Оптимальные и адаптивные системы

. 2003 . Contents . . .. . .. . .. . . . . ...

Author: Александров А.Г.

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2003 .

Contents . . .. . .. . .. . . . . .. . .. . . . . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . .. .7 . . . . . . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . . .8 1 10 1.1 . . . . . . . . . . . . . . . . . . . . 1.2 . . . . . . . . . . . . . . . . . 1.2.1

. . . . . . . . . . . . . . . . . . 1.2.2 . . . . . . 1.2.3

! # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 $ % . . . . . . . . . . . . . . . . . 1.3.1 & ' ! ! . . . . . 1.3.2 % % . . . . . . .

10 15 15 17

20 22 22 23

2

26

3

49

2.1 # ( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 )* . . . . . . . . . 2.1.2 # ( . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 , (* ( # ( . . . 2.2 % . . . . . . . . . . . 2.2.1 # ( ! % 2.2.2 - % . ( . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 , n - ! . . . . . . . . . . . . . . . . . . . . . . . . 2.3 0 * ( . . . . . . . . . . . . . . . . . . 2.3.1 # . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 1 (# . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 3

' * . . . . . . . . . . . . . . . . 2.3.4 & # ( . . . . . . . . . . . . . . . . . . . . . 3.1 # * ( ( . . . . . . . . . 3.1.1 * ( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 4 # . ( . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 3

' * * ( $((. 0 $ - , ( . . . . . . . . . . . . . . . . . . . 3.1.4 3

' $(( # . 3.1.5 5 %

% . . . . . . . . . . . . . . . . . . . . . 1

27 27 29 30 33 33 35 38 39 39 41 45 47

49

49 55 58 59 60

3.1.6 4 % . ( . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.7 7

' . . . . . . . . . . . . . . 3.1.8 )* . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.9 7( (#9 ) . . . . . . . . . . . . . . . . . 3.1.10 * (( . . . . . . . . . . . . . . . . 3.2 % 9 # ( 3.2.1 ( * ( ) . . . . . . . . . . 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 $(( . . . . . . 3.2.4 . . . . 3.2.5 & ( . . . . . . . . . . . . . . . . . 3.2.6 ; . . . . . . . . . . 3.2.7 ! . . . . . . . . . . . . . . 3.2.8 * (( . . . . . . . . . . . . . . . . 3.3 3 % ! . . . . . . . . . . . . . . . . . . 3.3.1 * % 9 . . . . . . . . . . . . . 3.3.2 5<99# * * . . . . . . . . . . . . . . . 3.3.3 = # %* ! . . . . . . . . 3.4 ,* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 ( * . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 . ( . . . . . . . . . . . . . . . . . 3.4.3 . ( . . . . . . . . . . . . . . . . . 3.4.4 * (( . . . . . . . . . . . . . . . .

4 #

61 63 64 65 69 70 70 70 72 73 75 77 79 80 80 80 82 86 89 89 90 90 92

92

4.1 ( % . . . . . . . 93 4.2 % 9 # ( . 96 4.2.1 & ( . . . . . . . . . . . . . . . . . 96 4.2.2 % . . . . . . . . . . . . . . . . . . . . . . . 98 4.2.3 , . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2.4 ;

! % ! . ( . . . 101 4.2.5 * . . . . . . . . . . . . . . . . . . . . . . . 102 4.3 !* ( ( . . . . . . . . . . . . . 103 4.3.1 ( . . . 103 4.3.2 % ( % ( ( % 9 5 > ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.3 , . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.4 * . . . . . . . . . . . . . . . . . . . . . . . 110

5 H1 %

110

5.1 5 % # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.1.1 3 * # . . . . . . . . . . . . . . 111 2

5.1.2 H1 - * % * % # . 5.1.3 ; * H1 - . . . . . . . . . . . . . . . . . . . 5.1.4 ( * H1 . . . . 5.2 . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 0 ( . . . . . . . . . . . . . . . . . . . 5.2.2 , * *

H1 - . 5.2.3 # . . . . . 5.2.4 * (( . . . . . . . . . . . 5.3 . ,* . . . . . . . . . . . . . . . . . . . 5.3.1 & . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 * (( . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

113 114 115 115 115 116 118 119 119 119 121

6.1 ! !. . . . . . . . . . . . . . . . . . . . . . . . 6.1.1

! ! . (. = (# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 9(# . . . . . . . . . . . . . . . . . . . . . . 6.1.3 @ 9(#

% . * ( . . . . . . . . . . . . 6.1.4 % . 1 (# - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 & ( ! . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 & ( ! . . . . . . . . . . . . . . . . . . . . . 6.2.3 ( * < ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

. . . . . . . . .. . . . . . . .. . . . . . . . . . . .. . . . . . . . . 123 6 & 125

7 ' ( ) *

7.1 B# "* !" ! . . . . . . . . . . . . 7.1.1 . ( . . . . . . . . . . . . . . . . . . . 7.1.2 ( * . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 B # . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 $ . . . . . . . . . . . . . 7.2.1 * % <

% . . . . . . . . . . . . . . . 7.2.2 B % . ( % % # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 B % . ( % % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 * (( . . . . . . . . . . . . . . . .

3

125 128 130 133 135 135 138 139

141 141 141 143 144 147 147 148 151 154

8 + , -

8.1 0 '! ( . . . . . . . . . . . . . . . . . . . 8.1.1 4 (

! . . . . . . 8.1.2 0 '! ( . . . . . . . . . . . . . . . 8.1.3 $ (

% '! ( 8.1.4 # ( &&- . . . . . . . . . . . . . . 8.1.5 # ( B$&&- . . . . . . . . . . . . . 8.1.6 & 9 # . . . . . . . . . . . . . 8.1.7 * (( . . . . . . . . . . 8.2 5 * -* 9(# . . . . . . . . . . . . . . . . 8.2.1 ( * ! ( ' . . . . . . . . 8.2.2 3 * . . . . . . 8.2.3 D( * ! 8.2.4 B 9(# . . . . . . . . . . . . . . . . . 8.2.5 ; * . . . 8.2.6 * (( . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

9 #

. . . . . . . . . . . . . . .

9.1 ( * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 B # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 '! ( 9.2.2 * (( . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

154 154 154 156 159 161 163 164 165 166 167 168 169 171 172 173

173 173 177 177 180

10 . #, # .0#180 10.1 0 (

! # ! . . . . . . . . . . . . . . . . 10.1.1 ( * . . . . . . 10.1.2 & . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 B # ' ! % . . . . . . . . . 10.1.4 B # ' ! ! . . . . . . . . 10.2 3 . . . . . . . . . . . . . . . . . . . . . 10.2.1 ( * . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 % # ( 9(# . () . . . . . 10.2.3 ; % #( 9(# ( % 10.2.4 &! # . . . . . . . . . . . . . . . . . . . . . . . 10.2.5 * (( . . . . . . . . . . . . . .

.. .. .. .. .. .. .. .. ) .. ..

180 180 182 183 186 188 188 190 191 193 193

. . .. . .. . .. . .. . .. .. . .. . . . . . . . .. . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . 194 .1 . . . . . . . . . . .. . . . . . 194

.1.1& ( * . . . . . . . . . . . . . . . . . . . . 195 .1.2=B00B-2$& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 .1.2.1& ($ *( ) . . . . . . . 196 .1.2.2& .($ *( () . . . . . . . . . . . 199 .2 .. . .. . . . .. . . . . . . . .. . . . . . . . . . . .. . . . . .. . .. . . . . . . . .204 4

.2.1 B * ( ( ! (7 ( 111) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 .2.2 (7 (113) . . . . . . . . . 206 .2.3 H1 { (7 ( 131) . . . . . . . . . . . . . 207 .2.4 & * . ( (7 ( 113)207 .2.5 & * . ( (7 ( 144) 208 .3 . . . .. . . . . . . . .. . . . . . . . . . . .. . . . . . 209 .3.15 * - * 9(# ( ( 111.3). . . . . . . . . . . 210 .3.2 @ 9(# '! ( ( ( 211) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 .3.3 3 ( ( 311) . . . . . . . . . . . . 212 .3.4 B 045 ( ( 411) . . . . 213 .3.5 B <

% ( ( 511) . . . . . 213

1 . .. . .. . .. . . . . .. . .. . . . . .. . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . .. . . . . . . 216 .1 . .. . . . . . . . . . . .. . . . . . . 216

7.1.1 & #

* .. . . . . . . . . . . . . . . . . . . . . .. .. .. . . . . . . . . . .. 216 7.1.2 #

* . .. .. .. . .. .. .. .. . . ... .. . . .. .. .. . .. .. .. .. 216 7.1.3 ; #

* (

* *(. ! <( ( D% ). . .. . . . . . . . . . . . . . . . . . ..218 7.1.4 D% - . . .. . . . . . . .. .. . . . .. .. . . . . . . . . . . . . . . . . . . . .. . . . . . .221 7.1.5 ; #

* # . . .. .. . . .. .. .. . . . .. .. .. .. . . 222 7.1.6 ; ! <( ( - ). . . . . . . . . . . . . .223 7.1.7 ; #

* % <( . D% - - . . .. .. .. . . .. ... . . . . .. .. .. . .. .. .. . . .. ... .. .. .. . . .. ... . . 224 7.1.8 )* 0% > #. . .. .. .. . .. .. .. .. . . .. ... . . .. .. .. . . ... . . . . . . .. .. ... ..227 7.1.9 7( - . . . . . . .. .. .. . .. .. .. . . .. .. . .. .. .. . . .. ... .228 .2 ! " . # $ . . . . . . . . . . . .. . . . . .. . .. . . . . . . . . .230 7.2.1 & (* (! * ( . .. . . . . .. 231 7.2.2 & # . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . ..232 .3 % " . .. . . . . .. . .. . .. . .. . .. . .. . . . . .. . . . . . . .. . . . . . . . . . . .. . . . . .. . .. . . . . . . . . .238 .4 ! . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 241

1 . . . . .. . .. . .. . .. . . . . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . .. . . . .245 .1 ' $ . . . . . . . . . . . . . . . . . . . . .245 .2 ' ( . . . . .. . . . . . . . . . . . . . . . . . . .. . . . .248

.3 ' ) * .4 . . . . . . . . .. . . . . . . . . . .. . . . . .. . . . . . . . 257 .5 ,) H1 . . .. . .. . . . . . . .. . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . . 261 .6 ( , */) H1 . .. . . . . . . . .. . . . . . . . . . . .. . . . . . . . . 261 .7 ' ) .. . . . . . . . . . . . . .. . . . .265

5

.8 ' ) . . .. . . . . . . . . . . .. . . . . .. . . . . . . . . 267

2 . .. . .. . .. . . . . . . . .. . .. . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . . . . 269

6

7 ( ! <99 ( ! * ( (&B)

! : ) , ) ( ( % *, %

, ) * <! , ( ( ! , ( ( &B ( * . 5 , , : , ! ( (

! * . * %: *( &B #, *! % "(* (%" * ( . 5 ! * % ,

!

, ( : (( , 7 7( . 3 * * . ! ( 5 . ; * (% * % ( 1- 6-%) , 9 * . ) , 2-% % 6-% , ( ! ! ! * ! ! . * <! , ! ! =B00B-2$& 0B,-B>. @ ( ! ( 7 ( % ! ( . > ! 7( . ;

* 7 . $ (( ( ( . < * , ! =B00B-2$&. $ *( &B % ( ( ( , (

! ( ( ! ( , ( ((. 4

( 1989.), (( * , ! < ( %

( (#%. &( ,

% ( , ' , ( ! * , , ( ! < * %' , (* ( , ( * . 7

. & ; ' ( * " %" * *'% ( ( <99 ( . ( ( *

%, ( (( ((%- , * ((- ' *' ! . 5 , ( ! (( ), * . D ( (* ( # , , < , , . ., ( <! ( ( (<99# . 4 " %" ( , ( * ( # ' (* * (! # < ( ( %) < (*

%

! (9 (# ). ,( ( (*

! ! '

% ( #

* . ; #

*, ( ! ! # , %! # % , 9 % . ; % * ( ( # ( -. &. * ( $. > ). D 9 ! . '% . 9 # ( . & % , <% . ; (* ( ( * ! ( . 7 , * * ( # ( ' ( % * 99 # ! % ( (

! * ! !).@ *

' ( ! * ( , * ! % ( ( . ( % * (% * %, ' ( % * #9* (!

% . (, # *(. D ( ( ( . (, ( ! ( * ( ' *

. ,( . ( ( . ( , % 99 # . D , *

B. 0. - $. 5 , % % % #: * ( ( ( . (( LQ { # % ) % 8

9 # < ( ( LQG {#).D 3 ( 3.1,3.2) 4. = 5 ( 5.1 5.2) H 1 { ,( LQG {#. (* ( LQ {, LQG H 1 {#% . 1.A ( ( * 9 (# , <! !. * < * '! '(, ( ( * (* ) 2. .D * , * ( * ! * % ( %* . D ( %' # (<99# ( * 9 (# ( , ( . (, ( ! .. D ( ( ! 3.2, 3.3 5.2.

9

1 "#" $#%& '$ " 1.1 $ . ( , (

x_ = '(x u t)

(1.1.1) x(t) {n- % (

! . (, u(t) { m - % ( %. ; % 9 (1.1.1)

dxi=dt = 'i(x1 xn u1 um t) (i = 1 n) 'i(x1 xn u1 um t) (i = 1 n) {

9 (#. ! * 99 # x1 xn , u1 um t . ; (1.1.1) 9 # , ( ! ! %. 1. ) * ( *

x(t0) = x(0)

(1.1.2)

x(t1) = x(1)

(1.1.3) . ( (1.1.1), t0 { *, t1 { ( * 9 (# . (. 2. D99 ( #

Zt1

J = '0(x u t)dt t0

(1.1.4)

'0(x u t) {

9 (# ! . 7

, * <99 ( ' , * ' * < . 3. 4

( * , *

! . 3 *

juk (t)j uk

(k = 1 m)

(1.1.5)

uk (k = 1 m) {

*. m = 2 *( ( u = (u1 u2) , ( ( <

, ' !

% (,

% . 1.1.1. 10

$. 1.1.1 ; * * , * ( (# % . ( <(# ( U

! u1 um ( % ' * < . )( U * , * ! ( , # ( , u1(t) = u1 ). 7 9 (# uk (t) = u0k (t) (k = 1 m) , * U , ( ! . ( (1.1.1) (1.1.2) (1.1.3) < 9 (# (1.1.4) ' * . $ (1.1.2),(1.1.3) .3 % : ) t0 t1 (1.1.2), (1.1.3) !

( I ) ( x(1) ( x(0) ) ( ( ) 9(

9(

t0 t1 )I ) (1.1.2), (1.1.3) ( xi 0 xi 1 (i = 1 m) ( x(0) x(1) , ! !

j0(x(0) t0) = 0I j1(x(1) t1) = 0 (j = 1 s nI i = 1 p n) ( 4 ). @ (1.1.4) ( ( : Zt1 J = q1 '0(x u t)dt + q20(x(1) t1) t0

0(x(1) t1) {

9 (#, q1 q2 { *. 5 ,

, (( , ( *

x 2 X

(1.1.6)

X { ( % x1 : : : xn . ; *

( * , ,

Zt1 t0

u2k (t) dt

Juk (k = 1 m) 11

Zt1 t0

x2i (t) dt Jxi (i = 1 n)

(1.1.7)

4 ' <! ! *!. 1.1.1. ' " - ". $ * < ( * ( " - " ( K

. 1.1.2).

$. 1.1.2 )' , # ! < ! 1.

A ddt2 = M ; M (1.1.8) A { # ( * ! ($. 0.), H m c2 I { , I M { , , % M = KI 2I I M { (, H . 2 ( % # 2

E ; E = IR E { < ( (;),

( I1 (B) ( % * E = '(I1) I E { < ( ,

( : I2 (B) E = CI 2_ , ( % C { (<99# # . < ( % # , * I = ('(I1) ; CI 2_ )=R (1.1.9) 3. # %

L1 dIdt1 + I1R1 = E1I L2 dIdt2 + I2R2 = E2 (1.1.10) 12

Ei , Ii , Ri M] Li M= ] (i = 1 2) { , (, ( #

. ; * * ! ,

, ( * * ! , ( % : )

% ( ) '

, )

! ! # ! ( % # . ( <! # * %, ( * ( ! : 1. ( , % ( % # , ( # ( ( <% # . , # * Zt1 Zt1 h'(I1) ; CI 2_ i2 I2(t)R dt = dt R t0

t0

, , * ' Zt1 h'(I1) ; CI 2_ i2 dt T (1.1.11) R t0 , {

*, ! ( . 2. 4 , ( ( ( , * * ( { E10 E20 :

jE1(t)j E10 jE2(t)j E20:

(1.1.12) 3. 0( * ( % ( % * % * * ! (9 , , , * ! 9 . D *

j_ j 1I j_ j 2

(1.1.13)

1 2 {

*. ;

! ' "" "" * (( <99 ( . D

Zt1

J = 1 dt: t0

(1.1.14)

7 % , (1.1.14) J = t ; t0 . 4* ( * " - " , * , ( (! * % (t0) ( * % (t1 = min t) . 13

(t0) = 0 _ (t0) = O0 I1(t0) = I10 I2(t0) = I20I

(1.1.15)

(t1) = 1 _ (t1) = O1 I1(t1) = I11 I2(t1) = I21I (1.1.16) ( % E1(t) , E2(t) , ! * (1.1.12), ( ! " " ! (1.1.15) (1.1.16) < 9 (# (1.1.14) ' * * (1.1.13), (1.1.13). 7 "", ( *% !

t1 ; t0 ! ! # ! ( % # , % 9 (# Zt1 J = (I2(t)R + I12(t)R1 + I22(t)R2) dt t0

< , <! # !. 7 ' " " * % 9 . ; < *

= x I _ = x I I1 = x I I2 = x I E1 = u I E2 = u (1.1.17) 1 O 2 I1 3 I2 4 I1 R1 1 I2 R2 2 , O , I1 , I2 - * ( ), * ( ;1) , ( (! (B) ( , * * * , O ). & * <! * % ' (1.1.8), (1.1.9), (1.1.10) % 9 ( Mc = 0 ): x_ 1 = x2 x_ 2 = a1'1(x3)x4 + a2x2x24I

(1.1.18)

x_ 3 = ; T1 x3 + T1 u1I x_ 4 = ; T1 x4 + T1 u2

(1.1.19)

1

1

2

2

'(I1 x3) I a = ; KCI2 2 I T = L1 I T = L2 : 2 I1 R1 a1 = KIAR I ' ( x ) = 1 3 2 1 I R AR R 2 R O 1 1

* (1.1.11) . . . (1.1.13) :

Zt1 t0

Mq1'1(x3) + q2x4x2]2 dt Jx

ju1j u1 ju2j u2 14

1

2

(1.1.20) (1.1.21)

jx2j x2I ja1'1(x3)x4 + a2x2x24j xO2

(1.1.22)

R1 I q = ; CI2 O I x2 = O1 I xO2 = O2 I q1 = ; I1 E10 2 E10 E10 I u = E20 : I u = Jx = TR 1 E10 I1 R1 2 I2 R2 * ( (0) , "") 9 (# u(0) 1 (t) u2 (t) , ( , * *% ! * % (1.1.20) ... (1.1.22) xi(t0) = xi0 (i = 1 4) , xi(t1) = xi1 (i = 1 4) , O x10 = 0 I x20 = O0 I x30 = II10 I 1 , O x40 = II20 I x21 = O1 I x11 = x31 = x41 = 0 2

1.2 %.*0 1.2.1 5 .0 6 % . D * , *

9 (# uk (t) = u0k (t) (k = 1 m) . < 9 (# (1.1.1)

' * (1.1.2) * 9 (# xi (t) (i = 1 n) , ( 4 ) ) ). $ (

) * * : ) * # * ! % (1.1.2), ) 9 # ' ! !, %! , ) * # . ., < 9 (#: (1.2.1) xi(t) = xi (t) + xi(t)I uk (t) = uk (t) + uk (t)I (i = 1 nI k = 1 m) xi(t) (i = 1 n) { ( ( ) 9(* ( I uk (t) (k = 1 m) { ( . 3 xi(t0) (i = 1 n) { * , *, *% ' #

! * ! % (1.1.2). <! ' ! * ' , * 15

n X i=1

x2i (t0) "2

(1.2.2)

" { *. 4 * (

), ( 9(* ( , ( / . 7 % , , * 9 (# (1.2.1) (1.1.1), * %

x_ i (t) + x_ i(t) = 'i Mx1(t) + x1(t) : : : xn(t) + xn(t) u1(t) + u1(t) : : : um(t) + um(t) t]

x_ i (t) = 'i(x1(t) : : : xn(t) u1(t) : : : um(t) t) (i = 1 n)

* / 4

x_ i (t) = 'i(x1 : : : xn u1 : : : um t) (i = 1 n) (1.2.3) 'i(x1 : : : xn u1 : : : um t) = 'i(x1+x1 : : :+xn+xn u1+u1 : : : um+um t); ;'i(x1 : : : xn u1 : : : um t) . P 9 (# 'i (i = 1 n) , % ( *( x1 : : : xn u1 : : : um , (1.2.3) x_ i(t) =

n X j =1

aij (t)xj +

m X k=1

bik (t)uk + oj (x1 : : : xn u1 : : : um t) (i = 1 n) (1.2.4)

@' @' i i aij (t) = @x I bik (t) = @u j k j * , * * * *( xi = xi , uk = uk (i = 1 n I k = 1 m) I oi(x1 : : : xn u1 : : : um t) (i = 1 n) { 9 (#, ( ! , % * * ( . (1.2.4) % * , * n m X X x_ i = aij (t)xj + bik (t)uk (i = 1 n) (1.2.5) j =1

k=1

16

1.2.2

% %.*0

$ ' (1.2.4) * ! ! (1.2.2) ( ( % . 7 (*

% ! ( ( <! ( * * ! Zt1 X n J= qiix2i dt (1.2.6) t0

i=1

( qii (i = 1 n) { *. @ (1.2.6) % '

(<99# qii (i = 1 n) %, *

! ( ( %

(%

% . ! ( " " " %" <! %. @ uk (t) (k = 1 m) <! %, uk (k = 1 m)

*/ . ,( , uk (t) = uk (t) + uk(t) (k = 1 m) ! ! %. < (1.1.5), * * :

;uk ; uk (t) uk(t) uk ; uk (t) (1.2.7) * juk (t)j juk(t)j (k = 1 m) . D . , * -

* ( ) , ' " " ( , *, t1 ! 1 , %* ( " ") * . ; < * * % (1.2.7), ! % " !" ( % , ( * ( * "< ")

Zt1 t0

u2k(t) dt Juk (k = 1 m)

(1.2.8)

7 * * % (1.2.8) (1.2.6) ( (* # # Zt1 "X n m X 2 2 J= qiixi + kk uk dt (1.2.9) t0 i=1

k=1

* kk (k = 1 m) * Juk (k = 1 m) . & * # (1.2.9). 5 , t1 ! 1 , < 17

* * ( %* ' (1.2.4). P ( (( 9 (# ( ), ( * (1.2.2) * ukMt x1(t0) : : : xn(t0)] (k = 1 m) , # ( ! !

t = t0 , ( (( * xi(t0) (i = 1 n) . 5 , 9 (# uk (t x1(t0) : : : xn(t0)) * ( xi(t0) (i = 1 n) (1.2.2). ; <

( (( 9 (# , (( 9 (#

!

uk(t) = rk Mx1(t) : : : xn(t) t] (k = 1 m) (1.2.10) ) , * <! 9 (#% * ! % (1.2.2). < . 7, * % u0k = rk0 Mx1 : : : xn t] (k = 1 m) , ( (1.2.9) ' * ! (1.2.4), * % * xi (t0) (i = 1 n) . (1.2.4) u0k = rk0 (k = 1 m) , ' < , ' (1.2.9), * * . * * J (t0 t1 x1(t0) : : : xn(t0)) ' ! * % (1.2.9) u0k 6= rk0 (k = 1 m) . 7 , * * (1.2.4) * 0 0 x i (t0) 6= xi (t0) (k = 1 n) , uk = rk (k = 1 m) * * (1.2.4), (( J (t0 t1 x i (t0) : : : xn (t0)) . D * ' * ( ! (1.2.4) uk 6= rk0 (k = 1 m) * ! ! x i (t0 ) (i = 1 n) . , (( 9 (#

! , ( ! ! (1.2.4),

! * ( (1.2.2), ( (* , (1.2.9), ' * . P (1.2.9) ! % t1 * , ( * * ( %* . 1.2.1. & , ( % * ( %, * ! ( : ! , % # ((

( ! ( ), ! . ) 99 # , , (* (1.2.4) 7 , (1.2.4) { < 9* ( . ( % . , xi(t) (k = 1 n) { ! , ! %, uk (t) (k = 1 m) { ! ! . (1.2.10) % # . 18

7 - 5 (1.2.4) % ( %) * , % . ( < , - (1.2.10) { ( ) * , ' % # .

1.2.1. %.*0 " - ". " - " %

u1(t) u2(t) . D 9 (# xi (t) (i = 1 4) , ( ! *

% (1.1.18), (1.1.19) D;0, ((%- *

( , $ - 5). < (1.1.18), (1.1.19) u1 = u1(t) , u2 = u2(t) . $ * * ! * % , ( , ( (! * * ! - ' % " ( " ( * % . < * * ( ). ! (

, , *

:

x_ 1 = x2I x_ 2 = a1'1(x3)x4 + a2x2x42I x_ 3 = ; T1 x3 + T1 u1I x_ 4 = ; T1 x4 + T1 u2: 1 1 2 2 ,( ((

(1.1.18), (1.1.19), x_ 1 + x_ 1 = x2 + x2I x_ 2 + x_ 2 = a1'1(x3 + x3)(x4 + x4) + a2(x2 + x2)(x4 + x4)I x_ 3 + x_ 3 = ; T1 (x3 + x3) + T1 (u1 + u1)I 1

1

x_ 4 + x_ 4 = ; T1 (x4 + x4) + T1 (u2 + u2): 2 2 *

, * x_ 1 = x2I

h i x_ 2 = a1 M'1(x3 + x3)(x4 + x4) ; '1(x3)x4] + a2 (x2 + x2)(x4 + x4)2 ; x2x42 I x_ 3 = ; T1 x3 + T1 u1I x_ 4 = ; T1 x4 + T1 u2I 1

1

2

19

2

(1.2.11)

; (* ( ( *

J=

Zt1

t0 q11 , q22 , 11 , 22

q11x21 + q22x22 + 11u21 + 22u22 dt

(1.2.12)

( {

*. uk (x1 x2 x4 t) (k = 1 2) < 9 (# ! (1.2.11) * ! ! 4 X

i=1

xi(t0) 2

(1.2.13)

' # * ! % xi(t0) (i = 1 4) . 1* ( # ! % ! ( , (

! . 1.1.2 ( . ; ! *! ( ( %

,

, ( ) . ( (%

% ' n X (1.2.14) i = nij xj (i = 1 m) j =1

i (i = 1 m) {

. 5 %, ( # < ( , ! Zt1 X m m X (0) 2 2 J= qii i + kk uk dt: (1.2.15) t0

i=1

k=1

1.2.3 % # %. - $ * %. ; < ( ! # <! % ( . 1.2.1), ( % . ( (1.1.1), (1.2.10). . ( *( , . ( - # . 4 . 1.3.1 ( %, ( * 1.2.1 (* . (.

20

$. 1.2.1 $* 9 (# # . 1. 7 % <! * (1.1.2) * ( , % * , ' , *

! !, ! (1.2.2). 2. ; * 9 (# , { 9 (# !

! ( * ). ,( , * ( #(, - # % . 3. D99 ( #

(1.1.4), ( 9 (# '0(x u t) 9* (% % . ( . ; # ( % (( ) (* (1.2.9) 9 (# * 9* (% % . ( , (<99# qii (i = 1 n) ! ! % ( ! #

( , < , ' % '( . .). ( ( % (1.2.9), * (1.1.2),

, (<99# ( * ( ) <% . 4. (1.2.10) * (1.2.5). D . , * * ' ( xi (i = 1 n) , ! * ! <! ( % (1.2.4) (1.2.5) ( ' , ( (( 9 (# oi (i = 1 n) ( , ( . . <! ( %, < < 9 (# . - % % ! ( %

# ! % (1.2.10). @ % , (( , .

21

1.3 7 . ) 1.3.1 ' %.*0 8# .0# * %

* % ' ! %! . ( . (1.1.1) * ' ! %

x_ = '(x u f t)

(1.3.1)

f (t) { - % ( ' ! %%. > , * < 9 (# : { fi(t) (i = 1 ) { fi(t) (i = 1 ) .

1.2, *

* ' ! %%. ; <

x_ i =

n X j =1

aij (t)xj +

m X k=1

bik (t)uk +

X =1

i(t)f (i = 1 n)

(1.3.2)

@ i i(t) = @f (i = 1 n = 1 ) ; . 9 # 9 (#! fi(t) * *: ) 9 # (< * , * 9 (# I , * , (* fi (t) (i = 1 ) * # . ()I ) fi(t) (i = 1 ) { *% % # * ( ! ( (I ) ((- 9 # 9 (#! fi(t) (i = 1 ) , ( , * * ( * fO(jfi(t)j fOi , i = 1 ) . ; . 9 # ' ! %! * ! : ) - I ) * ( I ) ( - M1.4]. & ! 9 (# (1.2.9) ' ! (1.3.2). ; ! (% # ' % ! ! (1.2.10) * (1.2.9), < (* <99 ( ! % * ( < 8Zt1 n ! 9 m < X = X J1 = M : qiix2i + kk u2k dt : (1.3.3) i=1 k=1 t0

22

1* (% * J1 , * *% % *% xi(t) (i = 1 n) . P * * (1.2.9) (% # *% " 9 * ( ", * * J1 . , ( J1 , , < #

). 9 # ' ! %! % ! ( . ; < ! 9 (# fi(t) (i = 1 ) * " " (# (1.2.9), uk (t) (k = 1 m) { #. D * *'% !' ' % M ( * 9 (# (1.2.9)], < ( - .

1.3.2 %0) ) %.*0 ; * * ( (1.2.10), 99 #

x_p = 'p(xp x t)

(1.3.4)

u = rp(xp x t) (1.3.5) xp(t) { np - % (

! % ( ), 'p(xp x t) , rp(xp x t) { np; m - (

. ; *

. (

. (

y1(t) : : : yr (t) ,

. ( '

y = w(x t)

(1.3.6) y(t) { r; % ( !

!I w(x t) {

% r; % ( . ; < *

x_p = 'p(xp y t)

(1.3.7)

u = rp(xp y t) (1.3.8) 7 ( ' ! (1.2.2) ... (1.2.5)), (1.2.9), (1.2.10), ! ( #. P 9 (# (1.2.9) 9 (# % '0 , . ( # % (( (* ) ! 23

# . D

, ( (( * (% *(

! <! %. @ * 9 , ' ( , , (1.3.2) (1.2.14) !

