Nonlinear and Adaptive Control Design (Adaptive and Learning Systems for Signal Processing, Communications and Control Series)

2.5

Stabilization with Uncertainty

The fuB power of backstepping is exhibited in the presence of ullcel't,ail1 n011linearitics and unknown parameters, because for such applicat.ions no other systematic design pl'ocedul'o exist.Ii, Wo now begin the st.udy of sueh design problenls which are tho main subject of this book. The first of the design tools that will be llsed to COllnteract ullcertainty is nonlinear dam.ping.

2.5

73

STABILlZA'l'ION WITH UNCERTAINTY r--

;r

.f

lL 'I'

"---

..:

cp(. )

'C)j'

.6.(1) Figure 2.9: A system with "matched" uncertaintya(t.).

2.5.1

Nonlinear damping

'Vc int.roduc-e nonlinear damping for syst.t"ms ,,,ith "mat.ched'· ullcertainty, in which both the ullcertainty aud the control appear in the same equation. The simplest example is the scalar nonlinear system depict.ed in Figure 2.D: .1; =

11

+ !p(J:).6.(t)

(2.240)

I

where cp(x) is a known smooth nonlinearity, and .6.(/) is a bounded fuuC"t.iol1 of t. Let. us first examine the case when .6.(l) is an exponentially deeaying disturbance: .6.(t) = .6.(O)e-l.-t. (2.2.n) Can such an innocent-looking unccrtainty eause harm? One might be tempteu to ignore it and usc the linear control·u = -C.l:, which results in the closed-loop system :[ = -ex + cp(:r}.6.(O)e- kl • (2.242)

While this design may be satisfactory when cp(.1.') is bounded by a constant or a lineal' function of :r it is inadequate if "'(:1') is allowed to be any smooth nonlinear function. For example, when cp(:1:) = :r2 we have J

,j;

= -ex + .1·2 .6.(0)e- kl

(2.243)

•

.As we saw in Chapter 1, equations (1.2D)-(1.32), the solution :z:(1) of this system can he calculated explicitly using the change of variable lIJ = 1/:r: (2.244)

which yields

w(t) = [W(O) - .6.(O}.] eel c + h!

+ .6.(O} e- kt • l'

+

(2.245)

fl'

The substitution w = 1/.1: gives

x(O)(c + k) :1'(1.) = [e + k - .6.(O).1.·(O)]ect + .6.(O).l:(O)e- kl

.

(2.246)

74

DESION TOOLS FOR STABILIZATION

From (2.246) we see that the behavior of the closed-loop system (2.243) depends criticAlly 011 t.lle initial conditions .!l(0), .x(0):

(i) If .!l{0).1:(0) < c + A", the solutions x(t) are bounded and converge asymptotically to zero. (ii) The situation cbanges dramatically when .!l(O)x(O) > c + k > O. The solutions x(t) which start from such initial conditions not only diverge to infinity, but do so in finite tim.e:

.1'(£)

-+

1 { .!l(O)x(O) } 00 as t ..... t J = c + A-: In .!l(O)a'(O) _ (c + k) .

(2.247)

Note f',hat this finite escape canllot be eliminated by maldng c larger: For any valnes of c and A-: and for any nonzero value of .!l(0) there exist initial condiUons .1:(0) wbich satis(y the inequality .!l(O)x(O) > c + k. This example shows tbat in a nonlinear system, neglecting the effects of exponentially decaying disturbances or nonzero initial conditions can be catastrophic. To overcome this problem anel guarantee that x(t) Willl'eDltlin boullded for all bounded .!l{t) and for all :1'(0), we augment the control law lJ = -ex with a nonlinear damping te1'1lJ. -s(x).v:

u

=

-ex - s(:r).1:.

'Ve design s(x) for the system (2.240), using the quadratic fUllction V'(.l·) whose derivative is

li = xu+.-z;tp(x).!l(t) = -c:r2 - x!!s(.r) + x cp(:r).!l(t) .

= !.r!.'! -

(2.249)

The objective of guaranteeing global boundedness of solutions can be equivalently expressed as rendering ,7 negative outside a compact region. This is achieved with the choice (2.250) which yields the control (2.251) and tbe derivative

(2.252)

2.5

75

STABILIZATION WITH UNCERTAINTY

Comparing (2.249) with (2.252) we see that the nOll1inear damping term (2.250) is chosen to allow the complet.ion of squares in (2.252). In more complicat.ed situations we can use Young's ll1cq'llulitll, which, in ~l simplified form, st.ates that if the constant.s p > 1 and q > 1 are such that (]J - 1)(1] - 1) = 1. then for all c > 0 and all (.-r,1/) E IR? we have 1

f:P

-1:zoIP + -qf:q lul q • p

.1'11 :$

Choosing P =

I]

= 2 and

f:2

(2.253)

= 2~, (2.253) becomes (2.254)

which is the incqualit:y lIsed ill (2.252):

(2.255)

Global boundedness and convergence. Returning to (2.252), we sec Umt li' is negative whenever Ix(t)1 ~ ::J:2:. Since d(t) is a bounded disturbance, we ('onclude that ,i' is negative outside the compact residual spt, (2.256) Recalling that. V'(x) = ~:l':'\ wc concludc that. 1:l'(t)1 decreases whellcver .1:(1) is outside the set 'R, and iwncc x(l.) is bounded:

IIxli oo

~

IId IlOJ } . ma.x { l:r(O)I, 2ViiC

(2.257)

~\'Ioreover, we can draw some conclusions about t.he asymptotic bchavior of :r(t). Let. us rewrite (2.252) as

~ (~.~2)

:$

-c.t~ + d:~t)

.

(2.258)

To obt.aill c..xplicit bounds on .7:(1), we consider t.he signal J;(t)e d , Using (2.258) we get

(2.259)

76

DESIGN TOOLS FOR STABILIZATION

Integrating both sides over the interval [0, t] yields

.r"( f ) e-"d

(2.260)

Mult.iplying bot:h sides with e-:!rt ;:md using t,he fact that. Jlp· we obtain an explicit bound for .J:(f):

+ c!:! :s; Ibl + Icl.

,.1'(1) I (2.261) l::.

Sillce snp IA(T)I ~ sup IA(T)I = O:::;T~1

IIAlltXlt (2.261) leads 1:0

O$T
(2.262) which shows that :c{t) cOlwerges to t.he compact set 'R. defined ill (2.250);

lil11 dist. {.1:(t), 'R.}

1-00

= O.

(2.203)

\\Te l'eitera,1,e t.hat. these propert.ies of bonndedlless (d. (2.257)) and COIlvel'gence (cr. (2.263)) arc gna.rantced far any bounded disturbance AU) and for any smooth nonlinearity lP{a;}, including
=

2.5

77

STABILIZATION WI'l'H UNCERTAINTY

Finally, we should note that" if the disturbance ~(t) converges to zero in addition to being bounded, then t.he control (2.251) guarantees cOIn-ergencc of .c(t) to zero in addition 1.0 global boulldedness. To show t.his, let. ~(t) he n continuous nonnegative m.onotonically dec1"(~a.si7).g function SUdl that 1~(1)1 ~ Ll(t) Md lilll , _ oo Li(l) = O. Then, st.arting wit.h the first inequality from (2.260) and using all argument. almost. identkal to thc proof of Lelllma 2.'24, we obtain I.l'(t) I ~ 1·1:(0}le-r.L +

Since limt-oo Li(t/2) =

~ (Li(O)e-~1 + .6.(t/2)) .

2v"-c

a, we se(' that. lim,_oo :r(t} =

(2.264)

O.

ISS interpretation.

For interpreting t.he effect of the nonliu(,al' damping term -1i..l·cp2(.r} in (2.251) from an input-output point, of vicw, it. is yel',\' COllvenient to use the eoncept of input-tn-state stability; (cf. Appendix C) This /i.-term renders the closed-loop system ISS with respect to the clist.urbauee input .6.(1.). To show that the ISS inequality (2.12) holds for our dmipd-Ioop system with v (T) replared by tIll" dist ur bance A ( T), we repeaL the tU'gll mell t that lerl from (2.259) to (2.261), t.his time integrat.ing over t.he interval llo, fl. The result is

I:r(t)l

~

l:c(to)lc-C(/-tu )

+ _1_

which is identical to (2.12) with .8(1', s) = s = t -lo.

2ViiC 1'e-

ca 1

[sup 1.6.(1")1], 'o~"'~1 1'(1') = 2~[''''

l'

(~.2G5)

= I:,.(lo)] a.nd

Operator gain interpretation, It. is also convenient t.o int.erpret. the ('.£fel't of nonlinear damping from an operator point. of view on t.he basis of (2.257) and Figure 2.10. For all initial conditions :1:(0) such that. 1:l'(O) I < ~~. we obtain

(2.266) which shows that the nonlinear operat.or 1\- maPl)ing the dist.urbau('e ~(l) to the output .7:(t) is bounded, and its .c~-inrluced gain is

(2.267) The nonlinear damping term renders the operat.or 1\- bounded {or any positive values of n. and c. Note, however l that. (2.266) does not: provid(' a complet.e deseription of this opera,tor b('c'ause, unlike (2.257), it hides the effect of initial conditions, which can be quite dangerous for nonlinear systems. The following lemma recapitulates the properties achieved wit.h nonJinear damping as a design tool.

78

DESIGN TOOLS FOR STABlLlZA'rION

PLANT

-

A(t) u

-

CONTROLLER Figure 2.10: The bounded nonlinear Opel'll.l.ol· [(:

~{t.}

-. xCi).

Lemma 2.26 (Nonlinear Damping) Let the sll.r;tcm (2.48) be pe1tm'bed as i11

(2.268) WhC7'C

cp(x) is a (p x 1) lJed-oT of known smooth nonlinem' functiol1s, and t) is a (p x 1) 1Jecto7' of'll'llcc7tul1l. TAo71linea1'ities which a.J'C uniformly

~(.t', ll,

bounded f01' all 'lJal1J.e.r; of .1:,11, t. If Ass'lL'ITl,ptian 2.7 is satisfied with lY(a:) po.r;iti·lJc definite and 'I'U.diallll unbou.nded, then fhe control (2.269) when a.pplied to (2.268), 7"enders the closed-loop system. ISS with respect to the distu,.bance inpu./. A(a', 'II, t) and hence !JlIurantees globaI1J.nifm"1n b01J.ndcil1l.ess of .r(t) lind COTlvergence to the residual set

(2.270)

where

')'11 "2, ')'3 (ITe

cla.s.Ij-IC oo functions such that 11

l' (:1:) $ 1'2 (1·1:1) 1'a(l:l:1) ::; lY(.1!) . 1', (I:r I)

$

(2.271) (2.272)

J I Since l'(x) lind H'(:l:) arc posit.ivc definite and radially ltllbouudcd ana l'(x) iii smoot.h, there exist class-lC;x: fuuctious 71, ')'::1,,3 satisfying (2.271) and (2.272).

2.5

79

STABILIZATION WITH UNCERTAINTY

Proof. The derivative of V' (J') is

av all' T ax [J + gu] + a.~~ gep ll.

,1' =

01' [J + au] 7J x

by (2.269) = by (2.50)

by (2.254)

< -1 V{:t·) -

n

:::;

-1F(.1') - n

:::;

-IF(.L') +

It"

r·

('wF9

r J

(lW -JJ D.I.

of 'r Ll Icp\- + {FOCP x

8lI

\cp12 + -. ax 9l{JTll.

8.1' \et'I:! + IOV IIcpllIll.lloo (DI'r Dx 9

9

II~";, . Ii

(2.273)

From (2.273) it follows that ,1' is ncgativc whenever IV(.l:} t.his with (2.272) we cOllt'lnde that

J:r{l}l > 13"'

("~~!.)

=?

> II~~;;.. Combining

Ii < 0,

(2.274)

This llleans that if l.r{O)1 .$ ''iiI (jl~!;"'), then

V{x{t)) ::;

')':10,3"'

("~!!.)

,

(2.275)

which in t.urn implies tlu1t (2.2i6)

If, on the other band, 1.1:(0)1 implies

> 13]

(!I7.ll?s )

I

then 1/(:r(t» :::; \l{:r(O))! which

-

(" ....,,--) (( Combining (2.270) and (2.277) leadR t,o the global uniform boU)ulcc1ness of

x{t);

11:"11", ::;

max {,,' 0 'Yo 0 1'3"

("~!;'

),

1'j"' 0 12 (I:r{O)I) } ,

(2.278)

while (2.274) and (2.271) prove the convergence of .r(t) to t.he residual set defined ill (2.270). Finally, the ISS property of the closed-loop system with respect to the dis~urballce input ll.(J:, lJ, t) follows from Theorem C.2. 0

80

2.5.2

DEs[GN TOOLS FOR STABILIZATION

Backstepping with uncertainty

Lemma 2.26 deals with the case where the unrerblinty is in the span of the control '1/., i.e, t.he matching condition is satisfied. Combining Lemma 2.2B with Lemma 2.8 allows liS to go heyond t.he mat.rhing rase, as t.he following example ill us t, l"t1t es.

Example 2.27 Consider the system j'

~

= ~ + :l'~ arctan e6()(/.) = (1 + ~2)U + e:re~D(t) ,

(2.279a) (2.279b)

where 6.0(1.) is a bounded tim('~varyillg disturbance. Clearly, the Ullcert·ain terms in (2.279) arc not in the span of the control 'lI.. Therefore, we will design n static nonlinear controller in two steps, combining nonlinear damping and badcsteppjng.

Step 1. The staltillg point is equation (2.279a) and the choice of a virtual ('ont.rol vRriJ:luJc. Clearly, { is tlle only choice. The lact that E. is also present in the U1u'ert.aill term docs not. present a problem, since it enters that term through the bounded fundion aJ'ctan(·). In the notation of (2.268), we 11avc (2.280) The uncert.ain nonlinearity /);.1 (~,/.) is bounded: (2.281) Hence, Lemma. 2.26 can bo lIsed to design a stabilizing function for~. The unperturbed syst.em in this case would be the integrator ± = ~, for which 1:t elf is given by "(x) = !x2 and the conesponding control is o(x) -CI'T.. From (2.269) we have (2.282)

=

which results in (2.283) with the error variable

z defined as in Lemma 2.8: (2.284)

The derivative of l,l(.:z') along (2.283) is

,i-

=

by (2.280) by (2.254)

(2.285)

2.5

81

STABILIZA"rrON WITH UNCERTAIN'l.'Y

which confirms that if :; == 0, that is~ if ~ were the actual control. tben (2.282) would guarantee global uniform bOl1ndeduE'ss of x.

Step 2. Using the error variable

=from (2.284), the system (2,279) is l'cwritt.eu

as :i:

=

-

::::

(2,286a)

=

(2.286b)

where the partial ~ is comput,ed from (2.280) and (2.282): 80') 8:.1.'1

= -c) - 1i1..i!-.(J:J:

[;l'tpi(:~')] ::::

-Ct -

5Iil:c1.

(2.287)

If the ~o(t)-terII1 "Were not present in (2.286b), then LeIllllltl 2.8 would dictate the Lyapunov [ullction

I"

112 ( x,) e = 2' .1;" +

1., '2:;-

=

(2.288)

and the following choice of control:

1:

" = li("" {) =

£;2

[-C2= + ';:' { -:r] .

(2.289)

To compensate for the presence of the ~o(t)-term in (2.28Gb), Lemma 2.26 is used again. From (2.269) we obtain (2.290)

which renders the derivative of 1l2(x, e) negative outside a compact set., thus gnaranteeing bounded ness of x(t) and ~(t):

V2 by (2.285)

by (2.286b) and (2.290)

=

Ii" + zz

::; "'x _ £.yr.2 ..,

=

-Ct X2

..

+ 1I~11l~ + --:.... 4lbl

+ II~I fI~ 4lil

+= { -c!!z + [e.( -

h~2:: [ e:re -

;::£2

arctan

a.r x 2 arctan e]2

80'1

e] ~(t)}

82

DESIGN TOOLS FOR STABILIZATION

(2.291)

by (2.254)

o The combination of Lemmas 2.8 and 2.26, illustrated in t.he above example, is now formulated a..., a.nother design tool.

Lemma 2.28 (Boundedness via Backstepping) C07uJidcT Ihe system :i' = 1(.1:)

+ g(J.')u. + F(J;)~I (:1', Il, t) ,

(2.292)

where :,. E IRn. II E JR., F{:r) is an. (n x q) ma.tri:r: oj known smooth nonlinecu' Junctions, c1.1l·d ~1 (:1.', II, t) is a (q x 1) vectol' oj uncertain 1wnlineu7i.tie.i which 11.7'f. uniformly bounded J07' nil vallle.~ oj x, tt, t. Su.ppose lha.t i.here C:L'ists a. Jcedback contl'Dl 'lL = o{l') that 7-enders x(/.) globall1J uniJ01'111.111 bounded, a.nd that this is est(J,blisheti 'uia positive definite and 'f'IJ.diallll U7l.bounded junclions ll(:l:),IF(x) and (]. constant h, such that all

D.'C (.1.') [f(:&)

+ g(.1·)(1(3:) + F(:r)d l (3:,11, t)]

~ -''I'(x)

+ b.

(2.293)

Now con.r;ider the aUflmentcci .'1y.r;tem .1: = E;.

=

j(.r} + g(x)~ + F(x).6 t (.1", u, i) 'U + cp(:z:, ~)1'~2(X, €, u, I.) ,

(2.294a) (2.294b)

when!
~2 (x, ~, 1l, t) is

u =

-c (~ - ll'(x)]

Bo·

+ a:c (.1') [J(.r) + g(.T)~J -

-" [( - 0:(x)] { I'P{ x. {W +

I::

(x )F(x)

guo.ronlees globlll 'lInijonn bOltTIde.dness of ,l;(t) and h'

> O.

~(t)

all'

n

a:r: (.r)g(J:)

(2.295)

'with any c > 0 and

2.5

83

STABILIZATlON WITH UNCERTAINTY

Proof. Using the error variable z =

~ -a:(xL

(2.296)

the system (2.204) is l'ewritten as

;i'

=

., =

1(:1') + g(:l.') [a:(J') + =] + F(;v).6. (a',t" t) 1I + tp(.7·, e)T 6.:d:1:, t)

(2,297a)

e,u,

ao' (.1') [I(.1:) + g(:r){ + F{.r).6.{x, Il, t)] --a x

.

