z)

+ K'~XP[ -

f !~:~

dZ]

(2.85)

where K is the integration constant. In general. A(z) can be represented as A(z}

=

-a.v(z - Otlyl(z -

Ot2)T, •••

m

= -a.\, II (z -

(z -

Otm)Tm

m

with I r i = N

IX./Yi,

t-l

+1

(2.86)

i-·l

Using equation (2.86), we can write

-f

B(z) dz = A(z)

-fi I td

1

t, J (j -

[d J- 1 B(z)(z - Oti)'i] 1)! dz H

A(::)

: a,

(/::

X --.-..... . (:: _ IX,)"

~ ~('/, III ", I

I' I

"I

rI.: "

, (:

1'1,)'1 ,I I

(~.H7)

z-TRANSFORM SOLUTION OF LINEAR DifFERENCE EQUATIONS

61

where

C _ II -

1 [di-l B(z)(z - «XI] (j _ I)! dzi-l A(z) '~"I

(2.88)

Performing the integration in equation (2.87), we have

:lnd

f

B(z) '" [rl-I C I ] -dz=~ ~_IJ -Kln(z-Cl) , A(z) '_I'_I,,-i(z-Cl,),1-1 I I where K, = Ciri (2.89)

B(z) J m [ f -.4(z) dz = II (z -

exp -

i~l

C·

• m rl-I

Oti)-1I:1 II

II exp

;~1 i-I

(Z -

Ii r. Ot;) ,-I

(2.90)

Where all the roots of A(z) are distinct, that is,

'I = '2 = ... =

rm

=

m=N+1

I,

(2.91)

we ubtain from equation (2.90)

[f

exp -

B(z)

]

"'+1

dz = II (z - Ot,)-r.i A(z) i~1

(2.92)

where Ci = [B(Z)(Z - Oti)] A(z) Z~"I IIII'

(2.93)

J

Ihis case, equation (2.85) reduces to ,\'+1

•:F(z) = II (z i~1

•

Oti)-(;/

D(z)·\·+1

-

.4(z)

.

II(z - Otit, 1dz

1=1

.\'+1

+ K II (z -

Oti)-C'i

(2.94)

i=1

I he inverse z-transform of equation (2.94) yields fen). the solution of IIII' '~~'lIcml Nth-order time-varying difference equation given in (2.79). II lIIay hI.' noled thut if CI is noninteger, as is often the case, .?(z) will be 1111 1IIIIIillnni function of z. In Table I of the Appendix many entries for , ( ) III' such n furm ure tabulated and thus can be conveniently used to ,,11111111 Ihl.' linal solution. nCAMI'I.I;

III 11m "X:llllple: we shall nn:lly'-c u sumpled-dutu feedback system with "II III hie' /~:lill. This syslem is shuwn in Fig. 2.4. ,0\·.·.11111111/'. I.~·I·II illilinl cnlUliliulll; :tIIt! lelting (0 .. = a. we h.lYC for the ,,\",11'111

1'(11

I I)

,,1'(11) -

I

/I

'1(" )

62 THEORY AND APPLICATION OF THE z-TRANSFORM 61(z)

C"L *' T-l

~ -j

~(z)

T-l

f--A:.-j .!:p H ..... t---ofr+-o.

c(ra)

1

0

'fI(z)

FIGURE 2.4 Time-varying sampled-data system.

and v(n)

= ~ [r(n) -

(2.96)

c(n)]

n+1

Eliminating v(n) from these two equations and denoting k

= (I

- a) kl' ex b = k - a, we have the following time-varying difference equation: (n

+ I)e(n + I) + (bn -

a)e(n) = knr(n)

(2.97)

Applying the z-transform to equation (2.97) we get using equation (1.34) z(z

+

b) d~(z) dz

+ a"'(z) =

kz d9t(z) dz

Assuming a step input. that is, 9t(z) =

b=~

f

-z- ~(z) = -k z+a (z

*' -

zdz

+ a)2(z -

1)2

(2.98)

\

1 arict for k = 2a, that is,

I

+K

(2.99)

where K is the integration constant. We now discuss the following three cases.

1. a ,t: -I, for this case (equation 2.99) yields

~(z) =

k

(I

+ a)1I

{Z(l -

a} + 2a z(z - J)

+

1- az

+ a In z -

t

z

+a

I}

z+a

+K z+a

(2.100)

z

Since c(n)ln =0 = lim 'if(z) = 0, K = O. Hence equation (2.100) becomes z-co

~(z) =

k

(I

+ a)2

a) + 2a z(z - 1)

{zC1 -

t}

+1-

a z + a In z 1+ a z z+a

The inverse of this equation yields ('(n) = (I

k

+

I

tI)

{< I -

(2.101)

tl)lI(n -

+1-

I ··f

l)

+ 2(III(n -

tI [R(II) (I

-t

2)

(111(11 .-

I

U}

(2.102)

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

63

where g(n) is defined as g(n) = 0,

=

~

for n = 0 for"

[(_a)" - 1],

n

~0

(2.103)

Equation (2.102) yields after using g(n)

= 0, c(1) = 0 k { c(n) = (1 + a) (l + a)2 c(O)

X [n(a

+ I) -

I-a 1 -- --1 + a n(n - I) I

+ (-a)"J},

for n

~2

(2.104)

2. a = -1, k = -2. For this case equation (2.99) becomes -z- <6(z) z- 1

=

-k

f

zdz

(z - I)

4

+ ko

(2.105)

= 0 = lim ~(z), we have ko = O.

Since c(O)

Equation (2.105) yields 3l- I fi'(l) =

(2.106)

k 6l(l- 1)2

Thc inverse of this equation gives e(O)

= 0,

e(n) =

e(l) = 0 k(2n-1)

6

for n ~ 2

,

.\. h = 0, a = 1, or (or. = 0, k inJlIII. e<'luution (2.98) becomes :1 d'6'(z)

dz

TIll'

~nlllti()n

+ ~(z) =

_

=

(2.107)

For this case, assuming a step

J).

z

(z _ 1)2

(2.108)

of this equation yields (2.109)

"11

nhtnin Ihe integmtinll constant III\' illili:11 vulllc:

Kt~.

we use the following limiting

111'01' 1'111

lim '("(.) I

dU),

I ·1· 1\;' = ()

(2.11())

I."

111'111'" 1\'II

(,'.111)

64

THEORY AND APPLICATION OF THE z-TRANSFORM

Substituting this in equation (2.109) and finding the inverse z-transform. we finally obtain: I c(n) = u(n) - (2.112)

n!

where u(n) is the unit step at discrete times.

Simplification of equations be/ore z-transjorm application7.l8.46 In certain cases the rational coefficients difference equation could be reduced to a constant coefficients equation by certain function substitutions in the original equation. This reduction simplifies the application of the z-transform to obtain the time solution. A typical unmixed difference-differential equation with rational coefficients which may be reduced to one with constant coefficients is

+ k + q)(n + k - I + q) ... (n + 1 + q)Y~".l.k + 0k_l(n + k - 1 + q) ... (n + 1 + q)y~t;':~1 + ... + ol(n + 1 + q)Y~~1 + 0oY" = /,,(1)

(n

where

d lr

Ik'

( )

(2.113) (2.114)

Y,,-u = dl k Yn+1 I

and n is a parameter. The substitution function is %

Yn = J"'(1

+ ~ + q)

(2.115)

Substituting this in equation (2.113), we obtain the following constant coefficient difference equation: 110' %n+k

+ 0k-I%n+ 1lt-1I /11 + 00%" .-. + • • • + 01%,,+1 = r(l + q + n)/"(I) =

g,,(r) (2.116)

By applying both the Laplace and z-transform to this equation. we can obtain the solution.

I. EXAMPLE Obtain the solution 11.. (1) of the following unmixed difference-differential equation:

!I:,':. +.

/I

a

I I

!I"

,0.

with !I"IU)

/I! •

a

l'U list illl I

1}..117)

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

6S

Let X"

+ I) =

JI" = I'(n

X"

(2.lIS)

n!

Substitutillg the above in (equation 2.117), we have Cll X,,+1

+ ax,. = 0

(2.119)

Applying the Laplace transform to equation (2.119), and using the initial value %,,(0) = y,,(O)n!, we get

+ aX,,(s) =

SX'Hl(S)

I

(2.120)

Defining the z-transform of X,,(s), oc

= J'[X lIes)] =,,=0 ! X ,,(s)z-"

.-l'(z, s)

(2.120a)

and applying it to this equation with s constant, we have sz.'l'(z. s) - szXo(s)

+ a.'l'{z, s) =

z -z- I

(2.121)

This equation reduces to

= szXo(s) +

.tr(z, s)

sz

+a

z (z - 1)(sz

(2.122)

+ a)

The inverse z-transform of this equation yields Xo(s) (- -a)"[ s

X,,(s) =

-I-]

s+a

1 +--

s+a

(2.123)

We can write for Xo{s) the following form: 1

Xo(s) = - -

s+a

+ K1(s)

(2.124)

where K.(s) is determined from the given initial condition. If we let K. (s)

= K. / s, the inverse Laplace transform of equation (2.123)

lives ;r ,,(1)

=

,r"' + K 1 (_~)n t" n.

(2.125)

I J!oill~ eq lIHlion (2.118), we obtain for the solution c.

,,1

f/,,(I) = - PI!

1

0'

Ih,'

,':lM' A·.(.~) I' '"

f/"I II

II!

+ K, (-")"1" ---.(,,,)2

(2.126)

() (or 1\. = 0). Ihis etlll:tlinll reduces tn (2.127)

66

THEORY AND APPLICATION OF THE Z-TRANSFORM

2.6 Time-varying z-transform and the system function 10 ,J3,25 In this section the solution of a general linear ~ime-varying difference equation will be indicated by using the z-transform of the system function. For the cases where closed form solution is not obtainable and the system is slowly varying, an approximate method based on the system function is used. The discussion follows closely Zadeh's work 1 J.12.26 as applied to time-varying differential equations and the use of the time-varying transform methods. Assume a time-varying difference equation as follows:

+ a,._I(n)£-11I-1I + ... + a1(n)E- 1 + ao(n)]cn = [bA(n)E-.l + bA_ 1(n)E- CA- 1l + ... + b1(n)E- 1 + bo{n)]rn

[a/n)E-"

(2.128) with p ~ i. and Cn , r n representing the output and input sequences of the system described by equation (2.128). The delay operator E-l is defined as E-'1n = In-/(' Equation (2.128) could be expressed in a condensed form A(E-I, n)c"

= B(E--l, n)rn

(2.129)

Since the system H is linear, we may define its impulse response hen, n - r) = h(II, k) by (2.130) where the Kronecker delta function 6nk is defined as if n =

k}

(2.131) if n ~ k Using the superposition theorem for linear systems, the output Cn is given as n Cn

= !h(n, k)r/(

(2.132)

k=O

It should be noted that hen. k) is sometimes written as hen, n - r) as indicated. The impulse response hen, k) is expressed from equation (2.132) as the amplitude of the pulse occurring at the output at the nth instant as a result of an input pulse of unit amplitude applied at the kth instant. In analogy to the one-sided z-transfnrm. we define II limc-vllrying z-transform of h(fl. fl - r):

.

.*'(11. z) ,:'-

",111(",

1/

,

~ ',(11, "

r)l ,

II

r).~

r

(2.13.\)

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

Setting k =

/I -

67

r in this equation. we obtain »

,~(n, z)

= L

n

hen, k)z-"+k = z-n 2. hen, k);;k

k=-»

(2.134)*

k=O

The summation on the right-hand side of equation (2.134) represents the response of H to an input signal having the sequence of amplitude :::(l, ;;1, .:;2..... Hence the system transfer function .~(n. z) may be alternately defined by

£(n, z)

= response ~f H to z" z

(2.135)

Input-output relationship

The output at the nth instant may be obtained from the consideration '(,',,(z), where ~ ft(;;)

= Yl(z).:tt'(n, z)

ot

(2.136)

where

:3I(z) ~ J[r k] =

to

L rkz- k

(2.137)

k~O

The output sequence is given by the following inversion integral e" = _1

21Tj

rf~(z).;;fl(z,

J1'

n)z,,-I dz

(2.138)

when r is a counterclockwise closed contour in the z-plane which encloses singularities of the integrand. II should be noted that the index n in equation (2.138) is treated as a parameter; thus C n could be obtained from the ordinary table of inverse .:-transforms, that is, Table I of the Appendix. Furthermore, if'tf,,(z) is l'''panded in power series. the coefficient of r k is not equal to Ck except for J.. .0 II. Thus equation (2,136) does not represent the z-transform of the OUlput for all sampling instants, The :::-tr:lOsl'orm of the output (sometimes referred to as the ",-transforl11)1U can can he ohtained by using the double-difference system function III analogy to the hifrequency system function for the systems described by dill"l'n..mlial elluations.20 In addition. as noticed. the method of the 1lIl'l'l'ding Sl'l'tioll yields in some cases the :-transform of the output. I he

• 1111'. l'll"allly huhb ht,,'mlsl' ill Ilwsc IIiSt'liSsinns wc arc :lsslll1lin~ Ihal nn inpul exisls

1''''1'1'11111/: "

n,

68 nlEORY AND APPLICATION OF THE z-TRANSFORM

Determination of ~(z, n) The system transfer function .JtP(z, n) satisfies a difference equation which sometimes is easier to solve than the original difference equation (2.128). To obtain the equation for Jf7(z, n), we take the z-transform of equation (2.130) with respect to the index k to get GO

I

A(E- I, n)h(n, k)ZTc

co

= IB(E-\ n)~ntZk

(2.139)

Tc-O

k~O

and noting that hen, k) = 0 for k A(E-I, n).Jt"(z, n)z"

> n, this yields

= B(E-l, n)zn

(2.140)

Before rewriting equation (2.140) in a more convenient form, we utilize the following identities: (2.141)

and (2.142)

since A(E-l, n) and B(E-l, n) are polynomials in E-l, using equations (2.141) and (2.142), the left- and right-hand sides of equation (2.140) become (2.143)

and (2.144)

Therefore

~(z, n)

is given from equations (2.143), (2.144), and (2.140) as (2.145)

Equation (2.145) is a ,uth order difference equation with nonconstant coefficients. By adding and subtracting A(Z-l, n)~(z, n) to equation (2.145). we may obtain ~(z n) =

,

B( -1 ) z, n A(Z-I, n)

[fa,(n)z-i - f ai(n)z-iE-i]~(Z' n) + .:=.:.;;;o_.:..O_ _ _-"i-;;;.:O~----==---A(Z-l, n)

(2.14() or II

..0°(.,.,.," ) --- /1(: I, '1)

1 a;CII):

i[-'po(:, II) - .tI'(::, II -

01

,. I

.R

A(:

I,

II)

A(.: 1,'1)

0.147)

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

69

The second term on the right side of equation (2.147) is small compared to the first if either .J?"(z, n) changes slowly with n or if the changes are small. This suggests that equation (2.l47) can be solved by iterative methods. Thus if .71'k(z, n) is the kth iterated solution we have

,.

'W'k+l(Z, n}

=

,7I'0(Z,

n}

+

I

aj(n)z-ipr'k(Z, n) - :It'iz, n - i)]

.:;;;1~;.o.I_ _ _ _ _-:-_ _ _ _ __

A(Z-l, n)

(2.148) where -Vi> (., .J(

0 -,

)

_

n -

8(%-1, n) A(z-t, n)

(2.149)

is called the Jrozen transfer Junclion. Equation (2.148) is very useful in obtaining an approximate solution for slowly varying discrete systems. In some cases the frozen transfer function will give a good indication of the response behavior. Such !oituations may arise in sampled-data systems where the sampling rate l'IHlIlges very slowly. Another application of this technique is in the analysis II\' automatic gain control where the pulse-frequency modulation varies according to a sinusoidal law. Such a system will be studied as an illustnttive example. Furthermore, if .11 o{z, n) has its poles inside the unit circle for al\ n, all the iterative solutions HI> H 2 • ••• will 'have their poles inside the unit I'Irde and vice versa. ILLUSTRATIVE EXAMPLE

The following example was studied by Tratakowskii l3 using a similar approach. A more detailed analysis of this system will be discussed and n:lt't conditions on stability as well as the response approximation IlIt· ... t·nted. The system under study is referred to as "pulse feedback with \',11 iahle pulse repetition frequency." This system is shown in Fig. 2.5 ,11111 its etjuivalent feedback system in Fig. 2.6. The automatic amplifiI al ion control in u receiver of pulse signals of constant amplitude when an

...... (h'III,1)

/I

c y L-_r-----'--,---o

r-

I II iI/IO ."

I

\OllIIl'h'

fur II,,· v;II'iahlc pulse I'eetlback system.

(Output)

70

T-Y

THEORY AND APPLICATION OF THE z-TRANSFORM

T

FIGURE 2.6

PFM

H;i, I

Block diagram of the variable pulse feedback system.

RC circuit is used for filtering in the AVe leads to the study of such a system as shown in Fig. 2.5. The pulse-frequency modulator (PFM) yields the following equation In

< t < In+1 (2.150)

where In is the time of sampling which is determined by In - 1._ 1 = T(l

+ Msin On)

(2.151)

where T is the average sampling period. When 10 = 0, h is a positive constant with the following constraint: h ~ T(I

-IMD

(2.152)

The relation between y and :r for Fig. 2.6 is given by the following differential equation:

y + fly = kx

(2.153)

The solution of equation (2.153) yields y(t)

+ ke-p'l'X(T)ePr dT '0 = Itc and '0 = 'n-I, since X(/) = 0, for '

= y(to)e-P(Ho)

In particular when t

In' equation (2.154) yields

y(t",}

= y(tn_l)e-fJ(I.-tn-l) + ke- fJt •

f'.-1

'·-I+h

(2.154) 1+ h


d'T

(2.155)

11 _

q(tn_l)e fJr

Using equation (2.151) in (2.155), we obtain y(t n)

=

y(t,,_l)e-PTU+.l l sin 0,,)

+ ~ q(t "_I)e- fJTU + ill sin 0,,)( eflh -

I) (2.156)

Let ex

= /0:

"

l' =,/"

kll

:--=

ii' "

!lU,,):,= !I(II), q(I,,)

=

q(II)

(2.157)

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

and assuming oc« 1 or yen)

=

fJh «

[I - oc(1

71

I, we have from equation (2.156)

+ M sin On)]u(n -

+ koOty[1

- oc(1

+

I)

M sin On)]q(n - I) (2.158)

From Fig. 2.6, we have for the feedback equation q(n) = u(n)

+ y(n)

(2.159)

Substituting equation (2.159) in equation (2.158), we get yen) = (1

+ k oocy)[1

- oc(1 + M sin !In)]y(n - I) + koocy[l - Ot(l + M sin Un)]u(n - I) (2.160)

The frozen transfer function is simply obtained by taking the z-transform of equation (2.160) and considering n as a parameter [this is seen from equation (2.149)]. Therefore

koocy[1 - oc(1 + M sin On)]z-l 1 - (I + koocy)[l - oc(1 + M sin On)]z-l If we assume a slow pulse-frequency modulation, that is, ,Yfo(z, n)

=

)

ocM« 0

(2.161)

(2.162)

Ihe system function £'(z, n) can be approximated by the frozen transfer fUllction or by .K1(z, n). The response of the system for a step input has h~'ell studied for different 0 and oc/tJ and the exact and approximate (using Ihe frozen transfer function) responses are shown in Figs. 2.7 and 2.8. As expected. the smaller the quantity T with respect to 2fT/!). the better " Ihe approximation. hluation (2.160) describes a first-order time-varying difference equation, which can be written as yen) = a(n)y(n - 1)

+ b(n)u(n -

(2.163)

1)

From this equation if a(n) and ben) are bounded (as is usually the case)

.11111 '\upposing a(n) ¢ 0.* then the system is stable if and only if I ,\' lim - ! In la(n)1 < 0 (2.164) N~'" N n l A ~ul1icient condition for stability requires that Ia(n) I < I for all n. In

('xamplc, ItI(tI)1 < I. is the same as the condition that the pole of the Ir:lllsfl'r function [see equation (2.161)] should lie inside the unit • lid,'. Ilence Ihe suflicicnt condition for stability (for M> 0) is 11110,

111I/l'1I

I

Il(

I - M~.- 2 < ko M) - 1

)'111)(1 , ..

• I h.,

\llIlIr l'lIlIIlitilili I'll.

Mlihilil.Y ellll he


pmyell IIll1n

I - M et() -

(2.165) M)

if u(,,) ". o.

72

THEORY AND APPLICATION OF THE z-TRANSFORM

For the plots of Figs. 2.7 and 2.8 the preceding condition is satisfied. Thus the system is stable. To obtain a better approximation, the first iterative solution from equation (2.148) could be used, and for this example ~

(

.:n 1

) _

.yp (

z, n - on

0

)

z, n

+ Ql(n)z-lpof'o(z, n) - .Yt'o(z, n ao(n) + a1(n)z-1

1)]

(2.166) where Ql(n)

= -(1 + k oocy)[1

ao(n)

= 1

- oc(1

+ M sin On)]

(2.167) (2. 167a)

0.25,..------..------,------.------,

"

I \

I ~lst order approx. I \

0.20

t-------+--+~---t-----.f....-4i_-----I

Oth order approx.

0.15

f------+---!--'~_i__--+--~__f+l_+-----I

0.10

f-----+-H---lH~-+-__f_+__!___f__1___+_----l

0.05 t--.",."----+------+-------I-------I a =0.2

ko"Y = 0.1 M=0.4

o

25

_____

~

_____

~

OL-_____

50

~

n= 7r/2

_ _ _ _. _ J 15

100

/I

FIGURE 2.7

i\XIICllllld

npprnxllllilio relll,ulIlIC ur .Ylllcmlll Fig. 2.b

rUf

II 11101'

Inpul.

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

73

0.15

0.10

~

Oth order approx.~ ~ !:"'Exact 1st order approx.

I

0.05

.-=. ~

a=0.2 = 0.1 M=0.4 0=11'/100

k{j'y

o

o

50

150

100

250

200

n

FIGURE 2.8.

Exact and approximate response of system in Fig. 2.6 for a step input.

and .,g> (

.n

0

)

_

z, n -

B(Z-l, n) _ ko«y[(l - «(1 + M sin On») A(z-t, n) ao(n) + a 1(n)z-1

= __b..!,;l(~n:.....)_ _ ao(n) + a1(n)z-1

(2.168)

It may be noticed that if the pole of '~o(z, n) is inside the unit circle for all n, so is the pole of £'1(Z, n) and similarly for other higher-order iterative solutions. The response using the first-order iterative solution is also shown in Figs. 2.7 and 2.8.

2.7 Double z-transformation* and solution of partial difference equations21 ,23,4.1,49 Partial difference equations arise in many applications such as probability theory, boundary value problems, lattice network, and transmission lines. The systematic solution of such equations can be performed using the duuble z-transformation (or two-dimensional z-transform). This transform is defined as follows: 00

U(z, w) ~ a'w[u(n, m») ~ ~

co

!

u(n, m)z-nw-m

(2.169)

n-O m=O

where ,)'". denotes the zw-transform of u(n, m). • AIN" rdcrrcd Itl ill Illllthclllllticnlliternture 05 bivariate generating function."

74

THEORY AND APPLICATION OF THE z-TRANSFORM

We apply the zw-transform to the following functions to obtain co

II11[u(n

GO

+ 1, m)] == .... I I u(n + I, m)z-"w- m 0 ...=0

I u(O, m)w'l11[u(n, m + 1)] == w[U(Z, w) - I u(n, O)Z-"J II11[u(n + 1, m + 1)] == zw[U(Z, w) - I u(n, o)z-n == z[U(Z, w) -

m]

(2.170)

.... -0

(2.171)

.. -0

..-0

-

m~oU(O, m)w- m + U(O,O)J

(2.172)

Similarly. we can extend this procedure to obtain the general zwtransform of u(n + a, m + b). lbis transform is given as follows and can be easily derived from procedure given, CIO

1",[u(n

CIO

+ a, m + b)] == I I ?I~O

.-0u(n + a, m + b)z-"w-

a-I

m

CIO

-I I

u(n, m)z-"w- m

.. =0 .... -0

(2.173) Equation (2.173) can be readily used to obtain zw-transformation of any order linear partial difference equations. By successive inversion with respect to wand z, the discrete time function u(n, m) can be obtained. This procedure is illustrated by the following example. ILLUSTRATIVE EXAMPLE

We consider in this example a partial difference equation which arises in probability theory. Obtain u(n, m) which denotes the probability of In successes in n trials and which satisfy the following partial difference equation: u(n

+ I, m + I) -

quell,

In

+ I) -

pll(n, m)

== ()

(2.174)

The boundary condition on Ihe n-Itxill ill u( n, 0) ." q" for n .- O. The number p denotes the prubllhility fur III1CCClIlI in independent scrics or

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

7S

trials, and q = 1 - P denotes the probability of failures. Applying zwtransform to equation (2.174),

zw[U(Z, w) -

I u(n, O)z-" - I u(O, m)w-

,,-0

m-O

m

+ U(O,O)J

- qw[U(z, w) - Jou(n, O)Z-"J - pU(z, w) =

°

(2.175) From the formulation of the problem it is evident that u(O, m) = m > O}. (0,l,m=O

Therefore equation (2.175) becomes

zw[U(Z, w) -

I (~)-"J ,,-0 q

- qw[U{Z, w) - ,,=0 I (~)-"J q - pU(z, w) = 0 (2.176)

III'

(z)-" q

(2.176a)

~o (;f" _ ~o (;f" 1-(zw -Pqw) I-!(-P) %-q

(2.177)

U(z, w)(zw - qw - p) = (zw - qw) I00

,,~O

lienee,

U(z, w) =

W

Tllk ing the inverse w-transform of (2.177) we obtain '1/(%, m) =

I (z)-"( - - p)tn = ,,-0 q z- q 00

(p/z)m I00 1 - 4/% ,,~O

(q)" -

(2.178)

Z.

(p)m( 1-;q)-m(1-;q)-1 =~(p)tn( 1-;q)-l-m

=~ "I"

'f/(:, m)

=

(;r[1 + (-I ~ m) (~)(-J) + (-I ;- m)~(_1)2 + ...J

(2.179)

IIII' \'(1"lIid,'nt III" Ihe:: " term is

,,"',," "'I (-"I.-- '""')1(-1)" ' ',

(2.180)

76

THEORY AND APPLICATION OF THE z-TRANSFORM

Therefore the inverse z-transform of equation (2.179) becomes u(n, m) = pmq"-m X

[(-1-

m)(-2 - m) ... (-n (n - m)!

m lI_m[n(n - 1) ... (m =pq

(n - m)!

+ 1)J

+ m - m>J(_I),,-m (2.181)

Finally, u(n, m) yields u(n, m) = pmqn-m

(:1)

(2.182)

REFERENCES

I. Doole, G., A Treatise on the Calculus of Finite Dijferenus, G.E. Stechert, New York. 1926.

2. Jordan, C., Calculus of Finite Differences, Chelsea Publishing Co., New York. 1960. 3. Milne·Thom~on, L. M., The Calcuills of Finite Difference, Macmillan Co., London, 1951. -"'''4. Goldberg. S., Introduction to Difference Equations with Illustrative Examples from Economics. Psycllolo,f{Y and SoC'iolo.IfY. John Wiley and Sons. New York, 1958. "l5. Pinney, E., Ordinary DiJjerence·Differential Equations. University of California Press, 1959. 6. Gelfund. A. 0., Differenzenrechnung. Moscow·Leningrad. 1952 (in Russian). 7. Levy, H., and F. Lessman. Finite Difference Equations, Macmillan Co.. New York, 1961.

~

8. Nor1und, N. E.• J'orlesu".f{l'n uber Differenzenrechnung, Verlag Von Julius. Springer. Berlin. 1924. 9. Samuelson, P. A., FOllndation of Economic Analysis, Harvard University Press: 1947. 10. Friedland, B., "Time·Varying Sampled·Data Systems," Proc. Second Midwest Symp(l.fium on Circuit Theory, Michigan State University, East Lansing. December 3-4, 1956. 11. Zadeh, L. A., "Circuit Analysis of Linear Varying·Parameter Networks," J. Appl. Phys., Vol. 21. November 1950, pp. 1171-1177. 12. Zadeh, L. A., "On Stability or Linear Varying Parameter Systems," J. Appl. Phys., Vol. 22, Apri119SI, pp.402-405. 13. Tratakowskii, G. P., "Application of Simulation in Analyzing Linear Pulse Systems with Variable Parameters," At'lomatica i Telemekh, Vol. 20, May 19S9, pp. 559· S(lf!. 14. Aseltine, J. A., "A Transform Method ror Linear Time-Varying Systems," J. AI'I,I. Phys. Vol. 25, June 1954, pp. 761-764. 15. Cooke, R. c., Infinite Matrices a/l(l Seqll('n('(' .\i'u(·(·.~, MlIcmillllll ('11., 1.1\lldllll, 1')~(). 16. Ho, E. C .• and H. Davis. "Gcncrali/cli Op~"alin'ml ('alclllu~ fill' Time-Varyinl: Networks," Dept. of F.ngineering Repnrt ~4 71. JIII.V 11)';4, IInivl''',ity nf ('lllifn,""', Los Angeles. 17. Bubh, I-'. W .• "A New l.inl'lli 01"'111""11111 (·lIk"I"·.... II.S.A I. h·,·hlli,·,,1 It"I"'1I Nil. MHI. May I ')~ I. J)uylnll. Ol"n_

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

77

18. Madwed, A., "Number Series Method of Solving Linear and Nonlinear Differential Equations," Instrumentation Laboratory Repon No. 6445-T-26, April 1950, M.I.T., Cambridge, Massachusetts. 19. Raymond, F. H., "Analyse du Fonctionnement des Systemes Physiques Discontinus (ct son Application aux Servomechanismes)," Ann. Teluomm., Vol. 4. July, August, October 1949, pp. 250-256. 367-314, 347-357. m. Poincare, H., "Sur les Equations Lineaires aux Differentielles Ordinaires et aux llilTcrences Finies," Am. J. Math., Vol. 7, 1885, pp. 213-217, 237-258. ~~1. I.awden, D. F., "On the Solution of Linear Difference Equations," Mathematkal (it/,-,·I1('. Vol. 36, 1952, pp. 193-1971. n. nelhnan, R., and K. L. Cooke, Diffcrential-Di.Q;rence Equations, Academic Press. N.·\\" York and London, 1962. .. I. Mcschkowski. H., DifferenZf!llgleichungen, Vandenhoeck and Ruprecht, GoUingen. 1"\11 • .'01 1luetsch, G., Handbucll der Laplace Transformation, Basel und Stuttgart. 1956. "" lIurnegel, R. G., "Analysis of Aperiodically-Sampled-Data Feedback Control ';y·,It·Ill'." Ph.D. Thesis, Dept. of Electrical Engineering, Cornell University. Ithaca, N.·\\ Yurk, 1954 . .'" /;lIlch, L. A., "Frequency Analysis of Variable Networks," Proc. IRE, Vol. 38, 1"',11, PI'. 2t}1 ·298 . .'1 1\;lpllln, W., Operational Methods for Linear Systems, Addison Wesley Publishing I II, Ih·;nling, Massachusetts. 1962. .'N .IIIIV. L I.. and F. J. Mullin, "A Note on the Operational Solution of Linear I hll"II'lIn' I.quations." J. Franklin Inst., Vol. 266, No.3, September 1958, pp. 189-205 • .., A·.•·lline. J .• Transform Method in Linear System~, McGraw-Hili Book Co., New \ .. ,~.

I·I~K.

Itl 1\111111,'1. I'.•

I.('t;ons sllr les Remrrences et leurs Applications, Gauthier-Villars, Paris,

1'" I

hn y. I . I.. SlIInpl('d-Data Control Systems, John Wiley and Sons, New York. 1958. I' ,..·111"",,', J.• D(lfital Processes for Sampled-Dota Systems. John Wiley and Sons, I 10·\\ \ "' J,., 11/(,2. II I "'1. ·1 .. "I incar Difference Equations and the Dirichlet Series Transform," I", """" "'II/II"~I" Vol. 62. No.9, 1955, p. 641. ... N'·lIh·hl. I., "On the Operational Solution of Linear Mixed Difference-Differential 1'1".11'"11":' ("I/I"";/{~/' "lri/. SIIC., Vol. 30, 1934, p, 289. " 111"\\11, II, IIII' MIIII,('Itltltical TI,e/lfY of Linear Systems, John Wiley and Sons, '11.\ \ "' J,., 1'11,1. ". 1',11. 1\1 A," I hc Operatiunal Sulution of Difference-Differential Equations Using 110. 1\1",h' .... 1 1':11,,1''''''11,'' IR/:, 1'(jA(', October 1962, pp. 124-126. I' I., ",111111". I., "Oh"1 ciuc Vcmllgcll1cincrung der Exponentialreihe," Math . . ,1/" I" ,\'111 1", 1'11.2. PI'. 2.1'1 242. II, I., h.III'II". I, "I "'·III'.,·"'I~' "'",ktiullen von Zugeordneten Kugelrunktionen," 11.1/1. .', ,/I. I" , V"I/H. 1')(,2. 1'1'. 1.11 1.\4. 1'1 I ,.10 1\ I, .0\ , II~I,""'I I I,IIIU·,'"d,·",,,II·illl/'ti,,,r.v, Me(imw-JliII nook C<, .. New York. II

7M

'1'1 II "It V "NI) A1'1'1 ... AIIUN III

1111' • I" ANllt". M

""""1",·"",, t" tl,,' •"",,,1,,, ,./ I,,,,,,,

41. Miller, K. S., All IJlI/,'/I'"", .111,1 11'/"'/1'11'" Equatio".f, Henry Holt lind ('11 •• Nt'w ~'II''', 1'''011. 42. Ku, Y. H., and A. A. Wulf, "l.lIlIrelll-( ·lIlId.y "'11'''/'111111\ , ... A,"I1V\I' "' IllIrlll Systems Described by Differential-Difference lind S\l1lI I ''1"11lilln\,'' I'll"'. IHI , VIII . • May 1960, pp. 923-931. 43. Jury. E. I., Discussion of the above reference, Pror. IRE, Vol. 4K, Nil. 12. Url'('I11''''' 1960. 44. Mikusinski, Jan, Operational Ca/rulus, Pergamon Press, Landltll, I')!I'). 45. Jury, E. I., "Contribution to the Modified z-Transform," J. Frallklill Illst., VIII. 21t1, August 1960, pp. 118-129. 46. Tschauner, J., Einfuhrung in die Theorie der Abtasysteme, Verlag R. OldenhulII1:, Munich, 1960. 47. Cypkin. J., Theory of Pulse Systems, State Press for Physics and Mlllhel1lllticlll Literature, Moscow, Russia, 1958 (in Russian). 48. Stone, W. M., "A List of Generalized Laplace Transforms," Iowa Statl' C"II':I{" Journal of Srience, Vol. 22, April 1948, pp. 215-225. 49. Feller, W., An Introduction to Probability Theory and Its Appliratiol/s, John Wiley and Sons, New York (second edilion), 1959, Chapter XI. 50. Wloka, J., "Ober die Anwendung der Operatorenreihnung auf Lineare DitTerentilllDitTerenzengleichungen mit Konstanten Koeffizienten," J. fur die Riene lind Augewandtl' Mathematik, Vol. 202, OClober 1959, pp. 107-128. 51. Fi4!vet M., On the Analysis of Feedback Systems with Transportation Lo,f Using the Modi/h'd z-Tran~rorm, Master of Science Project, Dept. of E. E., University of Californill, Berkeley, June, 1963. 52. Alper P., TWI} Diml'"sional z-Transforms, Technological University Eleclronics Labamtory, Delfl-Netherlands, August, 1963. 53. Gradstein, E. S., and E. M. Ryzik, Tables of Integrals, Sums and Products, State Press for Physical and Mathematical Literature, Moscow, 1962, p. 37, Formula 1.223.

'H.

S I A BII.ITY (,ONSIDERATION I'()I{ I.INEAR DISCRETE SYSTI~MS

"I IIII' main pl'Ohlcllls in the study of linear systems is the detel'stahility. For linear continuous systems, the problem of ·.I,lhlllly 1\'lh'l'~S til the necessary and sufficient condition for the roots of II" /'oY'oI\'1lI dHlnlt;tcristic equation) to lie in Ihe left half of the s-plane. 8 II ....... nllltlilinlls huve been studied in detail and as a result the criteria of IIIIIWIII 1~llllthH and of Licnard-Chipart8.38 completely solve the problem ,III'llylll:ally. For linear discrete systems the condition of stability reduces ", .. hlallling the necessary and sufficient condition for the roots of F(z) 1I1I·.I'II'll' system characteristic equation) to lie inside the unit circle in II ... ·pJ:lnc.!I,17 Although these conditions have been studied by Schur.28 C 'lIhll,~ and Marden,· the final stability criteria to be obtained from these !"IIWI'S arc mther complicated and not readily applicable for system anal~",I" and design. Therefore in this chapter a complete study of this problem I', IlIlIk,rtaken where the final solution of the stability criteria in the forms "IIII,terminunt,l table,l2 and division 14 methods are obtained. Siahility tests within the unit unit circle are very important in many IIl'lds, For instance, the stability of difference equations with constant or 1"'1 iotiically varying coefficients,21 the stability of limit cycles in nonlinear tI"l'l'cle systems, the stability test of linear systems with randomly varying parameters, the convergence of iterative computational rule using digital 1·lImplIlers. 21 the stability of periodic regimes in nonlinear systems with JlII:~ewise linear characteristics, and many 'others all reduce to stability 1l'~ls within the unit circle. In view of these applications. it is felt that I'IIncerted efforts should be directed toward stability tests within the unit !'irde as well as the conditions on the roots distribution of £(z) within the unit circle. 'ltll'

111111.111"11 .. I'

79

NO

IIlIORV ANI! AI'I'IIIAIlCtN III

1111 .IIlANlilltllM

3.1 ()eliniliull tlfllillhilllyll A linear discrete system is considered It) he "'"hl(~ it III 1111 hllll III h',1 inputs there always correspond bounded outputs.· The stability condition can be obtained if we sepunlh~ IIw 111111'"1 (system response) into steady-state and transient terms. Thi, t:UII h,' easily established if we write the output z-transform (6'(.::)
==

Er(z)~(z)

(.1.1 )

where 8(z) and .~(z) are the z-transforms of the input and the syslem transfer function respectively. The output related to the steady-stule p:lfl (or due to the input function) can be obtained from the inverse ::-trunsful'm formula as the sum of the following typical term: residue of lEr(z) . .Jf'(z)]l,.=.... z::,-l

(3.2)

where Z'II is a typical singularity of Er(z). Since the input is bounded, that is, Iz".1 ~ 1 (3.3) then the output part resulting from this input is also bounded. Similarly, a typical transient term of the output can be written residue of Pt"(z) . 8(z)]I._I", ZItn-l

(3.4)

where ZIt is a typical singularity of ~(z). From this expression, it is evident that in order for the ouptut to be bounded, the following conditions should exist.

(3.5) or all the poles (singularities) of .Te(z) should lie inside or on the unit circle. When one singularity is on the unit circle, that is, Iz,,1 = I, then from the physical point of view such a system should be regarded as unstable, since a very small change in the physical constants may throw the critical singularity from the boundary into the exterior of the unit circle, causing actual instability. Summarizing this remark, we may state the following fundamental theorem: If all the singular points of the discrete system transfer function ~(z) are located inside the unit circle, the system is stable. If at least one singular point lies outside or on the unit circle, the system is unstable. • These outputs are considered only at discrete intervals or time (sampling instants). For a continuous output or a discrete system, the modified z-transrorm or the output should be used instead of equation (3.1).

It I 41111 II ~ ' ....... 1111 It" 1111'" 11111 II NI AII III!I' III II ""'111 M'I

KI

I hi" I "IlIhll"" "I ·,1,1111"1" Il',ullly 111'1'''''' III "I""I"II'T 1"1"01111111\ Wllh ,,,",.1,1111 ,11111 1"'11,"111 1111'11" 11'111'., 11\1\\'\'\"'1, filII' '.hllllitl Ill' I'XI'II'I\Cd III .\,1111111". IIII' I h.II,I' 11,,,',111' l'll'lallllll hll ,'OIl'll III Ih,'''' 1'111'111\. I "I 111111' \ III \ III". ,II',I'II'I!' .. y.. h'JII!' (III II IIII'·VllI yi"~ di IIcl'cncc ell uations),

,, .. 1"11""1"" 1111 ,.llIhlhly ill'p"I"h, 'III the Irilnsfer function .Y/ (:::, II) for ,III \ ,11111", .,1 1/ I hi' 11,'1"1 IIlIlIallllll of stability for the time-varying case is 111"'11,,,,<1 separately.

I.llh"1 III\IIhl',1 illI" 1\

\.,' Stahility l'Onditiull for lincar 111m' Vll ryi ,,/!. d i!o.lTetc systems I h,' "1'1111111"" "I' .,Iahility for this case is the same as that for the time 1',1'>l' disl'll~sed in the preceding section. It can be presented as:*

111\111 101111

..\ "y',lI'lII I~ 1"II"wlII)~

'Iahk if and only if its impulse response 11(11, k) satisfies the I'OIl"itioll; for some N the following inequality exists:

"

~11r(",k)l

',,1

.111 II,

(3.6)

'k

IIIl' input is bounded, there is an M such that rhlls frolll the output relationship,

SlIh'I'

"

< N < 00,

"

"

kO

k"O

~ 2Ih(lI, k)lIl'A'1 ~ M 2111(11, k)l ~ NM

k,,1

Irkl

~

M for all

(3.7)

Ilw. proves the sufficiency of the stability condition, III prove the necessity of the condition, suppose it is not satisfied. II,,'" for any fixed P < 00 there is a 110 such that "0

2Ih(no, k)1 k ,0

> P.

(3.7a)

('hoose the inputs 'Ic' I'k

=

{+1

if h(no. k) ~ 0

no

no

,,=0

k=O

-I

if h(lIo, k)

(3.8)

<0

Wl' Ihen have C no

= 2 r"h(n o, k) = I

I'hcrcfore for all P < and an 110 such that en

00

Ih(lIo, k)1

> ,P

(3,8a)

we can find an input uniformly bounded by hence the system is not stable.

> P; o

• An alternate and more precise definition of stability is: A system is stable if for any input sequence uniformly bounded by I, that is, Ir.1 ~ I. for all k. there is a P < :tJ 'llch that 1('.1 ~ P for all n.

82

THEORY AND APPLICATION OF THE Z-TRANSFORM

If every hen, k) is bounded, it is necessary only to find the limit of

!" Ih(n, k)1

as n -+

00.

k~O

From the definition of Jf"(z, n), it follows that the preceding stability requirement is equivalent to the condition that ~(z, n) be analytic for Izi ~ 1, which can be shown as follows: From the definition of £(z, n) we have n

.;F(z, n)

== !h(n, k)zlr-n

(3.8b)

A.-O

Letting k

== n -

j this is equivalent to

~(z, n)

n

== I

hen, n - j)Z-1

(3.8e)

I~O

For

Izl

~ I we have n

.;>F(z, n) ~ ! Ih(n, n 1-0

n

i)llz-/1

~! Ih(n, n ;.0

i)1 < N <

00

(3.8d) Therefore if the system is stable, .Jt"(z. n) is analytic on and outside the unit circle. Furthermore, if Jt'(z, n) is analytic for Izi ~ 1, it requires that the radius of convergence of the series is strictly less than unity. From Cauchy-Hadamard 33 theorem we know that such a radius is given by lim sup Ih(n, n _ i)1 1/i (3.8e) It is also known that if this limit is strictly less than unity, the series

{Ih(n, n - j)l} is absolutely convergent and hence for all n there exists an N such that . n

!

1=0

Ih(n, n -

i)1

~ N

< 00

(3.S/)

Hence the necessary and sufficient condition for the stability of linear time-varying discrete system is that equation (3.8 f) be satisfied for all n, By a similar reasoning the stability requirement for the time invariant case discussed in the preceding section can be similarly established.

3,3 Tests for stability We shall.conccnlralc in Ihis seclion Ilnly nn Ihe varillus lInalylil' It"".

Uowever, we IIllly nWl1linn Ihal Ih,:n' ,'xi,1 !.I'vl'lal/~r:lphical h'~h, ~lIl'h a .. lilt! NY'I"i~1 tlia/\r:l1II alltl Ihe 1'11111-111"11'; 1I"'lhllll, which nrc Ih!.l'IIS~'~" in "cillil d!ll'wlll'l'c ill III,' 1I1\'llIllIu'.

STABILITY CONSIDERATION FOR LINEAR DISCRETE SYSTEMS

83

The test involved is the determination of the necessary and sufficient conditions for the roots of the following polynomial in z, which represents the system characteristic equation to lie inside the unit circle in the z-plane:

F(z) = ao + a1z

+ azz'l. + ... + a,l' + ... + aflz" with an

>0

(3.9)

The various analytic tests can be summarized as follows.

Bilinear Iransformation:5.9.17 If the closed region 1%1 ~ I is mapped into the region Re [w] ~ 0 of an auxiliary w-plane, the condition for stability becomes exactly the Hurwitz condition. The bilinear transformation

w+l

(3.10)

z=--

w-l

is the simplest possible mapping of the unit circle into the left half of the ,,·-plane. In this manner the nth-degree polynomial in z is converted to a ratio of nth-degree polynomials in "' and the numerator is tested for its fl'WS in the left half of the w-plane. Let the system characteristic equation in the z-plane be defined by (3.9) with il ll > O. When mapped into the w-plane by (3.10), the zeros of the "'-polynomial are given by

b .. The rclutionship between the a,. and b,. is

>0 (3.11)

(3.12) WIlI'II' (3.13)

IIII', plOl"I'llme involve!; a complicated ~\Igebraic manipulation since .1 ... hllllll'ar 'ransfCll'lIlatilln. and the limll constraints on the original )I"IYII""l1al I"lI'lIidl'nl~ :1I"l~ ralhl~" involved. Thus this procedure has not 1""," .... 1"11 ·.III"I·I· ..·.l'lIl1y fur dl'sil~1I ur panllllctcr c1ll1n~e fur systems higher .1It111 ,.1'1'1111""1 1111,,1 mdl·... lIC1wl~vt'r. il Ims the advanta~e thai we can

84

'l'lIIiOR\' ANI) AI'I'I.IC'A'I'rUN UP '1'1111 .·lMANIII'UMM

-2,0

FIGURE 3.1

A shifted unit circle in the C-plane.

apply the Routh-Hurwitz criterion or the simpler criterion of Lienllrd· Chipart to the transformed polynomial in the w-plane. In some cases the discrete system can be described in the '-planc'" which requires a stability test for the shifted unit circle of Fig. 3.1. A criterion similar to that of Hurwitz has been developed for the shined unit circle which offers an alternate method of testing the stability of linear discrete systems. 37.38

3.4 Stability test directly applied in the z-plane A logical and useful approach to the problem of stability is to be ahlc t(l obtain a stability criterion directly in the z-plane. thus avoiding ltny transformation as in (3.10). Such a stability criterion was first established by Schur-Cohn in 1922 and further discussed by Fujiwara3 in 1924. However. the original form of this criterion is quite complicated; Cohn'. has 2n - 1 constraints and requires the evaluation of two determinllnll of order n, whereas in the Hurwitz and Lienard-Chipart criteria, only n

z-plane

FlcnIlU~:\.2

1/"" c:lrdD In IIID '-rlnne, "

H~

N'AIIIIIIY C"UN,um UIIClN IUK UNIAK IIllIC"KIII lIVlIlI M!I

1"1'""11"111"1" IIl1ll 1111'" IIIh·cmltl· deterlnillll'" IIrc retluired. ThwI this "1 uri 11111

h'I_ IIClI rmvell

~IICCe!illl'lIl und

is very rarely used by engineers

ICII hl~hc,"-urdcr lIy!ilel1l!l. I he' IIri/tilllll 1'111'111 ClI' Ihe C'IIII

Schur-Cohn criteritm for complex coefficients hl' !illlh'ci lis I'lIlInws. The rolynomial /0"(.:) ....

+ ".: + Clz:1 + ... + akzk + ... + anz"

tI.,

h'I" 1111 il!! '"lIl1ls· inside the unit circle, as shown in Fig. 3.2, if and only if II", r"II"willl-t cnl\!ltrnints ure fulfilled.

IA.I < 0,

k-odd

1/\,1> 0,

k-even

(3.14)

wh"I\' Ih(' 1I11111'ix tlk represents·

fl., ....

a"

0

0

0

0-

0_-1

On-hi

al

00

0

0

0

0-

O_-hl

Ok I

Ok. "I

Dk._1

00

0

0

"a.

0

0

0

00

a

an

0

0

0

00

I

an

I' ••

-°t_1 Ot-I

k = I, 2,,, ., n ak.

=

complex conjugate

ofDt

-

On

00

(l.IS)

Itrcclllly. this criterion was significantly simplified, and three forms of ,,11I"ilily crileria were derived; the analogs of the Hurwitz-Lienard.Chipart, Mlllllh, lind Ihe continued fraction expansion methods for the continuous ,·.I"C'. These urc: formulated in the following discussions.

3.5 Determinant method 1,15.30 I hc' clrlrl'lninllnt method developed in this section is similar to the Hurwitz mel hod for the continuous case. Here the stability ,"11111" minl!lllre I'urmulated in algebraic equations in terms of the coefficients III Ilcr 1'"lynmniul /'l:). The derivation of this method is based on the 111111111 simplilicillinns of lhe Schut-Cohn determinants which arise by ' ..... 1111111\/1, Ihlll the cnef1icienls of F(z) are all real, an assumption which is III 1'~'"I1"(I.(,hipllrt

• III IIIf J'III1I1WhIK lll"I:"N"iUIiN. Ilia rCKll1I

\If II plllynomilllllrc also referred to as zeros.

K(,

1'111:11"" ANn Allllllc'AlinN nl' lilt: '-IIANlltnIM

satisficd in rhylliclllllyNI~'nul, Thr flvC' "'"IU1' ,,1"'rllll"lIth"'''' wlll"',lInlllly led to lhe concise fnrOlullllinl\ ttl' Ihe ~'nll!lll'lllnill lin' hi "!lIy 1111 follows. 1. The determinant I~tl givcn in equlltinn (3.15) elln he rreHelll~',II"

,11"",,",",,1

= IXt + Ykll Xt - Ytl where Xt and Yk are the following matrices: I~tl

aD al aa aD a1 aa

0

Xt

(.l.It,)

a,\:_l a"'_2

=

(3.17) 0

0

0

a1

0

0

0

aD

and

a..-l:+1

a .._1

a ..

a..-k-ta

a..

0

Yk =

(3.18)

a.._l 0

all

0

0

0

0

In the first simplification it is readily noticed that the Schur-Cohn determinant I~tl is reduced to the product of two k-order determinants which is considerably easier to evaluate than the 2k-order determinant I~kl. If the ak's are complex. this simplification is no longer possible. The determinants I Xt ± Ykl can be written in terms of Ale and Ble which we designate as stability constants as follows.

IXt

+

Ylel =

At

+ Bk

(3.19)

(3.20) In terms of the stability constants, the stability condition can be written

> IBtl. IAtl < IB",I. IAkl

k-eVen) k-odd

k =

1.2, ...• n

(3.21)

• The proofs of thes'e simplifications are available in the literature.I,II,II,1G For brevity of the discussions we only state the useful results.

",,11.·.1 1111 "rllllill I" "IWIII.·.. "'IIIIIIIp, Ih,' "llIhlhl,Y '·1111 .. 11111 .... WI' Ilhlllin Ih,' IIlh"1 'I111I1,lIlh"IIIIIIIII, ) I h.· """1111,1 11111I1,hll"1I111111 I" Ihe' elluivnlr4'cc ClI' Ihe IUsl cunslruint I f,,1 111,,1111 1\ !lilliI'll' III1Kih,"y ~·III1"'l'IIilll. This lIimplificaliun is based 1111 Ih,' I'lIl1uwillt'. ich'Jllily: (3.22) Sllin'

(A".

II" .)2 is II pusilive quantity, the last constraint can be

/1'1'1""1',1 hy I''(I)N -I)

> 0,

n-even

/.'(1 )/-l - I)

< 0,

n-odd

(3.23)

II' w,' 1,'1 II" he pOllitive and multiply if necessary, F(z) by minus one, lit,· lillilll'unsimint IA"I ?2 IB"I can be equivalently replaced by 1-'(1)

As

:111

> 0,

(-I)"F(-I)

>0

(3.24)

addition to this property we can also write

An

+ B/I =

F(I)(A"_1 - B n _ I ),

(3.25) (3.250)

All - B" = F(-I)(A"_1 - B"-I)

1 The third property which led to the formulation of the stability 1'I11lstmints for values of k of only half of equation (3.21) is given.

A! - B! = At-lAtH - Bt-IBH1 = ![(A k + l

+ (AH1 -

Bk +1)(AI<-I k = 2, 3, 4, ... , n - 1

+ Bk+l)(AA~1 -

B4_ 1)

+ BI<-I)], (3.26)

It is noticed from this property that by forcing certain restrictions on the A's and B's before and after a certain k as required by the stability condition, we can dispense with Ai ~ Bi. 4. The following property which can be obtained from the second and Ihird properties is A,,_1

+ B,,_I =

l[F(l)(A"_2 - 8,,_'1.)

+ F(-I)(A n _ 2 + B .. _2)], n ~ 3

(3.27)

The importance of this equation lies in the fact that we can terminate the last constraint by A .. _I - B"_1 ~ 0 only. This is evident since the constraint on A .. -I + 8 .. _ 1 is necessarily satisfied when the constraints

88 THDORY AND APPI.ICATIDN 0' THI '·TAANIPOIM

=

or

on F(I), F(-I), and A"..• B" .• a" •• tl.nod. Th, U.,ruhllil thl. property will be shown in the dl.cuilion of tho critical onnllrlh" •• will low.a,d,r as in the final formulation of the Itability con.t'ltinta systems. 5. The final property relates to the determinants IXII::i: Y"I - A" B". and states that by expanding the matrix I X 11_ I :t: Y" _" we can obtain III other determinants for other values of k from this expansion. Thl. ,. evident if the determinant is written in detail; we can notice thut It includes inside itself all the other determinants involved in the stability constraints. The matrices Xn _ 1 and Yn-l can be easily remembered Ir their form is noticed as written here in detail.

r",

=

01

Oz

°n-Z

0 ao

01

a.._3

0

0

ao

a.._4

0

0

0

00

00

X .._I

=

(3.28)

and

Yn- I

u.

0 ..-8

0 ..-1

a..

a8

0 ..-1

a ..

0

O.

a..

0

0

o

o

0

I:

a"

0

(3.29)

It is noticed that the matrix X.._I can be easily generated by starting with the first row from ao up to 0 ..- 2 and the second row the same as tho first displaced by one column. This process is continued until we obtain the n - 2 row and column. For the matrix Y..- 1• we start the first row from the extreme right by a .. and continue up to aa. and similarly, for tho second row displaced by one column. Continue this process up to the first column and (n - 2)th row which has the entry a". THE NEW STABILITY CRITERION. I •80 From the preceding simplificationll and properties the simplified stability criterion can be written in tho following form for n-odd and n-cven respectively:

F.?_-C Gt=~

c:

.'. . _: =)'-,_: =

=a»

~

±a.

~=Ds

=Il._: D&::do Ds=Q.

411

± a•

o. ± o.

0,

,.f,_:

011,-»

=....-:

&"-1

=..

k=II-3

=41, :a.

i

=!

. a. ± a, ±a,

k=6

±a.

±a,

410 ± a,,_s

411

± 41 ..-,

at

± 41.-3 a, ± 41,,_, a, ± 41,,-1

Ql

± Q.-I Q. ± 41,,-1 a» ± a.

k =4

±a7

±as

!

±a,,-4

:1::41,,-2

:1::41.-1

±a.-l

±a. 0

Do

± Q.-3

I

as

± 41"1 a,

k=2

Ql :I:: an

a.

a,

:1::41,,-1

:1::41"

410

Ql

at

:1::41"

0

0

410

411

±a.

0

0

0

0

410

±a. 0

0

0

0

0

:1::41'_1 "

±a"

11"-11

410 ± 41"-1

±an - 3 :1::41.. _.

>D

a,:: 0,

= II.

Q ... -s

±o.

I

00

± a,

~Q~

0

±an_1

I I

-.-"

~

0, ...

11..-5

Q.-I

II....

Q....

G .._,:,

Os

0

0

0

0

Do

a,

0

Qo

(lJlt

00

± 02 ±03 ±04

±Os ±Os

01

± a3

°2

± O~

03

± Os

O(

± 0.

Os ± 01

±

a1

± Os

Q2

± o.

a3

± a,

04

k=n-3 00

a4 ±05 ±Q& ±o,

k =5

lao ± °

71 -4

±Qn-3

± Qn-3 k =3 I 00 ± On-2 01

±Qn-2

±On-l

±an_ 1

±an

±On

0

Q2

± ° 71-2

± 0n-l k =1

01

100 ±On I

°3

02

± Qn-J ± On

0

00

01

0

0

°0

=a,._1 :

±a..

0

0

0

0

0

0

0

0

0

Qn-4.± On

I

° 71-2

± °n

On-a ° 71-4 ° 71 -5 ° 71 -6

(3.31)

°3

Q2

±On_l

± ° 71- 1

±an I ...

°1

±On-tl ::6,

I

04

± 08

0n-3

0

°1

°1

lie

a1

lie

,nt\llll.l.'V C'ONIIUJ'I,."'.'IUN "Uk UN'!",.

mlle'.u""

IIYS'r'!MS

91

Tilt' .'lIhlllly &!unilirulntH for II-odd IIrc 1-'(1)

> 0,

F(-I)

<

°

(3.32)

k - 2, 4, 6, ... , n - 1 (3.33)

The IIllIhility constraints for n-even are 1-'(1)

> 0,

F(-I»O

(_I)t(k 11)12 (At - Bit)

> 0,

(3.34) (_1)t(k+1I/ 2(A k

+ Bt ) < 0,

k - 1, 3, 5, ... , n - 1 (3.35)

.,'rum the form of these determinants it is noticed first that the IIrc symmetrical around the center. This phenomenon may be Ill1rihlllc~ll() the stability conditions for the unit circle which is symmetric 1I1111111d Ihc origin in the z-plane. A similar symmetry also exists if the "c'l'rliNIII'Y lind sufficient conditions are sought for the roots of any poly1111111 ill I In lie along the real axis in the complex plane. This will be 11t~"II!t:'l'd Illter in connection with the conditions that the roots of F(z) III' h,'lween 0 and I in the z-plane. The second point to be "bserved is Ihlll III "hlllin the stability constraints both determinants IX" ± Ytl are hI hc' l'lIpllndcd, Lastly, in the final constraints both An-I ± B n- 1 ~ nrc'llllI hl~ !mtisfied. This requires the solution of two algebraic equations .. ,. clrp.rce n - I. In the following discussion the last two points will be .11tI1"iIIl'd. 11II11,'i\'~1i

°

"VAl IIA'I'ION 011 TilE STABILITY DETERMINANTS. 30 To avoid the need of "I"II1!1illll of two determinants of order n - 1, we need expand only one ,16-11" lIIinllnl by using the following artificial procedure. Replace the III ", hy ",.'N und then expand only IXn- 1 + Yn- 1 1. From this expansion, WI' "1111 "blnin 1111 other determinants by a substitution process. For 111111 II 1I"l', I X" • + Y" .1 can be obtained by letting ao - 0, ai -+ ai-l for , - I. 1. •.•• , n in IXn-l + Yn-II and then dividing by (-l)"-Ib n. Simi111I1y. wr l~lI11 ublnin I XII • 9 + Yn-al directly from IXn_1 + Yn-II by letting III Ih,' Inlll'r (nner expansion) a o = 0, a l - 0, Q i -+ QH and dividing by ,,: All nlher delerminants can be obtained by a similar substitution IIIIII'I'~", Tn idelllify the A,,'s and B,t's from IXI: + Ykl, perform the 1IIIIIIWillPoI'fClcesIi, Afler expansion, examine every term which is a product III ":~ 111111 ".·s; if' it contains an even number (including zero) of b/s. 11_~1p.I1 II III A,; nlhcrwise, aNsign it to Bt . After collecting the terms of .4, 111111 II, ,'epillce 1111 lhe ",'N by (he a/s,

a,:s

92

THEOR.Y AND' APPLICATION OF THE Z-TR.ANSFORM ILLUSTRATIVE EXAMPLE

In this example we shall obtain the stability constraints for the fifthdegree case. that is

F(z) = ao We form X,

+

+ alz + a.z2 + aar' + a,z' + aszS.

as> 0 (3.36)

Y. by replacing the a/s of Y, by b/s: ao

IX. + Y,I =

ba

+ b. a o + bt

be

b"

+ b.

al

aa + b.

Q3

+ bs

+ b"

a2

bs

au

al

0

0

Qo

Q1

(3.37)

After expansion, let ao = 0, al = 0, a. - a i - Z and divide by -b~ to obtain IX. Y.I. To identify At ± B, and A. ± 8 2 we can perform the following manipu· lation. The constants At + B4 can be readily obtained from IX4 + Y.I by replacing all the b/s by a/s. Replacing the sign of all the terms which contain odd number of ba's by negative signs and replacing all the b,'s by a.'s, we readily obtain A4 - B.. Similarly, Aa ± B2 can be obtained from IX. + ral. The stability constraints for the fifth-degree case are

+

F(I)

> 0,

F(-I)

<0

(3.38) (3.39)

ALTERNATE FORM OF THE STABILITY CRITERION. To avoid having to satisfy for two inequality equations of degree n - I. we can write the stability constraints in different forms so that only one equation of degree n - I need be satisfied. This can be readily accomplished by noting the second, third, and fourth properties among the stability constraints. The final stability constraints for any n are given as

II-odd F(I)

> O.

(J.40)

F(-I)
(-l)"U'~ 1I/2(At -

Bt ) > 0, (_1)tCt+ll,2(AI; + B,,) < 0,

k=I.3.S ..... n-2 (JAI) PAl)

n-even F(I)

> 0,

F(-l»O

(JA "

k -. 2.4.6 •... . 1/'

7.