!:

x_ = A(t)(xp) + B (t)u + Q(t)f = N (t)x

(1.3.9) A(t), B (t) , Q(t) , N (t) { # , < ( ! 9 (# . D # n n , n m , n , m n

. & (1.3.6)

! . (

* * !

y = D(t)x + (t)

(1.3.10) (t) { r - % ( ! I D(t) {

# nr. % ( ) * (1.3.7) ... (1.3.8), %

x_ = Ap(t)xp + Bp(t)y

(1.3.11)

u = Dp(t)x + Fp(t)y

(1.3.12) Ap(t) , Bp(t) , Dp(t) , Fp(t) { # np np , np r , m np , m r

. 3 D;0. ; < * :

x M(k + 1)T ] = Rp(kT )xp(kT ) + Rp(kT )y(kT ) (k = 0 1 2 : : :)

(1.3.13)

u(kT ) = Dp (kT )xp(kT ) + Fp(kT )y(kT ) (k = 0 1 2 : : :)

(1.3.14)

u(t) = u(kT ) kT t (k + 1)T (k = 0 1 2 : : :)

(1.3.15) , { ( I Rp(kT ) Rp(kT ) Dp(kT ) Fp(kT ) (k = 0 1 2 : : :) { # * ! . ( ( (1.3.13) ... (1.3.15) * ( ' ( , 2, 3T . .,

(

( . ( (1.3.9), (1.3.10). ,( f (t) = (t) = 0

xM(k + 1)T ] = R(kT )x(kT ) + R(kT )u(kT )I (kT ) = N (kT )x(kT )I 24

(1.3.16)

y(kT ) = D(kT )x(kT ) (k = 0 1 2 : : :)I

(1.3.17) 0 # R(kT ) R(kT ) # A(t) B (t) 9 % 5'

Zt

x(t) = H (t t0)x(t0) + H (t )B ( )u( ) d t0

(1.3.18)

H (t t0) { . D # ( n n ) n ; ! ( ( % ( - < ' x_ = A(t)xp * ! ! x1(t0) = 1 x2(t0) = : : : = xn(t0) I % ( ' * ! ! x1(t0) = 0 x2(t0) = 1 x3(t0) = : : : = xn (t0) = 0 ..). H (t )B ( ) {< ! # . (. P * <( , ( ( ) ( ! . ( - . (1.3.18) t = (k + 1)T t0 = kT (1.3.15), *

2 (k+1)t 3 Z xM(k + 1)T ] = H M(k + 1)T kT ]x(kT ) + 64 H M(k + 1)T ]B ( )d 75 u(kT ) (1.3.19) kT

R(kT ) = H M(k + 1)T kT ]I R(kT ) =

(kZ+1)T

kT

H M(k + 1)T ]B ( )d (k = 0 1 2 : : :):

; ( * ( % (*

J=

N X x0(kT )Q(kT )x(kT ) + u0M(k ; 1)T ]uM(k ; 1)T ]

k=1

(1.3.20) (1.3.21)

Q(kT ) (k = 1 N ) {

-

# * . ; # * , ( . ( , (1.3.9) ((

x_ = Ax + B u + Qf = N x

(1.3.22)

xM(k + 1)T ] = Rx(kT ) + Ru(kT )I (k) = N x(k)

(1.3.23)

A , B , Q , N {

# * . 7( . (, (1.3.22), ( f = 0 )

25

R = eAT = E + AT + 2!1 (AT )2 + + 1! (AT ) + +I # " ;1 T 1 A 2 R = ET + 2! AT + + ! + B:

(1.3.24)

x_ = Ax + bu y = dx

(1.3.26)

(1.3.25)

& ' (1.3.23) ... (1.3.25) ( , , * # * ( %

% 9 % # H (t t0) = eA(t;t0) P

( 9 (# (r = m = 1) , . ( (1.3.9), (1.3.10) 5 . @(* ( x % * . ( 9 "!- !"

y(n) + dn;1 y(n;1) + + d1y_ + d0y = km u(m) + + k1u_ + k0u: (1.3.27) 4 % (<99# % (1.3.26) (1.3.27) 7 % , (1.3.26) -, * ! * ! ! x(s) = (Es ; A);1bu(s) . d(Esg ; A)b u(s) y(s) = d(Es ; A);1bu(s) = det( (1.3.28) Es ; A) (Esg ; A) { #: (Esg; A)(Es ; A) = E det(Es ; A) . & % , * 9 (# . ( 9 "!- !" (

(m) + k1s + k0 k (s ) = w(s) = s(n) +kmds s(+n; 1) + + d1s + d0 d(s) n;1

(1.3.29)

d(s) = det(Es ; A) k(s) = d(Esg ; A)b:

(1.3.30)

2 # ( # $#%& '$ " )* %,

% , * ( * #

* . 0 #

*

(* (

. 5 (* ( ,

! D% , - , T(, ; % ' , (

# ( * ( > . &

,

, ( 26

* . @! ( (* () * * %

, ' (% ( 9 (#% ,

* % ! ( . . 7 7 1

< (* ( #

* . ; x2.1 ' * # (. ( , * # ( < * ( ( % * (

! 99 # ! %. *

' . ; x2.2 ' * * ( . 0 * ( ( % * * ! !. ( *

' ( % *. # ( * ( .

2.1

-

$ , * % ( ! * *

! ! ! 40-50-! ! ( #

* ! . ; * (* ( #

* , ( ( , * % ! ( ( * , ( ( M2.19]), , ! %, . ., ! %' #

* * (% . 7 , * - * % ( , *

(* ( , * < > % . .) ( (* - 9 (# *( , * ( ! . D * (* ( #

* <( %. D ' % ( ,

1954 . ( @ ( ! ( B4 &&&$ 9. B. B. 1 (, ( (. -. &. . ; 1956-1960 . -. &. * ( * ( ! # ,

! % 9 M2.7]. <% " # (", ( % ! ' ( ( * .

2.1.1 9 % . ( 27

x_ = '(x u):

(2.1.1) u1(t) : : : um(t) ( t * ( ( U: ; (* ( , * ,

juk(t)j uk

(2.1.2) (k = 1 m): 4 uk (t) (k = 1 m) ( (* 9 (# * U: & ! %, ! . ( (2.1.1)

x(t0) = x(0)

(2.1.3)

x(t1) = x(1)

(2.1.4)

% ( , ( 9 (#

Zt1

J = '0(x u) dt t0

(2.1.5)

' * . ) , * x1.1 9 (# '0 'i (i = 1 n) t: (# . () , ( (( * ,

xn+1 = t (2.1.1) x_ n+1 = 1 * , * ( % t: ;

i(t) (i = 0 n) , ( ' % 99 # ! % n X j _ i = ; @' I 0 = ;1 (i = 1 n)I (2.1.6) @x j =0 i

i(t) (i = 0 n) * , (2.1.6) ! 4 ) ). )' (2.1.1), (2.1.6), (( % 9 . 7 < 9 (# H

! x1(t) : : : xn(t) 0(t) : : : n(t) u1(t) : : : um(t): n X H (x 0 u) = i'i(x u) (2.1.7) i=0

( (2.1.1), (2.1.6), (( @H (i = 0 n)I x_ i = @ i 28

(2.1.8)

_ i = ; @H @x (i = 0 n)I i

(2.1.9)

P * (2.1.2) , (* ( #

* ! <( @H = 0 (i = 0 m): (2.1.10) @uk

2.1.2

-

* * (2.1.2) . P # 9 (# uk (t) (k = 0 m) # (2.1.2) (* * juk (t)j < uk (k = 0 m) ! ' (2.1.10). ( * * * uk ;uk (k = 0 m) , (*( ! % # . ,( (* - 9 (# . # U ' (2.1.10), '. < * # ( -. &. ,

(

9

%

. ! ( <% , ( . ; u(t) * ! ! x(0) x0(t0) = 0 % ' (2.1.1): x1(t1) : : : xn(t): < ' u(t) (2.1.6), , ( ( ! ! * ! ! (t0 ) ' (2.1.6): 1(t) : : : n(t): 9(

! (

!) * ! ( x 9 (# H 9 (# % ( u 2 U: 0( <% 9 (# u * * M (x 0) :

M (x 0) = max H (x 0 u): (2.1.11) u2U 0( ( ' * ) % 9 (# H (x 0 u) (( *(! ( ( <% 9 (#, ( ! m @ 2H @H = 0 (k = 1 m) X uk ul < 0 (2.1.12) @uk l k=1 @uk @ul ( #! uk ;uk (k = 1 m) U:

29

: ( # ( -. &. M?]). u(t) t0

t t1

{ ( , * ' xi(t) (i = 0 n) (2.1.8), ! t0 (2.1.3), ! t1 * *( x(1) x0(t1): 7 ( ( x0(t1) ' * ) ! (! !

! 9 (#% 0(t) 1(t) : : : n(t) ! (2.1.9), * t (t0 t t1) 9 (# H (x(t) (t) 0(t) u)

u 2 U u = u(t) (

H (x(t) (t) (t) u(t)) = M (x(t) (t) (t)) < ( * % t1 '

(2.1.13)

0(t1) < 0I M (x(t1) (t1) (t1)) = 0 (2.1.14) P x(t) (t) u(t) (2.1.8), (2.1.9) (2.1.13), 9 (# 0(t) M (x(t) (t) (t)

t

< ( ' % (2.1.14) t1 % t (t0 t t1): 7( * , < 7( 1 ' ' (2.1.13) * ( # ( x(1) ) 9(

t1: & ' (2.1.13) (2.1.14) % 9 : max H (x 0 u) = 0 (2.1.15) u2U ,( , # ( (2.1.15). * , * u1(t) : : : um(t) { , x1(t) : : : xn(t) { ( ,

% (

0 < 0 ( ' 1(t) : : : n(t) (2.1.9), * 9 (# H (x1(t) : : : xn(t) u1 : : : um 0 1(t) : : : n(t)

! u1 : : : um ! t 2 Mt0 t1] ( U

! ! u1(t) : : : um(t): < < , ! *! , . , *

! *(! U (2.1.10), ( ! (2.1.15).

2.1.3 : - 5( (* ( (2.1.15), 9 (# x1(t) : : : xn(t) 1(t) : : : n(t)

0 ! < , ? ) : 9 (# H (x u 0) (( 9 (# m 30

! u1 : : : um 2 U *

x 0) , ' * (# 9 (# H ! 9 (#

u = u(x 0) 2 U

(2.1.16)

( % ' * 9 (# H: ; * 9 (# (2.1.16) . 4 , * (2.1.1) ( n X (1) 'i(x u) = 'i (x) + '(2) ik (x)uk (i = 1 n) k=1

9 (# (2.1.5) n X '0(x u) = '0x + '0k (x)uk i=1

U (2.1.2), ! n m X n X X (1) (2) H (x 0 u) = i(t)'i + i(t)'ik (x) uk i=0

k=0 i=0

(2.1.17)

< 9 (# M2.8] ' * U *( ( 8 n X > > u i(t)'(2) k ik (x) > 0I > > i=0 < uk = > n > k X > ; u i(t)'(2) > ik (x) < 0 : i=0 X ! n (2) uk (t) = uk sign i(t)'ik x(t) (k = 1 m): (2.1.18) i=0

1 (2.1.18) '% . 9 # ( : k - (k = 1 m) ( *% ((* -

%) 9 (# % * uk ;uk , < (* n X i(t)'(2) (2.1.19) ik (x(t)) = 0: i=1

@(, , * 9 (# (2.1.16) . $ 2n 99 # ! %

x_ i = 'i(x u (x 0)) (i = 1 n)I 31

(2.1.20)

n X _ i = ; @'i(x u@x(x 0)) j (i = 1 n): j =0

i

(2.1.21)

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!, ( ( ! % (2.1.3), (2.1.4). )* % (2.1.20), (2.1.21) ( ! ! (2.1.3), (2.1.4) ) ) (!* * % ( % * %). ,( , # ( ' * ( ' ( % *. , ' , * % (2.1.20), (2.1.21) " " , ( (( * i(t0) (i = 1 n): ! ! ( ' ( % * (* . ) ( (t0) = (0) (2.1.20), (2.1.21) ! * ! ! x(t0 ) (0) % 9 (# x(t) (t) t = t1 (2.1.4). P ' , ( (t0) = (1) , (2.1.20), (2.1.21) * ! ! x(t0) (1) * t = t1 ( x(t1): P

, # ! , ( % (% ( (t0) * (2.1.4) % * . < ! , ( (t0) " " x(t1)

( x(1): ; * % (

*

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(??) ( (2.1.15) * . D * #9* (!

%, ! ! *

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! # ( *!, ( % % % (2.1.16), ! ! ! ,

%* (2.1.20), (2.1.21). $ ' ( ! * # ( , , M2.12], M2.13]. (* , *, * *

' ( % * # (, # ' (% # < # % * (% * %, ' % (! % * % (, ( ( % . ( , * #9* (!

%, ! *' ! *

' ( % *. 32

2.2 % ) * ' %

! !-< ' # ' , < % ( ( * . ( # ( . ; 1935 . &&&$ * M2.27] ( ( , ( % ( * , * ( % . B * % # ( ( * ( # , ( % 9 &XB. ) * ( . ; 1953 . M2.28] # n - % ( n - 9 ) ( n - ! (* % # . (# # ( M2-29] ( ( (

% 9 ) #, % ( ' % * % . ; ' * ,

! # (, ( ! * % , ( (! .

2.2.1

- # % ) *

)* % ! % u 2 U ( ! . (

x_ = '(x u)

(2.2.1)

x(t0) = x(0)

(2.2.2)

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(2.2.3)

(x(0) x(1) t0 { , t1 { ), < 9 (#

Zt1

J = dt = t1 ; t0 t0

(2.2.4)

' * . 2.1.2 # (, ! %. @ (2.2.4) , * '0(x u) = 1 < 33

H (x 0 u) = ' + ; 9 (#

H (x u) =

n X i=1

n X i=1

i'0(x u):

i'0(x u)

' (2.2.1)

1 x_ i = @H @i (i = 1 n)I 1 _ i = ; @H @xi (i = 1 n) 9(

! x 41 9 (# % u . * * , *

(2.2.5) (2.2.6) (2.2.7)

M1(x ) = max (x u): u2U

M (x ) = M1(x ) ; 0: ,( , ! (2.1.15) % max H (x u) = 0 (0 0): u2U 1

(2.2.8) . " - ". $ * % " - ". * ( # (! , ' (1.1.18), (1.1.19) T1 = T2 = 0

x_ 1 = x2 x_ 2 = a1'1(u1)u2 + a2x2u22: (2.2.9) , ( * # ( 9 (#% u1(t) u2(t) ( ! ju1(t)j u1 ju2(t)j u2 ( ! *

x10 x20

x11 x21 ' . 1 (# H1 * H1 = 1(t)x2 + 2(t)(a1'1(u1)u2 + a2'2u22) (2.2.10) %

1(t) 2(t) 1 (2.2.11) _ i = ; @H = 0I _ 2 = ; @H1 = ;1 + a22u22: @x1 @x2 34

(2.2.9) ... (2.2.11) ( * # ( % .

2.2.2 2) %< $ % (( * % *% * %, ( (2.2.1) . ( %

x_ = Ax + B u: ; < * 9 (#

H1 =

n X i=1

(2.2.12)

2n 3 m X X i(t) 4 aij xj + bik uk 5 : j =1

k=1

&

(2.2.7) (: 2n 3 m X X @ 4 aix + bjk uk 5 n n X X =1 k=1 _ i(t) = ; = ; aij j (i = 1 n): i @x j =1

i

j =1

(2.2.13)

(2.2.14)

7 % ! . ( # ( ( ! , * %. ; (2.2.8) ! * , * 9 (# (2.2.13) ' * *

u: D 9 (# (, # # m "X n m "X n X X bik i(t) uk (t) = ju max b (t) uk : (2.2.15) (t)ju k=1 i=1 ik i k=1 i=1 m = 1 * ju(t)j u < k

n X

k

1(t)u(t) = ju max (t)u (t)ju 1 k

(2.2.16)

1(t) = bi1i(t): i=1 4 ! <( (% (% 9 (#,

% ( % , %. P 9 (# ( % , <( (( , (

# <% . ; * 9 (# H11 = 1u { % u u < u * , *( u ( % 9 (# H11 <( . P 9 (# H1 ( M;u u]

% u < ( #! ( . 3.1.1). 35

& ' , (( u * 9 (# H1 (? 5( . 2.2.1, u

$. 2.2.1 1 u = u sign @H @u = u sign 1: D ( , < n X u(t) = u sign bi1i(t): (2.2.17) i=1

; ( * (m > 1) * , * ( u1(t) : : : um(t) ( u ! !, < (2.2.15) , "X # n (2.2.18) uk (t) = uk sign bik i(t) (k = 1 m): i=1

,( , % ! . ( # ( % (2.2.18) , ( * ( (t0) ( '

x_ = Ax + B uI uk = uk sign B0k] (k = 1 m) (B0k] { k - % # # B )I

_ = ;A0

(2.2.19) (2.2.20) (2.2.21)

( (2.2.2), (2.2.3). ) , * ( ! ( * ( . ( (2.2.12)

% (2.2.14) , ( (. 7 % , ! ( * (% . ( det(Es ; A)

% det(Es + A0) , , . ( * ( %*,

%*. D ( *

' 36

( % *. ; < # (

,

, M2.30]) ' ( ! * (2.2.12) ... (2.2.15). , ' ( % * (2.2.12) ... (2.2.15), ( ( % * # (, ( ( M2.6],

4.4.5 (. D * % ! ! ( ( % , * % 9 (# . . . (

y +d2yY + d1y_ + d0y = bu: (2.2.22) , 9 (# u(t) ju(t)j 1 ( < . ( ...

y(0) = y10I y_ (0) = y20I yY(0) = y30

(2.2.23)

y(t1) = y_ (t1) = yY(t1) = 0

(2.2.24)

. ; * x1 = y x2 = y_ x3 = yY b = b31 , ' . ( 9

x_ 1 = x2I x_ 2 = x3I x_ 3 = ;d2x3 ; d1x2 ; d0x1 + b31u:

(2.2.25)

H1 = 1x2 + 2x3 + 3(;d2x3 ; d1x2 ; d0x1 + b31u)

(2.2.14)

(2.2.26)

1 (#

_ 1 = d03I _ 2 = ;1 + d13I _ 3 = ;2 + d23: @ (2.2.26) (* , * (

(2.2.27)

u = 1 sign b313(t): (2.2.28) $ ' ! % 3 , * 99 #

; 3 +d2Y3 ; d1_ 3 + d03 = 0 ...

9 (# 3(t) .

37

(2.2.29)

2.2.3 : % n - # @ (2.2.18) , * ( ( % (* -

9 (#, *( ( % *( 9 (# n X k (t) = bik i(t) (k = 1 m): i=1

4 . 2.2.2 9(, % <! 9 (#%.

$. 2.2.2 5 *( *(% (* . 3 (* ( % uk (t) (k = 1 m) * % 9 (# k (t) (k = 1 m) * '. & , (, % *%, ( * (* <! % ( * # (. D *% n - !.

: . (% n- #) P ( ! ( * ( .-

( (2.2.12) % , * (* ( % u1(t) : : : um(t) ' n ; 1: ( * * n = 3 m = 1: 5 , , * . ( % (2.2.25), < ( s3 + d2s2 + d1s + d0 = 0 . ( (2.2.25) * . ( ( *,

%. * * ;1 ;2 ;3 { ( ! ( * ( . (. , * , * ( ! ( * ( (2.2.29)

1 2 3 , , 9 (# 3(t) ' < ,

3(t) = k1e1t + k2e2t + k3e3t (2.2.30) k1 k2 k3 {

. ( ( * ( % ( %) 9 (# 3(t) * (* , ( , . 38

5 6 P 1 3 3 { * % *,

9 (# (2.2.30) ! % ! ( %.

1 . . n = 1 ( e t = 0 1

% ! ( %). , * ( *, ( (2.2.30) ' !, ( ! !. 7 , * 9 (# (2.2.30) ! % ! ( %. e;3t * 9 (#

k1e(1;3 )t + k2e(2;3 )t + k3 (2.2.31) ( ( ! % ! ( %. @ * ( ( $) , * % ( 9 (# ( %( % ( %. & , 9 (# (2.2.31) ! % ! ( %. & % , < k1(1 ; 3)e(1;3)t + ke(2;3)t(2 ; 3) (2.2.32) ( % * 1 ; 3 2 ; 3 * , , ,

% ( ( ' , * ( *, ( (2.2.30) ! !). *

* ( .

2.3 ; ! ! * ! ( ( * <( ( , %( .. (* , , %, # ). ( <99 ( *

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( ( ( $. > M2.20], , ( % 9 (# . $ ' < ( .

2.3.1

-

$ * . . ( , % 39

x_ = '(x u t):

(2.3.1)

, % (

u = r(x t)

(2.3.2) * ! (2.3.1), (2.3.2),

! * ( , 9 (#

Zt1

J = '0(x u t) dt: t0

(2.3.3)

< (2.3.2) * u 2 U: 7

* , *

;uk uk (t) uk

(2.3.4)

x_ 1 = '1(x1 x2 u t)I x_ 2 = '2(x1 x2 u t)I

(2.3.5)

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*. , * < * #

% * % ( # 9(

t1: 7 # * * * <% *, ( n = 2 m = 1: ; < * (2.3.1) (2.3.2) :

u = r(x1 x2 t) 9 (# (2.3.3) ' , t '0 (( Zt1 J = '0(x1 x2 u) dt: t0

(2.3.6) (2.3.7)

! ( # , , * (2.3.2)

% . D ( x1(t) x2(t) ( * , (2.3.1) 9 (# (2.3.2) (2.3.1) ( * x1(t0) x2(t0): D (

. 2.3.1.

40

$. 2.3.1. 0 ((- *( x % ( *( *(% x(0) = fx1(t0) x2(t0)g *(% x0 = fx1(t0) x2(t0)g ( ( 1), *( *( x0 = fx1(t0) x2(t0)g x(1) = fx1(t1) x2(t2)g *( ( ( ( 2).

: , (( (2.3.5) -

t0 *( fx1(t0) x2(t0)g ( 2. 7 , % *( % ( % ( %. D * , * , * *( x(0) ( t0 *( x0 <% *( ( % 2. # * * . 7 % , *( x0 ( 2, ( 20 < 9 (#

Zt1

J = '0(x1 x2 u) dt: t0

' * , * ( 2. , * 9 (# (2.3.7) ( 1 ; 20 ', * ( 1 ; 2 . D * u:

2.3.2 ? - 4 * * %, < * (% ! ( # , ( * 9 (# . ! ( , * * % 9 (# ! ( !:

Mx1(t0) x2(t0) t] = ju(min t)ju Mx1(t0) x2(t0) t0] = ju(min t)ju

Zt1 t0 Zt1 t

41

0

'0Mx1(t) x2(t) u(t)] dtI '0Mx1(t) x2(t) u(t)] dtI

( t0 = t0 + I { * *) 9 (# (2.3.7) 9 tZ0 + t0

'0Mx1(t) x2(t) u(t)] dt +

Zt1 t0 +

'0Mx1(t) x2(t) u(t)] dt:

7, * *( . ) * , ( 9 (# # < *(, Mx1(t0) x2(t0)]: 4 # 9 (#

8 tZ0+ 9 < = Mx1(t0) x2(t0) t0] = ju(min ' M x ( t ) x ( t ) u ( t )] dt + M x ( t + ) x ( t + ) t + ] 0 1 2 1 0 2 0 0 : t0 )ju : t0 * *

Mx1(t0) x2(t0) t0] = ju(min f'0Mx1(t0) x2(t0) u(t0)] + Mx1(t0 + ) x2(t0 + ) t0 + ]g : t0 )ju (2.3.8) 0 9 ! ((! u(t0) * *( . ( < 9 (# . ; < (2.3.8). @ , % , * i xi(t0+ ) = xi(t0)+ @x @t

t=t0

+o1i( ) = xi(t0)+'iMx1(t0) x2(t0) u(t0) t0] +o1i( ) (i = 1 2): o1i (') ;! 0I lim !0

Mx1(t0 + ) x2(t0 + ) t0 + ] = fx1(t0) + '1Mx1(t0) x2(t0) u(t0) t0] + o1i( ) x2(t0)+ +'2Mx1(t0) x2(t0) u(t0) t0] + o12( ) t0 + g = Mx1(t0) x2(t0) t0]+

@ + @x

1

@ + @x

2

t=t0 x1 =x1 (t0 ) x2 =x2 (t0 )

t=t0 x1 =x1 (t0 ) x2 =x2 (t0 )

'1Mx1(t0) x2(t0) u(t0) t0] + '2Mx1(t0) x2(t0) u(t0) t0] + @ @t

o3 ( ) ;! 0: lim !0 42

t=t0 x1 =x1 (t0 ) x2 =x2 (t0 )

+ o3( )

< (2.3.8), *

Mx1(t0) x2(t0) t0] = ju(min f' Mx (t ) x2(t0) u(t0)] + Mx1(t0) x2(t0) t0]+ t )ju 0 1 0 0

2 @ X + @x i=1 i

xi =xi (t0 ) t=t0

'iMx1(t0) x2(t0) u(t0) t0] + @ @t

9 = + o ( ) 3 : )

t=t0 x1 =x1 (t0 x2 =x2 (t0 )

&( Mx1(t0) x2(t0) t0] ! *! * ! 1 8 < =0 ; @ = min @t = ( ) ju(t )ju : '0Mx1(t0) x2(t0) u(t0)]+ t t x1 x1 t0 x2 =x2 (t0 )

0

9 2 @ = X + @x == 0( ) 'iMx1(t0) x2(t0) u(t0) t0] : i=1 i 21 = 21 ( 00 ) * , * *

% ! x1(t0) x2(t0) t0 ( "0" ' x x

t t x t x t

; @Mx1(t)@t x2(t) t] = (

) 2 @ Mx (t) x (t) t] X 1 2 = ju(min '0Mx1(t) x2(t) u] + 'iMx1(t) x2(t) u t] : t0 )ju @xi i=1 ; * , ( n > 2 m > 1 <

(2.3.9)

f' Mx : : : xn u1 : : : um]+ ; @Mx1 :@t: : xn t] = u u min ::: u 2U 0 1 1

2

n

(2.3.10) n @ Mx : : : x t] X 1 n + 'iMx1 : : : xn u1 : : : um]g : @x i i=1 P , * ! U * , (2.3.10) (( ( % * ! !:

X @ = ' 'iMx1 : : : xn u1 : : : um t]I ; @ 0Mx1 : : : xn u1 : : : um ] + @t i=1 @xi n

(2.3.11)

n @ @' Mx : : : x u : : : u t] @'0Mx1 : : : xn u1 : : : um] + X i 1 n 1 m = 0 (k = 1 m): @ u_ k @uk i=1 @xi (2.3.12)

43

,( , ' * % # ! ' , ( ! !

Mx1(t1) : : : xn(t1) t1] = 0 (2.3.13) #9* ( * ! ! (2.3.10) m +1 % * ! ! (2.3.11), (2.3.12). ; ' <! % * ( uk = uk (x1 : : : xn t) (k = 1 m) 9 (# (x1 : : : xn t) ( xi = xi0 t = t0 ' * 9 (# # Zt1 (x10 x20 : : : xn0 t0) = '0(x1 : : : xn u1 : : : um) dt (2.3.14) t0

( (2.3.13). 7 % , . ,, ! ( % %, (2.3.10)

@(x1 : : : xn t) ' (x : : : x u : : : u t) ; @(x1 :@t: : xn t) = '0(x1 : : : xn u1 : : : um)+X i 1 n 1 m @xi i=1 n

n @ (x : : : x t) @(x1 : : : xn t) +X 1 n 'i(x1 : : : xn u1 : : : um t) = ;'0(x1 : : : xn u1 : : : um): @t @xi i=1 (2.3.15) * , * < (( % 9

d(x1 : : : xn t) = ;' (x : : : x u : : : u ): 0 1 n 1 m dt @ ! t0 t1 (* , * Zt1 Mx1(t1) : : : xn(t1) t1] = ; '0(x1 : : : xn u1 : : : um) dt: t0

(2.3.16) (2.3.17)

* ( (2.3.13), * (2.3.14). t ! 1 ( * (% %*. P 9 (# '0 > 0 Mx1 : : : xn] ! x1 : : : xn , (2.3.1), (2.3.2) * ( %*. 7 % , (2.3.15) B. 0. - < * (% %* % * -

% 9 (# (x1 : : : xn) ( % 99 # ! % (2.3.1) # -

. 44

,( , '0(x1 : : : xn u1 : : : um) > 0 t1 ! 1 , 9 (# (x1 : : : xn u1 : : : um t) ! * ( ( 9 (# % - , < < 7 - 8 . ) ( , * * ( %*% % ( (2.3.13) * (. (* , * 9 (# # (2.3.3) %

Zt1

J = '0(x1 : : : xn u1 : : : um t) dt + 0Mx1(t1) : : : xn(t1) t1] t0

(2.3.18)

( (2.3.13) ((

Mx1(t1) : : : xn(t1) t1] = 0Mx1(t1) : : : xn(t1) t1]:

(2.3.19)

2.3.3 8 . 7, * % % 9 , ( 9 ! ((!, ! (2.3.10), :

u = u(x t vx)

(2.3.20)

@ : : : @ : vx { ( ( @x @xn 1 < (2.3.10), * % * ! ! ( n @ X ; @ = ' ( x u ( x t v )) + 'i(x u(x t vx) t): (2.3.21) x @t 0 i=1 @xi 3

' < ( ! ! (2.3.19) % *, * ' ( % * # (, ( (( * ' ( % * (

! 99 # ! %, ( % * % * ! !. D * % *

'

, ( (( * ( ' * %, (( # ( (( 9 (# . 5, < 9 (# * * ( ( * , uk = rk (x1(t) : : : xn(t) t) (k = 1 m) xi(t) (i = 1 n) ( . 7 ' (2.3.21) M2.10], M2.11] ' % * ! ! ( , ! ( (, ! ..), ( # %

*

' < . D 99 # ! % 45

(2.3.1) % 99 # - ! %, (2.3.3) - % ! ( (% 9 (# ( ! . &

, * ( ( 9 (# *

( ! . * ( ( ! 7 2. $ . ( ,

(

x_ 1 = x2 x_ 2 = ;x1 + u: , % u = r(x1 x2 t) ( , * 9 (# Zt1 J = (q11x21 + u2) dt + 1x21(t1) + 2x22(t1) t0

(2.3.22) (2.3.23)

( q11 1 , 2 {

*) ' * ! . (,

! * ( . 4 ( *

ju(t)j 1:

(2.3.24) ! ( <% *, ' 9 (# * ( : ( ) @ ( x @ ( x @ ( x 1 x2 t) 1 x2 t) 1 x2 t) 2 2 ; @t = jmin uj1 @x1 x2 + @x2 (;x1 + u) + q11x1 + u (2.3.25) (

Mx1(t1) x2(t1) t1] = 2x21(t1) + 2x22(t1): ; 9 ! ((! . , ( 8 > 1 @(x1 x2 t) 1 @(x1 x2 t) < 1I > ; > @x2 2 @x2 > < 2 1 @ ( x 1 x2 t) u = > 1 ; 2 > 1I @x 2 > > 1 x2 t) > ; 21 @(x@x < 1: : ;1 2 D ' 1 x2 t) 1 x2 t) ; @(x1@t x2 t) = @(x@x x2 + @(x@x (;x1 + u) + q11x21 + u2 1 2

(2.3.26)

(2.3.27)

( (2.3.26) ( * * ( . 46

2.3.4 '. - 7 M2.21] * ( # ( ' 9 (# (2.3.10) ( ( % 9 . ; <

xn+1 = t: * , * dxn+1 = ' (x : : : x u : : : u ) = 1: (2.3.28) n+1 1 n+1 1 m dt f(t)g = max f;(t)g & * < ' , ( * min t t ' (2.3.10) ( nX ) +1 @ max ; i=1 @xi 'i ; '0 = 0: (2.3.29) u2U # ( * 9 (# (2.3.3) ! (2.3.1) ( 0 = ;1) (nX ) +1 max ' ; '0 = 0 (2.3.30) u2U i=1 i i ( n+1(t) ' nX +1 _ n+1 = ; @x@'i i:

(2.3.31)

: : : xn+1 (t)] (i = 1 n + 1): i(t) = ; @Mx1(t)@x (t)

(2.3.32)

n+1

i=1

& (2.3.29) (2.3.30), ! * , ( , * ! ( % i

! ( ( < , *

! +1 ! +1 @ @ ! nX +1 @ 2 @ @ x_ = nX d @ = nX ' 'i (i = 1 n + 1): (2.3.33) i i= dt @xi i=1 @xi @xj i=1 @xi @xj i=1 @xi@xj & % , 99 # (2.3.29) xj (i = 1 n + 1) * ! ( %

@ ' ; X @ @'i ; @'0 = 0: ;X i @xj i=1 @xi @xj i=1 @xj @xi n+1

2

n+1

@ (2.3.33), ' < ! +1 @' @ ! @' d @ = ; nX i 0 dt @x @x @x ; @x : j

i=1

47

j

i

j

(2.3.34)

(2.3.35)

& <

(2.1.6), (* , * @ (t) (i = 1 n + 1) ( 99 9 (# ; @x i j

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(2.3.29) * ; @x i D *

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! < . ; 1962 . ;.1.5 M2.22] * , ( ! (( * % *% 9 (# * ( . D ' ' ( ' (* (! * M2.23], M2.24]. & <! % * 9 (# * ( . > ( , ( < , ( ! ( M2.25], M2.26]. & * ( * ' ( (* ( ( ( (, !* ( # , ( . (.

48

3

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% - ( * (% # %, % % (# % ' ' 1960 . ( ( ( $. 5 M3.27], ( % ' * # % !, # ! . (. ; 9 <% # * ( ( ( ) . 4 * ( ( , * # ! ( * 9 (# * ( ( ' (

! 99 # ! ! % ( % $(() * . ; x3.2 * , (

. ; , ( % % * ( , !

( % (

. (, ( ! . x3.3 # B5$ % ! * ! .