(2.297b)

The derhrative of (2,208)

along Ula solutions of (2.297) wit,b the control (2.295) is

\1;

=

8"

ax (f + go' + F.6.) + all a.'£ g=

+= [u + 1"1·60. - :: (f + g~ + FAI)] 8\1 ] < aF a.}; (/+90 +F.1d+= [aa' 'u- fJx(f+o{) + 8J,il

+= ['PTA, by (2.293)

:>

[v - :: (f + g~) + ~~g]

-W{.t) + b + =

+= [opT A. by (2.295)

=

:~ FAr]

- II' (.T) +

~~FAI]

b- c:' - ,,=' [1'P12 + I~: FrJ

+1=11'PIIIA21100 + 1=11:: FlllAtlioo by (2.25.1)

=

-IV(x) - c:: 2 + b +

1I.611l~ + 116.2 11!: . 4h'

-1 Ii

(2,290)

The radial ul1boundedness of li'(,}') combined with (2.2DD) implies lhat \~ is negative outside a compact set, which in turn implies that ,l:(t) and e(t) are globalJy uniformly bounded. 0

84

DESIGN TOOLS FOR Si'ABILJZATION

2.5.8

Robust strict-feedback systems

.Just as we generalized Lemma 2.8 to strict-feedback syst.ems in Section 2.3.1 and Lemma 2.25 to block-strict-fccdback systems in Sect.ion 2.3.3, we can generalize Lenmut 2.~8 to broader classes of Ullcertain nonlinear systems. We cOllsider syst.f'IDS ill the robu.st st1ict-Jeed1J(u:k J07"m:

= :r~ + cpT (xdd(~r:, lI, t) :1:2 :: x:i + cpI(,vl 1 x2)d(x, u., t) ";:1

(2.300)

xn + IPJ-l (.rl l •• • ,l'u_dA(.:r, 'lI, t) ;;:'1 :: {3(.I')'ll + IP;(:z:}d{x, 'u t) I

j'n-l

::

1

where (J(:J') ::F 0, V:1: E .IR", IPi(Xh"" Xi) is a (p x 1) vector of known smooth l10nlinear functions, anrl d( x, u, I) is a (p x 1) vector of uncertain nonlinearitic5 wbich are lJ.71ijoT11J.ly bounded for tul values of X, u, t.

Corollary 2.29 (Robust Strict-Feedback Systems) The .9tUte. :1'(1) oJthe 81Jstem (2.300) 1IJ'W be flloimJ,III1J.TJ.iforrnllJ bounded if the control is chosen as 1

u = {3{x) (J',,(x) , 'Ulhe7V~

i

the I.,m ctian an (.1:) i,&; defined by Ute fDllowin.g (yllhere 'we denote =0 == 0:0 == 0):

= 1, ... ,11

(2.301) 7"eC1tTlli1Je

C:L717Y!ssion.s JOT (2.302)

(2.303)

with C" Hi,

j ::

1. ... ,11 ]JOlJitive design CDnstants.

Proof. Using the definit.ions (2.302) and (2.303) and denoting Xo == 0'0 == ."l."n+l == {3(.J:)u., the derh'ative of the elTor variahle Zi, i:: 1, ... ,11, becomes

0,

2.5

85

STABILIZA'l'ION WITH UNCERTA1NTY

The choice of control (2.301) guanmtees that system can t.herefore be expressed 8S ":1

=

=n+l

== O.

The closed-loop error

-Cl=l + =:.! + cpT ~ - h"l=II'P11 2

Z2 = -C2=2 -

=1 + =:" + ('P'J - aa~~l ~I) T~ - h~2=2Icp'l - aa~J IPII~ .1'1

.ll

(2.305)

11-2

)=1

1

Zn

=

:CJ

T (

-C,I=J1 -

2

L -0-'-11')

-lin_1 =n-l 'PII-l -

•

00',,_2

n-1

+ 'P,,- L

Zn-J

0 0:1I _ 1

)

~-

- f ) "
j=1

.,

:r J

fi.nZn

CPn -

II-I

L

j=1

80,,_1 1- 8 ' 'P) "'J

Now we can use the quadratic Lyapllllov function \

1

n

_ ~_2 " -1.···' .._n ) ? L.J ':'i

' (_

.oJ

(2.306)

i=1

to prove global uniform boundedness, Indeed, the clerivat:ive of (2.306) along the solutions of (2.305) is

Vn

=

" L i=1 [

:::; L

') -CjZj

,,[

i=L

<

'1

-C:i=i

+ =i

(

+ I=il


i-I OOi-l) L - "'P) )=1 8a.)

'P; -

L

i-I ----a-:-tpj 8a:.;_1

j;1

T

~ - li.i=;
II~IIIXI

i-I 80';-1

"

- tii=;


.1 J

t [-c;..:; + /I~~I~l· ;=1

q]

;-1 OO'i-1 L - .. 'Pj )=1 OJ J

L

)=1

----a-:-If')

2]

"'J

(2.307)

41l'i

The lnst inequality implies that .:(t) is globally uniformly bounded. But from (2.303) we see that, since t,he a/s are smooth functions, Xj can be expressed as a smooth funct:ioll of Zit ... ,Zj: (2.308)

Hence, .1:(1) is globally uniformly bounded and: furthermore, COllVC.lrges to the compact residual set: 'R =

{

:1: E JRII :

L Cj;;; :::; L / I n

j=J

;=1

11~112 } ~ I

(2.309)

4""

whose size is unknowll since the bound 1I~1I1X1 is unknown.

o

86

DESIGN TOOLS FOR STABILIZATION

Notes and References Integrator backstepping is an idea whose origins are difficult to trace, berause of its simultaneous appearance, often implicit, in UIe worles of Tsinias [193]. Koditschek [84], Byrnes and Isidori [12], and Sontag and Sllsslllann [175]. Kokotovic and Sussmann [85] viewed the stabilization t.hrough an integrator as a special case of stabilization through all SPR transfer funct.ion, as in the early adaptive designs by Parks [150], Landau [109], and Narendra. et aI. (142]. This passivity view was extended to nonlinear cascades by Ortega [145] and Byrnes, Isidori, and Willellls [1~1]. Integrator backstepping as a recursive design 1.001 [85, Corollary 3,2] was employed ill the cascade design of Saberi, Kokot.ovic, and Sussmann [163] and further dC\'eloped by Kn.nellakopoulos, Kolrotovic, and iVlorse [73]. The passivity aspect of this design was pointed out by Lozano, Bl'ogliato, and Landau [116]. A tutorial overview of backstepping was given in the HJ91 Bode lecture by Kokotovic [88), Among the current. applications of backstepping al'e electric machines in the monograph of Dawson, HUt and Burg [2GI and steering and braking control by Chen and Tomizuka [17]. An important. question, not addressed in the above references, is whether backstepping designs can incorporate optimization with respect to some meaningful cost funct.ionals. It is cleat' that minimizing a part;ial cost: functional at each step docs not imp}Yl and lllay in fact contradict, tbe overall optimality. A framework for a backstepping-like recursive optimization was proposed in a Russian language paper by Kolesnil(Ov [89], which remained unnoticed in tlle English language literature. All the above references apply backstepping to systems without uncertainty. For s)'stems with uncertainty a robust nonlinear design was introduced for the makhed case in the works of Corless and Leitmann [20] and Barmish, Corless, and Leitmallll [G], and wId extended by variolls forms of "generalized matching condi1.ions·' [19, 20]. However, these extensions haven't led to a (.liscovery of hacksteppillg with uncertainty. After the development of adaptive backst.epping by Kanel1akopoulos, Kokotovic, and Nlorse {69, 87], backst.epping with uncert.ainty was pursued in the works of Qu [160], l\'lllJ'ino and Tomei [125], H'e~l11a.ll and Kolmtovic [37, 38]. and Slotille and Hedrick [168], Backst.epping designs for systems with unmeasured or unmodeled dynamics have been developed by Praly and Jiang [159], Jiang, Teel, and Praly (61], Khalil [82], WId Krstic, Sun, and Kokotovic [104]. Reeellt advances in backstepping, whidl are beyond the seope of this book, are due to Coron and Praly [21] and Praly, d'Andrea-Novel, and Coron [158]. The restrict.ioll of backsteppillg to pure-feedback systems has motivated alternative designs applicable to other classes of nonlinear syst.ellls, such as tbose by Teel [187], Teel and Pl'aly [192]. Qu [161], and Nlazenc and Praly [1261.

Chapter 3 Adaptive Backstepping Design The controllers designed in the prcceding chapter guarantee that in the presence of uncertain bounded nonlinearities the closed-loop st.ate remains boundecl. In this chapter. and in the remainder of the book, the III1(:ertainties are more specific. They consist of unknown constant. parameters which appear linearly in the syst.em equat.ions. III t.he presen('e of sueh paramet.ric uncertainties we will be able to achieve both bounded ness of the closed-loop states and convergence of the tracking errol' to zero. While all the controllers designed in Chapter 2 employ static fecdbaek, the controllers in this chapter will, in addition, employ a form of nonlinear integn\l feedback. The underlying idea in the design of this li1Jnamic part. of feedback is pa1nmeter e.r;tim.a.tion. The dynamic part of the controller is designed as a pammeter update law with which the static part is continuously mliJpteci to new parameter estimates, hence its name: AdaptitJf controllmn. In using this traditional terminology, however, we should keep in mind that so conceived adaptive controllers are but one type of nonlinear dynamic feE>dbaclc Adaptive controllers arc dynamic and therefore more romplcx than the static controllers designed in Chapt.er 2. \\That is achieved wit.h t.his additional complexity? As we will show in Section 3.1, an adaptive controller guarant.ees not only that the plant. state .1: rcmains bounded, but also that. it. t.ends t.o a desired constant value ("rcgulation") or asymptotically tracl\s a reference signal ("tracking"), The first results leading t.o a new syst.ematic design of adaptive cant-rollers 81'e presented in Section 3,2, which introduces aclapti've backstcPIJin9. The recursive design procedure for pam.met7'ic strict-feedback .r;lJ,r;tems is then developed in Section 3.3. In its basic form, t.he adaptive backstepping design employs Ollerpammetdzutiol1, that is, more t.han one estimate per unknown parameter. This means that the dynamic part of the controller is not. of minimal order. In Chapter 4, a more int.ricate backstepping procedure is developed-the tuning functions method-which employs the minimal number of parameter

88

ADAPTIVE BACKSTEPPIN'G DESICN

estimates. The e~1:ended-matching design, presented in Sect.ion 3.4, is of int,erest. as a transition between overparametrizcd and minimal-order designs. It ah~o cont,nins the first. adaptive performance results.

3.1

Adaptation as Dynamic Feedback

The difference between a static and a dynamic (that, is, adaptive) design will .first be illust.ro,t.ed on t.he simplest nonlinear syst.em:

.7: = 11 + 8cp(:r) ,

(3.1)

This is the special case of the system (2.240), where the uncertaint.y 8(1) is the unknown (·onst-ant. parameter (). Even if we do lIot know a bound for 0, we can use Lemma 2.26 to design a statk nonlinear control1er which guarantees global boundedncss of ~r(t). The nonlinear damping design (2.251) applies also here. The corresponding static controller is (3.2) 11. -c.r - /i,XCP-"( .1: ) J

=

and the result.ing closed-loop system is of first order:

x=

-cx - 1i.7:cp2(X) + Ocp(:c) .

According to (2.252), the derivative of V- =

Y~

'1

-c;r.-

4:r

2

(3.3)

satisfies

()2

+-

(3.4)

4n. '

which means t.hat 3:(t) converges to the interval

(3.5) This interval can be reduced by increasing the gains n. and c, but x(t) will not converge t.o zero if B is a nonzero constant. Excessive increase of these gains enlarges the syst.em bandwidth, which is undesirable. Our task is therefore to achieve lim/-ooo x(t) 0 without increasing Ii, and c. In fact, we will first accomplish this t,ask with n. = 0, and then use Ii. > 0 for further improvement of transients. To achieve regulat.ion of x(t), we design a dynamic feedback controller, that is, we employ adaptation. If () were known, the control

=

'U

= -B
Cl.1:, Cl

=

>0

(3.6)

=

2 would render t.he derivative of Vo{x) !x2 negative definite: Yo -CIX • Of coursE' t.he cOlltrollaw (3.6) CRn not be implemented, since 0 is unknown.

3.1

8D

ADAPTATION AS DYNAMIC FEEDBACK

Instea.d, one Ctl.ll employ its ce1'i.aintIJ-eqltillalence form in which 8 is replaced by an est.imate 0: 11 = -8cp(.1:} - Cl:r. (3.7)

Sllbst,jtuting (3.7) illt:O (3.B),

W~

j:

where

obtain

=

+ iil;'(x) ,

-CIX

(3.S)

ii is tbe pU'I'Umelel' e11"01'; (3.9)

The derhrative of ''0(:-1'}

= ~.1:2 becomes li"o =

-Cl x

2

+ O:rcp(:I:) ,

(3.10)

Since the second t.el'm is indefinite and contains the unknowll parameter ("rror ii, we can not conclude anything about. t;he st,ability of (3.6). 'Vc make the controller dynamic with A,n update law for To design t.his uprlate law, we augment Vo with a quadratic tP.l'fll in tbe parameter error 0,

e.

1 'I 1 -" Vi(x. , 8) = _;rM + -02' 21" lvhere /' > 0 is the adc/.IJlation gain. .

V'l

Tlu~

(3.ll)

derivat.ive of tbis func'tioll is

1 -:.

= .1::i- + -88 1 =

'I 1 -:. -CIX- + OXcp{.l') + -88

l'

=

-ClX~

l)

+8 -

[

"10

a'cp{;l'} + 1 :..]

(3.12)

e

The second term is sUll indefinii;e and conbuns as a fad.ol'. However, t,hc ~ituati(~)11 is much bet,tel' t.hall in (3.l0), because we now h~we t.11e dym:Ullics of

8 = -0 at

our disposal. With the appropriat,e choice of indefinite tcrm. Thus, we choose thc update law

jj = which yields

l~

-0 =

I :rcp(:c) ,

= -e)I?;!!::; O.

9 we can

callcel t.he

(3.13) (3.1<J)

The resulting adaptive system consists of (3.1) with the control (3.7) and the update law (3.13), and is shown in Figure 3.1. In Figure 3.2, t;hi~ system is redrawn in its closed-loop form consisting of (3.8) and (3.13), nAmely

+ 8cp(x)

;1' =

-C),T.

8

-")' xcp(x).

=

(3.15)

90

ADAPTIVE BACI(STEPPING DESIGN

PLANT

I I I

I I

···: I

I

I I ~

•

___


_______ M ______

~

•• ____

~

~

__ •

_____ . _ . _ . ___ •

______ M _______________

~

________

~

__ • _________

_

cp(x) ADAPTIVE CONTROLLER

Figure 3.1: The closed-loop adaptive system (3.15).

=

=

Because l~ ~ 0, the equilibrium :1; 0, 0 0 of (3.15) is global1y stable. In addition, the desired regulation property li1l1t_oo x{i) = 0 follows from the LaSalIe-Yoshizawa thC!orem (Theorem 2.1). The adaptive nonlinear controller which guarantees these properties is given by (3.8) and (3.13): 'U

=

8 =

-C1X -

8cp(x) (3.16)

'Y xcp(x) .

One may think that the above adaptive design is so straightfonvard because (3.1) is a first-order system. In fact, this is due to the matching condition: The terms containing unknown parameters in (3.1) are ill the span of the control, that is, they can be directly cancelled by u when f:} is known. To illustrate this point., let us consider the following second-order system, where again the uncertain term is "matched" by the control u;

Xl :;:2

= =

+ 'Pl(X1) 8!.p2(X) + u. X2

If (J were known, we would be able to apply Lemma 2.8: First view the virt.ual control, design the stabilizing function

(3.17) :C2

as

(3.18)

3.1

91

ADAPTATION AS DVNAM1C FEEDBACK

(J

x

tp(x)

J

1+--------'

Figure 3.2: An equivalent representation of (3.15). and thon form tbe Lyapunov function

V~(x}= ~:l:i+4(X!! -al(:t:d)2

(3.19)

t

whose derivative is rendered negative definite t~(x)

= -clxi -

c!! (x!! -

Cll)2

(3.20)

by the control

Since 8 is unknown, we again replace it with its estimate iJ in {3.21} to obtaiu the adaptivo control law: 'il

== -C2 (x:! -

0:1) - Xl

This results in the error system

80:1 + -a (:V2 + CPI) -

-

f}CP2(X).

Xl

(Zl

=

Xl, Z!!

=

X2 -

(3.22)

0'1):

(3.23)

Then we augment (3.20) with a quadratic term in the parameter error obtain the Lyapunov function: -

1 -.,

1.,

1..,

1

~')

l'i(z,6) = If.: + -0- = ;;;;; + -2=2 + ;;-0-. 21

-

-7

0 to

(3.24)

Its derivative is (3.25)

92

ADAPT[VE BACl(STEPPING DESIGN

/1+----Figure 3.3: The closed-loop adaptive sysLem (3.28).

The choice of update law (3.26)

plimint'l.tcs the jj~terll1 in {3.25} and reuders t.he derivativc of the Lya.punov fuuction (3.2"1) nonpositive: (3.27)

=

=

This implies tbo.t the z 0, jj 0 equilibriulll point of the closed-loop adaptive systenl consisting of (3.23) and (3.26) (see block diagram in Figure 3.3)

:, [ ~~] = ij

=

[~i -~~] [~~ ]+ [ \02~X) ] fl -'Y [0 \02 J[ ~~ 1

is globally stable and, in addition, :c{t) -;. 0 as t

3.2 3.2.1

(3.28)

-+ 00.

Adaptive Backstepping Adaptive integrator backstepping

The adapt:ive design iu the ltbove c..'\\:amples was simple because of the umtching: The pluametric nnccdainty wa.s ill the spall of the control. Vve now move to the more genera.l c.asc of extended matching, where the pnrametric uncertainty enters the system one integrator before the coutrol does:

.1:1 X2

= =

.1:2 'll.

+ 8tp{:I.'I}

(3.20a) (3.29b)

3.2

93

ADAPTIVE BACKSTEPPING

,\Ve use this e.xample to introduce ad{l.ptive 1m ckstepping. If (J were known, we would apply Lemma 2.8 t.o design the stabilizing function for 3:2 (3.30) 0'1 (.1;1,0) = -elI) - 8cp(J'd , wit.h the Lyapunov function {3.31}

whose derivative is rendered negative den.nit.(:'

(3.32) by the control II

=

-C2 (X2 -

(]'1(:J:l, 0))

-

D0

Xj

1 + ~(.1:2 + fJcp).