(.1.44)

('.4~'

STABILITY CONSIDERATION FOR LINEAR DISCRETE SYSTEMS

93

SUMMARY OF STABILITY CONSTRAINTS FOR LOW-ORDER SYSTEMS}·I&

In

the following the stability constraints in terms of the coefficients of the low-degree polynomials are presented. using the alternate form.

n=2 F(z) = a o + alz + aaZ2, > 0, F( -1) > 0 ao - 02 < 0

a2

>0

(3.46) . (3.47)· . .;

F( I)

(3.48) .'

n=3 F(z) = 00 F(l) > 0,

+ alz + aaZ2 + oaza, F(-I)

03>

(3.49) (3.50)

0

<0

100 1< aa 2

,,=4

I 3

ao - a

(3.51)

< aoal - al 0 3

(3.52)

alz + aaZ2 + aaZ3 F(-I) > 0 a! - a: - ooaa + ala. < 0 F(z) = ao F(I) > 0,

+

a: +

a: -

0oa 3

a~ + 20 0 a.a,

-

ala,

+ a,z',

(3.53) (3.54) (3.55) (3.56)

<0

+ alaaa. -

aoo: - a.a: - aoa~ - a:a.

- a:all - a~o,

+ a: + aOalaa > 0

(3.57)

I Ising algebraic manipulations the constraints for n = 4 could be more '1lIlIplificd so that instead of the two second-degree equations (3.55 and I. '1(,). we obtain the following two first-degree equations:

< 2(04 a3 > 2(ao -

03

01 -

0, -

"

5

(3.58) (3.59)

ao) a,)

+ alz + aaZ2 + a3z3 + o,z' + arl', 1:( I) > 0, F( - I) < 0 F(:;)

= ao

I(lul < or. (tI.II,(l6 (

(II:

Qoa: - a 2

a: + a: +

0 00 805 - 0 0 0:) =F

05>

0 (3.60) (3.61) (3.62)

2 2 »0 + OoOa + aOa& - °oo,a, < 0 (3.63) lI~)1 - (CloU, - al05)2 + (0 00, - 010&)(0! + 0 103 - ,,~ - U2U, - a: + 0:> + (a003 - aaos) x (11 11/,. - U,06 - UOU3 + 020,) - (0002 - 0a05) X 1(11: - ":) - 2(1/ 11114 - U.(16)] > 0 (3.64) 2

",(16 - ooaaOa -

3

05

• WI' III11Y IIl1tr tllllt 1"111111 chi\ rll"llllllll we IIlIv,," II. t·

II. ::.--

O.

04 '1IIIIlIIt\' ANn AI'I'I It AIII1N en 1'111:'" KANIII'UKM Theile cn".ln,lnlli rlUl 1,11111 he 1I1"'1,lIlIed hy IIIM"hllll,· lIu"ulmhuhlll_ Ihlll h'lItall" nf the twn third.daMr.:r "'IUllllm'lI (.UI.\) wr hllvo

1111

(1M)

In concluding this scclinn it shnuld be pointed nul Ilml thc dctcl'lnilUu" stability mclhou cnuld be useful for numcricllllcliting oJ' pnlyncuninlll flit stability and with its alternatc form they yicld the numhcr tlJ' runlH IIlIlIIldc the unit circle if the system is unstllhlc. This will hc i11l1strlltcd Intcr In tho discussion of the number of ruots outside Ihe unit cil'cle. If Ihe !itllhility constraints arc further simplificd as indicutcd for" ..., 4 lind" - S (1I11i1l" equations (3.58). (3.59) and (3.65» this informutiun un the nUlnh,'" til' roots is lost at the advantage of more simplified sttlbilily cunstrninlH. Fur system design this simplification is only of academic vullic. for with hi~hrr degree polynomials the final constraint. that is. A n_ 1

-

Bn _ 1

<00 >

is of major importance, as will be discussed in dettlil in the folluwlnM section.

3.6 Critical stability constraints for system dcsign211 In certain cases where the discrete system [whose chaructcristic cllllnlinn is F(z)] is initially stable, and if one or more parameters (for c'UIIUI,Il'. gain) are changed so that it becomes unstable, then only certain conslrllinlll need be satisfied. These constraints are called "critical construints" nnd are discussed in this section. Assume the characteristic equation of the linear discrete system III described as follows: with

II"

>U C:\'M)

This equation can be also written in terms of its roots

Zj

as

1,2, .•• ,n

F(z) =

Qn

,

II t

(z - z,)

We can readily ascertain Ihut F( I) ;> () und ( -- I)" F( - I) two of the critical cnnstruinf..'i, hCCIlII!iC Wt' "1111 Wl'itl'

(.U.1 )

>. () &:CllIlIt illll"

1,1•.... "

/o'(I) ....

CI"

II

(I

/,)

(HIM)

III A1111 II" I'tINlIllJlllA III1N 11111 IINI All UIIII'IlI II IIVNII MN

111111 1),"'(

I) - II"

'.. , ...• " II

+- :.)

(I

I)~

(3,69)

II " n'lIclily IIlIliw,1 I'rum these two equutions that F(l) = 0 and I) • n "I1I1!.tilllte the critic,,1 conditions when the real roots move "1I1~lIlt· Ih,' unil drde lit ;: = I and z = - J, l'ullht'l'lIIl1n\ the (Jther crilicnl constraint which yields the condition \\ 111'11 tht' l'IIlIIpll.'x runts lIIove outside the unit circle is given as20

N

1,2, ...• 11.

·1

'" I

.. /l , , 1-- (-I)"'"

IIIBa,,·1 "

IT

,<,t

(I -

z,z ) ,It

(3,70)

11", 'A' (cllnjugnte), Ihcn '::,z" = 1::,1 2 and An,_, - Bn ._ 1 changes its ~I/~II Wlll'lIl'ver /I pnir (or udd pairs) of complex conjugate roots move 1I1I ..... h· the unil circle. WI ... II tilt' "Ilmplc" ruots arc of even multiplicity, the equation A" • - 11" 1=0

h "h" nilkal although it docs not change its sign. In all these cases the h'·l'IlIllI.'S ullst:lblc whenever An .1 - B n_ 1 = O. Therefore we can "l1l1ll1llll'i/l' Ihe preceding critical constraints discussions as follows. 20 I III 1111 illilially !.tahle linear discrete system, a certain parameter is changed 'II' Ihlll I,tahilily ccuscs, the following are the critical conditions which vlI'hl Ih,' IlliIximum variation of parameters for stability limit.

',~',Ic'1II

1-'(1) ~ 0,

(-WF(-l)

(,.- 1)"1" 11I2(A" I -

B/I ,I)

i!:

0

~

0

(3.71) (3.72)

II .... t1i'l'ussinn shows that for analytic tests for stability when applied III Ihi' .h',i,'.11 III' cerluin systems, it is not necessary to consider all the ',I,IIIIIIIV l"IlI1!.lraillls hut only the critical ones. There the solution of the 111",11''',1 "c·/',rt·c c'luutinn is at most of (II - l)th degree; furthermore, this ''11111111111 lIt'cd he solved I'or only real roots, which can be easily done "I'll'hll'lIl1y. III a fullllwillg section an illustrative example indicates the npl'llI'lllillll IIIllI impnrlance ()f this point .

.t,7 Numher (If mots or a real polynomial i.. ,itll' the unil drde l ,4 IIII' ',lllhllily nih·rio .. cliSl'US!\l'd curlier in this chapter yields the necesv 111111 .. 1IIIidt·1l1 l'lIndilillll lill' al\ the mots of Fe::) In lie inside the unit

~,II

96 TIII:ORY ANI) APIII.IC'AIiUN 01· 1111; .·IIANltnll ..

,t"

circle. To obtain inl'ornultinn un Ihe nlllnhrr ul' mu'" Ihlll 11" Inilid. unit circle if the system is unstllble, we ~an mudily Ihe I"c\tllltli lll,· cussions to cover this casco Following the work uf Schur·Cuh" I'rr"c"lld in equation (3.14) and using some of the: pruperliell "I' the IIt .. hlllly constants At'S and B,:s we can state the foll()wing. The number of roots of F(t.) that lie inside the unit circle iA ellUII to the number of variations of the inequalities of the folluwing C:UI\straints: n-even

n-odd

1>0

1>0

IAll IAll ~ IBII 1.0411~ IB21 <=> AIAa ~ BJBa IAal ~ IBal 1.04.1 ~ IB.I <=> A8 A, ~ BaB,. IA,I ~ IB..I

IA .._II IA"I

~ IB_11

~ IB"I

¢>

IA.I ~ IA.I ~ IA.I ~ IAIiI ~ IA.I ~

1.0411-11 F(l)F( -1) ~ ~

~

IA"I

~

IBII ~ A.

>() < ()

IB.I IBal ~ A.A. ;.; B.B. IB.I IB,I <=> A.A. ~ B.B. IB.I

IB_II

~ IB"I ~ F(I)F( -I) ~ ~

It is noticed from the preceding that for n-even we do not have evaluate the even determinants but can use the relationship A: -

Bf = A.t-IA~l - B.t-lBI>+1 == ll(At+l + BI>+t)(A.t-t - Bt_l ) + (AI>+1 -

B~I)(A~l

'n

+ B._I»)

,h.

which are obtained from the calculated odd determinants. Similarly, odd determinants. are avoided for n-odd. It is assumed thlll in lid. counting of number of the inequality variations, none of the cunstllllill A, are equal to Bt . Thus the number of ruuts outside the unit cirt:lc I. equivalent to n minus the number inside the unit circle. To determine the number of roots insidc Ihe unil circle when A. Bt-l but A" == Bt • a speci .. 1 prucedllre is relillil'cd I"r thill Hingulllr CIINI.

I"

~I AI'" II V C'IINIUIIIIlA IIUN 'U" IINI'AM IIIIIC "tI II lIVlIlI-Mli

97

" ,II'IIl,h,.1 ,11""",,""11 III' Ih,' 1II1I,1iIil'lIli.H1" r\'tllli"'~,1 til ,I\'tcrmine Ihe 1IllIlIh,'r III 111111'1 ,h"'\'II",,,\',1 in "PIl\'nlli" 2.

'I

J,M Rclnt inn:o.hip hetween the determinant 1II"'lhlltl and lIurwitl. critcrion l ,20 I hi' 1'I'llIlllIn,lnp h'~lwc"1I sUllie uf the stability constraints and Hurwitz 1'1111,,11"111111 .. mill~~ Ih,' hilinear transfurmation can be established. If we \\ IIh' IIII' rllllllwilll~ lIurwit.l. pulynolllial obtained from F(z), with Q n > 0 hv lI .. illl'. Ih,' hilinc:lf Irnnsfurmation (3.73)

hu = (-I )nF(Z)/',,_1

(3.74) (3,75)

\\ h."", 1.\:,' II is Ihe next-to-Ielst Hurwitz determinant. The relationship II "I h:" I,..~'n wrilied by expansion for low-order systems and can be 1"'III',ally tI~'l1Iun~tmted through the use of Orlando's formula~,21) and the 1IIIIIII'al' Il'all~limnaliun .

.'.1) Table furm:!·ll.III,I:I.:l4.11:; In 1111'. 'I'~OI ill II an alternate form of the stability test for the unit circle ,', •h·.1'1 I ....,·, I. rhb furm is presented in a table form similar to the Routh III"'" 11'01'" IiiI' thc ~llIhilily in the left half of the s-plane. It requires "\lIlnallllll IIf unly secund-order determinants and can be easily pro1~""lIlIlI'tI 1111 a digilal computer. The table form is particularly useful III' 1111 II lI'ril'a I leslin~ uf polynomials as well as for design purposes. The 1IIIII1IIIalillll III' Ihis furm is bused on the early work of Rouche, Cohn, 1I1I.lllIh·1 1111 Ihl' work uf Marden. The derivation is presented in Appendix I II', \\1'11 iI" • hI' varillus reccnt mOllifications and their extensions. IIII' III·,·,', ..:!r}, alltl sullicient cllndition for the following polynomial Q II

>0

(3.76)

,II

I

all II ..

rll .. l~

im.itlc Ihc unit circle can be obtained by forming the lahl,'. (II !lhlluld be lIoted that the clements of row 2k + 2 I ""'.1·.1 III Ilu' 1'1"111"1110; IIf Ih .., ruw 2k + I written in reverse order. A II. I. .', .... 1 hI h,I\I'

111111',111\,1111',

98 THEORY AND APPLICATION OF THE z-TRANSFORM Row

to

J 2

00

3 4

bo bn - 1

Zl

Zl

an

---1--- - - - - - - - - - - - - - - - - - - - 5

6

Co Cn _ 1

Cn-s

Co

Cn_,

---1----------------- (3.77) ---1--- -------------------s, 2n-S ! s, i 2n - 4 i $11 So

$3

St

S3

$0

' . _ ••• 1

---1--- --- --- --- --- --- -----

where

'0 =

I

=

I

'2

So

S3

So S3

The stability constraints are

> 0, laol < an,

(-I)nF(-I)

F(1)

or

Ibol > Ibn_II leol > len-21 Idol> Idll-al

>0

(3.78)

a.. > 100 1

(n -

I) constraints

STABILITY CONSIDERATION FOR LINEAR DISCRETE SYSTEMS

99

It is noticed from the table that the dotted entry r 1 is not needed for calculation; hence the corresponding computation involving the dotted entries in rows 2n - 4 and 2n - 5 becomes redundant. The number of roots inside the unit circle, if none of the preceding constraints becomes equal, is given by the number of the products Pk , k = I, 2, ...• n, which are negative, where

(3.79) PII = [1001 - all] Ibol - [lbn-lil ... Ir21J(1r~ - ,:1 - Irorl - r,rall (3.80) The difference between n and this number yields the number of roots outside the unit circle. X

[lrol -

EXAMPLES

(I)

F(z) = 3 - 2z -

=z2 + Z3,

11

= 3

(3.81)

F(z>I, __

By applying the constraint 1'(-1) we notice that 1 > 0; thus Ihe system is unstable. Therefore the table need not be computed. However, to obtain information on the number of roots outside the unit circle we form the following table: Row

ZO

%1

3

-2 3

- 2

'2

3

31

i- 2i

L~~.!

3

(3.82)

5 2

8

1001 - an = 3 - 1 > 0 1'2 = 2[1hol - Ihll .. .11 = 2(8 + :> > 0 p .• = pus. pos. {l82 - (~)21 - 18(-:) -

P. =

(3.83) (3.84) :(~>I)

>0

(3.85)

11"'I'l'furc nu ruul exilits inside the unit circle and all three roots are 1IIII',i,h' Ihe unit circle. It may be noted in passing that the condition on II';; I'ia II'". nmy also be obtained from F(1)F(-I) and the IlIlIllIplklllilln IIrlhe IIlher cIIllslmints. For inlitance in this case the sign III Ihl\ h' .. ", i~ l'lluivulcnl III the liign or /0'( I)F( -I), Ihat is,

'.'lIlI

"Mn II":

- ':1

- 1"lIr. - r.,'.11 "'"

S~1l f( -I W( I)

(3.116)

100 THEORV AND APPLICATION 01' TIIH a-nAN.POIM

In view of the preceding relationship the dolled entry In lho lalll IIno need not be calculated. (2)

F(z)

= 0.0025 + 0.08z + 0.4126.;11 -

1.368.;3

+ .;'.

n-4 (3.87)

ZO

Row

1 2

0.08 -1.368

0.0025

I

4

-1 -0.0834

5

0.9936

0.4126 0.4126

................... 1.368 -0.41161 0.4116 1 1.368:

l

~

3

Z3

.;4

-1.368 0.08

0.0025

Z2

ZI

: ................... :

!':i402'i : ................:

I

-0.0834 -1

0.5256 (3.88)

Stability test F(I)

= 0.1271

laol < a .., Ibol > Ib'HI, ICol > IC n-21,

F( - 1) = 2.703

> 0, 0.0025

>0

<1

(3.89) (3.90)

1 > 0.0834

(3.91)

0.9936> 0.5256

(3.92)

The system is stable.

Simplifications of the stability constraints l 2.24

An alternate simplified form of the stability constraints given in equation (3.78) can be obtained by noting that • bo = a~ and

b~

- a! < 0

> b~_t,

or

(3.93)

Ibol > Ibn-II

(3.94)

These two constraints can be equivalently written as bo - bn - 1

< 0,

bo + bn-l

<0

(3.95)

Similarly, for

do = c~ and

C!_2 > 0

(3.96) (3.97)

NI'AlIII.II'Y t!UN.II)ItItA I'IUN I'UK LINhAlt UlS('KI! 1'1; lIYS"rIlMII

101

we can write tiD

+ d,. .3 > 0

(3.98)

Similar rclutioll!.hipli can ulso be obtained for the other constraints. Thus the stability conslrnint fur f,z) can be summarized as follows: /I-even F( I) 00 ('0 -

> 0,

F( -1)

< 0, (',,-2 > 0, an

>0

+ an > 0 Co + C,._2 > 0

00

eO -£>n-4>0,

So -

S2

> 0,

n constraints

So

+ S2 > 0

n-odd F(I)

> 0,

< 0, do - d n_ a > 0, /0 - /n-5 > 0, bo - bn- 1

<0 bo + bn _ 1 < 0 do + dn- 3 > 0 /0 +/n-5 > 0

F(-I)

(n - I) constraints

(3.99)

'0 - r > 0, 2

This form can be further simplified by noting that each of the above constraints is divisible by the preceding constraint with alternate signs; that is. ('0 - ('II 2 is divisible by ao + ant do - d n - 3 by bo + bn - 1 • etc. This property is utilized in the next section to show the equivalence between the table form and the determinant method. R('/tltiollship helll'e£>11 Ihe lab/e form tile' c/('/('rmiIlQIII method 24

(11/(/

The relutinnship hetween the stability criterion introduced earlier and 'he: '" hie form will now he established. The importance of this relationship

102 '1'lIIn\tY ANI) AI'I'IIC'A'IIUN UI' 1111: "'RANlltuIlM is thut the: eVllhllltinn (If hi~h~r urdcl' d,'lerrninllllh IIIlIy h" 1111111'1111",1 hy obtaining their value directly 1'rIlIlI Ihe 1IIIlIe I'urlll. To establish the identity between the A" ± II" uf Ihe prcccdin~ crilcrinn and the coefficients from the tuble, we may nute the rulluwill~ hy ,lircci examination of the terms Al ± Bl and All ± 8 2 lind the curresJlllllllinlt terms of the table, that is, (3, WO)

(J.IOI) Furthermore, by regular division, we can further identity the relationships.

A. =F Bol ==

do =F tl" (a: - a:')(Aa

I'ull(lwill~

3 _

_.

± II.) (3.102)

and

A =F B = 5

5

eo =F ell -5 (a: _ a~)2(b: - b~_I)(A3

± B3)

(3,103)

Using the induction method, we can show the following general relationship.

A =F B _ "

10 =F

" - (I ao - alla)1c-3(b0ll

-

III-Ir+l

bl11-1 )k-4(Co2 - CII_I I )"-5 ' • , X (h: - h:-k+3)(A/r-Z ± B". •) (3,104)

where the I's and h's are the appropriate entries in Table (3.77). By imposing the stability constraints on the coefficients of the tuble for n-odd and II-even respectively. we readily obtaih the constraintli on A" ± B" which are noticed to coincide with the determinant method. This will be illustrated for n = Sand n = 6, respectively.

n = 5: The stability constraints for this case in terms of the construints in equation (3.99) are (J.IOS) (J,IO()

1-'(1)

> 0,

,.~ -I)

< ()

(l.W7)

NIA"'IIIY C'UNIiIItl'MAIlIIN IUK IINI A" UilltlU II IIVIIINt!!'! 111111 lilt 111111 ....

II::

(till j tly )/I(f1ii _. "r.)(A w =F 112 )] I h"~,, \'I1I111lmlnl!o III lerms of A's nnd 8's CIIIl be rfllm "" f ". < () Ihlll hll < 0; we finally obtain for

I It, - "" t "., A. I II. -

,,~. wrllll'lI hy 111111111'

,,,,,I "" -

In.l

II-~

Au -t 11K

< ()

(3.108)

(3.109) F( I)

> o.

F{ - I)

<0

(3.110)

II .. (,:

III this casc the constraints arc (In -

(18

< O.

00

+ 0& > 0

(3.111) (3.112) (3.113)

F(I)

> 0,

F( -I)

>0

(3.114)

lJ!>ing the relationship introduced in equations (3.102) and (3.103), we ohtain (3.115) AI - BI < 0, AI + Bl > 0

> O. A3 + B3 < 0 Ar• - B.i < O. A5 + B. > 0 F(!) > 0, F(-I) > 0

A3 - B3

(3.116) (3.117)

(3.118)

Following the foregoing procedure and utilizing equation (3.104), the clluivalcncc for any n can be readily established. The stability constraints for n-even and n-odd from the table form could be written in an alternate form that coincides· with similar forms from the determinant form. This form can be obtained for n-odd by starting with constraints 0 0 - 0" < 0, 00 + an > 0, and thus the final constraint is > O. Similarly, for II-even starting with bo - b n_ 1 < o ~Ind ho + hIt I < 0, the final constraint is So - S2 > O. To show ihat + > 0, or (so + S2) > 0 is automatically satisfied, wc rewrite equation (3. \04) for k = n - I to obtain

'II - '2

'0 '2

'0

+ '2 = (An 1 + B n- 1) X [(a~ - CI!)"-'(b: - b!._I)"-s ... (An_a - Bn - 3 )]

(3.119)

104

rur.ORY ANn AJlI'I.IC'AI'IUN III' lltll

.·rltAN.I1,"M

Since all the constraint!! in the bracketed tern,,, "r" IIltth"I,," fur .,"hlilly from the constraints before '0 + '., the con!ltruill' '" .. '. iN tl"p""""'" only on A,,+l + B"_I' From the properly on p, K7, thuliN,

A,,_l

+ B,,_1 = i[F(I)(A""'2 -

B,,2)

+ F(-I)(An" + I," .>1, ,,~:\

0,120) we notice that A"_l + B.. _l is satisfied from F(l), fX-I), and All aT B,,_2 constraints; thus + or (so + S2) constraints are also siltisficd from F(l), F( -I), and the preceding constraints. Therefore we can state that the critical constraints for stability from the table form are (3.1200) F(I) ~ 0, and ~ 0 (3.120b) To facilitate the derivation of Ale ± B/c from Table (3.77), we can further simplify this table as foIlows:

'0 '2

'a - '2 Row 1 2 3

4

...

%,,-k

. ..

%"-1

°1 ... .. .

Dn-t

. .. ...

0,,-1

an

QI

Do

1.0

1. 1

Zl

00 an

°1 a n-l

an-t

bo b'l-!

b, b.. _t

bl b..- a

Co

C1

C.. _I

C.. _I

7 8

d'I

d;

d~_,

d~_.

d~_5

10

r~ e~_t

e~_1

2n - 5 2n-4

.I'~ .I'~

.1'; .1';

.1';

2n - 3

,~

"

"

9

where h/c and die'

Cle

r;

I

bn _ 1

bo

C. C,,_,

5 6

~

at

... ...

.1"1

Zn

('n-I Co

...

...

d~_.

(3.121)

d:

e~_,

e'•

.1'; So

--

t

are the same as in Table (3.77) and the change occurs in

_-I Co C,,_I

C,,_t-1c CI;

I..!. bo

' _I d~

e/c-

d~-~1c I.!. , ... ,

d~-3 dl;

Co

(3.122)

'11 Aim" y • ClN_IIII:U IIUN IHIt IINI AM 11I!l1"M1 II ToI\'ToIII MS

\()~

I hI' .11''''111111111111 .. III.' 11111111,.1 hv Iht' ,hvi!oiull uf the lirst terlll of two 111\\ ~ h,'IIIII' 11""',11111.1111', 1·"II ..tlllt·li,,1I the st:,hility clIllslraints from this Itlh'" Ill" II 1a'l1 I 11"111 III SI·h .... -( '"hll determinants.

< 0,

1~31 = d~ '" I4Jo n

< 00 _l 1 ='0'2 -'2'2 >

n-even n-odd (3.123)

hlllll

Ih,'

pn'~'t'dillg

t:lble we can easily identify Ak

A, ± B. =

As

± Bk. as follows:

do ± d~-3

bo T bll _ 1

± B" = (e~ ± e~_,)(ao ± Co

±

A 6

T

a,,)

Cn-Z

B = (f~ ±.f~-, )(bo ± b"_t) 6

(d~ T d~_3)

(3.123a)

Similarly. all the others Ak ± Bit. can be readily obtained from the form of the Table (3.121).

~illlplilied

Gt'Ilt'rotion oj the stability consl,ainlsz'I,Cl In the following discussion the stability constraints which are obtained frum the table form will be easily generated without the use of the table. This generation is based on a simple rule which can be easily remembered and applied, GENERATION RULE

Given the following real polynomial (whose coefficients are given by symbols) whose roots are to be tested for stability within the unit circle. F(z)

=

00

+ a1z + az%2 + ' .. + 0,,_2Z"-2 + O"_IZ"-1 + 0nz", an> 0

(3.124)

Perform the following manipulations. I, Form the first row from the original polynomial coefficients by replacing only a ll _l and an byb n_1 and b" respectively. The second row is formed from the first by reversing the coefficients and relabeling certain of the 01t.'S by bk·s. Form the third row by expanding the second-order matrix formed from the first column and all the other columns (starting

APPI.I(~A1·ION 01' 'rllil I-TIANlmIM

106 TIIIIORY ANI)

first with the column at extreme risht). ThoAO ara .umn1ltrI7.ed ft. fullow.: Row

z'

Zl

z·

1

a. b"

b.. _J

al

a. b".1

2

3

_

-- ....__I"a".,.•....... . .. -_ . _b...........

a: -b! aaOl - b".lb" a.b, - b"a".1

"all"

a.b" • -

I"

I" I

.

II" I

II"

".

fli

tlob" I - "I""

2. Starting with D~ - b! form a sequence of expressions, each fmm the preceding one, by the substitution given in row 3 with relabeling sume coefficients, that is by replacing

0: - b~

DO

-

01

-

DoDl -

De

-

ODDs -

b" - Dob,,_1 - Dlb"

b.._1b..

b.._1 - Dohn-a - Dshn

b.._zb,.

(3.124t1)

Continue this process to generate the required n - I constraints. After performing this process, replace all the bit's by DIt'S to obtain the tlnnl constraints. These n - 1 constraints with F( -I) > 0, (- 1)" F( -I) > 0 constitute the necessary and sufficient conditions for the roots of F(.) tn lie inside the unit circle. ILLUSTRATIVE EXAMPLE

By applying the preceding rule we shall generate the stability constraint" for the following fifth-degree polynomial: F(z)

= Do + D1Z + DaZ'J. + 03'1.3 + D.Z· + 0&'1.1,

Ofi

>0 (3.125)

1. Form the following three r!>ws: Row I 2

3

ao

a,

al

b,

b,

b,

Qi -

bl

aoQI -

b,b,

as bl

aob. - asb,

a.b.

aoba -

h.

hi

a,

flo

a.h, - n,".

2. Form the following four constraints by replacing the various coo",cients from the third row, that is, aD-aD -

"5 . ~ Do". - tI."a

01 -

h. ~ l'U"3

II , b' } aoal - h.ha

al -.~ aoo_ - "a"" aa .. ~

tlotl, - "_"a

-

11."11

•4'u". - 11.1,. I,. • "0". - "."11 "11

(lIH,,)

IIt'AhlUl'Y C'UNlllUltU'UUN t,(llt UNIIAI mAC'111I'11 IIYII'I'IIMII

.

107

Th. """.trAlnt" .,," "~a

- "',- IX

": - ,,: •(u: - ,':)' -

(aob, -

a,h.>· -

(' •(" - 1(,,: - ":)(1/0". -

Ct' -

(aob, - albrl =

P

a,b.)

- (aOul - b.b,)(aob, - a 1b,»)2 = "

,,:)1 - (aob. -

" .". - {[(a: -

a1 b,)I][(a: - b:)(aobll

a.b,)

-

- (uoa. - h.h,)(aob, - alb,» - [(a: - b:)(aoal - b,b$) - (uob. - a,ba)(aob. - a 1 b,))[(a: - b:>(aob. - a 2 b,) - (l,oU, - b.ba)(a.b. - alb,)W

- "I _ AI = l4y

re'rll,cin~

~ ,,~

II = "

~

:=

==

(3.126)

"lIlhe h/t'ft by a/t's we finally obtain the stability constraints

i"( I > > 0, IX

~

-

Ct l -

F(-I)

<0

(3.127)

u: < 0

(3,128)

(aoo, - alaS"

>0

(3.129)

(I' - [(a: - a:)(aOa3 - ala.) - (aoa l - a,a,)(aoa, - ala,»)1

>0

(3.130)

"I _ AI > 0

(3. t 31)

Thclle' cun!ilnaint!i could be simplified to obtain ','( I)

> 0,

(tI: -

a:>

F( - J)

±

<0

(aDa, - 0laa)

(3,132)

<0

(3. t 33)

"±A>O

(3,134)

The' I'rcc~'din~ !ihuw!i that for n-odd, we can obtain these expressions d'rl'~'lly f'rnrn Ihe polynomiul coefficients by certain substitution of two

""'1'11 III "limc, Thi!i cun be accomplished by forming rows 4 and 5 to "hlllin Ihc rClluired !iuhstitution,· Similarly, for n-even we can skip the Inll'l'Inrtlilllc Cnn!ilrllinl!i by using two steps at a time. Tlar rl'I"Iinn!ihip " ± Acan be also written as " I A ..... (,': - ":)(,,: - I':> T (aoa. - ula,;)](A, ± B.) (3.135) • I "' 11I~11"'l·". In Ih,. l'll\t ""

• (,,: -

":,. -

(n.,h. - a,".)',

"1 •I,,: '.:11",01', ".".' - (flJ.. . ".".)(".1'. - II,'''),

II.

~"

..

10K

I'IIWlty ANI) AI'Plle'AnUN

I'"

CJt

Iliit

'-IIAN.tUM,..

where A, andB, are given in ellflier dlll~u"!llnn. ""n~" Iho "IIIIIIIII,hll in equation (3.134) can be replaccd [noting "'Iulllinn (.'.1 .I~)j by A I I II.:> 0, A, - B. > o. Furthermore. the constraint A, % B, can be ailio ubillined Ihun lhe following equivalent form of,.. % A. that is, A,

± B. +

IXclI I

yCIII a IIt'"t (a~ - a:)[(a~ - a:) T (aoa, - a,QaH ..L ;;I;

(ll3t,).

The terms Xru and y~2) are obtained from (3.17). (3.18) for It. .. 2 by replacing the elements of X2 , that is, ao. Q10 by the two step suhstituliun using row 3 and replacing the elements of Ya• that is. b,. h~, by II !timilllr substitution as indicated earlier. After this substitution. the "It'S IIrc replaced by a,;5 to expand the matrix. Therefore we can replace (or obtain) any of the kth-generated cnllstraints by a certain relationship from the expansion of the IIIl1tr;II IXLj~j % Y~~)bt-lit' which indicates the relationship between the tnble furlll and the determinant method. The preceding equivalence could be Itlsn nl' advantage in numerical testing of polynomials where we are not retlui!'cd to expand high-order determinants. In conclusion, we discuss how to obtain the stability constraintli by It certain generating rule which can be easily applied. This rule is silllilltr to Routh rule number 2 for the continuous case. 10 The pmof nf thi" generation process is not presented, but it can be deduced from the identity of the table form and the determinant method.

lIIu.rlratiL'e design example The critical stability constraints developed in an earlier section are nnw applied to a design problem. For the sampled data feedback syslem shown in Fig. 3.3, we shall obtain the maximum value of the gain It. fur the stability limit. . For the parameters given, the z-transform of the open-loop tmnsler function is given as

~l(Z) =

<[1 -s

e-'l'· e-U&,'l'

k

s(s

]

+ 1)

= 0.2223k (z + 0.03)(% + 1.755) %2(% _

1)(% - 0,368)

• Another form of this equation can be written as IX :i: r"ll A. :!: B - - - -•.- ...... -

·

.'II . 6.-":-":

(3.Ll7)

1114111111V C'IINIIIIII:UIIUN l'IIM IINI"" 1)1111 MI II IIVIIIIM'I

Hf.' I () II •

••

X

.

-1";-';';'

_

lUI) ('(II) )

•

-I. ...._... _ _ __ 7'.1 b' • 1.25

l'I(lIIRI U

I hi' "Y'Ih'1I1

l'Iull"lu'l~'rislic

H.-) - I

A SlIlIlplcll-daC;I system with pure delay.

C4uution for stability investigations is

+ ~ql(z) =

Z, -

1.368::3 + (0.368 + 0.2223k)%2 + 0.3974kz + O,0123k = 0 (3.138)

I hI' 1II"I",,'tlilllt i). II fourth-dcgree polynomial and we notice that if II. A • , Ihl' ).y).ICIlI is stable, This can be readily ascertained from a _~"I .. h III Ihl' "lilli-locus us u function of k, Hence we apply the critical 111111111111111'1 for" .. 4 10 obtain the maximum gain. HI)

n

I)

~

k>O

0,

> n,

k.

< 16.75

I hi' IIlh"1 nilicnl <:ullslruint from the determinant method is

< -247

or 1111111 1111' IlIhle, form Ihc critical constraint k

('"

('2

~

-5.5

< k < 0.72

()

I hi .. 1'Il1I'llrninl clln he written as

",," !'\ltlll'

('~

= (II" + a4)(A 3 -

(3.139)

8 3) ~ 0

1111' l'l'ilklll (crm is A3 - 8 3, in both cases we obtain the maximum A11111_ :< n.72,

t!IIIII 11'1

.\, J()

Division mcthod'4,24

IIII' 1111111 rorlll or ohlaining a stahility test is using the simple method of 11111'\11111 Ill' polynomials, The derivation of this method can be obtained 1IIII'I'IIv I'l'lItn (Ill' IlIhlc form as follows. Slabilily

11',\'1

A IIC'C't'\~III'Y 111111 )'1I11k-icIII condition for the real polynomial n:)·

II"

~

til':

+ Cl~;:~ + . , . + ak::'" + . , , + a..z". with an

>0

(3.140)

110 THIOIlV AND A"LlrATION O' TN. "TIANlI'OIM to have all its roots Insldo the unit circle In tho follows. Form the inverse polynomial F·(z) as

I'

F·{z) - znF ( ~) - an

I· plano

t.

obtained ••

+ aft_Iz

+ an-tz' + ... + an_",1e + ... + a.,zn

(3.141)

Obtain the stability constraints ctle by simple division as shown. F(z) F*(z) - CI.o

Fl(z)

+ F:(z) -

FI(z)

CI.o

Fa(z) + Fa(z) -.... -

+ ctl F:{z) Oto

CXo

+ Otl + Ot,

,(z) + ctl + ct•••• + Ot,,_11 + 'F"..• ,,(,)

(3.142) where FI(z), FII(z) • •• are the remainder polynomials obtained from tho division and Fr(z), Ft(z) . .. are their inverses, as defined in equation (3.141).*

We now satisfy the following constraints for the stability test. F(I)

>00 > 0, F(-I) <

n-even n-odd k = 0, I, 2, ..• , n - 2

The proof of the test lies in establishing the identity between the the stability constants aD, boo Co•••• of the table as follows:

(3.143) Otle's

and

aD Oto - -

an

bn- I Otl--

bo

(3.144)

• To obtain the inverse of F,(z). we factor out the con\mon fllclnr , or Its ruwerll hornr. we apply (3.141); also. ror F.(,) and FI(a)•••••

I rAlllU rY r'ONIIUItRA I'IUN 'Oil 1.INteAl mlle'IUTI' IIYII'fIlMIi

III

"r"m Ihl ""IhUlly cnnltrlll"llI hnl""lcd prcvlnully on Iho ,,'II, h'" , , .• ~"I"'lrltl"llI In hi hnpullcd "n tho «.11 arc readily clltablillhed u!ling

Ih.

"I""I1"n (.tl44).

I h. n"mher nf' rnutll nf 1-'(:) l",deJe the unit circle is equal to the number

..I' prmhlL'l1I I't which nrc ncglltive, where Pit ill defined as

lit - (1«111- 11[_1 - IJ ... [_I - I] ... [_1 - IJ locI I loc,l locII--II k = 1,2,3, ... , n

(3.145)

\\lth

I', -

PI =

lI'Xol - 1J.

[locol -

1][_1 - IJ ' ... locil

(3.146)

."d I, .. ,I ill uhulinccJ from f" II:) . - .« ,•. - (.) " .. I "

I

+ 1-'" -

(3.147)

1,'*"

J;)CAMI'I.li

/0'(:) - 0.0025

+ 0.08: + 0.4126z

,..-(:) _ I - 1.368:

1.368z3

2 -

+ Z4

+ 0.4126z2 + 0.08z3 + 0.0025z'

(3.148) (3.149)

In thlll L'IU~C we: need obtain only OCo, OCIt OC2 as follows. 1 - L36H:

+ 0.41 26z1 + 0.08z3

./- () 002S-.) 0.0025 + 0.08z + 0.4126z2 - 1.368z3 + Z" (00025 .. 0.0025 - 0.0034z + 0.001z2 - 0.002Z3 ~ Oz" . .

+ 0.41 16:3

: - 1.36H.:1I

0.0834%

+ 0.4116z1 -

+ O.OH34:' ) 0.0834: _

__ 1.368z3 + z" ( 0.114%2 +\6~Oj42;~.+ 0.007z' 0.0834

....

---.+

-':f.:.~}~~+ 0.993Z'(0.53

U.993: 1 -~j .4()22.:~:' O.S2S6%')0.S256Z1

.

II .. hllIlM he: ,,,,tcd tlml in obtaining /xo, dult",1 h""11 need nut he: cilicuillted. III Ihlll l'lI!!l',

"'u -

_---,

. -- ......

().(K)2~

QC, -

0.0834

OCI'

and

ex. -

OC2.

the terms encircled in

0.53

(3.150)

112 'fllllORV ANI)

APPLJ(~A'rI()N or 1'1111 .·raANI..mlM

For stability constraints, F(I) = 0.1271 0.0025 < 1 0.0834 < 1 0.53 < I

>0

F( - 1) - 2.703

>0 (3.151)

Therefore the system described by equation (3.148) is stable. 3.11 Aperiodicity criterion for linear discrete systems 18 •13 An aperiodicity condition in discrete systems is obtained when ull the roots of the characteristic equation lie on the real axis in intervul (0. I) in the z-plane. Such an aperiodicity condition yields only a finite number of maxima and miliima in the system response (less than n. the order nf the system). This condition is important when excessive oscillations nrc not desirable as, for example, in instrumentation systems. For aperiodicity criterion, we obtain the necessary and sufficient conditions (I) for the roots to be real and (2) to require that these root!! be between (0, I). The necessary and sufficient conditions for the roots to be real can he obtained by using either the Romanov conditions,ls which require Ihllt the discriminants Do. D1 ••• • , D"_I (to be obtained from Table 3.1) be positive, or alternately, by using Fuller's conditions34 which arc bused on Trudi's determinants. In the latter method, for the roots to be relll it is necessary and sufficient that AI< > 0, k = 2, 3, ... , n, where Ao - I. AI == no" (always positive) and An is given by a"

Qn-l

0

AI

0.0

a"

Q,,-1

Q,,-t

°"_1

Q,,-1

Q .. - I

a._.

AI 0

0

a"

Q .. - I

",

(n - 2)a ••

"1

(n -

0

Al A.

=

0

0 na.

0

0

0

na..

na .. (II -

1)11.... 1

(n -

~

(n -

(n -

1)"'-1

1)11.. -1 (n - 2)a"_1

1)11"-1 (n - 2)11"_1 (n - 3)a""

3)/1""

(n - 4)a" •

0 (3.1~2)

In the second form we obtain" - I conditiuns to sutisfy for the rout!! to be real. The D,,'s of Romllnnv ure ubtuined frnm Tublc 3.1. To obtain the necessllry und lIuf1\ciellt C()lltiilitlllll for the roul!! ul' (.l.140)

III AlIIIII V I 'UNIUlltMA IIUN I'IIM IINI:AM IIlIlnnll: IIVIIII'MII

hi III' 11,,,1 1111,1 In II" h"twrl'''

Itrn

11.1

""el unily, thC' f'nlinwinJot IIelelitinlllll outlinccl cun he uscd,

I ",,"11 II In ... 1111 e'III"'1 1I1Il' "I' Ih,' IWtl I1Iclhtlllloo

1II1111f',,, ""II/,Y/flrll/(Illfm II 1111'

1"1'1"'IIlIl~

Irnn"l'nrllllllinn invulves mapping the real segment [0, 1)

II' III" ,."llInl' ""ltl Ihc IIcglllivc rcul axis of the II'-plane. It is given by

IIII' '"II"wlll~ I I'll lI!>fllrlllltt ion :

w

. -

::: w =--

'. or

z- 1 II 1'111111111111 (.l.I S.l) is suhslitutcd into (3.140). we obtain "'t(lI') _ /1,,11''' + R" IW" I + ... + Bn_kll'k + ' , . B t ", ,I

II'

(3.153)

+ Bo

(3.154) "'hllillmhip IlI'tween the (l's of (3.140) and the B's of equation (3.154) I, 1"\1'11 !ly

II,,·

(3.155)

111'111

(3.156)

r!/',! (r - k)!

\\ h"I"

or Fuller's conditions that the roots lie on the we rClluire thc additional constraints that

ItlllllllllllV'S

""l'IIII\I' 1I'1I11I\i!oo,

II"

A

> n.

k

= n, I •...• n

(3.157)

11",,,'1,,", Ihc IIcccloosary und sufficient condition that all the roots of I , '·11111,,· Iwlween l.ero and unity in the z-plane is given as /),>0, k.=0.1.2 •...• {1I-1) (orAk>O. k=2.3 •... ,n) (3.158) 111111

" (' 1')

= 0, J, ... , n 1'1"11111111' ('\'I~K) IInel (J.159) yield 2n + 1 constraints. I )A '~A II.

II" r

> 0,

k

(3.159)

I 'AMI'II' WI' IIhllllll

lilt' "llL'rillllil'ily condition for the following polynomial:

I II,.: I- tI:!~:! I "11,111111,,, fill' I'l'al mills Ie "'"I1I1I1V', \I('ll'rlllillants 'I)

lin

.lll~lIl1

+ U3::3,

Q3

=

1

> ()

'111ull~ - 211:

""n":

+

IOClllCl,Cl2 -

'4(1~ I IK"lIll,l1~ -

>0 271/~ > 0

9C1~

(3.160)

(3.161) (3.162) . (3.163)

114 'I'IWURY ANI' AI'PI.IC 'A I'IUN TABLE 3,1

()Jo

I'III~ '·I'RANIII'URM

ROMANOV'S <':ONDI'I'ION l'O OU'I'AIN I).,

k-O

4

2 •. _.4

i Q~

o+

-

2aoaz

a,a, - 3aoa3

la: - 4aoa,-

2a1Q.

2

4aua~

,a. - 6"a'"

ala, - SQoa,

Q

2asa. - Saoa. 3ala,

2a.a, - 6aoU, 4al a,

la.a, - 7a"", Sala,

3a: - 6aoa.4ala, - laaa,

3a,a, - 7aoal Sala, - 3a2a,

3aaa, - 8aoU. 6alal - 4a,a,

4a: - 8aoU.6ala7 - 4a.a, -

4a,a, - 9aoU. 7ala8 - Sa.a? 3a,a,

Qlaa -

,

3

laaas

-.

Sa: - 10aoa,o8a,a. - 6a.a. 4a.a7 - 2a,a,

4

:

-

S

6

7

8 9 This matrix is symmetric with respect to ttte majn diagonal. D~ is formed by the first k rows and columns, alf the system is of lower order than 10, put the higher order coefficients equal to zero 1

I

2, For aperiodicity conditions in addition to the preceding (through transformation)

+ al + aD > 0 a2 + 2al + 3ao < 0 al + 3ao > 0 aD <0 1 + a2

(3,164) (2.165) (3.166) (3,167)

In this case Fuller's determinants yield A2 = a: - 3a1 > 0 A3 = Q~a: - 4a~ - 4a oa:

+

18aoalQ2 - 27a:

>0

(3,168) (3,169)

11"lIll.1tY "'NlII'•• ArIUN tnll UN.. ". UIIIC'IlJ:IJt I""UMII 7

I"".,

II, ttl

'.'"

'1",,11_

••11.

~-

-1"", •.- ~t"UI

"I". - IOaoGlo

2"111 , -

2(a80. - 4a,as 500010)

2aaae - 9a l a l0

lusa, - 1000010 -

3aao. - 901010 7aaa,

3aaae - 80.0.0

3aa

40.0, - 901010 7a.a, - Saaa,

4a~a8

- Sasa.o 60.0,

40,0. - 7asal0

4a,

',"-

SOia, - Sa,alo60.0, - 4a,ail

Sa,a. - 7aaal0 Sa,a,

Sala, - 60,010

So,

60,0, - 70,0,0 50,0, - 3aaa,

60,0, - 60,010 40,0,

60,0, - Sasolo

60,

7a~ - 60,010 -

70,0, - 50,010 30,0,

70,0, - 40,0 10

70,

So: - 40,0 ' 0 20,0,

80,0, - 30,010

Saa

9a: -

90,

- H"..".

.. __ .._---

""ot1. 7a,u. ---

8a,ae - 60aa a

-- -

,II',11. - IlIt,,,II, H, - (,11.11. 411.U,

~

".u. - 9ao'l.

"''''

H"ull, b",II,

"

Il~

loa

-"~,'" ... 90,11,0 -

"I,',"-

- SII,a. 30,0,

': - 80,010 '",_". '" - 40,0,20,0,

40$0, - 20.0,

-.

-

20.010

10

Ilnd factor out from the nth column (and row) a. (finally a:).

Critical constraints Similar toclitical stability constraints, we can obtain the critical aperiodi. city constraints, which are given as or

An

~

0

(3.170) (3.171)

These equations indicate that when a parameter is changed for an initially aperiodic system, the determinant An is the first to become zero when two roots become complex.

1t6 "III!()RY ANI) API'I.U'A'rrUN ni' '1'1111 .·'I'UNltI'URM

When the roots remain relll hut mOYe Cllltsidt" the following critical constraint is satisfied:

re~itln

(n, I), tht" 0,172)

F(1) ~ 0

and F(-I)

~

(3,17.1)

0

Equations (3,t71) to (J.t73) yield the critical constraints for uperiodicity,

3,12 Theorems related to stability and number of roots2 •26,28.32 In this section, certain theorems are introduced which give a quick check on stability and on the number of roots within the unit circle,

Monotonic conditions If the coefficients of F(z) are such that

(3.174) then all the roots are within the unit circle; thus the system is stable, This can be easily verified since the constraints given in equation (3,7M) arc readily satisfied, that is, ho < 0, Co > 0, do > 0, ' , , . As a corollary to the preceding theorem we can state the following, If

o < a" < 0 ..._1 < , , , < 01 < 0 0

(3,175)

then F(::.) has all its roots outside the unit circle. This can be easily verified from the formula of number of roots inside the unit circle. Furthermore, if the coefficients of F(z} are all positive, the roots lie in the ring m ~ Izl ~ M, The values m and M are, respectively, the smallest and the biggest of the n-numbers38

(3.176)

Relationships between coejJicients2 •2S If the coefficients of F(z) are such that

lOki> 10,,1 + 10,,-11 + . , . + law I + IUk .1 + ... + Itl.l

-I

Ifl•• I

(3,177) then the polynomial is unstable,

IUAIIII" Y • ClNIIIII,IItA "UN WM UNI'AM UtIl"M'IIt' NYII ... MII

III 1'111\1" Ihl" 1111'111"'111, II-! ,J.( l) _ ",,,It .lII.1 VI(O:) - ""~" I ... I·

117

(3.178)

tlkl.:'·11 + tlk .:£0 • + ... + 0 0, (3. I78a) II WI' I'III1~id~r Ihis Cllullliun fur 1::1 = I. we have I"I(~)I .~ ItlA·1 > 1",,1 + ... + lawl + lak .. 11 + ... 1- Itlol ~ 1",.:" + ... + akHzkll + ak_1zk - 1

+ ... + aol = SUU'I' "1(.:)

Itp(z) I (3.179)

I VI(:) = /0"(:), we can write

,.., .• )

vl{::)

.." 1 -I- . --

(3.180)

if,(:)

"1(.:)

IIII\\"'VI'r.I~I(.:)1 < I; ,,1(." )

therefore in the mapping of equation (3.180) when

,h",,'rih,·s Ih,' hllundury of the unit circle, the origin is not enclosed. WI' k IIIIW 111111 ,"'::) = tI,.:'· has k roots inside the unit circle (at the origin); Ihll ... 1I"lillg '~'Illaliull (3.180). F(z) should also have the same number of 1111,1', III .. ill .. Ihe IIl1il circle. Therefore the remaining roots of F(z), that is, " A. ·,IlIlIIld lie uutside the unit circle, and thus the system is unstable.

AI ;II;m(l.\' pri"dple 23

IIII'. 1I1l'lh",I is II se I'u I in giving the stability conditions for certain I'"IYlllll1linls F(.:) whose coefficients (with an = I) are less than unity. II 1"- huscd IlII giving precisely the largest root of an equation which .1'"11111111"" Ihe furlll of F(%). For stability considerations the form of the .IIII ..lll1illl is givcn by

ItI.. kl::;;

I'"~

1_

p",

k = 1,2, ... ,n - I, an = I and 0

~

p

<1

(3.181)

111111

ItllIl

~ /1" ,

(3.182)

A "III hi lily dllirl fur different values of k as a function of p and also for ,II n""'11 I vullles tll'l/ up to 7 is given in Fig. 3.4. The calculation of the , 111111 ,'.111 hc l'xtcllded to higher-degree polynomials of F(z). Ut\MI'I.I!

I II .1I1I.,1.llh' Ihe applkatioll of the chart we test the following polynomial III. ·.llIh,hly: F( .. )

.,:4

I 11.2.:3

+ 0.01:: + 0.032z 2

0.024

(3.183)

11K '1'111'0"'1' ANU AIII'W'A'I'InN ClI' 1'1111 .·rUN.llfll,.. III (I."

OK 0.7

0.6

I

0.5

0:

Q,

i

!

:t:

0.4~ 0.3 0.2

0.1

0.9

0.8 0.7

...~

...

0.6

-

on 0.5 C ...

c:t

Q,

I

... I

'u

:r:

Q,

0.48 0.3 0.2

0.1

FIGURE 3.4 Stability chart for linear discrete systems.

IIV

III A"" II \' nIN_II'flU linN 111M IINIrAM Itllll HIli NYIUI Mil

"",I

<

1'111111 Ihr d,,"1 wr 111111 lUI \'rlllll" mit""·,, "' vlllul'S "I' " H.M 1111"1111111111'" (I.IKI) 111111 1 UK.') 11ft' 11111 i!lIl",1 , IIlIcI helll'e the system h"l1 "lIlh 11111'" 111,,111," Ih,' IIl1il l'lI'de,

Ih,'

NIIIII/"'" I~I

"""':i III.vitl,' tl drdl' tI/ radius < 124 (J

I II "hlllill illflll"lIlIIliulI UII Ihe number of roots inside a circle of radius ". I. WI' IIlIIy IIpply IIcerlain transformation on F(z), that is, z' = z/a so Ihlll Ihl' d""'I' "I' mdius /1 becomes equivalent to the unit circle. Hence Ihl' 1"lIl'l'l'Ilill~ ,liscIIssiul1 can be readily applied to this case. I 111'. !,1'1Il'l'clure is useful in testing the relative stability of the characterft

1,,111' plllynomial. The constraints for

F(z) =

I

apZlI

1'~C)

= 0 to have all its

.,

= (J < I will be obtained by replacing in the constraints '"I1I1c1 'ill' " ~.: I the following: 111111 .. lII~jllc

Izi

(3.184) /\\ an extension of this theorem, we can also obtain the stability 1"IIIIIIiliClIIS or the number of roots of a shifted unit circle. This is based 1111 Ih,' following theorem by Marden. To uhtain the !lumber of roots in the circle 12 - sl < I, replace the ",,'s "I' 1-'(::) by

~

(1" .... ~

It~p

(k)P

where thc binomial (k)

.p

a~

(3.185)

.~It-/>Ok

= k!/p! (k

- p)!

III particular. we obtain the number of roots in the unit shifted circle shown in Fig. 3.1 by letting s = 1 in the shifting transformation.

Theorem related to F(z) haL'ing roots on the eireli' Izl = I and reciprocal roots'l.,4.31

Ullit

In the preceding case when F(z) is such that in the table constraints certain ho = h n- k or Fit 01 == 0, the following theorem is useful. If for some k < n, Pi "#- 0 for all j ~ k, (that is, go"#- gn-ItH in the table form). but Fit., 1(%) == 0, then F(z) has n - k zeros at the zeros of I'~(::)' These zero!; are either on the unit circle Izl = 1 or occur in pairs that are inverse relative to the unit circle or are a combination of both. Furthermore F(z) has p additional zeros in the circle. where p is the number of negative Pi forj = 1.2, .•. ,k, and it hasq = k - P additional zeros outside the circle.

120 .... "'UKY ANn AIIIIIIC'AIIUN UI' 11111 '·llANII.... M liXAMIII.I!

Let F(z)

== 1 + lz + Zl + lzl 1.0

'1.1

1.1

Zl

-F(z) F*(z)

1 2

2 1

1 2

F1(z) Ff{z)

-3 -3

0 0

-3

F2{z)

0

0

2 1 (3.111(1)

-3

It is noticed from the table that 100 1 < la"l, FI(z) == 0; thus one rool is inside the unit circle and two roots are on the unit circle (since no reciprocal roots in F1{z) exist). To obtain the location of these roots we solve for (3.187) F1(z) == -3 - 31.· == -3(1 + '1.1) == 0

Therefore the roots are at '1.1 ,1 == ±j. When F(z) has no zeros on the unit circle, but ho == h,,_t, the number p of zeros of F(z) in the unit circle may be obtained by a limiting process as follows· We replace Ft{z) by the polynomial h(z) == Ft{rz) (3.188) which, because r == 1 ± e and e is a sufficiently small positive quantity, has as many zeros in the circle as does Ft{z). EXAMPLE

F(z)

==

-2

+ 3'1. + lz2

(3.189)

Here it is noticed that a~ - a! == 0 [and F(z) has no zero on the unit circle]; therefore we replace z in F(z) by rz == (1 - e)z, with e == 0.1; thus (3.190) h(z) == -2 + 2.7'1. + 1.6lz1 In the preceding case a~

- a! == 4 -

> 0, and Ib.1 < Ib,,_11

(1.62)'

(3.191) ftC -I) < 0, or Therefore PI> 0, PI < 0, one negative sign; hence one root exists outside the unit circle and the other root inside the circle. It should be noted that the same results are obtained if e is considered to be negative, that is, r == 1 + e == 1.1.

h{l) > 0,

III "'11111 ~

IIINII,,,,''''' IIIIN

... 11

IINI All 111"'"1

1],1

II "" .. 11 M'I

-"'\"'1111 1I .... htlll~ 1111' IIVllllllhll' Whl'l,'ltV "\1""'11"" III' 1IIIIII'"i"l'IWt' III' 1111 .111' 111111 ,lid" "'111 hi' ,11''''11111111,,1.1 h,''o'' 1I",lhll'l!. which ClIll be "1'1'111'11 1111 h'''IIII'~ Ih,' IIlIlIIh"1 "I /t'I'II'o elll Ihe IInil circle lire discussed III AI'I"'I"I" " "'III~

,4/'/"'11'//\ I. /)c'ril'tlli"" I~/'I/Il' l"hIt' flm" of sfobilily4

I" 1111'. I1l'l'e'mll'l WI' shull derivc Ihe table form of the stability test by first I'll' 1'C1II,liliullS nil the number of roots inside the unit circle. III 1111'. 1111111111· ... Ih,' slubility condition is obtained as a special case when "II Ih.· 1111'" Ill' 1-'(.:) = () IIrc within the unit circle, Before we derive the 'OIII,hllllll' 1'111' the Ilumber of roots, certain properties of F(z) and its 111\ I· ..... ,. +(,'1 us well as uther associated polynomials are introduced, I ,'I II", "I1I1 .. idercd real polynomial to be tested for the number of roots III' ,Ie',",Ie'" a, F(.:) and written as follows: f(.:) = lIo + tl l Z + a"z' + ... + 0/(1," + ... + anz n IIhllllllllll~

II

= ao IT (z - z;) I 1

III mltliliull, consider the sequence of polynomials Fo(z) = F{z), F1(z), 1 .•1 I.". I'~(::) whose properties will be studied in detail. If we associate III ":lI'h III' Ihe polynomials Fj(z) the polynomial Fj*{z) whose roots are the 11''''prlll'al with respect to the unit circle of Fj{z), Fil,(z) can be obtained hlllll I)::) as follows: F'H{Z) = a~JlF,(z) - a!:~JF:(z),

j = 0,1,2, ' , . ,n - 1

(A.I) h-I

Fi(z) =

2 o~ilzk,

with

,,~OI

written as a k

(A,2)

k~O

and j = 0, I, ... , n -

1

(A.3)

The following properties can be deduced from (A,I) and (A.3), I. If Fi(z) is of degree n - j, then FH 1 is at most of degree 11 - j - I in z. 2. The coefficients of F j ,.1 can be obtained from the coefficients of FJ as follows: (A.4)

In each polynomial Fj(z), the constant real coefficient by hi so that .I.

r1 i+l

-_

[awJ' 0

-

l [a IJ > J' n-i - a 0J+1)

aW>

is denoted (A.S)

122 l'lIIi()RY ANI) APltI.lC·A'I'IUN UI' 11111 "fUNII'URM 3. The zeros of "j-(I) arc the inverMe!l of' the 1.erUIi of' ,.~(.). r,hulv. In the circle Izl = l. In particular. it' ,.~(:) hll!! nl' lern nn tha 111111 ~lr,~I. and Pi zeros inside the unit circle, then I'i-(:) hall (II - J - p,) lernl the unit circle and no zero on the unit circle. 4. The absolute value of Fi(z) for z = eiO is equivalent tn the .lbllClluto value Fj-(e iO), that is,

' 'I'd.

IF Aell) I = IF:'(e") I

(A.6)

The preceding can be easily shown from the relationship F:'(z) =

zra-IF,(~)

(A.7)

When z = eiO, we get

F:'(e i') = eilra-i"FI(e-;') Since

(A.H)

le i (ra- 1II1= 1, then IF:(e I = IFI(e- II ) I = IFI(ell) I

(A.9) To illustrate the mentioned properties, we shall consider the following third-degree (n = 3) example. j

,)

F(z) = Z3 - 2.5z1

Here

0.=1,

7z

-

+4 =

a8=-2.5,

(z - 0.5)(z

01=-7,

+ 2)(z -

4)

ao=4

From (A.3), we obtain the inverse polynomial F*(z). P(z)

= Z3F(~) = z3(Z-· -

2.5z-8

=1-

2.5z - 7z· + 4z3 To obtain F 1(z), we use (A.I) for j = 0 FI(z)

= aoF(z) -

oraP(z)

-

7z-1

= 4(z -

= _3zt

-

+ 4)

2)(z

25.5z

+ 0.5)(z - U

+ 15

= o~llza + a~llz + 0~1I It is noticed that FI(z) is in this case one degree in z less than ,.~:). Ity noting (A.2), we readily identify, O~l) = IS,! = -25.5, a~1) - -3. These values can also be calculated from (A.4) for j = 0 as follow!!:

oP)

U) Oo

~Jll

l - 15 at0 - a 3 -

= 0oa 1 -

aaa. o~" = aoaa - a.o.

= -25.5

= -3

which are the same as those obtained from (A. I). From (A.5), we ellll calculate 61 as 61 =

0: - a: = J5 _

a~1I

IIAIUlII\' I'UNIIUUAIIUN tn.. UNIIAII UIIIC-1U1I ""IIIIMIl

12.1

Th. "mil .. I" N.) .r. 'I - +n.'. 'I - -2. 'I - 4. lhl' IcrUII uf ,.'.,.) IIro).. . •• I. which A." the hwcrliClN Clf Ihe I.crUII 01' f(:) with ",,,,,,,,'1 III the unit "lr~I,·. l,'urthcrmClrc. it hi noticed Ihal /-l::) hus one If,., hull"e Ih~ unit circle Ilntl ""(z) hus " - I 11:1 2 zeros inside Ihe unit ,'lr~I",

Nlmllllriy. We emn culculutc /'.(:.), FaC:.), ,sa. and 6a following the same ,,,'m·rdurr. Iluwever, II systematic: method for obtaining the Fs(z) and ",11 I" Iu "'uIINlruel the following table for the discussed example.