3.1 - 3.1.1 $ . ( ,

(

x_ = Ax + B uI x(t0) = x(0) t0 = 0

(3.1.1) A ; {

, # * n n n m

. , % # * &0 ( m n )

u = C 0 x 49

(3.1.2)

(, * * ( %* ! ! (3.1.1), (3.1.2),

! * ( x(0) , 9 (# Z1 J = (x0Qx + u0u) dt (3.1.3) 0

Q {

-

# n n (x0Qx > 0 ! x < * Q > 0): 0 # &0 ( (3.1.2) ) 9 . ! ( ' <% * % # * ( , * * * n = m = 1: ; < * 9 (#

x_ = ax + buI

(3.1.4)

u = cx

(3.1.5)

Z1 J = (qx2 + u2) dt:

(3.1.6)

0

, (2.3.11), (2.3.12) * ( ' (( 2 + u2 + @ (ax + bu)I = qx (3.1.7) ; @ @t @x @ b = 0 u = ; 1 @ b: 2u + @x (3.1.8) 2 @x ! <( % * (3.1.7). 4 , * < . 7 % , # " @ 2 2 2 d qx + u + @x (ax + bu) = 2 > 0: du2 D -

% <

(3.1.8). , (( ( , (3.1.7)

9 (# : D 9 (# %* (3.1.1), (3.1.2). @(* u (3.1.7) (3.1.8), * % * ! !: 50

!2 @ @ 1 2 ; @t = @x ax ; 4 @ @x b + qx : $ ' < (

(3.1.9)

Mx(t1)] = 0 (t1 ! 1)

(

= px2I p = const: < (3.1.9), *

(3.1.10)

0 = 2pax2 ; (pb)2x2 + qx2: (3.1.11) * ( (<99# p (3.1.10): 2pa ; p2b2 + q = 0

@ ! ' %

(3.1.12)

s 2 s 2 a a q a (2) = b2 + b4 + b2 I p = b2 ; ab4 + bq2 < ! 9 (# * * ( %* % , , ( Mx(1)] = 0: 4 (3.1.8) * p(1)

, ( , ( *

u = (;p(1)b)x

(3.1.13)

c = ;p(1)b: (3.1.14) ; * (n > 1 m 1) (3.1.12), (3.1.14) * ( ( PA + A0P ; PBB 0P + Q = 0I

(3.1.15)

C = ;PB (3.1.16) $ { * # * n n: ; <! % 7( 2. (3.1.15) ( ( ( ( ( ). 51

- ( %:() -

! #%: 1) ' $((, 2) <! ' % # P 0 > 0 (*

% ! P 0 ), 3) * (% # (<99# 9

C = ;P 0B:

(3.1.17)

, * # C ' (3.1.17), * * ( %* (3.1.1), (3.1.2). 7 %* x_ = (A + BC 0)x - . (* 9 (# - = x0P 0x > 0 * <% 9 (#:

d = x_ 0P 0x + x0P 0x_ = x0(A + BC 0)P 0 + x0P 0(A + BC 0)x = dt = x0MP 0A + A0P 0 + P 0BC 0 + CB 0P 0]x: * , * # & (3.1.17), *, * , * P 0 (3.1.15), d = x0MP 0A + A0P 0 ; P 0BB 0P 0 ; P 0BB 0P 0]x = dt = ;x0Qx ; x0P 0BB 0P 0x = ;x0Qx ; u0u < 0: P ( (3.1.1) Q > 0 ' % (3.1.15) %

-

# P 0: 4 , * ) 5 (3.1.1)

MB AB : : : A(n;1)B ] = n (3.1.18) ( (A B ): P # Q {

# -

# (Q 0)

Q = H 0H H { # n ( { # Q) . & ' % (3.1.15) M3.6]

# P 0 Q 9 (# (3.1.3)

# -

#, % (A0 H 0) :

MH 0 A0H 0 : : : A(n;1)0 H 0] = n: 52

, % (A B ) (A0 H 0)

P 0 > 0 , <! M3.6]. . . % < ( % (3.1.15), (3.1.16)) * ( ( . ' * 9* ( * # M3.3], ( ( ' <% * ,

<% ! !. $ !

% ( ( ( . 3 .1.1). P M?]:

$. 3.1.1 (J + J)Y + (J + J ; J)_ 2 sin cos + n _ + H _ cos = MxI

(3.1.19)

M(J + J) cos2 + J sin2 + J ]Y + 2(J ; J ; J)_ _ sin cos + n_ ; H _ cos = ;My I (3.1.20) { % OY I -

( # ( OX ( # )I J { # % (( #) OY I J { <( % # (I J J J { #

( # ( % OZ OX OY

, < J = J = JI H { ( * (% (I Mx My { % OX OY

I n n { (<99# 9 . = ( ( ( OY *( ) . ( ( , ( ) OY: ( - ! <% ( , . .) ( * " # " OX . . OZ

* * OY ( % ( . T #

(3.1.20), * % *, ( (( (( H J J J n): # , OX *( . < 53

, ( % % , % % ( . , # ( ( ! 9 (#. = ( ( % # # ). P ! . 3.1.2, 7 { *( # , 70 { *( ( ).

$. 3.1.2 )' (3.1.19), (3.1.20) 9 5'. * J J J J Mx = 0 * _ x1 = I x2 = 0 I x3 = _0 I ( = 1 I 0 = 1 = (I 0 = 1 = ()I

; J J+ J+J; J = R2I ; J +H J = a23I ; J +nJ = a22I ; 2(J ; JJ ; J) = R3I H = a I ; n = a I J 32 J 33 ' (3.1.19), (3.1.20)

; MJ y = b31u + m31f

x_ 1 = x2I x_ 2 = a22x2 + a23x3 cos x1 + R2x23 sin x1 cos x1I

(3.1.21)

x_ 3 = a32x2 cos x1 + a33x3 + R3x3x2 sin x1 + b31u + m31f: (3.1.22) $ * <! % , % ( *( x1 = x2 = x3 = 0 * x_ 1 = x2I x_ 2 = a22x2 + a23x31 x_ 3 = a32x2 + a33x3 + b31u + 31f (3.1.23) u # , *( , f # OY: ( f = 0 ( u = c1x1 + c2x2 + c3x3 54

(3.1.24)

( ! (

! * ( ) 9 (# Z1 J = q11x21 + q22x22 + q33x23 + u2 dt (qii > 0I i = 1 2 3): (3.1.25) 0

! ( ' <% *, ' (3.1.15), (3.1.16) # B5$. <! %

p11 p12 p13 0 1 p12 p22 p23 0 a22 p p p 0 a 13 23 33 32 p13 b31 ; p23 b31 ; p13b31 p33 b31

0 0 0 p11 + 1 a p p 0 a22 p23 p12 23 33 13 q11 0 p23b31 p33b31 + 0 q22 0 0 0 p23 p33

p12 p13 p22 p23 ; p23 p33 (3.1.26) 0 0 0 0 0 = 0 0 0 : q33 0 0 0

D * %

;(p12b31)2 + q11 = 0I p11 + p12a22 + p13a32 ; (p13b31)(p23b31) = 0I p12a23 + p13a33 ; (p13b31)(p33b31) = 0I 2p12 + 2p22a22 + 2p23 a23 ; (p23b31)2 + p22 = 0I p22a23 + p23a33 + p13 + a22p23 + a23p33 ; (p23b31)(p33b31) = 0I 2p23 a23 + 2p23a33 ; (p33b31)2 + q33 = 0:

(3.1.27)

(@- * # A * <! % n2 = 9 n(n2+ 1) = 6): 4 % (3.1.16) *

c1 = ;p13b31 c2 = ;p23b31 c3 = ;p33b31: (3.1.28) ,( , * ( ( ( # ) ( ' * (! % (3.1.26) ! ( ! (3.1.24) 9 (3.1.28).

3.1.2 @ - %< $ % # % . (, %

x_ = A(t)x + B (t)u x(t0) = x(0)

( A(t) B (t) Mt0 t1] # 9 (#%. ( % (* 55

(3.1.29)

Zt1 J = (x0Q(t)x + u0u) dt + x0(t1)P (1)x(t1) t0

(3.1.30)

Q(t) P (1) {

-

# 9 (#% *

. , % # C 0(t)

u = C 0(t)x

(3.1.31) ( % ! (3.1.29), (3.1.31),

!

* ( , 9 (# (3.1.30). ! ( ' <% *, * *% n = m = 1: , 9 (# # :

x_ = a(t)x + b(t)uI

(3.1.32)

u = c(t)xI

(3.1.33)

Zt1 J = (q(t)x2 + u2) dt + p(1)x2(t1):

(3.1.34)

t0

1 (# , ' * B5$ # . ( (3.1.29), ( = p(t)x2 . (3.1.9), * * ( (3.1.11) 99 # (

;p_(t) = 2p(t)a(t) ; p2(t)b2(t) + q(t) = 0

(3.1.35)

(3.1.36) p(t1) = p(1): (3.1.35) # 99 # ,

' ( * XVIII . ( ( T. $((, ( . ; * (n > 1 m 1) (3.1.35) ( (3.1.38) :

;P_ (t) = P (t)A(t) + A0(t)P (t) ; P (t)B(t)B 0(t) + Q(t): P (t1) = P (1):

(3.1.37)

(3.1.38) (3.1.37) (. P * ,

7( 2. 56

! ( ' (3.1.37), " " = t1 ; t * $(t) = $(t1 ; ) = PO ( ) . , (3.1.37) (3.1.38)

dPO ( ) = PO ( )A(t ; )+ A0(t ; )PO ( ) ; PO ( )B (t ; )B 0(t ; )PO ( )+ Q(t ; )I (3.1.39) 1 1 1 1 PO (0) = P (1): (3.1.40) ,( , ( * (3.1.37) ( ) ( * ' (3.1.39) * (3.1.40). 7 *

' % ! (

! 99 # ! % ( $ -5, D% . .). $ ' (3.1.40), % ( # C (t) = PO (t1 ; t)B (t): @ 9 (# (3.1.30) % Zt1 J = (x0Q(t)x + u0Q(1)(t)u) dt + x0(t1)P (1)x(t) t0

(3.1.41) (3.1.42)

Q(1)(t) { -

# m m . ;

u] = H (1)u

(H (1)0 H (1) = Q(1)) ' (3.1.29) 9 (# (3.1.42) (3.1.29), (3.1.30):

(3.1.43)

Zt1 J = (x0Q(t)x + u] 0u] )dt + x0(t1)P (1)x(t1)I

(3.1.44)

x_ = A(t)x + B] (t)]u

(3.1.45)

t0

B] (t) = B (t)H (1);1: ,( , 9 (# (3.1.42) . ( (3.1.29) (( u] = C] 0(t)x , C] = ;P (t)B] (t) u = H (1);1C] 0x = C 0(t)x , (

C (t) = ;P (t)B (t)Q(1);1 P (t) - ' $((:

;P_ (t) = P (t)A(t) + A0(t)P (t) ; P (t)B(t)Q(1);1(t)B0(t)P (t) + Q(t)I 57

(3.1.46)

(3.1.47)

P (t0) = P (1):

(3.1.48)

3.1.3 8 % 7 . 7 - : ; ( * * ( $((, ' * B5$ # ! . (, , * *

' % ! * (! % % %, * ' ( % * (

! 99 # ! % % * ! !. ( #9* (% ! ( (3.1.15) <99 ( ! *

! ' : $ -, ( M3.5], 4 -$9 M3.6], # M3.7]. ' % <! . ; < , * ! % 9 (# (3.1.3) ( * , 9 (# #

Zt1 J = (x0Qx + u0u) dt (t1 6= 1): 0

(3.1.49)

5 * % ! % ( , * n = m = 1 9 (# (3.1.10) ( = p(t)x2 . < ( (x(t1)) = 0 ( p(t1) = 0 ). ,,

* x 4.1, * 99 # (

;P_ (t) = P (t)A + A0P (t) ; P (t)BB0P (t) + QI P (t1) = 0 (3.1.50) ;, (( # * , = t1;t * P (t) = P (t1; ) = PO ( ) ,

' (3.1.50) (( dPO ( ) = PO ( )A + A0PO ( ) ; PO ( )BB 0PO ( ) + QI 0 t I 1 d

(3.1.51)

PO (0) = 0: (3.1.52) ! ( $ -, (, , * (

! M3.27], M3.5] ' O

lim !1 P ( ) = P

0

(3.1.53) M( (( ! 0 t1 , (3.1.53) , t1 * 9(

* , * t1 = 1 ]. @ ' (3.1.53) , * ! -

% # P 0 , % * ( $(( (3.1.15), * ' 99 # ! % (3.1.46) ! , 58

( ' ( PO ( ) , ), < ' ' ( # P 0 .

. 8 . % . * (3.1.23) 9 (# # (3.1.25):

a22 = ;300I a23 = 103I a32 = ;3I a33 = ;1I b31 = 10;3 I

(3.1.54)

q11 = 1 6 1012I q22 = 3 108I q33 = 5 109 : (3.1.55) * % (3.1.26) % ((, p_11 , { p_12 , { p_13 . .) ' *

' 99 # ! % $ -5, *: p011 = 53 4 109 I p012 = 147 106 I p013 = 12 6 108I p022 = 29 104I

(3.1.56)

p023 = 44 105 I p033 = 116 106 : @( * * (3.1.57):

(3.1.57)

c1 = ;0 126 107I c2 = 0 44 104 I c3 = ;116 103:

(3.1.58)

3.1.4 8 7 . - D * ( % ! ( * ( % # ,( 9 #,!! $(( " T # A ; BB == (3.1.59) ;Q AT D # 2n 2n . P ! ( * (% g(s) = det(Es ; =) * ! % s . D * , * { ( < (

* # = , ; ( ( . ,( , # = 1 : : : n

! * #

* ;1 : : : ;n

! *

*. @( # P 0 * 9

P (0) = P2 P1;1 P1 P2 { ( # , 9 : ). ; *

* # = . ). $ ' 59

(3.1.60)

(Ei ; =)ci = 0 (i = 1 n) 9

! ( ci (i = 1 n) 2n 2n # C . ). * n ( # C (( P1 , n - ( { P2 .

3.1.5 A ) %%0) % ; 1967 . B.B. 5 (% M3.8] # B5$ * % . 7 < 9 (# (3.1.3) , * ( 9 (# # ( % 9 ) 9 "X Z18 n m m X n @v #2= <X X 1 J = : qij xixj + u2k + 4 bik dt (3.1.61) @x i ij =1 i =1 k =1 k =1 0 ( * 9 v = x0Px -

# P , ' * * (

PA + A0P + Q = 0: (3.1.62) - 9 (3.1.16). 7 * <, * n = m = 1 . 1 (# (3.1.61) 9 Z18 < 2 2 1 @v !2= J = :qx + u + 4 @x b dt: (3.1.63) i 0

(3.1.9) qx2 !2 1 @v b 2 qx + 4 @x

* % * ( (3.1.12) % 2pa + q = 0 (3.1.64) (<99# ( * % 9 v = px2 . ,( , * ( ( (

% ' % * ( (3.1.62) * (% # C 9 (3.1.16). (3.1.62) 7 .

' P > 0 , (

* # A #

*. < ( , *

* ( %*. 7 % , - (* 9 (# - 9 (# v = x0Px > 0 , * , *, * 60

dv = ;x0 MQ + 2CC 0] x < 0 dt 1 (# (3.1.61) M3.9] / ) . D

Z1 , * (3.1.61) (( u0u dt , 0

( % % "< " (

) u .

3.1.6 @) %< $ . ( , % m X x_ i = 'i(x1 : : : xn) + bik uk (i = 1 n): k=1

(3.1.65)

* <! % , % ( *( x1 = : : : = xn = u1 = : : : = um = 0 . , (3.1.65)

x_ i =

n X j =1

aij xi +

n X jk=1

aijk xj xk +

n X j k =1

aijkxj xk x + : : : +

m X k=1

bik uk (i = 1 n):

(3.1.66)

, % (3.1.67) uk = rk (x1 : : : xn) (k = 1 m) ( ! ! (3.1.66), (3.1.67),

!

* ( , 9 (# (3.1.3). $ ' <% * * M3.10]. < ' , * * n = m = 1 . ; < * (3.1.66) ' ( * a111 = a(2) , a1111 = a(3) ..) (:

x_ = ax + a(2)x2 + a(3)x3 + : : : + bu: (2.3.11), (2.3.12) * ( * @v (ax + a(2)x2 + a(3)x3 + : : : + bu)I ; @v@t = qx2 + u2 + @x (3.1.68) @v b: u = ; 21 @x (3.1.69) @(* u (3.1.68) (3.1.69), * !2 @v b + qx2: @v @v 1 (2) 2 (3) 3 (3.1.70) ; @t = @x (ax + a x + a x + : : :) ; 4 @x 61

$ ' < (

v = px2 + p(3)x3 + p(4)x4 + : : : : (3.1.71) (3.1.70), *

(3.1.71)

(2px+3p(3) x2+4p(4)x3+: : :)(ax+a(2)x2+a(3)x3+: : :); 14 b2(2px+3p(3)x2+4p(4)x3+: : :)2+qx2 = 0: (3.1.72) ( (<99# ( ! ! x , * ! p , p(3) , p(4) : : : 9 (3.1.71). ,(, ( (<99# x2 2pa ; (pb)2 + q = 0 (3.1.73) ( (<99# !3 * (3.1.74) 2pa(2) + 2p(3)a ; 21 b2(2p)(3p(3)) = 0 . . (3.1.73) (3.1.12) ' s 2 a p(1) = b2 + ab4 + bq2 :

(3.1.74) ' % 9 * (3.1.14) D * (3.1.73) % (<99# p(3) 9 (3.1.71). $ ' < , a + bc 6= 0 . * (% %* x_ = (a + bc)x , ( 9 (# (3.1.3) % . ( (3.1.1). ( (<99# x4 , * 4p(4)(a + bc) = ;2p(1)a(3) ; 3p(3)a(2) + 41 b2(3p(3))2: (3.1.75) D , (( , % p(4) . . ; (3.1.69) (

u = cx + c(2)x2 + c(3)x3 + : : :

(3.1.76)

c = ;p(1)bI c(2) = ; 23 p(3)bI c(3) = ; 24 p(4)b : : : ; * (n > 1 m 1) 9 (# 62

(3.1.77)

=

n X (i j =1)

pij xixj +

n X (i j k=1)

pijk xixj xk +

n X (i j k =1)

pi j k xixj xk x + : : : :

(3.1.78)

P (<99# pij (i j = 1 n) ! ' * ( $(( (3.1.15), (<99# pijk (i j k = 1 n (* % ! 9 ' % ! * (! % - (3.1.62), ( ! # A # A + BC 0 ( &0 - # (3.1.2) % . (), Q -< #,

#, *

! ' ! 9 .

3.1.7 1 8 .0 $ . ( , %

x_ = Ax + B u + Qf I x(t0) = x(0)

(3.1.79) f (t) { - % ( ' ! %I Q {

# *

n . ( f (t) , *: 1) ( *

jfi(t)j fi (i = 1 )

(3.1.80)

lim f (t) = 0I t!1 i

(3.1.81)

fi (i = 1 ) -

*I 2) 9 (# fi(t) (i = 1 ) - * . D * , * 3) ( f (t) . , %

u = C 0x + 12 B 0L(t)

(3.1.82) M L(t) - ( # n n ], ( , * ! (3.1.79), (3.1.82),

! * ' , 9 (# (3.1.3): Z1 J = (x0Qx + u0u) dt: (3.1.83) 0

, * (3.1.81) ! ! (3.1.83). B * ( ( ' ! ! M3.12] #%: 1) * # &0 # % 3.1.1 * ( ( f = 0 I 2) ' 99 # 63

L_ = ;(A + BC 0)0L ; (P + P 0)Qf (t) (3.1.84) # L(t) , ! % ( (3.1.82). 7 ( *% n = m = = 1 . ; < * (3.1.79) x_ = ax + bu + f 9 (# (3.1.83) ' (( Z1 J = (ax2 + u2)dt: 0

* ( !2 @ @ 1 2 ; @t = @x (ax + f ) ; 4 @ @x b + qx :

(3.1.85) (3.1.86)

(3.1.87)

$ ' < (

= px2 + l1(t)x + l0(t) p { *, l1(t) l0(t) { 9 (#. 7 <! ! (3.1.88) (3.1.87): ;(l_1x + l_0) = (2px + l1)(ax + f ) ; 41 (2px + l1)2b2 + qx2: (<99# x2 x x0 , * : 2pa ; p2 b2 + q = 0I ;l1 = (a ; pb2)l1 + 2pf I ;l_0 = l1f ; 41 l12b2: , * (3.1.8) @ b = ;pbx0 ; 1 l (t)b u = ; 12 @x 21 (3.1.82) (3.1.84).

(3.1.88)

3.1.8 9 6 , * . ( (3.1.79)

( ( ( , m - % ( -9 (# x(t) , % Mt0 t]. 7 , x(t) ( " ") x(t) . 0 ( -9 (#% x(t) x(t) (( * 9 (# 64

Z1 J = M(x ; x)0Q0(x ; x) + u0u] dt: 0

(3.1.89)

,( , ( * , ( < 9 (# ' * . ( , * < * ( % M3.13]. 7 % , % ( e = x ; x , *, (3.1.79),

e_ = Ae + B u + f (1)

(3.1.90)

f (1)(t) = Ax ; x_ + Qf (t)

(3.1.91)

f (1)(t) { < n - % ( ,

1 (# (3.1.89) Z1 J = (e0Q0e + u0u) dt: 0

(3.1.92)

P f (1)(t) % (3.1.81), ' (3.1.82).

3.1.9 1 (-,) . ( , %

x(k + 1) = Rx(k) + Ru(k) (k = 0 1 2 : : :) x(0) = x(0)

(3.1.93) R R -

# * n n n m

. 5* ! ! # < . ( # %

J=

N X x0(k)Qx(k) + u0(k ; 1)u(k ; 1)

k=1

(3.1.94)

Q -

-

#. , % # C (k)

u(k) = C 0(k)x(k) (k = 0 1 2 : : :)

(3.1.95) ( 9 (# (3.1.94) ' * ! x(0) . B * ( ( ( ! . ( M?] #%: 1) * # P (N ; j ) j = 1 N (

'

P (N ; j ) = R0MQ + P (N ; j + 1)]R ; R0MQ + P (N ; j + 1)]R MR0(Q + P (N ; j + 1))R + E];1R0MQ + P (N ; j + 1)]R (j = 1 N )I 65

(3.1.96)

P (N ) = 0I

(3.1.97)

2) !

C 0(N ; j ) = ;fR0MQ + P (N ; j + 1)]R + E g;1 R0 MQ + P (N ; j + 1)]R (j = 1 N )I (3.1.98) 3) # (<99# C 0(k) = C 0(N ; j ) (j = 1 N ): (3.1.99) ; ' % (3.1.96) : : : (3.1.99) * # ( . ( 7 3. 7( < ' n = m = 1 . ; < * . ( (3.1.93) 9 (# (3.1.94) x(k + 1) = fx(k) + ru(k) (k = 0 1 2 : : :)I J=

N X k=1

qx2(k) + u2(k):

(3.1.100) (3.1.101)

7 !

u(k) = c(k)x(k) (3.1.102) # ,

% x2.3. ; < # , (( ' ( M(N ; 1) N ] ), (u(N ; 1)) ' ( , (' ! N ; 1 '). 3* , ( ! ' , J (N ;1) = qx2(N ) + u2(N ; 1) = qMfx(N ; 1) + ru(N ; 1)]2 + u2(N ; 1): @ ! <( <% @J (N ;1) = 2qMfx(N ; 1) + ru(N ; 1)]r + 2u(N ; 1) = 0 @u(N ; 1) * *( qfr x(N ; 1): u(N ; 1) = ; 1 + qr2 66

(3.1.103) (3.1.104) (3.1.105)

min

J (N ;1) = (N ;1)

=

"

qf 2

2f 2 r2 # q ; 1 + qr2 x2(N ; 1) = p(N ; 1)x2(N ; 1)

(3.1.106)

2 2 2 p(N ; 1) = qf 2 ; 1q +f rr2q :

(3.1.107)

Mq + p(N ; 1)]fr x(N ; 2): u(N ; 2) = ; 1 + Mq + p(N ; 1)]r2

(3.1.109)

! ( ! ' ( MN ;2 N ;1] ), ' ** , ( < : J (N ;2) = qx2(N ; 2) + u2(N ; 2) + (N ;1) = (3.1.108) = Mq + p(N ; 1)]x2(N ; 1) + u2(N ; 2) = 2 2 = Mq + p(N ; 1)]Mfx(N ; 2) + ru(N ; 2)] + u (N ; 2): (N ;2) @ ! @u@J(N ; 2) = 0 , * *( < ** (3.1.108) * min J (N ;2)

(

2f 2 r2 = Mq + p(N ; ; 1Mq++Mqp(+Np;(N1)]; 1)] r2 x2(N ; 2) = p(N ; 2)x2(N ; 2)

= (N ;2)

1)]f 2

)

1)]2f 2r2 p(N ; 2) = Mq + p(N ; 1)]f 2 ; 1Mq++Mqp(+Np; (N ; 1)]r2

(3.1.110)

(3.1.111)

< # , % j - ( ( #) *( ( MN ;j N ;j +1] ). 3* , ( u(N ; j ) ,

J (N ;j) = qx2(N ; j + 1) + u2(N ; j ) + (N ;j+1) = = Mq + p(N ; j + 1)]x2(N ; j + 1) + u2(N ; j ) = = Mq + p(N ; j + 1)]Mfx(N ; j ) + ru(N ; j ) + ru(N ; j )]2 + u2(N ; j ): Mq + p(N ; j + 1)]fr x(N ; j ) = u(N ; j ) = ; 1 + Mq + p(N ; j + 1)]r2

= c(N ; j )x(N ; j ) 67

(3.1.112)

(3.1.113)

Mq + p(N ; j + 1)]fr : c(N ; j ) = ; 1 + Mq + p(N ; j + 1)]r2

(3.1.114)

min J (N ;1) = (N ;2) = p(N ; j )x2(N ; j )

(3.1.115)

) * ** % (3.1.112) <

j + 1)]2f 2r2 : p(N ; j ) = Mq + p(N ; j + 1)]f 2 ; 1Mq++Mqp(+Np; (N ; j + 1)]r2

(3.1.116)

(3.1.98), (3.1.96) n = m = 1 , , * (3.1.114), (3.1.116)

. P 9 (# (3.1.94) ! % N ! 1 , (3.1.95)

u(k) = C0 x(k) (k = 0 1 2 : : :)

(3.1.117)

O + E ];1R0 PO R + Q = 0 PO ; R0PO R ; R0PO RMR0PR

(3.1.118)

C 0 { # * , %

h i O + E ;1 R0PO R C 0 = ; R0PR (3.1.119) D (3.1.96),(3.1.98), P (N ;j ) = P (N ;j +1) = P * Q + P = PO .

. (-,) . % #9 % u(kT ) = c1x1(kT ) + c2x2(kT ) + c3x3(kT ) (k = 0 1 2 : : :) (3.1.120) ( ! , % (3.1.23) ( f = 0 ), 9 (# NX !1 2 (3.1.121) J= q11x1(kT ) + q22x22(kT ) + q33x23(kT ) + u2(kT ): k=1

! ( *

' <% *, 9 * ( . 7 < 9 (3.1.26), (3.1.28), ( ! * # R ( R . * ! a22 = ;400 , a23 = 103 , a32 = ;10 , b31 = 10;2 * T = 0 015 0 19 10;15 1 0 192 10;2 0 279 10;1 (3.1.122) R = 0 ;0 477 10;1 0 192 10 I R = 0 279 10;3 : ; 3 ; 1 0 ;0 192 10 0 72 0 131 10 68

@ < # , ( * 9 (# (3.1.121), q11 = 1010 , q22 = q33 = 0 , * (3.1.96), (3.1.98), (??) ( *:

c1 = ;0 686 105I c2 = ;0 728 102 I c3 = ;0 414 104:

(3.1.123)

3.1.10 % - (: 111 (B * ( ( ! -

). @!

: ) . ( (3.1.1).( ! ! # % ( % * < Qf , f - ( * ! ' ! %). )0 # N !

! = N x . ($

! ! # ). ); # !

! Q0 .@ < # # N , % # Q = N T Q0N 9 (# # (3.1.3) ); ! ! # . $ : )0 # C 0 (3.1.2). )= 9( ! ! #

. @ ( 111 (( .2.1. # % ! ( % ( '! '( , , ) ! .$ < % 9(,% ,* % * ( %*% (3.1.1),(3.1.1),( % , * , "! " ( ) # % 9 (# ,* (3.1.3), Z1 J = (x0Qx + u0LO u + 2x0Lu) dt (3.1.124) 0

( # L Q ; L(LO );1LT 0 LO > 0 ), ( < %. -9 (#: 1) MC 0 P ] = lqr(A B Q LO L) - % (LQ-#).@!

: # . ( (3.1.1) 9 (# (3.1.124).$ : # $((, ( ( ()

! * # A + BC 0 ( % . ) ,*, 9 (# # (3.1.124), $(( #

PA + A0P ; (PB + L)(LO );1(B 0P + L0) + Q = 0I 69

(3.1.125)

C = ;(PB + L)(LO 0);1 (3.1.126) O ] = dlqr(Phi R Q LO L) - ( % 2) MC 0 P (3.1.93)- (3.1.95) N ! 1 . 3) MP C 0 rr] = care(A B Q)- ' $(( (3.1.18), rr { ' ' ( # P 0 ). O C 0 rr] = care(A B Q)- ' $(( (3.1.118) ( 4) MP

! , rr { ' ' . 5) MP ] = lyap(A Q)- ' - (3.1.62).

3.2

) , - 3.2.1 . ( %*) $ . ( ,

(

x_ = A(t)x + B (t)u x(t0) = x(0)

(3.2.1)

*

u = C 0(t)x:

(3.2.2) $ # < * , *

. (

, ' ( ( r - ( y ,

'

y = D(t)x:

(3.2.3) ; < ( * ( , # () ( x(t) y(t) Mt0 t] . (( ( , (3.2.2), %

( .

3.2.2 5 %* > * 9* ( %, ! (

% ( u(t) , %% . (, ! # ( ( . (. %' 9* ( %, 70

x^_ = A(t)^x + B (t)u x^ (t0) = x^ (0)

(3.2.4) { * -

x^ (t) { n - % ( ( !) , x^ (0) . (3.2.4) 9* ( . (, 9 (# . P * . ( , x^ (0) = x(0) , * ( ' % (3.2.4) (3.2.1), , * ( . (. 4* . ( * , < x^ (0) 6= x(0) . , ( '( e = x ; x^ .

e_ = A(t)eI e(t0) = x(0) ; x^ (0)

(3.2.5)

( * , (3.2.1) * (3.2.4). P . ( * ( %*, '( % tlim !1 e(t) = 0 , ! * ( ( . (. P . ( %*, , %

x_ = A(t)^x + K (t) My ; D(t)^x] + B (t)u x^ (t0) = x^ (0)

(3.2.6) K (t) { ( # n r , ) 9 * . ; <

* ( y

* D(t)^x . D (

(# '( # % K (t) . K (t) = 0 (3.2.6) %'. )'

x^_ = MA(t) ; K (t)D(t)] x^ + K (t)y + B (t)u

(3.2.7) ( , * ! ( ( !