(3.33)

UX1

Since (J is unknown and appeam one C{luatioll before t.hl:' cont.rol docs, we can not apply Lemma 2.8 because the dl:'pendence of (tl (~r:I) = -Ct.}:j - O'P(:CI) on the un}nlown parameter makes it impossible to contiuue t.he procedure. However, we can utilize the idea of int.egrator backstepping. Step 1. If .1.'2 were the control. an adaptive controlJer for (3.290.) would be given by (3.16): 0'1 (Xl!

t9d =

01

=

-Cl=l

(3.34)

-17 j 'P(J'd

(3.35)

1'=1CP(:Z:I)'

In the above equations we have replaced the parameter est.imatc

n

with the estimate th, which denotes the estimat.e generat.ed ill this design st.ep. As we will see, there will be another estimat.c generat.ed in the next. step. 'Vit.h (3.34) and t.he new error variable z:! = .l."::! - C\ I, t.he =I-eqnatioll becomes (3.36)

The derivative of the Lyapullov function

(3.37)

(3.38)

ADAPTIVE BACKSTEPPINC DESIGN

Step 2. The derivative of Z2 is

llOW

eJ..1l1·essed as

Substituting (3.29a) and the update law (3.35) results in =2

= =

oal

U'lL -

s;- (x!! uXI

oat

+ Otp) -

at) 7tp Zl !

00'1

00']

DCXl

aXl

OVl

aJ~l

-X:z - -7tpZl - 6-tp.

(3.39)

At this point we need to select a Lyapunov fUllction and design II to render its derivative non positive. Onr first, attelnpt is the augmented Lyapullov function

It;(z!, =2t nd = Vi(z), '19 1 )

+ ~z~,

whose derivative, using (3.38) and (3.39), is

The control u should now he able to cancel the indefinite terms in l~. To deal with the terms containing the unknmvn parameter 0, we will try to employ t.he e..~sting estimate '19):

From the resulting derivative

.

'1"

)80'}

V:! = -cl zi - c2 zi - (fJ - '191 -8 'PI:;!!, ·'1:1

we see that we have no design freedom left to cancel the (8 - iJ1)-term. To overcome this difficulty, we replace () I in the expression for u with a ne'W estimate t?!!: (3.40) \Vith the choice (3.40), the =2-e{luation becomes •

- = -C'Jz., - --

':'J

%1 -

(0 -

EJo'!

'{)'J)-Ill

- 8xI .,...

(3.41)

3.2

95

ADAPTIVE BACI<STEPPING

The presence of the new paramet.er estimate 192 suggests the following augmentation of the Lyapunov function:

(3.42)

The deriv{1,tivc of

"2 is

(3.43)

Now tho (0 - O!!)-term can be eliminated with the update law .

-02

an1

= -l'-a cpz'!. , Xl

(3.44)

which yields (3.45)

The equations (3.41) and (3.44) along wit.h (3.36) and (3.35) 1'01'111 I.ho error system representation of the result.ing closed-loop adaptive syst:om:

+ Z2 + (fJ - D1)cp

=1

=

-Cl=l

z~

=

-C'l':'1 - -

ill = 172

=

=1 -

(0 - O.,)~cp - Dx,

(3.46)

"Y CP=l - "'I thncp8x, -2,

The matrix form of this system,

d dt

['17) ] {)2

=

"[

~ -~,,] [ ~~ ] ,

(3.47)

makes its properties more visible: • The constant system matrix has negative terms along it.s diagonal, while its off-diagonal terms arc skew-symmetric, and • the matrix that multiplieR the parameter errors in the i-equat.ion is used in the update Jaws for the parameter estimates.

96

AOAPTIVE BACI(S'l'EPPING DESIGN

Tho stability properties of (3.47) follow from (3.42) and (3.45): The LaSalleYoshizawa theorem (Theorem 2,1) establishes tha.t =1! :2, {)I! 11'1 are bounded, and = -io 0 as i - 7 00. Sin<.'e':J :l'l, .1'1 is also hounded a.nd converges to zero. The boundedncs5 of J:2 then follows from the hounded ness of 0'1 (defined =2 + 0'1' Using (3.40) we conclude thai: the in (3.34)) and the fact. thai. .1'2 conf;rol 11 is also boullded. Finally, we note that the regula.tion of and :1:1 does nDt imply t.he regnlat.ion of :1'2: From =2 = X2 - al and (3.34) we see that :1'2 + t?1'P(O) will converge t.o zero. Thus, .1:2 is not guaranteed to converge to zero unless cp(O) O. Howevcl', .}'2 will converge to a constant value:

=

=

=

=

lim = -Hcp(O)

'~co

This can be seen .1:2

frolll

(3.29a): Since

Xl

b= :1'2 .

(3.48)

and j:1 (.'OllVerge to zero, so uoes

+ 8cp(0),

\Vith the above example we have illust.rated the idea of adapt.ive backstopping. To formulat.e it as it design tool analogous to Lemma 2.8, wo start wit.h the assumption thctt an adaptive l'ontroller is known for an init.ial system,

Assumption 3.1 COJ1,Sitlc1' the system .1: = f(x)

+ F(:r)O + g(J')u ,

(3A9)

whe1'f~

:,; E JR" is the ./itate., (J E HlP is a vector of unknown cOfl.st.a.nt p(l.7'Q.m.eter's, a.nd 11. E 1R is the r..ontml input, There exist (/.71. adaplil1e controller

= J = 'lJ.

0:(:,;,19)

T(x, tl),

(3.50)

with parnm.ete1· eBtima.te {) E IRq. and a 8mooth Junction V"(x, '19) : IR(r,+q) -io R wMch is positit1e definite a.nd 1'adia1l11 'Unbounded in the va1'iables (:1', {)-H) /:WI"/t that/oT all (.1:,,19) E lR.
81'

8:1: (:1.', l) [j'(:1:)

1IJhe1'f~

al'

+ F(:I.·)O + g(x)o{x, 19)] + atl (x, t9)T(:r;, tl) :5

H' : Dl,,+q

~

lR i.5 positivc semidefinite.

-H'(:1:,O) :5 0, (3.51) 0

Under t.his assnmption, the control (3.50), applied to the system (3A9), guarantees global boundedness of .r(t), 'O(t) and, by the LaS al1 e-YoshizRwa theorem (Theorem 2.1), regulation of IV (x(t) , iJ(I.)). Adaptive ba(~kst.cpping allows us to achieve the samp. properties for the augmented system.

Lemma 3.2 (Adaptive Backstepping) Let the systcm (3.49) be augm.ented b11 an integm.tor,

±

=

f(x)

~

=

u,

+ F(:v)(J + g(:c){

(3.52a)

(3.52b)

3.2

9i

ADAPTiVE BACl(STEPPINC

whe7'e

eE JR.

Consider for this .'f1Jste.m the dynamic feedback controlle7'

11. =

] + au a:t (:1', lJ) [/(3:) + F(;r)tJ+ g(:r:){

-c(~ - Q'(x, 17))

ao:

DV'

+ 8{)T(x, 19) - ax (;1:, t7}g(:l') , J

=

c> 0

(3.53)

T(x, iJ)

(3.54)

J = -r [:: (x, 11)F(Xf (t; - a (.r., 11)),

(3-55)

where {j is a neul cstima.te oj fJ, r = r 1' > 0 is the adaptation gain maida:. Under Assumption 3.1, this adll.ptive cont1TJller g1l.c1.mntees global bo1t7l.dcdness oj x( t), ~(t), D(t), 0(1) a.nd 7"C!J'ull1.tion of fI'(J:( l), t?(t)) and ~(t) -ll'(~l:(t), iJ( t»). These ProlJe7'tie..r; can be e.r;tablished 1JJith lile Lyapunov ju.nction 1 1 -" 1 l~(:l:, ~t iJ, 17) = "(x, t~) + 2' [{ - O'(:l~, ,a)]- + 2(8 - '0) r- (8 -19) . (3.56) I)

e-

Proof. vVith t.lm error variable == 0:(;1:,19), (3.52) is rewritten as x = J(:I:) + F(x)() + g(x) [0:(.1;,11) + ~] (3.57a)

z

=

u - : : (x, iJ) [J(.'l:)

+ F(;r)O + g(.1:) (O'(x, ·O) + :;)] - :~'T{:r, .[)).

(3.57b)

Note that in (3.Sib) the derh'ative of {) was replaced by t.he update law (3.54). Introducing a new parameter estimate 11, we augment the Lyapullov function:

=

F(a:,·tJ) + _?1:;2 + !(8 -- ii)Tr- 1 (8 - 11) . ... 2 Using (3.51), it is easy to Rbow that the deriva.tive of (3.58) satisfies l'o.(X'';1 0,19)

all

.

(3.58)

all'

Va = a3.~ (J + FIJ+ go + g:;) + {J.t) T +: [II -

=

81'" 8J.:

00' (J

8x

lY (x I 19)

+ cH) T

+ F8 + g(a: + z}) - a~'T + av' 9] _,nTr- 1 (fJ af) 8:r

+ z [u. -

- [:: F: + jT

:;T] - ilr-'(9 - ;i)

811

(f + FB + go')

+:: [u 5 -

~: {f + F9 + 9(0' + =)) -

r-']

i9)

oa (f + F';9 + g( a + .:») _ 80: T + at' gJ o.T at) ua' (9 - 0).

(3.59)

Tlle (8 - 79}-term is now eliminated with the update law (d. (3.55))

~

=

-r (ao~F)TZ' u:J.

(3.60)

98

ADAPTIVE BAC1{STEPPING DESIGN

and the control (3.53) is cbosen to make the bracketed term multiplying :- in (3.59) equal to -cz (ef. (2.54)): IJ

=

aa ( -

-c=+ 8J.' f+FtI+g(a+=)

) + 80' al' 81?T- a~.g·

(3.61)

This results in the desired l1on]>ositivity of l~:

\~ :::; -rV(:L',1?) - c:2

:::;

o.

(3.62)

From (3.56) and (3.62) we conclude that "'(3:,0), i9 and z are bounded. By Assumption 3.t this means that. x(t) and l'(t) are bounded. I-lel1t'e,'; = z + a(.l:, ·0) and It are bounded. By Thorem 2.1, the boundedness of all the signals combined wit.h (3.62) proves the reglllat:ion of H'(x(t), t9(t)) and =(1).

o

3.2.2

Adaptive block backstepping

We now extend the Adaptive Baclc.stepping Lemma (Lemma 3.2) byaugmenting the initial system with a relative-degree-one nonlineaT system whose zero dynamics subsyst.em is ISS, just-like we did in Chapter 2, Lemmas 2.8 and 2.25. The adaptive counterpart of Assumpt.ion 2.7 was Assumption 3.1. "Ve now formulate the adaptive counterpart of Assumption 2,21, with analogolls changes in the properties of l/(x, t?) from Assumption 3.1,

Assumption 3.3 Suppose Assumpti.on 3.1 is 'Volid, b,J.t l'{x,19) is onl,} positi'Ve semidefinite. and the closed-loop system {3.49} 'U1ith the atJ.apf.ilJC controller (3.50) h{t.s the lJropert1} that x{t) a.nd 11(t) are bounded if l!(x(t), tI(t)) is bounded. 0 Under this assumption, the control (3.50), applied to the system (3.49), guw'antees global bouudedness of .t·(t), t9(t) and, by Lemma A.6, regulation of

IV(x(t) , O{t)).

Lemma 3.4 (Adaptive Block Backstepping) Let the sJ}stem {3.49} be augmcnted by a 11.o71.linear system llJhich ill linear in the un/""1lo'Uln param.ete7' 'Vector 0, (3.63a) :1: = I(:,;) + F(x)8 + g(x)y (3.63h) € = 1n(:l:, {) + l\1(x, {)O + ,8(x, e)u, y = h(eL

e

where E :IRq, and suppose that {3.69b} has rclati11e degree 071·e 1.t.niforrnly in x and tllat its ze1TJ dynamics SUbsystem is ISS 1lJith. respect to y and x. Under' Assumption 3.3, the feedback C07l.trol tJ.

8h = [ a{ (';),8(:z:, e)

]-1 {

ah -C(l} - o'(x, 19)) - 8e (e) [m.(x, e)

+ l\l(x, {)i9]

] + aCt + 80: ax (x, 'D) [J(x) + F(x)tJ+ g(x)y af) T(x, ·0) - av ax (x, 11)g(x) } (3.64) I

3.3

99

RECURSIVE DESIGN PROCEDURES

with c

> 0 and 19 a ne'lIJ estimate of (J, along with I.he 1J.1Jda.te laws

iJ = T(:r,19) !..

f)

=

r

(3.65)

[8h a~ (~)}.J(X,~) -

with the adaptation go.in matrix r =

1 (1) -

80: u:r (.r, {})F(.r)

T

0:(:1:, 17)) ,

(3.G6)

r T > 0,

guam.ntees global bO'lJ.ndednes.fl oj .r(t), ~(t), 19(t), lj(t) and regulation oj l'F(:r(t), ·tI{t)) cwd ~(t) - a:(:r(t), {}(t)).

Proof. As in Lemma 2.25, we employ the change of coordinates (u, () = (h(~), 4>(:r:, ~)), with ~:/3 == 0, to transform (3.63b) into the normal form ;lj

= ah D~ U) [m.(:r:,~) + 1&1(:r:, ~)O + f3(x, ~)1I] a~

( = D:r (J',~) [1(.1') + F(x)O + g(:l:)'lJ) + l::,.

<po(x, y, ()

(3.67a)

a~ a~ (:t,~) [m(:r,~)

+ cJl(x, y, ()tJ .

Introducing a new parameter estimate " =

to rewrite (3.63a) and (3.67a) as x = f(x)

1j

(3.67b)

ii, we lise the feedback t.ransformat.jon

r {II - ~~

(~~p =

'U

[m +MD]}

(3.68)

+ F(.1·)9 + g(:I:)1/ fJh

+ AI(x, ~)B]

+ a~ (~)l'f(:r:, ~)(tJ -

(3.69a)

-

(3.69b)

17) .

We now apply Lemma 3.1 to (3.69). The only difference between (3.69) and (3.52) is the presence of the additional parameter error term ~~ 1&/(0 - 0) in (3.69b). This term can be eliminated in ,~'l by adding the term -r(~~llJP'(y­ ex) to the update law (3.55). Combining this modifi('ation with (3.68), we see that the resulting adaptive controller is given by (3.64)-(3.66). This guarantees the boulldedness of x, of), 19, z and the regulation of H'(:t,19) and z. Hence, 1J = Z + Q·(x, '19) is bounded. Then, from (3.67b) and the ISS property of the zero dynamics, ( is also bounded, and thus ~ and 1I are bounded. 0

3.3 3.3.1

Recursive Design Procedures Parametric strict-feedback systems

Through repeated application of Lemma 3.2, the bacl\stepping design procedure is now generalized to nonlinear systems which can be transformed! into IThe coordinate-free charncterizat.ion of I.bcse syslems in t.crms conditions is givcn in Appendix G, Corollary 0.15.

or differential geometric

100

ADAPTIVE BACKSTEPPINC DESIGN

Figure 3.4: Block diagram of it. third-order parametric strict-feedback systcm with

P(x) = 1. The nonlinearitics depend only on varia.bles which llre "fed back." the paramel1"i.c .drict-feedback j07m

:h =

3:2

.1:2 =

:t'3

+ V'I(xJ)O + I.{)I (x 1, x2)6 (3.70)

i:n-l

=

:1:'1

=

Xu + tp~-l (Xb" • ,xtl-dO ,8(x)u. + V'~'(X)O,

where j3 (x) -:F 0 for all $ E 1Rn. The reason for the name "parametric strictfeedback" can be deducoo from tIle block diagram jn FigtlJ'e 3. t 1, whel'e, except for the integrators, there are only feedback paths. For systems in the fonn (3.70)1 the number of design steps required is equal to the degree 11 of the system. At each step, an error variable ::i: a stabilizing fUllction ll';, and a parameter estimate fJ, are generated. As a result, if a system contains p unknown parameters, the overparametrlzed adaptive cont:l'OIler may employ as many as p 11 parameter estimates. A schematic representation of this design procedure is given in Figure 3.5, and the resulting expressions axe summarized in the fonowing theorem:

Theorem 3.5 (Parametric Strict-Feedback Systems) For the sy.'Jtem (3.70) with. {3(:c) :F 0 f07' all x E IRn , consider the adapU.1Je controller' (3.71)

i = 11 " when; iJ i E lRl' lIte mu.ltiple estimates

of () r

rT >

n1

'l'

(3.72)

0 is the adaptation gain m.at7ir., and the varia.bles ::; and the stabilizing fun.ctions Gi, i = 1, ... , 1l t 1

=

3.3

REcuRsIve DESIGN PROCEDURES

----r -=.

,

I

I

I I

I

101

I

_.J

0'1

'- _____

~

___

Zq

I

-~.J.

ll'::!

~a:'

-

11-

U

Figure 3.5: Thc design procedure for overpa.rnmetl'i~ed schcmc.c:;. Each step generntes an error variable =i, a stabilizing fUllctioll Q;. nnd n. new cstimate 1?i of the unlmmvll pILT8metel' vedOl' 8.

defined by the following recu1'si'ue e:r.p'ressions (with C; > 0 being design constanl.s, and 4:0 :: ao == 0 '1I.S(1.(1 f01' notat.iDnal c01wenience):

aJ-e

This overparamct'li.=ed adaptive cO'1l,t1'(Jller ,Q1£lI1'llnlees global bO'Ulldetlness oj x(t), 'l?(t), ... ,'l?n(t), and regulation oj:t'.{t) ancl:r'i(1)-,l:Y, i = 2, ... ,n. where ':1'1 = -OT tpi-l (0, J;~t·· . , a:l_I)' Proof. Using the definitions (3.73), (3.74) nnd denoting Xo == O:u :: 0, P(.l:)ll~ the derivative of the error variable =j, i = 1, , .. ,TI, becomes

:1'11+1

==

102

ADAPTIVE BACI{STEPPINO DrnSJGN

The choice of control (3.71) guarantees that system can therefore be expressed 88

':,1-1

=

Zn

=

fJ i

=

Zn+l

== O.

The closed-loop error

or, equivalently, in the matrix form =1

Z2

d

==

dt

-Cl

1

0

-1

-C2

1

.,.