,-"11,,,,,,,,,,,1 ,.'(:)

,.'.(,) ~I -

,",(1)

,.•

:0

zl

4

-7

I IS

-3

"'r(:) (:)

d.... 216 -4S9 d. ~ -164,000

Ft(I) Fo(z)

-2.S -2S.S -2S.S -4S9 216

zB

I

%3

-2.S

I

-7 -3

4

IS

.. 1_ noticed from the table that F3(Z) is a constant and thus has no roots Inllhtr Ihl' lIlIil circle. In addition, from the table we obtain 61 - IS > 0, I~. - ]U, > n, Ilnd ~3 ~ -164,000 < O. It should be noted from the lilllilwillf( dhicussiuns that the sign of 6H1 is important in determining 1I111h11lly ur the lIumber of roots inside and outside the unit circle. IItrlClIlI'M:

If "~(:.) has PI zeros in the unit circle C and has no zeros on 0, then FIH(z) has

(. And II' "111

I'" I

" -

Un - j

- [en - j) - 2pj] sgn 6H1 }

(A.IO)

' 'IU'' in (' And has no zeros on C.·

rruvc Ihis theorem, we assume first that 61+ 1 > O. From equation we CllII write

'I'll (~.I)

"'",..<:1- _ 1 _ IIIIIII~

(A.lI)

e'lulltinn (A.S) und the property of equation (A.9), we ha¥e

"", , '- (:.) I ' n "

"~" ,.',(:)

• II ..

a~'IF~(z) a~/'",,(Z)

1I!"I",(2)

l'IlR


whl'1I ,.'(" hlllller"!1

(A.l2) UII

the unit circle will be discussed in Appendix 2,

J24

THEORY AND APPLICATION OF THE z-TRANSFORM

Now if we apply the mapping theorem to equation (A. I I), we notice that FH1 (z)/aV'F;(z) does not enclose the origin when % moves on C. Therefore we deduce from Rouche's theorem (or from Cauchy's mapping theorem) that the zeros of F1+1(z) are of the same number PI as the zeros of F/(z). Since sgn 151+1 = I, this number is in agreement with (A.10). Furthermore, inequality (A.l2) makes it impossible for FmCz) to have any zeros on the unit circle. If we take the case 151+1 < 0, we have, from (A.I) and (A.19), the following:

and

F H1(Z) = a~IIFI(z) _ 1 a~.!.IF~(z) a~.!.IF:(z)

(A.13)

(z) I Ia~.!.iF:(Z)
(A.14)

a(IIF

By the same reasoning as before, this case shows that the polynomial F/+l(Z) has in C (noting property 3) the same number (n - j - Pi) of zeros as a!!~jF:(z). Since now sgn ,)i.tl = -I, this number is also in agreement with the formula (A. 10). Likewise inequality (A.14) makes it impossible for FiTl(Z) to have any zeros on the circle C. Thus we have proved that the theorem of (A. I0) is valid in both cases. If we apply the formula (A.lO) to each Fi(z) in equation (A.J), we readily obtain

PI

= Un -

(A.IS)

(n - 2p) sgn 61 ]

where po = P = number of zeros of F(z) inside the unit circle.

P2 =

~[(n -

1) - (n - 1 - 2Pl) sgn 62]

= H(n - I) - [en - 1) - n + (n - 2p) sgn

6d sgn ~z}

The expression for P2 is the special case of the formula

Pi = U(n - j

+ I) -

(n - 2p) sgn (~1~2

••• ~;)

+ sgn (~263 ..• ~j) + sgn (~3~1 ..• IIi) + ... + sgn 1i,1 (A.17)

Let U5 assume thnl Ih~ prl'l'~cJilll~ formula ha!t hl'l'lI Vt'rilit'cJ nlso fill j = 3,4, ...• k - I, ami We' will !thow hy illllllC'lIClII Ihal il hold .. lot all k.

STABILITY CONSIDERATION FOR LINEAR DISCRETE SYSTEMS

Letj

=k-

125

I in equations (A.IO) and (A.l7) to obtain

= Hn - k + I - [(n - k + I) - 2p,,_d sgn o,,} = H(n - k + 1) - [(n - k + 1) - (n - k + 2)

PI<

+ (n

- 2p) sgn (0 102 ••• bk - 1 ) - sgn (b 2b3 • , • 15k _ I ) - sgn (bab•... b"_I) - ... ~ sgn 15,,_1] sgn b,,} = Wn - k + I) - (n - 2p) sgn (0 1152 , , • 15k ) + sgn (b 2b3 ••• 15k ) + sgn (153 15• •• • 15,,) + ... + sgn b,,} (A.I8)

It is noticed that (A.IS) is exactly equation (A.17) for j = k. Thus by mathematical induction we have shown that (A.17) holds for all j.

2 ~j

~

n.

If we let j = n in (A. 17) and note that Pn I,~rm (see Fa for the case n = 3). we have

o= I

= 0 because Fro

is a constant

bn ) + sgn (0 2153 ••• 0,,) + sgn (153 15•••• an) + ... + sgn 15'1 (A.19)

- (n - 2p) sgn (15 1152

• , ,

Solving this equation for p, we obtain

= !(n - !sgn Pk )

p

2

wh,'rc

P"

(A.20)

"=1

= 01152 , •• 15k •

k

=

1,2,3, ... ,n

(A.21)

rhc formula (A.20) is the key relationship which yields information on Ih,' Ilumber of roots inside the unit circle. To interpret (A.20). let us '!c'lIole: by 'II the number of negative Pk , k = I, 2, ...• fl. Then as (11 - 'II) I'. Iht.' Ilumber of positive Pk • we may write (A.20) as

P

= Hn + '11- (n -

'II)] =

'/I

(A.22)

whirh Icads us to express finally this important theorem.· '11 for thc polynomial F(::)

= 00 + 0 1% + . , . + On%n

/' III Ihl' products P4' defined in (A.2I) are negative and the remaining " I' OIf" positive. then F(z) has p zeros in the unit circle 1%1 I, no 1111 Ihis drcJe. and 11 - P zeros outside the unit circle. 1,"111 Ihis J.:cllcral thcorem, we can deduce that the necessary and ·.,,'hlll·1I1 "olldilillns for F(:;) 0 to have all its zeros (or roots) inside 1111,1 l·ird,· is

=

/t., ....

=

,h.'

".' II, b2 >

n,

":1

> n, ... (i,. > 0

"It" It ~~'vc'!t, illlhi!t "11"",/' = II.

(A.23)

12Cl 'rIIltUkY ANn A""I,U'AIlIIN CII' 1'111' .-IRANII'UU. A convenienl wily In nblllin~, .md case) is to construct Tllble (A.24) TABLE A.24

Poly"omial

111111111", Ihlrd ,1"Mrl'"

I'~(') (aullluIlL'C'd

A STABILITY ItROCEI)URL FORM

Row ~.

_.. ,.

F(z)

I

ao

F*(t.)

2

an

FICz)

3 4

al

til" ,

".

II• •

... -_._. . I" alii .. a alII ... n-I ..

"-

an I

"

Fa·(z)

<S, = a~lI

all' 0 alII

n-a

,

tI~ll,

II~II

01 = a~·'

F._a(t.)

2n - I

F:_a(z)

2n

F.Cz)

+I 2n + 2 2n

F:(z)

"n-a

-_ ..

= a~n-U

<Sn = a~'"

_-

a~·-II

a:n-al

a:n- II

a~n-lI

-- ----

...

--_..

a~nl

a~'"

CA.24)

In Table (A.24)

a l2'

_I a~ll

o -

III Q .. -l

a~I~1 1 aol1l

' •••

(A.2S)

Furthermore, it should be noted that the elements of row 2k + 2 consist of the coefficients of the row 2k + I written in reverse order (k = 0, I, 2, ... , n). The number of roots inside the unit circle if lJk " 0 is equal to the negative number of the products of k = I, 2, 3, 4, ... , n

(A.26)

where PI = 151,

P 2 = 6 16 2 ,

• • •

(A.27)

The necessary and sufficient condition for all the roots to be inside the unit circle is that the negative number of the products Pk be n. This condition is satisfied if and only if the following exists: 61

< 0, 62 > 0, 63 > 0, ... , 15" > 0

(A.28)

1114HIIII\' CUNII""MAIIIIN IIIIl I INUit 111'11 MI'II '1V!lIIMli

127

III 11'1111111 'llI't'llIl tll~t" III III1I1"'II~'1I1 p"IVllllllllll" wh~'11 "u III l' III' the I I II' \It. "'III', 111 I I '1111' " ..... I,I ) III Inhk (A,~4) I" III' tle~ree less thull " I, \\ hll h 11I1.,.hl II II lim Il' Ihlll "llIlll' III' the '~A.''' arc Illissing, Ilowcver. III III...... Ul .... ~ \\t' 11111,1 IId.1 1111 the rC1luired zerus in thc tablc for F1(:::) _II Ihlll 1111 Iht' ,\" "ppl'lIr IIl1d thus Ihe dctermination of stability or root 11 ... 1111111111111 1'1111 jlron'l·tI ill the usual munner.!1tI Similar procedures ~hllllill h,· 1I~,·tI ",lwlI the!o.e cascs ()ccur for any of the intermediate poly1111111 hi h III ',.1,) III Ih~' elll rics of thc table, Exam pies of such special cases fill' 1111".\'111\',1 ill I'rllhlelll 3.9, "I~II'I 1111 A III IN or 1111: STAllIl.ITY CRITI:RION. 24 To simplify the criterion \\ 1111 h \\ III "lIable liS to wrilc Table (A.24) for the stability test, we relate Ihl' 1111111\,,'111111 betwcclI thc Ilk's and Schur-Cohn determinants /).t' The Ikll'llIlIll:llIl" '~A arc discusscd earlier, and they offer an alternate method 1111 IIhlalllill~ Ihe numbcr of roots inside the unit circle. The connection hd \\ ''I'll 1\ alltl D.k was first established and proved by Marden 4 and is 1"\"11 hy Ihe fullowing relation:

~" A

h"

_

-

\k ,2 -'1<-3

111

U2

(A.2

.I:

••. flk-2

9)

I hi, 1'llll:1Iiun could be used to calculate the /).,,·s from the b,;s, thus 11\ IIlding evuluation of higher-order determinants. When k = n. equation IA ..")) yields !:i

"

= -,n-2 -,,,-3 .l:n-4 CJ n U1

(J2

liS

(A.30)

.I:

••• U,,_2

III

(A.3!)

I:rom the discussion of equation (3.22) it is indicated that ~n

=

A~ -

B~

= (An-l

- B n_ I )2F(l)· F( -I)

(A.32)

Suhstituting this expression in (A.31), we obtain

n" = b~·-2 b;-3 15;-4, ... , CJ n _ 2(A /I-I

-

Bn_ 1)2F( I) , F( - I), n ~ 2. (A,33)

Since (A n _ 1 - B,,_1)2 is always positive when, 15 1, b2 , • • , , bn - 1 yI: 0, and for the roots to be inside the unit circle b2 , b3, ••• , CJ n _ 1 are all positive, the last condition ~n > 0 can be equivalently replaced by

~incc

15"

> 0<=>( -l)nF(I)' F( -I) > O.

(A.34)*

• This equivalence can also be shown from geometric consideration of the arrangement of the zeros without the mathematical verification using equation (A.29).

12M

IIIWllY AN.. 10'1'1." AllIIN I'"

If we Ict

II"

f~l)

11111 , IMANlltllMM

> u, "n > U ..edu~·,"" In > O.

/-'( -I)

,,-cvrll II-odd

> O. < 0,

It should be noted that the last constraint in CllulUinn (A.2M) \'lllIhl I", replaced by equation (A.3S) by letting an > O. We may utilize the relationship (A.33) to obtain the siMn nl' I'" wllhnlll having to calculate 6" as follows. We multiply (A.33) hy ,\.,\•... ,," I and use (A.26) and (A.27) to obtain

If the number of negative PIt (k = 1.2•...• n - I) is even, sgnPn = sgn [F(I)F(-I)] and if the number of negative PIt (k = 1.2, ... ,n - I) is odd, (A . .1~1')

sgnPn = sgn [-F(I)F(-I)]

Since the connection between 6,. and 6,. = A~ - Bi is obtained, we Ciln apply all the significant simplifications obtained on the A,.·s and B,.'s In 6,. or to the stability problem using this criterion. Utilizing equation (A.3S). we can rewrite (A.24) in the form of Table (3.77) by noting the following from (A.24) and (3.77):

,. - a,.,

bIt -- all) ,., .I

a(1I -

_

Ul -

.I

0

_

-

.I

_

un -

b0 -- as0

aln-1) -

U n-l -

0 (

2

-

,.

-

,.,

,,- "....

d - a lS)

(2 )

C -

and

e - a (4 )

.

(A 36)

at

n'

,a _ ,1 0

I)I!

O•

(

'0 -'a - '0'1 - '1'2)2 =

Inl

ao

(A.38)

Appendix 2. Singular cases in determinant and table !orms2.31.35.38.40 Sometimes the kth determinant 6" or the kth entry in the table 6" vanishes. making the method of determining the zeros of F(z) inside the unit circle appear inapplicable. Such cases of vanishing 6,. (or when A,. = B,.) or 6,. are called singular cases. One such case of the table form has been briefly discussed in Section 3.12. and the purpose of this appendix. is to present in detail the modifications required to enable us to determine

'11\1111 " ' '11"/'111111""111' 1111'

IINIAII

111'1'111 II '1''111''''

1."1

I 1IIIIh IIII' IIlIIlIh"1 .. , 111111'. 1I1·,I,h·. 1111. III 1I111·."h' Ih,' 111111 "11'1'1,·. I he " ,"11. ,III· ·.I.,It't! \\ /Ihlllli 11/11111',.111111 11'''''''11'',·, III 1111' ""l'iValillll will be I"" II I .. '.\"'."'11111111" II,,· '"·.'·11·.·.11111 lin' 1I11I"ili",1 IIIclhods to he applied 11111 .. '.1111'111111 'i".,". ,II,· 11/\111,'" 1IIII1IwlIllIailll'alagorics, the perturhation IIlI"lh,"IIIIIIIIII,' ,,",·,·1 1Ill'lhml. I 1'11'11'1111 ..\ "liN MI II II III. I his method is useful in certain cases, but /1. ,11'1"11.11/01/1 lIIay /1111 hl' a!l !ltraightforward as the direct method. The "U·.I· ... , 1111'0 1I1'1'1t"alillll lie~ in the fact that since both ~k and ()k are I'"h IIIIIIII./h i/l Ilw rll"lIkiellts of F(~) of equation (3.9), a coefficient of 'I I 1I1.IV hi' \O,,·i,·11 ill ,"ch a way that ~k and 15k become nonzero (if they \ .1 II 1·.It ... , I",f. In' Ihe variation). To he specific, let 0 •• 02' Q3• ••• ,0" be It,," 1111'11 ,\,. alld 1\. arc nonconstant polynomials in 00' Now if we "·1".11" "" h,v tilt we IIbtain the following new polynomial ( /..'(.;) .. ~ II"* + + 2 + ... + 0,,2 II A. 39)

".= "2=

I III Ih" polynomial ~A. and /)k are nonvanishing. Furthermore, assuming \, .11111 ,~! arc the first vanishing clements, the signs of o'i and OJ for j < k \\ III hi' the same for F t (=) as for F(z) provided 100 - a~1 is sufficiently '.1I1i111. IIIC' ,Iill'crenee between the zeros of F(z) and F'(z) is also small provided Ih.11 Illn .-. /I~I is small. Therefore if F(z) has no zeros on the unit circle I '( ) \\ ill have the same number of zeros inside ·the unit circle as F(=}. I hI' f"rc1!uing mcthod of perturbing 0u is useful in principle for handling '.111111' !'.ingular cases. Therefore

has no zeros on the unit circle, consider r(z) = a~ + 0lZ + ... Tlh.re exists an a~ (ncar ao) such that ~r and do not vanish 1111" ,...(.~), and the latter has the same number of zeros inside the unit !"Ill'll' as F(::;). ,,. 1-"(::) tI,,~".

,

or

h;

It should be noted that it suffices to compute ~j and for only j ~ k. Tu determine whether F(z) has any roots on the unit circle or not, which is important for this application, the following useful theorem which is stated in Section 3.12 is reintroduced.

necessary condition for F(z) to have a zero on the unit circle is for to vanish for some I ~ m ~ n and for the next (that is, m + Ist) row in the table to be all zeros. 1\

,~ '"

Thus when I)k vanishes, we can sometimes ascertain whether F(z) has :lny roots on the unit circle by computing the next row of the table. The determination of the exact number of roots on the unit circle is presented in the next section in a discussion of the direct method. The criterion for determining whether F(z) has a root on the unit circle when using

UO

1I110ltV AN., AI'I'IUAIIiIN

(110

1111'

I

IMAN!IIOIIM

the dctcl'minunt IIlcthml ili given hy Jury"': A lIl'l'l'~"1I1 Y \'1111111111111 hll F(z) to have a root on the unit circlc is 111111 1\.,," n. For a polynomial with real cocllicients. il is indknll',1 111111 1\,,(An-l - B,,_1)2F(I)F( -I) and thus thc vanishing of All ill SUIlI,' l·U!.l'1I could be easily determined from F(I) and F( -I), In gel\t~ral. All vanishl~s only if F(z) has roots on the unit circle or reciprocal pairs. This is evidenl from equation (3.70), where we notice that A n- 1 - B .. .1 = 0 when Z,Zk"'"

Iz;12 or Z;

=

..!... Zk

It should be noted that the computation of A l i i

-

/J"

I

«

is rather lengthy in comparison with that for A k • k n. If A"_l - B"_1 = 0, we can factor out the polynomial that contains the roots on the unit circle or the reciprocal pairs by obtaining D(z), the greatest common divisor of F(z) and F*(z). Sometimes this can be easily achieved by the division process (Euclidian algorithm) by using the stability table form as shown later. The remaining polynomial G(z) = F(z)/ D(z) can then be studied by using the perturbation method if required. Closely related to the perturbation method is the method used by Marden 4 to determine the number of zeros inside an ,-circle, that is, I Izi ::;; -. Consider the following polynomial if F(z) has roots on the unit

,

circle: (A.40)

It is evident that G(z) of equation (A.40) has as many roots in the ,-circle as F(z) has in the unit circle. Therefore, by examining the number of roots of G(z) in a I + e or in a I - e circle for e smail, we can. in principle, determine the number of roots of F(z) in and on the unit circle. An example has been indicated in Section 3.12. In concluding this section, it might be mentioned that the success of the perturbation method depends on the choice of a*. In some rare cases it might happen that one choice of a* would not make A; and Di nonzero; thus another choice is required. Therefore this method tells us that the perturbation procedure does work and nothing further. However, the direct method. which will be discussed. seems to offer a straightforward (although lengthy) way of dealing with the singular cases and thus appears useful for application. DIRECT METHOD. This method is only applicable to the table form and offers a convenient way for obtaining information on the roots of F(z) when 61: = O. As indicated in Appendix I, Table (A.24) represents a sequence of polynomials of descending degree. Any polynomial in this sequence may be

'II ... UII II'

I ""''' 1111

II ... III IN 11111 I I NI AII 111'11 III" '0 \ 'oil ,...1

I \I

11'\,111'1'11 ltv ,1111111,,'1 \\llh IIII' •... 1111· IIIIUlIt"1 "I "'1'" 1I1',,,h' Ih,' 111111 l'irl'le 11'111\1111'111111' 1,".1 III IIII' '.1"1111'1111' 1', "111111"111'" hlllll Iht' IIl'W JllllYIIUllli:l1 '1lIlIllhlll'. hI IIII' ',111111" 1I1h-") \\llhlllll dlalll'.ing Ihl' validity "I' the tablc 1111'1111111. 1111' Ind ,'. IIIl" Im~i!> "I' Ihl' dirl'l:1 melhod. SIII'I''''''' .~" ",,'

tl,,·

1"1"'I'dilll~ I'IIW~

i, lilt' I'Ir!>1 .\ {with k. = " - m ill the tahle :lrc

/'n

/"

"",

".11'

/)11

+ I} to vanish and that

'"1' = b~ - h~1I = 0, bob, - b .. _I"III' ... , bob",_1 - btb",

(A.41)

We notice thut (~" ""'" = 0 iff Ib",1 = Ibol. The polynomial to be assot'iatcd with the first row of Table (A.4I) is (A.42)

Our objective in the following is to try to replace g(=) by a polynomial with the same number of 'Zeros inside the unit circle and ·for which none of the Ilk vanishes. Two cases are distinguished. I. Some clement in the row of 15"_'"1) does 'not vanish. This is so itT h", = ±bo but for some q < m/2, bill_II ~ ±bll • We shall let q denote the smallest integer such that h "'_11 ~ ±bq is satisfied. For this case the following theorem due to Cohn 2 •4 enables us to construct a new polynomial having the same number of zeros inside and on the unit circle as g(=). THEOREM: If the coefficients of g(z) satisfy this case, g(z) has as many zeros in and on the unit circle as the polynomial

Gl (:) = BoG(z) - B," l.qG*(z} =

"'+'1

I

B~"Z1

(A.43)

1-0

where G(z) = (zq

b

+ 2..!?.) g(z) Ibl

= (b,"_,,, -

(bm/bo}b q ) b,,,

k>

In

(A.44)

(A.45) (A.46)

and (A.47)

1.12

IIIIClMV ANI) AI'I'IIC'AIIUN UI' 1m ,·lItANIIIClItM

hiliutiull (A.47) gUlIl'lllItccs tlmt rur Ihe new plll,Ynlllllllll f;.(:) Ihl replace }:(z)J. ,)" -Inll ducs nut vlInish IIl1d thus we 1.'1111 I.'lIlltlllll~· tht' Inhll' method to determine the cxact numbcr or zcros insidc the ullit I.'il'l'ic rill g(z). EXAMPLE

Let g(z) be given as g(z)

= I + 2z + 3z 2 + ;:3

(AAK)

To obtain the number of roots inside the unit circle, we form the table: g(z)

2

g*(z)

3

fnll()will~

3 2

(A.49)

This represents the singular case discussed previously. Hence to form Gl (;:), we first obtain from equations (A.43) to (A.47) q = \,

b= G(z)

= 3,

m

bo =

\,

b2 -(b)/bo )b.3-2

b)

1

b",

= b3 =

1

=l.lbl=l

= (z + 2)g(z) = 2 + Sz + 8z 2 + 5z3 + Z4

(A.SO)

(A.51) (A.52)

From equation (A.52) we have Bo

= 2,

B4

=

Brn

=

and GJ(z)

=

2G(z) - G*(z)

(A.S3)

I

= 3 + Sz + 8z2 + SZ3

(A.S4)

Now we can continue the table form starting from GJ(z). 3 5

G2(z) ~l = -16 Gi(z) -1

5 8

8 5

-2S

-1

-25

-16

5

3

G3 (z) ~2 = 255 384 G:(z) 384 255

-------<0

<53

(A.55)

.'AIII'''\' 'UNII.Uf:IIAIIIIN 111M IINIAII 1tllllRIli 11V!lIIM"

I.U

I h, 1111111111.'1 "I 11111'" III ,tt("11II11 fil(z) III!!Idt.! Ih~ 111111 dn:lt.! IIl'e the sume IIIIIIIIH' ,1~'lel'llllm',1 hy Ih,' II 11111 her ,,\" nC~lIlivc I'". Itere

I', '. . '~I < U, I h,'n'l'lIl'l' Ihe number of' roots of ~(z) inside the unit circle is two, Ihil'" remuining root should be outside, because if it were 1111 Ih,' IIl1il circle. the singular case as discussed in item 2 would have

1I11t1 111l~

m·~·urred.

We can readily establish for this example that no root exists on the IInil circle from the fact that the multiplication of the roots is unity Ifrom clJuution (A.48)] and g(l)g(-I);e O. Hence we can apply the perlurbation method for the sake of illustration as follows. In T~lblc A.49, replace the zero term by a small quantity (positive or negalive) and then proceed with the table.

I

-1 -1

(e 2 - 1) -(e - 1)

-(e - 1) (e 2 - I)

F.

In this case if we consider e

h,

=e >

0, b2 = (e 2

> 0, we have -

I)

< 0, 153 = (e2 -

1)2 - (e - 1)2> 0

As established before, two roots are inside the unit circle and one outside the unit circle. We obtain the same distribution of the roots if we consider e < O. In conclusion, we may mention that if the singular case occurs, we readily know that at least one root exists inside the unit circle because of the arbitrary choice of the sign of e. Thus finding that g(l) > 0 and g( - I) > 0, we readily' ascertain that two roots are inside the unit circle. This simplification can also be obtained for higher-degree polynomials by checking the sign of the constraints gel) and g( -1) at the start of the test. S9 2. The entire row of which c5"_"'H is the first element vanishes. This ±bo, b",_l = ±b1••••• bo ±bm or F"_"'+1 == O. For is so iff b... this case the following theorem due to CohnM gives the form of a new polynomial that has the same number of zeros inside the unit circle as F,,_ ...(z).

=

=

1.'4

nll'IIMY ANn AI'I'lle'AIIIIN ell' 1111' .·IIlANIII....MM

If the cucflicie:llhl ul' /.'" ",(:) !"Itj'll'y tla" 1''''''''llln" 1'1111111111111, then f'n_m(z) has as many 1-ems in the ullit "Ird,' It!! th~ !,lIlynn",".I.

TIU'.oRI!M:

(,,)]* -_. f,n-m+l(z) -- [F'(n-m)"

(I) _""'

..~'" 11"" 'n -.

_

.:

I ,,111.1 I·

(A ~/)

1-0

where _ dF n-m(z) F 'n-m() z --=~ dz

("l,".

Thus for this case, we replace the row (bo• b10 "2' ...• "111) Ily bP) .•• b~:~l)' If (btu, hpj, ... , b~~l) still satisfies the: preccdill~ elise, it can be shown that F,,_m(z) is of the form (z + ej~)'I."'. If it sntislies the first singular case, then we proceed in the method outlined carlier. EXAMPLE

Let

= -0.5 +

F{z)

1.65z - 0.8z2

-

0.35z3

-

1.8z.

+ Z6

(A.59)

Form the following table: F(z) F*(z) F.(z)

-0.5 I ~1

= -0.75

<5 2

-0.75 0

F~{z)

F'l.(z)

=

11=5

1.65 -1.8 0.975 0.975 0

-0.8 -0.35 0.75 0.75 0

-0.35 -0.8 0.975 0.975 0

-1.8 -1.65 -0.75 -0.75 0

I

-0.5

(A.60) From ~~ = 0 and F2(z) == 0, it is evident that the second singular case occurs: therefore we obtain/2(z) from equation (A.57)

+ 1.5z + 2.925z2 -3 + 2.925z + 1.5z2 + 0.975z3

h(z) = [F;(z)]* = [0.975

=

3z3 ]*

(A.6J)

Now we continue the table for the new polynomiaI/2(z)

-3 0.975

liz) 12*(Z)

2.925 1.5

13(Z) 13*(Z)

152

=

8.06 -7.4

-10.24 -10.24

~(z)

153

=

9.24 -152

-152 9.24

I.*(z)

1.5 0.975 2.925 -3 -7.4 8.06

(A.62)

IIII' !,lllvIIIIIIII,,1 1,1 I 11,,'1 IIII' '111111' 111111111,'1 III Ilioh 11I',lIk 1111' 111111 IIII11 11'1 '~I I 1111111 IIII' pll·,,·dlll~l. ,~( ) 1111 .. 1111" 1'11111 (/1'1'11) inside the 111111 1111 k (1"'lIIl1"" I~_ • II. I~:I • II, I~~ " II). Thlls FI(z) Iws also one root Imltll' Ihl' 111111 1'111'1". A~ 1IIIIIrl'" 1'1"11111 a Ih~orel11 on p. 119 for this form of ',,11/'.11111, nl·,I·. II 1'11111 i~ ~I .. il-Ily inside the unit circle if and only if there I " ..... II 1I"'ljlllIl'/l1 111'1 hili rool outside the unit circle, and, in fact, the two 1.... 1·. 111" Ihl' n'l'ipl'ocals or caeh other. Therefore, because F1(z) is a '1IIIIIh link .. pulynolllial, the remaining two roots lie on the unit circle. 11111111'1111111'1'. I'WI1I F(::) we found that 15 1 < 0; therefore the remaining 1,,"1 III 1'(.) lil'S inside Ihe unit circle. In summarizing, F(z) has two roots 111'.11 It· IIII' IIl1it circle, Olle' roof outside the unit circle, and the remaining '1111 ''',,(.\. arl' Oil the unit circle. The roots of equation (A.59) are i, 2, i, II /1 I ;O,K.

Allhllll!'.11 Ihe ahove procedure is straightforward, in some cases it 1.. ·1'11111· ... 'I"ite lahorious to calculate all the (jk'S from the table. Therefore II,' ,'all IIlilil:C the following two facts to simplify the determination of the 111111 di!'.lrihlltioll. I, "'rolll cquation (A.35c). we can determine the sign of Pit from the F1( - I).

"'t"'" III' F I ( I) and ,I. 1·'01'

rcal polynomials complex zeros appear only in conjugate pairs.

Ba"'l·d on Ihese facts we have for the example discussed FI(-I)

<0

lienee F 1(::) should have only one real root between (-I, I); another lI'al root (its reciprocal) is outside the unit circle and the remaining two ~holild be on the unit circle. It should be noted that three real roots rallllot exist between (-1, I), for this requires F I (::) to be at least a sixthdl'!~rce polynomial. Therefore, by determining only the sign of FI(I) and FI <-I) we can obtain the number of reciprocal roots with respect to the IIl1it circle and roots on the unit circle, thus avoiding the use of the table I'llr this example. Similar simplifications can also be obtained for other n:ciprocal polynomials. It should be noted that in some of these singular cases we can also obtain information on the number of roots on the unit eircle or the number of reciprocal roots with certain multiplicity.2,31.39 In concluding the discussion of the singular cases, one may mention that, in general. cases I and 2 could occur in combination for the same polynomial F(z), In this situation, we may count in the regular way for the sign of hk (or for the negative PA.) even though some of the intermediate polynomials are replaced by different ones, particularly for case 1. This becomes evident if we would consider the counting using equation (A. I?).

136 TltllOIlY ANIl APPI.IC'IITmN

ft,

TH' '·?lANlI'OIM

App"IICII:c 3. SU",,,,tlrJ' 'if 'hI' ",(,M")' (·rl,,.rlll In this appendix. we summarize the threc form II nf tho IItnhlllly crl,erla discussed in this chapter. For practical UllC. the lIummarl7.cd It"",. unly are of importance. DETERMINANT METHOD. This method, which is discussed in Seetiul1 l.5 and presented in equations (3.28) to (3.35). is presented here in 1\ mnr. convenient form so that the sign of the constraints for stability ill himllllr to Hurwitz constraints. This can be readily achieved if the columnli uro interchanged in equations (3.28) and (3.29) and also a minor change sign is made by noting that (_I)k(k+l)f2 = (_I)k I IH'/2( _I)k, If we denote by X~-1 and Y~-1 the following matrices.

or

X~_l

=

aft

13,,-1

13,,-11

0

a"

13 11_1

0

0

all

0

0

0

0

0

0

aft_a Y~-l

I

•••••

.....

Q,,_t

=

0

ao

0

I

••

as

I

..... ..... .....

138

at as

0

.....

Q,,_B

•

0

Q II

Qs

Dl

Do

a1

Do

0

Do

0

0

0

0

0

the stability constraints for n-even and n-odd are presented for the following polynomial: f{z) =

130

+ a1z + a,.zll + ... + a,;zk + ... + a"z",

I. n-even F(t)

> 0,

IX';"'1

±

Y~ll

F(-I)

=

A~

>0 ± B~ > 0,

k

= ].~5.7., . .• n -

I

....A.n "" C'aNIII.IIU IIUN I'nll UNUM UIIIC In II' IIVIIII Mil

137

III Itlll'rnllllvlly rur Ihl" d."rll1ll11"" A~

t

,,~

A~

I -

> n, k - 2.4 •...• n IJ:, I > 0

2

,. I/-mld

> O.

F( 1)

IX; ± Y;I = III

<0 A; ± B; > 0, F(-I)

k = 2, 4, 6, ... , n - I

IIlh·rt1nlivcly for this determinant k = I, 3, 5, ... ,

A;,

I -

B~_l

n- 2

>0

I h,' IIhcfnate constraints are useful for design in order not to solve for Iwn Ctllllltiuns of degree 11 - 1. It should be noted that all the IX; ± Y~I Itn' nhluined ffom IX~_I ± y;'-ti as indicated in equations (3.30) and ( I. 'I). As an illustration, let us consider the stability constraint for the 1"lIowing polynomial: Q

I,'Oflll

IX;' . I

s >0

T Y~-.1 as follows:

De

TO.

"5 T

D3

a. T

01

0.

T

as T a,

a2

A; T Ba

k=3

TO,

a. T au

Tal

TU,

=Fo, =Fao

T a, k=1 A, T

a. T

01

I

"0

00

"3

B{

I

a5

'"

'=F°o

0

a.

a5

0

0

0

".

". T

Ai =F B;

The stability constraints are

> 0, F(-I) > 0 A; T B~ > 0, A~ T B3 > 0,

F(I)

TABLE FORM. The table is useful mainly for numerical test of stability Qr for the root distribution. It is presented in two forms in Tables (3.77) and (3.121). The latter form is particularly useful for design purposes.

13K 'l'I"'U!tV ANn APPIWA'I'IUN III' nut '·I'UN.I't •• M

We summarile mllillly ill thill ApJlCII'Ux th" furm Itly,," In !'ahll' (Un Row

Z3

• ••

I

Ir

aa

., ,--..an

Ir

zJ

Zo

a~

0 0:

as

2

Qn

an - l

°n-2

3

bo

b1

b2

bll _ 2 bn- s

5

Co

C1

6

Cn _2

c n- 3

2n- 5

So

SJ

Sa

2n -4

rs~"1

S8

32

1 SI

'.- .... !,

So

To to

~-- .. I

-3

2n- 2

bn-t

"n 1

JI ..

D,

""'0 I

"0 cna

Ca

C2

,II' ."

"tI

ba

bll-

2n

•••

Q!:¥2:~~

4

1

.&

Co

r- --, : TJ :

r2

where hk

=

ro =

, ao aft

I

So S,

a" __ ak

I_ Ck

s. ,_ So

r,

=

, bo

=,

b._1

b.- I _ k bk

So

SI

s.

s.

dk

' '

I,

=,

Co C.. _I

to

=

I

C"-.-k' , ...

ro

r.

Ck

r. ro

I

The stability constraints are

> 0, (-WF(-I) > 0 bo < 0, Co > 0, do > 0, ... 30 > 0, To > 0, to > 0 F(l)

(71-1) cnn.trainl.

It is noticed from the table that the dotted entry in row 2n - 3 is redundant. Therefore the corresponding second-order matrix to be calculated using the dotted entries in rows 2n - 4 and 2n - 5 is also redundant. Finally, an easy way to remember the rule to form the table is indicated by the arrows of the first two rows.

"IAIIIIII\' • IINNIItI'U,IIIN IIIIl IINIAM 11111. 11111

nil II MlC

1.19

III"""IIN MI'IIIIIII, II", ,llvl"""I lIu'lhml i, ,imilu!' III Ihc tllblc form, 1'1 \\'1'1111'11 III II 1111111 Ihlll l'lIl1 hI.' rcmemhcrcd by using simple ,IIVI'IIIII III' plllylllllllilll", II i, IIIM,'usscd ill Section 3.10 and a summary

hili II

1"lIl1wo,. 1'\11'111 Ihe illvcrsc pulynomial F*(z) as

,...(::) = ::"

FG) = a" +

O,,_lZ

+ 0,,_2z2 + ... + O,,_,.zk + ... + 002", a" > 0

()hlilin the stability constraints (XI: by simple division:

Hz)

- - · ..~:lto

/:*(::)

(xo

F (z}

1 +-Fi(z)

F {z)

txo

2 + +- - ... F:(z) tx1

F"_l(Z} + OC1 + (7.2 + . , . (X,,_2 + -*-F,,_l(Z)

whcre 1-'\(::), F2(z}, . .. , F.. _2{z) are the remainder polynomials obtained l"nllll the division and Ff(z), F!(z), ... , F:_ 2(z) are their inverses.

The stability constraints are F(I)

l(Xltl

> 0, < I,

(-I)"F(-I)

>0

k = 0, I, 2, ... , n - 2

REFERENCES

I. Jury. E. I., "A Simplified S.ability Criterion for Linear Discrete Systems," University of California, Berkeley, ERL Report Series No. 60, Issue No. 373, June 1961. Also, IRE Pror(!('cli'Ws, Vol. 50, No.6, June 1962, pp. 1493-1500. 2. Cohn, A., "Ober die Anzahl der Wurzeln einer Algebraischen Gleichung in einem Kreise," Malh. 2, Vol. 14, August 1922, pp. 110-148. .\. Fujiwara. M., "Ober die Aigebraischcn Gleichung deren Wurzeln in cinem Kreise mler in einer Halbebeneliegen," Mall!. Z('ilscilr., Vol. 24, 1926, pp. 160-169. 4. Marden, M., "The Geometry of the Zeros ofa Polynomial in a Complex Variable," Amer. Malh. S"r., New York, 1949, pp. 152-157. 5. Samuelson, P. A., "Conditions That the Roots of a Polynomial Be Less Than Unity in. Absolute Value," Am. Math. Slati.t" Vol. 12, 1941, pp. 360-364. 6. Tschauner, J .. "Die Stabilitat der Impulse Systeme," R('gelungslechnik, Vol. 8, 1960, pp. 42-46. 7. Jury, E. I., "Discussion on the 'Stability of Sampled-Data Systems' by J. Tschauner," R~f('III'!lfsle(·hllik. Vol. 8, Munich, Germany, August 1961, pp. 340-342. 8. Gantmacher, F. R., Tire Th('ory of Matrices, Vol. II, Chelsea Publishing Co., New York, 1959. 9. Tsypkin, Y., Theory of Pulse Sysl('ms, State Press for Physics and Mathematical Literature, Moscow (in Russian), 1958, pp. 423-427. 10. Brown, B. M.• The Malh('maliral Th('ory of Lillear Sysf('ms, John Wiley and Sons, New York, 1961, pp. 119-152.

140

THEORY AND APPLICATION OF THE %-TRANSFORM

II. Rouche, E., "Memoire Sue la Serie de Lagrange," J. Ecole Polytech., Vol. 22, 1862, pp. 217-218. 12. Jury, E. J., and J. Blanchard, "A Stability Test for Linear Discrete Systems in Table Form." Proc. IRE, Vol. 44, December 1961, pp. 1947-1948. 13. Romanov, M. I., "Algcbraic Criteria for Aperiodicity of Linear Systems," Soviet Physirs Duklady, Vol. 4, March-April 1960, pp. 955-961. 14. Jury. E. I., "A Stability Test for Linear Discrete Systems Using a Simple Division," Proc. IRE. Vol. 44, December 1961, pp. 1948-1949. 15. Jury, E. I., "Additions to Notes on the Stability Criterion for Line Linear Discrete Systems," IRE Trans. on AutOn/otic Control, Vol. AC-6, September 1961, pp. 342-343. 16. Jury, E. I., and S. C. Gupta, "On the Stability of Linear Discrete Systems," Rf.lfelIIngstet"l",ik, Munich, part 4, No. 10, April 1962, pp. 157-159. 17. Jury, E. I., Sampled-Data Control Systems, John Wiley and Sons, New York, 1958. 18. Jury, E. I., "Proof of a General Relationship Used in the Stability of Test of Linear Discrete Systems, Addendum to Reference I. 19. Jury, E. I. and Pavlidis, T., "A periodicity Criteria for Linear Discrete Systems," IE££ PGCT, Vol. CT-9, No.4, December 1962, pp. 431-434. 20. Jury, E. I., and T. Pavlidis, "Stability and Aperiodicity Constraints for System DeSign." IEEE PGCT, Vol. CT-IO, No. I. March 1963, pp. 137-141. 21. Wilf, H. S., "A Stability Criterion for Numerical Integration," J. Assoc. Compo Marh., Vol. 6, 1959, pp. 363-365. 22. Arzhanykh. I. S., "New Stability Inequalitie~," AUlOmatics and Remote Control, Vol. 22, No.4, April 1961, pp. 436-442. 23. Jury, E. I., "A Stability Chart for Linear Discrete Systems," Proc. I.R.E., Vol. 50, No. 12, December 1962. 24. Jury, E. I., "On the Roots of a Real Polynomial Inside the Unit Circle and a Stability Criterion for Linear Discrete Systems," presented at the Second IF A Congress held August 27-September 3, 1963, Basle, Switzerland, published by the CUt!l{re.l.f Proct't'ditWs, 1964. 25. Thoma. V. M., "Ein einfaches Verfuhen Zur Stabilitatsprufung Von Lincllrcn Abtastsysteme," Rt'gt'ltmgstedmik, Part 7, Vol. 10, 1962, pp. 302-306. 26. Jury, E. I.. "Discussion of Reference 25. Regt'lungslt'C'hnik, December 1962. 27. Jury. E. I., "On the Generation of the Stability Constraints in Linear Discrete: Systems," IEEE PGAC. Vol. AC-8, No.2. April 1963, p. 184. 28. Schur. I., Uber Potenzreihen. die im Innern des Einheitskreiscs beschrankt sind, JOI//'nal fiir Mathematik, Vol. 147, 1917. pp. 205-232, and also Vol. 148, 1918, pp. 122-145. 29. Chow, T. S., and H. W. Milnes. "Numerical Solution of a Class uf lIypc:,.hnlicParobolic Partial Differential Equations by Boundary Contraction," S.I.A.M. i",,,,IIII, Vol. 10, No. I. March 1962, pp. 124-148. 30. Jury, E. I .• "On the EVlIluatioll of the: Stllhility De:tcrmi'llIl1ts ill l.ill~·i"· J)i'~''''h' Systems," IRE PGA\, Vul. AC-7. Nil. 4, July 11)(,2. pp. 'il 'i~. 31. Jury, F.. I., "A Nute Oil the Rcdpl"ll~'ill Z~·nt\ III" a Itl'all'uIY"lllllial \\ilh 1h"I"'l"t It, the lJnit Circ:le," /1-:/-:/': "(ie;-(·. til hi' pllhli!ihl'lI .111111' I'IM. 32. I'crrull, 0 .• A~~"I"'II, /1",,,/ II, "II,,'''' i,' .I," ."~,:",,, ,111'-/"." ("""'I''''~I:'''', W'III,', III'


STABILITY CONSIDERATION FOR LINEAR DISCRETE SYSTEMS

:\3.

141

Knopp, K., Theory of Function, Dover Publications, New York, p. 68. \4. Fuller, A. T., "Conditions for Aperiodici~ in Linear Systems," British J. App/. /'/lys., June 19S5, pp. 19S-198• .l5. Thoma, Von M., "Ober die Wurzelverteilung von Linearen Abtastsystemen," H,:It('/uII"fsrechnik, Part 2, Vol. 11, )963, pp. 70-74. \rl. Li~nard, and Chipart, "Sur la signe de la partie r~lIe des racines d'une equation 1I11:~hrique," J. Math. P""es Appl. (6), Vol. 10, 1914, pp. 291-346. 17. Jury, E, I., and S. C. Gupta, "A General Procedure for Obtaining Tschauner's SIIIhility Constraints," Regel"'''Kstechnik, November 1962. lH. Tschauner, J. "A General Formulation of the Stability Constraints, for SampledD,II:1 Control Systems," IEEE Proc. Vol. SI, No.4, April 1963. 1'1, Jury. E. I. Further remarks on the paper of Reference (40), to be published by the H':~"/II'!ltstechnik No.2, February 1964. ·111 nlUma, M., "Ober die Wurzelverteilung Von LinearenAbtastsystemen," RegelungsI,·,'/mik, Vol. II, 1963, Part 2, pp. 70-74. ·11 .I II I'y , E. I., "Proof of the Generation Rule for the Stability Constraints in Linear D".,·n·le Systems," Technical Memorandum M-39, Electronics Res. Lab., Univ. of c ,.111 • Uerkeley, California, December S, 1963. ·1.' .Jury, E. I., "A Memo on Marden's Investigations of the Zero-Distribution within IIII'I hili Circle," Technical Memorandum M-38, Electronics Res. Lab., Univ. of Calif., 1I,·,~,·lt·Y. California, December 4,1963.

4 CONVOLUTION z-TRANSFORM

The convolution z-transform is an extension of the z-transform from which a certain theorem called the complex convolution theorem is obtained. This theorem yields the z-transform of the multiplication of two (or more) discrete sequences in terms of the z-transform of the individual sequences. This important theorem of the z-transform has many applications. such as in summation of series. in obtaining the z-transform of special functions, in extending the z-transform method to the solution of unmixed differencedifferential and sum equations, and others which are discussed in detail in this chapter. The significance of the convolution z-transform lies in the extension of the z-transform to nonlinear discrete systems. Certain types of nonlinear difference equations could be solved by this method. Thus the z-transform method which is usually considered applicable to \incur discrete systems is extended to cover the nonlinear case too. This study is presented in detail in the next two chapters.

4.1 Complex convolution

theoreml-~

Assume we have the sampled function [f(t)h(t»)· with F·(s) and 11·(.\') liN the Laplace transform of [·(/) and h·(/) respectively. Tu ubtllin Ihe Laplace transform of the product we may use the complex convolution formulll in the a-plane (see Appendix for derivation) 1'5 fullows. (4.1)

1-17.

CONVOLUTION z-TRANSFORM

It should be noted that both F*(s) and 1I*(s) have an infinite number of poles in

143

1m

the s-plane because of the periodic form x x of' these functions, s-plane x x Assume as an example that F(s) has a pole in the left half plane at s = sa.. ~[X T X x lind lI(s) has a pole in the same plane at S "': .1",: (both S7. and sri are assumed for ~ake of illustration to be real). then the Re x x pille-zero pattern in both the s- and qSf3 plancs of F*(s) and H*(s) is as shown III rigs. 4.1 and 4.2. x x 2" I he strip of analyticity in the q-plane T x x I', ~IIown in crosshatch and the path of sa IIlh'/'.ration as a dotted line, It should be lIellt'd that because of the periodicity . FIGURE 4.1 Pole-zero pattern of III' hoth F*(s) and 1I*(s), the path of F'"(s) and H·(s) in s-plane. IIlh'/',ralion is taken along one strip and IIIe' illlcWIlI is divided by the period 21Tj1T as required. /lulll F*(s) and H*(s) are functions of e' " only; we can replace e T & II.,. and ("/',/ by p. so that F*(q) becomes a function of p. that is • .fF(p) 1111.1 II +(.~ - q) becomes a function of zip. that is • .:f( (zIp). As required by IIII', 111:1 ppi ng

X

e~

x

x X

2..

T

x

Re X

x

I.

~

111,1110 ,I,'

l(c','.1I1II1I1

'1IllIlytlnty

III

thl", plll1l1'

1111"

pllih "I' 1IIII'I:ratillll.

144

THEORY AND APPLICATION OF THE Z-TRANSFORM

___I_m-r-__ z-plane

FIGURE 4.3

Poles of ~(z) and Jf"'(z) in z-plane.

The mapping of the s-plane and q-plane into the z- and p-planes is shown in Figs. 4.3 and 4.4. Note that the strip of analyticity in the q-plane maps into an annulus of convergence in the p-plane. With this substitution equation (4.1) becomes

9' ([[
== 1[[(t)h(t)] ==

P == I Jr p-l§"(P).Y(>(=)d p

~f 217)

[flhll'l.-n

,,=0

(4.2)

where r is the dotted contour in the p-plane as shown in Fig. 4.4. Equation (4.2) represents the convolution 'I.-transform theorem which gives the 'I.-transform of the product of two time functions. knowing the 'I.-transform of each of these functions separately, (for 1[[(/)] ~ ~(Z),

Radius ,'"

(contour r)

FUIlIMF ~.4

1'llIh 111'1111"1\1111111" III/'-I,IH"".

CONVOLUTION Z-TRANSFORM

145

= .;f.,(z». Extension of this theorem to multiplication of more than two discrete time functions is evident.

J[h(t)]

Em/ualion of the complex COnl'o/ution inlegra/4 ,. Hy Cauchy's theorem on residues, we can evaluate residues at the singularities of rl~(p) (if :F(p) has no branch points as assumed here). In addition if the value of the integral along the infinite circle tends to zero, we can evaluate residues at the singularities of .;I(z/p) (provided the path ~,r integration is taken in a negative sense). The choice between the two llIethods depends on the number and complexity of poles of each of the fUllctions .
J[f(t)h(t)]

= ~ f p-l:F(p).Tt'(~) 2TTJ )1'

=

It

J (I')

~f

2TT)

P

dp

dp Jr p- :Yf'(P):F(!) p 1

=

I

=

-I residues at singularities of K(;)

residues at singularities of p-l~{p) (4.3)

hils simple poles only,

Jf/(t)h(t)] =

± A(~n)

,,~1 p,,8 (PR)

,Yt.(~) + :F(Po)J{'(-=-)/ P..

Po

(4.~)

1>0=0

"h"'I' I' ' ..

~(p) = ~~:~,

and

8(p) = 0,

for P = Po. Ph P2 • ...• P...

with Po

=0

till' I

/1'(/',,) ::;:: II,! /

,III /'

(4.5) /'n

,I..' ('ulllplcx cOllvolution theorem

rllr Ihe.' modified .:::-tmnsform4 '11111,1,11 III

IIIII.ltlh',1

l"II"'ilr",, we IlIlIy "hlain the ee.luivalcnt theorem for the 11,111',1""11 hy tlhlnil1i"~ firsl the I.nplllcc tmnsfurm of

lIlt. ", }hlt, ",) J It,,( t).

U"';,,,:
146

THEORY AND APPLICATION OF THE z-TRANSFORM

This is given by ~({f(t,

m)h(t, m)] <ST(t)}

T

iC+ihtlTl

21J')

c-/(lIITl

= -.

o~

F*(q, m)H·(s - q, m) dq,

m ~ 1

(4.6)t

By using the same mapping as for the z-transform case, we readily obtain .1m[f(t)h(t)]

=~ 21J')

=~

m) dp Jrr p-1.F(p, m)~ (~, p

r p-1Jf'(p, m).F(~, m)dP, p

0

21T) Jr

~m~

1

= h(t), we get .1m[f(t)2] = ~ rp-1.F(p, m).F(~ , m) dp

(4.7)

For the case f(/)

21T)

When h(t)

Jr

(4.8)

p

= u(t), equation (4.7) readily yields .F(z, m).

EXAMPLE

Find the modified z-transform of the following function: Im(if(t)]

(4.9)

In this case .1 [t] = mTz - mT + T m (z _ 1)2

(4.10)

and .1m[f(t)] = .F(z, m) From equation (4.7), Im[if(t)]

= _1

(4.10a)

r mTp(p- - mT+ T !.F(~, m)dP 1)2 P p

21Tj Jr

(4.11)

=

The integrand in this integral equation has a multiple pole at p 1 (and a pole at the origin; however, since its residue is zero in most case!! because .F(z/p, m) jp=o 0, it is ignored). The residue of equation (4.11) yields

=

.1m[lf(t)]

=.! [mTP op

mT+ T p

= :F(z. m>rf(m tit mlly

I»)

be noted thill/V. til) - /'(t •.• (I - m)1·).

.F(~, m)JI P

,,-1

+ '/' -~- :F(~- m) 1

ai'

P

.

11_1

(4.11.)

CONVOLUTION Z-TRANSFORM

147

We can easily verify that

i.~(~,m)1

op

P

=

-z!~(z,m)

(4.13)

iJz

11_1

Therefore

T[(m - l)F(z, m) - z:z F(z, m)]

Im[I/(I)] =

=§

Similarly,

I(t, m) wbere II (t)

= tl(t)

(4.14)

T[(m - 1)F (2:, m) - z:z :F (2:, m)]

Imrt~(t)] =

1

1

(4.15)

lIy iterative extension of the preceding we can obtain where k is a positive integer

;1 ".[1'1(/)],

4.3 Applications of the convolution modified or z-transform method 4 ,2 III Ihb section we shall introduce a few applications which are useful in 1II',ade system analysis as well as design. Im'use III n

Zm

-

!P trallsformatioll

l)J"l~ecding

discussion* we obtained the modified z-transform from Now we ',hullllhiain /'1s), knowing ff(z, m) Iht, I.lIplace transform without indicating the inverse process. ,..(.~)

= Z;,lF(z, m)

(4.16)

IIII' 1.lIplace transform of /(1) can be written as ,..(.~)

=

L ob

o

/(t)e- SI dt =

Nm\ II Wt'lcl, = (1/ -

1.111 III' wl"illcn n,v) "'"

I

00

."

+ m)T, 0 ;S; m :::;;

'J' }; I' .,,'1'. "

! f'''+1IT /(t)e- SI dt liT

(4.17)

,,~O

II

11 /(nT -

T

I, II = integer, equation (4.17)

+ mT)e l1- m )$T dm

(4.18)

0

III

(4.19) • " •.•. 'H'.II.1I1 I \,

148

THEORY AND APPLICATION OF THE z-TRANSFORM

However. by definition

eo

~ l(nT n=O

T

+ mT)z-n = ~(z. m) =

Therefore F(s) = Z-".'

~ (t. m) = T

f:

Z-I+", sr(%.

(4.20)

.1m(j(/)]

m)dm It = en

(4.21)

It should be noted that this transformation is unique. that is. for each m) there exists only one form of F(s).

.~·(z.

Im'erse

.~(z)

- F(s) or (z-s) translormation2,4

This transformation indicates how to obtain F(s) = 2[[(1)] when we know F*(s) = ..P[J*(t)J. To obtain the z-s transformation, we first find l(nT). This can be achieved by recognizing that F*(q) is a periodic function of q with the imaginary period 21TjIT; hence the value l(nT) is the coefficient of e 1[11'1' in the Fourier series representation Z of F*(q). Hence, T fC+J(or!TI (4.22) l(nT) = -. F*(q)e qnT dq, for/(O) = 0 21TJ c-i(~I7'1 where the path of integration lies to the right of the poles of F*(q). In order to obtain a certain I(t) whose sampled values coincide with l(nT), we change nT into t in equation (4.22) and then apply the Laplace transform to obtain, after interchanging the order of integration,

F(s)

=

I-.

21TJ

[CHlatTI F*(q)[ JC"J(/rITI

[eo e-(··-qlt dtJ dq,

Jo

for 1(0) We note thatioor(S-qll dt

T

(4.2.\)

I/(s - -q); hence equation (4.23) reduces 10

fC+iCorlTI

F*(q)

for/(O) = 0 21TJ c-j(IIITI S - q It should be noted that the right-hand side of equation (4.24) relm'sellts the Laplace transform of f*(/)u(t). Helice pes) can he also ohlairwd directly from ,27[f*(t)u(t)] using the complex convolution theorem disl,'\I!\st'd in this chapter. If F*(s) is written in terms of z = (,'I'M, the function /0'.(.\.) ht'l'OIllI'~ .:F(z), the z-transform of F(s). Thus equatioll (4.24) is considt'J't'd liS lit,· inverse z-s transformation, Ihal is,

F(s)

= -.

=

=0

- - dq,

CONVOLUTION Z-TRANSFORM

149

It may be noted that this inverse transformation is not unique, that is, many forms of F(s) have the same Z-transform F(z). This is evident since f(t) is forced to coincide in values with f* (I) only at sampling instants.

Skip sampling theorem 1 ,17,18 This theorem signifies that, knowing the z-transform with respect to the pcriod TIn (when 1/ is an integer is larger than unity), we can obtain from it the z-transform with respect to the period T. This theorem is important ill obtaining the limit cycle (or periodic modes of oscillation) of nonlinear discrete systems and in the study of multirate sampled-data systems. The fundamental equation that describes this theorem is

f(kT) = f(k:T) u(kT),

= nk

where k'

(4.26)

Applying the z-transform to this equation, and using the complex convolution flll'lllula, we get

,:F(z) wlll'n~

;:"

2'TTj

= e(T{II)s = ZI/".

WI' ~':111

. Ii

~ J[f(kT)] = _1 f .?'(Pn)p;;l

1 dp 1 - (zn/Pn)-n

Jr

(4.27)

evaluate this integral in a positive sense for the singularities of

(1' .. )1'" " to obtain one form, or, alternatively, integrate in a negative

'.1'1\'0('

with respect to the poles of

1-

( 1 I)

Pn z"

to obtain -n

(4.28) 1 IIllhl'l' application of the preceding theorem is in the analysis of . 1I1111111111tl'd discrete signals appearing in some systems,

'"tl'gmtli",/' fUI/Nioll 'I'onsjorm 4•1l J ... I.Cf(t)II(t)

.

till

. T

~/~/(::

-

. (' I)

f

Jo J1'

p- 1§(p,

m)£(=-, m)d Pdm P

m (Ip (1m '/' J'"'J' . I' I:F(p, m).Jf" (:.,p )

I ,

h~1

u

I

(4.29)

150 THEORY AND APPLICATION OF THE Z-TRANSFORM This relationship can be obtained from writing the integral in a summation form.

l o'f(t)h(t)dt

= TLi

i1f(k - 1

aO

+i

m

+ m)Th(k -

1

+ m)Tdm

0

fen - 1

+

m)T hen - 1

+

m)T dm ] (4.30)

The modified z-transform of the preceding, noting that when m

JJ"ff(k - I)T] ~-o

= 0,

= _1-1 ~(z), yields z-

1m [L'f(t)h(t) dtJ

=Z~1

f Jl(n -

+

n~/(n

Tim

1 + m)T hen - 1 + m)Tz- n dm

- 1 + m)T hen - 1 + m)Tz- n dm

(4.31)

but the infinite summations are the modified z-transform of f(t)h(t), thus

~

1m [ltf(t)h(t) dtJ = i1§"1(Z, m) dm o z-lo

+ Tim~l(Z, m) dm 0

(4.32) where

§"l(Z, m}

= 1m[f(t)h(I}} = ~ f p-1F(p, m)Jf'(-= , m) dp 21TJ

Jr.

P

(4.33) For the special case when her) 1m [itf(t) dtJ o

= u(t) = unit step, equation (4.33) yields

=~ ll§"(z, m) dm + Tlm~(Z' m) dm z-lo 0 (4.34)

where

§"(z, m)

= 1m[/(t»)

(4.35)

Equation (4.34) also yields the modified z-transform of £(s)/s. Extension to obtain Zm[£(s)/sk) by iterative process is straightforward. z- Tran.iform of special fUlIcticm.v4,9,lO One of the several applications of the ,,~onvoluti()11 z-trnllsfnrm is to cllublc us to tlbtnin the z-trnnsforrn (If spcdnl function" such ns .1V'U)/t"). k > n.

CONVOLUTION Z-TRANSFORM

151

This can be achieved by obtaining first the z-transform of (III) for t > O. The z-transform of I/t for t > 0 can be easily obtained by using the definition of the z-transform as follows: <X>

~(z) 1%_1>

= L f(nT)p-n

(4.36)

n-O

Multiplying equation (4.36) by;-1 and integrating both sides from z to co, we obtain

! f.
t

1-0

If we let f(t) be a unit step for

t

(4.36a)

t

> 0 whose ~(p) =

[1J 1

-1 dp1- =- f."" p t T z p - 1

for t

l/(p -

>0

I), we have (4.36b)

Choosing T = I, we have

1[!J t

= In-z z- 1

t

> O.

(4.37)

For any f(t) we have, applying formula (4.2),

l[f(t)J = ~ f I 21TJ Jl'

p-l~(p) In _z_ dp + lim Z -

P

n-O

fn n

(4.38)*

The function In z/(z - p) has a branch cut in the p-plane extending from I' = z to p = co. Since we are free to take the branch cut anywhere in the p-plane, we shall take it along the positive real axis. This is illustrated in Fig. 4.5. As before, the crosshatched area shows annulus of convergence and we have

LI= -fEA 1.:", + J.w + fIleD + fm; =

0

-i.:. . = f.w + fV1-: + fBeI> 1.1 = f.,,/ LI'; + fReD • III Ihl' 1'lIl1l1willl: It~rivali\ln we shall show the equivalence between this equation hn~I'a1 lin 1'lIlIvllhllinll .··trallsrorm and equation (4.300) obtained directly from the , 1...11.,1'111111.