!, ! # # % MA(t);K (t)D(t)] < # K (t) * (% %* . ; * (3.2.1) (3.2.6), * '(

e_ = MA(t) ; K (t)D(t)] eI e(t0) = x(0) ; x^ (0):

(3.2.8) P # K (t) , (, * * ( %*, '( % lim e(t) = 0 . t !0 71

& ( ( # K (t) , ( % * ( %*. $ (! , * , , # . (. ; < * . (

x_ = Ax + B u y = Dx

(3.2.9)

(3.2.6)

x_ = Ax^ + K My ; Dx] + B u x^ (t0) = x^ (0)

(3.2.10)

D0 A0D0 : : : (A0n;1D0) = n:

(3.2.11)

K { # * . > , * . ( (3.2.9) * * . D * , *

3.2.3

%* 7

b ( * (% (3.2.10)

D (s) = det kEs ; A + KDk: (3.2.12) )* , * # K (, * ( < #

*. ; 3.1.1 ( , * ' * B5$ * * ( %* (3.1.1), (3.1.2). < * # K ( #

% * B5$. ; < ' (3.2.12) (( D (s) = det jjEs ; A0 + D0 K jj (3.2.13) 9 " " < ! ( * (

_ = A0 + D0u

u = ;K 0

(3.2.14)

(3.2.15) (t) { n - % ( " ". D " " %

* . % (3.1.1), (3.1.2), A = A0 , B = D0 , C = ;K . $(( (3.1.15) <% 72

P A0 + AP ; P D0DP + Q = 0 (3.2.16) Q {

-

#, P { ( # n n . $ ' < , %

# P(0) (3.1.16) 9 ( # K = P(0)D0 :

3.2.4

(3.2.17)

%*

P # K * ' , * * ( %* ,

, % ( # K , * ( ! ( * ( D (s)

* 1 : : : n (Re n < 0 , i = 1 n) . * , * # K s n Y D (s) = det jjEs ; A + KDjj = (s ; i ) : (3.2.18) i=1

7 (% # K %

* ,

% % (3.2.14), (3.2.15), M3.25], M3.26] . ; (* . 7

. (

x_ = Ax + B u

(3.2.19)

u = C 0 x

(3.2.20)

% # C (

( , * ( ( ) ! ( * ( ( %

D(s) = det jjEs ; A ; BC 0jj (3.2.21)

* 1 : : : n . # # C % < *, 7 4. 7 # K , (3.2.19) # A B # A0 D0 , i = i i = 1 n) % # C 0 , ( # K = ;C 73

(3.2.22)

. - %* . -

(

! , % (??), (??), (3.2.10) :

x^_ 1 = x^1 + k11(y ; x^1)I

(3.2.23)

x^_ 2 = a22x^2 + a23x^3 + k21(y ; x^1)I

(3.2.24)

x^_ 3 = a32x^2 + a33x^3 + k31(y ; x^1) + b31u: (3.2.25) 4 k11 , k21 , k31 (, * ( ! ( * (

* 1 , 2 , 3 . ; < 9 * : ". ("

% " "

x_ 1 0 0 0 x_ 2 = 1 a22 a32 x_ 3 0 a23 a33

1 x1 x2 + 0 u 0 x3

(3.2.26)

u = c1x1 + c2x2 + c3x3 (3.2.27) ( ! ( * (% (3.2.26), (3.2.27)

D(s) = s3 + d2s2 + d1 s + d0

(3.2.28)

d2 = ; 1 ; 2 ; 3I d1 = 1 2 + 1 3 + 2 3I d0 = ; 1 2 3 : (3.2.29) ; % # % # 9 # ;1 0 0 1 1 0 0 Qy = 0 1 ;d2 (3.2.30) 0 1 a32 2 1 ;d2 ;d1 + d2 0 0 a23 d0 , d1 , d2 { (<99# ! ( * ( . ( s 0 0 D(s) = det ;1 s ; a22 ;a32 = sM(s ; a22)(s ; a33) ; a32a23] = s3 + d2s2 + d1s 0 ;a23 s ; a33 (3.2.31) 74

d2 = ;a22 ; a33I d1 = a22a33 ; a32a33: ; # ( *

;c]1 = d0I ;c]2 = d1 ; d1I ;c]3 = d2 ; d2:

(3.2.32) @ (7.4.14) # % (3.2.30), * * ci (i = 1 2 3) , (

k1i = ;ci (i = 1 2 3):

(3.2.33)

3.2.5 ' %* ; ( (3.2.1), (3.2.2), # ( (3.2.2) ( % , *

, , * < *

(3.2.6), ( (3.2.2) (

. *

( :

x_ = A(t)x + B (t)u y = D(t)xI

(3.2.34)

x^_ = MA(t) + K (t)D(t)] x^ + K (t)y + B (t)uI

(3.2.35)

u^ = C 0(t)^x:

(3.2.36) 4 . 3.2.1 ( ! ,

% (3.2.34) ... (3.2.36).

75

$. 3.2.1 @ %* (3.2.34) ... (3.2.36). <( <% . ; * (3.2.34) (3.2.35) (3.2.36) x^ = x ; e , * ( (3.2.36) (3.2.34) :

e_ = MA(t) + K (t)D(t)] eI e(t0) = x(t0) ; x^ (t0)I

(3.2.37)

x_ = MA(t) + B (t)C 0(t)] x ; B (t)C 0(t)eI x(t0) = x(0):

(3.2.38) P # (<99# K (t) (, * (3.2.35) * ( %* y(t) = u(t) = 0 , ' (3.2.37) e(t) ! 0 t1 ! 1 * e(t0) . # B (t) C 0(t) , ! (3.2.38), * e(t) ! 1 t ! 1 , x(t) ! 0 , * ( %*

x_ = MA(t) + B (t)C 0(t)] x:

(3.2.39)

; # * (3.2.34) ... (3.2.36)

x_ = Ax + B u y = DxI

(3.2.40)

x^_ = MA ; KD] x^ + K y + B uI

(3.2.41)

u = C 0(t)^x:

(3.2.42)

D( % , * (3.2.37), (3.2.38), ((

e_ = MA ; KD] eI x_ = MA + BC 0] x ; BC 0e:

(3.2.43)

0 D(s) = det E (s) ; A 0+ KD = BC Es ; A ; BC 0

(3.2.44)

b ( * (% (3.2.43)

= det (Es ; A + KD) det (Es ; A ; BC 0) : @ < , * ( ! ( * ( % ( % ! ( * ( Du (s) = det (Es ; A ; BC 0) % ( ( %

) ( % ! ( * ( Du (s) = det (E ; A + KD) . ,( , ( . 76

. B %* . $ f1 = 0 (3.1.23)

9 (# (3.1.25) (3.1.24) ; *

'

x1 , ! !

! (3.2.1) ... (3.2.3) ,

x_ 1 = x2I x_ 2 = a22x2 + a23x3I x_ 3 = a32x2 + a33x3 + b31u

(3.2.45)

u = c1x^1 + c2x^2 + c3x^3I

(3.2.46)

x^_ 1 = x^1 + k11(y ; x^1)I x^_ 2 = a22x^2 + a23x^3 + k21(y ; x^1)I

(3.2.47)

x^3 = a32x^2 + a33x^3 + k31(y ; x^1)b31uI (3.2.48) ( ! (3.2.46) ... (3.2.48) { , ( c1 , c2 , c3 ' * ,

4.1.2, k11 , k21 , k31 ! ,

4.2.3. b ( * (% (3.2.45), (3.2.46), (3.1.24) x]i = xi (i = 1 2 3)

;1 0 = s ; a22 ;a23 ;a32 ; b31c2 s ; a33 ; b31c3 = sM(s ; a22)(s ; a33 ; b31c3) ; a23(a32 + b31c2)] ; b31c1a23: s Du (s) = det 0 ;b31c1

(3.2.49)

; (3.2.28) ! ( * (%

D (s) = s3 + d2s2 + d1s + d0 ! ( * (% (3.2.45) ... (3.2.48) D(s) = Du (s)D (s):

(3.2.50) (3.2.51)

3.2.6 &.6 % %* > * ( %* ), ! * ( %* ! ! (<99# , ! 99 # ! %. ( % M3.24] %* 9 . @ %* (3.2.40) { (3.2.42) ( , * M3.14],M3.15],M3.16]. 77

@(* ( x^ , ' ((

u = C 0 (Es ; A + KD);1 MK y + B u]

h i E ; C 0 (Es ; A + KD);1 B u = C 0 (Es ; A + KD);1 K y: (3.2.52) (s) (3.2.40), (3.2.52) &9 * # ;W ( ( ! . () . 7 < ( ! . (

( % ( ;r (u = ;r , * * ( ! . (, ' #, ! u ( r . , *

h i (s) = ; E ; C 0 (Es ; A + KD);1 B ;1 C 0 (Es ; A + KD);1 KD (Es ; A);1 B: W (3.2.53) * , , * ( (m = 1) , ' * 9 (# 0 (Es ; A + KD);1 KD (Es ; A);1 b c (3.2.54) W (s) = ; 1 ; c0 (Es ; A + KD);1 b PK (( 0 Esg 0 Es ; g c ; A b D ( s ) + c A + KD bd(s) (s) = W (3.2.55) d(s)D (s) ; c0 Es ; g A + KD bd(s) g *

#, d(s) D (s) { ! ( * ( . ( . 7 % , # * * % 9 (# (3.2.54) (( (Es ; A + KD);1 KD (Es ; A);1 = (Es ; A + KD);1 (;A + KD) (Es ; A);1 + + (Es ; A + KD);1 A (Es ; A);1 * , * % ( % # M * (Es ; M );1 M = s (Es ; M );1 ; E M (Es ; M );1 = (Es ; M );1 s ; E * (Es ; A + KD);1 KD (Es ; A);1 = (Es ; A);1 ; (Es ; A + KD);1 ( * 9 (# (3.2.55). 78

4 * K * ( , 0 g

(s) = c Es ; A + KD bdh(s) , ( g i ( ! ( * ( 0 D(s) = d(s) ; c Es ; A b D (s) + (s) ; (s): P (<99# . ( * * !, (s) * *. ,(, * , ( b , !% * ( . (, ( b 9 %

. ; M3.24] ( , * * ! ( * ( ( '! * ! (<99#

(s) ) ( ! %* 9 . ,

! %* .

3.2.7

%# %* .

P ( . ( , (3.2.1), % 9 (# (3.1.3) * . 5( ( %* 9 % * (. * # ( % ( ! . () (% %

W (s) = ;C 0 (Es ; A);1 B: (3.2.56) ; % (% ( : # K (, * * 9 (# ( ( K * % # (s)

W (3.2.57) = W (s): ( , * -9 ! . ( < . . ( g(3.2.40) r = m -, ( det D Es ; A B #

*. > # K %(??), (??), ( ! # Qc

( , (

K = Pe(0)D0 (3.2.58) # -

# Pe(0) ' $(( Pe A0 + APe ; PeD0 DPe + R + BV B T = 0 (3.2.59) R V { # -

# , { * ' *, ( (<99# . 79

5 6. M3.15] .

! 1 * # W (s)

( * % # W (s) .

3.2.8 % - (: 113 ( ).

@!

: ) . ( (3.1.1).( ! ! # % ( % * < Qf , f - ( * ! ' ! %). )0 # D !

!,( : = y . ($

! ! # ). ); # !

! Q0 .@ < # # N , % # Q = N T Q0N 9 (# # (3.1.3) ) 4 # # V $(( (3.2.59) .(D # # % 7 ) ) % , ( < #. ); ! ! # . $ : )0 # C 0 K (3.2.41),(3.2.42) )= 9( ! ! #

. @ ( 113 (( .2.2. # % ! ( % ( '! '( , , ) . -9 (#: O K ) -9 (3.2.10), DO - # 1) est = estim(A B D D : y = Dx + DO u . O C 0 K ) -9 (3.2.41),(3.2.42 2) rsys = reg(A B D D 3) K = place(A0 D0 ) - * (<99# . 4) MPc K rr] = care(A0 D0 Qc) - * (<99# $(( (3.2.16).

3.3 ) # 3.3.1 5 ) , # * ( ( % <99 ( % , (* ( ! # ( . ( * ( , * 80

! * ( (

( ( , < ( * (<99# 9 (# #

( * (* . 7 ' ( % 9 (# #, % , ( (* ( , , %* 9 ) * ( ' '( ' ! !) { %. (% ) M3.17]. ! ( < ,

x_ = Ax + B uI

(3.3.1)

u = C 0 x

(3.3.2)

9 (# Z1 J = (x0Qx + u0u) dt 0

(3.3.3)

( Q { -

#. (3.3.1), (3.3.2) * , * #

C = ;PB (3.3.4)

- # P ' * ( $((: PA + A0P ; PBB 0P + Q = 0 (3.3.5) ( A B {

# , . (3.3.1), (3.3.2) - ! * ! !, ' * # <% ( W (s) = ;C 0(Es ; A);1B

(3.3.6)

s { ( ( *. * % * (3.3.5) sP *

B 0(Es ; A);1 , (Es ; A);1 ,

B 0(;Es ; A);10 M;PA + Ps ; A0P ; sP + PBB 0P ; Q](Es ; A);1B = = B 0(;Es ; A);10 PB + B 0P (Es ; A);1B + B 0(;Es ; A);10 PBB 0P (Es ; A);1B ; ;B0(;Es ; A);10 Q(Es ; A);1B = 0: (3.3.7) 81

; * H (s) = H (Es ; A);1B , H 0H = Q , * (3.3.4), ' (3.3.7)

;B0(Es ; A);1 C ; C 0(Es ; A);1B + B0(;Es ; A);1 CC 0(Es ; A);1B = H 0(;s)H (s): 0

0

( * * # * (3.3.6) * % # ( % , * ( * MEm + W (;s)0]MEm + W (s)] = Em + H 0(;s)H (s): s = j! , * )

(3.3.8)

MEm + W (;j!)]0MEm + W (j!)] = Em + H 0(;j!)H (j!): (3.3.9) D !

! ! * * # W (j!) % 9 (# #. ; %' ' *% ( (m = 1) . ; < * (3.3.1), (3.3.2)

x_ = Ax + buI

(3.3.10)

u = c0x u { ( I b c { n - ( -# . * 9 (# <%

(3.3.11)

w (s) = ;c0(Es ; A);1b: (3.3.12) (3.3.9) ( n X M1 + w (;j!)]M1 + w (j!)] = 1 + hi (;j!)hi(j!) (3.3.13) i=1

hi (j!) (i = 1 n) { - # 9 (#, ( ( H (Ej! ; A);1b .

3.3.2 A(,,- . 4% (<99# 9 (# ! Z1X n 2 2 J= qiixi + u dt 0

i=1

82

(3.3.14)

( Q = diagMq11 : : : qnn] , qii > 0 , i = 1 n) ( (3.3.10), (3.3.11), % , (<99# * k *% ! <% - %. 4 , *

k = w (0)

(3.3.15)

* (! 0 k = slim !0 s w(s) r { ( , *

(3.3.16)

jw (j! )j = 1:

(3.3.17) (3.3.13) ! = 0 , * , * * k 1 * n X 2 k h2i (0): (3.3.18) i=1

< . ( ( *

H = diag Mpq11 : : : pqnn]

i(s) , (s) { * , * ( ( (Es ; A);1b D i (s) ( (s) = (Es ; A)b D(s) = det(Es ; A) = sn + dn;1 sn;1 + : : : + d1s + d0 (Es ; A) {

#, * "X # n n X hi(s)hi(;s) = qii i (s) i(;s) D(s)D1 (;s) : i=1

i=1

,( , (3.3.18) n 2 X k 2 qii id(0) 2 : i=1

0

(3.3.19)

(3.3.20)

7 * (! * , * ' (3.3.19) s2r s ! 0 : n 2 X (3.3.21) k 2 qii id(0) 2 : r i=1

83

3 * * k . , *

(<99# * ( % ! (<99# 9 (# (3.3.14) (3.3.20), (3.3.21), ( ! i (0) (i = 1 n) d0(dr ) { *, . ( . 7 * (3.3.10), (3.3.11) 9 (# (3.3.14) ( * ! , n X qii i(;j! ) i(j! ) i=1 (3.3.22) D(;j! ) D(j! ) = 1 ( <( n X hi(;j! )hi(j! ) = 1: (3.3.23) i=1

5 , ' (3.3.13) (( 2Re w (j!) + w (;j!)w (j!) = 2a(!) cos '(!) + a2(!) =

n X i=1

n X i=1

hi(;j!)hi(j!)

hi(;j!)hi(j!)

(3.3.24) (3.3.25)

a(!) = jw (j!)jI '(!) = arg w (j!): ! = ! (3.3.25) * (3.3.23) 2a(! ) cos '(! )+ a2(! = 1 , ( q a(! ) = ; cos '(! ) + cos2 '(! ) + 1: (3.3.26) * , * ;1 cos '(!) 1 , * # * % * % ! ( ( a(!) % *( ! : 0 4 a(! ) 2 4 20 lg(! ) 8 >: (3.3.27) P ( 9* (% -* % ! ( ( (-B3b) ( ! 20 >/ (, (* , *

* ! * ! * 0,4 ( ( 2,5 ). ,( , (<99# qii (i = 1 n) 9 (# (3.3.14) (, *

! (3.3.22), * % ( 9 (# (3.3.10), (3.3.11) *

% * 2,5 . 84

. $ , x_ 1 = x2I x_ 2 = a22x2 + a23x3I x_ 3 = a32x2 + a33x3 + b31u % (

(3.3.28)

u = c1x1 + c2x2 + c3x3 (3.3.29) ( (3.3.28), (3.3.29) (<99# * ( % * , ( (

;k , ! . 7 ' <% * ci (i = 1 3) 9 (# Z1 J = q11x21 + q22x22 + q33x23 + u2 dt (3.3.30) 0

(<99# ( ' % (3.3.21), (3.3.22). 7 ! 9 (#% i (s) (i = 1 3) * ;1 s ;1 0 0 a23 1 ; 1 (Es ; A) b = 0 s ; a22 ;a23 b31 0 = D(s) a23s 0 ;a32 s ; a33 b31 s(s ; a22)

(3.3.31)

D(s) = s3 ; (a22 + a33)s2 + (a22a33 ; a32a23)s . ,( ,

1(s) = a23b31I 2(s) = a23b31sI 3(s) = s(s ; a22)b31: < (3.3.21), (3.3.22), * q11a223b231 (3.3.32) (a22a33 ; a32a23)2 = k I q11a223b231 + q22! 2a223b231 + q33! 2 ! 2 + a222 b231 = 1: (3.3.33) D(;j! ) D(j! ) * (3.1.55), k = 4 102 , ! = 100 c;1 . ) * q33 = 5109 , * 9 (3.3.32), (3.3.33) q11 = 1 61012 , q22 = 3 108 . $ ' * B5$ <! * ! (<99# 9 (# # (3.3.30), % c1 = ;0 126 107I c2 = ;0 44 104I c3 = ;166 103: * 9 (# ( % 85

ci i (s) a23b31 c1 + c2 ; c3 aa22 s + ac3 s2 w (s) = ; i=1D(s) = ; s3 ; (a + a )s2 + (a 23a ; a 23a )s = 22 33 22 23 32 23 (3.3.34) 7 92 10;4 s2 + 3 1 10;2 s + 1) : = 0 126 10s(0 (s2 + 301s + 3 3 103 ) 4 . 3.3.1 -* ! ( ( ( % , ( % , * ( k ! . 3 X

$. 3.3.1

3.3.3 B - . ) # @ * % (3.3.10), (3.3.11) ( % 9 (# (3.3.14). 7 , * < 9 (# * (<99# qii , (, % (A0 , H 0) . ( , * ! ( % (* ( %* 9 ' , %* L ( ( M ) ! ( M3.18],M3.19] ! # , (<99# 9 (# (3.3.14). 5 6 ) %* ( ( (3.3.10), (3.3.11)

' 60I L 2I M 2:

(3.3.35)

1 . (3.3.13). * , * n X i=1

hi(;j!)hi(j!) 0 86

(3.3.36)

' (3.3.13) M1 + Re w (j!)]2 + Im2 w (j!) 1: (3.3.37) $ M1 + Re w (j!)]2 + Im2 w (j!) = 1 ( 9 -9% ! ( ( (B13b) ( * # *( Re w (j!) = ;1 , Im w (j!) = 0 . D ( (

. 3.3.2. 4 (3.3.37) * , * 9 B13b %

( (< . 3.3.2 ' ! ), *

% ( * # *( (;1 j 0) .

$. 3.3.2 ( * ( # * % 9 (3.3.13) (3.3.36), ( ' (3.3.35) # * ! ( % (* . 7 % , * % ( * # * ( , ( %

! ( O1OK1 , O1OK ( <! ( (! ( %), ( , *

( ( (, < OKK1 120 . D * , * 9 ' 60 . ! ( (3.3.35), , * (

% M;2 0] , *

% ( ( (, ! % . D * , * %* ! B13b !, B13b - ( * (. , * *( M;1 0]

% ( B13b % . = # ( ( ! ! . 4 . 4.3.2 ' ! ( % % ( r = M = 0 66 # *( (;a j 0) , a = M 2 = 1 3 . D ( (M 2 ; 1) (M 2 ; 1) * ( * (, ! B13b ( ( M = 2 . ,( (( < ( ! % , ( 87

# <% , M 2 , ( , ( . , * ( (3.3.37), ( (<99# 9 (# #. , < , * qii 0 (i = 1 n) , ( (( < * (3.3.36), < # (3.3.35)

<! (<99# , ( , ( . , * * * ! ( M3.20],M3.21] . %* ( ( ( , *

4.3.1. * 9 (# ( (3.3.34). 4 . 4.3.1 9* ( -* ! ( ( ;20 lg a(!) 9 (# '(!) = 180+ '(!) ( '(!) { 9* ! ( (), * % 9 (# (3.3.34). 4 , * ' = 80 , L ! 1 . 4 <

( 9(

(j! )j M (!) = j1 j+WW (j!)j

( , * ( (

M = 0max M (!) = 1 1: !1 , * * 9 (# (3.3.14) xi u: X ! ! Z1 X n n J= qiix2i + 2 lixi u + u2 dt 0

i=1

i=1

* % 9 Z1 J = (x0Qx + 2(l0x)u + u2) dt 0

(3.3.38)

l { n - % ( . # % ( * % 9 n +1

Q ; ll0 0 . ;

u = u + l0x , Q = Q + ll0 9 (# (3.3.38) ( 9 (3.3.14) # B5$. ( * ! ( % (* , ! 9 (# (3.3.38), ( # (3.3.35). > , ( , * % ( * ( "!%" * ( ) (3.3.10), (3.3.11) # % 9 (# (3.3.38), ( < %.

88

3.4 : 3.4.1 . $ * ( %* ,

x_ = Ax + B u + Qf y = Dx = N x

(3.4.1)

x_ = A x + B u u = D x+F y

(3.4.2) D # * % (1.3.9) { (1.3.12). ( - ( % f (t) * ( 9 (#

fi(t) =

p X k=1

fik sin(!k t + ik ) (i = 1 )

(3.4.3)

) fik 9 ik (i = 1 k = 1 ) , ( * !k (k = 1 ) ( . ( , * ( * p X

fik2 fi2 (i = 1 ) (3.4.4) k=1 p { * (, fi (i = 1 ) {

*. ' '(

* i = tlim (3.4.5) !1 sup ji(t)j ; M3.23] . 7

. ( (3.4.1) % * # A , B , D F (3.4.2) ( , * (3.4.1), (3.4.2) ( * i i

(3.4.6)

i (i = 1 m) {

*. 3 * <% * M3.20], M3.22] $ ' <% * ! * i (i = 1 m) . ; . (, ( ! '. ) 5 (3.4.1), ( ! Q = B , y = x , (

), = m . ') 5 ! ( , * ( ! !

! ( = y N = D r = m) ( , . ( - 9 . * , * ( det kDadj(Es ; A)B k , adj {

% # , #

*. 89

3.4.2 5 %< . ( , ( ,

( ! . (

x_ = Ax + B (u + f ) = N x

(3.4.7) ( % , *% % (3.4.6)

u = C 0 x:

(3.4.8) 7 ' * # B5$, ( % (* 9 (# # % Z1 J = (0Q0 + u0u) dt (3.4.9) 0

Q0 = diagk k { # (<99# 0 (qii > 0 (i = 1 m) . ; < * # Q ! $(( (3.1.15) Q = N 0Q0N . 0 q110 : : : qmm

5 6. 7 (3.4.8), *

% ( * (3.4.6) * ( (3.4.3) * # B5$ . ( (3.3.1) 9 (# (3.4.9), (<99# , m X p fk2 qii0 k=1 (3.4.10) i2 (i = 1 m) 7( M3.24].

3.4.3 5 %< ,( (( <! . (! ( , ! , < (3.2.40) { (3.2.42):

x^_ = MA ; KD]^x + K y + B u u = C 0x^ :

(3.4.11) 4 , * # (3.4.2) *

A + A ; KD + BC 0 B = K D = C 0 F = 0: 90

(3.4.12)

7 ' * * . ( (3.4.1) ( (3.3.35). ; < ' (3.4.1)

y(s) = D(Es ; A);1B u(s) + D(Es ; A);1Qf (s) (3.4.13) ( Of (t) <( ! ' ! %,

! ( ! . (, ' Of (s) = hD(Es ; A);1i;1 D(Es ; A);1Qf (s) = Tff (s)f (s) (3.4.13)

(3.4.14)

y(s) = D(Es ; A);1B (u + Of )

(3.4.15) 0 - 9 . ( #

! * % * % # Tff (s) < *

( - 9 (# Of (t) , ' (3.4.14). (3.4.15) (3.4.7)

x_ = Ax + B (u + Of )

(3.4.16) # m C (3.4.11) 4.3.2. ( X < ! % # Ofk2 (k = 1 m) " " Of (t) . k=1 ; < , * % * ( , *

fOkT fOk = fkT TffT (;j!k )Tff (j!k )fk 2fkT fk 0 !k 1 k = 1 : (3.4.17) Ofk fk m - ( ! <( %. & < , * m m m X Ofk;T Ofk 2 X fkT fk = 2 X X fik2 = 2 X X fik2 2 X fk2 (3.4.18) k=1

k=1

k=1 i=1

i=1 k=1

i=1

0 # K (<99# 9

K = P0(0)D0 (3.4.19) P0(0) { -

#, ' $(( P!A0 + AP! ; P!D0DP! + Qc + BV B T = 0

91

(3.4.20)

5 6. 0 # C K (3.4.11), *

% (3.4.6) ( * , ! % $(( (3.1.15), (3.4.20), ( ! m X p2 fk2 Q = N 0QoN qii(0) k=1 (3.4.21) (i = 1 m) 2 i Qc V { # -

# , % (<99# { * (.

3.4.4 % - (: 113.1 (,* . ( ).

@!

: ) . ( * ' ! %. )0 # D !

!,( : = y . ); ( # ' ! %. ); ( # ! * % '! '(. ) 4 # # V $(( (3.2.59) .(D # # % 7 ) ) % , ( < #. $ : )0 # C 0 K (3.4.11). )= 9( ! ! #

. @ ( 113.1 (( .2.4. # % ! ( % .

4 $#%&( #+#) #%( $ # . (,

! ' , ( !, # ! !

!. ; ' ! ( *% # . ; x5.1 * ' ! ! " % '",

! ( . (,

( * ( !). ( , * < ( , *

% , ' ! %. ; x5.2 !* ( (

! ! !! # . 92

* ' # (<99# (, ( % 9 #

! . (. 7( , ( % *

* * ( , % % , (

# ( <!

!, * ! ( 9 ). ; x5.3 * !* ( ( (#9 ) . $ < 9 x5.1 x5.2 *% ( ! .

4.1 . ) $ # % . (

x_ = A(t)x + B (t)u + Q(t)f I x(t0) = x(0)

(4.1.1) f (t) { - % ( ' ! %, % ( *% # " % '". ) , * * (

M ff (t)g = 0: 5 #

# < #

(4.1.2)

Rf (t0 t00) = M ff (t0)f 0(t00)g = R(1)(t)(t0 ; t00) (4.1.3) R(1)(t) { -

# , ! ( " '" t0 . * x(0) ( ( n (0)o *% ( ,

' ! % M x = 0 ( #

# n o M x(0)x(0)0 = R(0): (4.1.4) $ ( % 8Zt1 9 < 0 = J = M : Mx Q(t)x + u0u] + x0(t1)P (1)x(t1) (4.1.5) t0

Q(t) { -

#. , % u(t) (( 9 (# ( % '% 9 # x(t) , ( (4.1.5) ' * . 93

,( (( ( 9 # x(t) *% % ! ( , 9 ) (" ) . 4

( 9(, * * " '" (4.1.1)

, ( * ( x 3.1) ' ! %. @ ' * ( . &9 < M1.4], M3.7].

5 6 !* ( . ( (4.1.1), ( 9 (# (4.1.5) ' * ,

u = C 0(t)x

(4.1.6)

C (t) = ;P (t)B (t)I P (t) { ' * $((

(4.1.7)

;P_ (t) = P (t)A(t) + A0(t)P (t) ; P (t)B(t)B 0(t)P (t) + Q(t)

(4.1.8)

P (t1) = P (1):

(4.1.9)

(

) * 9 (# (4.1.5) (4.1.6) 8 9 Zt1 < = tr :P (t0)R(0) + Q(t)R(1)(t)Q0(t)P (t) dt : (4.1.10) t0

(; < trA * ( % # A . , n X trA = aii i=1

ii (i = 1 n) { < # A .) $ # % *%, ( # , ! . ( (4.1.1), 9 (# (4.1.5)

, # " '" ! ( # % * R(1) . 4 ' * 9 (# # 8 9 Zt1 < = min J = tr :P (t0)R(0) + QR(1)Q0P (t) dt : (4.1.11) t0

94

; ! (* (! *! 9 (# (. , 9 (# # t1 ! 1 * 9 (# min J = trP 0R(0) + (t1 ; t0)QR(1)Q0P 0 (4.1.12) P 0 { ' ' * $(( (3.1.15). * , * t1 ! 1 * min J ! 1 . * % <% # *

< *% # " % '", < t1 ! 1 9 (# (4.1.11) 9 (# 8Zt1 9 < 0 1 0u) dt + x0(t1)P (1)x(t1)= : J = t1lim M ( x Q ( t ) x + u (4.1.13) !1 t ; t : 1

0

t0

7 # ! < 9 (# (( 8Zt1 9 < 0 1 0u) dt= : J = t1lim M ( x Q x + u !1 t ; t : 1

0

t0

(4.1.14)

. # ). $-

, (3.1.23), ( ! f (t) { # % (1) *% % # " % '" r11 = 103 . * % ( ' (* #, ( ( *% ! OY . , % ( , ( 9 (# 8Zt1 9 < 2 = 1 2 2 2 J = t1lim (4.1.15) !1 t ; t M : q11x1 + q22x2 + q33x3 + u dt 1

0

t0

' * . 9 (# # (3.1.55), *

31 = 10;3 (4.1.16) (3.1.24), c1 , c2 , c3 ' (3.1.58). ; * * 9 (# (4.1.15) . (4.1.12) t1 ; t0 t1 ! 1 ,

95

0 (1) (1) 0 0 min J = trQR Q P = tr 0 r11 0 0 31 31 0 0 0 = tr 0 0 0 0 0 r11(1)312

p011 p012 p013

p012 p022 p023

p013 p023 p033

p011 p012 p013 p012 p022 p023 = p013 p023 p033

(4.1.17)

= r11(1)312 p033 :

,( (( (3.1.57) p033 = 116 106 , ( * min J = 116 10: D * %

(4.1.18)

(4.1.19) q11x21 + q22x22 + q33x23 + u2 = 116 103 x2 (i = 1 2 3) { xi , u { u . @ < , * # ( (%

! . i

4.2 ) , - . 4.2.1 '

. ( (4.1.1)

, ( , !. , . (

x_ = A(t)x + B (t)u + Q(t)f I x(t0) = x(0)I

(4.2.1)

y = D(t)x + (t)

(4.2.2) , (( , f (t) { - % ( ' ! %, % ( *% # " % '" * (

% ( #

% # % R(1)(t) I A(t), B (t) , Q(t) {

# I y(t) { r - % ( !

!I (t) { < r - % ( ! ('), ( % *% # " % '" * ( ( #

% # %

R(2)(t0 t00) = M f(t0)0(t00)g = R(2)(t)(t0 ; t00) (4.2.3) R(2)(t) {

-

# r r . 96

7 , * ' ! % ( (

). 4( #, * h ih i0 (0) (0) (0) (0) M fx(t0)g = xO I M x(t0) ; xO x(t0) ; xO =R (4.2.4) , * * % !, ( xO (0) # R(0) n n . , % u , ( y , ( , * ( % 8Zt1 9 < 0 = J = M : Mx Q(t)x + u0u] dt + x0(t1)P (1)x(t1) (4.2.5) t0

Q(t) , P (1) {

-

# , ' * . $ , 9 % ( , (((

* ,

% ) ! * %: %, % ( (4.1.6), ( (

! # # ( #^ , % { . 5(

* ,

x^_ = A(t)^x + K (t) My ; D(t)^x] + B (t)u

(4.2.6)

( # K (t) 9 (#

J = M fe0q(t)eg (4.2.7) ( q(t) {

-

#) '( e = x ; x^ ( , 9 #). ( # K (t) (4.2.6) ) * ( ) ). ,( , # ! # K (t) (4.2.7) ( , ( % ! * %. , * % (4.2.6) % ! < * (! %. ; % * M4.6] $.5 > , ( %' B.4.5 4.; M4.4], M4.5] % 9 #.