0 -1 -Cn-l 0 -1

0 0 [ WI

+ ~

0

rlt

: tJ n

=n-1

1

Zn

-C n

T

- f)2 [ 8 9d,

W2

1

(3.77)

0

0

~ [:~ 1 = [1

:!:1 %2

0

Zn-1 "'71

0

0

WI!

8 - On 0

0

r 0

11

[I

0

1 [ -, 1 ~ft :

W2

0

z..,

;,,'

3.3

103

RECURSIVE DESIGN PROCEDURES

where we have used the convenient notation i-l

WI

=

'PI ,

'Wi

= 'Pi -

L

00'i_l

-a--'Pj

I

i = 2, ... , n .

(3.78)

Xj

j;::;;;1

Tbis system has two important properties: (i) The z-system matrix in (3.77) has negative diagonal and skew-symmetric off-diagonal t.erms, and (ii) t.he transpose of the matrix that mu1tiplies the parameter errors in the z-equat.ioll appears ill the update law. This structure is a result of the design procedure, and it allows us to use the simple quadratic Lyapullov function 1 '1 lIn (.::1, ... ,Zn, Dh .. , ,t?,,) = ry + (8 - OJ )'I'r-] (8 - iI;)] (3.79)

L [=;

... i==1

t.o prove stability and regulation. Its derivative along the solutions of (3.77) is

,i"n =

=T

z-

n

~,t9lr-I(8

-

t9 j )

1=1 n

= =

L

[-CiZ~ + Zjw;(8 - iJ j )

;=1

,.

-L:CjZ;.

-

zi w ?,(8

-19;)] (3.80)

i=I

The LaSalle-Yoshizawa theorem (Theorem 2.1) now guarantees t.he g10bal uniform boundedness of z(t), ill (t) •... , '!9 n (t), as well as the regulation of =(t). Since Zl = X], we see that :l'1 is also bounded and regulated. The hOl.lndedness of X2, •• • , Xn then follows from the boundedness of O'j (defined in (3.74)) and the fact that Xj = Zi + ai-I- Since x is bounded, {3(.-.:) is bounded away from zero. Combining this with (3.71) we conclude that the control 11 is a1so bounded. Finally, the regulation of .'Vi - xf is concluded as follows: Since Zj(t), i = 1, .. _,11 converge to zero, Jdt), i = 1, ... ,n also converge to zero and =i(t) is integrable over [0,00). Furthermore, the boundedness of all the signals and their derivatives guarantees the boundedness of =i(t) and hence the uniform continuity of Zj(t). From Lemma A.G, we conclude that lim/_ oo Zj(t) = 0, i = 1" .. , rI. Since x can be expressed as a smooth vector function of =1, • , • '=11 and '011' • , , {)n , we can express X as a linear comhination of Zj and 17i wit.h coefficients which are bounded because they are smooth vector functions of the bounded signals ZI," . ,Zn and 1)1,.' . I {In' Hence, the convergence of Z and tJ j to zero implies that :i: converges to zero. Combining this with (3.70) and the regulation of XI leads to the desired result. 0

3.3.2

Multi-input systems

The adaptive backstepping design procedure of Theorem 3.5 can be easily e}..'i;ended to nonlinear systems which have been transformed into the multiinput pam,metric strict-feedback 101m

104

ADAPTIVE BACKS1.'EPPING DESIGN

j.'I,1

';'1,2

= =

Xl,!!

+ cprl (3,'1,11'1"2,., ••• ·1"2,P2-PI+h"

·1'1.3

+ cpI:ll.·1.1,J~I':!1.l~2.Jl'" •• , !·T.'rn.J,·"

J" l,p,-]

., :L'm.lI ••• , ,l:m ,Pn,-PJ+J)O

.1;2.p'.J-PI+21

,X rn •p,,,-Pl+2)9

= ·&.l.P1' ", + cp1'l.Pl-1 (...."1.1,···, "l,Pl-h· :/. .... ,., 2.h··· "'2,P2-h L

, •• I

·'I'm,], • , • 1 ,t m ,pm-l)6

m

i'l,111

I: f31J (.x) lJ.j + 'PT./)1 (:1:)0

=

j=l

(3.81) !i:i.)

=

Xi,j+]

+ cpl,j(xl,h' .. , 3.~l'P!-P.+jl' •• ,3:;,1, ••. ,l'i,j, •.. , :t'lJI.l, ..• , X m ,p".-PI+i}8

;i'm,l

,i'm,2

= =

+ It'!.,1 (.t'1.1,' .. ,.1'I,Pl-Prn+b .:r2,h ' , • X!!,p:t-p",+b •.. .1'm.3 + fP;'.2(XI.lt., • ,3:1,PI-P,"+21 ;]=2.1, , •• X2.P'.!-Pm+2, .1: 111 ,2

• • • 1 :l'""l t

:i:m'PPI,-J

=

Xm,Pm

:r:m,I)O

:l:m ,2)0

+ tp~'PfIl-1 (.J;],., ... - •• I

1

1

1'l,PI-]' X'J,l, ' •••1:2,{I2-J 1

Xm.l1 ••• ,Xm,p,.._I)O

711

:i'm,p".

=

2: f3".J(:V)-Ui + lP~'Pl (X)O

1

j=J

wberp 'Uh'"

, Um

are the inputs, and tbe input matrix is llollsingulru.' 'V:I:

e m,n

(ll=PJ+"'+Prn): clet B(l:) ::F 0, 'V ;1: E

m.n

(3.82)

I

The design procedure for this class of syst.ems cOllsists of applying the design proeedure of Theorem 3.5 to the fil'st Pi - 1 equations of each of the m subs)'stems of (3.81), to obttUll the system -CiJZi,j -

=iJ-l

+ ZiJ+J + tl1i ,)X, 191 " , , ,t),.-l )({I- llt) '('

;-1

e= L:(Pk -1) + j ,I$. j S Pi - 1, k=. = r Wi.j(X, 1111 , ,. , tlc)z;J' 1 $. e$. In 1]. -

1 S i $. m (3.83)

3.3

105

RECURSIVE DESJGN PROCEDURES

where t.ho functions Wi,j, rI> and HJ:II _ m +1 are defined appropria t.cly. Now let -o"-JII+I be a new estimate or 0 and define the cont:rol1J. as

lL

=

CI,PI ZI.Pl

B- 1 (.1:)

{

[ Cm'PrrI-7R,Pm

-

+ ':J ,PI-l

_:

]

_

- ID(.1:, #1, ...• ,0 11 - " , )

+ -m,p... -l

W~~m+l (x. 19" ...• U

n -.n jU,,-m+ 1 } •

(3.84)

and t.he updat.e law for 01l-m+l as

(3.85)

The stability properties of the resulting closed-loop system are analogous to those listed in Theorem 3.5, and can be similarly established Ilsing the Lyapunov fuuct.ion

{3.86}

3.3.3

Parametric block-strict-feedback systems

Lemma 3.4 can be applied repeatedly to design adaptive controllers for nonlinear systems which can be transformed, after a change of coordinates, into the pa7·ameL1i.c block-strid-feedback form

Xl =

11 ttl) + FI (XI)6 + iit (Xl )Y2 11.1 (:\:1)

111

=

:\,'1

= h6:'1, :(2) + F':!(XIt X2)8 + 02 (X.I ,X:!)Ya

Y'l.

=

h 2 (X2)

X·

=

Yi

=

/;("Xl, ... ,Xi) + Fi(Xll ... ,Xi)6 + 9;(YI, ... ,Xi)Yi+l hi(Xi)

.

I

Xp-l Yp-l

:\,p YP

(3.87)

= j,l-l (Xb ... ,Xp-t} + Fp- J (Xb ... , Xp-l)9 + 9p-l h'l' ... ,Xp-l )yp = hp _ 1(Xp- J) = jp(xJ + Fp(x,lO + 9p(X)U = hp(Xp) \

Ir06

ADAPTIVE BACKs'rEPPING DESIGN

lvhere each of the p subsystems with state Xi E m.nl , output IIi E lR, and input HI (for convenience we denote Yp+l == u) satisfies conditions (BSF-1) and BSF-2) (see Chapter 2, equation (2.198)), t.hat is, it has relative degree one niformly ill All' .. ,Xi-h and its zero dynamics subsystem is ISS with respect oX}'···, Xi-I! Yj·

Using the change of coordinates which transformed (2.198) into (2.201) in ection 2.3.3, we can now transform the system (3.87) into IiI

=

Yp-l

=

11 (yt, (I) + CPI'1' (Yll (1)0 + Yl(Yh (1)Y2 Y2 = hhJl,(ldJ21 (2) + cpI (111 , (11 Y2,(2)(J + 92(1111 (ltlJ':!, (2)1/:-1 /p-l (y., (1, ... 1 IIp-b (p-I)

=

(p

=

I

1/p-l. (p_I)8

+gp-1(Yl, (h' .. , Up-II (,,-I )yP (3.BB) 1p(/l1, (1, ... , Yp, (,,) + qJ~ (Vll (I, ... , 1Jp, (p)O + 9p(YI , (1, ... 1 Up! (II)?/'

YP = (1

-1' + 'Pp-l (VI, (II ...

cI> 1,O(Yl, (1)

+i

l (Yl, (1)9

Then we employ anotber change of coordinates which replaces Vi by

t/JdUh (h'"

1

Yi-I! (i-lt

=

YI ~ 1/'1 (Y1)

·'t·2

=

-;;;-(/1 (.lUI

:ti+l

=

+ 91112) =

II(y.,(I) + Ol(11I,(t}~J2

i-I 8~J,

i-I

=A '~'2(11I'(1'Y~)

tPi+ I (Yl t

(3.89)

81/Ji _

L ~(fj + gjYj+l) + L ar.~J WitO + 91 ... Oi-I!i + 91'" i=:1 ul/j

l:l

=

Hi), where

Xl

a~tl

Xi

9iYHI

j=1

'It ... ,

YEt (i I YiH)

i = 2, ... ,p - 1 .

f

FinallYl we use the feedback transformation p- 1

v =

L

j=J

8VJp

-0. (h Y.1

p-I

8t/Jp -

j=1

J

+ gjYi+l) + L

8(. tI>j,O + 91'" gll-l!p + gl'" 9p U. (3.90)

CondHion (BSF-l) guarantees tbat UI,'" ,gp :F Q everywhere. Hence, the change or coordinates (3.89) relating [YI! ... dIp, (l t • • • , (JJT to [:CIt- .. ,.1:/1< cT (J]T is a global diffeOJnorp11iSnl, and the feedback transformation (3.90) relat.ing II to u is llonsingular. It is now straighLfonvard La verify that (3.S9) and (3.90) transform (3.88) int:o a form l'Cmilliscent of tJle I ' •• t

3.3

107

RECURSIVE DESIGN PROCEDUR.ES

parametric strict-feedback form (3.70): !i'l

X2

=

+ 'PT(Xl1 (1)0 == Xa + \O~'(.Th X2t (it (2) X2

X,,-l == XI' + cp~_] (:rl1 .•• , ·2:,,_11 (" ... t (P-l)O .i;p == v + 'P~'(XI ()6 (1 = Wl.0(l!I,(1)+ ID 1(Xlt(dB (" = y

=

tPp,O(a:h'"

(3.91)

,.l:p , (I," . ,(p-lt (,,) + tP p(:l!I •• •• toVPI (It ... ,(p-ll (,,)0

$1·

In (3.91) each (i-subsystem is ISS with respect to 3'" ... , Xi, (1, ... , (;-1 ~i,O, eDit i = 1, ... ,p are defined as

ru;

its

inputs, and IPi,

a'Pi L a.(YII (Jf .•. i

11,

j=l

I

Yi-I, (;-1, lIi)

~J(Ylt'l" .. ,Yj,(j)

i-Io,p.

+L

j=1

0(' bJh(j. .. ·,Yi-lt(i-h1Ji) J

~AY1t (1, .... Hjt (j)

(3.D2) (3.93)

(3.94)

It is now clear t11at UlC class of parametric block-stric1.-feedback nonlinear syst.ems strictly contains the dass of paramet:l'ic strict-feedback nonlinear systems, sill('e (3.70) can be obtained by setting ni 1, p 71, and v p(x)1J in (3.91). We now state and prove t.Ile generalization of Theorem 3.5 to block-strictfeedback systems of the form (3.91).

=

=

=

Theorem 3.6 (Parametric Bloc1c.-Strict-Feedback Systems)

F07'

the

slJstem (3.91), consider U."e adaplive controller (3.95)

i

= 1"",p,

(3.06)

108

ADAPTIVE BACKSTEPPING DESIGN

whc7'e tJ j E 1RP a.re multiple e.r;tima,tes of (), r = r1' > 0 is t.he a,daptation .qain matri.:c, and the vmi.ables Zj anti fhe sta.bili:.iug jlJ.nctions O'j, i = 1, ... , P. are defined by the following 7'Ccursil1f C3.']J1'fJ8,1iions (with. Ci > 0 being design constants, and Zn :::;;: 0'0 == 0 1/-sed fo'" 1J.ot(].tional convenience): (3.97)

D:j

=

-Cj=j -

+

=i-l -

8a i - 1 ..T), 8(, ':iJ,O

[

_

J

~ (DO;i-l T 80';-1 )] 0; + ~ {80'i-J - - ! . p j + --Wj L- --J.'i+l 8x 8(j j=1 tJ.1:j

IfJi'1' - L-

I

i=l

fJO'i-1

8 11 • /'J

r

j

'1 ?' _ ~

[ 'PJ

L-

k=l

(80;j-l a
XI.'

]1'} (

(T») _,

f)a:j-l f)r i:A' '-,1.-

-J'

3.98

)

Thi.9 overparamct7'i::.ed adapli1.e controller gua,rnntces global boundedncss oj .,ch, ... , iJp a.nd reg'ulal.ion oJ:lJ = Xl'

.1:1,'" ,.J:p , (1,' " ,(pI

Proof. As one would expect from the similarities between the systems (3,91) and (3.70), t.he e.'\:pressions (3.95)-(3.D8) are similar to (3.71)·-(3.74). Using the same arguments as in the proof of Theorem 3.5, we write the derivative of the error variable Zj, i = I, ... , Pr as

3.3

109

RECURSIVE DESIGN PROCEDURES

The closed-loop error system can thus be expressed in the matrix fonn ZI

d

dt

:2

=

-Cl

1

0

-1

-c:!

1

0

':::1 .:~

0

0

(3.100)

with the not.ation i-I "'"

[!'l

Uai-l

Wi = 'Pi - L." -.-.-
+

(!'l

UO:i-}

--.-(1).1

8(}

)or]

,

i = 2, ... , p. (3.101)

The struct.ural properties of this error system are once again apparent. The derivative of the nonnegative fUllction (3.102) along tlle solutions of (3.100) is p

l~

=

-Lei=;'

(3.103)

i=l

The LaSaUe-Yosbizawa tllcorem {Theorem 2.1} now guarantees the global uniform boundedness of z(t), -0 1 (t), .. . ,vp(f), as well as the regulation of z{t). Since =1 = :ll, we sec that. y .t'} is also bounded and regulated. Tile boundedness of x:!, ... , x pt (It ... ,(p and of t.he trallsformed control variable v is then est.ablished vitt an induction argument for i = 2, ... I P + 1 (with Xp+1 == 1J): If x), ... , .1'i-l and (h .. " (1-2 are bounded, the ISS property of the (i_rsubsystD111 guarnntees t.hat (;-1 is bounded. Hence, ai-l(Xl, .. " XI-It (1, .. " (i-I, -0., •.. , Oi-.) is bounded, which implies that :r.j = Zi + Q:i-l is also bounded, 0

=

I 10

~

ADAPTNE BACJ(S'I'EPPING DESIGN

\\Te should note that the adaptive controUer (3.95)-(3.96), when apI)1ied o the system (3.87) using the expressions (3.89) and (3.90), guarantees the Iobal uuiform bouncledllcss of X and il, as weH as tbe l'egulutioll of y 11] (:\'1). This follows from the fact that ~gj :F 0, which guarantees tha.t tile transformation relating (Yl t • • • , Yn, (11 •.• , (,,) to (X I, ... I J'rl , (II ... 1(n) is a global diffeomorphislll, and the feedback transfo11natioll (3.00) from 11 to 11 :is nonsingular.

,3.4

=

Extended Matching Design

The increase ill the llumber of parameter est.jmat.es ealiRed by ovel'paramet.rization cau be all undesirable feiltnre, since jt. rapidly inCl'E"8SeS tlu~ dynamic order of t.he rcsulting adaptive controller. In Chapter ~l, the oYerparallletdzat.iol1 will be elillliImt;ecl by the tuning functions l11et])od. As prelim.inar.v to this development:, we now show how the ovcrpanunetrizatioll can be avoided in the case of ext.ended matrhing, that is, when the llucprtaill parnmetCl'f; are Duly one integrator away from t.he control.

I

3.4.1

Reducing the overparametrization

\Vc consider again the nonline~u: system (3.29),

.1: 1 =

:l!~

+ Btp(x)}

and modify it.s t.wo-step design.

Step 1. Wit.h Z'1 = Xj and :t::! viewed as the virtual control hl the zl-equation, we define the first stabilizing function 0') as ill (3.3..:1): (3.104)

Comparing (3.104) with (3.34), we see UUtt the parameter estimate 19 1 has been replaced by the paramcter esl:imate 0. The differellce in Jlotl:ltion iudict:ltcs thcl.t ill this design procedure ouly one estimate 0 of the unknown parameter will bc used. The first. Lyapullov fUnction is now chosen as (3.105)

where 8 = () Witb Z2 = .r2 -

Bis 0'1,

the parameter error, and " the derivative of Vi is

>

0 is the adapt~tt:ioll gain.

(3.106)

3.4

111

EXTENDED MATCHING DESIGN

Vie postpone the choice of update law for () until the next st.ep. The first error su bsystt"m becomes

(3.107)

Step 2. The derivative of =2

= =

To design the ('ontro]

Z2

=

X2 -

ao: I 'U -

l)XI (X2

0'1

is aO'l :. .

+ OIP) - 00 0

-ao

au) 1 -80'1 DOl ':.. u- -:r..., -8-1.{) - (J-I.{) - - . B. ax! 0.1'1 OJ:. {)e 'U t

(3.108)

we consider t·he augmented Lyapullov function 1'11,)

-:j

2

1-...

+ -.:; + -(J- . 2 -

2"',

(3.109)

The only difference bet.ween (3.109) and (3A2) is the absel1ce of t,be new parameter error (8 - O2 ) in (3.109). In view of (3.106) amI (3.108), the derivative

of l'2 is

(3.110) In the last equation, all the terms containing 0 ba.ve been grouped f·ogether. To eliminate them, the update law is chosen as

B= " (tpZl -

80:1

ox)

CP':2) .