152

THEORY AND APPLICATION OF THE Z-TRANSFORM / ' - - - --........ p-plane Poles of

/ _;

C

1m

p-'F'(p)

~

" '"

"\

~___ PJJranch cut from , p=ztop_oo

~~~~

'-

E -~-};7 1,0

7

/ _/

FIGURE 4.5 Branch cut in p-plane and path of integration.

Along AB, P = Ipl ei ., Along DE, p = Ipl ei(21r-·>, In _z_ Z -

= In (_~) + In _.;;,.1_

P

1 - zIp

P

(4.39)

Expansion for In I l L is given by the following: -zlfJ

In

1 I-zip

= -Z

(I I )

(4.40)

(_~) + ~ + L + ...

(4.41)

P

z2 + . .. + -2pB -zp < 1

Therefore In _z_ = In z- p

Now

r

JBCD

= O.

p

P

2p2

Hence

(4.42)

CONVOLUTION z·TRANSfORM

Now In k (where k is a complex number) easy to see that equation 4.42 reduces to

= In \k\ + arg k.

153

It is then

(4.43) We can generalize this result for higher powers of t i'n the denominator.

~ b z-n ,,~1

n

multiplying by

Z-l

f ~n

=frr> p-l§(p) dp. •

and integrating from z to

00,

z-n =f."'q-Il'" p-I§(p) dp dq

•

n~1

q

(4.44)

Continuing this, we obtain

~ V~)} = J.rr> w-1L" v-11
i'"

p-l§'(p) dp ... du dv dw (4.45)

.Himes·

For k > I this equation is defined for t > 0; for k = 1, equation (4.38) is valid. Although the p'receding results appear to yield ~[f(t)ltk] for k > I in II closed form, actually it is not so because of difficulty in evaluating intcgrals in a closed form of the type S[ln (a + bx)jx]. Therefore it is not possible to obtain closed expressions for z·transforms of functions Iikef(t)/tk for k > 1. However, the usefulness of the result lies in evaluat· 'ing infinite series since numerical evaluations of definite integrals of the preceding type are available. This is discussed in the next section. To obtain the modified z·transform of [j(t)!t] for limf(t) = 0, we use 1-·0

unother method which avoids the complex integration, based on the following derivation. Let :F(z, m) be presented (notingf(nT)ln=o = 0) as follows: tr)

:F(z, m)/._"

= ~ f(nT "

.,.

= ~ f(nT 11-

T

+

T

+ mT)p-n

mT)p-li

II

I

(4.46)

154

THEORY AND APPLICATION OF THE Z-TRANSFORM

Where we multiply this equation by p-m and integrate from Z to infinity,

roo p-m!F(p, m) dp _ roo I

J&

_

J. n~l

f(nT - T + mT)p-< ..+ml dp

~1(nT-

T+ mT)z-
-n - m

.. -I

+1 0< m < 1

or

)d ~ 1(nT - T + mT)z-" -1 zm-If.oop-mdr( or p, m p - ~ T. (nT- T+ mT)

If ~~n ~ / :

;>;

(4.47)

..-1

z-O is added to both sides of equation (4.47), this term

= I, which is the lim f(');

has a contribution only at m side of the above equation is I

Im[f(I)J =

rf(')J L-t.

,-0

t

! zm-lf.oo p-m!F(p, m) dp + limf(t) ,

T

t

m

thus the right

•

''''0 t

o < m ~ 1 (4.48) For the case of z-transform we obtain, by substituting m = 1 in equation (4.48) l[l(t)J - ! foo p-1!F(p) dp + Iim1(') IT. '-0 t where !F(z) = !F(z, m)lm_l (4.49) Extension of equation (4.48) to obtain Im[f(t)/lk] is evident. Special cases that stem from this theorem are indicated in the following.

1[ let) ] _1 i'f.oo p-l1+A)!F(p) dp + hT T

(4.50) t & where h is a positive constant larger than zero. Equation (4.50) is obtained from equation (4.36) by multiplying both sides by p-(1+A) and integrating from z to 00. If we assume in equation (4.50), T - l,f(l) = unit step and if we denote h as a (integer),

J _ zaJ.Z[.!. + _1_ + ... +! __J -J dp 00 pa p p- J

1 [ -1- = I00 -1- z-" - t"f.oo p-(1+a) -p- dp n+a ..aO n + a • p- 1 pa-l

=

ZCl

Z fZ 1"'-1) Z.(.. {In - - - ---- +

II

-0.' -

:-1

a-I

a-2

+ ... + z.-..Il} I.

(4.51)

CONVOLUTION Z-TRANSFORM

155

or

1 ] .1 [ -n+a

=Z4

Z 4-1 1 ] [ In---2 -z-1I z - 1 1'~1 P

(4.52)

EXAMPLE

Obtain the following z-transform:

~{(n + a;(n + b)J From equation (4.52) we can write

"[(n + a;(n + b)J = "[b ~ aC~ a- n~ b) ] = _ 1 [(In _z)(z~ _ Zb) z-J

b-a

4-1

1

1·=1

P

,,~ -b -z~-z

If we let a

= I, b =

+

b-l)

]

b ~ -/J z~-z 11=1

(4.53)

P

2,

.1[(n + 1)(n1 + 2)]

= Z[1 + (1

- z)

In _ z J z- 1

(4.54)

Similarly, we can develop the z-transform of

"[(n

+ a)(n + Ih)(n + c) .. .J

(4.55)

Summation of infinite and finite series 5- 7 ,9,10

The convolution z-transform readily yields the summation of series, where Ihe sum exists. These summation forms are discussed as follows: . I f we let z = 1 in equation (4.2), we readily obtain

i

fllhn

= ~ J.:~(P) ,;Yf(p-l) dp

" n

where the contour

27f}

r

I

(4.56)

P

could be taken in a positive sense along a circle co

whit'h encloses the poles of ,~(p)lp. If the summation

I

fnh" exists, then

n ·.0

Ihi ..

l'011101lr

l'IIS,'

or "'I"aliol1 (4.56). when II" = /". we get

could be along the unit circle in the p-plane. As a special

.... ~

-'\ 1'" '." "

II

1 ~ ::&"(p) "L"( I) I ··.r I' I P 'J.rr', I' I'

(4.57)

156

THEORY AND APPLICATION OF THE Z-TRANSFORM

where the contour r in this case is along the unit circle where the poles of ~(p)lp are enclosed. A method for obtaining equation (4.57) without having to obtain the poles of fF(P)/p is given in Table III of the Appendix by letting m = 1 in I". The proof is presented in Appendix 2 of this chapter. Another important summation could be obtained by letting Z = I in equation (4.45), which gives

oo I l; = lim f.oo w- 1 &~l 1

n-on

V-I • ••

III

,J

Coo p-1fF(p) dp . .. du dw

Jq

(4.58)

k.th;;ea 00

To obtain the summation

I

n,zn, z

< I,

which often appears in the

,,~O

analysis of discrete filters, we can also apply the convolution z-transform as follows:

I n'z" = _1 I. (p - p z)p .Jt"(p-l) dp 2.".j

01=0

where

r

(4.59)

I'

encloses the poles of l/(p - z) in a positive sense, and (4.60)

The list of Table IV in the Appendix presents this summation for values of r up to 10. As a special case of equation (4.59), we can also obtain the finite sum of the following serieso,lO ,,-1

S,

=I

r = 0, 1,2, ...

j'zi,

(4.61)

j-O

for which z"-I with So = - - -

dS,_l S,=z--, dz

z-1

(4.62)

k

Finite summation of the form If" can also be obtained from equation (4.56) by letting ,,-0

h(t)I,=n = u(t)lt.n - U[I - (k + 1)]1,=" Therefore equation (4.56) reduces for this case to

I J,

,,-0"

(4.63)

C!F(p) p-l [1 _ pieHl dp 2.".j JI' P p-l - 1

= _1 1 =2.".j

I.

I'

,~(p)(pIcH --

P

P- 1

1)

(4.64)

dp

where the contour I' cnclllsl.'s the poles (If the

illtc~rnnd,

CONVOLUTION Z-TRANSFORM

157

EXAMPLES

Obtain the following summation: (1)

Using equation (4.58) for this particular case, we have

ao

I

! -2 ,,_tn

=

iaorlln~dp p- 1

1

With the change of the variables of integration from p-l = x and using tables of integral, we finally obtain

.i n12 = Jofix-lin (_1_) dx = "II I - x 6

.. zl

I (lk) for even values of k > 2 can be similarly

Summation of the form n-I n obtained. (2)

This summation can be obtained from equation (4.59) as follows: GO !nx"

n-O

!

= -1

21Tj

I"

P P4 dp (p - x)p (p-l _ 1)2

where r is the contour in the p-plane which encloses the pole of I/(p - x). Evaluating this integral, we get 00

:t

II~O

(1 - x)'I.

!nxt'l =

,x< 1

I<

.(3)

In .... 0

From equation (4.64), we have I< In = .. -0

It

pl<+1_1

dp 2"j r (p - 1)2(p - 1)

This contour integral yields

~ n = .!.. ~ [ k+l _ 1]/

L ,,·0

(4)

2'. d pII P

l:'''. . __(-1)" ......... _,,·-0 (II + 1)(" + 2)

p_l

= k(k 2+ 1)

158

THEORY AND APPLICATION OF THE %-TRANSFORM

This summation can be obtained from equation (4.54) by letting %== -I, which gives

i

(-I)"

n-II

(n

+ I)(n + 2)

==2In2-1 «)

Finally, summations of the form! fnhn+m, 0 ~ m ~ 1 can be similarly n=O

obtained from the convolution modified z-transform. Furthermore, from the integrand form we can ascertain the conditions on convergence of the series or the possibility of obtaining a closed form.

Unmixed difference-differential and sum l'quations1 ,B The convolution %-transform is readily applicable to the solution of equations of the following form: .11

K

n

I !ocremY~~m(t) +re=o !fJreYre(t) == x..(t) m-O re .. o

(4.65)

Assuming zero initial conditions, we proceed to solve this equation by applying first the Laplace transform and then the %-transform to obtain M

K

~OTc~OOCremsre.1[Yn+m(S)]

+ 1" Ln~!reY1o(S)] == .1[X ..(s)]

(4.66)

where ~[y~Tc~."I(/)] == sre Yn+m(s). Using the convolution %-transform and noting that .1

rl..t=o f fre] == ~I §"(%), %-

we obtain for

.1[IfJ/cY/c(S)] 10-0

==

(_%_)~ %-

I 2.".,

r 8{p) q1j(~, s) dp p p

Jr

(4.67)

where B{%) == .1[B.. ] and qv(%, s) == .1[Yn(s)] as defined in equation (2.120a). Using equation (4.67) in obtaining the %-transform of equation (4.66) we get

qv(%, s)

I

Iockmbm+ _%_~

m-OTc-O

%-

r

s)

B(p) qv(~, dp 1 2."., Jr p p = .f."(%, s) (4.68)

This linear integral equation cannot he sulved in terms of elclllcnlluy functions, but it cnn he solved hy Il l'ul1Vl'rgcl1t ilerntivc process.

CONVOLUTION Z-TRANSFORM

159

If we examine the special case where {J,. = {J, when {J is a constant, equation (4.68) reduces to M

K

tpj(z, s)! !OCkn/'z'" mcO/c~O

(J

+ _z_
Z -

~'(z. s)

(4.69)

This can be written

tpj{z, s) =

~'(z,

s)

~ £.. ~ OC,.",s/czm

£..

mmO/c=O

+ -z{J-

.f'(z, s) = £(z, s)

(4.70)

1

Z -

To obtain y.,(t), it is necessary to know the zeros of the denominator. Knowing these zeros, we can apply the inverse Laplace transform and then the inverse z-transform as follows: (4.71)

or (4.72)

El'aluQtion oj illtegrals oj time Juncliolls'~,'·11.22 We shall use the convolution modified z-transform to evaluate the following integral: LO)J(t)h(t) dt

(4.73)

We obtain this integral from equation (4.29) by using the final value theorem for the modified z-transform

io

O)J(I)h(t) dt =

I... Jo(1 J.r §(P.p m) .YI'(p-l, m) dp dm

(4.74)

2?r)

The contour r can be taken in a positive sense which encloses the poles ,if :F(p. m)/p or in a negative sense that encloses the poles of .%(,-1, m) ill the p-plane. As special cases of the preceding, let h(t) = J(I); then

s.

trJ

[J(tW dt = -T.

"

2fT)

lIt 0

p-l.:F(p, m)§'{p-l, m) dp dm

(4.75)

I'

whl'rc the contour I' is taken along the unit circle since all the poles of :,.. (1',111)11' ure inside the unit circle if the integral is finite. It should be IIlIh'd

thut in the evaluation of the integral I"

_ I

1rrj

J' I' 1:IJ(p, m):F(/' I'

I,

m)tlp

(4.76)

160

THEORY AND APPLICATION OF THE z-TRANSFORM

we do not need to fin8 the poles of ~(p, m) for the integral can be calculated as the ratio of two determinants in the coefficients of the numerator and denominator of ~(p, m). The evaluation of this integral is presented in Table 1Il and calculated up to values n = 4. It should be noted that the evaluation of the integral in equation (4.75) is simplified if the order of integration in p and m is interchanged. This is possible since the integrand is continuous along r and in the interval O~m~l.

Furthermore, if we let h(/)

Lo'"

t[f(t)]2 dt

= tf(t)

= T2. 2~)

(1 ( p-l[<m - l)§(p. m)

JoJr

- p :p .fF(p,

m)Jl.~(p-\ m)] dp dm

(4.77)

Similar relations can be obtained for higher-order moments. In conclusion, by utilizing equation (1.178) of the modified z-transform which states

Tf~(Z' m).)t"(z-l, m) dm

= Z[F(s)H( -5)].

we can rewrite equation (4.74) in the following simplified form: ("'f(t)h(t) dt

Jo

= _1. ( ZlF(s), H( -s)]I%~" p-1 dp 2~)

JI'

(4.78)

Similar relationship could be obtained for evaluating finite integrals of the following form:

(II

('I

Jo fl(t) dt =Jo f(t)h(t) dt

(4.79)

where 11 is finite integer. The evaluation of the integral of the square of the time function is very important in the design of sampled-data feedback system that is based on the minimization of the total square value of the error function. It is also important in the statistical design of a system that is based on the minimization of the mean square error function. The entries of Table III are important in obtaining the noise power gain in a discrete filter.

Two-sided z-transform 12.1~.16.17.18 In the preceding chapters, the z-transform of f(II) is defined fur positive values of n, that is. II ~ n, which we may refer tn as the olle-sided ztransform. III sOllie npplil'lltinns stich liS in tilt' stlitislit:1I1 study of discrete

CONVOLUTION Z-TRANSFORM

161

1m z-plane

Re Ring of convergence of 6(z)

FIGURE 4.6 The contour C and the ring of convergence of .:F(z).

systems, the statistical functions may also be described for negative n, thus the z-transform definition should be modified to cover such cases. The two-sided z-transform is defined as follows: ~(z)

'Ie""

= J[J(n)] ==

I

/(n)z-lI

(4.80)

where .~(::) is analytic in the ring shown in Fig. 4.6. The inverse of ".(:) is unique and is given by

f(n)

= .rl[§'(Z)] == ~ 2fT)

r

§'(Z)Z"-l

Jc

dz

(4.81)

where thc closed contour C lies in the ring of convergence of 9"(z) with the point:: = 0 in its interior. The definition of the two-sided z-transform requires that ~(:) converge in some ring (of nonzero area) about z = 0 in the complex plane. The contour C is shown in Fig. 4.6. If the functionf(lI) is bounded, the contour r of Fig 4.7 enclosing the unit circle will belong to the ring of convergence. Here we write /(11) =

rl[.~(z)]

= _1 2fTj

r§(Z)Z..-l dz,

Jr

(- 00

< n < 00) (4.82)

1m

---r----~----~--x~

A pole 01

[J'(z-l,

I It il IIU .\ 7

I h.,

fllllhllil

I'

ill

,hl' •

plmll"

16~

THEORY AND APPLICATION OF THE z-TRANSFORM

If the only singularities of .F(z) are poles of finite order, the inverse can be evaluated as usual by the residue method, that is,

I

fen) =

residue .F(Z)Z"-l,

(-00

nil poles

< n < (0)

(4.83) ,..,

Inside unit

circle

We may note that we could also evaluate fen) by considering the poles

outside the unit circle; then we will have fen)

I

=-

residue .F(z)z..-t,

(-00

all poles

uutsidethe unit circle

< n < (0) (4.84) "

Although both equations (4.83) and (4.84) are valid for all n for computational purposes, equation (4.83) is preferable for n ~ 0, and equation (4.84) is preferable for n ~ O. Since, in statistical study of signals in discrete systems, both boundedness and symmetry around the time origin are assumed, we could then write o

.F(z)

= I

n--oo

00

f(n)z-"

+I

f(n)z-" - f(O)

(4.85)

n=O

If we let m = -II in the first term of the right side and impose the condition fen) = f( -n), we will have ~(z)

= ~l(Z-l) + ~l(Z) -

f(O)

(4.86)

where §l(Z) has all its poles inside the unit circle and it is defined for positive II, and ~l(Z-l) has all its poles outside the unit circle and is defined for negative n. The inverse of equation (4.86) gives

fen)

= ,rl[§l(Z-l)] + ,rl[FJ(z)] -

f(O){)"o

(4.87)

where 6" o = I for n = 0 = 0 otherwise We may note that here we might use equation (4.83) for n ~ 0, substituting only ~l(z) for $I"(z). Similarly, we may use equation (4.84) for n ~ 0, substituting only F 1(z-J) for F(z). The general case where fen) is not symmetrical and boundedness condition does not exist can also be treated, However, in this situation care should be exercised in defining the ring of convergence of F(z). If such a ring is nonexistent the z-transform does not exist. For instance, if a pole exists inside the unit circle, then this indicates a decreasing exponential for positive n, or 1I I1OI17.em value for the timc function for ne~Hlivt~ II. The decision liS to which time function is represented hy the .:-lrunsfol'lII (It'pends Oil Iht, riny, of ctlllvcrgl'nl'e ill Ihc

163

CONVOLUTION Z-TRANSFORM

z-plane in which ~(z) is defined and on the manner in which the ring divides the poles of .9"(z) within or outside this contour. ILLUSTRATIVE EXAMPLE

Let !F(z) be given as follows: !F(z) =

-z(1 - q2)

,

q(z _ q-l)(Z - q}

with q

<1

To obtainf(n) for n ~ 0, we use equation (4.83) for roots inside the unit circle. In this case, we obtain

fen) For n

~

= residue of [~(z)z'a-J] Iz ~'1 = q",

n~O

0, we obtain from equation (4.84) the following:

fen) = q-n,

n~O

The total function fen) is represented by

len) = qft,

II ~

n s- 1

1;

and f(O) = I

Applications to correlation functions and spectral analysis l6 •17 One of the applications of the two-sided z-transform is in obtaining the power spectrum density of the discrete random signal. This discrete power spectrum is defined as the two-sided z-transform of the autocorrelation function of the signal. For instance, if the signal is represented as f
cf,(n)

= lim

1

.v-«> 2N

N

I

+ 1 /c",-lV

f(k)f(k - 11),

(4.88)

where n is an integer. The power-spectrum density of the random signal is defined as
n=co

I

cf,(n)z-n ~ 1[cf,(n)]

(4.89)

Since 1,(11) is an even function of time and bounded, it can be obtained from equation (4.89) as .p(II)

wh,-n' I'

i~

= J 11'11(::)1 = _1-

r

(I)(z)z"-I clz 2nj ) •.

th,- huundllr'y "f till! unit d ..de in the z-plam!.

(4.90)

164

THEORY AND APPLICATION OF THE z-TRANSFORM

It is seen from equation (4.88) that when n = 0, we get I Ic-N _ +(0) = lim ~ {f(k)P = N-- 2N + 1 k--N ~ mean square value of the signal

r

= 0 in (4.90) and

Substituting for n

using (4.91), we obtain:

Is

- = _1 /2

(4,91)

(Z)Z-1 dz 21Tj ('

(4.92)

Since <1>(%) is a symmetric function with respect to z, it can be written as (4.93) If we substitute (4.93) in (4.92), we can evaluate f2 readily by using the tables of total square integrals which we discussed earlier. /2

=

I n

= -'

r $(Z)jF(Z-l)Z-l dz.

(4.94)

21Tj )1'

Furthermore, if the stationary signal Ik having the power spectrum (z) is sent through a linear, time-invariant system with discrete stable transfer function ~(z), the output mean square value of the signal is given as (4.95) where EXAMPLE

1. We shall calculate the mean square value of the output of a discrete system shown in Fig. 4.8. Given the power spectrum density of the input

<1>,(.;:) = N2 where N is a constant (this represents white noise). To obtain the system transfer function (,,9(z) from Fig. 4.8, we use

_ ~9(z) =.~o(z) =

.~,(z)

_c_' z- I

[1 + (blc)Z] I

1+ _"_[I + (b/C)%] z- 1

=

%-

z- I

::(1) + c:) -- C' z:& ... (II ,. c' -- 2): I- (I ... c')

(4.9(,)

CONVOLUTION Z-TRANSFORM

Input+

165

Output

fin

fon

FIGURE 4.8 A feedback discrete system.

The mean square value of the output is given as

f~n = ~ f !J(Z)!J(Z-l)j(Z) z-Idz 27T)

Jr

N2 f =:"27Tj Jr Z2

z(b

+ e) -

+ (b + e -

2)z

e

+ (1

- e)

z-I(b+ c)- c

x Z-2

+ (b + e -

2) Z-I

+ (I

z-Idz

(4.97)

- c)

This integral can be easily evaluated by using Table III for the secondorder case and noting the following constants: bo = 0,

ao

=

I,

+ e, (b + e -

bi = b

a1 =

(4.98)

2),

Substituting these values in the 12 formula of the second-order case to obtain finally

j2 = N 2 [c(1 + b/c2 ) + b/2l When b

(4.99)

2 - e - (bt2)

on

= 0, this reduces to -

2 1on

.

-

-

eN? 2--e

(4.100)

Extension of two-sided z-transJorm to continuous outputI2.18.2S III Ihis application the two-sided z-transform is applied to the continuous c:IIl'rclation function as follows:

x .f.,(k

+ III .-

I)/,,(k -

'1 + m -

I),

o~

In ~

1 (4.101)

166

THEORY AND APPLICATION OF THE Z-TRANSFORM

If we denote the coefficient of z-n as tPo(n, m), the continuous spectral density is n-«>

~o(z, m)

!

=

z-ntPo(n, m)

(4.102)

nM.-CO

To obtain the mean square value of the form: f~

=

lim

outputf~(/),

!N 11f!(k + m + 1 k~-.V 0

1

.v~«>

2N

we use the following

1) dm,

o~ m ~ 1 (4.103)

This equation can be obtained from tPo(n, m) by Jetting n

f~ =

f

= 0 as follows:

tPo(O, m) dm

(4.104)

On the other hand, tPo(n, m) is obtained from ~o(z, m) as inverse ztransform relationship, that is,

<po(n, m)

= _1-

27Tj

ro(z, m)zn-l dz

(4.105)

Jr'

r

is taken along the unit circle for stable tPo(n, m) as is usually where the case. Therefore

tPo(O, m)

=

_I. 27TJ

ro(z, m)z-l dz

(4.106)

Jr

Inserting equation (4.106) in (4.104), we finally obtain

f: =

_1_ 27Tj

r

r\f)o(z, m)z-l dm dz = _1_

J,' Jo

27Tj

ro(z, m)z-J dz

Jr

(4.107) Evaluation of equation (4.107) can be readily obtained by using the tables of integrals which were derived in Table Ill. It should be noted that ~o(z. m) can be always written in the form (4.108) which is readily amenable to table form. Finally, we may note that the mean square value of the ouput for a discrete system could also be obtained from the Laplace transform theory. An illustrative exam pic of optimization and filtcring using Laplace and z-tmnsform methods is dis~'ussed in d~~tllil in Section K.J.

167

CONVOLUTION Z-TRANSFORM

Appendix I.. Proof of complex COnt'olulion formulatll.14.20 Suppose F*(s) and H*(s) are the Laplace transforms of sampled function f*(t) and h*(t), respectively, defined by pes)

1 = 2'[f*(t)] = -.

1 00

27T}

+1<0

F(q}

Qo-;W

I -TI.-o) dq I - e

+ If(O)

(A.I) with Re rs1 > qo > max [real part of poles of F(s)], and F(s) = 2'[/(/)], To find It'{[f(t)h(t)] . !ST(t)}, we use the following definition from (A.I): It'{[f(t)lr(t)] . bT(t)} = -I.

i

wo +JW

I -TIs-w) dw 1- e (A.2}t From the convolution theorem of Laplace transform,3 equation (A.2) can be written as 27T}

It'{[f(t)h(t)] . bT(t)} = -1. 27T}

2'[f(t)h(t)]

100-1«>

f.

+ IW 100-;'" WO

~T($-W) dw (A.3) I - e In equation (A.3) we interchange the integration with respect to wand q. Therefore [_1_.

X

27T}

1

iC+.J"; F(q)H(w -

q) ci q]

C-}a:;

Z{[f(t)Ir(t)] . DT(t)} I

= -. 27T}

c+;«>

c-;«>

1 F(q) dq - . 27T}

J.WO+i«>.

H(w - q)

100-;'"

1

I -

-Tis-Io)

dw

It

(A.4) Noting equation (A. I), we obtain for the right side of equation (A.4)

-

I

27Tj

f<+I

00

F(q)H*(s - q) dq

(A.S)

'-17>

It should be noted that H*(s - q) is a function of eT(.-q) only, In \'\luation (A.5) we divide the range of integration into intervals of length "'11 = 27T/T so that equation (A,S) becomes

.!I'{(J(t)h(t)]· ~T(t)}

t

k~
= -. 2

27T} k--a>

[fe+Ck+t);CIIO

F(q)H*(s - q) dq

]

<+(k-t);Cllo

(A.6) I AlIlllht'r tlt'rivalillll 1>1' Ihe cunvl>huiun formula can be directly obtained from the . 111111 ..1'''1'111 tlt'lillilioll .... 11 (See also l'rnhlclll 4.1.) I silllpikily WI' as,,, II It' 111111 ((lI)iI(O) . ,n. Howeyer, the sallle re~1I1t is Ilhlained if 11111/'(1) tI- II. TIll' nlll"l .1""YllliulI i~ Icflils 1111 l""c,'dse Ii,,· Ihe rl';ulc,·.

I''''

168

THEORY AND APPLICATION OF THE Z-TRANSFORM

Changing q to q + jkwo, and since H*(s - q) is a function of e7'('-Q) only, we get 9{[f(t)h(t)] . c5 T (t)}

1 le-oo ic+/fllO/I

I

= -.

2fT} 1<--00

F(q

e-/(A)o/8

+ jkwo}H*(s -

2fT

wo = -

q) dq,

T (A.7)

Equation (A.7) is further simplified as follows:

9'{[f(t)h(t)] . c5 T (t)}

d [.! leICX) F(q + j 2kfT)]

= ~ fO+lhlITIH*(S _ 2fT}

q} q

Tt--oo

o-/hl/TI

(A.8)

T

Noting that the bracketed term is P*(q), T fc+/h,ITI 9'{[f(t)h(t)] . c5 T(t)} = - . H*(s - q}F*(q) dq 2fT}

(A.9)

<-/IIIIT)

Similar derivations could be used to prove the formula for the modified z-transform convolutions.

Appendix 2: Derivation of total square integrals formula ll •22 The solution of the integral I" given as /" = !h(kya =

~.1

.;f(Z).;f(Z-l) d%

2fT} hDlt circle

11:-0

(A. 10)

%

is presented in this Appendix. The derivation is based on obtaining the right side of equation (A.lO) as a sum of residues. However, instead of using the explicit expressions for the residues, we obtain a set of 2n + I linear equations, with the sum of the residues as one variable. Because of symmetry this set reduces to n + I equations. Assuming Jf'(z) is given,

(A.11)

The integral of equation (A. 10) is solved by the method of the residues; thus we are interested in zA(z) z - Pit

~

,

- - = ~ (',."z. ,-II

k =

n.

I. 2•...• "

(A.12)

CONVOLUTION Z-TRANSFORM

169

Since we assume po to be the location of the pole at the origin of the z-plane, we have Co,,,

co,o

== 0 == aft

when k F 0

(A.l3)

We also have ZA(Z-l) -1

21 -

p"

ZA(Z-I)p

== ( 21

_~) == -zp"

p" -

21

Z-I A(Z-I) -1 21 -

PIc

"_ == -zp" I ci,,,z i i-O

(A.14)

and (A.1S)

By partial fraction expansion, ~

B(z)B(z-l)

~

Rpt

R;!

-"-'---'-..;... == " - - + " zA (z)A (Z-l) "-021 - Pic "-121 _

(A.17)

p;l

where RpA equals residue atp,,; R;"l equals residue at Pi" 1. In the preceding we have assumed only simple poles. Later we shall indicate that the results are also valid for multiple poles. For k F 0, we have zA(z)A(z-l) = z(z - p,,)(Z-l - p,,)[A'(p,,)A'(p;l)]Pk=Z

==

-p,,(z - p,,)(z - p;l)[A'(p,,)A'(Pkl)]"k~Z

(A.IS)

lind

R = PI:

_

B(p/e)B(p;;l) p,,(P/e _ p;;l)A'(p/e)A'(p;;l)

==

_R- l

(A.19)

Pt

('lIl1lbining equations (A. IS), (A. 16), (A.17), and (A.19), we obtain

ft

ft .. Iz-IRptc""aft_i-I

+I I

/e-O 1=1 t-O

ft

" ft

••

,.

+ zlr-,I/OI-O I I I zIR""p/eci,/eaft_i-J n

+ z~ ~ ~z· '·-1/-1,-0

IR".Prrl"II/,lra"

i

(A.20)

170

THEORY AND APPLICATION OF THE Z-TRANSFORM

We can also write n

n

ft

ft

B(z)B(z-l) = !zl! b"_Hlb,,_j + Iz-; I b,,_j_;b,,_j i-O j.O i-I j-O

Let

Mj =

!" Cj.kRpk

(A.21)

(A.22)

Ie-O

and

Qj

= I" Cj.lcRptplc

(A.23)

1e~1

From equation (A.12), we have

k = 0, I, ... ,n and

=

k

Clk

a", = -Pic

1,2 •... ,n

(A.24)

Then n

= ao!Rpt = I"ao k=O

Mn

(A.2S)

In addition, Ql

= -an!" Rpt = -a,,1 + a"Ro

(A.26)

n

1c=1

Since COIc

= 0,

when k:F 0

then (A.27)

QO= 0

Combining equations (A.20) and (A.21) and using the preceding relationshi ps, we have PI

tI

1-0

fJ

ft

!Zi! bn_l+lbn_1

+ !z-I 1=1

leO fa

fa

1=0

i-O

Ib"-Hb,,-, i-O n

ft

= !Zl I Ml+ja,,_j + !z-I I Mjan_i-I n

1=1 n

i-O

n

+ ;I.. 0zl+1 i-I I Qja,,-f-i + !

1=1

n

Z-I+l

I

Q/+jQ n-I

(A .28)

i=O

Equating the coefficients of ZO in equation (A.28), we get "n

!

i-O

b~ =

!

n

M n_jaj

1'.0

+ i-O I ai Qn-HI

(A.29)

We notice from equation (A.28) the coefficients of Zl is equal to the· "

• Note that

I

n

h,h" / -

i-'O

!

h,h/

/-11

-I'

CONVOLUTION

%-TRANSfORM

171

coefficient of %-1. Thus, by summing the equations for j and -j, n

" " = !M H7I- t t + !M 7I _.aH

2 !btbHJ

Q

t=o

i~O

i~O

n-I

"

+ 1=0 ! Q,,-t QH+l + i=O ! Q,,-Hi+1

Qi

" " = 1-0 ~ Qj(M J+n- i + Q,1-i+Hl) + ! Q;_;(M 7I-i + Q,,-i+l)' i-O where 0i = 0 when n < j and 0 > i. Equation (A.30) can also be written as

for j = J, 2, 3, ... , n (A.30)*

"n

2 ! bibi+1 = ~ at_/(M ,,+i + Q,,-i+l) i-O

i~O

n

= ~ (ai-l /=0

II

+ i-O ~ Qi_/(M n-i + Q,,-i+l)

+ Qi+I)(M n-i + Q,,-I+l)'

for j = 1,2, ... ,n (A.31) The n + 1 simultaneous linear equations (A.29) and (A.31) can be written as the vector equation

Om

=d

(A.32)

where m and d are the vectors !b~

Mn M n-l M n-8

2 ! bi bi +1

+ Qn

+ Qn-l

2! bi bi +2 and

(A.33)

Mo + QI and Q is the matrix Qo

• 1\

~h()1I1d

01

Q2

Q3

a"

o

o

be: nlllt'" Ihlll

00

+ Os

Ihi~

03

Q2

+ 03 Qo + 0,

01

Q=

2bob n

01

cqlllliinn

°2 al

1I1~o yield~

+ 0. + 06

(In

0,,-1 0,,-2

equal ion (A.29) when j .... O.

(A. 34)

172 THF.ORY AND APPLICATION OF THE Z-TRANSFORM

Since, from equ.ation (A.25)

Mn

(A.35)

/,,=00

Hence

I" =

J.!hl

(A.36)

QolOI

where 0 1 is the matrix formed from n by replacing the first column by the vector d. Since I" = In(ou, 01' •.• , 0,1' bo, bl> .•. , bll) is an analytic function of the coefficients Q j and b,t 0 ~ i ~ n, the result is also valid for multiple poles. Table III presents the values of In for n = 1,2,3,4. REFERENCES

J. Jury, E.

r., Sampled-Data C(lntrol Sy.vtems, John

Wiley and Sons. New York, 19S5.

2. Tsypkin. Y. Z., Theory of P"lse Systems (in Russian), State Press for Physics and Mathematical Literature. Moscow, 19S8. 3. Gardner, M. F., and J. L. Barnes, Transients in Linear Systems, Vol. '1, John Wiley and Son~, New York, 1942. 4. Jury, E. I., "A Contribution 10 Modified z-Transform Theory," J. Franklin Inst., Vol. 270, 1960, PI" 114-129. GO S. Lawden, D. F., "The Functions! nrz" and Associated Polynomials," ProCl't'din,1{1 ,,~t

o,'tl/e CUI/Ibrid..~e PI/i{os('phical Societ)', Vol. 47, Part 2,1954,1'1'.309-314.

6. Whcclon, A. D., "On the Summation of Infinite Series in Closed Form," J. Appl. Ph)'.f.• Vol. 25,1954, pp. 113-IIS.

7. Ku, Y. H., and A. Wolf, "Laurent-Cauchy Transforms for Analysis of Linear Systems Described by Differential Difference and Sum Eqllations," PrOf. IRE. Vol. 48, 1960, PI'. 923-431. S. Di~cussion of Reference 7 by E. I. Jury in Pr(lf. IRE, Vol. 4S, 1960,l>p. 2026-27. 9. Jury, E. I., and M. A. Pai, "On the Summation of

1: .q and Its Associated Ir.te-

.. on

grals," J. Franklin In.f/., Vol. 271, No.2, February 1961. 10. Stone, W. M., "A List of Generalized Laplace Transforms," Iowa Slal(' Col/eKe JOIITl/a{ (If Sc:iI'IlU, Vol. 22, April 1945, pp. 215-22S. 11. Whalen, B. A.. "Noise Power Gain of Discrete Filters," Master of Science Thesis, University of California, Berkeley, June 19S9. 12. Mori, M., "Statistical Treatment of Sampled-Data Control Systems for Actual Random Inputs." T,.ans. ASME, Vol. SO, No.2, February 1958. 13. Kaplan, Wilfred. Operatio"al Melhods /I" Lim'ar Sy.fl(!/1/l, Addison Wesley Publishing Co., Reading. Massachusells, 1962. 14. Pai, M. A., "The Analysis of Non-linear Feedback Silmpled-()i1ta Systems," Ph.D. Thesis. Dept. of F.lectricill Engineerin!~. University u«'alifllrni", Ilcrkeley. 1')1II. IS. Thellicr, P. L., Ooctnr of Science Thesis, lInivcr\ity III' (il"cnllblc, ....alll:'.. 1%2.

CONVOLUTION %-TRANSFORM

173

16. Nishimura, T., and E. I. Jury. "Contribution to Statistical Designing of SampledData Control Systems." Electronics Research Laboratory Report Series 60, Issue No. 210, August 5. 1958, University of California, Berkeley. 17. Ragazzini, J. R., and G. F. Franklin, Sampled-Data Control Systems, McGrawHill Book Co., New York, 1958. 18. Tou, J. T., Digital and Sampled-Data Control Systems, McGraw-Hili Book Co, New York, 1959. 19. Monroe, A. J., Digital Processes lor Sampled-Data Systems, John Wiley and Sons, New York, 1962. 20. Montel, Paul, Lerons sur les Recurrences et Leurs Applications. Gauthier-Villars, Paris, 1957. 21. Friedland, B., "Sampled-Data Control Systems Containing Periodically Varying Members," Automatic and Remote Control, Proceedingf 01 the First I.F.A.C. Congress, Moscow, 1960. Butterworths, London, 1961, p. 369. 22. Bharucha, B. A., "Analysis of Integral-Square Error in Sampled-Data Control Systems," Electronics Research Laboratory Report Series 60, Issue No .. 206, June 1958, University of California, Berkeley. 23. Nishimura, T., "On the Modified z-Transform of Power Spectral Densities," Trans. IRE, PGAC, Vol. AC-7, No.4, July 1962, pp. 55-56. 24. Sawaragi, Y., and A. Fukaw, "On the Sum of Squares Values of Sampled-Data," 'frans. 01 Japanese S.M.E., Part I, No. 155, Vol. 25, July 1959.

5 CONVOLUTION z-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMSl,3,6

In the preceding discussions, the z-transform method has been mainly applied to linear discrete systems. For time-invariant systems, the ztransform offers some advantages as has been explained in detail. However, for time-varying systems this advantage is no longer apparent except in a few particular cases. Nevertheless, the method is still applicable as shown earlier. In this discussion, the convolution z-transform will be applied for the solution of certain nonlinear discrete systems described by nonlinear difference equations. It will be shown that the method proposed is general in solving such difference equations. The form is r

Z ht>x(n + p) + F[x(n), x(n + 1) ... x(n + r

- 1)]

=0

(5.1)

JJ=O

where " is the discrete time variable and F[x(n), x(n + I) ... J is the nonlinear term containing terms in x(n + p) of degree two and higher. This form of the difference equation is fairly general in the sense that it describes many nonlinear discrete systems with saturation-type nonlinearities. This type is quite common in feedback sampled-data systems. The objective of this chapter is to obtain an approximate closed-form solution of equation (5.1) in a systematic manner using the convolution z-transform; a few examples arc; also discussed to explain in detail the method of solution and to compare it with the exact method using the difference equation as a recurrence relationship. In order for the method of solution to he applicahle, certain assumptions are to be imposed on the: tlilrercnce l'llulltion. These IIssumptions lire nol 174

2-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMS

175

restrictive, for in practical situations they are usually satisfied. It should be noted that there is no general method of solution for any nonlinear difference equation, .,6-10 but certain techniques are applicable to certain types of difference equations or noniinear discrete systems. In this chapter the application of only one such method is discussed in detail. This method is analogous to the application of the convolution Laplace transform 2 to certain nonlinear differential equations. Rigorous mathematical justification of this method was proposed by Wasow. 11 Similar justification for the method has been indicated$ and by applying Wasow's techniques a rigorous mathematical validity has been obtained. 12

5.1 Assumptions I. The nonlinear difference equation in (5.1) has a steady-state or equilibrium value Ao given by r

! h~Ao + F(A o• ••.• Au) = 11-0

(5.2)

0

2. The nonlinear function F[x(n) •.• z(n + r - I)] is analytic in its variables X(II), x(n + \) ... X(1l + r - \) around the equilibrium point. Only nonlinear functions involving polynomials are to be considered. 3. The equilibrium point is asymptotically stable in the sense that x(n) converges to the equilibrium point Ao as n -+ co. The initial values of the dependent variable must lie "close" to the equilibrium point in order that II convergent solution may be obtained. However, in some cases, as' will be explained later, by shifting the time axis (U!;ing the difference equation ~IS a recurrence relationship) by one or two sampling periods, quicker convergence can be expected. The basis of the solution of the difference equation is to assume :'1'(::) = J[r,,] to be a meromorphic function of the form .~(2)

GO

=! II~

A II

(5.3)

_I

01 - a,.%

(I ..'s are distinct with modulus less than unity, and ao = I. Ikl'OlIIse of Ihis assumed basic form of the solution, we have to obtain IIII' .:-Intnsform of functions of the form

Whl'I'C

JI/(" -/- k )'1 or 'yU(II + r),,/(n + q)I], 111111 ~,II 1111.

'1lwll'flln' Wl' !tlmll prm'l'ed first to derive this ::-transform by

Ihl' nllwllllllillll

·111111 ... 1"111'111 IIl('lhllllll' di""IIS!tl'" ill ('hilpll'"

4.

176

THEORY AND APPLICATION OF THE z-TRANSFORM

5.2 Convolution z-transforms of certain functions Assume fen) is a time function defined for discrete values of n = 0, I, 2• . . . , and its z-transform is of the form 00 A (5.4) « -1 1[/(n)] ~(z) !

=

=«=01 -

Q,.z

where 0 0 = I and the other Q'I:5 are distinct with absolute values less than unity. Using the convolution integral,

r

1[f(n)2] = _1 p-l!P"(P)~(=)dP (5.5) . 2~)r p where r encloses all the singularities of p-l!P"(p) in a positive direction in the p-p[ane. Therefore 1

l

[/(n)2J = _1 -1~ A" ~ A, d 27rJ. [. P ,,=0 ~ 1 - Q,.p-1 ,-0 ~ I - Q,z -1 P P (5.6)

Similar[y,

I[J(n)3] =

III

A,.ApA y

(5.7)

-1

,.=0,.0 y-o 1 - Q"O,QyZ

The extension to ,,[f(n)l] is evident. It is also easy to obtain the following transforms in a similar manner.

:~:[/(q)'lz-q] (5.8) For the nonlinear terms of the form!(n)!(" + k), we use the following: 1[J(n

+ k)2 =

1[/(n)J(n

zk[1[J(n)l] -

+ k)J = ~ 27rJ X

rp-lpk

)1'

[~(P) - ~>(q)P-q]~(;)dP

where r encloses the singularities of p-lpk[$'(P) Equation (5.9) is evaluated as

(5.9)

kf!(q)p-q] 11=0

(5.10)

Following the procedure outlined. we can, in general, find the ;:transform of nonlinear terms of the type fCIl + ,)It/ O. pruvidillA .JII
Z-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMS

177

5.3 Method of solution for secondand higher-order equations Consider a second-order nonlinear difference equation x(n

+ 2) + bx(n +

I)

+ cx(n) + kx(n)S = {3u(n)

(5.1I)

where u(n) is a sampJed unit step function and the coefficients b, c, k are constants. The constant Pis the magnitude of the step forcing function. I. Assume the z-transform ,q[(z) of zen) to be a meromorphic function of the form .q(z) =

f

II~O

=

1-

All OIlZ-1

+

Ao

1 - Z-I

+

Al + A2 1 - a 1z- 1 1 - a2z-1

As 1+ A41 I - a3zI - 04Z-

+...

(5.12)

The term O
\ > lOll > 10 1> 10 1> 1041 > ...

(5.13)

3

2

2. Take z-transforms of all the terms in equation (5.11) to yield 2"(Z)(Z2

Dividing by

Z2

+ bz + c) + kJ[x(n)S] = ~ + (Z2 + bz)x(O) + zx(\). z- 1

(5.14)

+ bz + c = (z -

.~"(z) +

k

a;)(z - a;) gives a~)

(z - a;)(z -

J[x(n)3]

= . t [~+ (Z2 (z - al)(z - U2) z - I I

I

+ bz)x(O) + ZX(I)J

(5.15)

Now if .f(z) is written in the form .'?f(z) = ~

I-

+

Z-1

Al

I - a l z- 1

+

A2

1 - a22- 1

+

A 20

1-

. 3 -I + ... + 1 - A02U2Z2 -I A03 + All + -~:-I -I + . . .

+t

A30

- a1z

I -

.,

a~z-l

Alii

I .

t - a 1a22

{liZ

a'.+

"."11:

A. 2

-1

-~-I+'"

1 .-

lIllla:

(5.16)

178

THEORY AND APPLICATION OF THE ;::-TRANSfORM

the z-transrorm or X(II)3 can be written as .\ J[x(n)' J =

A30 I - %-1

1 3A2oA2 + I 3A2oA + -..::.....:=- al%-I I - Q2Z-1

+ 3A~A20 + 3AoA~ I - a~z-I

+ 3A~A30 + A~ + 6AoAtA20 I - a~z-'

+ 3A~A40 + 3A~oAu + 6AoA,A30 + 3A~A20 1 - a~z-'

+ ... + 3A~A02 +2 3_~oA~ + ... 1- Q2Z

+ 3A~A21 + 3A~Az + 6AoAzoAz I - a~azZ-l

+ 3A~AII + 6Ao~lA2 + . . .

(5.17)

1 - o,azZ-

It is assumed that a;, o~ in equation (5.15) are distinct from the a's in equation (5.12). Comparison or coefficients is now possible in equation (5.15). 3. By multiplying both sides of equation (5.15) by % - I and letting :: --.. I. the steady-state relation involving Ao is obtained. Ao can also be obtained by letting x(n) = x(n + I) = x(n + 2) = Ao in equation (5.11). 4. By multiplying equation (5.15) by z - 0Ot. IX = 1, 2 and by letting in each case z - 0",. we obtain

+

AI,2(a~.2 + ba,,2 c + k3A~) = 0 For nontrivial solutions of AI, A2 in equation (5.18).

a~.2

+ bal,2 + c + k3A~ =

0

(5.18) (5.19)

°

which determines 1 and 02 in terms of steady-state amplitude Ao. All the 3) can then be evaluated as explained earlier in equation (5.12). 5. By multiplying equation (5.15) by % - 0fl.' IX ;;:: 3 and by letting in each case.: - a,.. the following relations are obtained:

Ofl.·s (IX ;;::

A3 = !1(A o• AI' A 2)

A4

= !2(A CI• AI.

A 2 • A:I)

( .....'(1)

;;-TRANSFORM APPI.IF.D TO NONLINEAR DISCRETE SYSTEMS

179

where fl' ./~, .... etc., denote functional relationships. The process by which any A,(1. ~ 3) in equation (5.20) is found in tcrm~ of previous A;s is possible because of the manner in which the a;s were ordered in e'luation (5.12). Thus all A~'s (~~ 3) can be expressed ultimately in terms of Au. AI' and A 2 • 6. Finally. by multiplying equation (5. I5) by z - a~, (1. = I, 2 and letting in each case:; -> a~. two nonlinear relation~ are obtained.

3A~AI + 3A~A2 ) iJ , + . .. = -'--I ao - a l a. a. +(a;+b).r(O)+:r(l) ,,;=1,2

Ai: k ('-'--I Q

+ ,

a-

"2

(5.21) The convergence of these nonlinear relations depends on the initial values of .r(lI) and the magnitude of the nonlinearity. If the nonlinearity is "small." then for a fairly large range of initial values convergence could be expected. On the other hand, when the nonlinear terms are comparable to the linear terms, then the range of initial values is necessarily restricted. At any rate, convergence is best judged from the particular problem on hand. With first-order difference equations, this does not pose a serious problem. However, for higher-order systems the numerical computations become too involved and it is also sometimes difficult to determine the cnnvergence. The procedure just detailed can be easily extended to higher-order equations. 5.4 Illustrative examples To illustrate the method discussed in the preceding section. two examples are introduced. The first example represents the solution of a certain nonlinear first-order difference equation. This equation can be solved by other methods which are limited only to first-order case. 4 The second l'xample represents the analysis of a first-order saturating sampled-data Il'cdbad system. 1 We limited these examples to first-order for ease of 1'l1l11putalion. However, as outlined in Section 5.3. the method is general and call he applied to second- and higher-order nonlinear difference \'qllatiolls or discrete systems. A study of a second-order system is 1I1l'IIIdni ill Chapter 1<. I'XAMI'I F

I

("'II~id\'r Ihc soilltioll

of the following difference equation.

"!lIII I I) 1111I.llitlll (', .'.') 1.111

H'/(I/ I

III'

H.'I(II)~

+1

!/(" f

I I

\C'\\Tilh'1I

II I H'/(I/ I

(5.22)

as

1)1/11/)"

H!lIIIf

l'i.~~.\)

180

THEORY AND APPLICATION Of THE %·TRANSfORM

Assume

l!V(%)

= i:

II~O

=

1-

All

-I Q,,%

Ao 1- %

---1

+ 1 - Alalz -1 + 1 - AeQl%e -I + . . .

(5.24)

By taking the z·transform of equation (5.23),

8%[!V(%) - yeO)]

+ 8J[y(n + 1)y(n)2] -

8J[y(n)2] =

~ % -

1 (5.25)

But, as shown in Section 5.2, (5.26) and

1. p-Ip LeoI.. 1 - AO"p..

+

J[y(n - 1)y(n)2] = -1. 21TJ ['

-I -

0

yeO)]

. (5.27)

where

r

encloses the singularities of

p-'pLi:=0 1 - AllO"p

-1 -

yeo)]

•

This equation becomes (5.28) Substituting this transforms into equation (5.25) and dividing by z, we get 8

f

".0

A"

1-

a,.%-I

+!f

I

A .. A,tA 7o y

zL.,,,,Y-G 1 -

o"O,tOyZ-1

A ] - ",/1=0 Ieo 1 - Aa"a,z-I ""

3 = - - + 8y(0) (5.29) %- 1

Comparing coefficients of 1/(% - l)andwriting%/(z - ex) = [ex/(% - ex) + I], we obtain

8Ao

+ 8A~ -

8A: = 3

(5.30)

The equilibrium solution Ao can be obtained, which is

A.. =0.5

(S.31)

Z-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMS

181

The solution around this point is obtained as follows. Comparing the coefficients of I/(z - Ql),

8Al

+ !. (A~AIQl + 2A~Al) - !. (2AoAl) = 0 al

(5.32)

01

Rearranging this equation,

+ A~al + 2A~ -

AI(ol

2Ao)

=0

(5.33)

For the nontrivial solution of At> the term inside the parentheses must be zero. (5.34) Substituting Au =! in this equation, we obtain Q I = §. Comparing coefficients of I/(z - aI), the following relation is obtained: 8a~A2

+

8(A2A~0~

+

2A2A~

+

2A~Aool

+ A~Ao) 8(A~ + 2AoA2) = 0

-

(5.35)

which, when simplified. yields A2 = -~A~

(5.36)

In the preceding manner. by comparing coefficients with the other terms, •• can be determined in terms of AI' To obtain the nonlinear relation for AI' compare the constant term on both sides of equation (5.29). This gives A 3, A 4••


I

A..

a~O

= yeO)

(5.37)

Thus II¥( ) _ 1 , z - 2(1 _

+ Z-I)

AI

!

1 - ~Z-I

_

iA~a5)"

+ .,.

3 (1 -

A~

246 Z- 1)

+ ...

(5.38)

which. upon inverse, yields yen)

= ~ + A (i)" 1

(5.39)

If !I(O) is specified. Al can be determined from this equation. providing (5.39) converges for the given value of yeO). EXAMI'I.!: 2 III this cXlIIlIple u sumpled-data system with a small nonlinearity in the f,'cdhal'k path is considcred. The nonlinearity is generally incidental and IIll1y rl'prcscnt a sli~ht dcviution from otherwise normal linear feedback. 'I'll" prllhll'llI is (if sllch a systcm is stable) how to gct an cxplicit solution for Ihc IIIIIPllt whcn Ihc furcin~ function is II stcp input, The stahility of

182

THEORY AND APPLICATION OF THE z-TRANSFORM

the system is determined from the stability of the "linearized" system, with nonlinearities neglected. By referring to Fig. 5.1, the difference equation relating the output to the input is first derived. An explicit relation is possible between the sampled output and input. The input is assumed to be a unit step function. The nonlinearity is represented as a finite power series, as indicated in Fig. 5.1. Let ret) = u(t), where u(t) is a unit step function E(s) = R(s) - .!l'(m1c

+ Imllcv)

(5.40)

11-2

where .!l' indicates the operation of taking the Laplace transform of the function in the parentheses. The sampled form of equation (5.40) is E*(s) = R*(s) - Z[.!l'(m1c

+o-a fmoc

lI ) ]

(5.41)

I~.f'·

where Z indicates z-transform operation. In equation (5.41) the .!l'transform operation can also be omitted. The output transform is given as C*(s)

= G*(s)E*(s) = G*(S){R*(S) -

Z[.!l'(m1c

+ Imllc

lI ) ]

CJ-=2

Now

}

(5.42)

.".".

(5.43)

C*(s)/.=er• = <;fez)

Therefore in z-transform notation, <;f(z)[1

+ m 1~(z)] + ~(Z)Z[.!l' C~2mvc")]

=

fR(z)~(z)

(5.44)

It is easy to see that in the absence of nonlinearity and ml = 1, we obtain the linear sampled-data system with unity feedback. Zero-order

hold c(t)

Gi.) N

1II1C

+ ~ 1119C9,..----:--:---,

,-2

FIGURE S.I

Ntlnlinelu' NIIIllI,lcd-dlllll r«dhm.:k lIy~lcll1.

Z-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMS

Choose a first-order linear plant with G1(s) = 1/(s

~(z) =

-

<[_1_ 1-se-'I'S] s+1

183

+ 1),

1 - e-'I' e-'I'

(5.45)

Z -

With T= I

~(z) =

0.632 0.368

(5.46)

z -

and ~(%)[Z

- 0.368

+ 0.632mlJ + 0.632Z[2(fmqcq)] q~B

= 0.632% (5.47)

z- 1

This corresponds to the nonlinear difference equation

c(n

+ 1) + (0.632m. -

N

0.368)c(n)

+ 0.632 I

mqc'l(n)

= 0.632r(n) (5.48)

If mq = 0 (for all q), in equation (5.48) and m. = I, it reduces to the linear system with unity feedback, as explained before. Equation (5.48) then reduces to

~(%)(z + 0.264) = 0.632z %-

(5.49)

1

Taking the inverse z-transform or making use of tables, the solution for linearized system is obtained as

c(nT) = HI - (-0.264)RJ

(5.50)

By inspection it is seen that for n approaches infinity, c(nT) tends to ~ and hence the linearized system is stable. For the nonlinear system, assume that mq = 0 for all q in equation (5.4K) except q = 3 and m. = I, and ma is "small" compared to J. Since Ihe linearized system is stable, assume*

=i

All (5.51) I - a,.z-·I wilh till ,.." I (from steady-stute conditions), 02 = of. 03 = a~• ... , aq = /I'f . . .. With '(,'(::) ~Issumed ;IS curlier, it readily follows from the discussion


II

•

TIJI~

0

rllnll IIr ~lIhlllllll ire the

bllsi~

IIr Ihe CUIIYlllutiulI %·trllllsrllrlll liSt'" ill

Ihi~ CXlIl1lplc.

184

THEORY AND APPLICATION Of THE Z-TRANSFORM

of the previous section that

:1

[c(n)3] = (

1-

A~

+ 3A~AI + 3A:Az + 3A~Ao 1-

-1 Z

1-

-I Ql%

I -1 Ql%

+ 3A~A3 + 6AlAaAo + A~ 1-

Q~Z-l

+ 3A~A4 + 3A~A2 + 3A:Ao + 6AIAaAo 1-

Q~Z-1

+ 3A l A: + 3A~Aa + 6A.A4AO + 6AIIAaAo + 3A5A.s a~z-1

13A~A6

+ 6AIAsAo + 3A~A, + A: + 6AIAIAs + 6AoAtA, + 3A:Ao

+------~~~~~~~~~~~~~

1 - a:z- 1

+ ... + .. .)

(5.52)

Take z-transform of equation (5.48) assuming zero initial conditions and use equation (5.52) to get

f{(z)

+

0. 632ms ( A~ z + 0.264 1 - Z-1

+ ...) =

0.632z (z - 1)(z + 0.264)

(5.53) Comparison of coefficients is now possible. Comparing the coefficients of z/(z - I), z/(% - 0 1), %/(% - a~) ••• z/(z - a1), and finally z/(z + 0.264), the following relations are obtained:

A

o

632m s A3 = + 0.1.264 0

+

A 1

A 2

0.5

(5.54)

0. 632m s (3AtA) = O' + 0.264 0 1

(5.55)

OJ

s (3A 2 A + 3A t A ) = + a~0.+632m 0.264 0 2 1 0

0.632m 3 ( - ~~ 1.2M

~~~.__

___ I'I

+ O.2M

-

0

(5.56)

(3-1.~~~±.~_~_~.!!1_ (II

+ 0.264

....-

().~

... ) (~.~K)

Z-TRANSFORM APPLIED TO NONLINEAR DJSCRETE SYSTEMS

185

Equation (5.54), when solved for Ao, gives the steady-state value of c(nT). Substituting this value of Ao in equation (5.55) gives A 1(al + 0.264 + O.632m33A~) = 0 For nontrivial solution of AI' it foHows that

(5.59)

+ 0.264 + 0.632m33A~ =

0 (5.60) from which 0 1 can be readily computed. If lOll < 1, the system is locally stable around the equilibrium point. The other equations can be successively solved for A 2 • A 3 • A4 ..• in terms of AI' Using equations (5.54), (5.55), (5.56), and (5.57), equation (5.58) can be written in the form al

Ao+AI+Az+ •.. =O

(5.61)

This is merely the initial condition requirement (assuming zero initial condition) which can be written by inspection of the form of ('6'(z) in equation (5.51). By knowing Ao and A 2 , A 3 • •• in terms of AI> equation (5.61) can be solved for Al for whatever accuracy desired. A 2 , A3 , ••• can also be then computed. By taking a few trial values of Al convergence can be easily ascertained. It is now a simple matter to take the inverse z-transform of equation (5.51) to get the time domain solution. The actual numerical calculatio.ns are now carried out. Assuming* rn3 = 0.1. solution of equation (5.54) yields

Ao

= 0.494

(5.62)

With these values of Ao and rna, a1

° is calculated from equation (5.59) as 1

= -0.31

(5.63)

Since lad < I, the system is stable for a unit step function input. From equations (5.56) and (5.57), A z• A 3 • •• are obtained as follows: Z A _ _A _ 1 A ___ A 31_ A _ A1 2-

As

=-

4.35'

3-

AS 10.;5'

Ae

=-

13.95'

4-

6. 14 '

A~

26.7' ..

(5.64)

Suhstituting equation (5.64) in (5.61) gives

0.494

+ Al -

Ai A~ 4.35 - [3.95

At

+ 6.14 A~ A~ - -10.25 - -26.7 + ... = 0

(5.65)

• hll III. (1.1. whkh i,"lkates a Salllnltiull type nonlinearity, it will be found that ,u"vl"I:I'IJ('I' 1'1' l'll"illi"" (~,44) ;~ 1'''''''' lIowever. hy shiftill!! time axis (lIsin!! the ,hlll''''''''(' (,IJlllllill" I" " "'1'111 n'm'(' !l'IIII;IlII) hy 1IIIt' '1IIIIplill/: Pl·ri,,&!. 111Ik-ker cunver· 1:(,111'1' 1""' II(' 1'~IlC'I'I.',1 'J hi' i, tllmllnl.',1 Inl .. r.

186

THEORY AND APPLICATION OF THE Z-TRANSFORM

Solution by trial and error of equation (5.65) yields Al

~

-0.461

(5.66)

Hence AI = -0.0489

Aa = 0.007025 A:; = 0.002015

Iit(z) = 0.494 1 - Z-l

0.461 1 + 0.31z- l

A. = 0.00735 A. = -0.000359 (5.67)

Therefore

+

0.0489 1 - 0.096z-1

0.007025 + 1 + 0.0298z-1

0.00735 + 0.002015 1 - 0.0092z- 1 1 + 0.00285z- 1 0.000359 1 _ 0.000884z-1

+ ...

(5.68)

Corresponding to c(nT) = 0.494 - 0.461(-0.31)" - 0.0489(0.096)"

+ 0.007025(-0.0298)" + 0.00735(0.0092)" + 0.00201 5( -0.00285)" -

0.000359(0.000884)" (5.69)

It should be noted that some of the modes are oscillatory decaying unlike the continuous case which are exponentially decaying. The nonlinear difference equation of this system is c(n

+ 1) + 0.264c(n) + 0.0632c(n)3 =

0.632r(n)

(5.70)

With c(O) = 0, this can be solved as a recurrence relation to get an exact solution. A comparison of this exact solution with the approximate closed form solution in equation (5.69) is given and it is seen that agreement is good up to the third decimal place. The output between sampling instants could be easily obtained by minor modification of equation (5.51). Sampled Output

Exact Solution

Closed Form Solution

c(0) c(l)

0 0.632 0.4491 0.50779 0.48970 0.4955 0.4935 0.4942

0.000131 0.6323 0.44925 0.50767 0.48975 0.4953 0.4936 0.4941

c(2) c(3) c(4) c(5) c(6) c(7)

Z-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMS

187

2a

EXAMPLE

If we take in Example 2 m3 = -OJ, the difference equation that describes the system is now c(n

+

1)

+ 0.264c(n) -

0.0632c(n)3 = 0.632

(5.71)

For this equation, the coefficients are

Ao = 0.5065

A. = 0.93A~

At = 0.356A~

As = J.7A~

Aa = O.64A~

As

=

(5.72)

3.74A~

It is noticed from the preceding that the convergence is rather poor. However, it can be improved if we shift the time axis by one sampling period so that the difference equation is the same but the new initial value is 0.632 (instead of zero); this can be done by using t.he difference equation as a recurrence relationship. For instance, if we let c(n)I,,~o = 0 in Equation (5.71), we readily obtain the new initial value as 0.632. The amplitude determining equation is 0.632 = 0.5065A 1 which gives

+ 0.356A~ + O.64A~ + 0.93A~ + 1.7A~ + 3.34A~

At = 0.1191

AI = 0.00506

Aa = 0.001

A. = 0.0001

(5.73)

(5.74)

The solution is then c(nT)

= 0.5065 + 0.1191(-0.2155)"-1 + (0.00506)(0.0465)"-1

+ (0.001)(-0.01)'1-1

n

~

I, for n = 0, c(O+) = O.

(5.75)

. It is evident that this solution is rapidly convergent. Comparing the output amplitude with the exact solution, we notice

Approximate c(1) c(2)

('(3) c(4)

= 0.632

= 0.4811 = 0.512

= 0.5053

Exact c(1) = 0.632 c(2) = 0.4809 c(3) = 0.512 c(4) = 0.5054

III cCllldusiun. we ,'(lIIld deduce th:lt bcttcr convergence is expected if Ih,' m~w inililll vllh,,' i ... IWIIC tlw \'(luilihriulll pnint. The new initial vullie

188

THEORY AND APPLICATION OF THE Z-TRANSFORM

can be easily obtained for the discrete case by using the difference equation as a recurrence relationship. Such an advantage is not possible in the continuous case. A compromise between using the difference equation as a recurrence relationship and the labor involved in the convergence is required in any problem amenable to solution. REFERENCES

J. Jury, E. I., and M. A. Pai, "Convolution .:-Transform Method Applied to Certain Non-linear Discrete Systems," IRE Trans. PGAC, Vol. AC-7, No. I. January 1962, pp.57-64. 2. Weber, E., "Complex Convolution Method Applied to Non-linear Problems;' Proceedings (1' lire Symposium on Non-linear Circuil Ano~l'sis, Polytechnic I nstitute of Brooklyn, Brooklyn, New York, Vol. 6, 1956. 3. Montel, Paul, Lerons sur Its Recurrences el Lel/rs Applications, Gauthier-Villars, Paris, France, 1957. 4. Levy, H., and F. Lessman, Finile Dijference Equations, Macmillan Co., New York, 1961. 5. Pai, M. A., "The Analysis of Non-linear Feedback Sampled-Data Systems," Ph.D. Dissertation. Dept. of Electrical Engineering, University of California, Berkeley, September 1961. 6. Torng, H. C, "Second-Order Non·linear Difference Equations Containing Small Parameters," J. Franklin Insl., Vol. 269, February 1960, pp. 97-104. 7. Norlund, N. E., "Nichtlineare Differenzgleichungen," Encyklopcdie dcr Mathematischcn Wissenschaflen, Leipzig, Germany, Vol. II, No.3, 1927. 8. Trijinsky, W. J., "Non-linear Difference Equations," CO/llPO.I·iIO MOlh('/IIotic/(, Vol. 5, 1937, pp. 1-66. 9. Tsypkin, Y., Theory of Pulse Systems, State Press for Physics and Mathematical Literature, Moscow, 1958. 10. Strodt, W., "Analytic Solutions of Nonlinear Difference Equations," Anl/ols Math., Vol. 44, No.3, July 1943, pp. 375-396. 11. Wasow, W., "Solution or Certain Nonlinear Differential Equations, by Series of Exponential Functions," "'il/oid. Malh., Vol. 2, No.2, June 1958, pp. 254-260. 12. Dewey, A. G., "On the Convergence of a Series Solutiun of Certain Nonlinear Difference Equations," Electronics Research Laboratory Technical Memo No. TM-18, University of California, Berkeley, July II, 1963. 13. Alper, P., "Higher-Dimensional Z-Transforms and Nonlinear Discrete Systems," Technological University, Electronics Laboratory, Delft-Netherlands, August 1963. 14. Alper, P.:,"A consideration of the discrete volterra series," IEEE Trans. Automat. Contr., Vol. AC10, July 1965, pp. 322-327. IS. Rault, A. and Jury, E. I.: "Nonlinear sampled-data systems and multldimensional z-transform," Electm. Res. Lab, University of California, Berkeley, Memo 1121, February 1965. 16. Lavi, A. and Narayan, S.: "Analysis of a class of nonlinear discrete system using multidimensional modified z-transform," IEEE Trans. Automat. ConlT. (shorl papers) Vol. AC-13, February 1968, pp. 90-93.