97

4.2.2 ) %* 5 6. 0 # K (t) (4.2.6), ( % (4.2.7) * ,

;1 K (t) = Pe (t)D0(t)R(2) (t) (4.2.8) Pe (t) { # n n, ' $((

Pe (t) = A(t)Pe(t)+ Pe (t)A0(t) ; Pe(t)D0(t)R(2);1 (t)Pe (t)+Q(t)R(1)Q0(t) t t0 (4.2.9) *

Pe (t0) = R(0): (4.2.10) 4* (4.2.6)

x^ (t0) = x(0):

(4.2.11)

7( < 7( 3. 4 (4.2.6), ( # K (t) * ' (4.2.8) (4.2.11), * : - 8* <! ' % M4.6]. 4 ! ' * (B5$) % 9 #. 7 % , (3.1.47), (3.1.48), (3.1.46), (4.2.8), (4.2.9), (* , * K (t) = ;C (t) , D0 (t) = B (t) , Pe (t) = P (t) , R(2) = Q(1) , A0(t) = A(t) , Q(t) = R(1)(t)Q0(t) = Q(t) (4.2.12) < * ( % ( ! %. (; * < ( * % t1 , * { * % t0 ). D ! %

( ) * . P , * # (<99# ' $(( (4.2.9) " " , (( * < ' " " . ; # * (4.2.1), (4.2.2)

x_ = Ax + B u + Qf I y = Dx + 98

(4.2.13)

*% # f (t) , (t) " % '" ! (

( #

# R(1) R(2) . 0 # K ((

x^_ = Ax^ + K My ; Dx^ ] + B u

(4.2.14)

;1 K = Pe D0 R(2) (4.2.15) P! { # * ( n n ) ' * (

APe + Pe A0 ; Pe D0R(2);1 DPe + QR(1)Q0 = 0 (4.2.16) ( ! (( ' ' 99 # (4.2.9) ( ( A(t) = A , D(t) = D , R(1)(t) = R(1) , R(2)(t) = R(2) t ! 1 . ,(%

9 (# 0 J = tlim (4.2.17) !1 M fe qe(t)g : , *, (( # * , # K # q 9 (# #. . % . ( (??), (??),

*% ' , * ! !. (??), (??) < *

x_ 1 = x2 + b11u + 11f I x_ 2 = b21u + 21f1I

(4.2.18)

y = x1 + (4.2.19) f (t) , (t) { *% # " % '" r11(1) > 0 , r11(2) > 0

. 4 , % 9 (# n2 o 2 (t) J = tlim M e ( t ) + e (4.2.20) 1 2 !1 (e1 = x1 ; x^1 , e2 = x2 ; x^2 (4.2.14) x^_1 = x^2 + k11(y ; x^1) + b11u x^_2 = +k21(y ; x^1) + b21u ( ! (<99# k11 , k21 ! ' %

pe11 , pe12 , pe22

(4.2.21)

1 k11 = pe11 pe12 (4.2.22) 0 pe12 pe22 k21 ' * (4.2.16) 99

0 1 0 0

0 0 1 0

;

pe11 pe12 + 11 r(1) = 0 0 : pe12 pe22 21 11 11 21 0 0 (4.2.23) ; % 9 < ' ((

;

pe11 pe12 pe12 pe22

pe11 pe12 + pe11 pe12 pe12 pe22 pe12 pe22

1 0

1 0

2pe12 ; p2e11 + 112 r11(1) = 0I

pe22 ; pe11pe12 + 1121r11(1) = 0I ;p2e12 + 212 r11(1) = 0: @ * q pe12 = 212 r11(1): < %, * rq pe11 = 2 212 r11(1) + 112 r11(1)I , (* , * rq q (1) (1) (1) 2 2 pe12 = 2 21r11 + 11r11 212 r11 ; 1121r11(1): @( k11 = pe11I k21 = pe12:

(4.2.24) (4.2.25) (4.2.26) (4.2.27)

4.2.3 : . ; ( * !* ( % 9 # (

! , , * ' ( # % ' * !* ( % 9 # (

! ' * . &9 < .

100

: ( ). 9 (# (4.2.5) !* ( . ( (4.2.1), (4.2.2)

u = C 0(t)^x(t)0

(4.2.28) C 0(t) { # (<99# , ' (4.1.7) : : : (4.1.9), ( * 9 (# (4.2.5) !* ( ( . ( (4.2.1)I ( x^(t) { < n - % (

! 9 (# (4.2.7) (4.2.6), # K (t) (<99# ( (4.2.8), (4.2.9). 7( 7( 4.

4.2.4 & # )# %< $ . ( , % %

x_ = '(x u t) + f I x(t0) = x(0)I

(4.2.29)

y = w(x t) +

(4.2.30) '(x u t) w(x t) {

n - r - ( -9 (# ! I f (t) (t) { *% # " % '" ( #

# (4.1.3), (4.2.3)I x(0) { *% % ( , ! ( % (4.2.4). ( y % ( . ( x . 7 ' <% *

% 9 '

% 9 , ( < * ( ( % 9 #) % ! . (. $ * ) . , * ( x(t) , u(t), ' (4.2.29)

( x(t) f (t) = 0 . P ( y = y ; y

* ( y = w(x t) ( x = x ; x , ( # ( x^ = x^ + x^ , x^ ' (4.2.6), (4.2.9) : : : (4.2.11), ( :

x^_ = A(t)x^ + K (t) My ; D (t)x^] I

(4.2.31)

K (t) = Pe (t)D0 (t)R(2);1 (t)I

(4.2.32)

P_e (t) = A(t)Pe (t) + Pe (t)A0 (t) ; Pe (t)D0 (t)R(2);1 D (t)Pe(t) + R(1)I Pe (t0) = R(0) 101

(4.2.33)

( ! < aij (t) , dkj (t) (i j = 1 n k = 1 r) # A(t) D(t) ((

@'i k aij (t) = @x (4.2.34) I dkj (t) = @w @xj x=x (i j = 1 n k = 1 r): j x=x u=u $ . > , * ! . ( u = 0 . ( % t * # ( x^ ( % . ( (4.2.29), (4.2.30). $ ( -9 (# '(x u t) w(x t) , % ( ! <% # ( * * < :

'(x t) (^x t) + A^(t)(x ; x^ )I

(4.2.35)

w(x t) w(^x t) + D^ (t)(x ; x^ )

(4.2.36)

< aij (t) , dkj (t) # A^(t) D^ (t)

@'i I d^ = @wk a^ij (t) = @x kj @xj x=^x (i j = 1 n k = 1 r): j x=^x & * <! % ' (4.2.29), (4.2.30)

x_ = A^(t)x + ^ (1) + f I y = D^ (t)x + ^ (2) +

(4.2.37) (4.2.38)

^ (1) = '(^x t) ; A^(t)^x ^ (2) = w(^x t) ; D^ (t)^x:

(4.2.39) % ". (" (4.2.38) (4.2.6) h i (4.2.40) x^_ = A^(t)^x + K (t) y ; D^ (t)^x ; ^ (2) + ^ (1) x^ (t0) = x^ (0) # K (t) ' (4.2.32) (4.2.33), ( ! D (t) D^ (t) , A(t) { A^(t) . , *, * (4.2.39), (4.2.40)

x^_ = '(^x t) + K (t) My ; w(^x t)] : 4.2.5

(4.2.41)

%

.

-9 (#:

1) Mkest K Pe] = kalman(A B D E Q R1 R2 ) - 9 9 5 (4.2.14). 102

2) Mrlqg] = lqgreg(kest C 0) -9 LQG - ( % -( * ( ) 9 5 kest # C 0 , *

% 9 (# lqr

4.3 # 4.3.1 . $ ( % . (

x(k + 1) = R(k)x(k) + R(k)u(k) + Q(k)f (k) (k = 0 1 2 : : :)

(4.3.1) f (k) { - % ( ' ! %, % f (0) , f (1) , f (2) : : : (

! !c* (! * # R(1)(k) (k = 0 1 2 : : :))I R(k) , Q(k) , R(k) (k = 0 1 2 : : :) {

# . ( % (X ) N 0 0 0 (1) J=M x (k)Q(k)x(k) + u (k ; 1)u(k ; 1) + x (N )P x(N ) (4.3.2) k=1

Q(k) (k = 0 1 2 : : :) , P (1) {

-

# . , % u(k) (( 9 (#

! , ( 9 (# (4.3.2) ' * . @( , (( * , , *

x3.1 ' ! %%. &9 < M3.11].

103

5 6. !* ( ( . ( (4.3.1), ( ( % (4.3.2) ' * ,

u(k) = C 0(k)x(k) (k = 0 1 2 : : :)

(4.3.3)

C 0(k) = ;fR0(k)MQ(k + 1) + P (k + 1)]R(k) + Em g;1 R0(k)MQ(k + 1) + P (k + 1)]R(k): (4.3.4) # P (k) (k = 0 1 2 : : :) ' *

P (k) = R0(k)MQ(k + 1) + P (k + 1)]MR(k) + R(k)C 0 (k)] (k = N ; 1 N ; 2 : : : 1) (4.3.5) ( *

A(N ) = P (1):

(4.3.6)

4 , * (4.3.4) (4.3.5) P (1) = 0 , R(k) = R , R(k) = R , k = N ; j , (4.3.3) : : : (4.3.6) (3.1.96) : : : (3.1.99). ; # * , ( # , ! . ( (4.3.1), 9 (# (4.3.2)

, * N ! 1 9 (# (X ) N 1 0 0 J = Nlim (4.3.7) !1 N M k=1 x (k)Q(k)x(k) + u (k ; 1)u(k ; 1)

x(k + 1) = Rx(k) + Ru(k) (k = 0 1 2 : : :)I

(4.3.8)

u(k) = C 0x(k) (k = 0 1 2 : : :)

(4.3.9) ( % # C , ((

* , ' (3.1.98) : : : (3.1.96).

4.3.2 ) ) %* ( ) , A D*) $ . ( , %

x(k + 1) = R(k)x(k) + R(k)u(k) + Q(k)f (k) (k = 0 1 2 : : :) x(0) = x(0)I 104

(4.3.10)

y(k) = D(k)x(k) + (k) (k = 0 1 2 : : :)

(4.3.11) f (k) (k) , (k = 0 1 2 : : :) { (

! ( ! !* (! *

# % R(1)(k) R(2)(k) . )

R(1)(k) = M ff (k)f 0(k)g I R(2)(k) = M f(k)0(k)g : x(0) { *% % ( , (

% ( f (k) (k) , < M fx(0)g = xO (0) M fMx(0) ; xO (0)]Mx(0) ; x(0)]0g = R(0) xO (0) R(0) . , % % ( , 9 #), ! ( # ( x^ (k) ( % x(k) . < ( % J = M fe0(k)q(k)e(k)g (4.3.12) M q(k) (k = 0 1 2 : : :) {

-

# , e(k) = x(k);x^(k) ] ' * .

105

5 6. % ( (4.3.12) ( %

, 9 #) . ( (4.3.10), (4.3.11)

x^ (k + 1) = R(k)^x(k) + K (k) My(k) ; D(k)^x(k)] + R(k)u(k)I x^ (0) = x^ (0)

(4.3.13)

( # K (k) (k = 0 1 2 : : :) (

' :

h i;1 K (k) = R(k)Pe (k)D0 (k) R(2)(k) + D(k)Pe (k)D0(k) (k = 0 1 2 : : :)I

(4.3.14)

Pe (k + 1) = MR(k) ; K (k)D(k)] Pe R0(k) + Q(k)R(1)(k)Q0(k) (k = 0 1 2 : : :) (4.3.15) *

Pe (0) = R(0): 4* (4.3.13)

(4.3.16)

x^ (0) = xO (0)

(4.3.17) 0 # Pe (k) n n # % '( e(k) = x(k) ; x^ (k). 7 *

M fe0(k)q(k)e(k)g = tr MPe(k)q(k)] (k = 0 1 2 : : :):

(4.3.18)

7( < * * M3.7]. 3 (4.3.13), ( ! y(k) y(k + 1) . ,(%

x^ (k+1) = R(k)^x(k)+K (k+1) My(k + 1) ; D(k + 1)R(k)^x(k) ; D(k + 1)R(k)u(k)]+R(k)u(k) (4.3.19)

h i;1 K (k + 1) = R(k + 1)Pa (k + 1)D0(k + 1) D(k + 1)Pa(k + 1)D0 (k + 1) + R(2)(k + 1) I (4.3.20) Pa(k + 1) = R(k) ME ; K (k)D(k)] Pa(k)R0(k) + Q(k)R(1)Q0(k): (4.3.21) ; # * . ( (4.3.10), (4.3.11) 106

x(k + 1) = Rx(k) + Ru(k) + Qf (k)I y(k) = Dx(k) + (k)

(4.3.22)

%

x^ (k + 1) = R^x(k) + K My(k) ; Dx^ (k)] + Ru(k)

(4.3.23)

h i;1 K = RPe D0 R(2) ; DPe0 D0 (4.3.24) Pe0 { # * ( n m ), ' Pe0 = MR ; KD] PeR0 + QR(1)Q0: 0 # $0e ! (( ' '

(4.3.25)

h i;1 Pe (k + 1) = RPe (k)R0 ; RPe (k)D0 R(2) + DPe (k)D0 DPe (k)R0 + QR(1)(k)Q0 Pe (0) = R0 (4.3.26) k ! 1 . @ . Pe0 = klim !1 Pe (k): . $ , (3.1.23): x_ 1 = x2 x_ 2 = a22x2 + a23x3 x_ 3 = a32x2 + a33x3 + b31u + 31f (4.3.27) 4.1.2 31 = b31 = 10;3 , f (t) { *% % (% # " % '" * ( ( #

% 9 (# r(1) = 106 . # x1 *( # *% % (% !% " % '" * ( ( #

% 9 (# r(2) = 104 . ,( , y = x1 + : *

(4.3.28)

x1(0) = x2(0) = x3(0) = 0: (4.3.29) , (9 )

! x1 , x2 , x3 , % * <!

! , , 2, , 3, , : : : , k, , , = 0 01 . < % '( 107

n o J = M Mx1(kT ) ; x^1(kT )]2 + Mx2(kT ) ; x^2(kT )]2 + Mx3(kT ) ; x^3(kT )]2 ' % ( kT . ! ( ' <% *, ' ( 9 x1(k + 1) = x1(k) + '12x2(k) + '13x3(k) + r1(u(k) + f (k))I > = x2(k + 1) = '22x2(k) + '23x3(k) + r2(u(k) + f (k))I > x3(k + 1) = '32x2(k) + '33x3(k) + r3(u(k) + f (k))I

(4.3.30)

(4.3.31)

y(k) = x1(k) + (k) (4.3.32) * ('ij , ri , (i j = 1 3)) ( % (3.1.122). ; (4.3.23). : x^1(k + 1) = x^1(k) + '12x^2(k) + '13x^3(k) + r1u(k) + k11 My(k) ; x^1(k)] I

(4.3.33)

x^2(k + 1) = '22x^2(k) + '23x^3(k) + r2u(k) + k21 My(k) ; x^1(k)] I

(4.3.34)

(4.3.35) x^3(k + 1) = '32x^2(k) + '33x^3(k) + r3u(k) + k31 My(k) ; x^1(k)] : 4 k11 , k21 , k31 <! ! (4.3.24):

k11 1 '12 '13 p0e11 h i;1 (4.3.36) k21 = 0 '22 '23 p0e12 r(2) ; p0e11 : 0 k31 0 '32 '33 pe13 7 ! * p0e11 , p0e12 , p0e13 (4.3.26) * Pe (1) ( < Pe (0) = 0 (4.3.29)), Pe (2) : : : ( * ', ( Pe (k + 1) Pe (k) , Pe (k) = Pe0

(4.3.37)

4.3.3 : . $ . ( , % (4.3.10), (4.3.11). % , ( y ! . (, ( , * ! . ( 9 (#

J=M

(X N k=1

) 0 0 0 (1) x (k)Q(k)x(k) + u (k ; 1)u(k ; 1) + x (N )P x(N ) 108

(4.3.38)

Q(k) P (1) {

-

# . 5( * , ' <% * # .

5 6. ( # ). 9 (# (4.3.38) !* ( . ( (4.3.10), (4.3.11)

u(k) = C 0(k)^x(k) (k = 0 1 2 : : :)

(4.3.39) C 0(k) (k = 0 1 2 : : :) { # (<99# , ' (4.3.4) ... (4.3.6), ( * 9 (# (4.3.38) !* ( ( . ( (4.3.10)I ( x^ (k) { n- % (

! 9 (# (4.3.12) (4.3.13), # K (k) (<99# ( ' % (4.3.14), (4.3.15). 7( < * *. ; # * . ( (4.3.22)

u(k) = C 0x^(k)

(4.3.40) # C 0 , ((

* , ' (3.2.4) ... (3.2.8), ( x^ (k) ! (4.3.23), ( # K

! ' % (4.3.24) ... (4.3.26). . % #9 % , % (4.3.31), (4.3.32). #9 , * ! , ( % < , 9 (# (X ) N 10 2 2 J = Nlim (4.3.41) !1 M k=1 10 x1(k) + u (k ; 1) : ; # ( % % (3.1.120), (4.3.33) ... (4.3.35). <! % : (3.1.120) * 4.1.3 ' * # ' ! %% ! x1 , x2 , x3 , (4.3.33) ... (4.3.35) 9 ' * !

! x2 x3 .

<% * * # * %: 1) ' % ! ( *% # " % '"I 2) ' % ! ( (

)I 3) # R(1)(t) R(2)(t) { (

) . . 109

5 <! * % M?], M?]. ,(,

!* ( , ( ( f (

, # R(2)(t) . 5 ' % ! " '", ( *% # (

#

#, 9 (#% (# ' ), ( # (( ! *% # " % '" * % * ( . ; * , * < * , *

f (t) = L(1)(t)z + ~f (t)I (t) = L(2)(t)z + ~ (t)I

(4.3.42)

z_ = A~(t)z + (t)

(4.3.43) ~f (t), ~ (t) , (t) { ( *% # " % '"I # L(1)(t) , L(2)(t) , A~(t)

(

#

# # f (t) (t) . (4.3.42), (4.3.43) */ . . (4.2.1), (4.2.2) 9 9 , *

x + B (t) u + ~f (t) I z 0 (t) y = D(t) L(2)(t) xz + ~ (t)

x_ = A(t) Q(t)L(1)(t) z_ 0 A~(t)

*% # " % '".

4.3.4

%

.

-9 (#:

1) Mkest K Pe Z ZO ] = kalman(Phi R D E Q R1 R2) - 9 ( 9 5 (4.3.23), Z ZO { ( #

# '( # . 2) Mrlqg] = lqgreg(kest C 0) -9 LQG - ( % -( * ( ) ( % 9 5 # C 0 , * % 9 (# dlqr .

5 H1 '-$#%& '$ )* * . ( 9 7. ) % M5.1] c H1 * % # . 110

7 ! H1 -< ( * % * % 9 (# * ( * . 4 ,( ( -< H1 * % 9 (#, %

% . @ H1 M5.2], M5.3] H1 , * <% , 9 (#% ( (

( 4 ! , # 4 (). M5.4] ,M5.5] H1 - , % 2-$(( !,( %

% *% , ( ' ! * 9 (# * ( ! ( (.D ! . ; 5.1 H1 9 * H1 , # ' ( % 5.2.$ 5.3 <% # * . ( , . ( 3.3, .

5.1 A ) . - 5.1.1 - $ #,

x_ = Ax + B u + Qf y = Dx + w = N x

(5.1.1)

x_ = A x + B y u = D x + F y:

(5.1.2)

(s) = P11(s)Of + P12(s)u y = P21(s)Of + P22(s)u

(5.1.3)

< * ( %* f = 0 w = 0 . # T f (s) ,h ( !

! (t) i T ( % Of (t) = f T (t) wT (t) ,

' ! %% !. . ( (5.1.1) (( ;1 ;1 P11(s) = N h (Es ; A) ;1QO i P12(s) = N (Es ; A);1 B O P21(s) = D (Es ; A) Q Er P22(s) = D (Es ; A) B Q = MQ On# ]

(5.1.4)

(( (( * . #: Q % # n w # D (Es ; A);1 Q * % # % r r ). 111

(5.1.2) '

u = K(s)y

(5.1.5)

u = Tuf (s)Of

(5.1.6)

K (s) = D (Es ; A );1 B + F . (5.1.5) (5.1.3) ( * u = K (s)P21(s)Of + K (s)P22(s)u

Tuf (s) = ME ; K (s)P22(s)];1 K (s)P21(s) (5.1.7) @ (5.1.6) ' (5.1.3) ( ((

(s) = T f (s)Of

( # T f (s)

T f (s) = P11(s) + P12 ME ; K (s)P22(s)];1 K (s)P21(s) (5.1.8) s = j! , T f (j!) { * * # (5.1.1), (5.1.2) 7 9* ( <% * % # (# (5.1.1), (5.1.2) * ( Of (t) = Of s sin !f t (5.1.9) f s {

% O - % ( * (O = + r) , !f { * . ; ' (t ! 1)

i(t) = ai(!f ) sin(!f t + 'i) (i = 1 m): (5.1.10) 4% ai(!f ) (i = 1 m)

! ( % < # T f (j!f ) . j! t ;j! t * , * sin !f t = e ;2je , * (# % ej! t e;j! t (( f

f

f

f

+ = T f (j!f )Of sej! t ; = T f (;j!f )Of s e;j! t: + ; 4 ( , * (t) = 2;j . ; ( q(!f ) + j(!f ) = T f(j!f )Efs ' h i h i + = qO(!f ) + j O (!f ) ej! t ; = qO(!f ) ; j O(!f ) e;j! t: f

f

f

f

112

(5.1.11)

(5.1.12)

@ < , * ! % q i (t) = qi2(!t) + 2i (!t) sin(!f t + 'i) (i = 1 m): ( ,

! ( % (5.1.10) < * % * % # (( q ai(!f ) = qi2(!f ) + 2i (!f ) (i = 1 m): (5.1.13) ) ( , * (5.1.12) (5.1.13)

a2i (!f ) = i+i; (i = 1 m):

5.1.2

(5.1.14)

H1 - ) ) -

; *

# * T m O . P < # ( (O = m) ,

* i MT ] (i = 1 m) ! (( ( d(s) = det(Es;T ) . P (O 6= m) ( (m O) , (% , < ! * iMT ] (i = overline1 O) , * ( ( % # , (( q iMT ] = iMT T ] (i = 1 ): (5.1.15) T { ( ( -

T

#. (P T = T1+jT2 , T = MT1 ; jT2]T = T1T ; jT2T ). # T 9 (# %qj! : T (j!) = T1(!) + jT2(!) , * ! : iMT (j!)] = i MT T (;j!)T (j!)] (i = 1 ) ; ( * % # , ' h i r h T i i T f (j!) = i T f (;j!)T f (j!) (i = 1 ) (5.1.16) H1 % * % # T f (j!) , * % (( T f (j!)1 , (( T f 1 , * h h i h ii T f (j!)1 = 1max sup T ( j! ) : : : T ( j! ) : (5.1.17) m f im 0!1 1 f 1* (% < * (* . (

(m = 1) . ; < * 1 T f (;j!)T f (j!) = 2 T f (j!) = a21(!) , a1(!) { ( % %

% ,

% fO1 = 1 sin !t , H1 * % 9 (# T f 1 = sup0!1 ja1(!)j ' ( * ( % * ! * ! * ! . 113

; * 9* (% H1 ( %. ') P (5.1.1), (5.1.2) * ( (5.1.9), ' ( ! ( ! m X a2i (!f ) T f (j!f ) i=1 (5.1.18) 1 X s 2 (fOk ) k=1

! * (0 !f < 1) . 7( < % 7( 5.

5.1.3 & H1 - $ (, * ( %* ( ~f = 0)

x~_ = A~x~ + Q~ ~f = N~ x~ :

(5.1.19)

P * #

T~(s) = N~ (Es ; A~);1Q~ : (5.1.20) 5 6 H1 * % # T~(s) T~1 < ( > 0) (5.1.21) ( # " ~ ;2 Q~ Q~ T # A = = ;N~ T N~ ;A~T (5.1.22)

! * % : Re iM=] = 0 (i = 1 2~n) . ' 3 % * = min ( * > 0 ,

! * = * !,

, ' , *

* # = ( ! , ( ( min ,

(, ( * Re iM=] = 0 (i = 1 2n) ,

kT k1 = min

( > 0)

(5.1.23)

,( , H1 * % # * min ! ( 9 (# iMT (j!)] (i = 1 n) ! * ! . 114

5.1.4

. H1

$ . ( (5.1.1), ( ' ! , *

% < %. * , * Z1 Of T (t)Of (t) dt < 1: (5.1.24) 0

O ; < * ( , h T *T iLT 2 9 (# f (t) * . ; ( ( z = u * # Tzf (s) , ( z Of , h iT Tzf (s) = T Tf (s) TuTf (s) (5.1.25) 9 7

. ( (5.1.1) % (5.1.2) (%, * H1 * % # Tzf (j!) <% ' * Tzf 1 = min: (5.1.26) * <% * * B5$ !* (% # , * Of (t) { %, % ( . ; * B5$ ' , * 9 (#, !* (% # < *% # * ( %. O 4 (5.1.24) * , * Of (t) { * 9 (# (tlim !1 f (t) = 0) . ( . D % ( % # ( H1 * % # ) < * % ' , *

.

5.2 5.2.1 * B5$ *

% < %. . ( (3.1.1) < *

x_ = Ax + B u + Qf :

(5.2.1) ,( (( f (t) 9 (#, ( !' 9 (# Z1 J = xT Qx + uT u ; 2f T f dt (5.2.2) 0

115

( {

*, Q { -

#. )* ( , * % u(t) , ( < 9 (# f (t) , ( . @( !'

u = C T x C T = ;B T P

(5.2.3)

f = Kf x Kf = ;2 FTP (5.2.4) - # P ' $(( PA + AT P ; PBB T P + ;2 P QQT P + Q = 0: (5.2.5) ; ' % (5.2.3) { (5.2.5) % (3.1.2), (3.1.15), (3.1.16),

% 7( 2. ! 1 (5.2.5) $(( (3.1.15) # B5$. # * <! % $(( , * * -

# P , ' (5.2.5). & ( * = min , ( P > 0 < min # P (

% < P > 0 * min 1 . 4 ( , * (5.2.1), (5.2.3), (5.2.4) * ( %*: # Ac = A+BC T +QK (s) { # ( * #

*) < !' { ! 9 (# *

% < %.

5.2.2 5 , % *0 * H1 - $ , %

x_ = Ax + B u + Qf + K (y ; Dx ) f = Kf x

(5.2.6)

u = C T x

(5.2.7)

C T = ;B T P K = E ; ;2Pe P Pe DT Kf = ;2QT P: (5.2.8) 5 6 P ( # -

# P Pe , $(( 116

PA + AT P ; PBB T P + ;2P QQT P + N T N = 0 APe + PeAT ; Pe DT DPe + ;2 Pe N T NPe + QQT = 0

(5.2.9) (5.2.10)

max MPPe ] < 2 (5.2.11) maxM0] { (

* # 0 , H1 * % # (5.1.1), (5.2.6) { (5.2.8) Tzf 1 < : (5.2.12) 7( 7( 6. ! 1 $(( (5.2.9) (5.2.10) (3.1.15) # B5$ (4.2.16) 9 5 ( R(2) = E , R(1) = E ) , (5.2.6), (5.2.7) (4.2.28), (4.2.6) !* ( . P ( x . ( (5.1.1) * , ((

u = C T x C T = ;B T P

(5.2.13) # P 0 (5.2.9), ( ( (5.2.3), (5.2.5). )' (5.2.6) { (5.2.8) 9 (5.1.2). @(* (5.2.6)

u f , (5.2.6), (5.2.8), * #

A = A ; BB T P + ;2QQT P ; KD

(5.2.14)

B = K D = B T P F = 0:

(5.2.15)

,

x_ 1 = x2 + b11u1 +

11f1

x_ 2 = b21u1 +

21f1

y1 = x1 + 1 1 = x1

(5.2.16)

x_ 1 = a 11x1 + a 12x2 + b 11y1 x_ 2 = a 21x1 + a 22x2 + b 21y1

(5.2.17)

u1 = a 11x 1 + a 12x 2 + f y1:

(5.2.18) ! "# , $ % &'( (5.2.16) ) * (5.2.17), (5.2.18) Tzf (j! ) , % $ ( z = +1 u1 ] ( % $ f- = +f1 1] * !# % 2 2 .

117

. , & * &'( (5.2.16) % ("//! # ("//! ) * (5.2.17), (5.2.18) (, & **0

Tzf 1 < :

(5.2.19) 2 (5.2.6) { (5.2.8) 4 "# % 4

p11 p12 c11 c12 = ; b11 b21 p p 12 22 p p 11 12 kf 11 kf 12 = ; ;2 11 21 p p 12 22 " k11 = 1 0 ; ;2 pe11 pe12 p11 p12 k12 0 1 pe12 pe22 p12 p22

# pe11 pe12 pe12 pe22

1 0

( * pij , peij (ij = 1 2) * 4 (( (5.2.9), (5.2.10). 6( ("//! 7 (5.2.13) ((

" 0 1 b c b c k k ; k11 11 11 11 12 11 f 11 11 f 12 ; 2 A = 0 0 ; ; 21kf 11 21kf 12 k12 b21c11 b21c12 a 11 = ;b11c11 ; ;2 11kf 11 ; k119 a 12 = 1 ; b11c12 ; ;2 11kf 129 a 21 = ;b21c11 ; ;2

21kf 11 ; k119

a 22 = ;b21c12 ; ;2

#

21kf 129

b 11 = k119 b 21 = k219 d 11 = c219 d 12 = c129 f 1 = 0:

5.2.3

-

5.2.2 , ' * 6.1.4 ! '! * % . ( * min) ! #% # 1. ) ( * = (0) . # 2. $ ' $(( (5.2.9) # # P . P P 0 , ! ( % #, * ( # 1 * . # 3. $ ' $(( (5.2.10) = (0) . P Pe 0 , ! ( % #, * ( # 1 > (0) . 118

# 4. (5.2.11). P , ( # 1 ' ( < (0)) . ; * ( # 2 * ( > (0)) . $ ' $(( #! 2 3 #. 7 < # .3.1.4, .6.1.3.

" ;2 QQT ; BB T # A =$ = (5.2.20) ;N T N ;AT " T ;2 T TD # A N N ; D = = (5.2.21) QQT ;A * ! * .. P <! * ! * , ' , ( % (Re i(=$) = 0 , Re i(= ) (i = 1 2n)) , < , * # P Pe ' #

.

5.2.4 % - (: 131 ( H1 - ).

@!

: ) . ( (5.1.1). )0 # D N % ! !

!. ) % . ); ! ! # . $ : )0 # C 0 K (5.2.6),(5.2.7) )= 9( ! ! #

. @ ( 131 (( .2.3. # % ! ( % ( '! '( , , ) .

5.3 . : 5.3.1 ' . ' * * ,

% 3.3, K ! . (. , . ( (( ' % ), * ! . ; < * (5.1.1) 119

x_ = Ax + B u + Qf y = Dx + G = N x

(5.3.1)

x_ = A x + B y u = D x + F y

(5.3.2) ( ! ( ( ! * ( 9 (#

fi (t) = wj (t) =

p1 X i=1 p2 X q=1

fik sin (!k t + 'ik ) k = 1

(5.3.3)

wjq sin (~!q t + '~jq ) j = 1 r

(5.3.4)

* !k !~q 9 'ik '~jq (k = 1 P1 , (q = 1 P2 , i = 1 , j = 1 r)

, *

p1 X k=1

fik2

p2 X 2 fi (i = 1 ) w2jq wj 2 (j = 1 r) q=1

(5.3.5)

p1 , p2 , fi wj (i = 1 , j = 1 r) {

*. B * ' '(

, ' * ((

uj = tlim (5.3.6) !1 sup juj (t)j (j = 1 m) )* * ! # (5.3.2) (!, * (5.3.1), (5.3.2) j i uj ui (i j = 1 m) (5.3.7) i ui (i j = 1 m) -

*. ) , * ' <% * . ; < *

% (5.3.2) * > 1 u > 1 ( , * j i uj ui u (i j = 1 m) $ , '% < *

x_ = Ax + B u + Qf +K (y ; Dx ) u = C T x f = Kf x

(5.3.8) (5.3.9)

;1 C T = R;1 B T P Kf = ;2LQT P K = E ; ;2Pe P Pe DT (5.3.10) # -

# Pu Pe $(( 120

PA + AP ; PBR;1BP + ;2P QPLQT P + N T Q0N = 0

(5.3.11)

APe + Pe AT ; Pe DDT Pe + ;2Pe N T Q0NPe + QLQT = 0

(5.3.12)

MPPe ] 2 (5.3.13) Q0 , R L { - # m m , m m {

. D ! ! # (, * ( ' % * , # L , , * %. $(( (5.3.11), (5.3.12) (5.2.9), (5.2.10) * ! ! #!. 5 6 P (<99# ! # Q0 R qij(0)

0 1 r X X 2 2 (p1 + p2 ) @ fi + wj A i=1

j =1

rij(0)

0 1 r X X 2 2 (p1 + p2) @ fi + wj A i=1

j =1

(i = 1 m) (5.3.14) ' (5.3.1), (5.3.9), # ( % * 9 (5.3.10) { (5.3.12) (5.3.13), % yi 2

ui 2

j i 2 uj ui 2 (i j = 1 m): (5.3.15) @ < , (

M5.7], , * * u ! (5.3.8) 2 < = < * = u = 2 . P 2 1 , * ' (5.3.7) ( * . ) , * ! # (5.3.1), (5.3.9), % ( * , *

. ; < # L , * * !, M5.7] * % .