(3.111)

Then. the last. bracketed term in (3.110) will be rendered equal 1.0 -C2':~ wif:h tl1e control lL

=

(3.112)

where for 9 we usc the analyti('al expression of tbe update law (3.111). Substitutlng the expressions (3.111) and (3.112) illto (3.110) we obtain (3.113)

112

ADAP1.'IVE BACKSTEPPINO DESIGN

(_lu lP ]

'1'

U.r:1

J

1+---------'

Figure 3.6: The closed-loop adaptive system (3.114).

and tile error system bec-omes (see block diagram in Figure 3.6)

~. [ ~ 1 = [ __~l _~] ~ [

iI =

1

]

+ ( _~~ ] 8

[~ - ~~ 1[ ~~ ] .

(3.114)

Comparing (3.114) with (3.47), we see that the system matrbc in (3.114) has preserved the important structural properties it had in (3A7): Its diagonal terms arc negative and its off-diagonal terms are skew-symmetric. Furthermore, we see that, as in (3A7), the matrix that mUltiplies the parameter error 8 in the z-equation is usecl (in its transposed form) in the update law for the parameter estimates. It is ruso instructive to compare tbe ex.pressions for the parametel' update laws ill (3.114) and (3.47): Even though the update law for iJ appears in the form of the sum of the update laws for t}. and il2 , the expressions (3.34) and (3.104) for 0'] depend on different parameter estimates (tJ 1 and 8, respectively), fUld thus Z2 and the partial derivative ~ will have different values in (3...:14) and (3.111). Due to the structure of the errol' system (3.114), its stability and convergence properties are derivE'd in a manner almost identical to those of (3.47) and are therefore omitted here. In the extended matching case we avoided the overparametrizati?D by postponing the choice of the update Jaw until the second step_ When 6 appeared in the second step, it was replaced by its known analytical ex.press~.on. Beyond the extended matching case we need more than two steps, so that iJ and bigher derivatives of 6 appear. Instead of the simple idea of postponing the choice

3.4

113

EXTENDED lVIATCHING DESIGN

of the update law, we need the more intricat.e tunin.q juncf;ions method, t.o be developed in Chapter 4.

3.4.2

Example: biochemical process

Extended matc:hing design is also applicable to pure-feedback systemH, int.radured in Section 2.3.2, provided that the unknown parameters appenr linearly. 'iVhile the general case of these parmn.etric pu.re-feedback s1}stc'rns is presented in Section 4.5.3, t.he extended matching design will be illustrat,ed on u simplified model of a biotechnological process which goes as far back as IvIonod [135]. In spite of its simplicity and somewhat unrealist,ic assumptions, this example is representative of several successful applications of adaptive nonlinear control to more complex processes described by Bast,in [7]. In a model of no fed-bat.ch process. S is the concent.ration of the growt.h limitiug substrate, X is the concentration of the growing microbial population, ". is the yield const.ant, D is the dilutiollrate: and the control u is the substrate feed rate. In a batch process, that is, when both D = 0 and u = 0: the rate of microbial growth i; is modeled as i = p(5).¥, where p.(S) is t.he "spedfic growth rate." The nonlinear function p,(S) is usually poorly known. and for our illustrat.ive purpose we parametrize it using unknown parametel's: (3.115) Note that .X, 5, and J.L(5) are nonnegat.ive quantities. \Vith this panllnetrization the fed-batch process operating at constant temperature is modeled by the following two rnass-balance equa.tions:

..:t

=

S =

[IPo(S) + ()IIP1(S) + ()2IP:!(S)].Y - D ..Y -k [IPo(S) + 81IP1(S) + 1J21P2(S)].\ - DS + 'U.

(3.116a) (3.116b)

The control objective is regulation of X to Ute set point X r • To furt.her simplify the system (3.116), we use the change of coordinates Xl = In ...\'", X2 = S, which is well-defined and invertible since ~y > O. Then (3.116) becomes Xl = IPO(:C2) + ()lIPl(X2) + 82 IP2(X2) - D .1:2 = -I;: [lPO(X2) + 814'1 (X2) + 82CP2(.l:2)] e:r'1

-

D.l:':]. + 'll.

(3.117a) (3.117b)

This system is clearly not in t.he paramet.ric strict.-feedback form (3.70), since the nOl1linearities in (3.117a) depend on the second state variable :r:2. However, we can still apply the design procedure illustrated in Section 3.4.1 to this system. In Section 2.3 we saw that our recursive design procedures can be applied not only to strict-feedback systems (2.165), but also to pure-feedback systems (2.180), whose nonlinearities are allowed to depend on one more state variable. The price to be paid is that the stability properties are no longer

114

ADAPTIVE BACI{S'l'EPPING DESIGN

Substrate feed (control -u)

Heating/ cooling

Figure 3.7: Fed-ba.tch stirred tank rendar.

global, bu t regional: They are guaranteed only for a compad set of iuitial condit,ions. The same is possible for our adaptive designs. For t.he system (3.117), our design pl'Oceeds by choosing tpO(3:2) as tht:' virtual ('ontrol variable ill (3.117a) and dcsigning for it the stabilizing rnlle-tioll

(3.118) where Zl

=

=1

=

Xl

= IPO(X2) -

-In X r • \'Vit.h Z2

-C1Z1

at,

the error system becomes

+ =2 + (0 - 81)IPl + (fJ - 92 )V'2

(3.119)

(aa~O + 0/a:,1 + 82 8aV(2) { - ~~ [V'o + OIIPl + lJ~tp2J eaJ

=2 =

X2

,1:2

X2

I

-

D:V2 + u}

+ct [-Cl=1 + =2 + (0 - 6d'P1 + (0 - 82 )V'2] + B1V'] + 02V'!! .

(3.120)

Following the development. of Sectioll 3.4.1, we choose the update law

~

r [ : }{ =, + [c,

=

= [Ol 021.

wherc 01' u

!

-

+

(3.121)

Tbe corresponding control law

= (!!!!! 8 ~ -I- ~,~) O:r:l

- k ( : : + 8, :~; + 8, ::) ] =, } ,

I 8:1::1

{ -

C2=2 - =1 - Cl [-C1Zl

+ z!! - D1V'l - 92V'2]

- DJ:'J

[~ .. v>ol r [ ;. ] {., + [ct - k ( : : + 8, :~; + 8

2 ::.: )

+k [cpo + 81lP! + 62 V'2] e + DX2, Z

'

1o.}} (3.122)

3.4

115

EXTENDED MATCHING DESIGN

is feasibile only in the region in which ~ + 01 ~ choicps, the derivative of the Lyapunov function

+ O:!~ =/: O.

'Vith thcsp

(3.123) is nonposit.ivp: (3.12-!)

" =

As we will see in Section 4.5.3, stability is guarant.eed for all initial conditiolls inside the largest level set of the Lyapunov function 3.123 (·ont.ained in the feasibility region.

3.4.3

Transient performance improvement

The llonlillem' damping with h:-terms introduced in Section 2.5 can easily bE' incorporat.ed int.o the adaptive design procedures we have discnssed so far. The resulting adaptive controllers guarantee boundedness even when the adapt.ation is switched oIT, and their transient performance can be improved in a systemat:ic way t.hrough trajectory) initiuli:;ation and the choke of design paramelers. To illustrate the design with Ii.-terms and t.1m process of trajedory init.ializatioll, we consider again the system {3.29} with the out.put /I = :1'1:

.1:1 = :1:2

:r2

=

+ 8cp( .1' J ) (3.125)

1I

Y =

.1'1'

The control objective is to asymptotically t.racl\: a reference out.put Yr(l) with the out.put JJ of the system (3.125). 'Ve assullle that not. ouly Yr. but also it.s first t.wo derivatives Yr, jjr are known and uniformly bounded. and, in addition, Yr is piecewise cont.inuous.

Step 1. The first errol' variable is now the t7'lJch71.g =1

= Y - Yr =

e7'1'07'

I1r ,

Xl -

(3.126)

whose derivative is ':-1 = :1:2 + 0'1'91 (.t·l) -

Viewing

:1:2 ~lS

.llr·

(3.127)

t.he virtual control we define the st.abiliziug fUlldion 0'1

= -c} =1

-

n.1 =1 cp2 -

Ocp + Yr .

(3.128)

Comparing {3.128} with (3.104) we note two new t.erms in (3.128). The term which is intended to cancel t.he corresponding t.erm in (3.127), is due to the t.racking objective. The nonlinear damping tE'l'lll -11:1 =1 cp2 is motivated

Yn

116

ADAPTIVE BACI(STEPPJNG DESIGN

by Lemma 2.26. It contains the square of the term (
,,+ =~ + (hp. The derivative of the Lyapunov function Vi = 1=r + '3; 82 .

=1

=

• - -2 - \ r.1 -_ -]

. -- - lil_] . -- cP CII I ) "

I)

+ (J -

Step 2. As in (3.108), tho derivat.ive of =2 = =2

= =

U. -

(3.120)

-CI=1 - Iil=lcp·

(

CP-l

X2 -

bec!omes

-,0 . 1 :..)

(3.130)

a:1 is

UOI aa:1 . aa:1 .. aa:l :.. -(x:! +(JcP) - -Yr - - .Yr - - ,(J

ao

aYr allr (J-aO'l oerl • .. -cP - -cP - ~Yr - Yr aJ:l aXI aYr

[):rJ

[)a:l

u - -X2 -

(JA{JO']

{);,:]

!Jf;

where ill the last equalit.y we have lIsed the ident.ity and (3.131), the derivative of the Lyapunov fUllction

1 ')

1.,

2

2 -

00'1 (}:.. -~

ao

(3.131 )

!

= 1. Using (3.130)

1-.,

-zi + -:;:; + -0-

(3.133)

2")'

is expressed as

Ii, =

=,::0 - 0,=; - Nl=i
=

...

{J01

aaal

-80:.1

aXl

OXl

aX1

8a:l .

..

-a::! - O-cP - 8-cp - -Yr - Yr .,.,

-(~);:;i' - n.1=jCP·

+=2 [ =1 + u -

OYr

00'] + (J- [ CP=l - =2cP 8

Xl

00'1

-8 ;t':! :1:1

~{)O'l - I J - cP -

oa:.

00'1 :..]

an

-A

(J

1 :..] l'

-f)

aO'1 • -a Yr Yr

•.

aO I

'!..]

Yr - - . IJ. {J(}

(3.133)

As in (3.110), all the terms containing 8 have been grouped together. To eliminate them, the l1pdat,e law is chosen as

~ = -y (
-c2zi -

1i2=~ (~tp)!!,

instead of just equal to

-£'2Z~

as in (3.112):

3.4

117

EXTENDED MATCHING DESIGN

f)

0

,.r-. 'I-'

-0

-

[-~~]

oJ

.f

1'1"\ ,,~

O:rL

-

[-~~r "

=2

r-r-+-

8.r1

-

[-<"-"'~' -1

1

-(',,-/':'1 -

-

(f!QJ. )~2 Or)

] f4

1

.-

-

I

Figure 3.8: The closed-loop lldaptive system {3.137}.

To implement this control law, we will replace iJ wit.h t.he analyt.ical expression of the update law (3.134). Su bst.ituting the expressiolls (3.134) and (3.135) int.o (3.133) we obt.ain (3.136)

while the romplete error syst.em becomes (sec block diagram ill Figurc 3.8)

,/ [ =, ] = [ -c, =1"'1" -c, _ n~ (~\O)' ] [ ~; ] + [ -;'1" ]H =2

dt

iJ

= .r[ ~ -~~ 1[ ~: ].

(3.137)

Comparing (3.137) with (3.114), we see that. the system matrLx ill (3.13i) is not constant: Its diagonal terms have been "fort.ified~' with additional nonlinear damping terms. These t,erms contain tho squares of the elcment.s of the vector that multiplies the parameter errol' O. Let us now study the properties of the error system (3.137):

Global stability and asymptotic tracking.

Using (3.132) and (3.136) we conclude that the (.:,8)-syst.em has a globall), uniformly stable equilibrium at. the origin, amI lim .:::(t) = O. (3.138) t-oo

In particular. t.his implies that Ule state of the syst.em (3.125) is globally uniformly bounded (since Ur, Yrl and fir are bounded) and t.hut. t.he tracking error Z1 Y - Yr converges to zero asympt.ot.ically.

=

118

ADAPTIVE BACKSTEPPING DESIGN

Boundedness without adaptation. It is also straightfonvard to see that. the designed controller guarantees global uniform boundedness eveu when the adaptation is turned off, that is, even with, = O. In that case, the closed-loOI> system (3.137) becomes

A candidate Lyapullov Cundion for this system is given

_ ! (_2 + ""2.,.2) \/{-).. -_ !1_12 2'" - 2 -1

.

b)1

(3.140)

It.s derivative along the solutions of (3.139) satisfies

(3.142)

It is clear from (3.141) that., for allY positive values of Co and li:o, the state of the error system (and hence the state of the plant) is uniformly bounded, since lr < 0 whenever \zl:! > 1012/4noco, where 8 = 0 - O{O) is constant. since adaptation is turned off.

Transient performance improvement with trajectory initialization. Let ns now investigate the transient performance of the adaptive dosed-loop system (3.137). The derivative of the nonnegative fUllction \1(.::) defined in (3.140) along t.he solutions of (3.137) satisfies the same inequality as in (3.141):

(1, dt 2

2

-d -:; I") - ::; -co Iz 1'-1 + -0

4n:o

.

(3.143)

8 has already been established from (3.132) and (3.136), we can strengthen the inequality in (3.143) by replacing ij'J with its

Since the boundedlless of

3.4

119

EXTENDED lvIATCHING DESIGN

bound 11811~. This bound is est,imated from (3.132) using t.he fact that ~ is nonincreasing:

:5

1 ., 1 - '1 2Iz(t)l- + 21,8(t)- = ''2(l)

:5 ~(O) = ~1=(O)12 + ')1 0(0)2. ...

(3.144)

....1'

This implies

IIOII~ $ 71=(0)1 2 + 0(0)2.

(3.145)

Combining (3.143) and (3,145) we obt.ain

d

at

(1;;12) :5 - 2col=l:! + _1 ["'11=(0)1 2 + 8(0)2] , 2nn

Multiplying both sides of (3.V!6) by results in

e'lcol

(3.l46)

and integrating over the int.erval [0, I]

(3.VJ7) The bound (3.l tl7) suggests that the transient beha.vior of the error system can be influenced through the choice of design constants Cn, n.o and 1', ~'hat is not clear, however, is that an increase of h~OCO alone may not reduce the ma..ximulll value of 1=(t)1 and \Vill certainly not reduce the comput.able .coo-bound of =. In fact, it may even increase this bound by increasing the initial value 1=(0)1. To clarify this point, let us recall the definitions of =. and =2: Zl

=

·r-l - llr

Z2

=

X2 -

0'1

=

;1:2

+ CI Zl + /i'IZ1cp2 + Otp -

lir·

Suppose now that ZI(O) is different than zero. In that case, all increase of and may increase the value of :~(O) and thus also the value of 1=(0)1. IvIOl'cover, this increase may more than offset the dccreasing efrect of the term 1/4n.ocn in (3.147), since Iz(O)1 2 will increase in proportion to ci and It would seem that the dependence of =(0) 011 the design constants Cl, C:Z, 11:1, n.:z eliminates any possibility of systematically improving the t.ransient performance of the error system through the choice of Co and 11:0' Fortunately, it is not so. The remedy for this problem is to use tmjector!J initialization to render =(0) = 0 independently of the choice of these design constants. The initialization procedure, present.ed for the general case in Section 4.3.2, is straightforward and is dictated by thc definitions of the =-variables: Cl

li,

h-i-

• Stal'ting with

z},

set.

=1 (0) =

0 by choosing (3.148)

120

ADAPTIVE BACKSTEPPING DESIGN

• Since :'1(0) = 0, (3.128) shows that

where we use t.he notation !p(O) = CP(:t:I(O)). From (3.149) it is ("lear t.hat we can set. .::AO) = 0 with the c:hoice

(3.150) With the t.rajcct.ory initialization defined by (3.148) and (3.150), we have set. .:(0) = O. III t.he case of model reference control, this is achieved by adjusting the init.ial conditions of the reference model. If, on t.he other hand, the reference trajectory is given as a prccomputed function of t.ime, t.hen it can be iuit.ialized t.hrough the addition of exponentially decaying t.erms whicll define the refcl'Cnce l1Yl1t.rlients. Vve note t.hat. (3.148) and (3.150) are independent of tht" deRign ('onstnnts c., C:!. h'lt 1i2' This means t.hat. different choices of Co and KO will stilll'esult in =(0) = 0 with the same values of Yr(O) and 1jr(0). Ret.urning to (3.147), we substitute .:(0) = 0 to obtain 2 1=(t)1

::;

1

-

'l

-8(0)~lli:oco

(3.151) j

which imp1ies

(3.152) Hence, the .coo-bound on the transient performance of the error system is direct.ly proportional t.o t.he initial parametric uncertainty and can be reduced arbitrarily by increasing the values of Co and 1\:0. In particular, this implies that the transients of the tracking error Zl = Y - Yr are directly influenced by the design const.ants Cj and Iii' This possibility of arbitrary reduction may seem peculiar, since it can be achieved for all initial conditions. vVe must. remember, however, that this elTor is defined with respect to the reference signaJs which have in tUrn been initialized to set =(0) = O. Hence, the effect of the plant iuitial conditions has been "absorbed" into the reference t.ransients. To provide some further insight into the process of trajectory initialization, let liS return to the Lyapunov function (3.132). When .:::(0) = 0, the initial value of this function is reduced to the initial value of the parametric uncertainty. 1£ we interpret the value of this function as a distance between the actual system trajectory and the reference trajectory, we see that trajectory initialization places I.he initial point of the reference trajeclory as close as possible to thc initial point of the system trajectory. If the parametric ullcertainty were zero, t.rajectory ini tiaJization would have placed the reference output and its derh'at.ives at the true values of the plant output and its derivatives. Tilis

NOTES AND REFERENCES

121

is easily seen if B{O) is replaced by () in (3.148) and (3.150):

Yr(O) = .1:1 (0) = !I(D) Yt(O) = l'2(0) + 81;'(0) = 1j(0) , However, since the paral11etpr () is unknown, trajectory initia.lizat.ion placed only the reference output at: the t.rut:' value of the plant output,. while ii,s derivo,th'es wore placed at the estimated values of the plant ont.put: derivat.iveF;. Through this process, the iniUI:11 value of the Lyapullov function is (3.153) \Vhich~

in the presence of parametric ullt"el't.ainty, is its smallest possihl(' valuo,

Notes and References Adaptive backstepping (Kallellakopoulos, Kokotovic\ and rvlorse [60]), first present.ed in a Grainger lecture [87], was a culminat;ion of an intellsivp eIrod of severol groups of ~1.uthors. The path to adaptive backst;cpping was not, as direct as it may appear from t.his chapter. It led ~,brough the makhed ease in Taylor, Kokotovic, rvIarino, a.nd f\:anel1akopoulos [186J, itnd then thp case of extellded mat,ching ill Kanellakopoulos, Kokot:ovic, and :Marino [65J, and Bastin and Campion [8]. Even t.hough nonadaptive backstcpping was ftvaHablc from Saberi, Kokotovic, and Sussmann [163}. the st.eps beyond the ext(:'uded matching case wore delayed lUld ruterlmtivo npproaehes were explored. The focus was on estimat:ion-based designs sllllullarizec1 in Pral)" Bast-in, Pomet'l and .Jiang [157]" The rIass of paramet.ric strict:-feedbal'k systems was characterizerl via coordinat.E"-fl'ee geometric conditions by Kanellakopo ulm; ct 0.1. [63, 69), and the class of pru"ametl"ic }lure-feedback systelIlR by Akhrif tlud Blankenship The multi-input design £01' parametric pUl'L.1..feedback systems was prcs('l1t,ed

rlJ.

in [B7]. Teel {lBB] increased the feasibility region for parametric pure-feedhack syst.ems by cast.ing the scheme [69J in an observer-baserl setting. Severa] extensions of [69] were proposed by Seto, AUllaswamy, and Brumettl [3, 167J.