6 PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

In this chapter a study is made of periodic modes of oscillation of autonomous discrete systems. The types of nonlinear systems to be studied are pulse-width modulated, relay, nonlinear gain amplifier, and quantized level systems. The method given is also applicable to obtaining the periodic modes of oscillation for nonlinear difference equations as indicated briefly toward the end. In addition, the stability study of limit cycles as well as the limitation on the period of limit cycles for relay mode of oscillations is presented. Finally. a certain analytical technique for examining the existence and stability of forced oscillation in a sampled-data system is discussed. A~ain the method presented is based on the z-transform, and the examples arc chosen from the widely used field of sampled-data and digital control lIystcms.

6.1 Limit cycle analysis of nonlinear discrete systems Wltl'n a periodic forcing function is applied to a nonlinear discrete system III' a spontaneous oscillation is excited within the feedback system, the '.y~Il'1ll will frequently move into a limit cycle, that is, sustained periodic IIwilial ions then arise in the system. ()I\l"l' lite feedback system shown in Fig. 6.1 is forced into a limit cycle, 1111" olltput fmm the nonlinear component e;.,h(t) will have a periodic It'allln' ~illlilar 10 I he other variables of the system such as the output /',,(1) til tlw (:olltml ('rmr (',,(t). This constitutes the basic assumptions for IlIc' nllllly~i .. whidl ill l"()IllIllCIIl III all ollll'r sludies or periodic oSl"illatiol\s. I Hel

190 THEORY AND APPLICATION Of THE z-TRANSFORM

In our study we shall also assume that these oscillations are periodic with period an integc:r multiple of the sampling period T. With the pulse width modulated feedback system, as will be discussed later, the signal e~it) will actually be a train of finite pulses with periodic features as shown in Fig.6.2. Therefore the application of the theory of the finite-pulsed system shown in Fig. 6.3 is immediately suggested for the analysis. To obtain the basic equations that yield the limit cycles, we shall review first some of the pertinent concepts of finite-pulsed systems. 15 - 11 Let e~.Ia(t) be the output of the finite pulse width sampler as indicated in Fig. 6.4c during the (n + I )th sampling period. This output is represented mathematically by the product of e~.Ia(t) and a unit pulse uli(t) as indicated in Fig. 6.40, b. e~.Ii(t)

=

e~(t)uli(t)

(6.1)

where for 0 S; t S; h elsewhere

(6.2)

The incremental response transform, ~C:'(s) is defined as the product of the transfer function KG(s) and the Laplace transform of the pulsed input to the linear system Y.le~.h(t)] AC~{s)

=

KG(s)£~.II(s)

FIGURE 6.2 SlImrling rnllcrn rllr II glll""1 rerill"

(6.3)

'/~/

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

E~

r---i

hIt

KG(s)

191

~S)

.

FIGURE 6.3 Open loop finite pulse-width system.

The term llC;'(s) represents a fraction of the response transform effected + I )th sampling period. This incremental response is shown in Fig. 6.4d. Let

by the input which is supplied to the linear system during the (n n = Mk, Mk

+ I, ... ,Mk + I . .. , M(k + I) -

I

,-'

(6.4)

in equation (6.3) where M is given by To = MT. We then add AC;'(s) from n = Mk up to n = M(k + I) - I, multiplied by the relative delay

L o

-------~

til

(b) Uh (t)

pulsed function along t-axis

.(e) e~. h (t)

pulsed input

(eI) 6(',,'(t)

Incremental re!>llUnsl!

-~.~

........ I Ie ;111(1· "."

•.

~t

192

THEORY AND APPLICATION OF THE Z-TRANSFORM

factor. The relative delay is the delay with respect to the Mkth sampling instant, and the delay factor is given by where 1=0, 1.2•... , M - 1

(6.5)

Then equation (6.3) gives I1Ck(s)

M-l

=

Ie-liT I1C MkH(S)

(6.6)

'~O

Thus I1Ck (s) gives the incremental response resulting from the input during the (k + l)th global period T G • If this incremental response I1Ck (s) is superposed for all k from 0 to ro with proper time lags. the summation will give the total response C(s). Hence 00

C(s) = Ie- kSTu I1Ck(s) k=O 00

= Ie- k• Tu k=O

.v-I e- I• T

I

lEO

I1C Mk+l(S)

(6.7)

The z-transform of this equation is obtained by using the notation

z

= e'T and Z = e'Tu = zM.

where the function Zd[ ] represents an operator that may be described as the sequence z-transformation and it is performed by multiplying the function of k by Z-k and summing for all k from zero to infinity. Similarly. the response between sampling instants is given with the aid of the modified z-transform as follows:
[,\(-1z- Z

= IZ-k l1
kaO

I

m

[KG(s)E;.lfk+ll.h(S)]

]

1=0

(6.9)

The general form of (6.8) and (6.9) can be readily applicable to the analysis of finite pulsed sampled-data feedback systems. It is used in the following to aid us in obtaining the fundamental equations for the limit cycles. After the system enters the limit cycle. the incremental response of each sampling period will be repeated for every global period Ta of the limit cycle, and thus I1Ck (s) of equation (6.6) will become identical for every k. Thus the incremental responses of each limit cycle become independent of k. Hence Mk at the subscript of (" and H' muy be removed. Iinwevcr, we add the subscript .~ tn indicate the skady-state (althnll~h pcrimlically varying) (If the variuhle. Thl'n the illl'n'lIIt'ntnl respnllsl' in (1m~ limit t'ydc

193

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

is denoted as t1Cs(s) and is given by the sum of t1C IS(s) from 1= 0 to I = M - 1 with proper delay factors (where I indicates the relative location of the specified sampling period within the global period T () of tbe limit cycle). It is given as ,\1-1

t1CN(s)

=I

e-I,T 6,C I.(s)

(6.10)

1-0

Using this notation, ~(z) of equation (6.8) will be divided into two parts, the transient-state and the steady-state (limit cycle)
where t1~$(z)

.\"-1

cc

k~O

k-.""

l: Z-k Mfk(z) + I = <7.[t1C.(s)]

Z-k 6,~.(z)

and

t1~k(Z)

(6.11)

=
Equation (6.11) can also be written as ~(z)

=

'\"-1

~ Z-k 6,lifk (z)

k~O

Z··.\·+!

+ - - a~s(z) Z -

(6.12)

1

At th.is point the skip sampling operation (introduced in Section 4.3) is lJSed to obtain the z-transform with respect to the period Tn = MT from '?J'(z). This operation is described as and its mathematical form is given as

<.[ )

1II - 1dp 1f(Z) = <.['if(z») = -1. ~ ~(p) 21TJ r 1- p" Z P

(6.13)

Applying the skip sampling operation to equation (6.12) yields 'ifo(Z) =

X-I

-X+l

k~O

Z- 1

1 Z-k U ~Ok(Z) + ~ t1~o.(Z)

(6.14)

where (6.15)

T

If the response at the next set of sampling instants, that is, T. T + Ta• + 2T(; . ... is desired, it is obtained by two different ways as follows: ~'I(Z) = Zs[zi{(z» =

;\'-1

Z-.\·+1

k=D

Z - 1

l: Z-k.('.[Z t1~k(Z») + - - a~ls(Z) (6.16)

where (6.17)

or hy the IIlternate and equivalent form (since no discontinuities at the ·.alllplillg inslanls exist) .\' I Z :v II (6.IH) ~ /. l·~.It\'(.;.(zU I ' N(.' (Z) J

II

Z

1

h

194

THEORY AND APPLICATION OF THE Z-TRANSFORM

where (6.19) and (6.20) The responses at other sets of sampling instants can be obtained in a similar fashion. The relationships (6.16) and (6.17) with 6.18 through 6.20 will be utilized to arrive at two equivalent forms of the fundamental equation for the limit cycles. At this point. the assumption is made that the transfer function does not contain any integrator and that all its poles possess negative real parts. This restriction on KG(s) will be relaxed and KG(s) with a single integrator will be discussed later. Then it is observed that neither 6.~Ok(Z) nor 6.~o.(Z) possesses poles at Z = I because neither KG(s) not E~is) has poles at the origin of the s-plane. Hence the first term under the summation in equation (6.14) has no pole at Z = I. and the last term has a single pole at Z = I. Now the final value theorem is applied to equation (6.14) to obtain the output co. as follows: Co.

= lim

(Z - 1) ~o(Z) = lim Mfo.(Z)

Z-l

In general. the final value of
(6.21)

7.-1

Cis

and is given by

= lim (Z - 1) ~I(Z) = lim 6.~i'(Z) Z-1

7.-1

i=0.1.2..... M-1

(6.22)

This is one form of the fundamental equation to be used for the limit cycle analysis. Another form which is very convenient consists of equations (6.18) through (6.20). Following the same manipulation used in (6.14) and (6.15), we finally obtain

i = 0, 1.2•...• M - I

It may be noted that H; I ,.,,(.f) = I~'; I

i

AI.,,(s). "hen I -i ;

> M.

(6.23)

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

195

The response between sampling instants is derived by applying the final value theorem to fI;(Z, m):

;=

°: ;

0, \, 2, ... , M - \

m ::;; \

(6.24)

When no discrete processing units are included in the plant, equation (6.23) can be more conveniently and simply written as

; = 0, \, 2, ... , M - I (6.25) We note that the skip sampling process is removed, and instead we use the regular z-transform with respect to the global period Ta = MT. The evaluation of equation (6.25) can be performed by using the integral theorem which gives '·i. =

lim

1._r·"r·_l


Z-l = t·"r'~1.-1 lim 21Tj

!

,U-l

e-lsTKG(s)E;+i.h(S)

]}

Z-1

,~O

i

,\1-1

e,l1,,7'KG(p) ~ e-'"TE' . (p) ~

I+ •• h

';0

1 _ e-.1ITI<-",

I'

dp,

.-0

i = 0, 1,2, . , , ,M - 1 (6.26)

The multiplication of the expression within the bracket by e·lh7' is done to insure the convergence of the integral along the infinite semicircle in the left half plane, This does not cause any change in the final result; we compensated for this by multiplying by Z-I. , We can interchange the limiting process and the integration if we take the path of the latter sufficiently close to the imaginary axis of the p-plane. lienee equation (6.26) reduces to ,\1-1

\

(./. -__ r 21Tj J..

f;',llpTKG(p)

!

,~O

e-I'PTE!+i.h(P)

I - e!l1'PT

dp '

; = 0, \,2, ... ,M - 1 (6,27) Furthcr simplification uf this equation is possible, when the oscillations sYlllllIt'trinli. SYIIIIIlt'trical (lsciliuliull rcsults whcn the same shape or IIsl'iliatioll:o. b rqwatcd lilr cwry 11lI1!' pcriod or thc limit l:ycle with an "PII'I',ill' ~il',II,

lll'l'

196

THEORY AND APPLICATION OF THE z-TRANSFORM

If we let M cycle) and

= 2f.t (f.t =

number of samples in the half period of limit for I = 0, I, 2, ... , f.t - t

then, since E/+II+U'(s)

=

(6.28)

-E/ H.is). equation (6.27) can be written as

or 1'-1

_I c - II 27Tj

r

)1'

el,uTKG(p) ~e-">T£i_;.II(P)

dp

1=0

1 + eP PT

(6.29)

We note that half the zeros of the denominator of the integrand which include the zero at the origin are canceled by the zeros of (I - rIll>'/') in the numerator by introducing the symmetry condition. In some cases the symmetry condition requires that the summation of e;.h(O) over one global period is equal to zero. It should be noted that the elimination of the pole at the origin simplifies consideration of the analysis of the system when KG(s) contains an integrator. The symmetry condition is a valid assumption for relay and PWM systems when the plant has an integrator and low-pass filter characteristic as is usually so. In summarizing the preceding, equation (6.26) is a general form to obtain the response at the sampling instants when the system is in the limit cycle. Tt could be used for either free oscillation or forced oscillations (when the forcing function is periodic with the period an integer multiple of the sampling period). Equation (6.29) is a simplified form of (6.26) when the symmetry condition exists. It yields the oscillations with zero doc components* (because the constant value of the limit of the summation in equation (6.18) is ignored).

6.2 Application of the fundamental equation to specific examples 22 In this section we shall apply the theory to three examples representing a pulse-width modulated feedback system, a nonlinear gain amplifier, and a quantized level system. In the first two examples, symmetry condition is assumed, whereas, in the last, asymmetrical oscillation is considered. • Thc d·c component implies the averagc vaillc during onc period or Ihe limil cycle.

(lr Ihe response at the s:lmplinl: ilhwnl'

PERIODIC MODtiS OF OSCI LLA liON IN NONLINEAR DISCRETE SYSTEMS r(t)

+

197

e(t)

FIGURE 6.S

Pulse-width modulated system.

Pulse-ll'idth modulated feedback system In this system as shown in Fig. 6.5, the signal pulse-width modulation is indicated in Fig. 6.6. I - e-&hr E:'h(s) = y(1) s = 9'[e;.I.(t)1 = 9'[y(l)u,,(t)1 (6.30) where el(O)

(and this takes only the values

1'(/) = le,(O)1

+I

and -I)

and h, = oc le,(O)1

=

~

Ir,(O) - c,(O)I,

for oc le,(O)1

::s; T

(6.31)

Substituting for E':h(S) in equation (6.29), we readily obtain for this case

-I

C IS

= -27rj

1.

I'

el'PTKG(p)

1 + ell"T

I'

I

1

y(l

+ i)e-",7'(I-e' "h, .. )

1=0

p

,,(t). ". (t): input and sampled input

10 controller ",. A(t): oulpul uf I:olllroller

Ht ilIIU

f •. f,

1111'"1 111111111111'111 IIrJ'WM '·IIIII'ull,· ...

dp

(6.32)

198

THEORY AND APPLICATION OF THE Z-TRANSFORM

c

T=1 To = 4T= 4 KG(s)

= s(s~J)

= 0.5 rn: +1. +1, -I, -1 oc

FIGURE 6.7

Phase plane analysis of PWM system (M en'(O)

(Jd = 1 ell (0)

-----'-------- -1

FIGURE 6.8 Chlll'llctcrillic!lf IlIlurllllng IIl1lplillcr,

= 4).

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

199

+

We assume for this example that KG(s) == Kls(s b), and the input has a period of four examples, ro, r l , -ro, -r1 • Since the input is symmetric and KG(s) has an integrator, we can assume that the output is also symmetric. So we have

p.

== 2,

yeO)

== y( I) == + I,

,,(2)

== ')'(3) == -

1

(6.33)

Using the equation for ell and noting that

,,(1 + i)

== + I, == -1,

for I for I

== 0, I, 2, ... , p - I - i == p - i, P - i + I, ... , p -

1 (6.34) we obtain for the response at the sampling instants on the limit cycle,

c = _ K{ho + hi _ e- bT [e-1J7'(~/)lIo _ I) + (e blll o. b 2 b(l + e- 2bT )

l)]} (6.35)

C

== _ Kfh l bl

10

2

e-b7'

ho _ b(1

+ e-2bT )

[e-bT(ebhl _

I) _

(ebhO _

I)]}

(6.36) From equation (6.31) we know that (6.37)

These equations can be solved for ho and hI and consequently for cos and ('

...

When b == I, K == I, ex == 0.5, T can be solved numerically to yield

==

I and ro

== r 1 == + I,

eO. and e..

co. == -0.238 CIs == -0.162 (6.38) The limit cycle obtained in the phase plane of Fig. 6.7 verifies these calculated values.

Nonlinear gain amplifier21 ,92 For the nonlinear gain amplifier, a saturating amplifier with linear zone is chosen, as shown in Fig. 6.8. The use of this nonlinearity in a discrete l't'cdback system is shown in Fig. 6.9. The description of the nonlinear gain is the following: (1;.(/)

==

I,

< e (0) -d < e.. (O) < d d

7l

= (lell(O), ... -I, (6.39) (1,,(0) < -d We cXlll11inc the mode M .... 4, where the output of the saturating IllIIpliUer iN the Ne'luence I, k, -I, -k. Fur this cUlle the resrnl1se falls in

200

THEORY AND APPLICATION OF THE z-TRANSFORM

FlGURE 6.9 Saturating gain sampled-data system.

the saturated region at the first sampling instant and in the linear region at the next sampling instant. The same feature is repeated with opposite sign. No input is applied. The plant is KG(s)

=

t

s(s + 1) The sampling period is T = 2, (J = 2, and d = 0.5. . The outputs of the saturating amplifier are assumed as follows:

= 1, e;'(O) = -k,

e~iO)

ei,(O) = k,

Ikl

(6.40)

e;,(O) = -1,

<1

(6.41)

For a saturating amplifier the pulse width h is equal to T, since there is no pulse-width modulation and therefore the pulsed outputs from the hold denoted as ej,h(t), j = 0, 1,2,3 will have the following Laplace transforms: t - e-,T EOh(s) = , , s (6.42) -IT

E2h(S) = .

I -e

s

,

This sequence of oscillations is symmetrical and equation (6.29) can be used for the solution provided T is substituted for h. We ultimately get the following response: Co.

k[ , = - -2b T(eo. + e lO ) + I

t - e- bT I -bT 2bT (eo,e b(l + e- )

+ e ts, )] (6.43)

Substituting the given numerical values. we get

= -0.885 - O.ISOk Ch = 0.1 SO - O,HHSk Co.

«(l.4S) «(I.4ll)

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

201

Since it is assumed that the system is operating in the linear region at the sampling instant corresponding to Cis' we get from equation (6.39),

(Jels = -pc IS = e;.(t) = k

(6.47)

or Clr

=-

k

p

(6.48)

Substituting this into equation (6.46), we obtain

k

= 0.390 < 1

Using this value of k, Co. and

Cis

can be calculated.

Co.

=

< -d

(6.49)

Cis

= -0.195> -d

(6.50)

-0.943

The results show that co, falls in the saturating region producing unity as the output of the saturating amplifier and ('10 falls in the linear region producing k (= 0.390 in this case) as the output. Thus the assumption of the sequence I, k, -I, -k is verified and the existence of such a limit cycle of the mode M = 4 is proved.

Quantized levt'l amplifier In this example, the nonlinear gain amplifier has a characteristic of two tJlHlntized levels as shown in Fig. 6.10. Such a quantization appears in "i~ital control systems where analog-to-digital converters are used. e,,'(O)

Jolt illlt! ',III

('hllllll'lrl'~lIl' IIr 1111111111/1 •• 1 1,'\1'1,11111'11111'1

202

THEORY AND APPLICATION OF THE z-TRANSFORM

i:

c

co -1 KG(a)

=O(.~

1)

T=2

a=3 + 1. O. -1

ell':

FIGUR.E 6.11

Asymmetrical limit cycle of quantized level amplifier system.

The characteristic of this quantization is represented as follows: e~(t)

< e,,(O)

= 1

d

=0.5

-d < e..(O) < d 2

=0

- -d2 < e"(0) <-d2

= -0.5

-d

= -1 wher~

d

< e..(O) < - '2 e,,(O)

< -d

(6.51)

«d = 1 and the linear plant is the same as the previous example. The existence of an asymmetrical oscillation with M = 3 can be ascertained for this case, We let M = 3, and assume that the output of the nonlinear component has a sequence of I, 0, - I: hence ,,(0) = I, ,,(2) = -I. SlIhstitliting

+

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

203

these values in equation (6.26) with the preceding characteristics and letting h" - T, we obtain

co. == -

K[2T

b "3 -

K[T"'3 -

b

==

CIs

= b "3 -

[T

+ e-b7' + e-2b7')

1 - e- b,.

Cli

K

1 - e-II~7'

b(l

(6.52)

]

b(l

+ ,-/}7' + e-267')

b(1

e- b7' - e- 26 7' ] e-b7' e-2b7,)

+

]

(6.53)

+

Letting T == 2, b = I. K = I, and d =

! (0(

(6.54)

== 3), these values yield

co. == -0.482, Clo = -0.083, Ch = 0.565 «(,.55) The outputs of the quantized level amplifier corresponding to ('1,•• ( .... and C2. are + I, 0 and -I. which satisfy the original assumption of Iht' sequence. This limit cycle is shown in Fig. 6.11. In another example of asymmetrical oscillations, if we let d = A (IX ,,~ ;") and assume the sequence + I, 0, -~, o. A. 0, -I as the outpul of Ih,' nonlinear component. then a limit cycle of M = 7 is sustained. us shown in Fig. 6.12. Extension of the method to other types of nonlinearities such liS n'lny sampled-data systems is feasible and could be handled by Ihe lillm" procedure. A discussion of the maximum period and the stability of Iht' limit cycles obtained follows. 6.3 Limitation on the period of limit cycles of relay mode oscillations22 In this section, a derivation of the fundamental equation that yicldli Iht' limits of modes of oscillation is obtained. It is based on the USSUl11plioli of symmetry of the oscillations and is valid for (autollomuus) I'dll)' discrete systems. The equation for the response during the limit l'yd,' c:ln be obtained from equation (6.32) of the previolls sect inn hy replllt'i"", all the hili by T. Thus

(', = __1_ ,. 2'".1'

f.

I'

e,·,,7·KG(p) ,. " y(1 1 + (''"''/' Il II

+ ;k

1117'(1 -

P

,: ,':")" ,

(fl, "Itl)

I

From the liymmctry cunditiun. (';.(0) is positivc fur Ihc lin,t half 111'1'1",1 alld I1c~ativc fur the olhcr hillf pCl'iml. This implies Ihlll Ih,' 1'1'\1'"11'0" (',,(0) is nCAalivc fur Ihe lirsl hull' periml 111111 pm.ilivc ful' Ihe swuml hlill o;illl'l' 1111 illplll is lIpplit·d. Th,'rt'forc )'(/) I I ,.. II. I. 2•...• I' I , - 1', /' , I,.... At

204

THEORY AND APPLICATION OF THE z-TRANSFORM

c

c

KG(a)

= a(,~ 1)

T=2 a=2 e,,': + 1,0, -0.5, 0, +0.5,0, -1

FIGURE 6.12 Asymmetrical limit cycle of quantized level amplifier system.

and

c" C,.

<0 >0

I = 0, I, 2, ... , I" - I I = ~, I" + I, ... , M - 1

(6.57)

By investigating the polarity of C,,_l,. which is the response at the last sampling instant contained in the first half period of the limit cycle, we can obtain the maximum number I""'"X or M IIlIIX • This is done by letting i = I" - I in equation (6.56). I'pTKG() (I -liT) p-l I c _ = - e p - e '" (I + _ 1)e- 11I7' d pi,. 2TTJ. r 1 + ~..MilT ~Y I" P P 1-0

f.

(6.58) Since Co. is negative by the original assumption, the polarity of C,•.. l,. is tested starting from I" = 2, that is, from C II • The response CI , 1 •• may be negative up to a certain I" = 1/' but it may become positive f()f all I" ~ ~' + I. Then this critical 1/ is tuken us /"11111.'

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

205

This general formulation and procedure is applied to the following examples. I

EXAMPLE

The plant transfer is of first order, that is, KG(s) = X/(s + b). The response C,._I •• at the margin of the first half period is obtained by letting i = P, - I in equation (6.34) to give

+I = -I

,,(1 + p,) =

for I = 0 fori 1, 2, .•. , p, - 1

=

(6.59)

and

cp _ 1 Equation (6.58) becomes Co' C1, .

C -I I'

C2 • ••••

_

o· -

-

<0

(6.60)

J.

1 e""TKG(p) J - e-"T(J 21Tj (. I + e""T p

K 1 = _ -

b 1+

-liT e e-pllT

p~1 -'fiT) d

-k e 1=1

+ e- IIT + ... e-lp-2111T _

[I

p

e- lp - lIbT ]

(6.61)

It is evident from equation (6.61) that C,._I.R is positive for all p, ~ 2. thus violating the assumption of equation (6.60). Therefore it can be concluded that the first-order relay sampled-data system can maintain only the relay mode of oscillations of two sampling periods (M = 2 or I' = I) and that longer periods cannot be sustained. This holds for any K. b, and sampling period T. O

EXAMPLE 2 We shall consider a second-order plant with integrator in this example. that is, KG(s) = K/s(s + b). Letting hi+; - T in equation (6.29) yields a set of solutions Co,. Ch • ••• , c,,_1.. for the responses at all sampling instants the limit cycle. C = _ K {p-I-l e-1p-I-lIIIT(l - e-bT )] i. b ,~ 2 b(1 + e-pll ',)

ur

[I _

_ 1'-1

I,'or I'

= I and i = P, -

K[T

('II.

= - h 2" _ _ 0 1\

2/,-

[I _

e-lp-l-mTo -

,J-i 2

1

b(1

= O.

I - rb'r ] h(l + eO h7')

,.".n I +

h7') -

I'

I

+ I'

02( 1 -:'. I! ,,',.

e- IIT )]}

+ e-pl'T)

0"7')1

(6.62)

206

THEORY AND APPLICATION OF THE z-TRANSFORM

It can be proved that the numerator inside the bracket is positive for all bT > O. Hence it may be concluded that COs is negative for all T. for I-' = 2 and i = I-' - 1 = I, Cls

= -

K (I _ e- bT )2 b2 1 + e- 2bT

< 0,

(6.64)

for all bT> 0

Therefore Cb is negative for any sampling period T. for /l = 3 and i = /l - 1 = 2

= K

C 28

[I _

(1 - e- bT ) (t b(1 e- 3bT )

b 2

+

+ e- bT _

e-2bT)]

(6.65)

Simple trial shows that c2• is negative for small T and becomes positive for larger T, thus violating the original assumption. We can find the critical Tc by solving equation (6.65) for T when C28 is equated to zero. Therefore the oscillations of /l = 3 or M = 6 can exist for T < To, but it cannot be sustained for T ~ Te' A similar observation is obtained for larger /l. To derive the fundamental equation that relates the maximum /l and the sampling period T for any second-order system, we let i = /l -1 in equation (6.62). C =_K e-(p-UbT(I - e- bT ) p-l .• b 2 b(1 + e-PbT )

{! _

_"-l[I _ .

~1

e-(p-l-l)bT(l - e-bT)J}

2

b(l

= K[bT( _ 2) _ b2

2 '"

+ e- PbT ) ebT

1 - e- IlbT 1 + e- IlbT

+ ebT -

IJ

(6.66)

By the original assumption, e,l-l. s < O. Hence /In,,,x is obtained by finding the maximum integer /l which satisfies the following inequality: bT t -pbT - ( - 2) - e- bT - e + e"T - 1 < 0 (6.67) 2 /l 1 + e- IlbT To conclude, we may state that limit cycles of two and four sampling periods can exist in the relay system for any sampling period T, but that for '" ~ 3 the maximum number of sampling periods or the mode of oscillations is restricted by the system parameters band T. figure 6.13 indicates /lmax as a function of bT. These discussions can be applied to any order plant and to other types of nonlinearities. 20 - 22 An important application of the maximum number of modes of limit cycles is in the stability study of nonlinear discrete systems. It yields the boundary stability curves for various nonlinear discrete systems.

PERIODIC MODES OF OSCILLAnON IN NONLINEAR DISCRETE SYSTEMS

10 9 8 7 6 5

KG(s)

=

af.!61

207

Relay mode oscillation

4

3 2 1 O~__~~~~~~~~~____~~__~~~~~~

0.1

4

FIGURE 6.13 limit cycle.

6

8 10

bT

Maximum number of sampling periods contained in half period of

6.4 Stability study of limit eyeles S,so,II7 In the preceding section a method was presented for obtaining the periodic oscillations (or limit cycles) of a system without information as to their stability. We shall now discuss the stability by first defining a stable limit cycle. A limit cycle is considered to be asymptotically stable if any small perturbation around the limit cycle approaches 0 as n approaches 00. Dy following this definition, the following two theorems can be applied for testing the stability. TnEOREM

1: Assume that the nonlinear difference equation in vector

form

z ..+11 = F(z..)1 (6.68) has a periodic solution s..1of period M, and that the functions F(z..>are single valued and possess continuous first partial derivatives. The first approximation of the difference equation for small perturbations about the periodic solution s.. ] is Y.. I1] = [A .. ] Y.. ] (6.69) lII,d the solution of this equation determines the stability of the periodic solutiol1 ''',,) if [A II] ill nonsingular at all the solution points. In this equation r,,1 is the pcrturhatiol1 ;lhout ..... ] and the components of [An] are

"".11 -

iW,(;l' .. )

I

o.r, ... 1 •• 1

(6.70)

208

THEORY AND APPLICATION OF THE %-TRANSFORM

2: The system of equation (6.69) is asymptotically stable if all the eigenvalues of the matrix

THEOREM

.11-1 [Ao] =

n [Ata] =

(6.71)

[A,1I-1][A.v - a] ... [Ao]

lie inside the unit circle. Then all the solutions tend to 0 as n becomes large. If one of the values lies outside the unit circle, the solution is unstable. Finally, if all the values lie on the unit circle, the linearized solution Y ta ] of equation (6.69) can be considered as stable. * Combining the two theorems, the condition for the asymptotic stability of limit cycles is reduced to the following statement. All the eigenvalues of the matrix [Aa] which is the product of all the matrices which consist of the first partial derivatives of F(xn) lie inside the unit circle at all the periodic solution points sn]. The application of this theorem is shown in the following discussion.

6.S Forced oscillations in nonlinear discrete systems4.23.27 In this section, a particular example of the nonlinear sampled-data system discussed in the preceding chapter is analyzed for a sinusoidal forcing function. Since only harmonic oscillations (oscillations having the same frequency as input frequency) are considered, it is necessary to put suitable restrictions on the parameters of the system in order for such oscillations to exist. Furthermore, the ratio of the sampling frequency to signal frequency is assumed to be an integer. EXAMPLEu

Consider the sampled-data system shown in Fig. 6.14, where G1(s) = K/s(s + I) and r(/) is a sinusoidal input. The nonlinear characteristic is described by (6.72)

where c' is the output of the nonlinearity and

~(z) = <[I -

e- T • 01(5)J

s

=

K(0.368z

+ 0.264) , + 0.368)

(z' - 1.368z

T= 1

(6.73)

• It should be noted that even if the linearized solution of the perturbation is .fluhle-, the limit cycle is not asymplolirQlly IIIQbl, since the perturbation will not decay in thill

case.

209

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

T..-

r ........ ~ r(!)

+

I

---<)

~(z)

crt)

G(8)

Nonlinearity F[G URE 6. [4

A nonlinear discrete system with nonlinearity in feedback path.

The difference equation relating output to input is easily derived as follows: e(n + 2) - 1.368c(n + J) + 0.368e(n)

+ I) + 0.264e(n)] = K{0.368[r(n + I) - e'(n + I)] + 0.264[r(n) -

= K[0.368e(n

e'(n)]}

(6.74)

Choosing rn3 = -0.1 and substituting for c' in terms of e, the following difference equation is obtained: ('(n

+ 2) + ('(n + 1)[0.368K - 1.368] + e(n)[O.264K + 0.368] - K[0.0368e(n + 1)3 + 0.0264e(n)3] = O.368Kr(n +

I)

+ 0.264Kr(n)

(6.75)

If the peak value of r(n) is not chosen very large, it can be assumed that the harmonic oscillations will be the dominant term in the periodic response. Let r(n) = B sin (17/3)n. For the periodic steady-state output the following form can be assumed: c(n)

= pen) sin:! II + q(n) cos:! 11 3

3

(6.76)

where pen) and q(n) are slowly varying discrete functions of n. We substitute for r(n) and c(n) in equation (6.75) and equate coefficients of sin (17/3)1/ and cos (17/3)n, noting the fact that sin 3 () = 1(3 sin () - sin 3() ;tnd l'OSa (J = .\(3 cos 0 + cos 30). Since pen) and q(n) are slowly varying wit h II, it can he assumed that .

+ 2) = 21'(n + I) - pen) (that is, !J.2p(n) = 0) knlls invlllviuA "'3' 1'(1/ + I) could he replaced by pen). p("

(6.77)

These and in "'lIIplitkatilllls arc It~J~itillwtc so lunA as t."onsidcration is restricted to the 1lIlI'IlIOllil~ IIM·illatillns. With these as.,lIl11ptillns. the comparison of

210

THEORY AND APPLICATION OF THE Z-TRANSfORM

coefficients of sin ('TT13)n and cos ('TT13)n in equation (6.75) yields the following two relations:

+ I )(0.184K - 1.684) + q(n + 1)( -0.547 - 0.319K) = p(n)( -0.868 - 0.264K + 0.0336A2K) + q(n)( -0.866 - 0.0239KA2) + 0.448B. (6.78) pen + 1)(0.547 + 0.319K) + q(n + I )(0. I 84K - 1.684) = p(n)(0.866 + 0.0239KA2) + q(n)( -0.868 - 0.264K + 0.0336A2K) + 0.3185B (6.79) A2 = p2(n) + q2(n). These are the key equations in the study of pen

where periodic solutions and their stability. After the transients have disappeared, the solution will tend toward one of the periodic states given by the substitution of pen I) = pen) = Ps and q(n + I) = q(n) = q.• in equations (6.78) and (6.79). Define 0.184K - 1.684 -0.547 - 0.319KJ [ (6.80) =C 0.547 + 0.319K 0.184K - 1.684 and let

+

[pen)] = yen)

(6.81)

q(n)

Equations (6.78) and (6.79) are then rewritten in a simplified notation as Cy(n

+ I} =

(6.82)

H[y(n)]

where H[y(n)] is the nonlinear term appearing on the right-hand sides of equations (6.78) and (6.79). Equation (6.82) is further simplified as yen

+ I) = C-IH[y(n)] = F[y(n)]

(6.83)

The equilibrium points of this equation are given by

; = F(;)

where;

=

[;:J

(6.84 )

which reduces to the following two equations:. P.,( -0.816

+ 0.448K -

p,(0.319K':"- 0.319 -

O.0336KA2)

+ q,(0.319 -

0.319K

+ 0.0239KA2) = 0.448KB (6.85) 0.0239KA2) + l/.(-O.KI6 + 0.44KA· - ().O:n6KA2)

,..~

O.JIII5KII (6.IIlI)

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

211

0.20 .---r-r-..--r--..----,r--....,...---,r--.....---,r---r---.

FIGURE 6.15 Amplitude characteristic curve ror the harmonic oscillation.

These equations when solved for p,. A2 yield the amplitude characteristic A2[(-0.816

q. and

using the fact that

p! + q! =

+ 0.448K - 0.0336KA2)2 + (0.3 19K - 0.319 - 0.0239KA2)2] = 0.448K2A2 + 0.3 I 85K2B2 (6.87)

For a given value of K the functional relation A2 versus B2 can be plotted. It will be seen that for some values of K the system has more than one periodic state for a certain range of values of B2. Figure 6.) 5 shows B2 plotted as a function of A2 with K = 2.395. Computation shows that for values of K greater than 2.25 multivalued regions are obtained. . Let us choose K = 2.395. This will make the poles of the linear part IIf the difference equation just lie on the unit circle. The characteristic "'Illation for the linear portion is Z2 -

0.488z

+ 1= 0

which ~ives

z=

1)('

1;0,

where p = 1 and

=

() = 217

5

(6.88)

th~ period as 271'/() 5 secnnds. The period of the sinusoidal is (, s""ulld!o alld since the dillerencc between the natural frequency IIlId "11~1I111 rn·'IIIt·n,·y h. !;mall, II Inckin~ phcnnmcnnn or 1I frc'lucncy "111 rni II II Il'II I IlIk,·., plm',' 1\ plW1l1l1UC1I1I1I !;illlilar 10 lhlll in cnlllillllnlls

rhi .. I!,iws

11111111

212

THEORY AND APPLICATION OF THE %-TRANSFORM

!o'ystems. Calculation of the response using the difference equation as a n'currence relation indicates a response of period 6 as expected. The nmplitude characteristic has a region, for values of A2 between 1..25 and 4, where dB2/dA2 < O. From the theory for nonlinear continuous 'y~tl'IlIS, in particular, works of Hayashi,38 and more recently, that of KIlIttcr and Pinney,2. we would expect the periodic solution in this region III h,: unstable. That this is true is proved by considering the variational "'Illalilln around the periodic state. 1,'1 the components of the vector F[y(n)] in equation (6.83) be denoted !ly

I/~

I:!

I

If the eigenvalues of the matrix

[~ ;~l] of2

(6.89)

of2

op oq

If("1

~~

III' III\j,k the unit circle, then the equilibrium point is stable; otherwise II 1\ IIn,lahlc. For the region in Fig. 6.15 where dB2/dA2 < 0, the equilib1111111 point is ullstable and if the system is slightly disturbed at this point, II Will IIltimalely ~ettle down to one of the other two stable periodic ·... 111111111',. III Ihc numerical example discussed in this section, and for "V. 111.' 'i from Fig. 6.15, three periodic states with A2 = 1.495, 3.25, 111111 oI.M art' IIhtaincd. From equations (6,85) and (6.86), p" q, are comI'"h'" 1111 "u\'h III' these values and then the characteristic equation for the \'111111111111111 "'lll:Iliull at each point is computed. Stability of this characterI~III" "'11111111111 is thcll examined. A summary of the results are given. ". /I

..

2.395 O.J54

I i(IIIII", illlll I·"i"'.~

/', /', 1'•.

I.UI.! 'I. -, -0.69 LUI ~ 1/•.. -1.5 IUI'IO'i

1/.. : -2.2

Chara(teristic Equation for Variational Equation ).2 _ ).2 ).'1. _

1.793). 1.738). 1.688),

+ 0.809 + 0.73 + 0.706

Stable or Unstable Stable Unstable Stable

I hl'~I' ,·,'M.lls Wl~rc arrived at through lahorious numerical computation. '.1"'11 that t h(: milts (If the characteristic equation lic very ncar the 111111 1"111·1,', IICIIl't: it is til he cxpccted that the periodic solution will setth: .1"wlI III .1\ !lIt'ally-stOlte vOllllc aftcr II vcry ImlA lime. Thc c\ulIcness uf the 1II11II(',"'al valill's i~ lllll'ihlltCtl tn thl' sllIall slope tlrtht~ amplitude churactcr."1",, 1;111' "XlIlIIl'lc, fill" Ilw inl'"t mllfl,llitllde of II ..,. U.294, which IlIlI; Ullly II I',

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

213

one stable periodic solution. the characteristic equation is A2 - I.816A 0.86 which is clearly stable.

+

6.6 Direct z-transform for determining true oscillation 23 In addition to what has been discussed in the preceding section. other methods exist for determining oscillation in nonlinear discrete systems. These methods constitute both graphical procedures such as the phase plane (space) and the describing function. and analytical procedures such as the use of orthogonal discrete frequency functions or Fourier polynomials. the state-vector approach. and the direct z-transform. In the following. we shall discuss only the latter method, which is closely related to the contents of this book; however. the other techniques are important and have been extensively elaborated in the literature. , In the direct z-transform method, the type of nonlinearity assumed is either a relay or a saturating gain amplifier with straight-line characteristics, as shown in Fig. 6.8. The method is based on specifying a certain mode of oscillation for zero input and obtaining the conditions that sustain these oscillations. This technique will now be explained and later applied to a specific example. Let the output of the nonlinearity be assumed as in Fig. 6.16, where the period of oscillation, for sake of illustration, is assumed to be four sampling periods, the sequence being mo, mil m2 , m3 • In the steady state the z-transform of this repetitive sequence can be defined as mo

1-

Z-4

+

m1z- 1

1-

Z-4

+

m~-2 1 - Z-4

+

m az- 3 1 _ Z-4

The output of the system with a repetitive sequence similarly a steady-state z-transform of

(6.90) Co. Cl l C2' C3

has

(6.91)

"I( iI 11(1, f, If,

1111'111 11111 pili 'It-~clil)liun u" Iim'lIr 1'111111 IInder limit cycle npcrntinn.

214 THEORY AND APPLICATION OF THE z-TRANSFORM

FIGURE 6.17

Mathematical equivalent of system in Fig. 6.18.

Next take the z-transform of the difference equation describing the linear plant and substitute the z-transform expression (6.90) and (6.91) for the input and output, respectively. Comparison of coefficients of like powers of z will yield four linear relations involving Co, Cl. C2. C3. It is found that not all these relations are linearly independent. For the limit cycle with period four. only three relations are linearly independent. The fourth relation appears as a constraint which should be satisfied so that the specified output of the nonlinearity is maintained. The solution of these equations will yield bounds on the initial values of the output which are the conditions that are necessary (but not sufficient) to sustain the specified mode of oscillation. The following example will illustrate the procedure. EXAMPLE

Let Fig. 6.17 be the mathematical equivalent of the sampled-data system shown in Fig. 6.18. Let Gl(s) be

+ 0.695 + 0.695)

Gl(s) = 0.3055 s(s

(6.92)

and the nonlinearity be an ideal relay as shown in Fig. 6.19. The ztransform of Gl(s) is

~l(Z) =


se- TS Gl(S)]

where T= 1

0.5z (z - 1)(z - 0.5)

=------

(6.93)

Hold

circuit

~~----~-----------

FIOURI'tl.IM A !lUtllilll"lI!' dl\l·l'l"ll'

eft)

~y!lh·1I1.

215

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

Thus the difference equation for the linear portion can at once be written as

c(n

+ 2) -

) .Sc(n

+ 1) + O.Se(n) =

O.5e'(n

+

1)

(6.94)

where n takes on integer values 0, I, 2, .... It is required to find initial conditions corresponding to periodic solutions of period two. The z-transform of equation (6.94) in the steady state is

I.Sz

(z2 -

+ O.S)C/f(z) -

Z2C(0)

- zc(l}

+ I.Szc(O)

= 0.5z4"(z) - O,5ze'(0) (6,95) In the steady state, let the relay output be -I, + I, and output of the system be Co, c 1 • Then the z-transform of relay output in the steady state is given by

+

-1

1-

Z-2

Z-1 Z-lI

1-

-z

=Z+I=

+1

I'() (6.96) %

The z-transform of the output C(/) is

(co

+ C1Z-1)/(1

- Z-2) = tjf(z),

-1

FIGURE 6.19 Ideal relay. We substitute these expressions in equation (6.95), clear the fractions, and equate coefficients of like powers of z. Coefficients of Z3 and Z2 drop out and the coefficient of z and the constant term yield identical relation

(6.97) If the relay output is to be maintai ned -I, + I , it is necessary that - Co < 0 nnd -C 1 > 0; that is, co> 0, C1 < O. Combining this with the relation (6.97) between Co and c" it is immediately seen that - ~ < C1 < 0 and O· < Co < l, which agrees with the results obtained using other methods,

6.7 Periodic solution of certain nonlinear difference equations 27 ,37 111 the preceding sections, the z-transform method has been applied to nhtain the pcriodic oscillations of nonlinear discrete systems. The systems d iscusscd lIrc those lIsually encountered in control engineering applications. Thc mathclllatical description ()f these systems represents the field of 1I11111illl'ar dif1i.on~lll·c t:tluatinlls. Hcnce the ::-transform as shown in this dmpler call he readily lIsed to ohtilin the pcriodic solution of certain nnnlilU'l\r tlilll'n'lIt't' t"IIIOIlillll!l. The Iype nf dil1'crence Ctl"ution that cun

216

THEORY AND APPLICATION OF THE Z-TRANSFORM

be solved by this method is ,

'I,ptc(n

t-O

+ k) -

I,

'I,«t!(n

t-O

+ k),

with 11 ~ I and

n - 0, 1,2, . ..

(6.98)

In equation (6.98), the nonlinear term/en) represents

fen) - 4>[c(n)]

(6.99)

where 4> is a nonlinear single-valued symmetrical function and «t' PIt are constants. In applying the z-transform method, we can establish a set of nonlinear algebraic equations which give the general solution for any period. In particular, solution for the periodic oscillation c.(n) can be determined. If we restrict the symmetrical solution to that for even periods only, we can obtain the exact solution for periodic modes with half-periods less than or equal to two. For higher harmonic oscillations approximate solutions could be obtained. A detailed discussion of the solution of equation (6.98) is beyond the scope of this text; however, the method that can be used is similar to that applied in the preceding sections. Furthermore, the stability study of the periodic solution of equation (6.98) can be readily studied as discussed in Section 6.4. In concluding this chapter, it should be indicated that the z-transform method is also applicable for obtaining the periodic modes of oscillation in continuous nonlinear feedback systems. As mentioned earlier, other methods exist for detennining the periodic modes of oscillation. The purpose of this chapter is not to discuss and compare all these methods but to propose in detail the application of one particular method, the z-transform for obtaining a solution to these problems. REFERENCES

1. Kalman, R. E., "Nonlinear Aspects of Sampled-Data Control Systems," Pro~eetJ­ ings of the Symposium on Non-linear Circuit Analy.fis, Polytechnic Institute of Brooklyn, Brooklyn, New York, April 1956. 2. Nease, R. F., "Analysis and Design of Nonlinear Sampled-Data Control Systems," WADD Technical Note 57-162, Servomechanism Laboratory, M.I.T., June 1957. 3. Mullin, F. J., "The Stability and Compensation of Saturation Sampled-Data Systems," AlEE TraM., Vol. 78, Part 1, July 1959, pp. 270-278. 4. Mullin, F. J., and E. I. Jury, "A Phase Plane Approach to Relay Sampled-Data Feedback Systems," AlEE TraM., Vol. 78, Part 2,1959, pp. 517-524. S. lzawa, K., and L. E. Weaver, "Relay-Type Feedback Control Systems with Dead 49-54, and discus.~i()n by Time and Sampling," AlEE TraM., Vol. 78, Part 2,1959, K. lzawa in reference (4). 6. Cypkin, J., Thrnry of Pulsr S,I'slt'm., (in RUlIslnn), Slnte "re!!s fllr "hysic!l and Mathe· matical Literature, Moscow, lISSR, 19!'iK.

rr.

PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS

217

7. Nelson, W. L., "Pulse Widlh Control of Sampled-Data Systems," Technical Report T-35/B, Columbia University, July 1959. 8. Kadota, T. T., "Asymptotic Stability of Some Nonlinear Feedback Systems," ERL Report Series No. 60, Issue No. 264, University of California, Berkeley, January 1960. 9. Pyshkin, I. V., Self-Oscillation in Pulse-Width-Modulated Systems, Theory and Application of Discrete Automatic System~ (in Ru~~ian), Academy of Science, Moscow, USSR, 1960, pp. 134-1 SO. JO. Andeen, "Analysis of Pulse Duration Sampled-Data Systems with Linear Elements," IRE Tran.t. Automatic Control, Vol. AC-5, No.4, September 1960, pp. 306-313. 11. Andeen, R., "Analysis of Sampled-Data Systems with Nonlinear Elements," Automatic and Remote Control, Proceedings First Int. Con.r:ress IFAC, Butlerworths, London. 1961, pp. 314-320. 12. Torng, H. C., "A Study of Nonlinear Sampled-Data Systems Containing a Relay, a Saturating, or a Backlash Element," Ph.D Thesis, Cornell University, February 1960. 13. Torng, H. C., and W. C. Meserve, "Determination of Periodic Modes in Relay Servomechanisms Employing Sampled-Data," IRE Transacliems 011 AutClmatic Cmllml, Vol. AC-5. No.4, Septemhcr 1960. 14. Meserve, W. E., and H. C. Torng. "Investigation of Periodic Modes of SampledData Control Systems Containing a Satul'ating Element," TrailS. ASME. Series 0, J. Basic En./f., Vol. 83, 1961, pp. 77-81. IS. Jury, E. I., and T. Nishimura, "Analysis of Finite Pulsed Systems with PeriodicallyVarying Sampling Rate and Pulse Width," AlEE, CP 60-866, June 1960. 16. Nishimura, T., "Operational Analysis of Finite Pulsed Sampled-Data Systems," ERL Report Series No. 60, Issue No. 279, University of California, Berkeley, May 1960. 17. Jury, E. I., Sampled-Data Cemtrol Systems, John Wiley and Sons, New York, 1958, Chapters I and 9. 18. Jury, E. I., "A Note on the Steady-State Response of Linear Time-Invariant Systems to Genercll Periodic Inputs," IRE Proc. Vol. 48. No.5, May 1960. 19. Jury, E. I., "Contribution to the Modified :-Transform Theory," J. Franklin Inst., Vol. 270, No.2, August 1960. pp. 114-129. 20. Jury. E. I. and T. Nishimura, "On the Periodic Modes of Oscillations in PulseWidth-Modulated Feedback Systems," ERL Report No. 60, I~~ue 328, University of Culifornia, Berkeley, November 29, 1960; also, Trans. ASME, Series E, J. Belsic E'\I{., March 1962, pp. 71-84. 21. Nishimura. T. "Stability Analysis of Pulse·Width-Modulated Feedback Systems," LRI. Report Serics No. 60, Issue 353, University of California, Berkeley, April 4, 1%1. 22. Jury, I're~el1lc:d

r.

I., and T. Nishimura, "Stability Study of PWM Feedback Systems," Ht the JACC mceting in Minnesota, June 1963. To appear in the 1'raIlJ· •

.-ISMI:, 1%4 . •~ I. I'Hi. M. A., "Osc:iIlHtinns in Nllnlincilr Sampled-Datil Systems," AlEE T,.all.t., V,.I. HI. l'illI IIIAl'pli~'illinn' and Industry), JallIlilry 19(.3, pp. 350-355 .

.'.\, ('hnl\'. ('. K., "( 'nnla~'I'" S~'I VII"'t'chnni'l1I I "'pinyin!! SlIIlll,lc:d-[);ItIl," A lEE 1111'" , VIII. II, 1'111 I II, 1''',1, PI', 'I M.

218

THEORY AND APPLICATION OF THE %-TRANSFORM

25. Klouer, K., and E. Pinney, "A Comprehensive Stability Criterion for Forced Vibrations in Nonlinear Systems." J. of Appl. M~ch .• Trans. ASME. Vol. 75. 1953. pp. 9-12. 26. Oa-Chuan, Shao, "On the Possibility of Certain Types of Oscillations in SampledData Control Systems," Automation and Remot~ Control, Vol. 10. June 1959. pp. 85-89. 27. Tsypkin. Y. Z.• "Periodic Solutions of Nonlinear Finite Difference Equations and their Stability," International Union of Theoretical and Applied Mechanics, Institute of Mathematics. Academy of Science of the Ukranian SSR. Kiev, 1961. 28. Delfeld, F. R., and Q. J. Murphy. "Analysis of Pulse-Width-Modulated Control Systems," IRE Trans. A.C., Vol. AC-6, No.3. September 1961. pp. 283-292. 29. Tsypkin. Y. Z.• "Elements of Theory of Numerical Automatic Systems." Automatic alld Remote Control. Proceedings First Inl. Congress IFAC, Butterworths. London, 1961, pp. 286-294. 30. Jury. E. I.. "Contribution to the Modified z-Transform Theory." J. Franklin Inst .• Vol. 270. No.2. August 1960, pp. 114-129. 31. Monroe, J., D~f(ital Processe.f for Sampled-Dala Systems. John Wiley and Sons. New York, 1962. 32. Torng, H. C., "Complete and Exact Identification of Self-Sustained Oscillations in Relay Sampled-Data Control Systems," AlEE Trans. Vol. 81, part II, 1962. pp. 355-36\. 33. Jury, E. I.. Discussion of Reference (32). 34. Quichet, P. L., "Osci11ations Periodiques dans les Systemes Echantiltonnes NonLineaires," AUlomatisme, No.6, June 1963, pp. 229-239. 35. Kuo, B. C., Analysis and Synthesis of Sall/pled-Data Control Systems. PrenticeHall, Englewood Cliffs, New Jersey, 1963. 36. Hayashi, C., Forced Oscillations in Nonlinear Systems, Nippon Printing and Publishing Co., Japan, 1953. 37. Pasteur, Franceline, "Self-Sustained Oscillations in Nonlinear Sampled-Data Systems," Master of Science Project, Dept. of Electrical Engineering, University of California, Berkeley, June 1963. 38. Kaplan, W., "Stability Theory." Proceedings of th~ SympOSium on Non-linear Circllit AnalYSis, Polytechnic Institute of Brooklyn. Brooklyn, New York, Vol. 4, April 1956. 39. Kalman, R. E., and J. E. Bertram, "Control System Analysis and Design Via the Second Method of Lyapunov," Parts I and II, Trans. ASME(J. Baliic Eng.). Vol. 82, June 1960, pp. 371-400. 40. Tsypkin, Y. Z., "Investigation of Stability of Periodic States in Nonlinear Pulse Automatic Systems," AUlomatic and Remote Control, Academic Press, New York. Vol. 22. No.6, June 1961, pp. 614-623.

7 z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

The z-transform method discussed in the preceding chapters was applied for the most part to the exact solution of difference equations or to discrete systems which are describable by difference or difference-differential equations. One of the basic advantages of the z-transform method is that the time response can be obtained by power series or long division method. This procedure circumvents the evaluation of the poles of the system transfer function or the characteristic roots of the difference equation. This advantage could and has been utilized for the approximate solution of differential equations or continuous systems. Furthermore, the %transform could also be used for time domain synthesis of networks to yield certain responses, or for identifying the linear plant transfer function. The material of this chapter deals mainly with the various approximation techniques which utilize the z-transform theory.

7.1 Approximation methods Approximating a differential equation by a difference equation is not new. Mathematicians have been in the past and are at present engaged in doing extensive work toward this approximation, which is called numerical analysis. With the advent of digital computers, approximation is becoming increasingly important for programming the solution of differential equutions. In the following discussion this subject is only briefly indicated hy the irnportunt techniques for approximating a continuous system by a discrete mudel. Tn illustrate nne aspect nf the approximation procedure, assume that thl' l'nntilillUII!o. !o'y''''1II "hown in Fi~. 7.1 ili represented by a continuous 219

220

THEORY AND APPLICATION OF THE :t-YRANSFORM

input and a transfer function G(s}, and that the output is obtained at all times. An approximate system using sampled-data techniquesl - 3 is presented in Fig. 7.2. By using the z-transform or the modified z-transform, the approximate output of the system of Fig. 7.1 can be obtained. It is noticed from Fig. 7.2 that an interpolator x(t) y(t) (or a mathematical hold) is used. The pur0x(s) , G(s) )0 0 . . y(s) pose of the interpolator is to reconstruct as closely as possible the continuous function FIGURE 7.1 Continuous syswhich has been fictitiously sampled. lern. Shown in Fig. 7.3a to e are the various forms of the interpolator function commonly used. The approximation x,,(t) is the sum of a number of impulse responses, that is,

I

I

xG(t) =

!

at.

(7.1)

x"q(t - nT)

.. =0

where q(t) = .2'-l[Q(S)]. For the cases a, b, d, and e, equation (7.1) applies, whereas for case c, in which the values of x(/} at midpoints of the interval are used, the following is used: CD

xG(t)

= ..!x..+lq(t -IT- nT)

(7.la)

~O

If we impose the condition xG(nT) = x(nT) [or N'G(z) = N'(z)] in a, d, and e, by applying the z-transformation to equation (7.1) we obtain (7.lb)

1[q(t)J = J

The same relationship also holds for case c with the imposed condition + T12) = x(nT + TI2). These conditions are not imposed on b where the pulse heights are the average of the ordinates x«() on either side of the pulse. If we put . x,,(nT

.2'[q(t)] = Q(s) = Fk+l(Z-l)

(7.le)

Sk+l

x(t)

o

X

x· (tJ

Interpolator Q(s)

I

I

Sa (t)

G(s)

FIGURE 7.2 A slIl1l(lled-dlll8 1I1'I'roxilllllliul1

I

"r l'ig. 7.1.

YaW

0

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

1.0 q(l)

(a)

q(t)

-T

0

T

2T

-T (e)

(b)

(d)

1:I(jtJRE 7.3

Approximalion of a lime funclion.

221

222 THEORY AND APPUCATION OF THE z-TRANSFORM

Fk+l(Z-l) (a certain function of (Z-l» can be determined by applying the z-transformation to equation (7. Ie) and using the results of7.lb to obtain (7.ld) and (7.1 e)

The interpolator transfer functions Q(s) which result from equation (7.le) will be termed normal interpolators of order 0, I, 2, by putting k = 0, I, 2. For instance, the zero-order normal interpolator transfer function can be readily obtained by letting k = 0 in equation (7.le) to obtain

(7. I/) Similarly, higher-order normal interpolator transfer functions can be readily obtained. To summarize the various interpolation schemes shown in Fig. 7.3, we use Table 7.1, introduced by Cruickshank,' to indicate the transfer function of each interpolator. TABLE 7.1

CLASSIFICATION OF INTERPOLATIOW

Interpolator

Tran.ifer Function

Zero-order normal (a)

Qo(s)

=

First-order normal (d)

Ql(S)

=

Second-order normal (e)

Q.t<s)

= 1'2(1 + z-l)r

(J - z-l)

s 21(1 -

21-1)2

TS2 2%(1 -

Zero-order midinterval (c)

Q(s)

=

Zero-order mean (b)

Q(s)

=

z!-{(l -

Triangular Parabolic

21-1)

z(1 - %-2)

2$

Rectangular

%-1)3

s

~--

Name of Approximation

Midinterval rectangular Mean rectangular

......--..

It should be noted that the second-ordcr norntlll intcrpoilltiol\ C:I!iC (. gives exact representlltion of plImhnllls, whcreas case tI gives cxact filling

223

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUF..5

of straight lines. The mean rectangular and midinterval rectangular approximations give a relatively coarse fit to any curve except stepped functions. Having obtained the transfer functions of the interpolators, we can write the approximate output z-transform from Fig. 7.2 as follows: f{!f ,,(z)

(7.1 g)

= !l'(z)Z[ QG]

For integrating operators of any order n, we can readily obtain from tables of z-transforms the transform Z[Qr"]. These transforms are given in the second column of Table 7.2. By using the division method, the approximate output sequence yo(nT) can be readily obtained. The normal interpolators may be combined to represent accurately functions with given initial conditions, which is explained in the next section.

7.2 Initial conditions nonzero· The problem of initial conditions can be incorporated by combining the normal operators. Thus we can have linear or quadratic interpolation, depending on the number of terms used in the Taylor's series expansion of x(t) of Fig. 7.1 around the origin. This is illustrated as follows. Let the Taylor's series expansion of x(t) be presented x(t) = x(O)

+ IX'(O) + -t

2

2!

x"(O)

+ ... +

xm(t)

(7.2)

where xm(t) is the remainder after m terms. Taking the z-transform of this equation, we obtain

~(z) =

x(O)

1-

Z-1

+ x'(O)Tz(1 _

+

1

Z-I)2

x"(O)T2z- 1(l + Z-l) 2(1 _ Z-I)3

+ ... + ~

(z) (7.2a) m

We can replace '?V,,(z) from Fig. 7.2 by the following approximation:

(!VaCz) =

X(O)_l Z[QoG] 1- z

+

x'(O)Tz- 1 "[Q G] (1 _

Z-I)2 '"

+ ~"(~)T2Z

\(1,

1

~I Z-'l) Z[Q2G] + ... + ~m(z)Z[Q",G]

2( I - z )'

(7.3)

N

~

;!

TABLE 7.2 INTEGRATJNG OPERATORS

Function G(s)

Zero-Order Normal Interpolation TZ-l

s

J -

i

TZZ-1(1

~ 1

:;a

%-1

+ %-1)

2(1 - %-1)2 T3z-1(l + 4z-1 + z-2) 6(1 - z 1)3

T'z-l{1

:;i

+ Ilz-l + J Iz-2 + z-3)

~

~

Linear Interpolation T(1

+ Z-1)

'2 (l

-

C

Tx(O)

%-1) -

2(1 -

1'2(1 + 4z-1 + Z-2) 6 (l - %-1)2

T'(I

> "II "II

%-1)

!:

+ llz-1 + Ilz-2 + z-3)

T'(I

T'(I

z

+ 8z-1 + 3z-2~0)

24(1 - z-I)3

24(1 - Z-1)4

~

1'2(1 + 2z-1)x(O) 6(1 - %-1)2

24(1 - Z-1)3

+ 26%-1 + 66z-2 + 26z-3 + Z-4) 120(1 - z-I)' T'(l

+ 22%-1 + 33z-2 + 4z-3~O) 120(1 - z-I)'

1

~

T5Z-1(l

+ 26%-1 + 66z-2 + 26z-3 + z-') 120(1 - z-I)5

T5(l

+ 57r-l + 302%-2 + 302z-3 + 57z-' + z-5) 720(1 _ ;-1)5 T5(1

+ 52z-1 + 198z-2 + 104;-3 + Sz-')x(O) 720(1 -

%-1)5

o ." ..oj

:c III

~

> z

(I>

~3:

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

225

Using only the first three terms on the right side of equations (7.2a) and (7.3) and neglecting the remainder terms, we obtain the following relationship:

As an illustration. if we let G(s) 7.1 we obtain

= I/s, then from (7.4) and from Table

':"1 (z) = .'¥·(z) T(t + 4z- 1 + Z-2) .. 3(1 + z-I)(I - %-1)

+

x(O) [~_ T(I

I-

Z-I

1-

Z-I

3(1

+ 4z-' + z '2)J + Z ')(1 - Z-I)

+ x'(O)Tz- 1 {T(I + Z-I) _ TO + 4%·-1 + Z"2)} (I - Z-I)2 2(1 - Z··I) 3(1 + z-I)(I _ Z··I) (7.5) Combining thesc expressions. we finally obtain the quadratic operator of lIs with initial conditions as follows: !/f

(z) = .(·(z) T( I

3(1

1/

+ 4z- 1 + Z-2)

+ z.. I)(1

+

_ Z-I)

x(O)T(l 2z- 1) 3(1 z-I)(1 - z .1)

+

+

x'(O)T2z- 1 (7.6) 6(1 - Z-I)(1 Z-I)

+

which is seen to be identical to the entry in Table 7.3 for I/s. . Similarly. we can obtain the linear or higher-order interpolation as the situation rcquires. Table 7.2 gives zero-order normal and linear interpolations [using only Qo and Ql terms in equation (7.3)] for G(s) up to IIso. These are also known as point slope and trapezoidal rule interpolations. Table 7.3 lists these forms ofG(s) for quadratic (Simpson's rule) interpolation. EXAMPLE: I In this cXlImplc we apply Table 7.1 to a linear system des.cribed by the fllllnwing differential cquution with constant coefficients:

tll.l· rl/~

t. .,

r

,I.r /11

I· 2./' =

411( t)

(7.7)

226

THEORY AND APPLICATION OF THE z-TRANSFORM

TABLE 7.3

s

INTEGRATING OPERATORS (quadratic interpolation-)

T(l + 4z-1 + 21- 2 ) TO + 2z-1):I:(O) 3(1 + % lXI - 2:-1) - 3(1 + % 1)(1 - z 1) T2(1

+ IOz-1 + z-2) 12(1 - 21-1)2

1

sa

TaO

TI(I + 5z-1)x(O) 12(1 - 21-1)2

-

-~:-:---":":"':"'......;.