5.3.2 % - (: 131 (,* . ( ).

@!

: ) . ( (5.3.1) ); ( # ' ! % !. ); ( # ! * % '! '(. )X . 121

$ : )0 # C 0 , K Kf (5.3.9). )= 9( ! ! #

. @ ( 131 (( .2.5. # % ! ( %.

122

. &

! ! . ( % # ! * ( . P ( , , * ( % 4%(,

% 1932 ., %* , ( ! , ' * ! ( ( ( % , * <( . ; (1939-1965) ' (<99# M6.1],

% ( % M6.2], ! M6.3], ( ' % . ( ( ! ' ! %. ( ( ( < ' ( % % 9 !) (<99# ( % # , < * " !"

! , % ! 9( , . ; * ' %* . ,( , *

! . (

(<99# . D ( ! , ( ! ( , ) (, * ! ! . ( * (*

. & ( M6.4] M6.5]. 4 <% * ( , %, (( * , ! . ; * ( 6) ( : # (9 (# ) 9(#

# ( * (- ). 1 * !

( < ' . ; % <

% . , ( ! <

% .@ * (,< .; ! #. 9(#

# , ( ! 9(# . ( '! ( (045). D 8.2 8,

% 9(# . ( . ; 9 9(#

, 045. 123

; ! ,

! ! 7 9, , * ' ( ! <

% ) *% # * ( ! ( (.D ( (( , ' ! {9 (# * ( %,* ( * !, *

! % !. D ( ( 12, ( ! !: (

! # ! * . % { %, %{ 9(#

%,

% ( * {* % 9(#,

8.2.

124

6 $# '$ 6.1 % # #. 6.1.1 # # %< . B . . - $ . ( ,

(

x_ = '(x u f )I x(t0) = x(0)I

(6.1.1)

y = w(x u )

(6.1.2) x(t) { n - % (

! . (I y(t) { r - % ( !

! . (I f (t) (t) { r - ( ' ! % !

I (t) { n - % ( ! . (I ' , w { ( -9 (# ! . ; % (6.1.1), (6.1.2) * ,

x_ = A(t)x + B (t)u + Q(t)f I

(6.1.3)

y = D(t)x +

(6.1.4) A(t) , B (t) , Q(t) , D(t) { # , < ( ! aij (t) , bik (t) , i(t) , dli(t) (i j = 1 n , k = 1 m , l = 1 r , = 1 ) ,

, ( ! ( 0 = jj(1)(t) (2)(t) (3)(t) (4)(t)jj0

(1)(t) = jja11(t) a12(t) : : : a1n(t) a21(t) a22(t) : : :jj I

(6.1.5)

(2)(t) = jjb11(t) b12(t) : : : b1m(t) b21(t) b22(t) : : :jj I

(6.1.6)

(3)(t) = jj11(t) 12(t) : : : 1(t) 21(t) 22(t) : : :jj I

(6.1.7)

(4)(t) = jjd11(t) d12(t) : : : d1r (t) d21(t) d22(t) : : :jj :

(6.1.8)

! * %: ) * * (% . (I ) 9 # ,

125

, * , ( ! . ( % I ) ! ! * (! (I ) " " < . ( .. . % ! . ( * . P * < ! , * *. 1. 4

, *

. ; < * 9 (# i(t) (i = 1 n) { 9 (#,

ji(t)j i

i (i = 1 n) {

*.

(6.1.9)

2. . ( *% 9 (# ( , < (

. 4 , , * ( - (%,

(

#

# # . 3. . ( *% 9 (# ( < ( . 4. 1 (# i(t) (i = 1 n) , ( * # . ( (6.1.1), (6.1.2). 5. . ( i(t) (i = 1 n) { * 9 (#. D *%

% * ( . 5 % ! * ( z ! * % ( (t) , ( ! . (. * . (

, *

, < Mt0 t1] 9 (# . ( , * ( ! . ( *

. ( , '

i(t) = const = i(RT )I RT t (R + 1)T (R = 1 N I i = 1 n) ; t0 { (# . (. T = t1 N & '

(6.1.10)

(6.1.11) ti ! T (i = 1 n) (ti ! (i = 1 n) { ! ! ! # (%

! ) (6.1.10) , 126

( % # , ( . ( , " " (

! ) "

" ( ). ,( , ( . ( (6.1.3), (6.1.4)

x_ = A(R)x + B (R)u + Q(R)f I RT t (R + 1)T (R = 1 N )I

(6.1.12)

y = D(R)x + I RT t (R + 1)T (R = 1 N )

(6.1.13) A(R) , B (R) , Q(R) , D(R) { # * , (6.1.5) ... (6.1.8) (

! . ( (6.1.12), (6.1.13). ; ' ( D;0 # ! * ( . ( (R + 1)T I (R = 1 N )I xM(k + 1)T ] = R(R)x(kT ) + R(R)u(kT ) + Q(R)f (kT )I RT k T T

(6.1.14)

(R + 1)T (R = 1 N ): y = D(R)x(kT ) + (kT )I RT k T T

(6.1.15) 7 <% * ( . ( (( , u , f { ( ),

x_ = Ax + bu + f I y = dx +

(6.1.16)

x(k + 1) = Rx(k) + ru(k) + f (k)I y(k) = dx(k) + (k) (6.1.17) ( ! b , d0 , , r { n - ( -# * , ( R -

* ! # ( ( * % , ( , * < . ( ( (# . 4 < * = 0 9 "! !" . (

X ;1 X ;1 nX ;1 y(n) + aiy(i) = kj u(j) + k(1)f () (6.1.18) i=0

y(k) +

n X i=1

'iy(k ; i) =

j =0

X ;1 j =1

=0

rj(1)u(k ; j ) + 127

X ;1 =1

rf (k ; ):

(6.1.19)

ai , 'i , (i = 1 n) , kj , (j = 0 , ; 1) , k ( = 0 , ; 1) , i(1) , r (i = 1 , ; 1 I = 1 , ; 1) <! % * # ( % (6.1.16), (6.1.17).

6.1.2

% , -

$ # % . (, % (6.1.16), . 7 ! ( 9# ) . ) * *: - !, ( ' ! ( , f (t) = (t) = 0) , , !, ( ! ' # ! ! * % * ( ! ( ( (( ). ; * , * ' ! . , . (

x_ = Ax + buI y = dxI x(t0) = x(0)

(6.1.20) ( ) ! u(t) . B y(t) ! . (. * , (( < . 7 , * ' * { # A ( b d - ! u(t) ! y(t)

. 7 % ,

(6.1.20) %

x~_ = M ;1AM x~ + M ;1bu~I ~y = dM x~ I x~ (t0) = M x(0)

(6.1.21)

0 { , (det M 6= 0) #. P ! % u~(t) = u(t) , ! ! y~(t) = y(t), ! # ! * . ; ! ! , (6.1.21) - *

y~(s) = dM (Es ; M ;1AM );1M ;1 bu~ + dM (Es ; M ;1 AM );1M ;1x(0) = = d(Es ; A);1bu~ + d(Es ; A);1x(0) = y(s) u~(s) = u(s) . ; < ( : , ( %

"!- !"? ,( ! ! . ( ai (i = 0 n ; 1) , kj (j = 0 ; 1) . ( 9 (6.1.18). < 9(# % . ( 9 "! - !". ; 10 9(# . ( (6.1.18). P ,

<% , # 9(# (# ) 128

^ M(k + 1)T ] = (^(kT ) y(kT ) : : : y((k ; 1)T ) u(kT ) : : : u((k ; 2)T )) (k = 0 1 2 : : :)

(6.1.22) ^ (kT ) { # ( ( kT I (^(kT ) ... y((k; 1)T ), u(kT ) , ... u((k ; 2)T )) { ( -9 (#, ( 9(#. P

, * ' (6.1.22) %

lim ^ (kT ) = ( = jja0 : : : an;1 k0 : : : k ;1jj # 9(# (

* . (. , " 9(#" ( (( * (% (6.1.16) . (, ( ( % ( % % ! ( 99 # (6.1.16), # , (

! n ) . ; ' ( 9(# (* ! ! . ( ( * (% , # ( ) (

! . & ( 9* ( ( , ( . ( (( 5 !9, 0( , ( ! , < , ( (* < ). @ <! ( % 99 # * ! !, ( , ( # ) (

! 99 # ! %, ! ( . D 9 # . ( . . ( % ! ! . (. @ * 9 #. &! 9(# ( ' ( ) . 6.1.1. k!1

129

.

$. 6.1.1

6.1.3 + , -) . ! ( . ( (6.1.12), (6.1.13), < * ( , ( % (

! ! . (. ; < *, ( ' ( ( # < . (: 1) 9(# ( ) . ( I 2) ( ) ! ! . (, * (* I 3) ( ,

% . ( ( . ( (6.1.12), (6.1.13) , < * ' # . (, ' * (, * * (. 7 , !,

! ! . ( (

* ' # ( , ! ! '

! !

. ( (" " . () . (. D * , * # , ( , ) T ( . ( (, * (*

. 7 ( ' . ( (6.1.18)

130

x_ = Ap()xp + bp()yI u = dp()xp + fp()y

(6.1.23) xp(t) { np - % ( , Ap() { #, bp() dp() { ( , fp() { ( , ( () . ) (6.1.23) . ( (6.1.18) (( * ( (

9 ), ( * (. , * ( # ), ( ( 9(

( % # Ap ( bp , dp ( fp . ; (* (! , * , # ! ,

! 4, 5 6. P 9(#

* ( , , (6.1.23) = * ( % . ( ! # *, ( ( * 9(# * ( 9 # ) # Ap() , ( bp() , dp() ( fp() (6.1.23). P

, ( * # 9(#, # ( ^ , (6.1.22), (6.1.23)

x_ p = Ap(^ )xp + bp(^ )yI u = dp(^ )xp + fp(^)y

(6.1.24) (6.1.22), (6.1.24) ) . & 9(#

. , * (6.1.24) ( 9 "! - !": nX

X p ;1 p ;1 ( n ( i ) p) u + api(^ )u = kpj (^)y(i): (6.1.25) i=0

j =0

; ( * np X p ;1 X u(k) + 'pi(^ )u(k ; 1) = rpi(^ )y(k ; 1): i=0

j =0

(6.1.26)

. . ( ( % !(- ! * (% # , (% ( - ( M?]. ; 0 , , , 2T , ..., kT , ... ( , f (k) ( , , (( , ( ), ( u(k) . 5* (

(# y(k) ( # # * x(k) :

y(k) = dx(k) * x(k + 1) * x(k) , u(k) , f (k) : 131

(6.1.27)

x(k + 1) = ax(k) + bu(k) + f (k) (k = 0 1 2 : : :): (6.1.28) ; * y(k) f (k) (k = 0 1 2 : : :)

, ( u(k) (k = 0 1 2 : : :) % , ( ! # . 5<99# a , b , , d ' % (6.1.27), (6.1.28) ( ( , ( ( (#, ( (# ( . # ! ( y(k)

g = const . P * a , b , , d * , ( , * # D ) ; df (k) : u(k) = g ; ay(kdb 7 % , (6.1.29) (6.1.28), *, *

(6.1.29)

y(k + 1) = g: (6.1.30) ; ! ! 9( , ( ! a , b , , d

( , ( ( , ! ( .). < !

, ( ( (6.1.29)

. ! ( 9(#

, * 1 = a 2 = b 3 = 4 = d (6.1.31) ' ( (6.1.29) (( 9 (#

! i (i = 1 4) : u(k) = ; 1 y(k) ; 34 f (k) + 1 g: (6.1.32) 2 4 2 4 2 4 7 (<99# ( (6.1.32) 9# . ( (6.1.27), (6.1.28), ( *

! * % y(k + 1) = 1y(k) + 42u(k) + 34f (k) (k = 0 1 2 : : :): k = 0 1 2 * ! * (! %: y(1) = 1y(0) + 24u(0) + 34f (0)I y(2) = 1y(1) + 24u(1) + 34f (1)I y(3) = 1y(2) + 24u(2) + 34f (2) 132

(6.1.33)

(6.1.34)

' ( % * 1 , 24 , 34 . < * (6.1.32), * , * # (6.1.30). (6.1.32) # % ' * (! % (6.1.34) 9(#

. ) , * % f (k) ! y(k) . P f (k)

! , < ( # 9(#, (6.1.22).

6.1.4

) . ? - -

@ 9(#

% (6.1.22), (6.1.23) # ( , . (, 9(#

* < ( ( (* ( # ( ), * *. 7 , * 9(# # , ! . ; < ( : 9(# ( ( (6.1.23) !

( ) # % ?. 7 , (6.1.23) * ( (* ( 9 (# ). ,( , , < , - . ,(, . ( (6.1.18) < : nX X p ;1 p ;1 u(np) + p+i (t)u(i) = j y(j)I (6.1.35) i=0

j =0

_i = (0 : : : p+np ;1 y y_ : : : yp;1 u) (i = 0 p + np ; 1) (6.1.36) i (i = 0 p + np ; 1) { ( ) I i (i = 0 p + np ; 1) { 9 (#, ( (* (# ). (6.1.36) %( . 7( u(k) +

np X i=0

p+i(k)u(k ; i) =

p X j =0

j (k)y(k ; j ):

i(k + 1) = i(0(k) : : : p+np (k) y(k) y(k ; 1) : : : y(k ; p+1) u(k) u(k ; 1) : : : u(k ; np) (i = 0 p + np): 133

(6.1.37) (6.1.38)

. % !(- ! -

* ( # ,

6.1.1. ; (6.1.29) < #

u(k) = 0(k)y(k) + 1(k)g + 2(k)f (k) (k = 0 1 2 : : :) (6.1.39) 0(k) , 1(k) , 2(k) { ((<99# ). , % ( <! , ( # (6.1.30). 7 ! ( ( ( % (* J (k + 1) = (y(k + 1) ; g)2 (6.1.40) # (( # 9 (# (6.1.40). 7 # % , % ! , 9 (# J (k + 1) . ; J (k + 1) * < , * J (k + 1) = May(k) + df (k) + db(0(k)y(k) + 1(k)g + 2(k)f (k)) ; g]2 :

(6.1.41)

; * * 9 (# (6.1.41) 0(k) , 1(k) , 2(k) , ! ( (6.1.38) %( : 9 0(k + 1) = 0(k) ; 2a1(k)(y(k + 1) ; g)dby(k)I > = (6.1.42) 1(k + 1) = 1(k) ; 2a1(k)(y(k + 1) ; g)dbgI > 2(k + 1) = 2(k) ; 2a1(k)(y(k + 1) ; g)dbf (k)

a1(k) > 0 { (<99# # . < (<99# lim J k!1 (k+1)

= 0: (6.1.43) D * , * % (6.1.39), (6.1.42) !( ! * ( # * # (6.1.30). , < # ! ( (! '!, (( 9(#

, * ' * ' . * ! y(k) (6.1.30) # 9 (( lim J (k + 1) < { (6.1.44) * { > 0 !. & ' (6.1.44) * , * % ( (6.1.27), (6.1.28), (6.1.39), (6.1.42) k , * ( J (k + 1) < { . k!1

134

; * , ( ! !* (% ! ( , # " ": lim M fJ (k + 1)g < {:

k!1

(6.1.45)

6.2 ' # 6.2.1 H $ . ( , %

x_ = '(x u f )I y = w(x u ) t > t0

(6.2.1)

( ! { % ( * . , % . ( (6.2.1), ( # . D *, ( % * ( , * , ( # . 7 # # * 9 (#

J (t) = J (e(t)) (6.2.2) J (e) = q(e(t)) { ( ( 9 (#, * J (e) = e2 I e(t) { '( ( (), ( % # . ; '( * . ( <! % ! ( ! . 1. . ; < *

e(t) = y(t)I e(k + 1) = y(k + 1):

(6.2.3)

2. . , * !

y(t)

, ( %

y(l) +

l;1 X i=0

aiy(i) = 0 (l n)

(6.2.4)

( * ai (i = 0 l ; 1) (, * ' (6.2.4) * ( %* . , *

e(t) = y(l) + ( 135

l;1 X i=0

aiy(i)

(6.2.5)

e(k + 1) = y(k + 1) +

Xl i=0

'iy(k ; i):

(6.2.6)

3. !. P % g(t) , (

e(t) = y(t) ; g(t)I e(k + 1) = y(k + 1) ; g(k):

(6.2.7)

4. " # $. & <

% ' % ( ! , ( ! <

% , % 9* ( %,

x_ % = A%x% + %gI y% = d%x%

(6.2.8)

* %

x_ % = %(x% g)I y% = w%(x%)

(6.2.9) ( ! x% { n% - % (

! <

% I A% {

# * I % , d% {

( * , ( * ! . D ( ( -9 (# % 9 (# w% . ; ! y%(t) <

% (# )

( ) % g(t) . (

e(t) = y(t) ; y%(t)I e(k + 1) = y(k + 1) ; y%(k + 1):

(6.2.10)

) # * % 9 (# (( (* ) (6.2.2) '( e(t) * % . 9 # ' ! ! !!, %! . ( (6.2.1). P * ' ! ! !! 9 # !, * : )

, *

' !, ( f (t) , (t) 9 (#, f~ , ~ {

*I

jf (t)j f~ j(t)j ~ 136

(6.2.11)

) *% ' % ! ( ,

*

* ( :

jfOj fOI jOj OI

fO = M ff (t)g I O = M f(t)g :

(6.2.12)

h i2 n o f2 = M f (t) ; fO f2I 2 = M M(t) ; O]2 2 fO , O , f , {

*I

(6.2.13)

) ' % ! - *% # , ( ( ! , <! ( . D (* z ( ! . ( %I ) ' % ! - *% # (

<! ( . P ' % ! -

, *

, #

q(e(t)) {I q(e(k + 1)) { (6.2.14) { {

*,

! ' ! %%. *% ! %! # M fq(e(t))g {I M fq(e(k + 1))g {: (6.2.15) P

, * - ( 9 # ! . ( # (6.2.14) (6.2.15) * < 9 (# . (, < , * # * ( { * ' t ( k ) t ! 1 (k ! 1) . ,( , ! ( # ! lim q(e(t)) {I klim !1q(e(k + 1)) {

! %! t!1

(6.2.16)

lim M fq(e(t))g {I klim (6.2.17) !1M fq(e(k + 1))g { *% ! ' ! %! !!. , * "( " ( (6.2.14) (

! t!1

137

Zt

J = q(e) dt: t0

(6.2.18)

6.2.2 ' # @ 9(#

% % . ( (6.2.1)

x_ = ' (x y g )I u = w (x y )I x (t0) = x(0)

I

(6.2.19)

_ = ( y u g) (t0) = (0)

(6.2.20) (t) { n - % ( ! ( * 9(#

(t) # (% (

! )I ' , { n n - ( -9 (# ! , , (( 9 (# w , !

! # % (6.2.16) (6.2.17). (6.2.19) , (6.2.20) #. %, #, M?]. ,( , % . & ( ! % . 6.2.1.

$. 6.2.1 $ ,

% < ( , ! * %: % ( (

( ( ) % # % ( (

( ( ). * , -

. 3 , * . ( - < ) , *

( )). P # <

% , ( ! .

% . 6.2.2. 138

$. 6.2.2

6.2.3

. . ( 8

)* ( (6.2.1) . (, z # (6.2.16) (6.2.17)) (6.2.19) # (6.2.20), (, * ( * (0) # 2 z ! * ! % x(0) , x(0)

, (6.2.16) (6.2.17). ; ( ) ' <% * # , ( % ! * (. ) % 9 # . ( !. )

* " " { . 5 * , ( ! * {

9( , * # (( { > 0 . ,( * (, ( 9 # ' ! % ! , . ( %* ' * *

! y(t) . 7 % ( % % *% ( , ( * { , % % * (6.2.16), (6.2.17). ,( * * . 5 (! *! 9 (#

J = tlim q(e(t))I !1 jf jsup f~ jj~

(6.2.21)

J = tlim (6.2.22) !1 M fq(e(t))g : , *

% 9 ( * 9 ( * , (( % * ,

% x6.1, 139

< ( . D * . ;- !, ( * 9 %

(t) = ~ (1)1(t) + ~ (2)2(t) + + ~ (N )N (t) (6.2.23) i(t) (i = 1 N ) { 9 (#I O (1) (i = 1 N ) {

( * , . ( ! ( * .

;- !, # ! (6.2.16) (6.2.17) * , * tx ( (0)) , (%, *

j{ ; q(e(t))j " ! t > tx

(6.2.24) " > 0 { ( *

*, ! ( * # . P

tx( (0)) < T (6.2.25) # ( * " ) * (# . (. ) , * 9 ( * , * * (6.2.20) % lim (t) = { ( * , (%, * t!1

(6.2.26)

x_ = ' (x y g )I u = w (x y )I x (t0) = x(0)

I

(6.2.27) , ( % * , ' * . ( (6.2.1), ( (

* . ; * 9(#

' (6.2.20) % lim (t) = (6.2.28) ' 9

% *

* < M6.10]: t!1

1. ( ( ) . 4 < < 9 (# ' , w % (6.2.19). & <! 9 (#%

! % . ( ( ' . ,(, 9(#

! ( % ( (6.1.23) 9 "! - !" (6.1.25), * ! (6.1.35). 140

2. ; ( . ) ' 9(#

. 3. ; # (6.2.20). 4. # % ! # # # #.

7 #%( 0#1 % &2 $ * (% . (, % % 99 #

. > , * ' ! , . ( * ( () % . ; <% # * <

, ( 9* ( %, ! ( %, * . ( , ! . ( <

% .

7.1 - . " #" .# 7.1.1 5 . %< $ 9* ( %, <

% , (

y%(n) + d% n;1y%(n;1) + : : : + d% 1y_% + d% 0y% = k% m g(m ) + : : : + k% 0g

(7.1.1)

y%(t) { % !, g(t) { % - , % . 4 <

%

y(n) + dn;1 y(n;1) + : : : + d1y_ + d0y = km u(m) + : : : + k0uI

(7.1.2)

dp np u(np) +dp np;1u(np;1) +: : :+dp 1u_ +dp 0u = kp mp y(mp) +: : :+kp 0 +lpg(p) +: : :+l0g: (7.1.3) (<99# . ( (7.1.2) , % (<99# (7.1.3) ( , * ! . ( <% <

% , % g(t) % lim e = tlim !1(y(t) ; y%(t)) = 0:

t!1

141

(7.1.4)

,( , <

(7.1.2), (7.1.3). $ , ( ! < * ! <

% , '. ; < ! ! * ! !

y(s) = w(s)g(s) y%(s) = w%(s)g(s) (7.1.5) w(s) = d(s)d (ks()s;)l(ks()s)k (s) , w%(s) = kd%((ss)) . p p % ;

! , w(s) = w%(s) k%(s) : k(s)l(s) = (7.1.6) d(s)d (s) ; k(s)k (s) d%(s) & * % * < , (( , ' ! % *, < ! ( * <! . 4% , ( ! ( ( . k(s) = k(s)k;(s) , k; (s) { , ( ( % ( ( % (Re si < 0 si (i = 1 m { ( k;(s) , ( k+ (s) % ( #! (

! ( %). P k+(s) k%(s) , ! ( * ( (7.1.2), (7.1.3) D(s) = d(s)d (s) ; k(s)k (s) , * , ( (( < * ( %*%. ; < k%(s) (7.1.7) k%(s) = k+(s)kO%(s): D * , * %* . ( (( k+ (s)) ! (* k%(s)) . ,( (( k;(s) D(s) , d(s) (7.1.3) ( d (s) = k; (s)dO (s) . ,( (7.1.6) k+(s)kO%(s) k; (s)k+(s)l(s) = k; (s)(d(s)d (s) ; k+(s)k (s)) d%(s) l(s) kO%(s) = (7.1.8) d(s)d (s) ; k+(s)k (s) d%(s) (<99# (7.1.3)

l(s) = kO%(s)I 142

(7.1.9)

d(s)dO (s) ; k+(s)k (s) = d%(s): (7.1.10) $ ' > (7.1.10), ! dO (s) k (s) * d (s) = k;(s)dO (s) . *

% * ( !, ! * ! ( ! <

% (7.1.2), (7.1.3) ! !: y(t) = y%(t) . @* '( e(t) , % (7.1.4), ( * ! % <! .

7.1.2

.

$ . ( (7.1.2) (<99# . > , * n m , ( , * 9 . D * , * k(s) { # , ( ( % ( k(s) = k;(s) , k+ (s) = 1 ). )' * 9 (# m : : : + k1 s + k0 ) kk(s) : w(s) = sn +kd(s + = (7.1.11) n ; 1 + : : : + d1s + d0 d(s) n;1 s B * ' * 9 (# <

% (7.1.1), n% = n , m% = m m : : : + k%1 s + k%0) k%k%(s) w(s) = sn +kd%(s + = (7.1.12) d%(s) %n;1 sn;1 + : : : + d%1 s + d%0 k%(s) d%(s) {

# . )* , * % %( (<99# (7.1.3) (, * ! (7.1.2), (7.1.3) <

% ( ( (7.1.4)). & ( ! % . 7.1.1.

$. 7.1.1 ' <% * * ! ( . ( .

143

7.1.3 - $ . ( , % (sn + dn;1 sn;1 + : : : + d1s + d0) = u + hn g: 4 , * -9 % k(s) = 1 . D

(7.1.13)

(sn + d%n;1sn;1 + : : : + d%1s + d%0) = u + h%g: (7.1.13), (7.1.14) ((

(7.1.14)

x]_ = A]x] + h] gI y = d]x] I x]_ % = A]%x] % + h] %gI y% = d]%x] %

(7.1.15)

0 E n ; 1 I A] = ;d d] = 1 0 : : : d% = d%0 : : : p p p p

p

p

p

p

p

p p p p

p

p

p

p

p

p

p

p

p

p

p

0 b] = 1 I h] = h0 n 0 I d] % = 1 0 d%n;1 :

I A]% = 0 En;1 I h] % = 0 I ;d% h% : : : 0 I d = d0 : : : dn;1 I (7.1.16) p p p p

p

p

p

p

p

p p p p

p

p

p

p

p

p

p

p

p

p

p

p

p p

p

p

p

p

p p

! ( , , * * (* ) * ! !

! y y% n ; 1 - ( (* . , ( nX ;1 u = i(t)y(i) + n(t)g (7.1.17) i=0

y(i) { i - y(t) I i(t) (i = 0 n) { . (7.1.17) (7.1.13) * (7.1.13) (7.1.14), * '( e = y ; y% : nX ;1 i! nX ;1 n s + d%is e = (d%i ; di + i) y(i) + (;h% + hn + n) g: (7.1.18) i=0

i=0

144

5 6. B %( (7.1.17), ( # # (tlim !1 e = 0), 0n;1 1 X _1 = ; j;1 @ lj e(j)A y(i) (i = 0 n ; 1)I j =0 0n;1 1 X _n = ; n;1 @ lj e(j)A g j =0

(7.1.19) (7.1.20)

i > 0 (i = 0 n) ( l = kl0 : : : ln;1k0

l = P]]b

(7.1.21) ( -

# P] ' -

A]0%P] + P] A]0% = ;Q (Q { -

#).

(7.1.22)

B (7.1.19), (7.1.20) * ( %

% <

% . * ! M7.5], M7.2]. ! ( ( , ( 9 > ]e = ke]1 e]2 : : : e]nk e]1 = eI e]2 = e_ : : : e]n = e(n;1)I = 0 ' = kd%0 ; d0 + 0 : : : d%n;1 ; dn;1 + n;1 ; h% + hn + nk I > (7.1.23) 0 n ; 1 ! = y y_ : : : y g ( ' (7.1.18)

e] = A]%]e + b] '] 0!] :

(7.1.24) 7 %* (7.1.13), (7.1.14), (7.1.17), (7.1.19), (7.1.20)

% e 9 (# - (7.1.25) = ]e0P] e] + '] 0=]' > 0 ( % -

# P] ' (7.1.22), = = diag k 0 : : : nk . 9 (# (7.1.24) (7.1.25) 0 _] 0=]' + '] 0B']_ = _ = A]%]e + b] '] 0! P] e] + ]eP] A]%]e + b] '] 0! + ' (7.1.26) 0 ]] 0 ]0 ] 0 0 0 _] B '] + '] !] b P e] + B']_ : = ;e] Qe] + e] P b! + ' * , * 145

_ = ;e]0Qe] < 0

(7.1.27)

']_ = ;B;1(l0]e)]!: (7.1.28) , * a%i , i (i = 0 n ; 1) . (

, (* , * (7.1.28) (7.1.19), (7.1.20). . . ( , %

yY + 1y_ + 0y = u + h2g (7.1.29) ( 0 , 1 , h2 , % %( u = 0(t)y + 1(t)y_ + 2(t)g (7.1.30) ( ! y . ( ( * ! %

% <

% , % (7.1.31) yY% + a%1y_% + a%0ym = h%g

. 4 (7.1.19), (7.1.20) * ( %

%( 9 _0 = ; 0;1(l0e + l1e_)yI > = (7.1.32) _1 = ; 1;1(l0e + l1e_)y_ I > ; 1 _2 = ; 2 (l0e + l1e_)g ( 0 , 1 , 2 {

*, l0 = p]12 , l1 = p]22 , p]11 p]12 p]12 , p]22 { < # P] = , % ' * p]12 p]22

0 ;1

;a%0 p]11 ;a%1 p]12

p]12 p]22

+ p]11 p]12 0 p]12 p]22 ;a%0

1 ;a%1

= ; q11 0 : 0 q22

) q11 , q22 { *. & ( ! % .7.1.2.

146

(7.1.33)

$. 7.1.2

7.2 7 . 7.2.1 ) ( )

' # . (, ! * (7.1.13) * * ! %

%. % ( # * . ( (7.1.1) <

% (7.1.2). D "* !" ! %

%. (s) n;m;1 , (%, * * 9 (# (s)w%(s) %. (4 , * * 9 (# w(j!)

%, 2 Re w(j!) > 0 !lim !1 ! Re w(j!) > 0): , % % , (%, * #

lim (y ; y%) = 0: (7.2.1) ' * # ( *, ( m = n ; 1 , { * m = n ; 2 M7.3]. t!1

147

$ % * * % 9 (# % < *

# , - {.

7.2.2 ) %< ) * ) - * , * * (s) = 1 , ( ( k%(s) n ; 1 (, * w%(s) %. & ( ! %

. 7 .2.1.

$. 7.2.1 ; <% !

v_ (1) = F v(1) + b] yI v_ (2) = F v(2) + b] u

(7.2.2)

1 , 2

1 = 0y + (1)0 v(1)I 2 = (2)0 v(2) (7.2.3) v(1) v(2) { n ; 1 - (

! ! I 0(t) (1)(t) (2)(t) { , (1) = k1 : : : n;1k0 , (2) = kn+1 : : : n+2k0 I F {

# * , ( : 0 E n;2 F = ; n ; 1 - % ( - (I = k 0 : : : n;2k I p p p p p

p

* , *

p

p

p

p

p p p

p

p

p

p

p

p

p

p

p

p

p

b] 0 = k0 0 : : : 1k : 148

1(s) = 0

)

y + (1)0 (Es

" (1)(s) # ; 1 ; F ) b] y = 0 + (s) y(s) = m1(s)y(s)

(7.2.4)

(s) = sn;1 + n;2 sn;2 + : : : + 1s + 0I (1)(s) = 1sn;2 + : : : + n;2s + n;1I (2) 2(s) = (2)0 (Es ; F );1 b] 0u = (s()s) u(s) = m2(s)y(s):

(7.2.5)

(2) (2)(s) = n+1sn;2 + : : : + 2n;2s + 2n;1I m2(s) = (s()s) :

(7.2.6)

4

.7.2.1 , *

u = 0y + (1)0 v(1) + ng + (2)0 v(2): ; 2n - ( (t) = k0 1 : : : n;1 n n;1 : : : 2n;1k = 0 (1)0 n (2)0 I

(t) = y v(1) g v(2) ' (7.2.7) (( % 9

(7.2.7) (7.2.8) (7.2.9)

u = 0 : (7.2.10) 4 ( % ! (* , * * 9 (# . (

* ( ! w(s) nkk(s) (s) wa(s) = yg((ss)) = 1 m (s)n+ = (2) m1(s)w(s) M (s) (s)] d(s) kk(s) M (1)(s) + 0 (s)] : 2 (7.2.11) 7 % , .7.2.1 , * u = n g m2(s)u m1(s)y: (7.2.12) * , * y = w(s)u , (7.2.12), * (7.2.11). ( ! ((! * % 9 (# (7.2.11) ( (( ' > i h (2) i h (s) (s) d(s) kk(s) (1)(s) + 0 (s) = n(s) 149

n(s) { % 2n ; 1 (<99# s2n;1 1. ; ' * ( .

n(s) = k(s)d%(s)I (s) = k%(s) (7.2.13) (* , * * 9 (# . ( wa(s)0 = knk w%(s): (7.2.14) % ,( , (7.2.13) ( % # F % ! . 7 ' % ! %( ( . , * 9 *%, ( k(s) k%(s) { n ; 2 . ; < * ' (7.2.13) n(s) = (s + 0)k(s)d%(s)I (s) = (s + 0)k%(s):

(7.2.15)

5 6. B %( (7.2.10), (-

# (7.2.1) . ( % * % 9 (# % (7.1.11) m = n ; 1,

_ = ;=;1 e

(7.2.16) = { -

# * 2n 2n. 7( 7( 7.

. . ( * % 9 (# % 2 1 s + k0 (7.2.17) w(s) s3 +s d+s2k+ d1s + d0 2 ( % . , % , ( ! . ( y ( ! %

% <

% * % 9 (# % 2 + 4s + 3 79) ( s w%(s) s3 + 6s2 + 11s + 6 : (7.2.18) ! ( , , *, (( , * 9 (# (7.2.18) %. (7.2.13) % !