Chapter 4 Tuning Functions Design

The adaptive baclcst.epping solution to the problem of nonlinea.r stabilization and tracking in the presence of unknown parametel's is a starting point for more elaborate adaptive designs which lead to new properties of the designed controller and the resulting feedback system. One of t.he improvcmcnts t.o be a.chieved with t.he tuning functions design in this chapter is the redudioll of t.he dynamic order of the adaptive controller to it.s minimum: The number of parameter estimates is equal to the number of unknowll pammet.ers. This minimum-m'der design is advant,ageous not only for implement.ation, but also because it gunrantees the strongest achievable stability and convergence properties. In the tuning functions procedure the parameter updat.e law is designed recursively. At each consecut.ive step we design a tuning function as a potential update law. In contrast to adaptive ba("kstepping in Chapter 3, t.hcse intermediate update laws are not implemented. Instead, the controller uses them to compensate for the effect of parameter estimation t.ransients. Only the final tuning function is used as the parameter update law. We start this chapter with Section ~1.1, which introduces a g(meral framework for Lyapunov-bnsed adaptive design via adaptive control L;l/ap U7/. 0 II functions (adf). VVe depart from the certainty equivalence principle and approach the problem of adaptive stabilization of the original nonlinear syst.em as a problem of nonadaptive stabilization of a modified system. In this setting, the tuning functions design is a method for recursively genera.ting aclf's. The design procedure is presented in Sections 4.2 and 4.3, which are independent from Section 4.1 and can be read first. In Sect.ion 4.4 we derive transient performance bounds on the error state of the adaptive system. An essential part. of t.he technique for improving the t.ransients is t.he trajectory initialization presented in Section 4.3.2. Several e..xtensions of the design are presented in Section 4.5. In Sect.ion 4.6 the tuning functions design is applied to suppress the wing rock jnst.ability in aircraft flying at high angle-of-attack

124

TUNING FUNC'rIONS DESIGN

4.1

Adaptive Control Lyapunov Functions

The basic: iclea of t:he Lyapunov approach to adapt.ive c:ollt.rol is t:o design a control law and a paramet.er update law to gnal'ant.ec that. t:lu' derivative of a suit.able Lyapunov function is nonposit.ive. \¥e are therefore sent. to search for a triple: Lyapunov function, ('ont.rol law. and update law. For a class of nonHnetu· systems caIled paramctric'-st.ric~t.-fceclback syst.ems we will be able to make til is search systel1la tic. To hegin with. let. liS investigate the possibility of adaptive design for the syst.("lll (4.1) i: = f(:l') + F(.l:)O + g(:t}u, :[' E JR.". 'U E 1R,

n

where E JR.]) is a vector or unknown constant. parameters, and ,((:1:), F(3;) and g(.1") are smooth. For simplicity let j(O) o. F(O) = 0, so that. x 0 is an equilibriulll of the uncontrolled plant..

4.1.1

=

=

Departure from certainty equivalence

Ivluch of the traditional adaptive cont.l'ol employs some form of "certa.inty equivalence" thinking. FoJlowiJlg t.his path one first performs a design fol' t.he C8.Cie when the exact. value of 0 is known. Suppose that. t.his noutrivial ta.."k is compJeted and that its result is a fecdb~lck contra} 'U = (JA~I:, 0) wliich stabilizes t.he equilibrium x 0 with respect to a known Lyapnnov function \~(x, 0). The suhscript 'r:' st.ands for "certaintyequiva]cllre.'· \Ve 1mow that V';,.(;v,lJ) is posit.ive definite and radially unbounded in .1' for all 0, and thnt there exists a funetion l.f!(x, 0), which is also positive definite in .1' fol' all 0, such that l

=

a,~

ax' [f(,T) + F(x)B + g(x)Q:r.(:c: 8)] ~ -TV(:z:, H).

(4.2)

Hmv can we c>..1Jloit the knowledge of ll:c(:l', 8) and 1~(.1·1 0) for adaptive design when 0 is 110t. known? The certainty equivalence idea is to replace () by an estimate 9(1) obtained from a parameter Ilpual.e law (4.3)

r

where t.he adapt.ation gain matrb:: is posit.ive clefillit.e. \Ve want. to select 11and T 1.0 guarantee that the derivative of a Lyapullov fUllction is nonpositivc. For t.he syst.em (4.1), (4.3), a Lyapunov function candidate is

(4.4) 1 Throllghont. the chapter, we will drop t.he arguments in lJ\~.8) wId O\I/~~·/j) , nod write shortly ~~ and ~~. However, we will keep the flrgument.s in f(x), F(X)l g(x), and a(x. 0).

4.1

125

ADAPTIVE CONTROL LYAPUNOV FUNCTIONS

where the "certainty oquh7a]el1ce" form of \~ is augmented by a term quadratic ill the parameter estiIllation error

(4.5) Upon tbe substitution of F(:t)(J = F(x)9 along thE' solutions of ("1.1), (4.3) is

.

l'

+ F(.l')O,

the derh'at.ive of l'(.J:,8)

D1~ ( J(:I') + F(:z·)(J~ + g(x)u ) + -~'rT D1{, -. (D'~. )T_ = -0' + Or ~F(.r) - liT T. ~ 00 ~

(...t.6)

To eliminate t;be indefinite c1epclldell(~e of ,:r on the unknowl1 parameter error 0, we select r to cancel the last ';wo terms ill (4.6):

)'1'

8V r(x,O) = ( a.l~ F(;.. )

(4.7)

With this choice of T, the expression (4.6) is reduced t.o

81~ (f{3.) . 1'F -_ -8 X

•

...

. . al~ r (8\~ + F(.l)O + y(.t)lI) + -. ~F(.l) )'T . 88 v.l'

(4.8)

=

Our nc..xt task is to select a control law u a'(x,8) to make 1> llonpositive. The "certaintyequiwtlenceH controll1. a·c (.t',8) fails to acllieve i.bis hecause then (4.2) and (4.8) yie1d

=

.

•

\f S -1'li(:c,6)

O\~ (D\~

+ D9 r

lJ.r F(:r;)

),r •

Clearly, l' is not nonposith'e because a. sigll-illclelinit;e term is added to - (.t!(:r, 8). In search of a bettel' control law 0'(.1',8). we augment aA,'~, iJ) by 0-.,. (x, 0), (4.10) The substitution of (·tl0) into (4.8) shows that the desired nonpositivit~t -IV(x,8) will be achieved if aT CRn be found to saUsfy

al~

. + -~ aVe an r

~g(:t)o'l'(X' 0) vX

(81' c ) -0 F(:.:)

T

.1:

= O.

if s

(4.11)

This condition (or a'.,.. demonstrates tbe difficulty of adapt.ive designs for a general nonlinear system (4.1). It is casy to S(,E' that a r satisfying (4.11) is unlikely to c..xist: The scalar quantity ~g(:c) may be zel'O at: a set of points. StiU, the condition (4.11) is of iuterest because of an hUJJOl'f.ilUt. special caso, which will be the starting point of our recursive design. This special elISe is

126

TUNINC FUNC1'(ONS DESIGN

the "extended matching" studied in Section 3.4. In this case, a smooth vectorvalued fUllction cp : R n +1J --+ JRP is known such that ~ can be fadored as follows: a\~ aVe (}.'1: () 'P(.?", (j~)T . (~1.12)

ao = -ax'

-A

Then, irr~pective of the zeros of t~~.t}(X), an A

A

07"(:1',8) = -
T

ll:T

which sath;fies (lL11) js

T

a,~

r ( ax F(.];) )

A

T

A

= -
(4.13)

\Ve observe that, in addition to it.s "certainty equivalence" part G e , the adap:tive controllaw 0' contains a part aT which is proportional t.o i, that is, to iJ (see (4..3), (,1.10), and (~!.13)). In this way the adaptive control law t,akes int.o acconnt t.he parameter est,imation transients. \Vhcn t.he parameter estimate iFi constant, the control law reduces to the "certainty equivalence" cont,rol. Let us examine t\D example of a system for which (4.12) is satisfied. Example 4.1 Consider the problem of designing an adaptive controller for t.he system

=

:1: 2

(4.14)

It,

where () = [0 1 , 02J T is an unknown constant parameter vector, and tJle vectorvalued funct.ion cp(Xt) [cpt (.1:1), CP2(XI)F is known and smooth. VYe dealt with t.his system in Section 3.4.1. If the parameter f) were knowll, backstepping would result. in the 6-dependent change of coordinates

=

Zl

'<::2

= =

Xt X2

+
(4.15)

and the (·ont.rol law

v with

= aAx,O) = -=1 -

el, C2

c,,,, - (~~~ + c,) (x, + 'I'(XI)TO) 8

(4.16)

> 0, which results in the closed-loop system z=A=,

(4.17)

Due to the structure of A, an appropriate LyapllDov function is

t~(.l:, f))

=

~=(x, 8)T :;(a:, 8) ,

(4.18)

Observing from (4.1) and (4.14) that

j(x)

=[i

],

(4.19)

4.1

127

ADAPTIVE CONTROL LVAPUNOV FUNCTIONS

and evaluating

al~ -_

-;-

.T [

..

fJ:J:

1

aT ~8+cJ

0 1

1,

(4.20)

with (4.19), (4.20), and (4.16), it is ensy to show that (4.21)

Let us now evaluate tIle partial derivatives ::tppearing in (4.11): al~

aD = 81{ -g = ax

:;'re2lPT

;:;"e2

= =2tpT

= =2,

(4.22)

(£1.23)

where (~1.23) is immediate from (4.20) and (4.19). A comparison of (4.22) and (4.23) reveals that ~ = !!J:gcpTf so that 0'1' is given b~r ("1.13):

{4.24} Taking for simplicity U

r = I, the resulting adaptive control law is

=0:( •.,9) =

-2, -

0,"2 -

-<,0"<,0

[I.