T2z-1x'(0)

+ 6(1 + z

T 3z-I,r'(0)

+ -:'7":~--= 12(1 - 21- 1)3

+ 26z- 1 + 66%-2 + 26%-3 + 21- 4) 60(1

+ 21- 1)(1

-

% 1)3

T3(1

+ 17.2:-1 + 332:-2 + 9z-3)2'(0) 60(1 + z 1)(1 - z 1)3 +

Tf(l

~

T4 z-I(3 (I

360(1 -

1)(1 - z-I)3

+ 422:- 1 + 123z-2 + 14%-3):r(0) 21-1)4

+ j5

+z

21 1)4

360(1 -

TOO

+ 14z-1 + 32:- 2 )x'(0)

+ 56%-1 + 246:1:-2 + 56z-3 + z-t) T4(1

1

1)(1 - z 1)

Toz-IO + I3z-1 + z-2)x'(0) 180(1 - z-I)4

+ 120z- 1 + 119lz- 2 + 24162--3 + 119lz-- 4 + 120z--5 + z-6) 2520(1 + z-I)(1 - 2:-- 1)5 T°(l + looz- 1 + 724z- 2 + 1208:1:-3 + 467z- 4 + 20z-5}x(O) 2520(1 + z-I)(1 - z-I)5 T6z--1(5 + 17621-1 + 47821- 2 + 176z-3 + Sz-4)x'(O) +---~-~~~-~~-~--~~ 5040(1 %-1)(1 _ Z-I)5

+

where U(/) is a unit step and the given initial conditions are x(O) x'(O) = -4. The following steps are indicated for the solution. I. Apply the Laplace transform to equation (7.7) and divide by general, by sft, where n is the order of the equation) to get

( 1 + -3 + 2) - X(s) S

S8

= -4S3 + -S25 + -3S

= 3, S2

(in

(7.8)

2. Substitute integrating operators for the left side. In this case we choose quadratic operators from Table 7.3. We note equation (7.11:) and apply the z-transform to the right side of c(luatioll (7.8) by using Table II or Table I of the Appendix.

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

227

3. Carry out the algebraic manipulation to bring the form of .q'o(z) to a ratio of two polynomials in Z-l as follows

(18 + 18T+ 3Tt) + (-18 + 30T+ 39T 2 + 4T3)z-1 + (-18 - 54T - 3T2)z-2 q' . ,,(z) = (6

+ (18 + 6T + 9T2 - 4T3)z-3 + 6T+ T2) + (-12 + 12T+ 101'2)z-1 + (-36T)z-2 + (12 + 12T- tOT2)z-3 + (-6 + 6T- T 2)z-4 (7.80)

4. Choose a certain T; let T = 0.1. The initial choice of T will be discussed later. Divide the numerator by denominator to obtain an expression in powers of Z-l. We then obtain ~t1(Z)

= 3.00

+ 2.646z-1 + 2.373::- 2 + 2.1 64z- 4 + . . .

(7.9)

The coefficients of z-n yield the time response at the sampling instants. By use of the modified z-transform, we can extend the preceding methods to obtain the response between the sampling instants if desired. EXAMPLE 2 To illustrate the application of the z-transform method to the approximate analysis of continuous feedback system,1 consider the system shown in Fig. 7.4 where

_ s + 0.3 GJ() S S2

(7.10)

,

and

e- 6 ' 4 = t = e-'

(7.11 )

This example has been analyzed exactly for ramp input in Section 2.3. In this discussion we shall compare the approximate response with the exact one at the sampling instants. If the ramp response is determined by analyzing the sampled-data system shown in Fig. 7.5, the first step is to choose an approximate sampling period. The Laplace transform of the output is R(s) times the

LG".l<-'" 1111111( I "1.4

("UUIIlIlIIlIlS

1:"'·

dust'II"lnll)' :,yslt'JIl wilh IIt'lay"

228

THEORY AND APPLICATION OF THE %-TRANSFORM

O(s)

FIGURE 7.S

= O. ("e -

{loS

Sampled-data system approximation

or Fig. 7.4.

overall transfer fUl'\ction; therefore, for a unit ramp input, C(s) =

+ 0.3)e-· + (s + 0.3)e-~]

(s S2[S2

(7.12)

The relative real-frequency composition of the output can be obtained by replacing s by jw in equation (7.12), which yields ' ) C(}W =

(jw _(/I2[ _(1)2

+ 0.3)e- Jw + (j(l) + 0.3)e- .

(7.13)

J"']

As will be explained in Section 7.5, for obtaining an initial value of T, we let I C(j(l»I be approximately 0.01 or about -40 db. Thus equating the absolute amplitude of C(j(l) to 0,0] yields

1j(JJ + 0.311 e-i"'l An exact solution of equation (7.14) could be difficult, especially in complicated systems. Fortunately. an exact solution is not necessary since any convenient (" yielding an absolute value nearO.OI will be satisfactory. Such a (IJ can be obtained by recognizing that for large w equation (7.14) becomes approximately IC(j(l)1 ~

~=

0.01

(7.15)

CIJ

from which an upper frequency limit of 4.64 radians per second is obtained. Evaluation of equation (7.14) for II) equal to 4.64 yields an absolute value of C(j(l» of 0.0128, and therefore the approximation made in equation (7.14) is satisfactory. As explained in Section 7.5, a sampling frequency of twice 4.64 radians per second (twice because of the sampling theorem) corresponds to a sampling period of 0.675. This sampling period can be used. but a T of 0.5 seconds is more convenient (also more accumte) since the delay is I second. It is not necessary for the delay to be nn integer value of the sampling period since a sampled-data systcm can bc lInlllyl.cd for Hny ~lrbitrary delay.

Z-TRANSfORM METHOD IN APPROXIMATION TECHNIQUES

229

With a period of 0.5 second, the approximate transient response at the sampling instants to a ramp input can be found from the following expression, if we use the fictitious triangu.l~lr h91d J?.<s) in the feedback path as shown in Fig. 7.5.

~/I(z) = .~t6(Z)[

1

I

]

(7.16)

+ ~lt6(Z)

where

&l~(Z) = Z[R(s)G(s)] = 0.125(2.1z2

+ O.4z - 1.9) 2z(z - 1)4

(7.17)

ZlQl(S)G(S)] = 0.25(2.tz2 + O.4z - 1.9) 2Z2(Z _ 1)2

(7.18)*

and

~lrl(z) =

Substitution of equations (7.17) and (7.18) into (7.16) yields

tt' ( ) II

Z

= (z _

(1.05z2 + 0.2z - 0.95)z 1)2(8z4 _ 16z3 + lO.lz11 + 0.4% - 1.9)

(7.19)

Long division of the denominator into the numerator yields the power series
7.3 Integrating operators 2 ,3,27 The interpolating schemes indicated in Fig. 7.3 represent also the integrating operators used by mathematicians in numerical analysis. These integrating operators can also be classified as follows. (a). INTEGRATION BY RECTANGULAR RULE. Here the interpolator form is

as shown in Fig. 7.30.

.'

)

230

THEORY AND APPLICATION OF THE z-TRANSFORM

(b). INTEGRATION BY THE TRAPEZOIDAL RULE. This is similar to linear

interpolation given in Table 7.2. (c). INTEGRATION BY SIMPSON'S ONE-THIRD quadratic interpolation shown in Table 7.3.

This is similar to

RULE.

In this case the

(d). INTEGRATION BY SIMPSON'S THREE-EIGHTH RULE.

interpolation transfer function for I/s is

3T I 8

--+-

S

+ 3z- t + 3z- 2 + Z-3 I -

(7.21)

Z-3

(e). INTEGRATION BY WEDDLE'S RULE.

-

-.~

S

I + 5z- 1 + Z-2 + 6z- 3 + Z-4 + 5z- 5 + Z-6 -3T ----'-----'----'--......;.----10 1 - Z·6

(7.22)

Comparison of the relative amplitude-characteristics curves of cases b,

e, d, e, and the ideal integrator is shown in Fig. 7.6. It is seen that b is of simple form and can be used effectively because it also introduces high attenuation at high frequency, thus minimizing the effect of an external disturbance. The discussion and derivation of these integrating operators are available in any standard text on numerical analyses,27 and thus only a brief mention for comparison is described in this section. 3.0

I

2.5

I V«)

(d)1

2.0 1.5 1.2 1.0

0.8

./

-- .- .~ '"""'- -.... --

0.6

/

.........

(0)-

Ideal trIlegralor..;!"

N"

---'- 1 -~

0.4 M

0.3

o

V

/

w,/6

~-

... - -.--- ....

'"

\

--

(b)

\

\

2..

w'='T

FIGURE 7.6 Relative amplitude-charaeteristie curves of the integration operators. (From J. M. Sal7.er, "Frequency Anillysi~ of Digital Computers Operating in RClil Timc," p,.o('. IR/~, Vol. 42, I'chnHlry 11)54, p. 4C!.1,)

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

231

7.4 z-Forms and modified z-formsl,6,l1,23 The z-form method of analysis is similar to the integrating operator method discussed in the preceding section, This numerical method is useful in obtaining the responses of continuous linear or nonlinear systems. It is very useful for digital-computer approach to the solution of automatic control problems. This method was proposed by Boxer and Thaler and, in recent years, Wasow22 has put this joo method on a rigorous mathematical s-plane basis. We shall discuss this method both as proposed originally and with the justifications indicated by Wasow. - - - - H - - - - -__ I1-axis A closely related method to the z-forms is that proposed by M. Cuenod. 7 • u The latter algorithm deals directly with the sequence f(nT) as an approxi-joo mation to f(t), whereas the z-form FIGURE 7.7 Path of integration in deals directly with the Laplace trans- s-plane. form, ff.[f(t)] = F(s). Wasow has shown that the resulting two methods have the same accuracy as far as the order of magnitude of the error of approximation is concerned. z-Forms

relation.~hips

The inverse Laplace transform of a rational function of s with higherdcgree denominator in s than the numerator is given by the following contour integral: /(t)

1 i<+;oo = .z-llF(s)] = -. F(s)e 'S ds 271')

(7.23)

<-joo

We assume for simplicity that c is zero (that is, all poles of F(s) lie in the left half of the s-plane with the permissible location of a simple pole at the origin). This condition assures that the response of a physically realizable system is bounded. The contour integration can then be performed as shown in Fig. 7.7. Equation (7.23) can be written as the sum of the following three integrals: 1 f(l) = - . F(s)e tl ds

f'Ir,'T)

271')

/1,,1'1')

(7.24)

232 THEORY AND APPLICATION Of THE z-TRANSFORM If T is chosen sufficiently small, the first integral will yield a good approximation for J(/), and the second and the third integrals may be ignored. The error involved in this approximation will be discussed later. Provided T is chosen very small and letting t = nT, we obtain 1(1) :::::./u(nT) = -1.

211')

i

llw17')

(7.25)

F(s)e"T6 ds

-iI_IT)

By substituting for 5 = (lIT) In z (with z = eTI ) in this equation and noting that by this change of variable the imaginary axis becomes the unit circle, we obtain

1,.(nT) =

_1

r 1. F(1.1n Z)Z"-1 dz T

(7.26)

211'j JI' T

The contour r is the unit circle in the z-plane. Equation (7.26) will be recognized as similar to the relation for the inverse z-transform, with the exception of the factor liT. Indeed, Wasow has shown that if F[(IIT) In z] is considered as the z-transform of J(nT), the difference between the I.aplace transform F(5) and the z-transform F(z) times T (which is referred to as the Riemann sum corresponding to 2 f) can be expressed by the Euler-Maclaurin formula. A pertinent form is given. 2•

fiJI - T11

=! Pv! BO) {d: dt v~l

1 1

(/(t)e-"l}

t~O

+ R",

(7.27)

where m is an arbitrary integer, BO) is the vth Bernoulli number, and R", is the remainder that can be expressed by a Bernoulli polynomial. Furthermore, it can be shown that there exists a positive number A such that

I!l'f - T;yJI

~

AT

(7.28)

Thus the error involved in the discretization of the Laplace transform is proportional to the sampling period T. A better approximation is ubtllined if a refinement of the z-transformation is used as follows. Let (7.29) Then Wusow has IIhown that (7.30) The transfurmation .1* IlmOuntli to approximating the Laplace trans""!'lIIatiun hy means of the tnapezoidal rule. !la!>t'" un this ll1otivatiun, lIuxer and Thaler hllvc pmposcd still another IIIl1llilkatioll whkh a!> will he shuwn later, yicldsevell better appruximlliinn. It i!> nntkt~tI I'rlllll t~tJlmtiulI (7,211) that /0'1( 1/7') In :1 is a lransc~ndentlll fllm'liCl" uf :, whkh IIlllkt· .. it illlpll ..sihlt' til t'xpancl in puwt'rs IIf liz by

z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

233

synthetic division. Thus an approximation is made for In z. Of the several possible, the following approximation will be chosen: In z

= 2(u + tU3 + ius + ...)

(7.31)

where

u=

JI

Z-1

(7.32)

+ Z-1

The series in (7.31) converges rather rapidly and the approximation yields a minimum phase error. fn the final application of this method, F(s) is expanded in descending powers of s, that is, F(s)

=

Q1S- 1

+ Q 2S-2 + Q~-3 + . . .

Thus a substitution for equation (7.31) as

S-k

(7.33)

is required. This can be obtained if we rewrite

1

TI2 - = -T = ---~~--In z u + u3/3 + u6/5 + ...

s

(7.34)

By synthetic division, the Laurent series is obtained. S-1

=

f(~ _~ _~~3

-

: ' ••• )

(7.35)

Series expansions of s-" can be obtained by raising both sides of this equation to the kth power. Retaining the principal part and the constant term gives S-k

~

N,,(Z-I)

(1 _ Z-I)"

= F (Z-I)

(7.36)

"

where Nk is a polynomial in powers of Z-I. These approximate expressions for S-k are called z{orms and are given in Table 7.4. If we denote the ::-forms as Tl**, they are related to the z-transform as follows: It-I t"-I B T'¥**' = T'¥ + T".2 (7 37) t? (k _ I)! t7 (k 1)! k! . Based on these notations, Wasow has shown that

I

!R'

It-I

(k _ I)!

,k-l

I

- T ** < A Tk+l 1 (k _ I)! - k

(7.38)

where Ak is "constant. Hence the z-forms or (Tl**) leads to a still better approximation to the I.aplace transformation for these special functions, provided k > I. In applying Ilw :-1'111"111 IIIclhod 10 th~' solutioll of lincar dill'crcntial 1'111111 t iOIl!> wit h '"ClII,llInl l'Cll"lIidt·llh. Wa 'ClW 22 ha!t shown I hili I hl~ crror

234

THEORY AND APPLICATION OF THE Z-TRANSFORM

TABLE 7.4

z-Forms Ft (Z-I)

T 1 + z-1 2" 1 - z-1 T2 1 + )()Z-1 + :-2

12

(I -

T3 Z-1

+ .r2

2" (I

z-I)2

- z-I)3

T4 Z-1 + 4:- 2 + z-3 6 (I - Z-I)4 T5

(z-1

24

+ Ih- 2 + llz-3 +

Z-4)

(1 - z-I)5

(z·-1 + 26%-2 + 66z- 3 + 26:- 4 + z-5)

T6

120 T7

T4 720

(1_2- 1)6

T6

+30,240

(2-1 + 572-2 + 3022-3 + 302:-4 + 572-5 + 2-. 6) (l -

720

2-1)7

where !Ft (:) =

Z [~J t

magnitude between the exact solution at sampling instantsf(nT) and the approximate solutionf,,(nT) for (n > I, or t> 0) is proportional to T2, that is, (7.39) I~nl = If.. (nT) - f(nT)1 ~ kT2 Furthermore, if F(s) is written in the following form: F(s) =

P(s)

(7.40)

QTS-T

(7.41)

"" b,s-·' Q(5) = ~

(7.42)

where

pes)

1 + Q(s)

... =~ T~1

r -I

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

235

and if per) = ,fR-IP(S) and q(t) = g.>-lQ(S) have their first k (k > 0) derivatives equal to zero at t = 0, then the error of approximation is proportional to T2+k. This indicates that by using the z-forms a better approximation is obtained than with z-transform or its modified form 1*.

z-Form procedure The following steps are involved in obtaining the approximate response Ja(nT) from F(s) = !e[J(t)]. I. Express the function F(s) as a rational fraction in powers of I/s. 2. Substitute for each S-k a rational fraction in Z-1 obtained from Table 7.4 and arrange ~a(z) as a rational fraction in Z-I. 3. Divide the resulting expression by T. The initial choice of T will be discussed later. 4. Expand the fraction as a power series by using the synthetic division or other methods to obtain Ja(OT)

+Ja{l T)Z-1 + JR(2T)z-2 + ... + J,,(nT)z-n + ... (7.43)

The coefficientsJa(nT) are the approximate value ofJ(r) at the sampling instants. To obtain the approximate response between the sampling instants, if required. the method to be discussed on p. 236 can be used. If F(s) has one degree higher denominator than the numerator in s. we can improve the degree of approximation using the z-forms by writing F(s) in the following form: A

F(s) = -

s

+ F 1(s)

By obtaining first the inverse Laplace transform of A/s, AU(f). and then applying the z-forms into F1(s) and carrying out the procedure given. we can improve on the approximate response to J(t). Finally, if initial values do exist, the z-forms could be easily modified to cover this case as well. t See Table IV for higher values of k. The values of B•. are the Bernoulli numbers. lIome of which are given below: 21 Bo

","

= 1,8. = -A. B~ = "~' B. = -.'0. B. = II"'., ".,.:

-N.•. B••

=+....

*. B.

= - ...

with B~ • •

=O.

for k

> O.

236

THEORY AND APPLICATION OF THE z-TRANSFORM

Modified z-forms l1 As with the z-forms, we can also obtain approximate relations for the continuous response by using other relations; these relations are called modified z-forms and are derived as follows. Assume the Laplace transform of a continuous output is given as

F(s) =

Co

do

+ CIS + c'Jil.a+ ... + c"s: + dis + d,.s + ... + dlts

(7.44)

If this equation is multiplied by e-'1', the inverse Laplace transform will be /(1 + mT). By letting m vary between zero and unity. we obtain the response between the sampling instants if /(/) is approximated by /o(nT) as discussed earlier. The use of these relations requires rewriting equation (7.44) as F( ) = coO/sit) + cl(I/SIl- 1) + ... + c,,(l/slI-") S It It 1 do(l/s ) + d1(1/s - ) + ... + dk

(7.4S)

The numerator of equation (7.45) is multiplied by the Taylor series expansion of e· mT ; that is, by

em.'1' -_ I

+ sm T+ (smT)t + S3 m 3T3 + 2!

3!

(7.46)

...

If each term in the numerator is multiplied by the series and we carry terms that yield only the principal part plus a constant term, then

F(s)e· m '1' _ go(1/SIl, s, mT)

-

+ gl(1/SIl-1, s, mT) + ... + gll(I/i" + d (I/S + ... + "It

do(l/Sk)

1

", x, m'l')

II - 1 )

(7.47) Substituting the z-forms for I/slI, we obtain the modified z-forms.* These modified z-forms are tabulated in Table 7.5. • For instance, ir F(s) FCs)e,"·T

= I/s, we have from equation (7.46), I

CmT)1

= -s + mT+ .\-2' + ... .

To obtain the mudified :·furl11s, we: oblain T: .. I

sllh~lillll('

('1.-1'1,,) Ihe

.~·I'III'III' 1'111'

'I'll I 2m'- I (I

1,'(11/, I) ,. .•. I ", I - ., 2, - I I.

I

2""

th,' lir,t

IWII II'IIII~ hi

237

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

TABLE 7.S MODIFIED z-FORMS" sft

F(m,z)

TO + 2m)z + (I s

z- 1

2 T2 (I

+ 6m + 6m2)z2 + (10

- 12m2)z (z _ 1)2

12

ra

- 2m)

(m

+ (l

- 6m

+ 6m2 )

+ 3m2 + 2m3)Z3 + (6 + 9m - 3m2 - 6m3)Z2 + (6 - 9m - 3m2 + 6m3)z + (-m + 3m2 -

12

m3)

(z - 1)3

If m - 0, the modified z-forms reduce to the z-forms. The accuracy of the calculations is improved if m is allowed to vary in the range ... ~ ~ m ~ ! to obtain the total response. The procedures to be followed using the modified z-forms are I. The modified z-forms are substituted into the numerator of the I.aplace transform expression. 2. Thc z-forms are substituted into the denominator of the Laplace 1IIIIIsfnrm expression. J. Divide the resulting ratios of polynomials in Z-1 by T. To obtain II,,· limc function, carryon the synthetic division method by first choosing II I'"rticular value of T. II.I.USTRATIVE EXAMPLE FOR z-FORMS

III Ihis example we consider the feedback control system shown in Fig. I K. A!>SIlIllC the input as a unit step, R(s) == I/s; the output transform e '(.,) i!> ,~ivcn by ('(s) =

I

(7.48)

.~3+.~2+S

1"111' rnllnwill~ procedure is used to obtain the approximate response IIII' .... I'nI'IIIS,

.... 111'·,

eft)

C(.,

I Ie IURI' 7.N

('UIIIIIIIIIIII'I rl't'dhllC."k

"Ylll(,lI1,

238

THEORY AND APPLICATION OF THE z-TRANSFORM

I. Divide the numerator and denominator by the highest power of s; then (7.49)

2. Substitute for each term the to obtain

~(I(z) =

(12 + 6 T +

~-form

T a) -

+ (36 -

from Table 7.4 and divide by T

6T2(z-1 + Z-2) (36 + 6T _ 9T2)z-1

6T-

9rt)Z-2

+ (12 -

6T+ T 2)z-3 (7.50)

3. Choose the sampling period T on the basis of frequency response. Hcre T is chosen as 0.5, based on the criterion that frequency components below -30 db are negligible. Therefore ~(I(z) becomes 1.5z- 1 + 1.5z-2 (I z = 15.25 _ 36.75z- 1 + 30.75z-2

~( )

(7 51) _

9.25z-3

•

4. Dividing the numerator by the de!,\ominator, we obtain the time licquence response c(I(n) as the coefficient of z-n. Hence ~(I(Z)

= 0.0984z-1

+ 0.335z-2 + 0.610z-3 + 0.853z-· + ... (7.52)

In comparing the approximate with exact response, we would find that Ihe difference is negligibly small.

7.S The choice of the sampling period 1,10 The crror of approximation depends to a large extent on the sampling period. For instance, if the sampling period T is zero in the limit, the ;Irproximate system is exactly equivalent to the original system before ;Irrmximntion. On the other hand, if Tis large, the error of approximation is increased although the number of computations required to obtain the lime response is considerably less than for small T. Furthermore, if a di~ilal computer is used in connection with z-forms, as the number of cnmrutnlillns increases the round-ofT error also increases. Thus thc choice Clf Ihc sampling rcriod rCllllircs " cmnpromisc betwecn the crror of ;Ipprnxill1;I(ion "lid thc lIul1lht~r nf l'nmpulatiolls relluircd. The critcrinn fur Iht" illiliul dlOk,! nf T is rathcr difficult. Scvernl nih'rill IIfC "dvm'nlctl, nl\lClI\~ whit'h is Ihe olle cnnllidcred hcrc bascd on

Z-TRANSfORM METHOD IN APPROxiMATION TECHNIQUES

239

the frequency response of the output. For instance, if the output magnitude as a function of real frequency is written as

Icejw) I

(7.53)

we can choose a value of frequency W k that gives a small value of lC(jw)l, that is, around -30 db or about 0.01. By obtaining such a frequency, then from the sampling theorem, the value of T is given byl

T= !!...

(7.54)

WI<

To obtain an approximate value of W k , lcejm)/ is approximated for large (I), which avoids any exact calculations. This procedure is explained in Example 2. Another procedure for choosing T (as used in obtaining the transfer function of a system or network when both the input and the output /;(1), /,,(1) are given) is based on the fact that both /,,(1) and /;(1) should change on the average by about 10% of their maximum values within successive intervals of the chosen sampling time.

7.6 Analysis of the

error6,22,~,2l;

The error entailed in such an approximate analysis, discussed in the preceding sections, cannot be formulated exactly. The formulation of the error bound which can be obtained for linear systems is generally quite elaborate. The error bound becomes significant only when many samples are considered, as is done in the analysis using digital computers. However, the error can be readily estimated if the sampling period is halved or subdivided to calculate a few transient terms. If by so changing the sampling period the original solution is essentially unchanged or the change is small for engineering approximation, the original solution is satisfactory. This, of course, is based on the fact that round-off error is not considered, that is, when the computation is carried on without digital computers. If the initial choice of the sampling period is unsatisfactory. this procedure has to be repeated until the change in the response is insignificant. Both WlIsnw 22 ,and Bridgland 24 have discussed the error of approximation invulved in the use of the z-forms and other rules indicated earlier. The \'olllplete discussiun of this subject is beyond the scope of this text. In some cases Simpson's (quadratic) rule is numerically unstable when u!.t'd fur Ihe intc~mti()n of the dilTercntial equation d!lldx = AY when the n'al parI uf A is Iwgalive, This is because the churucteristic equation in z II:!!. a 1'11111111' IIllldulus ,',n':llt'r 111:111 unity. The stnbilily criterion estahlished III ('hlll'tt'" \ \':111 Ill' n':lIlily lISt'cllu ch'h'l'lIlim' sudl II situation. To :Ivuid Mle'h II tlillh'ulty, wC' IIIlIy liSt' II !.1II1111 inh'l'vlIl IIVI'I' tI", initinl alld sle\'I'

240

THEORY AND APPLICATION OF THE z-TRANSFORM

part of the response and continue with a much longer ihterval over the more slowly changing part. Thus the adjustable sampling interval plays an important role in minimizing the error of approximation.

7.7 Low-pass transformation for z-transforms2o ,21 (n certain applications, in which the output of a linear time-invariant system !o(I) as well as its input h(l) are given as experimental data. the z-transform method could be easily used to obtain the system transfer function ,"'(z) as follows: Jf(z) = fFo(1.) fFi(1.)

(7.55)

Using the synthetic division method .Jt"(z). the impulsive response at discrete intervals of time, can be easily determined. The data of the input and output are usually not exact in practice but are subject to error or noise and the division method of the z-transform will yield erroneous results. Error in the series for Jf'(z) is introduced if the leading (or initial) term of !Fo(z), that is'/D(O) is much smaller than the subsequent terms. To avoid this difficulty. the z-variable may be replaced, before performing the synthetic division, by the variable A, where

A+ I =

Z-l

'(7.56)

It is done by simple substitution of A + I for I/z, in the expressions for !Fo(z) and .~iz), followed by expansion and regrouping of terms to form a power series in it; thus 'o(z) and !Fi(Z) become .'!io().) and .#i().)' By this process, known as low-pass transformation. the leading term of :j;D(it). that is, .#o(A)IA~O is much larger and thus the division method is less sensitive to the errors in the individual terms. By dividing /f;o(A) by !F,(A) and transforming back to the z-plane by this low-pass transformation, we readily obtain J(.'(z) or the system transfer function. It should be noted that only when we consider noisy data do the advantages of A-domain become apparent. Solution of Problem 7.3 illustrates the procedure and the advantage of this transformation. The A-transform leads to a match for H(s)

=

Fo(s) Fi(s)

(7.57)

at low frequency, whereas the z-transform leud!l to a mulch for the !ly!ltem transfer function ul higher frequencies und therefore ex:tg~er:lte!l the en'cct

::-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

241

of the noise. Another kind of transformation. known as the two-sided :-transform, has also been advocated for this particular case.!1 However, we shall not elaborate on this technique.

7.8 Application to time-varying differential equations6 •1l Boxer and Thaler have advocated use of the z-forms method for the approximate solution of time-varying or nonlinear differential equations. Wasow has shown in a rigorous way the validity of this method and the underlying assumptions required to obtain a fairly good approximation. In this section, we shall briefly indicate the procedure to be applied to the following:

d" dl" [!,,(t)y]

d n- 1

+ dl"-I l!"_I(t)y] + .. ·!o(t)y =

f(r)

(7.58)

where 1,,(/), I" _1(1), .•• ,/"(t) are functions of the independent variable I. The procedure for the approximate solution is I. Obtain the Laplace transform of (7.58) after substituting for /,,(1) == • • • ,/;(1) = c, and inserting the initial condition. 2. Proceed with the division as in the constant coefficient case. However. change the values of C '" C .. _ I •••• , Cn at each step in the division process to the values of the functions at the corresponding times.

C II •

The minimum limitation on the form of the functions in (7.58) is that they be Laplace transformable. The procedure for inserting the initial conditions is illustrated in Problem 7.4. EXAMPLE

To illustrate the procedure, we choose the following example: (II/

-'- + t1/ == I, tit

with yeO)

== 0

(7.59)

hllllltinn (7.59) is written as

..,/1/,- + ,II

Wh('11

Wl'

!I( II)

n,

('1/

'

== I,

where (.

==

I

== nT

(7.60)

lakl' Ihl' I.aplacc Iwnsfurlll. cnn!oilicring c' as constant and l'( \ I

(7.(,1)

242

THEORY AND APPLICATION OF THE z-TRANSFORM

or Y(s)

=

S-3

1 + cs-

1

By substituting the z-forms of Table 7.4 and dividing by T, we obtain

rr?la
T (z-1 _ Z-2) = -------~-'-------':..-..____.,____.,----­ (2 + en - (6 + eT)z-1 + (6 - CT)Z-2 - (2 - CT)Z-3 2

(7.62) Letting T = 0.4, ~

0.16z- 1 + 0.16z- 2

o

(z)=------------------------------------(2+0.4c)-(6+0.4c)z-I+(6-0.4e)z-2_(2-0.4c)z-J (7.63)

The long division process is carried out letting c = 0.4 during the first step of the division, 0.8 during the second step, and so forth, yielding 9i,,(z)

= 0.074Iz-1 + 0.266z- 2 + 0.503z-3 + 0.714z- 4 + ... (7.64)

The approximate solution can be compared with the exact solution for this example. The latter solution is

Y = I - e- t2 / 2

(7.65)

This procedure is also applicable to differential equation of the following form if the j;(/) are slowly varying as in some physical systems.

tin

dn - 1

In dIn - Y + ... + In-l-l y + '" + loY =/(1) dt n-

(7.66)

This can be justified if we notice that (7.67) where

(Z)

are the binomial coefficients. If In is slowly varying, the con-

tributions of the derivatives ofIn are negligible, so that the approximation

dnUnY) '" d"y dIn -In dtn

(7.68)

can be justified. It should be cautioned that with this procedure information on the stahility of these equations l'anno\ he ohtained.

243

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

7.9 Application to nonlinear differential equations 8 ,l1-13,18 In the preceding sections the methods of integrating operators and the z-form have been applied to obtain the approximate solution of linear differential equations with or without initial conditions. In this section, the method of sequential computation, which consists in approximating a given differential equation by its discrete version, is employed. Furthermore, the method of integrating operators and the z-transform will be employed to obtain an approximate solution of a nonlinear differential equation. EXAMPLE I Consider, for illustration purposes, the nonlinear differential equation -di

+ I. +'21 =1,

dt

with i(O) = 0

(7.69)

We approximate this equation by means of a difference equation as follows. Let where T = sampling interval

(7.70)

Then, ill+l - in

+ Tin + Ti!

(7.71)

= T.

If we choose T = 0.2, we get

5i,.+1 - 4i"

+ i~ =

I

(7.72)

The series solution of this equation for T = 0.4 by the method of complex convolution discussed in Chapter 5 does not converge. However, we obtain a convergent solution for T = 0.2. The solution as a recurrence relCltionship compared with the exact solution of the differential equation is shown in Table 7.6. TARLE 7.6 COMPARISON OF EXACT AND APPROXIMATE SOLUTION FOR F.XAMPLE I t

= 117'

Rc'currt'1Icc' Equation Sollllion

()

()

~

o.:mo o..mm

III l'i

11,4710

,In

II. 'i IH.I

-\I)

lI.t.I)!! I

Exact Solution of Differential Equation _ _ _ _ •._._. _ _ _ _._00

o 0.217 0.370 0.467 II. 'i:'I) lI.tlllt.

244

THEORY AND APPLICATION OF THE z-TRANSFORM

EXAMPLE 2 In this example we employ the linear interpolation operators to solve the following nonlinear equation:

-d 2x + 2 -dx + x = -0.9 -dv , 2

dt

where

L'

dt

= f(x)

and 1~0)

= 0, x'(O) =

with x(O)

dt

1 (7.73)

=0

I. Apply the Laplace transform to equation (7.73). (S2

+ 2s +

= -0.9[sV(s) -

I)X(s)

L~O)]

+ 2.r(0)

+ x'(O) + sx(O) 2. Divide by

S2

(7.74)

and insert initial conditions

(.1 + 2-s + -1) X(s) ~

V(s) = -0.9 -

s

+ -~t

(7.75)

3. Substitute linear interpolation operators from Table 7.2 and the z-transform for l/s2 in the right-hand side.

!?l" (z)

[I + 2T(l +

Z-I) 2(1 _ Z-I)

n

+ T2 (I + 4z- 1 + Z-2)J (1 _

6

Z-I)2

= -0.91""(z)TI + Z-1 + 21-z-

4. With T transform is

= 0.4,

8.56x n

Tz- 1

(7.76)

(I-Z- I )2

the regression (or difference) equation of the above 11.36xn _ 1

-

1

for n = I,

+ 3.76xn _2 = -

= -1.08v"

+

1.08vn

1.08vn_ 2 •

+ 1.08vn _ 2 + 2.4, for n > I (7.77)

If we let l'n = x~, we can solve this equation as a recurrence relationship for each value of n starting with n = I and putting lIn = = 0 , for n < 1. The response is shown in Fig. 7.9. In conclusion it should be pointed out that the division method using the z-forms can also be used to obtain an approximate solution of such

r;

0.6 r---,--,---,---,--...----.

0.4

4.8 !'I(HIRE 7.1)

Rt'Spllll~C

III' I

~lIl1ll'lt' :'

I'll' lIIililll Vt'llIl'ily .• ··(111·- I

Z-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

245

equations. This method is based on adjusting the divisor at each step of the division, as illustrated by the following example. EXAMPLE

3

dY + y2 =1 dt ' This equation is written as dy

+ cy =

dl

with y(O) = 0

(7.78)

I

where c is considered constant in this operation and adjusted at each step of the division. The Laplace transform in powers of S-1 is given as S-I

Y(s) = I

+ cs- 1

(7.79)

By substituting the z-forms and dividing by T,

TO + lOCi + Z-2) ~ (d = - - - - - - - - - - (12 + 6cn - 24z- 1 + (12 - 6cT)Z-z

II

Letting T = 0.1,

0.1(1

o}j (z) _ •

R

-

(12

+ 0.6c) -

+ 10%-1 + Z-I) 24z- 1 + (12 - 0.6C)Z-1

Carrying out the long division and letting c adjusting it after each division, we get

tW'/J(z) = 0.00833

=0

(7.80)

for the step first and

+ 0.100z- 1 + 0.199z-1 + 0.295z- 3 + 0.390z- 4 + . .. (7.81)

The exact solution of the original equation is y = tanh t (7.82) The approximate solution compares favorably with the exact one. This method can be used for the approximate solution of the following differential equation:

d"

dt"

[J., (y, citdy , ...) yJ + clt"d"-1 [/,,-1 (y, dy , ...) yJ cit 1

+ ... + lo(Y, dy , .. .)y = ,It

I(t) (7.83)

wilh

'( ,/I,

,', . ./. /I. ' .'" . ,It

)

(7.R4)

We IlI'lII"'l'" I"m Ih,' III'PI'U!(illllllinll liS ill Ih,' "1111:.111111 ctll'lIkicIII case. ""Wl'VI'I, ill 1"'lflll'lIl1l1l! Ih,' IIIIII! tlivisiull PI'III""", WI' 1'111111,." Ihl' vlllm's

246 of

THEOIlY AND APPLICATION OF THE Z-TRANSFORM

CII ,

••• ,

Co

at each step in the division process to the most recent

values available of Ci

=

/;(Y, ~~ ,.. .).

7.10 Other numerical techniques 8 •9 .l&.J7-19 To obtain an exhaustive study of all the numerical techniques is beyond the scope of these discussions, and can be readily studied in any of the standard books on numerical analysis. In this chapter, we have used several of the integrating operators which are closely related to the subject matter, that is, the z-transform method. The main emphasis has been in using these to solve systems (linear or nonlinear) which are usually encountered in engineering applications. Rigorous application of the theory to approximation problems in mathematical terms is not attempted. In trying to limit the discussion, it was necessary to ignore other numerical techniques. Among those not discussed in detail are the works of Madwed,9 Cuenod,7·14 Tustin,' and Heizman et al. 13 However, the techniques discussed are closely related to these and other methods. The main emphasis of these numerical techniques is to obtain approximate solutions for systems in which exact solutions in a closed form are not possible. This is often true in a complicated nonlinear or time-varying continuous system. It is for solving such problems that this method could be advantageously used. Furthermore, the use of digital computers to solve differential equations requires the formulation of the problem in terms of difference equations. Hence the z-transform method plays an important part in obtaining an approximate solution. REFERENCES

I. Jury, E. I., Sampled Data Contf'fll Systems, John Wiley and Sons, New York, 1958. 2. Ragazzini. J. R.• and G. F. Franklin, Sampled-Dala Control Syslems, McGraw-Hili Book Co., New York. 1958. 3. TOll, J. T., Digital and Sampled-Dala Conlrol Syslems, McGraw-Hili Book Co., New York, 1959. 4. Cruickshank, A. J. 0., "Time Series and Z-Transform Methods of Analysis of Linear and Nonlinear Control Systems," Aulomali~ Qnd Remole Control, Pro~. of firSl /",. IFAC Congress, Moscow, 1960, Butterworths, London, 1961, pp. 277-285. 5. Robertson, H. H., Discussion of Reference (4), p. 285. 6. Boxer, R., and S. Thaler, "A Simplified Method for Solving Linear and Nunlinear Systems," Prod. Insl. Radio En.t:ineers, Vol. 44, 1956, p. 89. 7. Cuenod, M., "Contribution a L'etude de Phenomenes Transitllires :\ I':lide des Suites de Temps," BIIlI. Tech. SlIi,fse ((1m .• Vo\. 75, 1949, p. 201. 8. Tustin, A.• "A Method (If Analysing Ihe I'Irccl or Cerlain Kinlls of Nlllllilll:atity ill Closed Cycle Control Systcm!>," J. I".v/. IJ/'/'I. I':'l~"'v., Vul. "4. 1'.u1 II. 1')47, p. I 'i~.

%-TRANSFORM METHOD IN APPROXIMATION TECHNIQUES

247

9. Madwed, A., "Numerical Analysis by the Number Series Transformation Method," Proreedin,gs Symposi"m on Nonlinear Cirrllit Analysis, ",ew York. 1953, Edwards Publishing Company, Ann Arbor, Michigan, p. 320. 10. Truxal, J. G., "Numerical Analysis for Network Design," Proceeding InSlillile of Radio En,lfineers, Professional Group on Circuit Theory, CT-I, 19S4, p. 49. 11. Boxer, R., and S. Thaler, "Extension of Numerical Transform Theory," Report RADC-TR-S6-115, Rome Air Development Center, New York. 12. Naumov, B. N., "Eine Niherungmethode zur Berechnung der Obergangsprozesse in Selbsllatigen Regelungssystemen mit nichtlinearen Elementen," Rep. Con[. Alllo, m.llie Control, Heidelberg, 19S6, Oldenbourg, Munich, 19S6, p. 184. 13. Heizman, C., J. Millman, and A. Vagant, "Comparison of the X· Transform Method with Other Numerical Methods," Technical Report T-8/C. E. R. L., Columbia Uni· versity Engineering Center, New York, September, 1956. 14. Cuenod, M., "Principe de L'analyse impulsionelle et de son application ala Theorie des Servo·mechanismes," Onde elerl. Vol. 37, 19S7, p. 723. IS. Jury. E. I., and F. J. Mullin, "A Note on the Operational Solution of Linear Difference Equations," J. Franklin Insl., Vol. 266, 1958, p. 189. 16. Mishkin, E.• M. S. Goldstein, and J. G. Truxal, "Nonlinear Feedback Systems with Sampling." Research Report R-4SO-5S, PIB-380, Polytechnic Institute of Brooklyn, New York. 17. Hellman. 0 .• "On the Iterative Solution of the Equations of a Single· Loop System with One Non·linearity," J. Eleclronics and Conlrol, Vol. 6. 1959, p. 186. 18. Brown, B. M., The Mathemalical Theor)' of Linear SYltems, John Wiley and Sons, New York. 1961. 19. Raymond, F. H.• "Regimes Transitoires et Techniques des Impulsions," aI/de EIC'tri(l',es. Vol. 28. 1948, p. 222. 20. Huggins, W. H .• "A Low Pass Transformation for Z·Transforms," IRE Trans. Circllil Thellr)" Vol. I, September. 1954, pp. 69-70. 21. Thomson. W. E.• "On Two Sided. Z-Transform," IRE Trans. Circuit Theory, Vol. 3, Jure 1956. p. 156. 22. Wasow, W., "Discrete Approximalion to the laplace Transformation," Z. a/~few. MUIII. II. Php., Vol. 8, 19S7 pp. 401-407. 2:1. noxc:r, R., "A Note onNurnerical Transform Calculus," Pror. IRE, Vol. 45, 1957. 24. Bridgland. T. F .• Jr .• "A Note on Numerical Integrating Operators," J. Soc. IlUlllstri,,1 and Appl. Math .• Vol. 6, No.3, September 1958, pp. 240-256. 2~. (jihson. J. f.., Nonlinear Alilomatic Conlrol. McGraw·HiII Book Co .• New York, 1%.1.

2". Jllnhlll, C. Ca/cull/.r (if Finile Diffrrenc(!s, Chelsea Publishing Co., New York (2nd 111.). 11)(,f).

n.

Sl·:ll'hnrllllgh. J. n.. N,,,I/("'irtl/ Malilematical Analy.ri.f. The Johns Hopkins Press, lIallilllllrc. III~II. Chapler 7. ,III Salll·I' • .I. M., "Fl'ctlllcncy Analysis of Oighal Computers Operating in Real Time," ,., ..... flU. VIII. 4~. Nil. 2. 11)54. pp. 4~7 461•. •"1 WI'''~. I .•

;11111 It. N. Md)ullnlll:h. "1'ruIlY's Methnll. Z-Tmnsforrns, and Pad~ ~. Nil. 2, Aplil 1'1t•.\, 1'1" 145 141).

"1'1 ....'"111111 .. 11 ....'i/"M H,',·., VIII.

8 APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

In the preceding chapters the z-transform theory has been extensively discussed and developed. Several examples from physical systems were considered to indicate the applications of the theory. This chapter is concerned mainly with the application of the material developed. The examples chosen show the wide spectrum of application of the z-transform method. To limit the material, only six examples are discussed in detail. The problems section at the end of the book will give the reader a greater indication of the extensive applications. The examples are chosen from the following areas of system theory: sampled-data feedback systems,I.II.5 antenna theory,7-' economic systems,n-II information and filtering theory,87-aI sequential circuits,lt-17 and finite-state Markov processes. II-eo The examples are formulated mainly in terms of difference equations, both linear and nonlinear, and the z-transform with the other theorems developed earlier are used to obtain the required solutions.

8.1 Nonlinear sampled-data feedback systems3.6.1.23 Because the later examples are concerned for the most part with linear theory, it is advisable for the sake of completeness to choose a saturatingtype, nonlinear sampled-data system as an illustration. In the following discussion the convolution z-transform theory discussed in Chapters 4 and 5 will be applied to a second-order nonlinear system. 6 Consider the sampled-data system shown in Fig. 8. J. where the nonlinearity consists of a saturating amplifier in the fecd·forwIlrd p"th. We will obtain the response of this system lIsing the cnnvnJutiun :·tmnsfnrm. 24K

249

APPI.ICATIONS TO VARIOUS AREAS OF SYSTEM THEORY Zero-order hold

T.l. .... /l6'(z)

Saluratins

r-'''' ---0

I

e(,1)

I I

e(l)

T=l

FIGURE 8.1

Nonlinear sampled-data reedback system.

We now derive the difference equation of the system. Since hen} is constant over a sampling interval (in view of the zero-order hold), we obtain the pulse transfer function for the linear part as

~(z)

- e- T• G1(S}]

= <[I

=

S

0.716z - 0.284 1.135z 0.135

Z2 -

+

=

'if(z} ,;;f(z)

(8.1) This equation gives the following difference equation: c(n

+ 2} -

I. I35c(n

+ 1) + 0.135c(n) =

0.716h(n

+ I}

- 0.284h(n)

(8.2)

Now hen) is given as 40 hen) = { e(n)

e(n)

-40

-40

~

~ 40 e(n}

~

} 40

(8.3)

e(n} ~ -40

The power series representation of this nonlinearity over -65 is hen) = l.llle(n) - 0.1234 x 1O-3e(n}3

~ e(n) ~

65

(8.4)

From Fig. 8.1 we note e(n) = r(n) - c(n)

(8.5)

Uy substituting for e(n) in equation (8.4) from equation (8.5) and inserting the results in equation (8.2), we obtain c(n

+ 2) -

+ I) + 0.135c(n) = 0.795['(11 + 1) - ('(n + I)] - 0.0884 X 10- 3 x [,en + 1)3 - C'(n + 1):1 + 3r(n + I)c(n + 1)2 -- 3r(I1 .. 1)2C"(1I + I)] - O.J 16[,(11) - c(Il)] + 0.035 x

1.1 35c(n

J() :1 If(II):1

1'(11):1·1 .1,(,,)1'(11):1 _.. -"(11):.11'(11>1

(H.6)

250 THEORY AND APPLICATION OF THE Z-TRANSFORM If we assume that r(n) = 60u(n), where u(n) is a unit step function, then k) = r(n) and equation (8.6) becomes

r(n

+

+ 1) + 0.197c(n) + 10-3 x [-6.3c(n)* + 15.9c(n + 1)* - 0.0884c(n + 1)3 + 0.03Sc(n)3] = 0.2865 x

c(n

+ 2) -

1.295c(n

60u(n)

(8.7)

If the system is stable around the equilibrium point as assumed, then
A

00


I -1 ' 11-01 - a,.z

(8.8) (8.9)

ar

where a,. for at ~ 3 is of the form <4, where ". lJ are positive integers (including zero) and" + lJ ~ 2. Taking the z-transform of equation (8.7). we obtain

~(z) + Zl -

(15.9z - 6.3) {,1[c(n)'] _ 0.00555,1[c(n)]3} 1.295z + 0.197

= (0.2865) 60z

z- ]

+ zc(l) + (IS.9)10-azc(0)8 - 1O-3(O.0884)zc(0)3 z'l. 1.295z + 0.197

c(O)(z· - 1.295z)

+

(8.10)

The demoninator of the last term of this equation can be written ZB -

1.295z

+ 0.197 = (z -

where

ai}(z - a~)

(8.11)

a.

ai = 1.12, = 0.175 (8.11a) That la;1 > 1 should not be of concern because it is due to simplification

of terms of the form ,(n + l)c(n + 1)2 to 60c(n + 1)2 and r(n + I )2c(n + I) to 3600 c(n + I). This stems from the fact that ,(n + k) = ,en) = 60. From equation (8.8) we assume the folJowing solution for
Ao

1 - Z-I

+

+

Aao

1-

O~Z-l

+ ... +

+

AI

1 - alz- I

+ ... +

Au -1 1 - ala.z

As

1 - a.z- I AOB

I - a:Z- 1

+

+

+

1-

A20

a~z-l

Aoa

1-

O:Z-1

A:I .. 1 1 - ala2z (IU2)

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

251

Similarly, from equation (8.8), the %-transforms of c(n)i and c(n)3 can be written as [c(n)']

;y

=

A: 1 - Z-1

+

2AoAI 1 - OIZ-1

+

2AoAa 1 - Qr- I

+ 2AoAao + A~ 1-

a~z-I

+ 2AoAao + 2AIA20 + 2AoA2 + A: + ... 1 - a~z-I

1 - a~-I

+ 2AoAn + 2AIAa +

2AoA~n

1 - 01Qr- 1

+ 2AIIA. + 2AIAu 1-

a~Qr-1

+ ...

(8.13)

and

A~

+

+

1 - %-1

3A:A 1 1 - az1- 1

+ 3A:A20

+. 3~oA~ + ... +

;y[c(n)3] =

3A:A. 1 - ar- I 3A:A o2

1 - 0IZ

+.3~oA: + ...

1 - Or

+ 3A:A 21 + 3A~A2 + 6AoA20Aa 1-

o~ar-l

+ 3A:An + 6Ao~IAa + ...

(8.14)

1 - ala,.,r

Substitute equations (8.12 through 8.14) into equation (8.10) and compare coefficients of like terms of z/(% y, 6 = 0, I, 2, . . •. The steadystate amplitude is given by

ar a:>,

+ (lS.9 -

A o

6.3)10-3(A: - 0.OOS55A~) = (0.2865)60

(8.15)

1.197 - 1.295

which gives Ao = 60. This is expected since the system is stable and the steady-state error is zero. Comparison of coefficients of z/(z - al)' z/(% - aJ yields

A1,.{a:,. - 1.295Ql,I

+ 0.197 + (15.901.8 -

6.3)(6O)10-3} = 0 (8.16)

For a nontrivial solution of Al and All we must have the term inside the parentheses in equation (8.16) equal to zero, that is, 0:,2 - 0.34at.2 - 0.181 = 0

(8.17)

a 1 = 0.6275 " t = -O.2K72

(8.18)

or

252 THEORY AND APPLICATION OF THE %-TRANSFORM Comparison of coefficients of other terms gives Aao

= (0.0645)10-aA

I ;

Aoa

= (0.215)10-3A~.

A10

= (0.0143)10-t A:.

Ao,

= (O.t265)10-8A:.

Aio = (0.0249)10-8A~ All

ASI = Au

=

=

(1.04)IO-3A~A2

(8.19)

(0.6l)IO- 9 A:A 2 ; All = (0.465)10- 3AIA:

(0.511)10- 8A I A:•...

Next. we consider the coefficient of terms %/(% - a;). %/(% - a~). where a~ and a~ are given in equation (8.lla). We then obtain the two relations 20

+ 60A I +

60A 2

1407.2

A30 33.4

+ ASD +

492.5

60.5

AD3 33.5

+

+

+

+ ...

+

An

+

31.7

Au

44

= l(0.2865)60 0.12

_

A14 36.3

A70

39.8

Ani 34.2

+

+ Au

Aos 35.4

28

+ c(0)(1.I2 - 1.295) + c(1) } + [15.9('(0)2 - (0.0884)c(O)3JI0-3 -

I

11.5

and -2.91 _ 60A 1 452.5

+

A41 5.97

+ ... =

+ 60A 2 + 462.2

A3D 2.76

+ Aso +

A03

4.6

5.81

+

(8.20)

A21 6.55

{(0.2865)60 + c(0)(0.175 _ 1.295) -0.825

+ c(I)[15.9c(0)2 -

0.0884C(O)3JI0- 3} _1- (8.21) -3.52

Using relations (8.19). we reduce the preceding two equations into two nonlinear relations in Al and At. The series on the left side of equations (8.20) and (8.21) must be absolutely convergent. For this problem. we know c(O) = 0; therefore e(O) = 60 and h(O) = 40. Also we know that c(-I) = h(-I) = 0, so from equation (8.2) with n = -I we get ('(1) = 28.6. A few trial values of Al and A2 (AI and A2 initially chosen around Iintar solution values) indicate divergence of the series. This indicates that the initial values are far away from the steady-state value (equilibrium point). However. if the time axis is shifted by one integer vlllue (sampling interval). we must use the new values of £'(0) = 28.6 lind ('(1) = 43.6. the latter being calculated from equution (8.2) by using the dilTerencc ctlunti(ln as recurrence relation. It sh(luld be p(linled Clul th,,' since this shiftin~ can always be used in such ,·i\ses. :1 mnn' rapid l'unvl~r~em'e is nhlainahlt'.

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

253

A similar modification does not exist in the application of Laplace transform convolution to nonlinear differential equations. By this time axis shifting, convergence is obtained with the right-hand side of equations (8.20) and (8.21) now equal to -3.26 and 2.5, respectively. Numerical values obtained for the various coefficients are

= -26, At = -1.4, Aa = -1.13, Aao = -0.298, A70 = -0.115, AZ1 = -0.985, A41 = -0.54. ASI = -0.265, Al

An = -0.0239 .. , (8.22)

The z-transform of c(n) for n ~ I can now be written from equation (8.12), using the calculated A's and Q's: 60 -z

~(z) = -,- - - 1

26 1 - 0.6275z- 1

1.13 1 - 0.247z- 1 0.985 1 + O.l131z-1

] -

1.4 1 + 0.2872z- 1

0.298 0.0974z- 1

0.54 1 + 0.0445z- 1

0.0239 1 - 0.0518z- 1 •••

0.115 I - 0.038z

1

0.265 I + 0.0175z- 1 (8.23)

The output c(n) for n ~ I is readily obtained from the inverse ztransform of equation (8.23). c(n) = 60 - 26(0.6275)" -

1.4( -0.2872)" - 1.13(0.247)" - 0.298(0.0974)" - 0.115(0.038)" - 0.985( -0.113 I)" - 0.54(-0.0445)" - 0.265(-0.0175)" n ~ I (8.24) - 0.0239{0.OSI8)". . .

A comparison between the response of equation (8.24) and that obtained by using equation (8.7) as a recurrence relation is tabulated in Table 8. J. Agreement is satisfactory and could be improved if higher terms are considered. This example has demonstrated that the convolution z-transform can be systematically applied to the solution of higher-order discrete systems. Ilowever. the computation becomes exceedingly involved as the order increases. In practical cases for obtaining the response for a few sampling intervals, the difference equution yields the response if used as a recurrence rl'lilticmship. Whcre "'onvcrgence is mpid hy choosing the appropriate initilll vllhlt~s, the l~(lllIltiun fur ,.(,,) l'oJ1silds only of /I fcw tcrms or response IUndcli. This flldlitlltCli the lIynthl~lIill pfell'cdure by ()btllinin~ II lIystcm

254

THEORY AND APPLICATION OF THE Z-TRANSFORM

modification to cancel the troublesome modes in the response. It is from this viewpoint and for obtaining the response for larger n that the convolution z-transform procedure becomes very important. Thus the system designer should possess all the possible methods of solutions so that he can use his judgment as to the method best suited to his problem. TABLE 8.\

COMPARISON OF EXACT AND APPROXIMATE RESPONSES

n

Cont'olution t- Tran~rorm

Recurrellce Relationship

43.88 49.56 53.596 55.96 57.47 58.415

28.6 43.31 49.32 53.43 55.88 57.45 58.43

o I 2

3 4 5

6

8.2 Analysis of discrete antenna array by z-transform method 8 ,It In this section, a discrete linear array will be treated as a sampled-data system and thus the z-transform method is readily applicable. Using this method we may express the array polynomial in a closed form, even for nonuniform arrays. Furthermore, the analysis of the important characteristics of the radiation pattern and other useful characteristics can be conveniently performed using the z-transform theory. This is explained in the following. Consider the linear array of n equally spaced, identical elements in Fig. 8.2. The polynomial associated with such an array can be expressed ,,-I

G(z)

= 00 + 01Z- 1 + oaZ-a + ... + a.._1z-(,,-1I = I

a,.z-1c

k~O

(8.2S)

where 'P =

fJd cos ~ + ex

fJ = 21T - = phase constant ,t

(8.26)

and ex represents the progressive phase lead of the excitation in an clement compared to that with its neighbor on the left.

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

Jdk-

!II

--0--10---0---0------------0 1

2

FIGURE 8.2

3

n-I

~

255

"

A linear array of II equally spaced elements.

The coefficients Qu, Ql' ••• , Qn-l represent the excitation of the elements. For most practical arrays, a progressive phase shift (l exists along the array and the a-coefficients are used. When all the a's are equal, a uniform array results. Suppose that the envelope of the amplitude distribution of the excitation in the n-elcment array can be described by a continuous function fez) within the range 0 ~ % < (n - I)d. Then the a-coefficients in equation (8.25) can be written as

ao =/(0) Q 1 = fed)

an_I

(8.27)

= f[(n - I)d]

nnd the array polynomial of equation (8.25) becomes t."(z)

= 1(0) + f(d)z-l + .. .f[(n -

l)d]z-ln-ll

..-1

=

I

I(kd)z-k

(8.28)

k~O

This can be written also as GO

CJ)

1,'(1.) =

I

f(kd)z-It -

k-,O

I

It-n

f(kd)z-It

(8.29)

E
.~(::)

-

z-'''~9(z)

whl'rc ~q(::)

1111(1 "is thl'

lilllllplill~~

Iwriutl.

(8.30)

= :1rl(1It1 + :r))

256

THEORY AND APPLICATION OF THE Z-TRANSFORM

For most practical arrays the amplitudes of the excitation in the two end elements are equal and thus

I[(n - I)d + x]

==

%[(x)

(S.31)

By using equation (S.31), fI(z) [from (S.30» becomes co

fI(z)

== 3[f(nd + x») == I

I(nd

+ kd)z-k

k-O

co

== I

[/(n - I)d

+ (k + I)d)z-Ic

k-O

co

== :±: I J[(k

+ l)d)Z-k

(S.32)

/e.O

From the shifting theorem, we have

+ d)] == z[~(z) -

3[/(x

(S.33)

/(0)]

where /(0) is the excitation in the first element. Substituting (S.33) in (S.30),we finally obtain 8(z) == [I T

Z-("-II~(z)

±/(O)z-(II-1I

(S.34)

where the plus and minus sign follow from (8.31). Since the z-transform of many useful nonuniform envelope functions/(x) exist in a closed form, &(z) in (S.34) is expressible as a closed function of z instead of a polynomial of n terms (if this problem was treated in the classical way). Note further that increasing the number of elements n in an array does not increase the complexity of the expression for C(z).

ApplicQtions UNIFORM ARRAY OF n ELEMENTS. The amplitude of the excitation can be assumed to be unity with no loss of generality. Then,

1.4.(x)

== u(x)

I.~{(n

- I)d

!F..t(z)

(S.35)

+ x] == I~l(x)

== 3[/....(x)] == I

(S.36)

1

(S.37)

-I

-z

In this case C(z) in (S.34), using the proper sign, yields 1 -(11-11 l - tl ~ (z)

....

== -

1-

LINEAR ARRAY OF BUTION.

+ z-(n-l1 == -

z

n

Z-I

1-

Z

Z-1

(S.3S)

ELEMENTS WITH SINUSOIDAL AMPLITUDE DISTRI-

For the distribution shown in Fig. S.3, /ll(X)

== sin fJx

(8.39)

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

257

"' ....... - - - - - -............. (s(%)

"

/' / /

",

"--~--o--o--o- -------~

OJ

.~

1

2

3

(II -

~

)C

~

d

FIGURE 8.3

linear array with sine amplitude distribution.

In this particular case, (n - l)fJd = lu[(n -

1)

I)d

+ x] ==

7r,

and (8.40)

-IB(x)

The .:-transform oflu(x) is _

•.1-" u(z)

== J[[/I(x» =

z sin

IJti

-2---=-~~­ % -

2% cos fJd + 1

(8.41)

Substituting in equation (8.34) and using the proper sign, we obtain

+ %-11I- 1I JZ sin fJd %2 _ 2z cos (It/ + 1

. [1 Iff,,(z) ==

(8.42)

In practical problems other properties of the urray puttern are required, and among these the array factor thut describes the power pattern of the array is important. The array factor 1«9(;:)1 2 can be obtained from 6(.:) by noting that (8.43) Hence

1~'(z)12 == C(z)t'(Z-l) = C~:QkZ-k) C~:a~)

(8.44)

Using equation (8.34), we can obtain the general formula

16(:)12

=

- /(0):[1

[2 T ::111-11 T z

+

.:111

(II

lIJ.:F(.:).~(.:-l)

IlW(:) + [I T ;;.

(II

lI).F(.:-I)] +.(2(0)

(8.45)

To determine the array fuctor for the eX
1,.,(% I) ==~

(8.46) (8.47)

(HAH,

258

THEORY AND APPLICATION OF THE Z-TRANSFORM

For the linear array of n elements with sinusoidal amplitude distribution with [I + Z-(,,-lI]% sin (Jd t!H(z) = ~.:....--~~~ (8.49) Z? 2z cos {Jd + 1 we get [2 + zhl-lJ + z-, .. -1I] sin? fJd It!II(Z)1 2 = --:--''--~----'---~--'-----:(2 + Z2 + z-2) - 4(z + Z-I) cos fJd + 4 cos2 (1d (8.50) Other properties such as nulls, maxima, and beamwidth can also be obtained from the z-transform approach to this problem.