0 = 3 79I 1 = 4: (7.2.19) & ( ! % % 7.2.2. 150

$. 7.2.2 B %(

u = 0y + 1v1(1) + 2v2(1) + 3g + 4v1(2) + 5v2(2)

(7.2.20)

_0 = ; 0;1eyI _1 = ; 1;1ev1(1)I

(7.2.21)

_2 = ; 2;1ev2(1)I _3 = ; 3;1egI

(7.2.22)

_4 = ; 4;1ev1(2)I _5 = ; 3;1ev2(2) i (i = 0 5) { *.

(7.2.23)

7.2.3 ) %< ) * ) ,( (( % * * % 9 (# <

% ' # , * * % 9 (# w%(s) , < * * 0 , ( , * * 9 (#

(s)w%(s) = (s + 0)w%(s) %. ,( * 0 . & ( ! % . 7.2.3. 151

(7.2.24)

$. 7.2.3 @ ! , * u = 0 + _ 0

(7.2.25)

{ 2n - % ( , 0 = 0 (1)0 n (2)0 = k0 1 : : : n;1 n n+1 : : : 2n;1 k

% ( (t)

_ = ;0 + :

(7.2.26) (7.2.27)

5 6. B %( (7.2.25) .-

( % * % 9 (# % (7.1.11), ( % m = n ; 2,

_ = ;=;1 e

(7.2.28) = { -

# * 2n 2n. 7( 7( 8.

.

. * 9 (#

w(s) = s(s2k+(s d+sk+0) d ) : 2

1

D

* 9 (# 152

(7.2.29)

6s + 4 8 : w%(s) = (s + 15)( s2 + 3s + 2)

(7.2.30)

s + 3)(s + 0) w%(s)(s + 0) = (s1+6(5)( s2 + 3s + 2)

(7.2.31)

! ( , % * 0 , ( * 9 (# %. D *

0 = 1: (7.2.32) , (7.2.15), ! ( * ( % !

(s) = (s + 0)(s + 3) 0 { *. 0 = 4

(7.2.33)

0 = 12I 1 = 7: B %(

(7.2.34)

u = 0y + 1v1(1) + 2v2(1) + 3g + 4v1(2) + 5v2(1) + _01 + _12 + _23 + _34 + _45 + _56 (7.2.35)

_i = ; i;1ie (i = 0 5) i (i = 0 5) ' %:

(7.2.36)

_1 + 01 = yI _2 + 02 = v1(1)I _3 + 03 = v2(1)I (7.2.37) _4 + 04 = gI _5 + 05 = v1(2)I _6 + 06 = v2(2)I (7.2.38) v1(1) v2(1) v1(2) v2(2) { ! ! , ! : v_ 1(1) = v2(1)I v_ 2(1) = ;12v1(1) ; 7v2(1) + yI v_ 1(2) = v_ 2(2)I v2(1) = ;12v1(2) ; 7v2(2) + u:

153

(7.2.39) (7.2.40)

7.2.4 % - (: 511 (B <

% ).

@!

: ) d(s) , k(s) m(s) . ( (7.1.2) * ' ! %. D * ,* . ( : d(s)y = k(s)u + m(s)f . ) d%(s) k%(s) <

% (7.1.1). ); ' . ); %. ); # #. $ : )= 9( # #. @ ( 511 (( .3.1. # % # # # =

%( (7.2.16) ' ! %.

8 #45" @ 9(#, , (( , ( ((( . ( ), < ( . 5 , . (. 4 ' !, ( 9(#

! . ; * '! ( (x10.1), ( ' % ' ' *% # . x10.2

( * - * % (#. 9# . ( ! * ( ! ( (.

8.1 8# 8.1.1 @ # $ . (, %

y(kT ) + '1z;1y(kT ) + : : : + 'n;1 z;(n;1)y(kT ) + 'n z;n y(kT ) = = r0f (k) + r1z;1f (kT ) + : : : + r;1 z;(;1)f (kT ) + r z;f (kT ) (k = 1 :::)

(8.1.1)

7 * ( %* ! # ( 1 1 X X y(k) = h(l)f (k ; l) = hlz;lf (k) (8.1.2) i=0

i=0

( % * hl 154

r0 + : : : + r;1z;(;1) + rz; = h + h z;1 + h z;2 + : : : (8.1.3) 2 1 + 'z;1 + : : : + 'n;1z;(n;1) + 'nz;n 0 1 # % * < % * ) . 7 * ( %* ! # h(t) ! 0 t ! 1 , < * ( * * (q) ! (8.1.2). , q X y(k) = hlz;lf (k): (8.1.4) i=0

D

, % y k * f q , ' ! k . 0 (8.1.4) * / ,,- ). ,

"( " , * (8.1.4) (q + 1) * f ( , < q X hl = 1 , hl > 0 ! l ). i=0 (8.1.1) ri = 0 (i = 1 ) , r0 = 1 , n X y(k) = ; 'iz;(i)y(k) + f (k): (8.1.5) i=1

D (

% , % * y k * % y , ' ' k * f (k) . ; (8.1.5) ) * (%(- ). D , * (8.1.5) y(k) ' * y . @ ( #, (8.1.1), ( ((

y(k) = ;

n X i=1

X 'iz;iy(k) + riz;j f (k)

j =0

(8.1.6)

) * / (%(,,- ). % (8.1.5), (8.1.6) , * ! ( , ' < n X y(k) = iz;iy(k) + f (k)I (8.1.7)

y(k) =

n X i=1

i=1

X iz;i y(k) + n+j+1 z;j f (k):

j =0

(8.1.8)

, ( %) * y(k) (k = 0 1 2 : : :)

% ( .

155

8.1.2 8# (8.1.7) n = 2 , y(k) , f (k) (k = 0 1 2 : : :) * 2 1 (8.1.7), (

y(k) = 1y(k ; 1) + 2y(k ; 2) + f (k) (k = 0 1 2 : : :): (8.1.9) ) < k = 2 k = 3 , * * (! % ) 1y(1) + 2y(0) = y(2) ; f (2)I (8.1.10) 1y(2) + 2y(1) = y(3) ; f (3)

' ( % ( * 2 , 1 . 7 , * f (k) (k = 0 1 2 : : :) ' . , (% % (8.1.10),

% * ! k ( k = 4 , k = 5 , k = 6 , k = 7 ..), * * * ( ! 2 , 1 . ; ( 2 , 1 (, * ( () % % * (8.1.9) k = 2 : : : N

' %. 7 < 9 ( ( LN =

N X k=2

My(k) ; 1y(k ; 1) ; 2y(k ; 2) ; f (k)]2 :

(8.1.11)

4 ! * LN ! * (! % N @LN = 2 X My(k) ; 1y(k ; 1) ; 2y(k ; 2) ; f (k)] y(k ; 2)I (8.1.12) @2 k=2 N @LN = 2 X My(k) ; 1y(k ; 1) ; 2y(k ; 2) ; f (k)] y(k ; 1) = 0 (8.1.13) @1 k=2

' ( , % ( * 2 , 1 . $ (8.1.7), ( f (k) (k = 0 1 : : :) { 9 (#. )'

(8.1.7) ( % 9

y(k) = 0 y (k) + f (k) (k = n n + 1 : : :)

(8.1.14)

= k1 2 : : : nk0 I y (k) = ky(k ; 1) : : : y(k ; n)k0 : (8.1.15) ; (8.1.14) * (8.1.7) * * k = n . D , * k n ( (k) ( %, (( 156

* * y(;1) , y(;2) .. ( ( 9 (# f (k) (k = 0 1 : : :) , ( ( # ( ^ ( , * ( " ("

LN =

N h i2 X y(k) ; ^ 0 (k) (N ; n > n)

k =n

(8.1.16)

%. 799 # (8.1.16) ( ( ^ , * "X # N N X

(k) 0y (k) ^ = y (k)y(k): (8.1.17) ; *

k=n

k=n

PN =

"X N

#;1

k =n

y (k) 0y (k)

(8.1.18)

% (8.1.17) ( % ( (8.1.18)

^ = PN

N X k =n

y (k)y(k):

(8.1.19)

; <( (8.1.19) 9 # ( ( '! ( . ; < N ; n - ( v , ( # U : 0 y(n) y (n) f (n) y(n + 1) 0y (n + 1) f (n + 1)

= .. I U = .. I v = .. : (8.1.20) . . . y(N ) 0y (N ) f (N ) <! * ! (8.1.14) k = n N ; n 9 (# LN

= U + vI

(8.1.21)

LN = M ; U ^ ]0 M ; U ^ ] : (8.1.22) 799 # (8.1.22) ( ( ^

, * U 0 M ; U ^ ] = 0 ,

^ = MU 0U ];1 U 0 :

(8.1.23) . * ( %* % . ( , % 157

y_ + a0y = 0: (8.1.24) ( a0 % f (t) . % ! . ( 0 T 2T : : : (T = 0 08) *

y(0) = 1 5I y(1) = 0 6I y(2) = 0 56I y(3) = 0 236: (8.1.25) , a0 . ! ( ' <% *, ( (8.1.24) y(kT ) + '1yM(k ; 1)T ] = r0f (kT ) (k = 0 1 : : :) < ((

(8.1.7): y(k) = 1z;1y(k) + f~(k)I

(8.1.26) (8.1.27)

(8.1.28) 1 = ;'1 = a T1+ 1 I f~(k) = a TT+ 1 f (k): 0 0 ; * 9 (8.1.18) * PN;1 . ; * (8.1.15) y (k) = y(k ; 1) , ( ,

PN;1 =

3 X

k=1

y2(k ; 1) = 1 52 + 0 62 + 0 562 = 2 92:

4 (8.1.19) (* , * ^ 1 = 2192 (y(0)y(1) + y(1)y(2) + y(2)y(3)) = 12 36 (8.1.29) 92 = 0 47 , , # ( ( * ^0 = ;^T1^+ 1 = 14: (8.1.30) 1 P ^ 1 9 (8.1.23), ( # U : y(1) 0 6 1 5 = y(2) = 0 56 I U = 0 6 y(3) 0 236I 0 56 *

^1 = 0 47:

158

(8.1.31)

8.1.3 7 ) 8# % 9* (% # , %

% (8.1.7) i (i = 1 n) . 9# < # . D * , * # (

! * ! . (. @ '! ( , (: N + 1 - * (8.1.18)(9.2.22) * PN +1 % # ( ^ (N +1) 9 (8.1.19), PN +2 - , (8.1.18), (8.1.19), % # ( ^ (N +1) .. ,( , ( ! # 9 (8.1.18) * # ( (8.1.19). ; < ( : % % 9 # (% i - , % , # (% i ; 1 - i - { %. ,( (

'

.

5 6. $ (

% ( %) '!

( % # (

% (8.1.7) : h i ^ (i) = ^ (i;1) + k(i) y(i) ; 0y (i)^ (i;1) (i = n n + 1 : : :)I (8.1.32)

h

i

k(i) = Pi y (i) Pi;1 y (i) 1 + 0y Pi;1 y (i) ;1 I

(8.1.33)

h i;1 Pi = Pi;1 ; Pi;1 y (i) 1 + 0y Pi;1 y (i) y (i)Pi;1 (8.1.34) ^ (i) { # ( ( i - ! %

% y. ; (* * ! %

^ (0) = 0I P0 = aEn

(8.1.35)

a { * ' *.

% & . 7 % , (8.1.17), (8.1.18)

'

PN;1^ (N ) =

N X k =n

y (k)y(k) =

NX ;1 k=n

y (k)y(k) + y (N )y(N ):

) y(k) # (% 0y (k)^ (N ;1) , * 159

(8.1.36)

NX ;1 ( N ) ; 1 PN ^ = y (k) 0y (k)^(N ;1) + y (N )y(N ) = k=n # "X N =

y (k) 0y (k) ^ (N ;1) + y (N )y(N ) ; y (N ) 0y (N )(N ;1) = k=n

h i = PN;1^ (N ;1) + y (N ) y(N ) ; 0y (N )^(N ;1) ( PN (8.1.32). ! ( ' (8.1.34), ' (8.1.18) PN;1 =

N X k=n

y (k) 0y (k) =

NX ;1 k =n

y (k) 0y (k) + y (N ) 0y (N ) = PN;;1 1 + y (N ) 0y (N ): (8.1.37)

< PN PN ;1 , * , *

PN ;1 = PN + PN y (N ) 0y (N )PN ;1 :

(8.1.38)

h i PN ;1 y (N ) = PN y (N ) 1 + 0y (N )PN ;1 y (N )

h i;1 PN y (N ) = PN ;1 y (N ) 1 + 0y (N )PN ;1 y (N ) : < 0(N )PN ;1 * (8.1.38), * (8.1.34), ' % (8.1.33), * . ,( , ( . , * (

, * # #, ( (( ! (8.1.34) M1 + 0(i)Pi;1 y (i)] ( . $ (

%, %, ( # (, %. 1. P f (k) (k = 0 1 : : :) % ! (

! *% ! * , # ( ^

% %. 2. P f (k) (k = 0 1 : : :) (, # ( <99 ( . 10. (8.1.32) ... (8.1.34) # ( 1 (8.1.27) 10.1.1. @(, % * y(0) = 1 5 I y(1) = 0 6 . 4% * * p1 9 (8.1.34). , * * y (i) = y(i ; 1) p1 - ( , ' (8.1.34) (8.1.39) pi = 1 + p2i;(i1)p : i;1 y 160

5 , (8.1.35)

^ (0) (8.1.40) 1 = 0I p0 = 1 (9.2.43) 1= 1 * p1 = 1 + (11 5)2 1 = 0 31: 4 (8.1.32) (* h (0) (0)i ^ ^ ^ (1) = + p y (0) y (1) ; y (0) (8.1.41) 1 1 1 1 = 0 276: * y1(2) = 0 56 . , # ( (8.1.41) * . 7 < * p2 = 1 + p21(2)p = 1 + (006)312 0 31 = 0 28 1 y h i ^(2) ^ (1) (1) (8.1.42) 1 = 1 + p2 y (1) y (2) ; y (1)^ 1 = 0 336: ) * * y(3) = 0 236 . ; * # ( (8.1.42), % 28 p3 = 1 + p22(3)p = 1 + (00 56) 2 0 28 = 0 26I 2 y h i (2) y (3) ; y (2)^ ^(3) ^(2) (2) (8.1.43) 1 = 1 + p3 y 1 = 0 394: D # ( ( # ( (8.1.29), *

% (

'! ( .

8.1.4 - ''- (8.1.4) f (k) = u(k) - ( ) 9 (# y(k) * ! (k) , (

y(k) =

q X i=0

hlu(k ; l) + (k):

(8.1.44)

D ' , ! % % % # % (( '

' ! * % ! , ( % 9

y(k) = h0 0u(k) + (k) 161

(8.1.45)

h0 = kh0 h1 : : : hnk I u (k) = ku(k) u(k ; 1) : : : u(k ; q)k0 . 7 # ( ( h 9 (#

LN =

N h i2 X y(k) ; h^ 0 u(k) :

k=1

B * (8.1.16) (* , * h ' "X # X N N 0 ^

u(k) u (k) h = y(k) u (k): k=1

k=1

(8.1.46)

(8.1.47)

) , * ( # ( h

%, (* *, ( (k) u(k) (

. < .

y(k) = h0u(k) + (k) (k = 1 2 3 : : :): , (8.1.47) * N X

h^ 0 =

y(k)u(k)

k=1 N X

k=1

u2(k)

:

(8.1.48)

(8.1.49)

(8.1.49) (8.1.48) y(k) , * N X

(k)u(k) ^h0 = h0 + k=1N : X 2 u (k)

(8.1.50)

k=1 h0

, * h0 ( N ! 1 , (* *, ( M f(k)u(k)g = 0 . , * , * , , ( (k) (k = 0 1 : : :) " '". @ < ( ! " " # (k) (k = 0 1 : : :) . ; < ' ! (k) (k = 0 1 : : :)

%

(k) =

N2 X i=1

(2) (2) i (k ; i) + f (k )

(8.1.51)

f (2)(k) (k = 0 1 : : :) -(% " % '" . , # (2) i (i = 1 N2 ) . 0 , (8.1.44), (8.1.51), ((

y = h(z;1)u + I = (2)(z;1) + f (2) (2) ;N ;1 h(z;1) = h0 + h1z;1 + : : : + hq z;q I (2)(z;1) = (2) 1 z + : : : + N2 z 2 . 162

(8.1.52)

@(*

, * *

y = (2)(z;1)y + M1 ; (2)(z;1)]h(z;1)u + f (2): M1 ; (2)(z;1)]h(z;1) = b(z;1) =

qX +N2

(8.1.53) 9

y(k) =

N2 X i=1

(2) i y (k

i=0

(8.1.53)

biz;i

(8.1.54)

qX +N2

; i) + j=0 bj u(k ; j) + f (2)(k)

( (

y(k) = 0 (3)(k) + f (k)

(8.1.55)

(2) (3) 0 = jj(2) 1 : : : N2 b0 : : : bq+N2 jj (k ) = jjy (k ;1) : : : y (k ;N2) u(k ) : : :u(k ;q ;N2)jj:

, , ( q N2 , *

! <99 ( ! # ( (

% (8.1.32) : : : (8.1.34),

%, (8.1.54),

h^ 0 = ^b0I ^h1 = ^b1 + ^(2) 1 ^h0 : : :

(8.1.56)

8.1.5 - 7''- ! ( # (

% ( , ' (8.1.8) ( % 9 :

y(k) = 0 (4)(k) + n+1f (k) (k = 0 1 2 : : :)

0 = jj1 : : : n n+2 : : : n++1 jjI (4)(k) = jjy(k ; 1) : : : y(k ; n) f (k ; 1) : : : f (k ; )jj0:

(8.1.57)

)

(8.1.58)

1 (8.1.57) <( (8.1.14), < ( (

% (8.1.32) : : : (8.1.34). ( ( (4)(k) * f (k ; 1) : : : f (k ; ) . ; < # , (8.1.57),

f (k) , n+1 = 1 : 163

f^(k) = y(k) ; ^ 0 (4)(k) (k = 0 1 2 : : :): (8.1.59) ) * y(;i) = y(0)(;i) , f (;i) = f (0)(;i) (i = 1 n j = 1 ) ( ( (4)(k) ! # ( 9 ^(k) = jjy(k ; 1) : : : y(k ; n) f (k ; 1) : : : f (k ; )jj . ,( , % # ^ (i) = ^ (i;1) + k(i)My(i) ; ^0(i)^ (i;1)]I (8.1.60)

k(i) = Pi ^ (i)I

(8.1.61)

Pi = Pi;1 ; Pi;1 ^(i)M1 + ^0(i)Pi;1 ^0(i)];1 ^(i)Pi;1: (8.1.62) D (

# ( ^ . 7 *

! # ( (8.1.58) n * n0 > n . 1( ( n0 (8.1.60) : : : (8.1.60), ! ^ 9 (8.1.59) f (k) (k = 0 1 2 : : :) , (

, <

# ( ^ . P f (k) (k = 0 1 2 : : :) (

, * * n0 ! , ( < <% ( .

8.1.6 '. , - $ . ( , %

% : n X y(k) = iy(k ; i) + f (k): (8.1.63) i=1

i (i = 1 n) < , f (k) - (! *% ! * % % r11(1) . @ * ! i (i = 1 n) (( * 9 #. ; < ". (", % :

(k + 1) = (k)I (0) = (0)I

(8.1.64)

y(k) = 0y (k) + (k): (8.1.65) ! 9( ( k ) (8.1.63), - (8.1.63), * (8.1.15) * 164

f (k) = (k) . < y (k) - % (* % # ) ( . 7 ". (" (8.1.64), (8.1.65) % (9 #) ( (k) y(k) , ( % 0(k) ! (k) . 7 < 9 (??) : : : (??) % 9 #. * , * R(k) = En , R(k) = 0 , D(k) = 0(k) < 9 :

^ (k + 1) = ^ (k) + k(k + 1)My(k + 1) ; 0y (k + 1)^(k)]I

k(k + 1) = Pa(k + 1) y (k + 1)M 0y (k + 1)Pa(k + 1) y (k + 1) + r11(1)];1I Pa(k + 1) = ME ; k(k) 0y (k)]Pa(k): (8.1.67) (8.1.68), * Pa (k + 1) = Pa(k) ; Pa(k) y (k)My0 (k)Pa(k) y (k) + r11(1)];1 0y (k)Pa(k):

(8.1.66) (8.1.67) (8.1.68)

(8.1.69)

& (8.1.66), (8.1.69) (8.1.32) : : : (8.1.34), (* , * ' (8.1.32) : : : (8.1.34) 9 #, ( M9.1]

Pa (k) = Pi r11(1):

8.1.7 % - (: 211.3 (@ 9(# '! ( -

). @!

: ) d(s) , k(s) m(s) . ( * ' ! %. D * ,* . ( : d(s)y = k(s)u + m(s)f . P (

y(k) = ;

n X i=1

'iz;iy(k) +

n X X riz;j u(k) + ciz;j f (k) j =0

j =0

(8.1.70)

)B * 9 ! * ( n X u(t) = k sin !k t (8.1.71) k=1

)@ ( , , % * % ( % * (8.1.71) 165

); ' . )7 # 9(# * % * (8.1.71) $ : )# ( (<99# 'i ri ( % (8.1.70) )= 9( # 9(#. @ ( 211.3 (( .3.2. # % # 9(# ' ! ( ) -9 (#: 1) th = armax(p nn) -# ( (<99# ( % (8.1.70) '! ( B 9 (#: p = My u]- # <( !

!I * u # % * # * !. nn = Mn' nr nc] - z1 ( % (8.1.70) $ : # ( (<99# ( % (8.1.70) 2) thm = rarmax(p nn adm adg) -# ( (<99# ( % (8.1.70) (

'! ( .) adm adg , # 9(# , , * adm = ff adg = lam (

% 8.1.3. ; ( <! 9 (#! y(k ; 1) : : : y(k ; n)

u(k) : : : u(k ; n) < * y(k) ( (8.1.55)

8.2 A- , - 3 9(# ! . ( .; ! "(* (%" * % ( * -* % 9(#. ; M8.8],M8-9],M8-10] . ( ,( % % ( * * ! (.5<99# * % 9 (# . ( ! # % ( ! * ! ! ( ( * ! ! ( ( , ! ( (<99# .; ' # (

% # % '! ( . ; ( * -* M8-11] % * ( ( ( % . () <

' ( % !, ( ! ! * 9(#. 166

8.2.1

. # 8*

$ % * ( %* % . ( , %

y(n) + dn;1 y(n;1) + + d1y_ + d0y = kmu(m) + + k1u_ + kou + f (8.2.1) ( (<99# ai , kj (i = 0 n ; 1 (i = 0 m { *, n { , m (m < n) { , f (t) { *

9 (#

jf (t)j f

(8.2.2)

f { *. )* 9(# ! # ( d^i k^i (i = 0 n ; 1) (<99# (8.2.1) (!, * '( 9(# {di = di ; d^i , {kj = kj ; k^j

j{dij "di j{kij "ki (i = 0 n ; 1) (i = 0 n ; 1) {

*.

(8.2.3)

( ! "di , "ki * <% * ! , *

!* ( ! ( ( f (t) . % ( , * ! . ( y(t) ! (<99#

, <, * 9(# * # , # 9(#. 4 ! !, ( ( ! (<99# . 7 <% # % n X u(t) = k sin !k t (8.2.4) k=1

( k (k = 1 n) * !k (k = 1 n) {

*. ; < * ! . ( ((

y(t) = y$(t) + y0(t) + yf (t) (8.2.5) y$(t) { ,

(8.2.4), y0(t) { * ! %, yf (t) f (t) . & y$(t) < ! ! 9(#, < % * (8.2.3). ,( 9 1 , ( ! # ( * ! . 167

8.2.2 # 2n *

k = Re w(j!k ) k = Im w(j!k ) (k = 1 n)

w(s) = kd((ss)) . 4% * ! (<99# . ( * , *

k(j!k ) = + j (k = 1 n): k d(j!k ) k % k(j!k ) ; (k + jk)dO(j!k ) = (k + jk )(j!k )n (k = 1 n) dO(s) = d(s) ; s = dn;1 sn;1 + + d0 . ) <% * ! # ( ^ k , ^k (k ( , (( ( , ! 9 1 , % 9 , * nX ;1 nX ;1 (j!k )ik^i ; (^k + j ^k ) (j!k )id^i = (^k + j ^k )(j!k )n (k = 1 n): i=0

i=0

(8.2.6)

(8.2.7) (8.2.8) = 1 n) , (8.2.9)

D 2n % ! * (! % 2n ! k^i d^i (i = 0 n ; 1) .

5 6. P . ( , * !k

(k = 1 n) { * * * (^k = k (^k = k k = 1 n), M8-11] * (8.2.9)

' d^i = di k^i = ki ( kn;1 = = km=1 = 0) < ' ! *.

. )' * * n = 2 . ; < * . (

yY + d1y_ + d0y = k1u_ + k0u: (8.2.10) 3 # ( (<99# < . ( k^0(j!k )k^1 ; (^k + j ^k )(d^0 + (j!k )d^1) = (^k + j ^k )(j!k )2 (k = 1 2): @! (( 4 %: 168

(8.2.11)

k^0 ; ^ k d^0 + ^k !k d^1 = ;^k !k2 (k = 1 2) !k k^1 ; k d^0 ; ^k !k d^1 = ;^k!k2 4 ! d^1 , d^0 , k^1 , k^0 .

(8.2.12)

8.2.3 I # ! . ( (8.2.1),

(8.2.4) ! 9 1 , Z 2 k ( ) = y(t) sin !k t dt k 0 (k = 1 n) (8.2.13) Z 2 k ( ) = y(t) cos !k t dt k 0 { 9 #. * ^k ^k (k = 1 n) ! 9 9(

. , * (f (t) = 0) ! 9 ! ( * : lim !1 k ( ) = k lim !1 k( ) = k

(k = 1 n): (8.2.14) & ! . (,

% ( y(t) % (! %) 9 (# w$(t) : y$(t) = y(t) + w$(t) . ; n X yb(t) = k Mk sin !k t + k cos !k t] : (8.2.15) k=1

(8.2.5) ! . ( * f (t) = 0 ' (8.2.15) (8.2.13), *, , ! 9

Z 2 1( ) = M1(t) sin !k t + 1 cos !k t] sin !k t dt + 10(t) + 1$(t) = 0

Z Z 1 1 = 1 ; cos 2!1 t dt + sin 2!1 t dt + 10(t) + 1$(t) 0 0

169

(8.2.16)

Z 2 10(t) = Mw$(t) + y0(t)] sin !1t dtI 1 0

(8.2.17) "X Z n

k Mk sin !k t + k cos !k t] sin !1t dt: 1$(t) = 2 1 0 k=2 ,( (( * ! % * , (8.2.16) lim !1 1( ) = 1 . 7 ! ! 9 # * < ' (8.2.14). 5 f (t) 6= 0 , < ' ' . 4 , f (t) (, * ( ! *, . ; < * " " * ! ( . ' , ( ! ! 9 ! ( * . ; <( * 9 (# Z 2 `k ( ) = yO(t) sin !k t dtI k 0 (8.2.18) Z 2 `k ( ) = yO(t) cos !k t dt k 0 ! 9 1 , ! ( "

%" ! . (, ( u(t) = 0 , (Oy(t) = y0(t) + yf (t)) .

. ; f (t) 11-9 (9

9 1 )

! * !k (k = 1 n) 9 # ( , *

j`k ( )j "k

`k ( ) "k (k = 1 n) ( ! "k "k (k = 1 n) {

*. P ( , * lim !1 `k ( ) = lim !1 `k ( ) = 0

(8.2.19)

(8.2.20)

11-9 .

4 , *, , n1 X f (t) = k sin !kf t k;1 170

(8.2.21)

n1 , k , !kf { *, 11-9 , !kf 6= !i ((i = 1 n) k = 1 n1) .

5 6. P f (t) 11-9 , 9 # ( , * '( 9 # {k ( ) = k { k ( ) = k ; k( ) (k = 1 n),

; k( ),

j{k( )j "k j{k( )j "k(k = 1 n)

(8.2.22) P { 11-9 , < '( { * 9 (# lim !1 {k ( ) = lim !1 {k( ) = 0

(k = 1 n)

(8.2.23)

7( 9 (# # ( * !

k ( ) = k + ek( ) + `k ( ) k ( ) = k + ek ( ) + `k ( )(k = 1 n) (8.2.24) ( ek( ) ek ( ) (k = 1 n) { * ! 9 (#% w$(t) y0(t) < lim !1 ek ( ) = lim !1 ek ( ) = 0 (k = 1 n) . * (8.2.19), * (8.2.22) (8.2.23).

8.2.4 , - 7 , * ( % * !' . D * , * !k = ck !' (k = 1 n) , ck (k = 1 n) {

# *. > ! 9 1 ( = qT' , % T' = 2! , q = 1 2 : : : . ; < * , (( ( , '( ' 9 # ', ( (( ! (8.2.17) . 7 9(# ! ! 9(#

jdi(qT') ; di M(q ; 1)T')]j "di jki(qT') ; ki M(q ; 1)T')]j "ki (i = 0 n ; 1) (8.2.25) di(qT') ki(qT') (i = 0 n ; 1) { # ( (<99# . (, * -

' * ! % (8.2.9), ( ! ^k = k (qT') , ^k = k (qT') q = 1 2 : : : . . ( ( * -* % 9(#):

! . ( (8.2.1),

(8.2.4), ( ! 9 1 (8.2.13)I 171

! k (qT') k (qT') (k = 1 n) < 9 = (qT') q = 1 2 : : : I

% # ( (<99# . (, ' ( = (qT') * (8.2.9)I

!

(8.2.25) ( q ! , ( < ( q = q1 .

; = q1T' * # . P # (

% (8.2.3) ( * 9(#. & 10.2 ( * 10.2.3). P 9(# , 9(# ! , ( ! * . 7 # 9(#, % (8.2.3), % . P 11-9 , # 9(# ! (

* (<99# . (: qlim !1 di (qT') = di qlim !1 ki(qT') = ki

(i = 1 n):

(8.2.26)

D (8.2.1) (8.2.2). P ( 11-9 , % (8.2.3) # "i "i (i = 1 n) ! (8.2.19).

8.2.5 &% ; ' , * * (8.2.4) . ; % ! < 9(#, <( M8.2]. P * , (

! " ":

jy(t) ; yO(t)j "y

(8.2.27)

( "y {

*. D * , * % "

%" ! . ( y(t) ' !

( "y . * % ! * % * %. @ , * < * *, 9* ( -* ! ( ( (-B3b) . ( . 4 % ( , * < * 8.3.2, ( ' * ! % ! *. (, % , < ' * ! * ! !, * ! ! ! # (, < 172

9(# ! *. ; M8-12] <( ! *.

8.2.6 % - (: 111.4 (5 * -* 9(# ).

@!

: ) d(s) , k(s) m(s) . ( (8.2.1), ( % : d(s)y = k(s)u + m(s)f . ); ' . )B * (8.2.4) )7 # 9(# * % * )@ ( ,, % * % ( % * . $ : )# ( (<99# di i = 0 n kj j = overline0 m . ( (8.2.1) )= 9( # 9(#. @ ( 111.4 (( .3.1. # % # 9(# '

9

$# ( #+#) #%(

; <% * ( . B <! , (* 9(#

# # '! ( .

9.1

.