(01+ ~~~ II) (x,+ <,O(:r,)TO)

~~~ 11 + e1) : ,

(OS)

and the correspoDcUng parameter update law (4.7) is (~L26)

Note th::tt in (.£1.25) and (4.26) we llse =(x,9) instead of z(:v,O}. With the choice of a and T given by (4.25) and (4.26), the derivath'e li" of the Lyapunov fUDct-ion 1'(:1:,8) 4z(x, O)T :(x, 9) + ~8Tjj is gllarallte~d to be llonpositive: V = -Cl c2=i. This assures that both x and iJ are bounded. A standard argument llsing the LaSalle-Yoshizawa t,heorem {Theorem 2.1} proves that also x{i) -+ O. 0

=r -

=

In the above example the desired factorization (4.12) of ~ is 8. consequence of a. particular fea.ture of the system (4.14). The unknowll parameter R}'pears in the first, while the control appears only in the second equation. It is not

128

TUNING FUNCTIONS DESIGN

hard to sec tl1at. the same factorization (4.12) would be possible for ~t higherorder plant., provided that the unbwllm. par(l.mete7' i.s separated from the c071t1YJl inp1lt by a.t most one intcgroto7'. So {:he factorization (4,12) is not. a fmt.uitolls (went, hut a structural property. For systems wit.h this "extended matching" property, the above simple adaptive design is feasihle. However, most. systems fail to possess the "extended matching" property. A bellchmark example is the third-order syst.em

= = =

i' I j'2

i:a

:1'2

+ I;.7(Xl)'J'fj

.I,·j

(4.27)

u

which has t.he form of (4.14) augmented by an illt.egnttor. In {:his system, 0 and 1/ are sepantted by two integrators and we are unable to find 0 7 which satisfies (4.11). 'vVc will solve this problem with a recursive design which will drcnlllvent. the obst.acle posed by the rest.rict.ive condition (4.11).

4.1.2

Certainty equivalence for a modified system

Condition (4.11) was dictated by Olll' choice of t.he Lyapllnov function '~(.1~, 0) as I.he "certainty equivalence" form of '~(:r, O). The only good thing we know about. "~~(.r, 8} is that it \Vorles when the factorization (4.12) is possible. Othenvise we do not know how t.o remove the indelinite term preventing the n011positivit.y of Ii in (4.9). Having recognized t.hat a cause of our difficulties is '~(J:,8), we now embark on a search for Lyapunov functions mo~'c suitable COl' adaptive control. The key idea is to counteraet. the effect of 0 and thus prevent the parameter estimate transients from destroying the nonpositivity of t.he Lyapullov derivative.

We say that the syst.em

±=

J(x}

+ F(:r.)8 + g(.r}'U

(4.28)

is globally adaptively stabilizable if there exist a function 0:(.1.',8) smooth on (JR." \ {O}) x HlP wit,h 0:(0,0) == 0, a smoot.h function r(:I:, O}, and a positive definite symmetric p x 11 matrix r, such that t.he dynamic controller 'U

=

0'(:1:,8}

(4.29)

6

=

rr(.'l" O),

(4.30)

gua.rantees t.hat t.he solution (x(1), n(t)) is globally bounded, and .1:(t) --+ 0 as t --+ 00, for all () E IRP. Our approach is to replace the problem of adaptive st,abilization of t,he original system (4.28) by a problem of nonadaptive stabilization of a modified system.

4.1

129

ADAP'l'IVE CON'l'ROL LVAPUNOV FUNc'rrONs

Definition 4.2 A smooth Junction l~ : JRII x lR" -+ R+. positive definite and rudially unbounded in:1' for each 8, i,fj called an adaptive control Lyapunov function (aclf) for {4.28} if fJu~re exists a positi2Je definite s.lJrnmet1'ic mat,.i.:,: r E lRPxp such lIwt j07' each f} E lRP, '~(.l~, O) is a elf JOl' the modified S~Jlit.em ;i:

that is,

l/~

= /(.1:) + F(J:) (0 + r

(a; )T) +

g(:r)1I,

(4.31J

sati.#Jjies

We now show how to design aclf is known.

Theorem 4.3 The jollowing

all

t1t1O

adaptive ('ont.roller (4.29)-(4.30) \Vhell an

stalemeni.s are equivalenl:

1. There exi.4Jts a t'li.plc (f1', \;;.., r) s'lJ.ch that a(J:, 8} globally (l.sym'1)loticaliy stabilizes {4.31} at :,; = 0 f07' ea.ch 0 E JRII 'with resIled to the Luap1J.nov junction ';;"(:1:, O). 2. The1"e e:J.i.9i.s an adf l~{J.', 6) jor (4.f!8),

}.{01"Cove1', if an aclf a.ble.

\~(.:l', O)

existS r then (4.. 28) is globtl.Uy adaptil.ely slabili:;-

Proof. {1 => 2} Obvious because 1 implies that: t.herc exists a conthmous funct:ioll lV : lRlI x JR/I -+ 1R.+, positive definite in :1' for each 8, such that

8\/ [ 8.1~

f(x}

+ F(x}

((81/ )T) + +r 6

8~1

]

g(x}o(x, 6) :5 -H'(J', 0) ,

(4.33)

Thus l~(x, 8) is a df for (4.31) for each 8 E lRP , and therefore it is an aclf for

(4.28). (2 => I) The proof of this part is based 011 Sontng's cOllstructive proof [171J of Artstein's theorem [4J. "Ve assume tbat 1~1 is an adf for (4.28), tha.t is, a elf for (4.31). Sontag's formula (2.19) appJied t.o (4.31) gives a rOllt.rollaw smooth all (JR" \ {O}} X lRP :

.I

a:(x,6}

+ = - !!Y.a.! 8:1'

(4.34)

o

!

Qiag(x Ox • t 6)

= 0I

130

TUNING FUNCTIONS DESIGN

where

(4.35) 'Vitb the c110ice (4,34), inequality (4.33) is satisfied with t:he ('Old.inuous function

IF(:Z',8) =

_ )2 + (tn~ ).a ax 1(3:,8) Dxg(J.·,O) , ( al~

(4.36)

which is positive definite ill x for each 0, because (4.32) implies that ~ j(:r, 8) < 0 whenever IlJ:g(~v, 8) = 0 alld 3' i= O. We note that tbe COlltrollaw o'(:L',8) will be continuous at .1' = 0 if and only if the aeIf \fa satisfies the following property, called the small controi,}roPC7"ty [171]; For each () E B" and for any E: > 0 there is a 6 > 0 such that, if x =1= 0 satisfies 13:1 S 0, then there is some 'u, witb luI ~ E: such that

all axu [J(:r:) + F(:r.} ((all 0 + r ao )T) + g(~l:)ll ] < o.

(4.37)

Assuming the existence of an aclf we now show that (4.28) is globally adaptively stabilizable. Since (2 => 1), there exists a t~rjple (a, Va, f) and a function ll' such that (4.33) is satisfied, that is,

~~ (f(x) + F(·l:l8+ g(xla{x, 8)1 +

:"r (~

F{''I:'f

~ -W{x, II) .

r- 1(0 -

8) .

(4.38)

Consider the Lyapullov function candidate 1

~

4T

lI{.T.,O) = l~(:r:, 0) + 2(6 - 0) A

~

(4.39)

'Vitb the help of (4.38), the derivative of V along tbe solutions of (4.28), (4.29), (4.30), is .

V

= 81~ ax =

l)8'~ x

[

cJl~

-]

-;;J'

,.

f + FO + .qa:(x, 8) + 80 rr{x,8) - 0 r(x, 8)

[J + PO + ga(x, 8)] + 81~ rT(X, 0) + ~~ F8 88

eTr(x, 8)

uX

,. al~ :::; -H'(;r,O) - -,. f

80

"

+0m' (al~) ax F -

(aVa )T Dl~ ~ -;;-F + - . fT(X,O) 80

uX

° T{:C,O). nT

-

(4.40)

Choosing

r(x, 0) =

al~ (

T

ax (x,6)F{x) ) A

,

(4.41)

4.1

131

ADAPTIVE CON'rnOL LVAPUNOV FUNCTIONS

we get

11 ::; -1"'(.1',8) ,

VB E ffiP.

(4.42)

Thus, the equilibrium x = 0,0 = 0 of (4.28), (4.20), (4.30) is globally stable, and by the LaSalle-Yoshizawa theorem (Theorem 2.1), :c(t) ~ 0, that is, (4.28) is globally adaptively st.abilizahle. 0 The adaptive controller const;l'ueted ill the proof of Theorem .4.3 consists

=

of a control law lJ. = a(:r,8) given by (4.3 l1), and an update law iJ fr(:1.',8) with (4.,U). It. is of int.crest to inl;erpret this controllcr a.1Oj a certaint.y equivalence COIltroller. The control law 0:(:1:,8) given by (4.34) is stabilizing for the modified syst.em (4.31) but may not; stabilizing for the original system (4.28). However, as t.he proof of Theorem 4.3 shows, its cel'tainty equivalencc form 0:(:1:,8) is an adaptive globally stabilizing control law for the original system (4.28). Hence, if ~t certainty equivalence approach is to be applied t,o a nonlinear system, the system is to be modified to require a cont.rollaw whi{'h anticipates the parameter estimation transients. In the proof o[ Theorcm 4.3, this is achieved by incorporating thc t'uning JUTlction T in t.he control law 0', Indeed, the formula (4.34) for a: depends on r via

ue

av;. -. _ aVa .. T a,T. f(:,;, 0) - D."C J + r(.l, 8)

avo (0 + f ( 80 )T)

'

which is obtained by combining (4.35) and (4.41). Using (4A1) 1,0 rewrite the inequality (4.38) as a~

ax

[j(x)

+ F(x)8 + g(x)o'(x, 0)] + 8~ ao fr(x. 8) ~ -H'(,l.', 8)

I

it is not. difficult. to see that the control law (4.34) containing (4.43) prevent.s

r fl'om destroying the nonpositivity of the Lyapunov derivativE'. Remark 4.4 A relevant question remains unanswel'ed: If there exists an aclf for (4.28), is this system globally asymptotically st,abilizable for each f) (and vice vel'sa)? In other wOl'ds, does the existence of a pair (a, l/~) satisfying (~J.33) for some r > 0 imply the existence of it pair (0:°, l~O) satisfying (4.33) [or r 0 (and vice versa)? Adaptive Lyapunov designs available in the literature [59, 65, 69, 94, 156, 157, 186] are all for systems whirh are not. only globally adapt.ively stabilizable, but. also globally asymptot.ically stabiliznble for each O. 0

=

As is always the case in adaptive control, in the proof of Theorem 4.3 we used a Lyapunov function li(.1\ 8) given by (4.39). which is quadratic in the parameter errol' (J - O. The quadratic form is suggested by the linear

132

TUNING FUNCTIONS DESIGN

dependence of (4.28) on 8, and the fact that (J cannot be used for feedback. VVe will now show that the quadratic form of (4.39) is both necessary and sufficient for the existence of an adf. Vve say that systcm (4.28) is globally adaptively quadratically stabilizable if it is globally adaptivellJ .litabili::able and, in addition, there exist a smooth function 1'1I(.1·,9) positive definit.e and radially unbounded in x for each 9~ and It continuous function lV(x,O) posit.ive definite in .T for each 0, such t.hat fat' all (x(O),8(0)) E lR"+P tl.lld all () E JRP, the derivat.ive of (4.39) along the solutions of (4.28), (4.29), (.t.30) is given by (4.42).

Corollary 4.5 The 51JS tern (4.28) is gio bally lI.dap ti1Jei1J ljualiratically stabilizCLble if and only if there exists an adf Va (x, 8). Proof. The 'if' part is contained in t.he proof of Theol'em 4.3 where the Lyapullov function F(x,O) is in the form (4.39). To prove the 'only if part, we start. by assuming global adaptive quadratic sta.bilizability of (4.28), and first show that 1"(:1:,0) must be given by (4.41). The derivatjve of l' along the solutions of (4.28), {4.29}, (4.30)1 given by (4.40), is rewrit.t.en ru;

This expression has to be nonpositive to sat.isfy (4042). Since it is affine in 0, it can be nonpositive for all (x,6) E lRn +p and ~Il (J E lll.P ouly if Ule last term is zero, that. is, only if T is defined as in (4.41). Then l it is straightforward to verify that

a;;

[J(Xl + F(x)

(0 + r (a;) T) + Y(X}l>{X,Ol]

=V+(6T-~ir) (T-(~:Fr) ~ -lV(x,8) for all (~'I 0) E IR"+p. By (1

(4.46)

=> 2) in Theorem 4.3, Vo(:r, 0) is an aclf for (4.28). o

The above analysis applies also to t.he ('ase where t.he unknown parameters entcr the control vector field:

.i· = f(x)

+ F(x)8 + [g(x) + G(:r)O]u..

(4.47)

4.1

133

ADAPTIVE CON1'ROL LVAPUNOV FUNCTIONS

In this case the existence of an adf Va is equivalent to the existel1cc of a elf for the syst.cm

±=

/(x)

+ F(,,')

(OH (lJ~y) +

[9(X)

+ G(x)

(6+ r (~ f)]

II.

(4.48) Tl1e ext.ension 1:0 t.he multi-input case is also st.raightforward.

It. is of intercst. to examine the itl}lllt-output Pfopclties of I.he system resulting fro111 tbe application of the adaptive control Jaw Q·(.r.9) to the plant (4.1):

± = I(x) + F(x)9 + O(J:)O'(:I', 0) + F{a:)6.

(4.49)

In early Ly~tpUllDV designs for linear systems of relative dcgree DIU', an important pl"opert:y was the strict positive realness of t.he transfer I'ullctioll between the parameter error and tbe out;put error [142J. For an analogous passivity pl'operty of the nonlinear system (4.49), let us consider that it.R input is 0.

Corollary 4.6 (Passivity) Suppo.'Je a j'll1lcLion l~,(.r: 0) is hU:J1Im. to be an ad! 1lritll. (1.11. associated control la'lJJ o·{ X t 8). Then the By/stern

'llnth

+ F(:J:)O +g(:t)o'{J:, 0) + F(.")ii

j:

=

f(:L')

T

=

81'0 ~ )T ( 03: (:1:,O)F(:d

0 lIS the input and T

as the D1Jlp'Ut is

(4.50)

l;#'riCtly

passizlc.

Proof. Along the solutions of (4A9) we have .

1~1

= ~

01~

-8

x

(

~

o\~

.

a\'~I-

f + FO + go·(:r, 8) + -. rT(:!:, 6) + -8 FO A

,

]

W

- T-

-ll"(:r:,O)+T(X,8) 0,

7

(4.51)

wbich, upon integration, yields

10' TT(J=(S), O(s)}6(s)ds ~ V:1(.t'{f), 6(t)) - \~I(X{O), B(O)} + 10' l·{l(.r(s) , O(.~»d8. (4.52) Using V;,(.l~, 0) as a storage functioll and TV(x, 8) as fL dissipation rate, and r noting that. l~ and al'e positive definite in 3: [or each B~ tlle inequality (4.52) estnblishes strict passivity by Definition D.2. 0

n

Hence, our closed-loop ada,pt.ivc system represents a llega.1.ivc feedback connection of the strictly passive system (4.50) and tbe int.egl'ator

- -T, r

- 9=

S

(4.53)

134

TUNING FUNC'fIONS DeSIGN

which is passive (positive real) because

!!.. (!orr- 1o) = dt 2

_8'r T

(4.5~1)

implies that (4.55)

For snch a feedback connection, Theorem DA establishes that the equilibrium x = 0,9 = 0 is globally stable, and x{t) -+ 0 as t --I> o. Thus, t,he problem of adapt.ive stabilization can be approached as the problem of finding an output T with respect to which it is possible to achieve strict passivity from 8 as the input.

4.1.3

Adaptive backstepping via aclf

\Vith Theorem 4.3, the problem of adaptive stabilization is reduced to the probJcm of findhlg an adf. We }lOW address the problem of systematic COllstruction of an adf. Our aim is a recursive approach because we already know how to find ac1f's for systems with the extended matching property, and expect to recursively enlarge this initial class of syst.ems with repeated use of backstepping. So, we assume that an aclf is known for an initial system, and construct a new aclf for the initial system augmented by all integrator.

Lemma 4.7 If the system

± = f(x)

+ F(l')6 + g(x)u,

(4.56)

is g{oball1J adaptivcl1J q1l.adratica1l1J stabilizable with a: E Cl, then the augmented system :i; = /(:1') + F(x)6 + g(x)~

{ =

(4.57)

'I/.,

is also globally a.daptively quadra.tically stabilizable. Proof. Since system (4.56) is globally adaptively stabilizable, then by Corolhuy 4.5 there exists an aclf {~(x,O), and by Theorem 4.3 it satisfies (4.33) with a control law 'U = 0:(.'&,6). Vve will now show that

l'i (:I:,~, 8) = V'a(.'&, 6) + ! (~ - a:(x, 8))2

(4.58)

2

is an ac1f for the augmented system (4.57) by showing that it satisfies aVI

8(x,~)

[f + (8 + r (~)T) + F

0:)

(.1:, ~ t 6)

lJf. ] $ _ HI _ ({ _ a

f

(4.59)

4.1

135

ADAPTIVE CONTROL LVAPUNOV FUNCTIONS

with the contra] law

=

aa (/ + Fe + g~) ax

oVa - - 0 - (~- 0') + -

ax

+ 80'r (8Vi F)T + av;,r (BIl'F)T olJ

OJ'

08

o:r

Let us start by introducing for brevity:; = t -0:(3:,8). With (4.58) we compute

alii [ J + FO + gf. ]

o{x, f.)

01 (x,

{, 8}

alii

=

81't 8x (/ + FIJ + g{)

=

(av0; - =ax ao:) (J+ FO+gf,)

=

8\'0 8d~ (1 + FO + go:)

+

o{

0:1 {x,

{, iJ}

+ZO:l

81~

Oet

+ ax gz - ;; aa: (J + FO + .Qf,) + =°1

80: ) = 01'a ax (/+F8+go:)+z ( 0]+ tn! 8:9- ox (J+F8+gf,)

. (4.61)

On tbe other hand, ill view of (4.58), we have

8111 O(x, {)

--

[Fr (!:!!i)T ] = -Fr alii (alii) 80 -8e l' 0 ox

(en;;. _:; au) Fr (8\!a _; an) T

=

ax

ax

ae

89

a)T

8V'", Fr ( 8V

=

ax

-z

ao

(oar (8Viox. F)T + ollar (OO·F)"). 8S

80

83.:

(4.62)

Adding (4.61) and (4.62), with (4.33) and (4.60) we get

alii

[f+

F(0 + r (~ )T) + DC; ] 0'1 (x,

8(x f.) t

=

{,O)

aVa (/ + Fe + go:) + 81'", Ff

ox

a.r:

+Z

(0:

1

(aVn)T {)f}

+ al'a g _ 80' (/ + FO + gf.) ox 8x

_ Bar ( 81"i 00 ax ::; _HT(X,O)

F) T_tH~, r (00' F) T)

_.::;2.

01J

ax

(4.63)

136

TUNING FUNC'l'IONS DESIGN

This pI'oves by Theorem 4.3 that \'1 (x. {, 0) is an aclf for syst.em (4.57), and by Corollary 4.5 this system is globally adaptivcly qnadrat:ically stabilizablc.

o The new tuning function for system (4.57) is determincd by the new aclf

Vi and given by T] (:1', ~~O) =

[F ])T = (~Vj F)'f = [(aVa _(e _ a,}aa:) F]T ( B(~ .1.:, 0 o:r B.T. a3: ~)

(4.64) \Xle note that the new tuning function

I)

is obt.ained by augmenting the initial T

tuning IUllct:ion T with the term - (~~F) (~- 0') which accounts for the faet t hat the ac1f 1~, is augmented by ! (~ - a(.1: I 8))2. The form of the controllJ:l.w Cl:l~(3:1 {, £J) in (4.60) is of part.icu1ar interest. It consist.s of two parts~ 0:1 = O'ltC + O'I.T' TIll" first part., ltl,c(Xt(,IJ)

a\~ = --a' 9:r;

(~- 0')

ao

+~ cr + FIJ + g~) v.1.

,

{£l.GS}

would becomc the "certaintyequivalencc" controlla,w for the augmcnted system (4.57) if we were to set r = 0. 2 The set'olld part consists of t;wo terms,

aJ

,,(X,e,8)=:r(:Ff +~;r(~;Fr

Their role is to produce ~Fr (~)

'f

(4.66)

in t.he aclf inequality (4.59). Observe

that the first. term ill (4.66) incorporates TI = (~F) T. The controlln.w 0'1 (x, €, 0) in (4.60) is only one out of many possible control given by (4.58) is an ac1f for (4.57), we can laws. Once we have shown that uset for example, the CO control law 0'1 given by Sontag's forllluia (4.34) witl1 ...illi.. g - - "'lld O(.:E'.~I 1 - '" u

1'.

~

I

I

It can he shown that the following function, used as a. elf ill [1581, is a 1110re general aclf than (4.58): {tj (x,~, 6) :!Note, however, I.hat are also fUllctions of

r.

al.e

(-a(.:E'IO)

= "11('1', 6} + .L n

is not. obtained by setting r