8.3 Application to information and filtering theory 21-32 In some practical cases, for efficiency and economy, It IS necessary to transmit continuous signals over a sampled-data link. For instance, accurate transmitting of signals by short wave is reliably done in pulsecode modulation. Here the sampling operation is part of the pulse-code modulation. The sampling action causes frequency aliasing and mixing of high-frequency noise into the signal band. This effect can be remedied by using a low-pass filter F(s) before sampling to filter out most of the high-frequency noise. At the receiving end, a wave-shaping filter G(s) (or hold circuit) is used to reconstruct the continuous signal from the sampled signa\. Such a sampled-data transmission link is shown in Fig.8.4. In this section. we shall "indicate the method of optimizing the system in Fig. 8.4 by applying both the Laplace and z-transform theory. We can also use the modified z-transform, but for this case the former procedure is used because of ease of application. n(l)

--+f

ret)

! I

L _____

,..--....,

-----~--\

FIGURE 8.4 Block di;lgmm

L(s)

G(8)

:

dO)

~

r------------JI c'lm

ur IIptimi/alillll ur Nmnpl\."(I.llata u';I/lNmissilln lInk.

APPI.IC'ATIONS TO VARIOUS AREAS OF SYSTEM THEORY

259

Tilt' 0p,imi=alion procedure 27 In this procedure, given the spectral density ¢Jr",(jM), the networks F(s) and G(s) are determined to give the minimum value of (e(/)]2, where the input signal 1'1(1) = 1'(/) + n(l) and the desired output is cI(l). Three cases can be considered; (I) F(s) is fixed and G(s) is to be optimized. (2) the contrary. (3) the general case where F(s) and G(s) are to be optimized simultaneously. For brevity. we shall discuss only the optimization procedure where F(s) is fixed and G(s) is to be optimized. The other cases can be handled in the same fashion. We shall assume that the reader is familiar with the optimization procedure for the continuous and the discrete cases, and therefore many of the detailed derivations and explanations are omitted; references, however. will be made to standard texts on statistical design theory. From Fig. 8.4. the mean-square error is given as

[(>(,)]2

= _I. 27T}

1

i «. -if¥;

er(.s) ds

(8.51)

where the spectral density rc(s) is obtained as follows: (8.510) In terms of z-transform the mean-square error at the sampling instants is obtained [from equation (4.92)] as follows:

.!.

[e(nT)]2 = _I

27T}

1:11'1

,,(Z)Z-I dz

(8.5Ib)

('j", ...

We can also obtain the mean-square error at all times by using the modified z-transform. However, as mentioned earlier, this procedure for this example will not be used. but instead equation (8.51) will be utilized. The various spectral densities of equation (8.510) can be obtained from Fig. 8.4 by following certain statistical definitions and concepts, which are27 (8.52) (8.520)

(8.53)

260

THEORY AND APPLICATION OF THE z-TRANSFORM

In equations (8.52 through 8.530) the following symbols are used:

L = L( -s), etc. (FFct>rlr)*

(8.54)

= Z(FF<1>Tlr)lz=eT.

(8.540)

Substitutil1g equations (8.52) and (8.54) in equation (8.510), we obtain

<1>,. = LL<1>TT -

!

T

FGLcP rTI -

J. FGLrlT + 1. (FFr.,)*CG T

T

(8.54b)

If we let F be fixed, we can obtain G such that the mean square error is minimized. To find this optimum G, we can use a theorem from calculus of variations which states the following. The function G must be such that the partial derivative of " with respect to G is analytic in the left half of the s-plane (LHP). Hence we obtain from equation (8.54b) (8.54c)

where Xl is an unknown function which is analytic in the LHP. From equation (S.54c) and the requirement on Xl' we notice that the partial fraction expansion of a,.• joG corresponding to poles in the LHP is identically zero. that is,

[-FLT1T

+ (FFrlrl)*G]+ =

0

(8.55)

Because of symmetry, we can write

(FF<1\T,)*

= (FFcJ>rlr)*-(FFcJ>rlr)*+

(8.550)

where ( )*+ is analytic in the RHP (or outside the unit circle). Since G is analytic in the RHP (due to physical realizability), we can obtain a function Q defined by (8.55b)

which is also analytic in the RHP. Using equations (8.550) and (8.55b) in (8.55), we have

[-FLct>r 1r + (FFcJ> .. I rI )*-Q]+ = 0 This equation requires that

(8.5Se)

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

261

Utilizing this equation, we can write from equation (8.53a) ce =

.! (FF<1>'lrl)HG{ FF<1>"r,}*-a = .! QQ T

(8.56a)

T

To obtain a relationship for the mean square error, we have from equations (8.53), (8.53a), and (8.54c)

~ (ia) (cci J-ia)

_ cc) ds =

21TJ

~ (ia)

!

L,» T

27rJ

x [FGL<1>r,r - (FF<1>rIT.)*GG] ds

= _1 (iCO 21Tj

J-ia)

(-G 8<1>,.) iJG

ds

~ (ia) GXlds

= _

21TJ

(8.56b)

L,co

Since both G and Xl are analytic in the LHP, this integral vanishes. Furthermore, changing s into (-s) in equation (8.56b) we obtain the following expression:

_I. (i"" (<1>C1C _ cc) ds = 0 21TJ J-i

(8.56c)

From equations (8.51) and (8.51a) we have

[e(t)J2

=~ 21TJ

=

[iQ)•• ds

Lia)

~ [iCO 21TJ

Li""

(elClc - <1> cc l

+ .c) ds

(8.57)

This integral can also be written as

-

~ L;:;o (ioo ( eel

21TJ

-

cc) ds

(8.57a)

Noting equations (8.56a) and (8.56c), we finally obtain [e(/)]2

=

_1 21Tj

ji<'- (<1> j'

_ QQ) ds rlrt

T

(8.57b)

1I.1.IIS ntA'IIVE EXAMI'L);

Thi" l~xall1plc illustrate)' a prohlclll of optimulll filtering and prediction in thc prl'loCl1l'C "I' IlIll'orrl'latl'd l1oisc.

262

THEORY AND APPLICATION OF THE Z-TRANSFORM

Referring to Fig. 8.4, assume that e1(t) = r(t

+ IX)

(8.s8a)

L(s) = e"' Cl>r.r(s) =

(8.58b)

2

20

(S.sSe)

2

a - s 2a Cl>r,rl(s) = II II a -s where b2 = 2a/a2 + 0 2•

+

a'l.

u2(b2 - Sll) 2 'I. a -s

=

(8.sSd)

b-a

(8.sSe) F(s) = - s+b We shall determine the optimum G(s) and mean square value of the error as well as the error due to sampling. From equation (S.56) and the expressions (S.sSa-e), we have

Q(s) = [(b - a)/( -s + b)e"'(2a)/(a [ (b - a)2a2]*a2

l - S2)] +

S2

_

(S.s9)

From tables of z-transforms (see Table I) we obtain

]* ~ Z[(b -

S [ (b 2- a)la a

_

a2

S2

= a 2(b _

a)2a'] _

S2

z(e- aT _ e+IIT )

0)2

2a

22 -

(e"T

I

+ e-aT)z + 1 z=eT.

This equation can also be written in the following form:

(8.s9a)

(8.59b)

Substituting the last bracketed term of equation (S.s9b) into the expression for Q(s) in equation (S.s9), we get Q(s)

= [ ==

2ae'" ,J2a

a(s - b)(s - ales + a)

J2a(1 - e ,r(CI

2<.7')" 1111 •. ....... . /J)(.~ t')

+

+

1-

e-aTeTI]

JJ -

Ir'laT

I.

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

263

O(s)

FW

ret)

+ n(t) ..

I~

T

I

~ ~1-e-aTz-1H 7ii

Prefilter

FIGURE 8.S

Digital filter

,

I

Shaping filter

Block diagram for the optimized sampled-data transmission link.

From equation (8.55b) we finally obtain the optimum G(s) as follows: Gopt(s)

= a (b 2aI

II

2'

e-tl.a

(1 - e-a e-2")-a) S +a B

(8.60)

but 2a/a2 = b'l. - all; hence -tJa

GuPI(S)

= (1 _ e-tl.2'z-l>!.•••". _e__ s+a

(8.60a)

It is noticed from the preceding that G(s) is composed of a digital filter and a shaping filter_ The block diagram of the optimized system is shown in Fig. 8.5. The mean-square error is given by equation (8.57b), which gives --

[e(I)]2 = 1 _

(b

)(1 -taT) -'l.tl.R - a - e e 2aT(b a)

+

(S.60b)

As T approaches zero, equation (8.60b) gives the result of the Wiener filter. that is.

[ ]21 T.,O = I - b -a e-'I.tI. e(l) b+a

(8.60c)

The error due to sampling is given by [e(1)]2 _

[e(I)]2IT~O =

(1 _ 1-;:;'l.tl.2')(b ; :);-2za

(8.61)

8.4 z-Transform method applied to problems of economics National income, consumption, and investment are the three basic fundamentals of any economy_ These can be expressed at any given time, t = nT, by mcans oCa difference equation and certain constants, Solution uf Ihese C(IU:llions is Ihen found by the ::-trunsCorm method, Three models of cl'unulIlil' systems havc hecn considercd and they involve first-. secondand fuurth,"nler dint~n'l1l'" '~'IUUliClllli with l'nnstunl l'ucflicicnts, ('on· wr.~'·IU:l· nf 111,· snlulinn IIlId slllhility IIrt~ nlsn disl'usscd in Ihe fllllnwin~,

264 THEORY AND APPLICATION OF THE z-TRANSFORM Model I

Let y, e, i denote the national income, consumption, and investment, respectively. Considering that time is divided in periods of equal length, say years, we define that at any time t = nT (where n can have integer values 0, 1,2, ...) y, e, i have respective values Yn' en' in. Since national income is made up of investment and consumption, we can write (8.62) Now we assume that consumption depends linearly on national income but has a constant value even when there is no national income. This is quite logical; therefore we can write n

= 0, ],2, ...

(8.63)

where K ~ 0 and 0 < m < I. We must also consider the effect of investment on national income which can be expressed in terms of what the economists call the growth factor r. We can assume the following relationship:

Yn+1 - Yn

where r

= rin

(8.64)

> O.

I{clations (8.62), (8.63), and (8.64) enable us to determine the desired clill'crence equation. Eliminating in and en, we get (8.65) 01'

Yn'

J

+ (mr -

r - J)Yn

+ rK =

n = 0, 1',2, . . .

0

(8.66)

Suilltioll of this difference equation can be easily found by taking the ,·-tmnsform

,Y[!ln ,d

+ (mr -

r - I )J[Yn]

+ J[rK] = 0

(8.67)

Ill'

z[!V(z) - 110]

+ (mr -

r -

I)lt?/(z)

rKz

+ -- = z-l

(8.68)

0

whit'll Aives ,!I/(::)(o:

+ mr -

r - I)

rKz = zYo - -

(8.69)

%-1

Ill'

:I/(Z) :-:

• Z -

- - -

Z?J.u

(I

+r

..

-

",r)

-- .

rK:::------

-.,,,,~.

(::: - I)f:: - (I

+ r _..

",rll (H.7())

265

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

Now I

+ r(l

- m)

> O.

~(z) = Z1Io

So let I

+

z- a

=

Z1l0 -

+ r(l

- m) = a.

rKz (z - 1)(z - a)

rKz/(a - 1) z- a

-

rKz

(8.71)

(1 - a)(z - 1)

Taking the inverse z-transform of equation (8.71), we get

rK )" 11,,= ( 110--a - -rKa-I I-a

(8.72)

or

11" = 110[1

+ r(J

K

- m»)" - - - ([I I-m

+ r(1

- m»)" - I} (8.73)

Rewriting,

11" = [1

+ r(l - m»)"(1IO - ~) +~ I-m I-m

(8.74)

This gives the national income for any n = 0, I, 2, ...•

Model/l In this modeJ, we define at any time t = n (n = 0, I, 2, ...),

11" = income received Sn = total savings ;n = desired investment We make the following assumptions: I. Savings during any period is proportional to the income anticipated during the next period which can be mathematically expressed as (8.75) where K is the proportionality constant 2. Desired investment during any period is equal to a constant multiple X of the increase of the income of that period over the income of the preceding period. This can be expressed mathematically ;,,11

J.

Savin~s

= g(tl" I 1

-

1',1)

(8.76)

in any perind is cllLIll1 to desired investment (M.77)

266

THEORY AND APPLICATION OF THE z-TRANSFORM

Equations (8.75), (8.76), and (8.77) can be combined to yield the following difference equation:

g g K 1In+l + K1In = 0,

1In+1I -

n~O

(8.78)

Taking the z-transform of this equation, we obtain

ll9'(z) =

- (gl K)zyo + 1.111 1.'1 - (g/K)z + glK

1.1110

(8.79)

Using the inverse z-transform integral or tables, we obtain

1In

J(~J- ~r+1 - ~~ -J(2~J- !T+1}/

= 1Io{G~ +

J(g/K)2 - 4(g/K)

r

! yo){[ii + J(gI2K)2 - glK - [2~ - J(g/2K)1I - glKT}/J(gIK)1 - 4(gIK),

+ (Yl -

n ~ 0 (8.80) For stability or convergence we obtain from the denominator of equation (8.79) '-(1.) the following conditions: 1.

'-(1) > 0, or

2.

'-(-1) > 0, or

3.

'-(0)

< I,

or

1>0

(8.81)

1+ 2g >0

(8.82)

g<

(8.83)

K K

Mode/Ill In this model we obtain, as shown by Tinbergen,13 the numerical equation for the analysis of economic fluctuations. The numerical equation which is determined from statistical consideration is a fourth-order difference equation, given asla YnH

+ DYn+a + byn+l + CUn+! + dun =

where

a = -0.398

b

= 0.220

c

(8.84)

0

= -0.013 d = -0.027 (8.85)

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

267

The characteristic equation for (8.84) which can be obtained by taking the z-transform can be expressed as F(z) = Z4 - 0.398z3

+ O.220Z2 -

O.Ol3z - 0.027 = 0

(8.86)

To determine stability for the system of Model III, we require for the fourth-order case that is, I - 0.398 + 0.22 - 0.013 - 0.027> 0 F( -I) > 0, that is, I + 0.398 + 0.220+ 0.013 - 0.027 > 0 (8.87) which are both satisfied. I.

F(l) > 0,

2.

(0.027)2 - 12

<-

10.027 X 0.398 + 0.013 x II, that is, -0.999 < - 10.023751 (8.88)

which is satisfied. 3. (-0.027)3 + 2( -0.027)(0.220) + (-0.013)(-0.398) - (-0.027) - (0.220) - (-0.027)(-0.398)2 - (-0.027)2 - (-0.027)2(0.220) - (-0.013)2 + I + (-0.027)(-0.013)(0.270) > 0 (8.89) which is also satisfied. Therefore the system as described by equation (8.84) is stable.

8.S Linear sequential circuits14-17.37 In this section the z-transform method is used to analyze the behavior of linear sequential circuits. Such devices find application, for example. in coding theory, radar detection systems. and computer theory. The underlying mathematical field for analysis is not the field of real or complex numbers; thus the z-transform theory is carefully applied in the following. This discussion is limited, for the most part, to the autonomous behavior of such circuits. Sequential circuits are devices whose inputs are sequences of numbers belonging to a finite field. When they are linear and have a single input and output. the sequence x(n), its input, and yen) the output, may be related by the following difference equation: yen + k) + (J.11f(n + k - I) + ... + (J.oJ/(n) = /10:.'(" + k) + (1.:r(I/ + k - I) + ... + /f...r(,,)

(8.90)

In clillatioll (H.II0) IIIl1ltiplil'ation and additiull arc 1II(1(Iulo,I' alld the Ill' rt" {I,. !I. nlld .•. 1'I\1I'~l' OWl' tl1I' illh'I~"I'~ n. 1,2 •. l ..... " I.

vah,,'!.

268

THEORY AND APPLICATION OF THE z-:TRANSFORM

where p is a prime number. The set of numbers 0. 1.2•...• p - I with modulo-p arithmetic operations is called a modular field. The following shifting property of the z-transform remains valid when arithmetic operations are carried out modulo-p, that is, .1[x(n

+ k)] =

Tc-l

zt,i"(z) - zt Ix(l)z-&

(8.91)

1-0

Using this equation in obtaining the z-transform of equation (8.90). we get (z'" + «IZ"'-1 + ... + «... ]W(z)

= fPoZ" + (J1Z"-1 + ... + fJ. . ]!l'(z) + /(z)

(8.92)

where I(z) is a polynomial in z resulting from the initial conditions. Its exact form is not important for the following discussions. To calculate yen), we solve for '?V(z) and then calculate its inverse. The usual inversion integrals or tables are not directly applicable in this case because the modular field is not the field of real or complex numbers. Thus it is more convenient to resort to the power series method of finding the inverse z-transform. * The autonomous behavior of the circuit is determined by letting x = 0 in equation (8.90). Thus from equation (8.90) with % = O. it is easy to notice that any k consecutive value of y(n). say (Yo, YI. Y2.··" Yk-I), completely determines Y for all subsequent n. We may observe that the number of distinct k-tuples (Yo, Yl •.•.• Yk-I) is p". Consider the sequence of output symbols Yo. Yl' Y2' . .. Since the number of distinct k-tuples is finite. there exist indices m and m + L such that (y m' Y mU ••••• Y m+k-l) is identical to (y m+1.' Y." ~ L+I' ••• , Y rn U.+k-l)' Since k consecutive values of y determine all subsequent values of y, the output following Y m+L+k-l is identical to that following Y m+.l:-l; hence the output sequence is periodic after Y m' The smallest integer L is called the period of the sequence. The exact period of the sequence may be determined from equation (8.92) with ~(z) set equal to zero.

11/(z) _

I(z)

I(z)

- z. . + «IZ"'-1 + «aZIt-B + . . . + «" =-P(z)

(8.93)

Three cases of P(z) may be distinguished: (I) P(z) is irreducible, (2) P(z) factors into distinct irreducible polynomials, and (3) PC;:) factors but not into distinct polynomials. Each of these cases is discussed in the following. • However, with due care in interpretation, the inver~i()n integrals and tahles be used. Thus a solution in clo~d form cnn he.found.

II1I1Y niNO

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

269

P(z) Irreducible A polynomial with coefficients belonging to a field is said to be irreducible if it is not factorable into polynomials of smaller degree with coefficients belonging to the same field. In this case the coefficients of P(z) belong to a modular field of characteristic p. It is well known that an irreducible polynomial with coefficients in a modular field is a factor of a polynomial zL - I for some L that must be a divisor of pk - I, k being the degree of the polynomial. Using this fact, ZL -

1 = P(z)R(z)

(8.94)

where R(z) is a polynomial of degree L - k in z. The smallest L for which equation (8.94) is satisfied is called the period of P(z). Solving for P(z) in (8.94) and substituting in (8.93), we obtain ·!p(z)

= z-L /(z)R(z) = z-LI(z)R(z)(J + z-L + Z-2/. + ... ) l-z- L

(8.95)

From (8.95) it is clear that y(k) = y(k + L) for all k, therefore the period of yen) is at most L. (It can be shown that the period must be exactly L.) When pl- - I is prime, L = pk - I. Then, the circuit is said to have maximal period, for pI.; - I is the largest value L may assume. Except for the all-zero k-tuple, every possible k-tuple appears once and only once during each period.

PC:;;) factors into nonrepeated faclors Suppose P(z) factors into distinct nonrepeated factors

Then the period of yen) is a divisor of the least common multiple of the periods of p.(::), P2(z), ... , P,C::). Tn sec this, make a partial fraction expansion of equation (8.93), :1/(::) = 9'(z) 1'.(::)

+ qz(z) + ... + qlz) 1'2(z)

P,Cz)

(8.96)

1::II:h It'rlll "'(")/1',('), i .. 1,2, J" . , ,I in ~'(Iu:lti(ln (R.9(,), corresponds a Sl'llUt'm'(' wilh p,'riml Ihal is a divisor Ill' Illt~ pl'rilld "I' 1',(;:). The !'t'riml Ill" !/(,,), whlt'h i~ I hl' !lIIIII (If' I ht' !i("IIJ('lIl'l'!I ""rr('s!,lllldin~ tn III

270

THEORY AND APPLICATION OF THE z-TRANSFORM

q;(z)/P,(z), as a consequence, is a divisor of the least common multiple of the periods of P 1(z), Pa(z) • .••• P,(z). The maximum period of yen) if P(z) factors is always less than pk - I. This is so because the period L of yen) (supposing the degree of P;(z) is k i )

satisfies the inequality L

~

(pka _ I)(p'" _ I) ... (Pkl _ I)
= pk _ 1 (8.97) P(z)

Has repeated factors

To compute the period of yen) for this case. the following lemma which is proved by Elspas14 is needed. If [q(z)J is irreducible. [q(z)]m has period kpTm where k is the period of q(z) and r m is the smallest integer satisfying pTm ~ m. Now, suppose

LEMMA:

(8.98) Thus. just as for the nonrepeated factor case. the period of yen) is a divisor of the least common multiple of the periods of each factor pNz). In the preceding. it has been tacitly assumed that in equation (8.39) /(z) and P(z) have no common factors. This is always so when P(z) is irreducible. In the last two cases discussed. only upper bounds on the period were established. The reason more precise statements. in general. are not possible is that the periods depend on I(z). that is. the initial conditions.

8.6 Application to discrete Markov processes18 - 21 The z-transform's application in probability theory is well known and often used. This method is referred to as the generating function method and was used by De Moivre and Laplace as early as the nineteenth century. In the foHowing discussion, the z-transform method is used in the solution of the vector difference equation arising in discrete Markov process. Certain features of the process are discussed using this method. We shaH consider the problems of finite-state. discrete Markov process and restrict the discussion to an irreducible. regular Markov process in which there are neither transient states nor absorbing states and the state transition of all persistent states is aperiodic.. We define a set of states

S={1.2.3, ...• m}

(8.99)

APPLICATIONS TO VARIOUS AREAS OF SYSTEM TBEORY

271

Also, we define a column vector Pen) and a square matrix p(u, I) such that PIC,,)

P3(1l) P(n) =

(8.100)

P",(n)

(8.101)

p(u, n) = [... P'J(u, II) •.. ],

where P,(n) is the absolute probability of state i at time nand Pij(u, II) is the trunsilion probability from a state i at time II to a state j at timc n. (Here u and n are taken to be intcgral numbers.) They have the following properties:

'" !Pi/(u, II)

=

for all ; s and for II,

I,

1=1

" integers with

II

~ :t ~

0

p;;(u, n) ~ 0

(8.102) (8.103)

and (8.104) Using the matrices of equations (8.100) and (8.101), we can write pen)

= p'(u, n)P(u)

(8.105)

where the prime indicates the transpose of a matrix. If the process is time-homogeneous, the stochastic matrix p(u, n) is time-invariant and is given by p'(II, n)

= p'(n -

(8.106)

u)

Since the process is a Markov one, the Chapman-Kolomogorov equation must be satisfied, that is, p'(u, n) = p'(u, u ..... T)p'(U

This clJuation for

II

kllnwll·t1'~I·

1,'(11)

n~

T

+U~ u

(8.107)

time-homogeneous process reduces to

p'(,,) = p'(r)p'(n - r)

wilh Iht'

+ T, n), n

~

r

~

0,'

T

= integcr

(S.I08)

Ihal "kill/IV Illall'i\

(H. lUI))

272

THEORY AND APPLICATION OF THE z-TRANSFORM

(fwe define p(l) = p, by using equation (8.108) as it recurrence relation we have or ,'(n) = p'" (8.110) pen) = p", Therefore, from equations (8.105) and (8.106), we obtain

pen

+ '7') = P'(T)P(n) = p"P(n)

(8.111)

In particular. letting T = 1 in equation (8.111), we obtain the following first-order vector difference equation: Pen

+ 1) =

n = 0, 1,2, ...

p'P(n).

(8.112)

Application of the z-Iransjorm We shall apply the z-transform to the vector difference equation in (8.112) ns follows (with the understanding that the z-transforms of vectors and Il1ntriccs are obtained by taking the transform of each component in the army).

1[P(n)] =

D'(z) ~

."

IP(n)z-"

(8.113)

,,=0

By the time-translation theorem,

,},[P(n

+ 1)] =

z[&'(z) - P(O)]

(8.114)

Tht'rl'lilrc, substituting equations (8.113) and (8.114) into equation (8.112), (:I - p')D'(z) = zP(O) D'(z)

= z(d -

p')-lP(O) £ ~(z)P(O)

(8.115)

whl'rc ~(.:) is the fundamental matrix of the stochastic matrix p' such that ~(::)

=

z(d _ p')-1

(8.116)

I.nter we shall show the relation between ~(z) and pen). We assume here that there is no multiple pole. Then by the partial fraL'linll cxp~tnsion of equation (8.116), we have ~(::)

II: =I"I --G",

k I

z - Ale

whcre the il,.'s lIrc the eigenvulucs ofp', that is the roots l'tillal inn 1}.1 - p'l ,,= 0 Then

Wl'

haw Ihl'

filllnwinl.~

thcnrclII.

(8.117)

or the characteristic (8.11 8)

APPLICATIONS TO VARIOUS AREAS Of SYSTEM THEORY

273

I: For any regular stochastic matrix p of a finite-state, discretelime Markov process

THEOREM

(i) There exists an eigenvalue of p', which is equal to one. (ii) No eigenvalue of p' exists outside of the unit circle in the complex z-plane. Proof: Let us consider a determinant Pll - I

P21

PfI,l

PI~

P22 - I

P,"2

Plm

P2m

Ip' -II =

If we add all the rows from 2 to first row become zero because

...

Pmm - I

m to the first row, all the elements of the

711

IPii - I = 0,

Vi

E

S

i-I

Therefore we have

Ip' - II = 0

(8.119)

Comparing this with equation (8.118), we see that there exists an eigenvalue which is equal to one, which thus proves (i). The proof of (ii) is shown as follows. Let X be an eigenvector of p and M be the absolute value of the component of X of greatest magnitude. Then from the relation

AX =pX we have 1ft

l..ll M

~ M.

Therefore we get

I;=1 Pii =

(8.120)

M

(8.121) Thus, no cigcnvulue of equation (8.118) exists outside the unit circle. In fact, the term in equation (8.117) corresponding to .ill = I is the stationary solution of the Murkov process. We can verify it in many ways. II" 111l~ prm:l'!oos i!oo slalillllary, VII;

illh',.,t'J"

274

THEORY AND APPLICATION OF THE Z-TRANSfORM

and equation (8.112) can be rewritten as

(I - p')P(n) = 0

(8.122)

where P(n) is no longer dependent on n. From equation (8.119). we can say that there always exists a nontrivial solution for P(n) corresponding to the eigenvalue .ill = I.

Fundamental matrix We will now study more about the fundamental matrix defined by equation (8.116). If we assume that .ilk·s are the simple roots of the characteristic equation, W(z) from equation (8.117) is expanded by

z

m

<9(z) =

I -- G

k~l

.ilk

Z -

k

where G k is the residue of <9(z) and is

Gk = lim (z - Ak)(zI _ p,)-l

(8.123)

z-At:

We further assume that for all k = 2, 3, ... , m

(8.124)

From the final value theorem of the z-transform G(n)I .. _a:

= lim

.-1

(z - l) <9(z)

(8.125) We see here again that the first term G 1 corresponding to A1 = I is the limiting fundamental matrix of the stationary Markov process. Since 1.il,,1 < 1 for k > I, the state transition is always stable and the terms other than the first converge to zero as n increases. This is quite natural because Pen) itself is bounded by one. By the inverse z-transform of W(z), we can find the fundamental matrix in time domain

G(n) = r1[~(z)]

= _1_ f ~(Z)Z"-l dz 27Tj 11' =1 27Tj

f

m z" I--Gkclz -,t"

I'k I Z

(!U26)

APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY

275

where r is taken to be a counterclockwise closed contour on a unit circle in the complex z-plane. This is given by the sum of residues m

G(n)

= l:'~:Gk

(8.127)

kal

Since IAkl < I for all k(2, j, ... , m), the terms from 2 to m converge to zero as n - .. 00. When Ak is a complex root, the kth term is an oscillatory attenuating term, that is, denoting

Ak =

ak

+ jbl< = e-I«r.-iUlk).

the attenuating constant

OCI<

(XI<

= - ~ In (a:

(I)"

=

and the frequency

+ b:) =

tan -1 (01<) b,.

(8.128) (I),.

are given by

-In IA"I

= ang Ak 1

(8.129)

Since A.k is in the unit circle, (Xk should always be positive. By the inverse z-transform of equation (8.115), we have

= G(n)P(O)

Pen)

(8.130)

In general, Pen

+ T) =

G(T)P(n)

Comparing this with equation (8.111), we note that G(n) = p'(n) ~(z)

= J[p'(n)]

(8.131) (8.132)

The initial value of G(n) is G(II)ln=o

= lim ~(z) = I

(8.133)

,"co

Furthermore, from equation (8.125), G1 = lim G(n) = lim p'"

(8.134)

Therefore G 1 is the limit of the nth power of the transpose of the transition matrix. REI'ERI:NCI:S

I. Jury. L I.. SIII"I,/cIC/.I)CIICI Clltum/,C,)·.rlt'",." John Wiley and Sons, New York. 1958. 2. Tsypkin. Y. Z. 17,c'(,ry IIf' I'ul.w' S,'s/('/II.\·, .·hysicn·M:llhc,muiclIl '.ilcrlIlurc. Muscow, I'J'iK. I. .Iu,.y, I. I., "C '11111111mlilln 141 II,,· Mmlilh',I:·TI'I1I1,1'1I1'111 Ml·thllll," .I. ,.hlt/H,)' III.~/., VIII. UII, Nil. J, AII/:II',I 1%11,1'1" II"J I.t'J.

276

THEORY AND APPLICATION OF THE z-TRANSFORM

4. Jury, E. I., and M. A. Pai, "Convolution z-Transform Method Applied to Certain Nonlinear Discrete Systems," IRE, PGAC, Vol. AC-7, No. I, January 1962. 5. Pai, M. A., "The Analysis of Nonlinear Feedback Sampled-Data Systems," Ph.D. Thesis, September 1961, University of California, Berkeley. 6. Montel, P., LefonS SlIr les Recurrences et lellr Applications, Gauthier-Villars, Paris, 1957. 7. Schelkunoff, S. A., "A Mathematical Theory .of Linear Arrays," Bell System Tech. J., Vol. 22, January 1943, pp. 80-107. 8. Cheng, D. K., and M. T. Ma, "A New Mathematical Approach for Linear Array Analysis," IRE Trans. Anlennas and Propagation, Vol. AP-8, No.3, 1960. 9. Cheng, D. K., and M. T. Ma, "A Critical Study of Linear Arrays with Equal Side Lobes," IRE llIternatioflol COllt'l.'IItion Record. Part I. 1961, pp. 110-122. 10. Stone, W. M., "A List of Generalized Laplace Transforms," Iowa Journal of Science, Vol. 22, April 1948, pp. 215-225.

SIOI('

College

11. Goldberg, S.,llItrodllctioll to Difference Equalions, John Wiley and Sons, New York, 1958. 12. Samuelson, P. A., "Conditions that Roots of a Polynomial Be Less than Unity in Absolute Value." Allnals of Math. Slat., Vol. 12. 1\141, p. 360. 13. Tinbergen, J., Business Cycles in the United States, 1919-1932, League of Nations, 1939, pp. 126-150 (Economics Intelligence Service, Gcneva, Switzerland).

14. Elspas, B., "The Theory of Autonomous Linear Sequential Nctworks," IRE TrailS. Cirmit Theory, Vol. CT-6, No. I, March 1959, pp. 45-60. 15. Friedland. B., "Lincar Modular Sequential Circuits," Ibid.• pp. 61-69. 16. Harlmanis, J., "Linear Multivalued Sequential Coding Nctworks," ibid., pp. 69-74. 17. Huffman, D. A., "The Synthesis of Linear Sequcntial Coding Networks, Proc('ed· of the Symposium Oil b!formation Theory, London, England, 1955, pp. 77-95.

i'\~s

18. Feller, W., All Introdllctio" to Probability Theory and its Applications, John Wiley and Sons, New York, Vol. I, 1960. 19. Sillier, P. W., "Systcm Analysis of Discrete Markov Process," IRE PGCT, Vol. CT-3, No.4, December 1956, pp. 237-266. 20. Howard, R. A., Dynamic Programming and Markov Processes, MIT Press and John Wiley and Sons, New York, 1960. 21. Bellman. R., Introduction to Matrix Analysis, McGraw-Hili Book Co., New York, 1960. 22. Van der Waerdon, B. L, Modern Algebra, Vol. I, F. Ungar Publishing Co., New York,1945. 23. Mcschkowski, H., Dijji.·rellzengleicllullgen. Gotlingcn, Vandcnhoeck and Ruprecht, 1959. 24. Murphy, G. J., and J. F. Egan, "The Analysis of Demodulating Compcnsating Networks," IRE Trans. Automatic Control, Vol. AC-4, Dccember 1\159. pp. 71-79. 25. Brule, John D .• "Polynomial Extrapolations of Sampled-Data with lin Analog Computer," IRE Trans. AUIO/llalic Comrol, Vol. AC-7, January 1962, pp. 76 77. 26. Cattermolc, K. W., "Enlcicncy and Recipf"llcity in l'ube-Alllplitmlc Mudulati,ltI," Proc. lEE, Part a, Septcmher 19511, PI" 449 4(.2.

APPLICATIONS TO VARIOUS AR.EAS Of SYSTEM THEORY

277

27. Chang, S. S. L.. Synf"('.~is of Oplilllllm COlllrol S.I'SIt!ms, McGraw-Hili Book Co.' New York, 1961. Ch. 6. 28. Jury. E. I., "Optimization Procedure for Sampled-Data and Digital Control Systems," 5c'it!nfiu Ekclrico, Zurich, Switzerland, Vol. VII, Part I, 1961, pp. 16-26. 29. De Russo, P. M., "Optimum Linear Filtering of Signals Prior to Noise." AlEE Trolls., Part II. Vol. 79, 1960, pp. 549-555. 30. TheJlier. P. L., "Optimization et Auto-Optimisation des systemes de Commande it Donnees [chantillonnces, en Presence de Saturations," Doctor of Science Thesis, 1962. Facuhc des Sciences de L'universitc de Grenoble, Paris Impremiere Nationale, 1962. 31. Nishimura. T., "On the Modified ,,-Transform of Powcr Spectral Densities," IRE TrailS. AIIICJfI1atic C()IIlrel. July 1962, pp. 55-56. 32. Nishimura, T. and E. I. Jury, "Contribution to Statistical Design of Sampled-Data Control Systems," Institute of Engineering Research Report, Series 60, No. 210, July 1958, University of California. Berkeley. 33. Budnicki. Zdzislaw, "Input Impedance and Transfer Function of Ladder Network," U:F.f:, PGCT, Vol. CT-IO. No.2. June 1963, pp. 286-287. 34. Urkowitl. H .• "Analysis and Synthesis of Delay Line Periodic Filters," IRE. PGCT, No.2. Vol. CT-4, June 1957, pp. 41-53. 35. Rohinson, E. A., "External Properties of the Wold DecompOSition," J. Math. AII(I~\'sis tlIIII Appliw(iolls, Vol. 6. February 1963, pp. 75-85. 36. Murphy G. J .. "The Analysis and Design of Production and Inventory Control Systems". Proc. Ntuiollal Elec(ronic.l· Conferellc('. Vol. 19. 1963. 37. Iwens. R. P.. "Application of the :-Transform to Sequential Circuits," Master of Science Pr(lject. Dept. of Electrical Engineering. University of California. Berkeley, JHnuary 1964. 3!!. Higgins, T . .I. and J. K. Shah, "Basic Theory. and Application in Circuit Analysis of the Finite Alternate z-Transform," IEEE Conference paper No. CP 63-1467, Presented at Chicago. Illinois, October 28-30, 1963. 39. Higgins, T. J. and Robert E. Oslerlei, "General Theory of Finite z-Transforms and Applications to Analysis of Discrete Systems," IEEE Conference paper No. CP 63- 970, Presented al Toronlo. Ont., Canada, June 16-21, 1963. 40. Christiansen, Peter L., "On lhe Closed Form of the Array Faclor for Linear Arrays," 1£££ T/'{/lls. Oil AIlIt!t1/1QS olld Propagalioll, Vol. AP-J J, No.2, p. 198, March 1963.

APPENDIX TABLE I. z-TRANSFORM PAIRS

1.-Transform .F(z) = 1[f(n)), 11.1

Number

Discrete Time-Function f(n), n ~ 0 un ()

2

e-o.n

3

n

4

n2

5

nS

6

n'

= If(n)1.-n n=O

n ~0 =0,Ifor otherwise

1. 1. - 1 1. z - e-e& z (z - 1)2

+ 1)

1.(1.

(z - 1)3

1.(1.2

+ 4z + 1)

(z - I)'

%(zS

+ 11%2 + liz + 1) (z - 1)5

%(%4

7

"'

8

""*

9

>R

GO

+ 261.3 + 661.2 + 26% + I) (z - 1)8

=z~

<_I)kD'«_1._); D z - I

dz

u(n-k) z - 1

10

F(e"1.)

z 2 "'"(1.-----,1""")3

,,(2)

= lI(n

,,(3)

= nO -

13

nCk )

= n(n

14

n(kr(n),I1(k)

15

(-l)kll(n - 1)(n - 2) ... (n - k

16

11

- 1)

z 3! (z _ 1)4

1)(n - 2)

- 1)(11 - 2) ... (11 - k

1.

+ 1)

= n(n + l)(n + 2) ••. (n + k

k! (1. _ l)k+1

- 1)

+ lif"-k+lt

dk

( _1)"zk d1. k [.F(~)]

- (II - lifn-l

.F(l)(t)

17

(-I)k(n - l)(n - 2) ... (n - k)fn-k

.FCkl(t)

18

nf(n)

19

n"l(n)

~o

nSl(n)

-z.F(1)(z) .

+

. z2'{lr)(~) z.Fci)(~) ."" '..:. ~ly(3)(z) ":";-3z 2,Fi2)(z) _ z.FU)(z)

n!

.,.,

(In c)"

n! • Table IV represents entries for k up to 10. + It may be noted that In is the same as

= d1.d k .F(z) k

%.Fc'")(z), 3OCI:I(1.)

(n)

TABLE I (Continued)

z-Trans[orm §(z) = J[[(n»), Izi

Number

Discrete Time-Function fen), n > 0

23

('

24 25

26

(~)

n)ena k - n,

=

t-.)

00

>

0

R

'~"'

<Xl

:t

1 [(n)z-n

=

n=O

k! (k - n)! n!'

n~k

">

-<

+ el

(a z

:z:

zk

(n : k)cn

0

zk+l

> "Cj

(z - c)k+1

(')

"Cj

C

>

cn

(n

I n. '

= 1,3.5.7, ...)

::!

sinh (;)

0

'cosh (;)

"" :t '"'

:z: 0

en (n

I n. '

= 0,2,4,6, .•• )

III

27

sin (IXn)

%sin oc z2 - 2% cos ex

+1

~ >

28

cos (exn)

z(z - cos ex) z2 - 2z cos ex + 1

'"~

sin (IXn

30

cosh (exn)

31

sinh (a.n)

36

37

sin 'Xn n cos exn

2z cosh a.

+1

z2 -

z sinh a. 2z cosh a.

+1

(II

+

l)(n

+ 2) ... (II + k

ex

+ tan- 1

«>0

sin x , z - cos 0(

(1 - -1)-'-,

- 1)

k

%

" I !-

0(>0

z

z

%-1

z-1

= 2,3, ...

--In--

1

ell:

I, m=om.

z - 1

for n ~ p and n - p

2nt ; p)! (n ; P)!' = O.

for n 1/

n n

+ 1n--. z- 1

In -;=~~===",:, .Jz2 - 2% cos ex + 1

(k - I)!

(lIIp

a.

z

n>O

n

( -1 )(11-1»/2

41

z2 -

z In-z - 1

n

n -1

39

"3:

z(z - cosh oc)

lII=lm

38

:z:

z - e-7.

33

35

z2 -

n>O

II

I

+ z sin (ex - '1') 2z cos 0: + 1

z2 sin'll

+ lJI)

29

to

(x)

= mk. ;06

(d'"

(m

< p or

= even

n - p = odd

~ 0,1,2.. 00))

>

~

III

mk

n = 2nn! ~ -) (.t'2 dx

Z

S! _

1)"

><

z

.JZ2 -

2xaz

+ a2

t-.)

00

TABLE I (Continued)

Number 42

43 44

z-Transform 9>(z) = "[f(n)1, Izi

,,-0

~ >

z(z - az) z2 - 2zaz + a2

2:

c

>

-=I (n)<-_z)r

""C

_Z_ e-IC a-ll

CIO

'U

z -I

n!

r=O

r

H,.(x)

["/2)

(_I),,-kx,,-ak

7

-i

Sl0

= If(n)z-

= a" cos (n oos-1 z)

L,.(z)

>R

CD

Discrete Time-Function fen), n ~ 0 anT,,(z)

N co N

r!

=k~ok!(n -2k)!2A:

(')

g

e-ta-l/hl

2!

0

'II

45

a"P::(x)

46

L::(x)

=

a"(1 -z2)m/l(!fP,,(x),

m

= integer

(2m)! z"'+1(1 - zl)"'/la'" 2"'m! (z2 - 2zaz + a2)"'+1/2

:c 111

( -1 )"'z e-lf/l:o-ll (z - 1)"'+1

~ >

-i lit

·47

48

49

SO

51

n!

= (~\". L,,(z)

--,-I

dX}

n!

m '

= integer

[~'( z)

{I'(z) ] . Z ,where '-(z) and !fez) are rational II .!"(z) {I(z) polynomials in z of the same order 1

---z--

m(m + I)(m + 2) ... (m + n)

sin (exn)

cos (~n)

~

(m - I)! z'" [ ell" -

I

... -1

e

k=O

J

1 I"""'i k. z

/foa fl./z • cos ('in -z-«)

n!

I"

lIII

In ~(z)

in «) eCOB fl./z • sin -z-

n!

9>
!i.gn-k

k~O

31't:I

0

9>(z)

n

~:

I k!kg,,_A: k=O

!3

I

-

9>1l)(z)~(z),

d9>(z) 9>(I)(z) = _ _ dz

to

9>111(z)~(z)

k'f,.g"_k

k~O

~4

ex" + (-«)" 2«1

z2 _ «2

~5

ex" - pn --«-fJ

z (z - ocXz - fJ)

56

(If + k)lkl

57

(If - k)lkl

58

(n T k)'ml

m!

k

k! z (z _ ,-k

k. z

1 'IT -sin-n n 2

60

cos ex(2n - 1) 2n -1

(z _

z

J)k+l

zl =F kemfl.

e"',,-kl

59

Z I)k+l

(z -

efl.)tII+1

'IT 1 - + tan-I2 z

n>O

1 z+2v';cosoc+1 -_In _ 4v'z z - 2v'zcos« + I

>

ii!2:""

!a

>< N

co c...o

Number

Discrete Time-Function I(n), n ~ 0

61

"n + -n- ---:: ,,+ao (y _ ])2 yR

63

a'i cos."n

65

(.

0(

>

Z C

(z - ,,)(z - ])2

>

z(z + ao) (z - y)(z - 1)2

1 aO +l) 1 _ " - (1 _ y)2

"'\I "'\I

z

!: (') > -!

%+ a

z

(5

..,0

z(z - e-ex cos a)

e-"'" cos an

z2 - 2ze-ex cos a

+e

-!

2ex

:t

m

+ ze-ex sinh (a - 'P) 2ze-'" cosh a + e 2« •

z2 sinh 'P

+ 'P)

e-"'R sinh (an

z2 -

(~2

+ f/l +

(y. - ox)2

~

Z

(1 _ ,,)2

+~ n +

yR

·66

:t

·n=O

l+ao

62

-!

ro

2: f(n)z-n

=

---

1- "

(" - ])2

~

= .1[/(n»),lzl > R

F(z)

64

N 00

z- Transform

TABLE I (Continued)

+

+ 'P)

fJ2)n/2 sin (nO

(I[( a - ')1)2

+

fJ2]V2

0= tan- 1 ~ «

68

ny"- 1 (y -

3')1" 1)3 - (y _ 1)4

fJ

Lk (-1)" (k)(n

.~O

+k

1 [1I(n - I) 4n (1 - 1')2 - (1 - ')1)3

-

v)lk)

k!

v.

+ (1

(z - y)[(z - «)2

+

P2]

(z - y)2(% _ 1)3

%(z - 1)1.-

eexln- .1

(z - eot)k+J

i

«-

p-1/F(p) dp

10 = 0 1 + ao (1 - ,,)[(1 - «)2

(" + ao>yR + P2] + (" - 1)[(" _ at)2 + P2] [a 2 + p2]"/2[(ao + «)2 + {J2]1/2 . + P[(at _ ])2 + {J2]112[(a _ ,,)2 + P2]112 sm (nO + (J 'PI = -tan· J --1 at-

+ 'P2,

A = tan-1 _P- I ao +«

+ 1)e«n

72

(n

73

cos «n (_1)R - -

74 75

(n

+ k)! n!

'P2

- 2ne"'(n+l)

n

11I+k.

fen) n + h'

In

."

-

11

=

+ DO)

(z - 1)(z - y)[(z - at)2 'II

+

A),

P

I

{}

= tan-1 -IX

fJ = - tan-J at-"),

+ e<x(n-21(n -

1)

( ~y e<X'

%

11>0

h>O

-nan cos- n 2

z(z

z -

'

fen)

ziCX).F(P)dP

11 =0

'P = 'PI

+ lim n-O

l

7]

0, forO"

In 1-7=======

.Jz2 + 2z cos « + I

11


...,z ;:Q

z

6 ] _ 1')4

69 70

>

0 3:

Z

'P = tan-l--

+2

I

;:Q

CI>

«-I'

67

'"

-!

dl.: ( -1 )',%2k dz k [.F(z)]

i"

Zh

p-IHh).F(p) dp

2a2z2 (z2 + a 2)2

+ P2]

TABLE 1 (Continued)

z- Transform

3O(z) Number

Discrete Time-Function fen), n ~ 0

77

nan

78

I + cos 7Tn an sin - n . -"""'"""""--

79

an (

2

a 2z 2

z4 + 0 4 202z2 z4 - a4

2

') + cos 7Tn 2

7T

-cos-n

)

2

P,,(:t·)

n! P::"'(x)

m

+- m)!'

(n

82

1 (n +ex)i1' ex

83

an (

7r

n

II

./

5-

P~"

= 0,

for n

< In where 41(1, p, ex) = Wi, ex) = generalized RiemanZeta function

> 0, Re fJ > 0

2

(II

85

> 0,

+ cos rrn + cos- n

I

c"

84

)'

2

= 1,2,3,4, ... )

In z - In (z - c)

= 2,4,6,8, ...

cz(z + c)

n2c"

(z - C)3

cz(z2 + 4cz + c 2) (z - C)4

113C"

/

,W(zle)

n~c"

£("-2'14( - n .1112 ) a,,-2-4i(a 2 ;=02/+1 ~

,;yc;,),

g,:. ./

(II -

.fIO(z)

-~

-.

5;'

91

202z2 (z2 _ a2)2

4

81

85

n=O

7T

80

56

00

= ~f(n)%-n

+ cos 7Tn

1

= J[f(n)], Izi > R

k

> 0 and

4 _

z4

+ 2a2z2 + b4

d -z dz Ft(z), Ft(z)

IXII - 2)(11 - 3) ... (n - k + 1)

(k _ I)!

z2

b4 )i

integer

I)

= ,1[nl.:-1J

=

;,-[nl;-l j'(n)]

'I-I;

a

(z - a)1;

( Ir

J

k(k - l)(k - 2) ... (k - n + n!

93

./

nan cos bll

[(z/a)3 + z/a] cos b - 2(z/a)2 [(z/a)2 - 2(z/a) cos b + 1]2

94

"./

lIa n Sin bn

(z/a)3 sin b - (zla) sin b [(z/a)2 - 2(zla) cos b + IJ2

9:

95 96'

J'

/

"a n (II + I)(n + 2) ( -a)n (n

+ 1)(2n + I)

1

+-z

z(a - 2z)

a2

In

(

;I>

"Cl "Cl

a) 2 I - - - - Z z a

a

- tan- 1 val - z - z In ( 1 2 \',z/a

tTl

z 52 )(

+;a)

N

00

-...J

TABLE I '(Continued)

z- Transform 9"(z) = }[f(n»). 1z1

oc oc

>R

-l

:c

co

Discrete Time-Function fen). n ~ 0

Number

N

a

If(n)z-n

=

n=O

~

-<

z cos :I: a sin a. - - tan- 1 - - - a z-acos:I: % sin a. z2 +--In 2a

97

1

99

-

2az: cos :I: + a2

oZ o." -l

:c rn

100

~O)n.

.101

- . "'co~n -, 2 (~

\z/(z - a)

N

~> z

z

(/I

o

2

~

102

~.:ii:~'BernOulli polynomials n.

103

Wn(X)

I

/

104

Q ..(x)

m

+

Z2 -

2.t·z

Z2 -

2z cos ../m + I 1 -

Z2 -

2xz

1

1r In

= 1.2•...

= sin (n cos- 1 x)

.;'

TABLE II.

3:

eZ' :

.~ Tchebycheff polynomials of the second kind

sin : \ .

105

~

n > -l

cosh (%-1/ 2)

. / (2n)!

z o ?; C

z2

Z z2 + 2az sin a. + a2 - In "7""--:::---:----: 4a z2 - 2az sin a. + a 2

+ 1)

an cos ( ../2) n sin a.(n ./ n+ I

98

>

+

+ z-m z-m

1

PAIRS OF MODIFIED z-TRANSFORMSt ~(::.

G(s)

m)t

Simple Forms of G(s) (A.I)

z - I e- alliT

(A.2)

z - eaT

I-a

Quadratic Denominators of G(s)

.

mT z - 1

~

T

+ (z

Te-IJIIIT[

m

z - e aT

_1_( .e_1_[(ao-

amT

4';

~

s I j ....

a)(s

+ b)

+ ao a)(s+b)

b - a z - e b -

0

e":x,,,T

f3

(A.3)

- 1)2

z -

_

aT

+

e- aT

]

(A.4)

(z - e- aT )2

e-bIllT ) e- bT

(A.5)

z -

o)e- alllT

b)e- bIllT

_ (00 -

e- aT

z -

J

(A.6)

e- bT

z sin mf3T + e-:xT sin [(1 - m){JT] z2 - 2ze orT cos {JT + e- 27.T z cos (mpT

+ t/»

-

e-'%T

(A.7)

~

cos [(1 - m)f3T - t/>]

e-"'''' T sec t/> ---""";;"'-~~;;;--=____=:---%2 _

2ze

"'7'

cos

> ~

(A.S)

PT + e-2:xl'

a. - 0u tan 4> = - (J

rn

z S! x N

oc >D

t-.)

TABLE II (Continued)

\0

0

~(z,

G(s)

+p 1

S2 -

p2

S

+ p2

S2

-I

::t:

1 z sin mpT + sin (1 - m)pT {J z2 - 2z cos {JT + 1 1 z sinh m{JT + sinh [(1 - m){JTJ p z2 - 2% cosh {JT + 1 z cos mpT - cos [(1 - m){:JTJ

2

S2

m)

c

(A.l2)

0

z -I

::t:

"'

P

-I :0

N I

> Z

CI>

Cl :0

+ (-l)k+l

z

3:

sin m7r z + 1

+ :x)2 + (7rjT)2 (s + :x) + %)2 + (1T/T)2

(s

c

> ::!

ao

sin mk1T

e-<xmT

(s

"II

"Jl

(A.l3)

Some special forms of G(s)

1TjT

z

0

tan", = -

. hiT s2 + (k1TjT)2

-< >

(")

%

s2 + (J2

(A.10) (A.l1)

cosh mpT - cosh (1 - m){JT z2 - 2% cosh PT + 1 vQf+P2 z cos (m{JT - 4» - cos [(1 - m){3T + 4» {32 z2 - 2z cos {3T + 1

s

l - {J2 S + ao

0 :0

> "Ij

+1

2z cos {IT

z2 -

"'

(A.9)

sin m7r

+ e-rr.T

z

cos m1T

e-a.m T

+ e-a.T

z

s+a z

e-rr.'lIT [T(a e-xT 1T

+

CIt)

]

+ cos m1T

sin m7r

Cubic Denominators of G(s)

T2[ m2

"2

2m

1

% -

+ (z

+

1

- 1)2

+ (z

2 ] - 1)3

(A.14)

(A.15)

[e-

1

XmT

+ (z

;;: o:(z - e- aT )

.; - ao

1 [ao -

-

~..;.; ~ x)

".;;. - a1s

CIt

CIt

--( CIt

e-ulIIT

z-e-:1<'1')

T - 1)2

+

mT - 1/I1.J

(A.16)

(z - 1)

aoT

(

+ (-1)2 + aomT +] z-

ao) ] ] ()( z-1

- -

+ ao

(A.18)

n.s + :x)

1 ab(z - 1)

S'S -

a)(s

+ b)

s-. -:-.' als + ao s(s

+ a)(s + b)

(A.17)

+

ao ___ + ab z - 1

e-o. mT o(a - b)(z - e- a'1') 0

-

0

a

e-amT

a(a - b) z - e- uT

ao 1 + a 2 ___ ab z - 1

e- bmT

+-----.....,.-::bT b(b - a)(z - e-

bmT a _ - _ b _e+ -:--:-,-0 _----.'"

b(b - a) z - e- bT

a I a + a 0 e- alllT a(a - bl z - e-tlT -

)

+ b2

bmT a 1b + a o _ e-_ -:-=. b(b - a) z _ e-bT

(A.19)

(A.20)

> "II "Ij

"'z S! x

-

(A.21)

t-.)

\0

N

-c

TABLE II (Continued)

N ~(z,

G(s)

m)

'0:z:m"'

Cubic Denominators of G(s) (Col/ti"ued)

1

+ a)(s + b)(s +

(s

c)

+ a)(s + b)(s + c)

(s

(b - a)(c - a) z - e liT

°o -a

e-"mT

+ O)(S + b)(s + c) 0)2

-rt.2 - - e '''" Z - 1

~ {~ + e

s2 + 0IS + ao

1

s(s

«2

+ ~)2

+

(c-o)(b-a)z-e aT

+ 00 s(s + :1)2 s

+ (a

e

ao

1

+a

bmT

XIIlT

h)e- bm7'

(00 -

•

e bm7'

0

- c)(b - c) z - e- rT

°

+

-c

e

c

tmT

:--~o-:-:-_ _ ----:c.:;;

(A.23)

(a-c)(b-c)z-e- r1'

+ c2 -ca 1 +a0

+ (ao

>

e- rm7'

•

(A.24)

(a-c)(b-c)::-e- c7

(A.2S)

- ot)«Te

21111.[(b -

+e

z - e b7'

-<1.TJ}

(A.26)

1

00)

::!

0

z

x2 -

-e-

0o)e

(A.27)

"1')2

(<< - b)Te- 2 TJ}-

+ -:--~""'::(z_e- xT)2

- ~)(b - x)mT

+ (ao

;!

"'';' 'z"' ~

(I)

-a.TJ} + ----"--:------::m:-::-(XT(al~ (z

=_e- x1'

r-

0

- «2 - ao)

[",T(X - b) -

."

(;

(z - e 21')2

+ mTx(a1ot

> ."

."

z _e- a1'

e

(b - «)2l

e-h",T

(o-b)(c-b)::-e b1'

-I

_r

-b

0(

(A.22) > z

J}

- -........ + e ('%-b)2l.::- -b7 s + On

0

+ (a

e bT

"

e-~,.. T

7.[1 + rt.mT + :------::= rt.Te 27' z - e «1' (z - e - ~T)2

f-ao- + e _ ~m 1'[ot2 1

=-

(a-b)(c-b)z-e- b1

<1.IIlT[m'%T(Oo - :It) - ao :: - e -,,1'

z _ 1

-02lz

°

1

e-'m.T

- b)(c - b)

+ b2 -ba1

e-""·1'

_ _--"1_---'-::-0

1{ 1

s(s +

..

(b-a)(c-a)z-e a1 a 2 -aa +a

(s

1

e -alllT

~ ::u ~

(A_28)

+ (uo

- 'X)(b - «)TIt a.TJ 1

z: - e- 27'

f

(z _ e 2T)2

(A_29)

r - als + 00 IS - bl(S

+ x)2 + (<<2 )

-

0 1 :1

+ oo)(b

()

_

- ot)Te-,,"·T [ T

- - - - - - -e "Ill sec ,%2 + p2lz - 1

Z COS

", z - e- 2

7'

+

e

<1.1' l' ]}

(A.30)

(z - e- x )2

"']1

+
(mflT

J -!It

tan." 00 f 1 ---l-«2 + {J2 Z - 1

--:",,7'..L

e

sec

Z

'+'

cos (m{JT + "') z2 -

e-:x.T cos [(1 - m){JT -

2z:e-- 2T cos fiT

+ x)2 + (J2]

1 (<< - b)2

+ {J2

"']}

=

«2

+ fJ2 -

;::. -

0I S

+ 00 + fJ2]

+ fJ2

z -

+ (<<2 +

{12 _ a )e-2I1lT sec

1

0

." :: cos (m{lT + .,,) =2 _

tan s + 00

ao - b (e- bIllT ---r= (cc - b)2 + p2lz - e- bT

-

r

211•

01

= -- fJ

"'l}

b-cc

(A.33)

=-{J-

cos [(1 -

2::e 21' cos fiT :1 -

cf>

e- .. r

(A.32)

aofl

{ e- bll.7' -21117' ..L z cos (m{JT + .,,) - e- 2 7' cos [(1 - m){JT z - e-bT - e sec't' z2 _ 2:;:e aT COS {JT + e 2xT

__1_ {~ «2

«00

---:--~

tan." '::'! - x,2

(A.3l)

+ e-2x7' tan'"

: - bl[(s

= -{J-

m){JT -

+ e- :!d'

0o(h -

0 1)

-:-:-~--::,..--:~ (J(~2 p2 - 0 0)

+

cf>]}

(A.34)

>

." ."

cos (mfJT + cf» - e-"1' cos [(I - m){JT - ..L]1 sec '" --~_.~=__::.;;_____,~:..:...---.".__;,..:-=-_~'I' z2 - 2ze--"T cos PT + e--;!27' J

T::

tan ."

(00 -

:1)(b - «) - p2

= --'---;----:-:-=---'(0 - b)P 0

m Z

C

x

(A.H)

N

-c w

N

TABLE 11 (Continued)

~

!fez, m)

G(s)

+ b)[(s + «)2 + P2 ]

(s

(

1

(X-

+ p2

b)2

z cos [mpT

{ 2 (b - alb

+ ~] -

--l

e-blllT

+ 00) z-e-bT + (<<2 + fJ2

+ alb

== ~

- ao)e- XIIIT sec ~

lIG

-< >

z

e- rxT cos [(1 - m)pT - ~]\

2ze- rxT cos PT + e 2.. '1'

z2 -

- 2rxb

c

J

> "II "II

C (1

.!. {_l__ z COS m{JT fJ2 z - I 00

I

p2 \z

+

1

1 { ao

- I + (fJ Pz z-

--l

X

,

co --l

z

T

-rx

- ao)e

III

sec 4>

z CPS (m{JT - 4» - e-«T cos [(1 - m)pT z2 - z2e «T cos {JT + e 2rx'1'

-

+= (00 -

b)P

~..I..__.~··~··~<. . t~·rn . ·.. .$.'....I.··.F. .I.J.t.~ljS~"....~.-.~...·~·.'5~~. .,.~~~·~·le~G.r_·~·n~'·_~~~·i~~

(s

+ b)(r + PZ)

1

{ (bZ

-

alb

fIl

0

lIG

Doh - p2

+ b2

2:

"II

(A.39)

z cos (mfJT + ~) - cos [(1 - m)PT - ~)} sec ~ ------:zZ:----::2z:--c-os-p'::':T::--+---:-.- - -

b { e- b•n2'

tan

pz

>

(A.38)

3:

alfJ -aD

-PZ-+-b2 z _ e- bT

lIG

+ .p]}

tan+=-p:l

r + 01S + 00

"II

111

P

00 -

0

z cos (mpT - ~) - cos [(1 - m)pT + ~]} _ 1 - sec ~ z2 - 2z cos PT + 1 aD

+Qo

(A.37)

1

tan 4> = -

S

~ ~

cos [(1 - m)PI1}

2z COS fJT

Z2 -

(A.36)

(A.40)

,

.........

. ...::: ...;..:

~-.-

e-bIll'1' e bT

+ 00) z _

1l2

+ ( P- + °1

b

) ao sec

-

z cos (mpT

+

+ +) -

cos [(1 - m)pT -

z2 - 2:clCOS PT + 1

+]}

(A.41) Biquadratic Denominators of G(s)

I

s' I ~s+a)

r(s

+ a)2

6"

T 3 [ m3 z - 1

+

~[T2m2/2

-

a

+ 3m + 1 (z - 1)2

3m2

mTla z - I

1 [mT - 2/a z - I

-

02

02

+ p2

iii

+ (z

+ l/az + T2(m + 1/2)

6

]

T

+ (z-- + 1)2

- Tla

+~

2

Z

p2 + mT -e aT

+

_.!.

(z _ 1)3

e-aIllT(mT + 2/a) Z -

(A.42)

- 1).

(z - ])2

(-II '1.[a ~

-1- e

6m + 6 - 1)3

+ (z

Te-GT(m+U]

+ :----=:;:;:-:: (z _ e-4T)2

e-4 7'

J+ _____ 1 cos fJ

e-4mT ] a2 z _ e-aT

(A.43)

(A,44)

~»)

[(1 - ---::---=-__ m)pT + .pI ---:-::-----:-_ - z cos (mpT _ _---...,;",;,. z2 - 2z COS PT + ]

Te- aT (z - e 4'1')2

(A.4S)

2. {mT + __T_ p2

z - 1

(z - 1)2

_ ~ z sin pmT

P

2:2 -

+ sin [(1

> "II

~

!2 - m)PI1}

2z cos fJT + I

><

(A.46)

~

va

TABLE II (Continued) ~(z,m)

G(s)

COS (mpT + .p) - e-«T COS [(1 - m)pT - .pJ} -~1 {P~ +z DofPmT + (z-Dop:r - - - p21e-a.m 7' sec.p z----;:---:::--::';;;--=:-----:=;---- 1 - 1)2 z2 - 2ze "'T cos {JT + e 2",T

s + DO

P: = oc tan.p

2

+ p2; ocp~

=-

When Do

p~

=~ -

2OCQo

+ DOP~ pp~

= oc, tan .p

= -

20cP

-p2 2 -oc

E. I. Jury, "Additions to the Modified z-Transform Method," I.R.E. Wescon Convention Record, Part IV, August 21,1957, pp. 136-156. Similar tables of modified z-transform are presented by R. H. Barker, "The Pulse Transfer Function and Its Applications to Sampling Servo Systems," Proc. I.E.E. (London), Vol. 99, Part IV, 1952, pp. 302-307. t A few entries of Barker's table (British Crown Copyright) in a modified form have been included in the table above for which the permission of the Controller of Her Britannic Majesty's Stationary Office has been obtained. : It should be noted that ~(z) can be obtained by multiplying ~(z, m) by % and letting m go to zero.