l1{s)ds,

(4.68)

= 0 in 01 since o'(x, O} and Va(xlO)

I

}

4.1

137

ADAPTIVE CONTROL LYAPUNOV FUNCTIONS

where TJ is a CO function sneh that 811(8) 7J f/. £1 « -00,0]) U £1 ([0, +(0)).

> 0 whenever s

The following example illustrat.es t.he use of Lemma

=f:. 0, 7]/(0)

> 0,

and

~L7.

Example 4.8 Let us consider the system Xl X2 ·'Va

=

x:!

=

X:i

=

+ cp(:r:} )T (1 (4.G9)

'/I.

"Ve will treat the state 3'a os an integrator added to the (:1:1, .r~)-subsystem from Example 4.1. In that example, we have already designed an adaptive control law for the syst.em

Xl ·1:2

= =

.l'2

+ cp( X1 ) 'I'fJ

(4.70)

Xa,

considering X:i as a control input. With (4.18), (4.10), (4.20), (4.22), it ean bc shown t.hat

=

which means that l'a{xt, :1:2, 0) = '''=(XI, 3:2, 0) ~(zr + =i) is an aclf for the system (4.70) considering X:i as a control input. Therefore, Lcmma 4.7 is directly applicable. \Ve define z = X3 - a(x, 8). By Lemma 4.7, the function

Vi (:r; fJ ) = '12 ( =j + =2 + '::3 'I

'J

.')

I

(4.72)

is an aclf for the system (4.69). With (~L60) and (4.64) we obtain 0'1

(.1',0)

= (4.i3) (4.74)

'\Tith the following adaptive cOlltrollaw and the paramet.er upclat.e law:

(x, 0)

(4.75)

B = T}(x,9),

(4.76)

·u. =

0'1

138

TUNING FUNCTIONS DESIGN

it is straightforward to verify that the closed-loop adaptive system is

where ;;1, -=2,'::3 are used with (j as all argument. The global stability of this system is established using the Lyapunov fuudion l'{x,O) = lIt(r,O) + !(JTO. ~

0

While in Lelllma "1.7 the initial system is augmented only by an integrator, a minot' modification is sufficient t.o obt.ain an ~Ulalogous result for the more general system

x =

f(x) + F(x)8 + g(.T,)c;

t =

u + F1 (x,c;)8.

(4.79)

Corollary 4.9 The Junction Vi (x, ,;,8) defined in {4.58} is an acl! Jor the system. (4.79) wit}, the control law and the tuning function gi'uen as (4.80)

(4.81) A repeated applica.tion of Corollary 4.9 will furtber extend t.he class of nonlinear systems for this type of adaptive design. With the knowledge of Va, i, alld 0: for the system (4.79), it is llot bard to see that by applying and 0:2 for the system Corollary 4.9 twice we can find 1'2,

'2.

.1:

{l {!!

= = =

f(:v) ~2 'U

+ F(x}8 + g(X)(l

+ Fl (3:, €l)8 + F2(X, elt ~!!)8.

{4.82}

4.2

139

SET-POINT REGULATION

In fact, it is clear that an n-fold appJication of Corollary 4.9 ",ill provide us with \;;' , Tn, and Q' n for the system :i;

=

1(:1.') + F{.1:)6

~i

=

~2

+ g{J:)~l

+ FI (J:,~,)6 (4.83)

~r'-l

=

~11

\Ve

WillllOW

4.2

€" + F,,-l {x: {b' .. , {1I-l)6 1/. + Fn(~t:,~1., ... '~II)8.

develop a detailed design procedure for such systems.

Set-Point Regulation

\Vith repeated use of Corollary 4.9, we can design an adaptive cOllj:!'ollel' to globally stnbilize a desired equilibrium .-,;C of the paramel1i.c sf.1'icl-Jeedback system (3.70): :i:) = ·1:2 + !PJ (xd T 8

x!! =

X3

+ Y'2(X1, l':SJ'e

;r:n

+
(4.84) :i:,,_] = xn =

B(x)u + 'Pn(J·)T(J

where 0 E RP is a vector of unknown constant pru'ameters, {3 and

(·1.85) are smoot.h nonlinear (unct.ions taking arguments in [til, a.nd ,8(:1:)

m.n,

"f 0, 'rI:I.: E

In this section we develop a procedure I'or adaptive regulation o[ thc ontput Y = Xl to a given set-point. Ys. \Vi1.h a constant control u~ the first. n 1 equilibrium equations of j.e = 0 in (4.&1.) can be successively solved for x; ,. , . ,X~I as fUllct.ions of xi and 8: t

x~ = XC:J =

-
x~

-tp7l-1(.1:j, . , . t x~,-J )T(J

=

= =

',.e fl'0

-t.p:.! (:1'(! • '1" 2

(4.86)

.i·~, 0 yields n rclat.ionship between .i:j I u'\ and O. \Vhen (J is known, then :~ 0 can be 50hrpd for U C needed to keep :l'j at a desired set-point :z:Y = 1Is' The corresponding values 3:21' . , ,.T.~ will be dictated by (4.86). Therefore, for each value of 0 and a prescribed lis, the equilibrium :l'c

Then tlle 71th equation

140

TUNING FUNCTJONS DESIGN

and tIle corresponding control value lI. C are uniqucJy defined. In the special case whel'c tpl (0) = ... = 1,0,.-1 (O) = 0, t.he choice Us 0 resu1ts in the equilibrium being .te = 0 for all values of B. Our problem now is to globally si'alJilize this equilibrium when Dis unlmowll and also to achicvf" set-point regulation: J·(t) --+ :t.e as t -.. 00. Comparing the syst.ems (4.84) and (4.79), w(' observe t.hat if Xa WCl'e the control variable, then Corol1lU}" 4.0 wouJd provide tlle desired adaptive control for the subsystcm made of the first t.wo cquations of (4.84). Thereforc, we can init,iate our recursive design procedure by augmenting this subsystem by the third equal.ion, as in (4.82). For convenience, we wi11 do l:llis in a self-coutajuf>d fa.shion, independent, of Sed.ion 4.1. An 8.dditional feattu'e of the procedure in this scrt.ion is a set. of error coordinates in which the stability properties of the resulting closed-loop ada.ptive system are c1en.rly displayed without an explicit, lise of the aclf concept,

=

4.2.1

Design procedure

Vve will start by adaptively stabilizing t.he first equation of (4.84) considering to he its (·ontrol. At c.:'lch subsequent step we will augment the designed subsyst.em by one equation. At the ith stepl rul ith-order subsystem is stabilh:ed with respect to a Lyapunov function V; by the design of l:\ stabil7:zing junction D:; and a tuning ju.nction. ri. The uprlate law fol' the pal'ameter estimate O(t) and the adaptive feedba.ck cont.rol '/l are designed at the final step. .1'2

The third step is crucial for understanding thp general design procedure. Step 1. Introducing the first two error variables (4.87)

=

4.2

we rewrit.e :;:1

.1'2 -

0'1 ,

(4.88)

= :1:2 + 'PI {.1:!lTO, the first eqnation or (4.84), as :.

";'1

--

- + o· I + 'W 1·1 {a' )'rfJ ,

...2

(4.89)

whol'e, for uniformity with subse(Juent steps, we have defined the first regressor vector as (4.90) Our task in this step is to stabilize {4.89} with resped to the Lyapunov function

(4.91) whose derh'ative along tIle solutions of (4.89) js (4.92)

4.2

141

SET-POINT REGULATION

efrom l~ wi til the update law (j ==

We can pliminate

rTtl

where (4.93)

If X2 were our actual control, we would let. ma,)re l~ = -CI we lVould choosc3

=2 ;:

=r,

0, that is, :r!! ==

0'1.

Theil, t.o

(4.94) Since J.'!,! is not our control: we h.c1.ve ':;21= 0, and \ve do nol USE' fJ = rn as an update law. Insf,cad, we retain 1') as our first. tun.in.g fU71ction and tolerate tbe presence of 6 in 1:"1:

(4.95) The second term =1=2 ill 1~ will be cAllcelled at the next step. \\lith a:l(Xl.6) as ill (4.94), the =rS)1St.em becomes (4.96) Step 2. Vile now consider that: :ra is the control variable in the second equation of (4.84). Introducing =:-1

we rewrite j:~

=

X3 -

(4.07)

0'2 ,

= a';.:j + 1f'2(:J:t,X2)TB a..c;; =2

=

Z3

00'1

+ ct2 -

DXt x:!

+ tlJ2(Xl1 X:.!

~ T

1

9)

80'L :.

(J -

DO

(J:

(4.98)

where the second regl'Cssor vector w:! is defined as (4.99) Our task in this step is respect to

tlO

stabilize the

ti... = VJ -

(Zh

=2)-S)'st.em (4.9G), (4.98) with

+ !..:;; 2 .'

(4.100)

whose derivative along the solutions of (4.96), (cl98) is •

1'2

=r + =2 I)

=:

-Cl

+8T

(Tl

[

=1

+ Za + 0'2 -

+ 'W2=:J -

r-Jo) .

80'. l ax} X2

T•

80: J ~J

+ W2 H- ali ()

(4.101)

=

3The cbaructcr of the con(.I'Ol Jaw OJ is ilccrtnintyeqnivalence" because the ndf .\.:i l(J!l - YII)2, with respect to whid. t.he design is I)erformed, is illdependent. of 0. so it is also ;;, dC.

142

TUNING FUNCTIONS DESIGN

We can eliminate jj from

\i2 with the update law " = fT21 where {4.102}

were ou)' actual control and, hel1('e, =:. == 0, we would achieve li:! = c2=i by designing 0'2 t.o make t·he bl'ac1\eted term multiplying =2 in (4.101) equal to -C2=2, namely4

If

X3

-Cl='t -

Q'2(J:1, a:2,

We ret.ain

T:!

, 0) = -:1 -

{'2=2

+

80 1 {}J:I J:2 -

T' w:! (J

+

an

lJal

rT2.

(4.103)

a.c; our second t.uning ft1l!ction in the term rr2 which repJaces

in (4.103), However, we do not use resulting \~ is

8 ;;: rr2

iJ

as all update law, so that the

Tile first. two terms in \~ are negative definite, the t.hird ~erm will be cancelled at t.he next step, while ill€' discrepanc,V bf'twE'(,]) 1"72 and iJ in t.he last two terms TQnUlins. By substituting (4.103) into (4.98), t.hc (=1, =2)-subsystcm becomes

Step 3. Proceeding to the t.hird equation in (L1.84) we int.roduce (4.106)

(~1.107)

where tho third regressor vect.or IVa is

d~fincd

as

(4.108) "'\Vhile 0 I WRS .l IIcerletiut.y ~qui\'alcncc" ('ontrol, I he ('oot.rolla,w 02 is not. or the "L"eI'I.aint)' eqllh'fllence" type because it is designed wit.h respect. t.o the O-depcndcnt. nclf k:i+~':::;5' The

term ~rl'.! in Q!! corre.'iponds to the first lcnn ill th
III

i.:r

4.2

143

SET-POINl' REGULA"rION

Our task is to stabilize the (z\, Z2t Z3)-system with respect to

1 'l V:... = If..- + --:: 2 ....... ' wbose derivative along (4.105)

V:i. =

Rlld

').,

-c'l:j - c2 zi +'::.1

(4.109)

(4..107) is

lJa:t (

:.)

8iJ rT,2-8 BCt'!

8Q''1

Va .. :.)

+=:i [ .":2 + .:., + a':~ - ax,-,1'2 - aX2-:l'!i + wJ8 - ---;;:.(} ao +OT (1;) + W:~Z:i - r-1o) . Vva can ciimi.nate

efrom

~a with the update hnv

A

B = rT:",

\Vbel"e

(4.110) T3

is our

tuning fund.ion

If x:-\ were OUf actual contl'ol, we would have ':., == 0 and achieve l~ = -Cl =r C3Z~ hy designing 0:3 to make the hra.elected term multiplying =3 equal to -ClZ3, Dft.nlcly

C:!=5 -

(4.112) where "'J is a eorrectioll term yet to be chosen. 5 Substitutiug (4.112) into (4.110). and noting t.hat

jj -

1'T2

=

0-

rT3

+ rT3 -

==

e-

rT3

+ rUJa=a,

rT;z

(-!.113)

(4.110) is rewritten as

Ii;,

= -<,=; - ""~ + =. (,';] - ~'r'»3'" ) +-=3=.. + (="1 8C:1 + =3 aO:2) (rT3 - 08

DB

0) + 0"(T3 - r-10),

(4.114)

ftJt. is of iuterest to compare tMs cOlltl'ollaw with I.he cOllLrollaw (4.60) in LeU11Ua 4.7: The hlst two torms ill {J.1l2} aro analogous to the last two terms in (4,60).

TUNING FUNC'l'IONS DESIGN

and the (Zl, z:,h z3)-subsystem becomes

=

0, and wi(;h the up
If ;r.1 were our control, we would have Z,I

.

.

We again postpone tIle decision about [) a.nd do nOE use 0 = law. The resulting is

Va

rT:. as an updatE'

(4.117) and the (=h Z21=3)-subsystem becomes

(4.118) The 'system matrix' in (4.118) has a significant property: the sl,e\v symmetry of the nonlinear term ~ fWa ac.hieved by the choice of Va in (4.116). This term is analogous to the second term in (4.66) and tlle skew symmetry is crucial for stabilization.

Step i. Introducing • Xi • we rewr.1te •

Zj

=

Zi+l

= XHl + 'Pi (XJ r ••• , .ri)PJ'(J as + (Xi -

i- 1 8 """ O'i-l L.; -ll--X'k+l k=l UXk

+ UJt ( ~1'"

. , xh

o")T(J

-

aa:i-l (J'."

--,,-

86

,

(4.120)

4.2

145

SST-POINT REGULAT(ON

whet'e the ith regressor vector is defined as

(.:1.121) Our objective is to stabilize the

,Zi)-systelD with respect to

(ZI,'"

1 .,

V1 = V, - 1 +-Z=" 2! ,

(4.122)

\vhose derivative is

+.t'i [ Zi-l

+11""J' We can eliminate

+ ::i+l + 0i -

'T i-I a0:1-1 a0l-J 'J L ~a. Xk+l + Wi 0 - --A-(J 3'1' De A

A

k::::1

( T;_1

+ tlJI=i

-

r -I:") (J

(4.123)

•

lii with the update law {j = rTiJ whel'e

8 from

= [w" •.. , IUil

=1 ] [

~

(4.124)

tii

Then, in tbe absence of Zi+l, we would achieve = - L~=l CI..=l, by designing a; to make the bracketed term mUltiplying =i equal to -CjZit llRme]y

(4.125)

wbere

Vi

is a correction tenn yet to be chosen. Noting that

6-

fTi-l

=

+ rTj - rTi-l == 0- Pii + rfJJjZi fJ -

fTj

(4.126)

I

we rewrite

\1 as i-I 2 -L Ckzk +

[ Zj

k=l

a0:;-1 (rr; - 8). ] + Vi - --AA

Zi+l

~ ZI':+l-" oak) (r1i-l + ( Lk=l

80

:.

89

ilT

0) + (J

(Ti -

r

-J :.

fJ)

146

TUNING FUNCTIONS DESIGN

(4.127)

0"2.;-1

o o

1 + 0"1-2.1-1

o

-Ci-l

1

o

U:]

-1-

0"23

=

o o

-1 -

-0'2,i-l

o

o +

WI] : 0+

[ Wi'J'

O'i-2.i-l

o

-1

~1 ~1

+[

+[

_ -HI

O'j-l,i=j

'Ii

(fT; -

9),

(4.128)

8n._t

---no

where A

O"jI..(x: 0) = -

80';_1

88 r·w,..

(L1.129)

Now the correction term is chosen as ~ II; (Xl , ... , Xi,

i-2

6) =

L

lJll"

ao

k=J

Because we do not use iJ = •

Vi = -

~2

i """

~ Ck":'k

rTj

i-I

Zk+I--;-:-PWi = -

L O'k.iZk.

as an updat:e law: tlle resulting li, is

(i-J 8)'8) + 6

_ _ + "'i-i+l +

""" _

O'k

L- ""l"+I-A

k=l

(4.130)

1.:==2

k=l

80

(rTi -

•

ilT

(Ti -

.

.. r -1 OL

{4.131}

and the ('::1, ... , Zi)-Sllbsystem becomes

U:]

=

-CI

1

-1

-C2

o o

-1 -

o

0 1 + 0'23

0'23

-1- O";-I,i

{4.132}

4.2

147

SET-POINT REGULATION

Step n. At the final step, we introduce =

;;n

and rewrite the last. equat.ion :?n = /311

71-1

L 1..=1

+ cp~'(J -

:j"1

(4..133)

0'n-1

:L'n -

+ IPn{.r)TB as

= (3{."C)u

ao'

a~-1 (.1'k+1 ,'lk

+ cfJIo) -

ao' . ao

~o,

(4.13-1)

where the last regressor vector is defined as ~

II-I

aO',,_1

L -D" 'P"" 1..-=1 .[1.-

'Wn(x,B) = 'P" -

(4.135)

In this eqnation, the act.ual control input. is at: our disposal. 'Ve are finally in the position to design our actual update law 0 = 1"TII and feedback cont:rol 1.1 to st.abilize the full =-systelll with respect. t.o 1~,

=

, 111-1

]

_2

+ 2""71

= !=T =+ !oTr-1o, 2 2

(4,136)

Our goal is to make l~ nonpositive:

l~r

= -

L

II-I

'1

CI.:;;j;

+

(JI-2 L

1.'=1

,.

ao'' _) (frll_1 - tJ)'.

=k+l-

A -

ae

k=l

+=n

[ =,,-1

+8-T

(

T,,-1

+ f311. -

+ W n -_n

aOn-l 1 + 'Ill"T'(I - --~-O a 'J L -a--.:rI.·+ .7:1._ ao Q'n-l ~

n-I 1.:=1 -

r -1 B') .

(4.137)

:::

To clhninate

efrom l~, we choose the updat.e law o= =

rTn{:,O)

= rr"_1 + rw".="

rl-V(.:, O)z,

(4.138)

where the regressor matrix H' is composed of t.he regressor vectors WI, •. , , W,,: (4.139)

'Vc choose the control u to mal{f t.he bracketed term multiplying .;;" equal t.o

u

= -(31 ( -=n-l -

C'.=1l

f){Xn_1 +~ L- ~.1.'I.·+1

~'=1

k

-

T /IJ"':,,

aO'II-l I'" ) + ---Tn + l.1n

ae

1

(4.140)

148

TUNING FUNCTION'S DESIGN

whcre

1)'1

is a rOn'€ction tel'm yct to be chosen. With (4.140), '{I becomes (4.141)

TllCll, noting that. (4.142) we rewrite l~1 as

l~1

=

11-1

L

-

~I

Now the correction term

(n-2 L IJn -

~)

EJ Zk+l

G.J.· fwn

)

(4.143)

.

W

is chosen as

I)"

_ 1.111 (.1',0)

Cl.'=~ + =11

11-2

EJo'I.'

II-I

1.'=1

88

I..=:.!

= ~ ="'+1-.-. rWn = - L

(4.144)

O.,.·.f.Zk·

\Vc have thus reached our goal: 71

'7.1 - L" Ck-1...· I

_

_

_-

(4.145)

k=l

The overall closed-loop system is .:.

=

o=

A:(=, 8)= + H'(=, iJ)'I'o

(4.146)

fH'(=,8):,

(4.147)

where

A:(:::,O) =

-Cl

1

-1

-C2

0 0

-1-

o

0 1 +0"2:)

.

0"2:i

-1 - 0',,-1,"

-U2n

(4.148)

1 + un-I,u -en

The syst.em (4.14G) will be referred to as the enYJT system. It is important to note that a major portion of the design effort was invested into achieving

A.(z,li) +A,(=,ti)T = -2

[Cl ".

C

n

l'

'0'(=,8) E JR."'"

(4.149)

which yields (4.145) with the simple quadratic Lyapunov function (,1.136). 'Ve observe t.hat, as desired, the system (4.146)-(4.147) has an equilibrium at (=,0) = (0,0). The stability properties of this equilibrium will be established in Section 4.2.2.

4.2

149

SET-PoIN'r REGULA'nON

Remark 4.10 Slrew symmetry is not the only tool we can llse for achieving nonpositivity of l~. For instance, instead of (4.130) we can use (4.150)

which results in

A:(.:, iJ) =

-Cl

1

-1

-C2

0

-1

o

0 1 + 0"2~t -C;i -

o

11-2

2

~0"2:l

o

It can be shown that with (4.150) we obtain

V(z,B) E lR"+'~

which yields

. ~ l~r

""

1 "

(4.152)

(4.153)

-? L- CkZ'k. ')

- k=l

We can carry this idea a step further. By replacing the st;abilizing function (4.125) by

(-1.154) we arrive at the error system (4.146) with -Cl

1

-1

=

1 -1 -en

[

q~.l.

+ [

O"n,l

0'2,n

o

..•

1

q'

n.1I

o +

. (4.155)

150

It

CRn

TUNING FUNCTIONS DESIGN

be shown t.hat A: satisfies

'9'(=,6) E rn,n+p (4.156)

which yields (4..153). Instead of cancelling the tuning funct.ion Ti, the st.abilizing function OJ in (4.154) dominates it, \Ve ('onsider (4.130) preferab1e over (4.150) and (4.154) because the latt.er two lead to a much faster growth of nOlllincaritics in the ront.rol Jaw. 0

Exalnple 4.11 In applications of the t.uning functions procedure wc do not need to repeat t.he Lyapunov argument. All we need [01' a specific design al'e the nnal analyt.ical expressions provided by thc procedure. Let liS now iIlust.rate t.his by designing an adapt.ive cont.roller for the benchmark system from Example 4.8: Xl = .1:'2 +
(4..157)

·T.'3

11.

The design objective is the regulation of t.he output y =

.1:'1

to the set-point

:'19' The first. threc exprcssions provided by the procedure are the definitions

(4.87), (4.88), and (4..97) of the error variables

where

0'1

and

02

=1

=

.1'1 -

1Js

Z:!

=

:1:2 -

at (:1:1,0)

Z:i

=

:Z:3 -

02(Xl,

(4.158)

x~, 8) ,

are the stabilizing functions given by (4.94) and (4.103): TA

0'1

=

-CI=t -

'P 8

0'2

=

-C2=2 -

=1

80'1

+ 8:1:1 (X2 +
T

A

8)

aa1

+ 80

T!,!.

(4.159)

The tuning funct.iol1s, determined from (4..03), (4.102), and (4.111), are T1 T2

= =

Ta =

=I