•• <

.....

TlF."

III :l Q.

;. t ... ~

~

:::-

'!;. ~ ::

-::

N

N

t1>

::=>

3

III>

...

M M ii"M.. ...III ;:C. 0- 0~

T

to

Q.

...

.j:"

00-

w"to

IQ :0

0' ... 3

n

Jl

;. n

~

;

..

..Q

::;>

0

3

~

0

t:T

+ + Jl

~

Jl

n -0

~

..Q

...

n

:r

0

+ + 1:1

(7Q

'"

;.

Jl

..1:1

~

3

::I t:T

'<

n ~ 31 - !l.

(7Q

3

~

...;:c.

:O'=--

;so..

~ '-'

g

..c":l

E.O

1M;! aM" 0Q 1.;s

.. ..

~ A

n ::I

I

:IA

II> ::I Q.

-III

:T:l t1> Co

:;.

~o..... ere

t1>

.

0-

",'

..

2-1A

i

..1:1

Q ::s I

to

1:1

::s

.!.

~

::s

II>

n t:T

~.

Q.

-

I

~

'II. 0

,!

:T ;;;

~I~ ~~

...

II

:T

Q

...

-J

~I- S· > ...:. t""" S tI) S! ,,~ ~ ... !D !. t:) c: t1>

as: .

~ :! :$

.'1 ,! to

.!.

~

> ~ tTl

~

~ ~ > r-'

tI)

2'

:l

a o· :l

iA~

Il '" ~ ~. 020-<

III

'" cn

:l

IA -:!

:::J

:!

?

.

S·

t1>

::I

:l 0 III :l ::I

0

~

~ II

:-.

Q.

.to

tTl

-=

c

(til>

~ ..,

1.;s

~

r-'

S· ::I 0 ...,

0

::I Q.

~

~

~

00-

... (i

:n .., n 0

II

2c:

-1l n

(to

;so..

-J > > '" tD

:l

°IA fi";z

t1>

e-

(7Q

d

0 .!I

~

'<

ii>

:i"

+ ...Q

;;;.

~~

:l n

~ ~I 0' -::=> c;;.;: ~It1> Q ~

~

n

:T

n :T _.n

iii·

~

0

Q.

:0

-16:T ..

d ::=>

iii'

N

II>

~ :T

n

.P

:-~-

IA

>-

"CI "CI

"'z

~

N \C ~

298

THEORY AND APPLICATJON OF THE Z-TRANSFORM

TABLE III (Conlinut'd) Tobll/ott'd vollies of the integrol I ..

I

n

1 ~ B(z) B(z-l) ---z-Idz 2."j. unit A(~) A(z-I)

=-

circle

I.

.9"(z)

= =

I

+ bl + 01

boZ 00'l.

+

b~)Qo - 2bob1o l ""':'--=-"":"'-::-:-"'-'-":

(b:

oo(o~ - o~)

1

+ bIZ + hi + 0lz + Q2

boZ'I.

2.

.9"(z) =

12 =

B(z) =A(z)

Qot-2

Boooe, - Blooo l

+ B2(0~

0 0 [(0 01& -

(0 0 0 1

2 0a)el -

- O~I) 0 10 2) 0 1]

Bo = b~

Bl

+ b~ + b~ = 2(bohl + bl b2 )

B2 = 2bohz

3.

.~"{'l.)

bot-a

+ bIZ! + bIZ + ba --=--..::: + 03

= -=-=a:--""'::"""2 QoZ + 0lz + Or

ooRoQo - QOB 1Q 1 + ooBIQI - BaQs = ~:-----::~~:""""--=;"":"~-:-7-":";;;'~-'::"":"=--""""'-=[(Q~ - o:)Qo - (0 00 1 - OeDa)QI + (00"2 - 010a)Q 2lao Bo = b~ + b~ + b~ + b: Bl = 2(bohl + bib. + b2bs) BI = 2(bohz + blba) 13

B3 = 2boha

Qo

QI

= (Ooel = (0001

Q2 = (oJel

Q3 eJ

= (al =

00

e2 = 01

-

- 0ael) - 0ffla)

02el)

-

03)(e: - e~)

+ 02 + 03

+ 00(00e2

-

0ael)

APPENDIX

299

TABLE III (Continued) b~

4,

'-(z) = are

14 = 80 Bl

B2

+ biz! + b"z2 + baz + b4 + D1ZS + DeZ• + DSZ + D.

ao[(D~ -

o:)Qo - (D oa1

-

aaa.)Ql + (0002 - D.D.)Q2 - (0008 - DJa.)Q81

= b: + b~ + b: + b: + b:, B3 = = 2(bobl + bJb2 + b2ba + bab4), = 2(bob2 + b1bs + bab.)

Qo = Doele4 - Do03e2

+ a.(Dl e.

2(boba

+ bib.)

- eae.)

= aoOJe. - 000aDa + a4(DID2 - osee) = DoOJea - DoOsel + D.(aae, - Dsea} Qa = a1(aJell - eae.) - D2(D1e J - D3ea) + Da(ele. - D3e2) 0.· = Do[e 2(D Ja4 - a003) + e6(D~ - 0:)) + (e~ - e:)[DJ (01 - 0 3 ) + (Do el = DO + 0'1. ez = aJ + 03 e3 = a2 + 04 e. = ao + D. QJ

Q2

es

D.)(e. -

O2 )]

= DO + O2 + 04

0 3 of 13 can be respectively written 0., = - (14Q'1 + (I:lQ, - (l2Q2 + (I.Q:I 0:1 11:100 ":!('II (II (J~

• For elise of calcuilitions Q. and

1'11\' :l\lIh.1I " /:11111'1'111 I .. 1.1, 1'11111 I" I "VI" III' 1';11 i~, I nll1n' 1'.... p.. inlin~ oUI-thesc:

II'lalulII,hll"

300

THEORY AND APPLICATION OF THE z-TRANSFORM

00

TABLE IV CLOSED FORMS OF THE FUNCTION

I

nP:rf',

:t;

<1

,.~O

r

o

1 -x

(I - x)2

+ a:

xl

2

(I - x)3 :r3+~2+X

3

(I -

;lA + Ilr + Ilx\! + :r

4

(l - x)5 :1:6

S

6

8 9

10

+

2~

+ 66x3 + 2fix2 + :I: (I - x)'

",6

7

:l~)4

+ 57 x' + 302",3 + 302x· + 57 Xl + X (I - X)7

X'

+ 12Ox' + 1191.1:3 + 241fix' + 119la.3 + (l .t.8

12OJ~\!

+X

-:1:'1

+ 2473:7 + 4293.t:' + 15619'):6 + ... + ,I: (l - ;r)8

.1,9 + 502z8 + 14608.r7 + 88234.r1 + 156190.,05 + , .. + ,I' (l - X)10 ,t,10 + 1013.1,9 + 47840.1.8 + 4SS192x' + 13103S4.I,8 + ... + ,I: (l - :r)1l

The missing terms are apparent since the numerator polynomials are symmetric in the coefficients.

PROBLEMS

Chapter 1

Prove equations: (LSO), (1.51), and (1.52). Prove equations: (1.97)-(1.105) and (1.107)-(1.109). 1.3 The basic Fibonacci sequence In has the following recurrence

1.1 1.2

PROBLEM PROBLEM PROBLEM

relationship where 10

= O. II = 1.

"'n = /in and 'I.. = Izn+l' Obtain the z-transforms of (a) ~(%) J["'.. ), and (b) 'fez) = "['I.. ] (e) Find the relationship between 'fez) and ~(z).

Let

=

PROBLEM

1.4 Show that

[ +I] =

Z

(s

u)n

..r[1 + .,.n

2

r n- 1 (n - I)!

+ u~(.r. + J:n

3)

+ Q~(.r.1 + Xn

.

4)

+ ... + u:(;/:" + x:

(I - x)n

~

£0)1 .

where .,' _, c' 11'1':.

I

II;: _":(/'

I I)"

;I"d I

,.!""

I

I ... '(

II~"~("

q' II"

I

, ... ,t II",·:

,"'"

"I ,,1111

I'" 1111

302

THEORY AND APPLICATION OF THE Z-TRANSFORM

1.5 A sampled-data control system is shown in Fig. 1.5. Given

PROBLEM

= _1_ e-1.35I7'

G(s)

$+1

H(s) = _1- e-2.1I.2'

$+2

T

= I,

1

R(s) =$

(a) Obtain '6(%, III). (b) Find the continuous response. (e) Indicate how to obtain from (b) the response at sampling instants.

Rts}

=1/.

+

~(:,

m}

FIGURE PROB.I.S PROBLEM 1.6 For the delayed sampler operation of system in Fig. 1.6, obtain the modified a-transform of the output and the actual output c(n, m)T ... c(t).

1

i(iibj

FIGURE PROB.I.6 PROBLEM

1.7 If BII(x) _

eZ /l

J - - - a[ell• - 11'

show

where B,.(z) are the Bernoulli polynomials

"I

(a) BJ.z) = O! (I) - 1 B1(z) - I! (a: -

~

i) .. a: - i

Ba(a:) = 3'• (-6 - -4

~

Z)

+ -12 -

~

- -3:x; 2 :.;1 + 2

PROBLEMS

(b) B2n : 1

= 0, I'

X".....

=".11 IBn '( _ n. P

p.

where B,. == B..(O) = Bernoulli numbers

" = 1,2,3, ... ,

= 0, 2, 3, 4, ..•

n B,,(:I:) (d)-,-

303

I ", )' = p! 11-0

I

n.

(p) B,.XIIIf

Ii

1.8 Show that

PROBLEM

1

.V -I

!:r;1J = P--1 [B" ....(N) z-I +

B"+I(O)

when p = S, we get .\' ··1

!xS = UN· -

3N6

+ tN4 -IN! + ,,12

-

.\)

:1'=1

1.9 Show that

PROBLEM

co

I

,,~nzp =

(271')1!p( _1)"-1

2(2p)!

BIP'

P

= I, 2,

3, ...

Hint: (a) Show first that 2y coth y = 2y/(e211 - I) + 2y/(1 - e-2v) (b) Using the generating function of Bernoulli's numbers, obtain· co

11 coth y = PROBLEM

(2y)!"

! BIP (2:p)'. ,,=0

1.10 Show that

B..(I - x) = (-I)"B,,(x) PROBLEM 1.11 Obtain the steady-state response v" of the voltage across the resistor for the initially unexcited circuit shown in Fig. I.Ila in response to a periodic input shown in Fig. \.lIb.

(hi

(II'

FI(illlt F I'ROII. 1.11

."

I 'fll't It-II

• Tho ,Ollorl" II,. runllllnll ur ,fn) I. ".dll.d by

ll •

304

THEORY AND APPLICATION OF THE z-TRANSFORM

1.12 Show that

PROBLEM

f"[(/) dtJ = T f.F(Z, m)12= 1 dm 1""[(/)2 dtJ =

2~j

frfp-l$ m) (p,

$(p-l, m) dm dp

1.13 Show that

PROBLEM

~kJ.Q)F(jk 2;)

= .F(z,

m)lz~l.m=I'

for [(0';-) = 0

1.14 Show that

PROBLEM

,(m[F(s)I,=ol =

Tf

:F(l,m)dm

1.1 5 Show that

PROBLEM

,(m[F(s)G(s)] = Tz

Lm'(z, m -

A)~(z, A) dA +T

f:,-'(z, 1 + m- A)~(z, A) d;'

and ,(m{F(s)G( -s))

= T Lm'(z, m

-

A)~(z-l, 1 -

A) dA

+ TZ-1f 1,-(Z, 1 + m - ).)~(z-l, 1 -

).) d).

III

Chapler 2 PROBLEM 2.1 In the example of Section 2.1, let V be a d-c voltage source of 10 volts and R = 1 ohm. Find the current in the 10th loop (Le., n = 9), if the end loop (Le., 10th) is short-circuited. PROBLEM 2.2 For the ladder network with capacitors shown in Fig. 2.2, assuming zero initial conditions, (a) Write the equation of the (n + 1)st loop. (b) Obtain the expression for /(z, s) ~ J[/(n, s)), and I(n, s) = ~t{;(n, m. (e) Obtain I(n, s) (d) Show how to obtain ;(n, I).

R

R

R

R

R

v

FIGURE PROB. 2.2.

Ladder network with cup:lcilors.

PROBLEMS

l()~

PROBLEM 2.3 Consider a series in which the first r terms are all positive. the next r all negative, the next r all positive, and so on. This can be expressed as follows The sequence u.. is

1,1, ... ,1,

-J,-I •... ,-I.

r terms

r terms

1.1 •...

Obtain the factor u.. which must be introduced into the nth term to allow for such ftuctuation of sign. PROBLEM 2.4 Given the Fibonacci's numbers as follows: 0, I. 1.2,3, S. 8, 13,21.34•... (Each of these numbers is the sum of the two numbers immediately preceding it.) (a) Obtain the difference equation which describes the general terml... (b) Solve the differ~nce equation in (D) by the z-transform method and show that the solution forI.. can be presented by the following formula:

PROBLEM

2.S Given the following constant coefficient difference equation:

a,]/(n)

+ D'_l1/(n

-

J)

+ ... + a~/(n

-I)

=b,/:(n - / + 'I) + b'l_l:c(n - / + 'I

- 1)

+ ... + bo:r(n

- I)

For zero initial condition show that yen) can be represented as follows:

PROBLEM

2.6 Find the solution %(/) for the following difference-differential

equation: x'(/)

+ 2:c(1

-

J)

+ :c(/)

=/(1)

if:r; == 0 for 1 ~ 0, and/(I) = unit step and unit ramp. PROBLEM 2.7 For a periodically varying sampling rate system shown in Fig. 2.7a obtain the response for a step input. T.. is represented as follows (shown RCII':: l/s ,..

Cis'

!-IenlRI I'ROII.2.711

306

THEORY AND APPLICATION OF THE Z-TRANSFORM

graphically in Fig. 2.7b):

n

O.S second

Tn

= 0.5 second

= 0, 4, 8,

12, ...

n ... I, S, 9, 13, .. .

1.0 second

" - 2, 6, 10, IS, .. .

1.0 second

n =

3, 7, ll, 15, .. .

The total period T = 3.0 seconds.

[II I lit LlL o 1<

0.5 1.0

2.0

T

3.0 3.5 4.0

,IE

5

6

T--~~I

FIGURE PROB.2.7b The sampling pattern of system of Fig. 2.7a. PROBLEM 2.8 For the time-varying sampled-data system shown in Fig. 2.8, given the following values:

dn =l+e-n . '.J(z)

1

= --1 ' T == I, BI(z) z-

z .. - - I

z-

(0) Obtain the equation for the z-transform of the output 't'(z). (b) Show that the solution e" of the equation obtained in (a) is c(n) = I - ( -1 )ne-(1/2)n(.1-11

d"

~

--i

<9(z)

r~"J c(n)

flGURE PROB. 2.8 PROBLEM 2.9 A square of side n units is subdivided into n2 (see Fig. 2.9) unit squares by parallels to the sides. The diagonal OA is drawn and also the diagonals which are parallel to OA of all the unit squares. Obtain the number of paths by which one may travel from 0 to A kccping to

PROBLEMS

307

FIGURE PROB. 2.9 A square of side n units.

the lines of the figure. It is understood that at junctions such as P, a path may proceed in one of the directions PQ, PR, or PS only. PROBLEM ~.I0 Find the eigenvalues of the following matrix:

o

o 010

o

o

o

A=

o 010

o by uNing nllntl,

II

010

recurrence rellltinnHhir in the comrutillion of the rcquircd-dctcrmi-

308

THEORY AND APPLICATION OF THE %-TRANSFORM

Chapter 3 PROBLEM 3.1 Check for the number of roots inside the unit circle for the following polynomials:

(0) F&/:) (6) F.(2) PROBLEM

= 122' + 1625 + 20z· + 14z3 + 922 + 32 + I = 6z6 + 821> + 19:. + 19:3 + 1522 + 6: + 2 +

3.2 If the determinant IXA• 00

Ykl is expanded as follows:

+ bn _ k + J

. oJ

+ bn _ ktl

bn _"+2

00

+ 6"_"+8

111

Then if ~J is expanded. the expansion of the second determinant • .:1•• can be written from the first. ~J' by replacing Ok by 6n•k and ble by On.'" PROBLEM 3.3 Given the following fourth-order real pOlynomial. with

04

=I

(0) Show that the conditions for the roots to be real are

Do

= o~

DJ =

- 2000 2

I

>0

o~ - 200"2

0 102 -

3000 3

DJ D2

=

..............

400 2azOa -

3Ql

and Ql

Da

D.

=

202 30a

°1

-

2010 a Ol Qa

................. Q10 3 -

I

ala. - 3000a

20. - 400

20. 30a

4

>0

- 400

2ozOa 30~

-

>0

301

20.

>0

PROBLEMS

309

(b) For the real roots to lie between zero and unity we require the following additional conditions:

+ a3 + a2 + a1 + ao > 0 + 202 + 30 1 + 400 < 0 "t + 3al + 600 > 0 "I + 400 < 0 I

03

0u

>0

PROBLEM 3.4 Show that for an aperiodic system (discrete or continuous), the maximum number of extrema is n - I, where the order of the system is II. Hint: Show this for a second-order system and by induction for the nth order system. PROBLEM 3.5 Obtain the number of roots inside, outside, and on the unit circle for the following real polynomial:

F(z)

= z8 + 0.7'& + 0.4,·

- 8.5,3 - 5.6%2 - 3.2:

+4

PROBLEM 3.6 Show that a necessary condition for the roots of F(z) lie inside the unit circle is:

n! (n _ r)!

=0

to

(r = O. 1.2•... , n - I)

lanl > r! 10rl.

For a fifth-order polynomial this yields with as

=I

Hint: Show that every circle K (or polygon) encircling all roots of a polynomial F(z) = 0 also encircles the roots of the derivatives FI,..'(z) = O. PROBLEM 3.7 If the roots of F(z) (with On = I) are all inside the unit circle. show that the maximum modulus of the roots, r, can be obtained from the following relationship: (A n_ 1 - B n _ 1)2F(l)F( -1)( _1)n(n-lI

!l

r

<

1-

2,,(n-l)

or

Hint: Use equations (3.68) through (3.70) to verify this inequality. PRUIII.I:M 3.8 Obtain the root distribution of the following polynomial: 1-"(:) I'RClIlJ.l;M 3.')

(tl' /o"{:) (h) /o'(1) -

:II:

z& - 3.5714z"

+ 5.7755z3

-

5.303z2

+ 2.6405z

- 0.5414

()htllin the mill distribution of the following polynQmials:

4 I 3013 f 2:111 ··2 I 4. I ~.' .-

().rl

of- 2,' of 6111

.-

4:' - 2,7 ...

,I

310

THEORY AND APPLICATION OF THE %-TRANSFORM

Chapter 4 PROBLEM 4.1 If §(z) = J[J] and (f(z) = J[g] are both analytic for Izl > Ro, show that the function Jf"(z) = ][/g] is obtainable from .1"'(z) and ~(z) as follows:

.)F(Z) =

_1..1 ~m'Y(~)d' 2TTJ l(1 ~ R. , ,

for IZ, > RoRv where Rl > Ro. (The path of integration can be chosen as a simple closed curve r.) PROBLEM 4.2 For a multirate error-sampled feedback system shown in Fig. 4.2, obtain the modified z-transform of the output '6(zTI ' m); n is an integer larger than one.

FIGURE PROS. 4.2 PROBLEM

4.3

Let F,.

co

=

~

nr:r;"; show that

71~O

(a)

Fr' =

. 1 with Fo = -I- - , x

dFr = Fr+1 , dx x

-x

<1

(b) Show by induction that

S,(x) F, = (I _ xy+l •

where

x

<

I

, S,(x) = ~ S"i"'"

r>O

;=1

(c) Using Sr(x) in Fr of (b), differentiating and using (a), show that

.+1 r i '" Sr·t-l,ixi -- .~ - " S r+l , [ix i-1

+ (r + 1 -

i=1

(d) From (c) by equating coefficients, show S'+I,i = is,,;

+ (r + 2

- i)Sr,i_l

(e) If r = 3, obtain the value of

for

x


i)xi+l]S r.,.

PROBLEMS

.-1

PROBLEM 4.4 Let Fr,,,(x)

311

-

= ~i rx' and

Tr,II(x).

DO

'-0

~i rx'; show from the '-II

results of the preceding problem that (Q) Fr,II("') - Fr(x) - Tr.,,(x) (b) By a change of variable show that

=I(n +Wx"+'= I I (~)ilnr-Ix"+'=.i (~)nl'-i'l:"F/(X) J J.

Tr"(x)

'jaO

,=OJ-O

j"'O

(e) Show that

Fr ,,(x) ,

= Sr(:l')

-

f (~)nr-j:C"Sj('l:)'

j-O )

for all

:l'

71-1

(d) Show that the finite sum

numbers.

I

+ bIz

zll

PROBLEM

i' is a known result in terms of Bernoulli

i=1

4.5

Let .JF(z)

= z2 + Ql% + Q2 = (z

zt + b1z - P1)(% _ p,~.>

where Using the residue method, show that (a) N = _1 • 27fj

1.. Jt"(%).~(z-J) ~ = (b~ + 1)(1 + P.P2) + 2b1(P1 + P2) Jr z (I - pf)(I - p~)(l - PIP2)

(b) Show that N. in (Q) is the same as obtained from Table III of the Appendix when n = 2 and bo = 1, bl! = 0, Q o = 1. PROBLEM 4.6 Given the following expressions for the sampled spectral density cJ)1.2(z).

z+I

1 (a) (1

+ 0.5z- 1)(1 + 0.5%) ,

(b) (z

+ 0.5)(%

- 2)

Determine "'J2(nT) for eath case. PROBLEM 4.7 For the error-sampled data feedback system shown in Fig. 4.7. (Q) Obtain the total squared error for a step input. (b) Obtain the discrete square error (that is, the sampled square error). G(.f)

27.8(s + 1) = s[(s + 3)2 + (27f)2]

ret) = u (I) + L~(t) a JI. -

T=l

v-

~.......,.

------------

He n IIU IIIU III. 4.7 1·lTllr 11111111,1"" liot'''hlll'k "YIII"IlI.

312

THEORY AND APPLICATION OF THE Z-TRANSFORM

ChapterS PROBLEM 5.1 A saturating amplifier sampled-data feedback system is given in Fig. S.l; also given is the form of h(n) as follows: 4O e(n) ~ 40 ( hen) = e(n) -40 ~ e(n) S 40

-40

e(n)

S -40 T

Ty-

r~ -.Yf(z) .. .../, o~(z) r(l)

I

I I

+

1

1-+1 - - - 0 cit)

FIGURE PROB. S.l Saturating amplifier sampled-data feedback system.

The power series representation of this nonlinearity over the interval -6S

<

e(n) ~ 65 is h(n) == 1.1IIe(n) - 0.1234 O.IO-a[e(n)p r(n) = 60 volts e(0) 60, e(1) = 32, T = 1

=

(a) Obtain the nonlinear difference equation which describes the error e(n). (b) Solve the preceding equation using the convolution z..transform to obtain e(n). PROBLEM

S.2 Given a nonlinear discrete system as shown in Fig. S.2. (f

hen) = m1e(n)

+ mae(n)8.

with

ma = 0.1, ml

= I

and r(n) = unit step. and T = 1 obtain (a) The nonlinear difference equation which describes the output c(n). (b) Solve the preceding equation using the convolution z-transform. and obtain c(n).

r(l)

=II(/) +

1-+---oc(tJ

FIGURE PROB. S.2 Nonlincllr di~rclc

~y5ICI1l.

PROBLEMS

313

(c) Compare a few values of c(n) from (b) with the exact solution using (a) as a recurrence relationship.

Chapter 6 PROBLEM 6.1 Given a nonlinear sampled-data feedback system as shown in Fig. 6.la and the type of nonlinearity shown in Fig. 6.1h. (a) Show that the limit cycle +1, +1. -I, -1. exists. (b) Show that the limit cycle 0, +1. -1,0, does not exist.

r,,(t)=O +

c,,(t)

FIGURE

PRO~ 6.ta

e,,'(O)

... e,,(O)

----~...j..:~----

FIGURE PROB. 6.1b

6.2

PROBLEM

-I,

For the example in Section 6.6, if the relay output is -I,

+ I, + I, obtain the possible values of co. c.' "2, C3 to maintain a limit cycle

of period 4. PROBLEM 6.3 The nonlinear difference equation describing a PWM system which is in a limit cycle of period two is given as follows: Cn H

k

= y(n) IJ2 (bh" = f(c n•

.

r" II

- e

·b(T -It.>

= yen) hk [t"

If" ..

/I

1

e.. ) b( T

.It.> -

t" •• hT]

If. ;tnd

+ e -hT) + c" +

Ir" - t·"I.

r"

~

I

+ e-bTc" = g(e", en)

314

THEORY AND APPLICATION OF THE %-TRANSFORM

(a) Obtain the elements of the matrix [Ao]. (b) Find the characteristic equation of the matrix which can be tested for the

roots inside the unit circle. (c) Obtain conditions on the stability of the limit cycle. 1 + e -117' (d) Show that the solution for h ... 0, and a.k < (a.k)c = 2b 1 -liT' is always stable. - e PROBLEM 6.4 For a relay sampled-data system given in Fig. 6.1a and b, the following are given: b = 1, k = 1, T = 1 (a) Show that the continuous response at the limit cycle is given by co(mT) = -

{!(1

-

2m)T

+

I

+ e-22' + e-mT _ 1 + e- 2T

,,-(1 +

In)2'}

'

o ~m Hint: Note that when m Cl ..

I

= 0, the response at sampling instants is obtained.

(b) Show that the postulated sequence +1,0, -1,0 can exist if Co

and

~

= -0.245

0.166 and half the dead zone A - 0.2. Chapter 7

PROBLEM 7.1 (a) Find the approximate solution of the following differential equation using the z-forms method and assuming T = 0.2 second. day d2y ely dIS + 2 dt 2 + 2 dl + Y - I, yeO) - Y(O) = yeO) = 0

R(s,:II lIs

+

1-_..,-_-<> e(t,

Cis}

(0'

R(s'. lIs

+

t--,---<> Y(s}

(b,

FIGURE PROD. 7.2

PROBLEMS

315

(b) Compare this solution with the exact solution and comment on the choice

of T. PROBLEM 7.2 Compare the exact step response of Fig. 7.2a with the approximate model of Fig. 7.2b for the following values: (a) T = ls second, kl = 3.2, k2 = 555, Q = SO radians per second. (b) Repeat (a) with T = ,Ill second. PROBLEM 7.3 The z-transforms of the input and output of a discrete filter are given as follows:

SOj(z) = I SOo(z) = 4

+ 3z- 1 + SCi + 142:-1 + 27z- 2 + 13z-3 + 5z- 4

(a) Determine the z-transform of the filter Jt"(z). (b) If a small disturbance is made in the measurement of the output such that

the leading term of .'Fo(z), that is, 4, becomes 3, determine the z-transform of the filter for this case .Jt"·(z). (c) Using the substitution z-I

= l

+1

obtain Jt"l(A) for (a). (d) Determine .}f'tO) for (b). (e) If we truncate the series for Jf"r(l) after the second (or after the third) term, determine .}f'·(z) for these cases and compare with (b). (f) Comment on the value of the transformation in c. PROBLEM 7.4 To obtain the z-forms for a double integrator (l/SZ), the following second-order difference equation is used to approximate iz(t) = JIy(t) dt: Aiz(t

+ 2T) + Bi2(t + T) + Ciz(t) = Dy(t + 2T) + Ey(t + T) + Fy(t)

(I)

where the coefficients A, B, C, D, E, and F are to be determined. (a) Using the following relationship in equation (I), .sf[i(t

+ nT») = e"T'9'[i(t)]

and

obtain the Laplace transform of the difference equation. (h) If the exponential operators in (a) are expanded in a Taylor's series, and we equate the same powers of s on both sides of the difference equation, show that difference equation for a second-order integration is

;.J, + 2"1') -

2'2(1

+ 1') + ;2(1) .,.~

-

IIlT~

'/(1

I.' '

I .''f')

.,.2

I.' ,lIlt I 1') I I.' !I(t)

(2)

316

THEORY AND APPLICATION OF THE z-TRANSFORM

(e) Show that the z-transform of (2), neglecting initial conditions, is given by the following z-form of l/s2

.I"W 'dI(z)

T"z"+10z+1

= 12

(z _ 1)2

(d) If initial conditions are not zero, and if yeT) and i,,(T) are expressed as

follows:

T2

y( T)

= yeO) + Ty'(O) + 2!

yW(O)

+ ...

iz(T)

= yH'(O) +

+ 2! yeO) + ...

T2

T,II(-1'(O)

show that Ja(z) is given by

T2 z2

J 2(1.) =

12

+ 10% + 1

T2 z(z + 5)

-12 (1.

z

+ %_

(z _ 1)2&(%)

T3

- 1)2 !J(O)

+ 12 (z

1 yH'(O)

+ (z

To

z

%

Tz _ 1)2,111 1'(0)

_ 1)2 y'(O) - 180 (1. _ 1)2 //"(0)

+ ...

(e) Similarly, for one integration, show that

.1 1(:)

= -T: -+-I :f(z) 2Z - 1

-

Tz 2(1. - I)

yeO)

z z- I

+ _-!II -11(0), where yl 1)(0)

=J!I JII,

CJ

PROBLEM 7.5 Using the :-forms obtained in (7.4) and the 1.-transfoflll, compare the approximate solution of the following differential equation with the exact solution:

d 2y

ill"

(Iy

+ 3 dl + 2!/ = 0,

yeO)

= 3,

y'(O)

= -4

= 0.1

Choose T

second. 7.6 If the exact solution of the following time-varying equation

PROBLEM

d2 -

~2

[(/ 2 -

d

1)!I] - -~ [2/y] - 6y

= 0,

yeO)

=

-0.5,

!I'(O)

=0

is yet)

= 0.5(3/ 2

-

1),

OS,IS,I

Obtain the approximate solution for T = 0.1 using the z-forms and the ,:-transfOfm and compare with the exact solution. Chapler 8 PROBLEM 8.1 In the analysis of one-way transmission ch.mnels. sampling uf information is used. Figure 8.10 shows the elements of a connectiun cllllt:lining a unilateral repeater which isolates the two media (m()dern~) .lIld pmvidcs each with a matching termination.

PROBLEMS

317

(II)

2

2

(b)

FIGURE PROB.8.1 (a) If the pulse length h is assumed much less than the sampling period T, show that the circuit in Fig. 8.10 is equivalent to Fig. 8.lb, where x

I,

= c ~ ejUJtF,(jw)!J(t

- nT)

11 -II

and Fr(jw) is an impedance ratio function of the frequency (w). (b) If the voltage ratio 2v21v\ produced by insertion of the whole system is denoted as T(jw) and we assume that the intervening network (1-2) is eliminated, show that the modulus of transmission is 4exe-«(cosh « - cos y) IT(J·w)1 = It 2 ex + y where « = TI RIc, and y = wT. (c) Plot the modulus of transmission IT(jm)1 for various values of « as a function of !".If, = ratio between modulating and sampling frequencies. Hilll; See reference 26 of Chapter 8. PROBLEM 8.2 The circuit of a periodically switched network which produces a phase lag is shown in Fig. 8.20. The actuation of the sampler is such that the left-hand loop is closed for k T ~ t < (k + ,)T and open for (k + ,)T ~ t < (k + I )T, and that the right-hand loop is closed for (k + O.S)T ~ t < (k + 0.5 + ,)T lind open for kT ~ t < (k + O.S)T and (k + 0.5 + h)T ~ I < (I.. + I )1~ wherc () < ,) ~ 0.5 and k = 0, 1,2, .... The input to the network is a carricl"-lIupprcssed iUnplitudc-mudulatcd sinusoid given by ('I(t)

. (',(I) CII!> (UI

,

'M,

318

THEORY AND APPLICATION OF THE 2-TRANSFORM

N

00

~ ...

~

.....

e

8: I.IJ

gc:

~ <:)

[i:;

.!

-: ~

..

~

"3

"0

0

::Ii

~I e 8

ai»

~

I

~

~

PROBLEMS

319

(0) Show that the circuit system in Fig. 8.20 is equivalent to the block diagram Fig. 8.2b. where I; is an infinitely small quantity

I

I

= R 1cs + I •

G1(s)

G.Js)

= R"cs + 1

(b) If et(t) is a step function of magnitude

kEo[l -

Eo. show that the response is

e-(k+1)6TIR~][e-(t-(k+O.ii)TJ/Rle]

ea(t) = -~----I:--_-e--T.6T;r,/~R~~c:-----~ (ult - (k

+ O.5)]T -

lI[t - (k

+ ~ + O.5)]T)

where lI(t) is a unit step and I

Rt

= l/R) + I/R2 URIc sin (!MT

+

tf.)

+ cos (O~T + tf.) - (ORlcsin tf. 02R~C2 + I

+ cos tf.)e-6TIRI~

k=---------=-=-....,.....;,,.......;:....-----=.----=..;--~

PROBLEM 8.3 Sometimes inventory control problems can be represented as staggered sampled-data feedback systems. One such problem is that concerned with the maintenance of a certain quantity of parts in a storage location. This system can be represented in a simplified form as shown in Fig. 8.3. If

oT=

1. Gain eft)

FIGURE PROB. 8.3 (a) Determine the value of 61 (the sampler phasing) that yields the maximum gain k for stability limit. (b) Compare the gain obtained in (0) with the maximum gain for 61 = 0 and "I

= 1.

8.4 THE TOYMAKER PROBLEM· The toymaker is involved in the toy business. He may be in either of two states. He is in the first state if his toy is in favor and in the second state if his toy is uut of favor with the public. Suppose that when he is in state I there is a 50% chance of II transition to state 2. When he is in state 2. he experiments with new tuys und he Illuy return til stute I after a week with probability I or remain unpmlitable in !ilute 2 with probubility g. PROBLEM

• Thill pl"llblcm I, uhlllillcd rmlll Itclcrolll."O (ZU). ('l1ul,lol' II.

320 THEORY AND APPLICATION OF THE z-TRANSFORM (0) Find the transition matrix P which describes the discrete Markov process

P

= [Ptil = [:::

::J

Define a state probability 'trt(n), the probability that the system will occupy state i after n transitions if its state at n = 0 is known. (b) Write the vector difference equation which describes the probability that the system occupies each of its states after n moves; that is, the probability to be in state 1 after n weeks if we know he is in state I at the beginning of the n-week period, n = 0, 1, 2, .•.• (e) Apply the z-transform to the vector difference equation of (b). Assume P is given as in (0). (d) Find H(n) the inverse transform of (I - zP)-I. I is the identity matrix. (e) Obtain the state-probability vector at time n, knowing the initial state probability vector. ({) If the toymaker starts in the successful state I, obtain ,,(0), the initial state. Also obtain (e) for this particular initial state. (g) Show that when n -+ 00, the state probability components approach the limiting state probabilities of the process.

INDEX

Abscissa of absolute convergence. 20 Absolute probability. 271 Absolutely convergent series. 252 Absorbing st:lte. 270 Algorithm. 231 Alper. P.. 78. 188 Amplitude. 66 Amplitude characteristic. 211. 212 Amplitude distribution. 255 Analog-to-digital converters. 20 I Analytic function. 10 Analytic. in the left half plane. 260 in the riglit half plane. 260 in the ring. 161 Andeen. R_. 217 Annulus bf convergence. 144. 151 Antenna theory. 248 Aperiodicity condition. 112 Aperiodicity criterion for linear discrete systems. 112 Aperiodic persistent states. 270 Approximate solution. 69 Approximating a continuous function. 15 Approximation. mean rectangular. 222 midinterval rectangular. 222 paraholic, 222 rcctOlngulilr, 222 tri'lI1glllar. 222 Approximation methods, 219 Army pCllynulIlilll. 2~~ Al'/hulIVkh, I. S.. 1·111 A!oc1Ii11l', J. A., .1.1, 71., '1'1

Attenuating constant. 275 Auto-correlation function, 163 Automatic gain control, 69 Autonomous behavior, 268 Autonomous discrete systems. 189 AVe (automatic volume control), 70 Backward difference. 4 Barker. R. H .• 43, 296 Barnes. J. L .• 38, 42. 172 Beamwidth. 258 Bellman. R.• 77. 276 Bernoulli. number. 232. 235, 303, 311 polynomial, 232. 288, 302 Bertram, J. E., 218 Bharucha, B. A., 173 Bifrequency system function. 67 Bilinear transformation. 83 Binomial. 119 Bivariate generating function. 73 Boole. G., 76 Boundary condition. 52 Boundary value problems. 73 Bounded. 81 Bounded function. 161 Boundedness and symmetry. 162 Boxer. R.• 231. 232. 241. 246. 247 Branch cut. 13. 151 Branch points. 9. 145 Bremmer. H .• 43 Hl'idlli;md. T. F .• Jr .• 42. 247 IImlllwitch Inll:IIr11I. 20 IIrnwlI, Ii. M., 411, 42, 44, 77, I W, 247 ~21

322

INDEX

Brule, John D., 276 Bubb, F. W., 76 Budnicki, Zdzislaw, 277 Cattermole, K. W., 276 Cauchy,.82 Cauchy's integral formula, 11 Cauchy's mapping theorem, 124 Cauchy's theorem on residues, 145 Chang, S. S. L., 277 Chapman-Kolomogorov equation, 271 Characteristic equation, 81 Characteristic roots, 54, 219 Cheng, D. K., 43, 276 Chow,C. K., 217 Chow, T. S., 140 Christiansen, P. L., 277 Classical method, 45 Clockwise direction, 22 Closed-form solution, approximate, 174 Coding theory, 267 Cohn, A., 79, 131, 139 Commensurable time delays, 56 Complex convolution formula, proof, 167 Complex convolution integral, evaluation, 145 [145 theorem for the modified z-transform, Complex convolution theorem, 142 Complex differentiation, 7 Complex integral formula, II Complex multiplication, 6 Complex roots, 95 Complex scale change,S Complex z-plane, 275 Computational rule, 79 Computer theory, 267 Computers, digital, I Conjugate, 30 Conjugate pairs, 135 Consumption, 263 Continued fraction expansion, 85 Continuous feedback system, 227 Contour integration, 24 Controlled process, 51 Convergence, 179, 266 Convergence of solution, 263 Convergent infinite series, closed form, 33

Convolution Laplace transform, 175 Convolution modified or z-transform method, applications, 147 Convolution modified z-transform, 158, 159 Convolution z-transform, 253,312 applied to nonlinear discrete systems, 174 of certain functions, 176 method, 175 theory, 248 Cooke, R. c., 76 Correlation function, 163 Correlation functions and spectral analysis, applications, 163 Correlation functions, continuous, 165 Critical aperiodicity constraints, liS Critical constraints, 94, 115 Critical To' 206 Crosshatch, 143 Crosshatched area, lSI Cruickshank, A. J. 0., 220, 246 Cuenod, M., 231. 246, 247 Cypkin, J. A., 39, 42, 43, 77, 216 Set also Tsypkin, Y. Z. Da-Chuan, Shao, 218 doc component, 196 Davis, H., 76 Delay, 228 fictitious negative, 16 pure, 16, 51 time, 51 Delay factors, 193 relative. 19 I, 192 Delay operator £-1,66 Delayed sampler operation, 302 Delfeld, F. R., 218 DeMoivre, I, 42, 270 Derivatives. 51 De Russo, P. M., 277 Describing function, 213 Determinant method, 10, 136 and table form, 101 relationship, 101 Determinant method and Hurwit7. criterion relationship, 97 Determinant methods, 79 newey, A. G., HIli

INDEX

Difference equation, 264 fourth-order, 266 Difference equation with noncon5tant coefficients, 68 Difference equations, 219 linear, I, 248 nonlinear. 248 simultaneous (with periodic coefficients). solution of, 58 Difference equ:.tions whose coefficients are periodic functions, 16, 48, 57 Difference-differential equation, 16, 219, 305 mixed,51 and sum equations, 142, 158 unmixed,64 Differential equation, 243 Differential equation in the variable z, 59 Differential equations, approximate solution, 219 first-order, 52 time-varying. 241 Differentiation with respect to second independent variable, 9 Digital computers, 79,219 Digital-com.p.~!!!.r.approach, 231 Digital control systems, 28, 189 'Ufgii3i filter. 263· Dirac (delta) function, 37 Direct method, 130 Dirichlet series transform, 38 Discontinuities. 27 Discontinuity, 3 Discrete antenna array, 254 Discrete filter. 315 Discrete filters, 156 Discrete Markov process, 320 Discrete Markov processes, 270 Discrete square error, 311 Discrete systems, 36 higher-order, 253 time-varying, 81 Discrete time function, 2, 278 l)i~til1ct im:ducihle (1olynomi:lls, 268 Division, synthetic. 10 llivisiClII IIlI'thllll, 7'1, III'). I W Hurlsdl. (i. "1 .., I. n

323

Double-difference system function, 67 Double z-transformation, 73 Economic fluctuations. 266 Economic systems. 45. 248 Egan,1. F., 276 Eigenvalues, 208. 307 Elspas. B., 270, 276 Equilibrium point. 2S0 Equilibrium points, 2 10, 212 Equilibrium solution, 180 Equilibrium value. point, 175 Erdelyi, A., 77 Error, analysis of, 239 approximation, 239 Euclidian algorithm. 130 Euler-Maclurian formula. 232 Evaluations of the integral of the square time functions. 160 Extrema, 309 Feedback continuous system with lime delay, S3 Feller, W .• 78, 276 Fibonacci's numbers, 305 Fibonacci sequence, 301 Fictitiously advancing delay, 17 Fi<:vet, M., 78 Filtering theory, 248, 258 Final value theorem, 51. 194,274 Finite field, 267 Finite pulse, train of, 190 Finite pulsed sampled-data feedback system, 190, 192 Finite state, discrete Markov process, 248,270 Finite summation, 5 Forcing function (sinusoidal), 208 Fort. T., 38, 42, 77 Forward difference, 4 Fourier polynomials. 213 Fourier series, 32, 148 summation, 32 Franklin, G. F., 42, 173,246 Freeman, H., 43 Frequency 1Iliasing, 258 Frcqucncy entminment, 211 Frieillami. It .. 4~, 76, 173,276 Frnlt'll trlll1!1ft'I' function. (,'I, 71

324

INDEX

Fujiwara, M., 84, 139 Fukaw, A., 73 Fuller, A. T., 141 Fuller's conditions, 112, 113 Functional relationships, 179 Fundamental malrix, 272, 274 Gallieri, C. A., 43 Gantmacher, F. R., 139 Gardner, M. F., 38,42, 172 Gelfund, A. 0., 76 Generating function, 1,270,303 Generation rule, 105 Gibson, J. F.., 247 Global period Tn' 192, 195 Goldberg, S., 76, 276 Goldstein, M. S., 247 Gradstein, E. S., 78 Growth factor, 264 Guichet, P. L., 218 Gupta, S. C., 140, 141 H N (Hermite), 282

Hadamard, 82 Hartmanis, J., 276 Hayashi, C .. 212,218 Heizman, C., 246, 247 Hellman, 0., 247 Helm, H. A., 42 Hermite polynomial, 13 Higgins, T. J., 277 Ho, E.C., 76 Hold,200 mathematical, 220 triangular fictitiolls, 229 Hold circuit, 258 Howard, R. A., 276 Huffman, D. A., 276 Hufncgel, R. G., 77 Huggins, W. H., 247 Hurwitz cor,dition, 83, 136 Hurwitz-Routh, 79, 84 Hybrid systems, IS Impedance ratio, 317 Impulse function, 21 Impulse response, 66 Income received, 265 Incremental response transform, 190

Independent variable limit value, 9 Infinite converging series, 16 Information theory, 248 application, 258 Initial and final values, 5 Initial conditions, 53, 185,223,268, 270,316 Initial state probability vector, 320 Initial values, 235, 252 Input-output relationship, 67 Integral time function transform, 149 Integrating operators, 229, 246 Integration, rectangular rule, 229 Simpson's three-eighth rule, 230 trapezoidal rule, 230 Weddle's rule, 230 Integration with respect to second independent variable, 9 Integrator, 194 Interpolator, 220 Interpolation, linear, 223, 224 point-slope, 225 quadratic, 223 second-order normal, 222 trapezoidal rule, 225 zero-order normal, 224 Interpolators, first-order normal, 222 higher-order normal, 222 second-order normal, 222 zero-order mean, 222 zero-order midinterval, 222 zero-order normal, 222 Inverse !J(z) - F(s) or (z - s) transformr-tion, t 48 Inverse Laplace and modified z-transform, equivalence, 31 Inverse Laplace transform, 23 I, 235 Inverse <m-,c-transformation, 147 Inverse z-transform, 232, 253, 265, 266, 274 tables, 46 Inversion with respect to wand z successively,74 Investment, 263 desired, 265 Irrational function of z, 13, 6 t Iterative methods, 69 Iterutive process, 150 convergent, ~ ~K

INDEX

Iwens. R. P .• 277 Izawa. K .. 216

III (Bessel). 286 I", (Bessel>. 286 Ip (Bessel). 281 James. H .• 42 Jordan. C .. 76. 247 Jumps. 17 Jump-function method. 38 Jury. E. I.. 42. 43. 77. no, 139, 140, 141. 172. 173, 188,216,217,218, 246. 247, 275, 276. 277, 296 Kadota. T. T., 217 Kalman, R. E., 216, 218 Kaplan, W .. 44, 77. 172.218 KloUer. K., 212, 218 Knopp, K .• 141 Kroneker delta, 66 Ku, Y. H., 42. 73, 172 Kuo, B. Coo 44, 218 ),-transform. 240

Ln (Laguerre), 282

. Ladder network, 46, 304 ' Lago, G. V.. 43 Laplace. P. S., 42, 270 Lapluce and modified z;-transform, relationship. 26 Laplace and z-transform, 64 relationship. 20 Laphlce-Sticltjes integral, 36 Laplacc transform, 1, 142, 141. 182, 190.226,231.315 convolution. 253 convolution theorem. 167 discrete, 39 discretization, 232 generalized, 37 of a sampled function, J67 one-sided. 20 theory. 166.258 I.attice network. 73 I.uurcllt-( 'illlchy's trunsform. 39 I.anrcllt IOcl'ics. 11 I ",wllt·n. I). II .. 4 \,77. 172 I "FI'VII', 1' .• ~')') I '·'Mlllln. V. "". I KH

325

Levy. H .• 76, 188 Lienard-Chipart. 79. 84. 141 Limit cycle, 149, 313 existence. 201 fundamental equation, 194 limitation on period, 189 period limitation. 203 sampling instants. 205 stable. 207 Limit cycles (stability study). 207 Linear array. 256 Linear difference·differcntial equations. 51 Linear differencc equations with constant coefficients. 4S Linearized system. 182 Linear partial difference equations, 74 Linear space. 3 Linear time invariant circuits. 32 Linear zone. 199 Linearity, 3 Load,46 Long division method, 219 Lowenschuss, 0., 43 Low-pass filter, 2~8 characteristics. 196 Ma, M. T .• 276 Madwed, A., 77,246.247 Marden. M., 79, 127, 139 Markov process, stationary, 274 Matrix, 208 method,45 Maxima, 258 and minima points. 20 Maximal.period, 269 Maximum value of the gain. 109 Mean square error, 259. 261 Mean square value, 164 theorem, 29 Meromorphic fUnction, 175 Meschkowski. H.• 42. 77, 276 Meserve. W. E .• 217 Mikusinski. Jan, 43,78 Mikusinski's operational calculus, 40, 41 Miller. K. S., 43. 78 MillnlUn, J., 247 Milnt· ·/lullm(III.I .. M., 76

326

INDEX

Milnes, Ho Wo, 140 Minimax principle, 117 Mishkin, Eo, 247 Mixed form, 32 Modified l-forms, 231, 236 Modified l-lransform, 15,48, 192,220, 259, 302 complex scale change, 18 differentiation with respect to m, 18 division by r, 18 finnl value, 18 initial value, 17 integration with respect to r, 18 inverse, 19 maximum or minimum points of /(1), 19 multiplicution by tAO, 18 of a kth derivative, 20 pairs, 289 real translation, linearity, 18 summation of series, 19 Modular field, 268 Modulated discrete signals, 149 Modulating frequency, 317 Modulo-p, 268 Modulus, 17.5 Monotonic conditions, 116 Monroe, Ao Jo, 77,173,218 Montel, Po, 42, 77, 173, 188,276 Mori, Mo, 43, 172 Mullin, F. J., 42, 77, 216, 247 Multirate error-sampled feedback systern, 310 Multiple poles, 272 Murphy, O. J., 218, 276,277 National income, 263 Naumov, B. N., 247 Nease, R. Fo, 216 Necessary and sufficient condition, 106 Nelson, Wo L., 217 Network theorl, 45 Neufeld, J., 42, 77 Nichols, N., 42 Nishimura, T., 173,217,277 Noise power gain, 160 Nonlinear difference equation, 207, 312 periodic solution, 215 vector form, 207

Nonlinear difference equations, 179, 189 second-order, 177 Nonlinear differential equations, 175, 253 application to, 243 Nonlinear discrete system, 142, 174, 189,206, 312 forced oscillations, 208 limit cycle, analysis of, 189 stabilily study, 206 Nonlinear functions, 175 Nonlinear gain amplifier, 199 systems, 189 Nonlinear sampled-data system, 208 Nonlinear sampled-data feedback system, 248, 313 Nonlinear system, second-order, 248 Nonlinearity, incidental, 181 magnitude of, 179 small, 179 Nonsingular solution, 207 Nontrivial solution, 18S Nonuniform arrays, 254 Nonuniform envelope function, 256 Norlund, N. E., 76, 188 Nulls, 258 Numerical analysis, 219 Numerical equation, 266 Numerieallechniques, 246 Nyquist diagram, 82 Operator, shifting, 37 Operators, 36 Operators exponential, 315 Optimization procedure, 259 Optimum filtering, 261 Orlando's formula, 97 Orthogonal, 31 Orthogonal discrete frequency, 213 Oscillations, 196, 202, 203 excessive, 112 forced, 189, 196 free, 196 harmonic, 209 periodic, 207 periodic mudes uf, 141), I HI) rch,y mude nf, I K9, 2() l, 2()~ ~ymml'lrk·nl, 11)5 Osh''''''i, R. K, 7.77

INDEX

p-phane, IcCt half of, 23 right hllif of, 22 P n (Legendre), 281 Pnm (Legendre associated function), 282,286 Pai, M. A., 77,172, 188,217,276 Partial derivatives, 208 Partial fraction expansion, 10 Partial difference equations, solution, 73 Pasteur, Franceline, 43, 218 PavJidis, T., 140 Period of the sequence, 268 Periodic coefficients, difference equations,57 Periodic forcing function, 189 Periodic inputs, 16 Periodic regime, 79 Periodic sawtooth voltage, 33 Periodic solution, 207 Periodic states, 210 Periodically switched network, 317 Periodically varying steady state, 192 Periodicity of I,(t) and g,(t), 48 Perron, 0., 140 Perturbation, method, 129 small. 207 Phase lead, 254 Phase plane, 198 space, 213 Phillips, R., 42 Physically realizable system, 231 Physical systems, 45 Physiological systems, 45 Pinney, E., 76, 212, 218 Poincare, H., 77 Polarity, 204 Poles. 69, 194.219 multiple, 12 simple, 11 Polynomial, array. 254 in z. 60 Polynomials in z-l, 10 Powc r pallern. 257 I'uwcr licrics, 219 finilc,IR2 IIIrlhllll ••, I'uwl'"

,.,r\" II 11111

"I

t1l'lI~ily,

l'lt'llil'! lilli, I 1'1 illl" IIIIIUhrl, 'tlR

It. I

327

Probability theory, 73 Programming, 219 Pulse, of unit area, 21 width of, 21 Pulse feedback, 69 Pulse-code modulation, 258 Pulse-frequence modulator, 70 Pulse-frequency modulation, 69 Pulse transfer function, 249 Pulse width h, 200 Pulse-width modulation, 200 Pulse-width modulated feedback system, 190 PWM (pulse-width modulated) system, 189, 196,313 Pyshkin, I. V .• 217 Quadratic operator, 225 Quantization. 201 Quantized level amplifier, 20 I Quantized levels, 201 Quantized level systems, 189 Radar detection systems, 267 Ragazzini. J. R., 42, 43, 173.246 Random signal. 163 Randomly varying parameters, 79 Rational coefficients, 59, 64 Rational function. 10 Raymond, F. H., 77, 247 RC circuit, 70 Real convolution, 6 Real polynomial, 105 Reciprocal pairs, 130 Rectified sine wave, 35 Recurrence method, 45 Recurrence relationship. 174. 175, 243, 252,272,307 Relationship between coefficients, 116 Relay output, 215 Relay sampled-data system. 314 Relay system, 196. 213 autonomous, 203 Relay systems, 189 Residue, 274 method,311 RCliitlues, II, 145 ~1I111

nr. I(IR, 27.~

Rr~I"III~"

IIf 1\ linrllr circllit for periodic hllllll, "

328

INDEX

Riemann sum, 232 [286 Riemann-Zeta function, generalized, Ring of convergence, 161 RL circuit, 33 Robertson, H. H., 246 Romanov, M. I., 140 Romanov conditions, 112, 113 Root distribution, 79, 309 Root locus method, 82 Roots, 308 characteristic, 54, 219 complex,9S distinct, 61 of the characteristic equation, 272 on the unit circle, 119, 120 maximum modulus of, 309 number inside a circle of radius

.,<1,119 number of, 95 reciprocal, 119, 120, 135 simple, 11,23 Rouche, E., 97, 124, 140 Routh rule No.2. 108 Ryzik, E. M., 78 s-plane, left half of the, 79 mapping of, 144 Salzer, I. M., 230, 247 Sampled-data control system, 302 Sampled-data feedback system, 108, 160,248 configuration, 29 with variable gain, 61 Sampled-data system, 189,214.227 multirate, 149 time-varying, 306 with a small nonlinearity, 181 Sampled-data systems, application, 28 Sampled-data transmission link. 258 Sampled spectral density, 311 Sampler, actuation of, 317 Sampler phasing, 319 Sampling frequency, 208, 317 Sampling interval, 243 Sampling period, 51, 190, 206, 228, 255 choice of. 238 Sampling period T, 190 Sampling rate system, periodiclllly time-vuryinM, lOS SamplinM theorC'IIl, 22K. 2111

Samuelson, P. A., 76, 139, 276 Saturating amplifier, 199,248 Saturating amplifier sampled-data feedback system. 312 Saturating gain amplifier, 213 Saturating region, 201 Saturating-type nonlinear sampled-data system, 248 Saturation-type nonlinearities, 174 Sawaragi. Y., 173 Scarborough. J. G., 247 Schelkunoff, S. A., 276 Schur, 1.,79. 140 Seal. H. L .• I, 42 Sequence, 2 Sequence t-transformation, 192 Sequential circuits, 248 linear, 267 Sequential computation, method of, 243 Shah, I. K., 277 Shaping filter, 263 Shift operator E, 39, 60 Shifting the lime axis, 185 Shifting theorem. 3, 256 Shifting transformation. 119 Sc:hur-Cohn determinants, 84, 85, 96, 127 Signal band, 258 Signal frequency, 208 Simplification of equations before ttransform application, 64 Simpson's rule, 225 Single-valued nonlinear function, 216 Singular case, 96 Singular cases in determinant and table forms, 128 Singularities, essential. 9, 12 isolated, I I regular, 23 Singularity, 80 Sittler, P. W., 276 Skip-sampling,operution, 193 process, 195 theorem, 149 Slowly vnryinll discrcte funclions, 2111, Slt.wly varyinM ~y'll'IIl, f,f. S,'cl'lml ,ll'l\\iIV, .:!w St"hilily, f,I" !f.I, '2"" hlllllllllil V \,111 vt'~,

Jilt.

INDEX SI:lhility, consideration, 79 con5111nls, 86 dcfinition, 80 Ileterminants, evaluation of, 91 of limit cycles, 79 sufficient condition for, 71 test, J09 tests for, 82 directly applied in z-plane, 84 Stability and number of roots, theorems related to, 116 Stability constraints, 87, 88, 91 critical, 94 for low-order system, 93 generation of, 105 simplification, 88, 100 Stability criterion, alternate form, 92. 93 new, 88 simplification, 127 summary, 136 Siable, asymptotically, 175, 207 state transition, 274 system, 251 Staggered sampled-data feedback system, 319 Siaircase function, 36 State probability, 320 Stute vector, 213 Stationary process, 273 Stationary signal, 164 Statistical design, 160 theory, 259 Stelldy state, the, 193 Stelldy-state amplitude, 178, 251 Steady-stute error, 25 I Stelldy-st:lle output, periodic, 209 Stcady-stute part, 32 Stcady-slate relation, 178 Stcady-stllic I'esronse, 303 of a lincnr time-invariant circuit, 35 St~':uIY'~latc valuc. 17!1 Shll'lH"li~' lIIalrix, 271 Slunt·. W. M .• 17.01.'. 7M, 172,27(, Siril' IIf illlnlvlll'ilv. 1·11. 1·1·1 SIIIIIII, W .• IKK SlIlIIllIlIlIulI ur ~"I h·~. 1·1.'

1111111111' 1""llIlIlIr, I H SIIIIIIIIUlhlllllll'.II,·III. ~III,.IIII.

"

3'29

Symmetric function. 164 Symmetry condition. 196 System theory, various area appliclItions, 248 System transfer function. 67, 68 Til (Tchebychetr), 282 Table form. 97. 137 and determinant method, relationship, 101 of stability. derivation, 121, 123 Table method, 79 Tamburelli, C., 44 Taylor's series, 315 expansion, 9, 223 Tchebychetr polynomials. second kind, 288 Thaler. S .• 231,232, 241, 246, 247 Thellier. P. L. 172.277 Thoma. V. M .. 140,141 Thomson, W. E., 247 Time-axis shifting, 253 Time-homogeneous process, 271 Time-invariant, linear, 240 Time-invariant systems. 174 Time lags, 192 Time-varying ditrerence equations, 59 Time-varying discrete system. 81 stabilily condition, 81 Time-varying equation, 316 Time-varying systems, 174 Time-varying transform method, 66 Time-varying z-transform, 66 Tinbergen, J .• 266, 276 Torng, H. C., 188,217,218 Total savings, 265 Total squared error. 3 I I Total square integrals, derivation, 1(,8 formula, 168 tables, 164, 297 Tou, I. T., 42, 173,246 Toymaker, 319 prohlem. 319 Tl'lInwclltlenlal equations. !l6 Tr:llls\'~'ml"1I1:I1 fum·lion. 232 I. :1II,r,'" 1'I11l1'lillll •.~ I II lin, ... " III II,,'lhllll,. 1(".1 ~ III",' VIII viII". fit, IllImhllllllllloll., 11111 1111111111', 1·"1 111I1I~h'lIl ~h.h·,

I'll

330

INDEX

Transition matrix, transpose of, 275 Transition probability, 271 Translational form, 45 Transmission channels, one-way, 31"6 Transmission line, 73 Transportation lag, 5 I Trapezoidal rule, 232 Tratakowskii, G. P., 69, 76 Trijinsky, W. J., 188 Trudi's determinants, 112 Truxal, J. G., 43, 247 Two-dimensional z-transform, 73 Two-sided z-transform, 165,241 extension to continuous output, 163 Tschauner, J., 42, 77, 139, 141 Tsypkin, Y. Z., 39, 42, 43, 139, 172, 188, 218, 275 see also Cypkin, J. A. Tustin, A., 246 Uncorrelated noise, 26\ Uniform array, 256 Unilateral repeater, 316 Unique inverse of !f(z), 161 Unit circle, 69, 232, 275 inside, 79, 208, 308 on. 21 I Unit pulse, 190 Unit pulse of duration one, 38 UrkowilZ, H., 277 Van Der Pol, Balth, 42 Van Der Waerden, B. L., 276 Vagant, A., 247 Vector difference equation, 272, 320 z-transform, application to, 320 Vector equation, 171 Volgin, L. N., 44 w-transform, 67 inverse, 75 Wasow, W., 175, 188, 231, 232, 233, 247 Weaver, L. E., 216 Weber, E., 180 Weiss, L., 247 Whalen, B. A., 172 Wheelon, A. D., 172

White noise, 164 Widder. D. V.• 42 Wiener filter, 263 Wilf, H. S., 140 Wilts. C. H., 43 Wolf, A. A., 42, 78. 172 Wloka,J .• 78 I-transform method, 40 z-forms, 23 I, 233 illustrative example, 237 method, 241, 314 procedure, 235 relationship, 231 table, 234 z-transform, 192, 220, 243. 248, 253, 278, 316 alternate, 44 application, to probability theory, 270 to vector difference equation, 272 convolution, ]42 dofinition, 1 direct, 213 inverse, 9 low-pass transformation, 240 one-sided, 160 properties, I regular, 195 theory, 254, 2S8 time-varying. 66 two-sided, 160 z-transform method, 174, 21S. 267 applied to economics, 263 approximation techniques, 219 direct, for determining true oscillations, 213 discrete antenna array analysis, 254 solution of linear difference equations,45 z-transform pairs, 278 z-transform expansion, 41 z-transform of special functions. 150 zw-transform, 74 Zadeh, L. A., 43, 66, 76, 77 Zero initial conditions, 304 Zero-order hold, 249 Zeros, in the unit Circle, 122, 123 of .. polynomial, 121