$ % ( % . ( , %

y(k) + '1y(k ; 1) + + 'ny(k ; n) = r1u(k ; 1) + + r(k ; ) + f (k)

(9.1.1)

y(k) { ( ! )

. (I f (k) { ' % , *% # * (*% # " % '"), *

M ff (k)g = 0I M ff 2 (k)g = f2 (9.1.2) ( f2 { ( *), 'i , rj , (i = 1 nI j = 1 ) { *. 173

, % , ( # , ( lim M fy2(k)g < {

(9.1.3)

k!1

( { {

*. ; ( < , , * 'i , ri , (i = 1 nI j = 1 ) . ( (9.1.1) . (9.1.3) * { = 1 . A lim M fy2(k)g < 1 (9.1.4) , * * ( %* . ( * (% %*, * ( ! ( * ( ( %

* . ,( , * . ( (9.1.1). k!1

' 0u(k) + ' 1u(k ; 1) + + ' nu(k ; n ) = r 0y(k) + + r y(k ; )

(9.1.5)

' i r j (i = 0 n j = 0 ) { ( *. (9.1.1), (9.1.5) - ! * ! !, ' . (

'(z;1)y = r(z;1)u + f (k)I

(9.1.6)

' (z;1)u = r (z;1)y + r 0y

(9.1.7)

n X X '(z;1) = 1 + 'iz;iI r(z;1) = riz;i I

(9.1.8)

z { ( ( *I

' (z;1) = ' 0 +

i=1 n X i=1

i=1 X

' iz;i I r (z;1) =

i=1

r iz;i :

(9.1.9)

7 * z;i = i (9.1.8) = n , (<99# ! ! z . B * , (9.1.9) = n ( (9.1.5) , * = n . <! ! ! ( * (% (9.1.6), (9.1.7)

X ! X ! X ! X ! n n n n i i i i D() = '()' () ; r()r () = 1 + 'i ' i ; ri r i : i=1 i=0 i=1 i=0 (9.1.10) 174

| % ( % Y 2n !;1 Y 2n D () = i (i ; i ) = d22nn2n + + d1 + 1 i=1

i=1

(9.1.11)

i (i = 1 2n) {

*, jij 1 (i = 1 2n) . 7 (9.1.5) , *

r 0 = 0:

(9.1.12)

5 ,

' 0 = 1: (9.1.13) 7 ! (9.1.5) (9.1.10), (9.1.11). ,

1+

n X i=1

'ii

!

1+

n X i=1

' i

! X n i

;

i=1

rii

! X n i=1

! X 2n r ii ; di i ; 1 = 0: i=1

(9.1.14)

(<99# ( ! ! , * % ! * (! % ! ' i r i (i = 1 n) . D

N ( ) = d (9.1.15) 0 = k' 1 : : : ' n r 1 : : : r nk0 I d = d2n d2n;1 : : : d1 I N () { # *

2n 2n , < ( % . ( (9.1.1),

(( ( ( = k;'1 : : : ;'n r1 : : : ;rn k . ; M?] ( , * . ( (9.1.1) M< * , * '() r() ! ( %] d2n 6= 0 , (9.1.15)

' ( ! ' i r i (i = 1 n) .

5 6.. # (9.1.5), (-

! ( * (% ( % (9.1.1), (9.1.5)

* i (i = 1 2n) (* : 1) 9 (<99# di (i = 1 2n) (9.1.11)I 2) (9.1.14) # * N () (9.1.15)I 3) ' (9.1.15) % (9.1.5).

. ).

$ , ( ( % (4.3.31), (4.3.32).

x1(k + 1) = x1(k) + '12x2(k) + '13x3(k) + r1(u(k) + f (k))I 175

(9.1.16)

x2(x + 1) = '22x2(k) + '23x3(k) + r2(u(k) + f (k))I x3(k + 1) = '32x2(k) + '33x3(k) + r3(u(k) + f (k))I y(k) = x1(k) + (k): , %

(9.1.17) (9.1.18) (9.1.19)

u(k)+' 1u(k ;1)+' 2u(k ;2)+' 3u(k ;3) = r 1y(k ;1)+r 2y(k ;2)+r 3y(k ;3) (9.1.20) ( , * ( ! ( * ( ( % (9.1.16) ... (9.1.20)

* 1 2 3 4 5 6 . ! ( ' <% *, (9.1.16) ... (9.1.19) ( (9.1.1). 7 < ' '

y(k) = x1(k) + (k) = dx(k) + (k)I d = k1 0 0kI y(k ; 1) = dx(k) + (k ; 1) = dR;1x(k) ; dR;1r(u(k ; 1) + f (k ; 1)) + (k ; 1)I y(k ; 2) = dR;2x(k) ; dR;2r(u(k ; 1) + f (k ; 1)) ; dR;1r(u(k ; 2) + f (k ; 2)) + (k ; 2): $ ' < ! % ! ( x(k) *

x(k) y(k ; 3) = dR;3x(k) ; dR3r (u(k ; 1)+ f (k ; 1)) ; dR;2r(u(k ; 2)+ f (k ; 2)) ; dR;1r(u(k ; 3) + f (k ; 3) + (k ; 3) , * 9 "!- !" (9.1.1): y(k) + '1y(k ; 1) + '2y(k ; 2) + '3y(k ; 3) = r1u(k ; 1) + r2u(k ; 2) + r3u(k ; 3)+ +r1f (k ; 1) + r2f (k ; 2) + r3f (k ; 3) + r0(1)(k) + r1(1)(k ; 1) + r2(1)(k ; 2) + r3(1)(k ; 3): (9.1.21) ( ( ' ! ' <

y(k)+ '1y(k ; 1) + '2y(k ; 2)+ '3y(k ; 3) = r1u(k ; 1) + r2u(k ; 2) + r3u(k ; 3): (9.1.22) ; # % . ( 9 % ( %

D () = M123 456];1 ( ; 1 ) ( ; 2) ( ; 3) ( ; 4) ( ; 5 ) ( ; 6) = = d66 + d55 + d44 + d33 + d22 + d1 + d0: b ( * (% (9.1.20), (9.1.22) 176

(9.1.23)

D() = 1 + '1 + '22 + '33 1 + ' 1 + ' 22 + ' 33 ; (9.1.24) 2 3 2 3 ; r1 + r2 + r3 r 1 + r 2 + r 3 : & (<99# ( ! ! (9.1.23), (9.1.24), * * (! % (9.1.15): 9 '3' 3 ; r3r 3 = d6I '2' 3 + '3' 2 ; r2r 3 ; r3r 2 = d5I > > > = '3' 1 + '2' 2 + '1' 3 ; r3r 1 ; r2r 2 ; r1r 3 = d4I (9.1.25) > '3 + '2' 1 + '1' 2 + ' 3 ; r2r 1 ; r1r 2 = d3I > > '2 + '1' 1 + ' 2 ; r1r 1 = d2I '1 + ' 1 = d1: $ ' < ' % ! %, * ( * ' 1 ' 2 ' 3 r 1 r 2 r 3 (9.1.20).

9.2 - 9.2.1 8# ! ( 9(# . ( (9.1.1), 9

y(k +1)+'1y(k)++'ny(k ;n+1) = r1u(k)++r u(k ; +1)+f (k +1) (k = 0 1 2 : : :): (9.2.1) ; * )

(k) = ky(k) : : : y(k ; n + 1) u(k) : : : u(k ; + 1)k0 I (9.2.2) = k;'1 : : : ;'n r1 : : : rk0 ' (9.2.1) (( y(k + 1) ; 0(k) = f (k + 1): ; ( ! ( ( ( N X LN = N1 My(k + 1) ; 0(k)]2 : k=0

; '! ( < , ! # ( < ( , ' (??), (??). ( ! 9 , ( ( ( (k) <! ' ! *

% y * , ( (k) , % (9.2.2), (* !

177

u . P u(k) ! (9.1.5), ( # ,

% 9.1.1 , * (9.1.15) # (%, , ! ( ( ( . ,( (

' : (k + 1) = (k) + P (k) (k) My(k + 1) ; 0(k)(k)] h(k)c(k)I (9.2.3) (0) = (0)I P (k + 1) = P (k) ; P (k) (k) 0(k)P (k)h(k)c(k)I (9.2.4) nX !;1=2 + 2 h(k) = 1 + i (k) I c(k) = (1 + h(k) 0(k)P (k) (k));1 (9.2.5) i=1

( ( % 9(# % ' % (??) ... (??) h(k) = 1 * * % .

5 6. . ( , % (9.1.1),

. B % , *% # (9.1.3) { = 1,

u(k)+' 1(k)u(k ;1)+: : :+' n(k)u(k ;n) = r 1(k)y(k ;1)+: : :+r n(k)y(k ;n) (9.2.6) ( ' % * (

N ((k)) (k) = d (9.2.7) (k) { ( # ( . ( (9.1.1), * ! (

! ' % (9.2.3) ... (9.2.5). ; <! ' ! ( (k), % (9.2.2),

. * '! * ! k ( j(k) ; j E" (" { * *) ( ! ( * ( (9.1.11), (9.2.6) ( (

* i (i = 1 2n). & 9 ( < ( M10.3]. ; ( M6.5] < *%, ( { 6= 1 . , * ( ( J = lim M fy2(k)g t;+ 2 2 J = klim ;+ M fy (k) + u (k)g I ( , M6.5] ( . . ). $ , ( ( % (9.1.16) ... (9.1.19).

. * <! 178

% * . ,(, , ! ( * (% ( . < ( ( * ( ( ! # , < (# . D * , * (9.1.16) ... (9.1.19) 'ij ri (i j = 1 3)

,

* . (9.1.16) ... (9.1.19) ( (9.1.21) , * ( ' ! % ! % * (9.1.21) # " % '". $

u(k) + ' 1(k)u(k ; 1) + R 2(k ; 2) + ' 3(k)u(k ; 3) = (9.2.8) = r 1(k)y(k ; 1) + r 2(k)y(k ; 2) + r 3(k)y(k ; 3) ( ! (( ' %,

!

(9.1.25):

9 > ;3(k)' 3(k) + 6(k)r 3(k) = d6I > > ;2(k)' 3(k) ; 3(k)' 2(k) ; 5(k)r 3 ; 6(k)r 2(k) = d5I > ;3(k)' 1(k) ; 2(k)' 2(k) ; 1(k)' 3(k) ; 6(k)r 1(k) ; 5(k)r 2(k) ; 4(k)r 3(k) = d4 I >= > ;3(k) ; 2(k)' 1(k) ; 1(k)' 2(k) + ' 3(k) ; 5(k)r 1(k) ; 4(k)r 2(k) = d3I > > > ;2(k) ; 1(k)' 2(k) ; 4(k)r 1(k) = d2I > ;1(k) + ' 1(k) = d1:

(9.2.9) # ( 1(k) (k = 1 6) . ( (9.1.1), ! (9.2.9), (

' :

26 32 3 6 X X i(k + 1) = i(k) + 4 pij (k)j (k)5 4y(k + 1) ; (k)(k)5 h(k)c(k)I =1

j =1

P (k + 1) = P (k) ; P (k)(k)0(k)P (k)h(k)c(k)I (i = 1 6) X !;1=2 6 2 h(k) = 1 + i (k) I i=1 2 3;1 6 X c(k) = 41 + h(k) pij (k)i(k)j (k)5 i j =1

(9.2.10) (9.2.11) (9.2.12) (9.2.13)

P (k) { * # 6 6 I (k) { ( (

1(k) = y(k)I 2(k) = y(k ; 1)I 3(k) = y(k ; 2)I 4(k) = u(k)I 5(k) = u(k ; 1)I 6(k) = u(k ; 2): 179

(9.2.14)

9.2.2 % - (: 411 (B '!

( ). @!

: ) d(s) , k(s) m(s) . (,( % d(s)y = k(s)u + m(s)f . P ( (9.1.1) )@ ( , . ); ' . )5<99# * (9.1.5)(; (* <! (<99# *). )| % (9.1.11) ( % . )7 # #. $ : )= 9( # #. @ ( 411 (( .3.4. # % # # ' .

10 $# '$ $ 7 #(+, )(+ 7%':"+ 9 10.1 9 (# - ! 9 ( . ( (

! # ! M10.1]. ; x10.2. * ( - , * M10.3] ; (* 9(#

( * - * % 9(#.

10.1 # -# 10.1.1 . . $ ( % . ( , %

y(k + 1) + '1y(k) + + 'n y(k ; n + 1) = r0u(k) + + ru(k ; ) + f (k) (10.1.1) (k = 0 1 : : : I < n)

y(k) (

, ' % f (k) *

% ! *

jf (k)j f 180

(10.1.2)

f {

*. '1 : : : 'n , r0 : : : r . ( . @ ' , * . ( (10.1.1) -9 . D * , * ( r0 +r1+ +r % jij 1 (i = 1 ) . , ( , ( (<99# r0 ! # ( jr0j { * cr jr0j cr . #

jy(k + 1)j {

(10.1.3)

{ {

*,

' ! %%. 7 , * { f : (10.1.4) , % , !

( u(k) * . ( (10.1.1) # (10.1.3). , * 9* (% . ( * # , * (10.1.3),

e2(k + 1) {2 ( e(k + 1) (( e(k + 1) = y(k + 1) + { #

% (%

y(k + 1) +

l X i=0

Xl i=0

'iy(k ; i)

'iy(k ; i) = 0 e(k + 1) = y(k + 1) ; g(k)

(10.1.5) (10.1.6)

(10.1.7)

{ % g(k) ,

e(k + 1) = y(k + i) ; y%(k + 1) { <

% ,

%

(10.1.8)

y%(k + 1) + '%1y%(k) + : : : + '%ny(k ; n + 1) = r%0g(k) + : : : + rnl g(k ; l): , * #

% # (

e(k + 1) = y(k + 1) (10.1.3) (10.1.5). 181

(10.1.9)

10.1.2 ' . ( * ! < , , * ! ! 'i rj (i = 1 nI j = 0 ) . ( (10.1.1) , ( # (10.1.3), * . 7 % , <

u(k) = r0;1 M'1y(k) + : : : + 'ny(k ; n + 1) ; r1u(k ; 1) ; : : : + ru(k ; )]

(10.1.10)

, (10.1.10) (10.1.1) *

y(k + 1) = f (k): P 9 (# f (k) (10.1.2),

jy(k + 1)j f {

(10.1.11) (10.1.12)

, , # . > , ( (10.1.10) 9 (#

J1 = klim jy(k + 1)j (10.1.13) !1 jf (sup t)jf ( (( < ' * , f . P # (10.1.5), ( ( e(t) (10.1.6) ... (10.1.8), ( , ( ! < # , : ) #

% (% 2n;1 3 X X u(k) = r0;1 4 ('i+1 ; 'i)y(k ; i) ; rj u(k ; j )5 I (10.1.14) i=0

j =1

) % 2 3 nX ; 1 X u(k) = r0;1 4g(k) + 'i+1y(k ; i) + rj u(k ; j )5 : i=0

j =1

(10.1.15)

( (10.1.10) (( % 9 . 7 < n + - ( :

= ky(k) : : : y(k ; n + 1) u(k ; 1) : : : u(k ; )k0I 0 = 'r 1 : : : 'r n ; rr1 : : : ; rr : 0 0 0 0 182

(10.1.16) (10.1.17)

, (10.1.10)

u(k) = 0 (k):

(10.1.18) D ' , ( ( * # . )( (10.1.14), (10.1.15) ( (10.1.18), (10.1.14) '1 ; '1 ' r r n ; 'n 1 0 = r : : : r ; r : : : ; r (10.1.19) 0 0 0 0 (10.1.15)

(k) = ky(k) : : : y(k ; n + 1) u(k ; 1) : : : u(k ; ) g(k)kI = 'r 1 : : : 'r n ; rr1 : : : ; rr 1 : 0 0 0 0

(10.1.20) (10.1.21)

* (10.1.18), ' . ( (10.1.1) % <( % 9 :

y(k + 1) = r0Mu(k) ; 0 (k)] + f (k): (10.1.22) )( ( 9 , * % (10.1.18), % ( ( ! (k) , , ( , u(k) = 0(k) (k) .

(10.1.23)

10.1.3 - %. 8# .0) ; < % , ( % . $ * *% ' ! %

f (k) = 0: ; < * #

(10.1.24)

(10.1.25) lim e2(k) = 0: ; ! (k) # 9 (# h i2 n o2 r0;2e2(k + 1) = u(k) ; 0 (k) = M(k) ; ]0 (k) (10.1.26) . D * , * k!1

183

n o2 @ M(k) ; ]0 (k) 2 i(k + 1) = i(k) ; a~1(k) r0 : (10.1.27) @i(k) * (10.1.22), , * n ]0 (k)o2 n @ M ( k ) ; ]0 (k)o i(k)r02 = y(k + 1)r0i(k) r02 = 2 M ( k ) ; @i(k) , ( , ( % i(k + 1) = i(k) ; ~a1(k)r0y(k + 1)i(k) (i = 1 n + ): (10.1.28) ! ( (* ) < , a~1(k) (10.1.28), * a1(k) = a~1(k)r0:

(10.1.29)

9 i(k + 1) = i(k) ; a1(k)(sign r0)y(k + 1)i(k)I = i(0) = i(0) (i = 1 n + ):

(10.1.30)

, (10.1.28)

5 6. a1(k) # (10.1.30), ( -

% (10.1.23), (10.1.30) * . ( (10.1.1) f (k) = 0 # (10.1.25), " nX #;1 + 2 a1(k) = cr i (k) (0 < < 2): (10.1.31) i=1

7 ( < 9 (# -

(k) =

nX + i=1

(i(k) ; i)2 > 0

(10.1.32)

( " " ! (10.1.23) ( (10.1.18). 4% * ((k)) ( % (10.1.22), (10.1.23), (10.1.30). 7 < {(k) = (k + 1) ; (k) = M(k + 1) ; ]0 M(k + 1) ; ] ; M(k) ; ]0 M(k) ; ] : (10.1.33) * (10.1.30) (10.1.26), ' f (k) = 0 , * 184

M(k + 1) ; ]0 M(k + 1) ; ] = M(k) ; a1(k)(sign r0)y(k + 1)(k) ; ]0 M(k) ; a1(k)(sign r0)y(k0 + 1) (k) ; ] = M(k) ; ]0 ((k) ; 0]; ;a1(k)(sign r0)y(k + 1)f2 (k)M(k) ; ] ; a1(k)(sign 2 r0)y(k + 1) (k) (k) = = M(k) ; ]0M(k) ; ] ; a1(k)(sign r0)y(k + 1) r y(k + 1); 0 ;a1(k)(sign r0)y(k + 1) 0(k) (k)] : (10.1.34) < (10.1.33), * sign r0 0 {(k) = ;a1(k) 2 r ; a1(k) (k) (k) y2(k + 1): (10.1.35) 0 4 , * {(k) 0 , 0 a1(k) jr j0(2k)(k) c 0(k2)(k) : (10.1.36) 0 r 7 * * {(k) ( y(k +1) 6= 0) , * a1(k) = c 0(k )(k) (10.1.37) r 0 < < 2: ( * a1(k) *

(10.1.38)

2(k + 1) {(k) = ; y (10.1.39)

0(k) (k) < 0 { ( *. ,( (( 1 1 2 (0) (10.1.40) ; 1 X {(k) = X y0((kk)+ (k1)) = ( ) < 1 k=0 k=0 2(k + 1) y < * , * 0(k) (k) ! 0 , , , # # lim y(k + 1) = 0 , ( * 0(k) (k) *

. k!1 *

( (k) , * % ! ! . ( * , * %. 4 , %* * ! * ! , ! '! # #

. ( . D * % %( , , ' ! y(k +1) ). ( M6.5], * . ( -9 , 0(k) (k) * , ( , ( .

185

10.1.4 - 8# .0# $ . ( (10.1.1) <( % . ( (10.1.22) * ' ! %, ! (10.1.2). 4% " " (10.1.32) ! % . 4 , * * (10.1.35) " ! # f ( k ) 2 {(k) = ;a1(k) jr j 1 ; y(k + 1) ; a1(k)0(k)(k) y2(k + 1): (10.1.41) 0 D * , (10.1.34) 0(k)M (k); ] y(k+1);f (k) , y(k + 1) , (( . P * y(k + 1) , ' % f (k) % (, * {(k) % ( # #. D99 ( # % % * . @ (10.1.41) , * (k) ( jy(k + 1)j > { (, *

8 > < i(k) jy(k + 1)j { (i = 1 n + )I i(k + 1) = > (k) ; (sign r0)y(k + 1) (k) jy(k + 1)j > { : i c 0(k)(k) i

(10.1.42)

r

2 ! f 0< <2 1; 2 (10.1.43) { , (( f (k) , (10.1.40) , , *

0(k)(k) # (10.1.3) . *

# ! #

% # ( #

% (% . 7 % , ( e(t) (10.1.6) (10.1.8), . ( (10.1.1) % <( % 9 :

e(k + 1) = r0Mu(k) ; 0 (k)] + f (k) (10.1.44) ( (k) ' (10.1.17) ... (10.1.21)), # (10.1.42) ! ' !, * (10.1.28), y(k + 1) e(k +1) . ,( , (

.

186

5 6. -9 % . ( , -

% (10.1.1), ( ( ' ( r0 * cr jr0j < cr , ' % . (

, * * f ). B % , *% # (10.1.5) M e(k + 1) % (10.1.6), (10.1.7), (10.1.29)],

u(k) =

nX + i=1

i(k)i(k)I

8 > i(k) je(k + 1)j { (i = 1 n + )I > < (sign r0)e(k + 1)i(k) je(k + 1)j > { i(k + 1) = > i(k) ; nX + > c i2(k) : r

(10.1.45) (10.1.46)

i=1

2! f (10.1.47) 0 < < 2 1 ; {2 ( i(k) ( (k) ' % (10.1.16), (10.1.20). B # (10.1.42) * (

! # ! ,

;. B. T(* M10.1]. 7 (

(

# ! , ( # # (10.1.42) . 4 <

M6.5] *% ! . (, ( *% , * % . . % !(- ! * (! # ,

% ! 6.1.1, 6.1.2, D #

x(k + 1) = ax(k) + bu(k)] + f (k)I

(10.1.48)

y(k) = dx(k) (10.1.49) ( ! a , b , d . ; * 6.1.1, 6.1.2 , * f (k) { . <

jf (k)j f :

(10.1.50) , % , ( # My(k + 1) ; g]2 {2: 187

(10.1.51)

; (10.1.50) (10.1.51) f , g , { {

*. )' * (10.1.48), (10.1.49) 9 (10.1.1):

y(k + 1) + '1y(k) = r0u(k) + f (k)

(10.1.52)

'1 = ;aI r0 = dbI f (k) = df (k): 7 , * # (

(10.1.53)

jr0j cr

(10.1.54)

( * r0 . ! ( , (10.1.20) (10.1.21) ( (k) (k) ( 1 = 'r 1 I 2 = 1I 1(k) = y(k)I 2(k) = g: (10.1.55) 0 , ( (10.1.45)

u(k) = 1(k)y(k) + 2(k)g: (10.1.56) B # < ' (10.1.46) ((

8 > < 1(k) jy(k + 1) ; gj {I 1(k + 1) = > (k) ; (sign r0)My(k + 1) ; g]y(k) jy(k + 1) ; gj > {: : 1 c (y2(k) + g2)

(10.1.57)

8 > < 2(k) jy(k + 1) ; gj {I 2(k + 1) = > (k) ; (sign r0)My(k + 1) ; g]g jy(k + 1) ; gj > {: : 2 c (y2(k) + g2)

(10.1.58)

r

r

10.2 10.2.1 . $ %, * ( %* % . ( , %

y(n) + dn;1 y(n;1) + + d1y_ + d0y = kmu(m) + + k0u + f (10.2.1) ( di , kj (i = 0 n ; 1 j = 0 m) { *, f (t) { , * 9 (#. 188

)* , , * tN , 99 #

d ( ;1) u(n;1) + + d 1 u_ + d 0 u = k ;1 y(n;1) + + k 1 y_ + k 0 y_ (<99# ( ( , * ! ( * (%

(10.2.2)

D(s) = d(s)d (s) ; k(s)k (s) = D2n;1 s2n;1 + + D0 ( (

(10.2.3)

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(10.2.4)

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(10.2.7)

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(10.2.8)

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(10.2.17)

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(7.1.33)

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,

* (x2.1), * * (7.1.7)

! ! # Zt1 "X n @' m @' X d @' d @' 0 0 0 0 (7.1.43) @x ; dt @ x_ xi + @u ; dt @ u_ uk dt = 0: t0 i=1

i

i

k=1

k

k

( ( < #

* , ( (( # xi uk (i = 1 nI k = 1 m) ! ( ! xOi(t) = xi(t) + xi(t) , uOk (t) = uk (t) + uk(t) (i = 1 nI k = 1 m) (7.1.23). * , *

x_ i + x_ i = 'i(x1 + x1 : : : xn + xn u1 + x1 : : : um + um) (i = 1 n): 228

(7.1.44)

; * <! % (7.1.23), *,

x1.2, #% M

, * uk (t) (k = 1 m) , <( ]: n @' m @' X X i x_ i = @x xj + @u i uk (i = 1 n): (7.1.45) j =1 j k=1 k 3 <! ! * <( % xi(t) , uk (t) (i = 1 n k = 1 m) . ( <! % (7.1.45) i(t) , (i = 1 n) , * 2 3 Zt1 m @' n @' X X i i _ i(t) 4x_ i ; @x xj ; @u uk5 dt = 0: (7.1.46) j =1

t0

j

k=1

k

@ * * , * xi(t)jt=t0 = xi(t)jt=t1 = 0 * 3 Zt1 2 n @' m @' X X i i 4;_ ixi ; i 5 (7.1.47) @x xj ; i @u uk dt = 0 (i = 1 n): t0

j

j =1

k=1

k

& *

(7.1.47), 0n 13 Zt1 2 X n n m @' X X X @' i 4; _ ixi ; i @ xj + @u i uk A5 dt = 0 @x i=1 i=1 j =1 j k=1 k t0

3 Zt1 2 X m X n @' ! n n X n @' ! X X i i 4; _ j xj ; i uk5 dt = 0: i xj ; @x @u j k i =1 j =1 j =1 i =1 k=1 t0 & < (7.1.43), * ! !# Zt1 X n " @' n d @' 0 X @'i 0 ; @x i ; dt + @x_ x dt+ @x =1 i=1 t0 " ! !# Zt1 X m n @' d @'0 u dt = 0: 0 X @'i + ; i ; k @uk i=1 @uk dt @ u_ k t0 k=1

(7.1.48)

(7.1.49)

; i(t) (i = 1 n) (, * 99 # ! n @' @'0 ; X d @ ' i 0 (7.1.50) @x @x i ; dt + @ x_ = 0:

i=1

229

7 , i(t) (i = 1 n) ' (7.1.50). D ' n !

!. ( % - (7.1.49) ! !# Zt1 X m " @' n d @'0 u dt = 0: 0 X @'i ; (7.1.51) i ; k @u @u dt @ u_ t0 k=1

k

i=1

k

k

) # uk (t) (k = 1 m) , <, ( (7.1.51) #

* , * n @' @'0 ; X d @'0 = 0 (k = 1 m): i (7.1.52) i; @uk i=1 @uk dt @ u_ k ,( , ( .

;$ 2

# %) $ %% " " #(+ #%. = 4'5& ' " 7 * * * * <% *, ( . ( ( x_ = '(x u) . 5 ,

< % . ( ( . ; < (t0 t1) N ! *( * %

T ' ( * x(t) = x(kT ) u(t) = u(kT ) , t0 t = t , t = t + T , t = t + 2T , : : : , (k = 0 N N = t1 ; 0 0 0 T t = t0 + (N ; 1) , t = t0 + N T

( t0 = 0 ). 799 #

( * ! !

xM(k + 1)T ] ; xMkT ] = 'MxMkT ] MkT ]] T

x(k + 1) = 'Mx(k) u(k)] (k = 0 N ) (7.2.1) ' = x(k) + T'Mx(k) u(k)] , ( , % ( * ' , ( * T . 4* < : x(0) = x0: (7.2.2) 7 '0(x u t) = q(x(t)) + (u(t)) , q(x) , (u) {

9 (#. )

(2.3.3) % 230

JN = T

N X

fqMx(k)] + Mu(k)]g = T

k=0

(

N X

)

qMx(o)] + MqMx(k)] + Mu(k ; 1)]] + Mu(M )] : i=1

, * u(N ) = 0 , x(o) T ( , ' 9 (# (*

JN =

N X k=0

fqMx(k)] + Mu(k ; 1)]g:

(7.2.3)

9 . 9 (# (7.2.4) u(k) = rk Mx(k)] (k = 0 N ; 1) (, * ' ! (7.2.1), (7.2.4) ! * ! ! (7.2.2) (7.2.3). < ( 9 (# * u 2 U . 7

, * < *

ju(k)j 1 (k = 0 N ; 1):

(7.2.5) &9

* , , * % <( 9 (# (7.2.3). 4

* u(0) , u(1) , : : : u(N ;1) .

1.2.1 ' . # . @ (7.2.1), (* x(1) , x(2) , : : : , x(N ) (7.2.3),

JN = Mu(0)] + qM'Mx(0)] u(0)]] + Mu(1)] + qM'M'Mx(0)u(0)] u(1)]]+

(7.2.6) + Mu(2)] + : : : + qM'M'M: : :u(N ; 2)] u(N ; 1)]] + Mu(N ; 1)]: P % <% 9 (# N

! u(0) , : : : u(N ; 1) ju(k) < 1 , . . U , ! <( 9 (# (7.2.6) % N % ! * (! % @JN = 0I @JN = 0I : : : @JN (7.2.7) @u(0) @u(1) @u(N ; 1) = 0: 5 , % ! % ' '! N = 10 : : : 20 . ( * (! ( *

. N = 10 . > 9 (# (7.2.6) * * * ! * ! u(0) , u(1) , : : : u(9) . 231

$ M;1 1] ! * % u(k) 10 * % * * ( *

! ( 10 * % u(0) , : : : u(9) . P * % ' 1 * 9 (# (7.2.6) % *( , * * % 1010 2,77 . * ( 10 ). ,( , < ! ( ' * (7.2.1). > , < ( * u(0) , u(1) , : : : u(N ; 1) , ( ! 9 (# (7.2.6) ' * , < * ' * (7.2.1), ( ( %

% 9 (# (7.2.4), (! * % x(k) . *

* u(0) , : : : u(N ; 1) ( x(0) .

1.2.2 ' . - > , * N 9( , * * * 7 2.1, * * N , ! ( , * ( (* # . D * * N -' # . * N * 9 (# (* (7.2.3) N -' # . ( N = 1 . 7 < ' # (7.2.1) 9 (# (7.2.3)

x(1) = 'Mx(0) u(0)]I

(7.2.8)

J1 = qMx(1)] + Mu(0)]:

(7.2.9)

4 , *

1Mx(0)] = umin fqMx(1)] + Mu(0)]g = umin fqM'Mx(0)] u(0)]] + Mu(0)]g: (0)2U (0)2U

(7.2.10)

' * # u(1)(0) = r(1)Mx(0)] ' # . ( 1. x(0) = x(0) , x(0) ( 9(

*. ,

J1 = qM'Mx(0) u(0)]] + Mu(0)]: (7.2.11) $ M;1 1] ! % u(0) i ( % 2i * * 9 (# (7.2.11) *(! . ( * 2 j i = j < 9 (# ' * J1 = Mx (0) i ; 1 . ) 2j u (0) = i ; 1 . 232

x(0) = x(0) . , J1 = qM'Mx(0) u(0)]] + Mu(0)] . ( u(0) = 2ik ; 1 < 9 (# ' * . ,( , * * x(0) , * * u(0) . D ! * % u(0) * ! % x(0) (% 9 (# %

u(1)(0) = r(1)Mx(0)]

(7.2.12)

% * . @ < 9 (#, * ' * 9 (# (7.2.9):

1Mx(0)] = qM'Mx(0) r(1)Mx(0)]]] + Mr(1)Mx(0)]]: $ &(*+, &" N = 2 . ; < * x(1) = 'Mx(0) u(0)]I x(2) = 'Mx(1) u(1)]I

(7.2.13) (7.2.14)

J2 = qMx(1)] + Mu(0)] + qMx(2)] + Mu(1)]: (7.2.15) J2 = J20 + J200 , J20 = qMx(1) + Mu(1)] I J200 = qMx(2)] + Mu(1)] . ) * J2 u(2)(0) u(2)(1) . < J20 u(0) , J200 { u(0) u(1) , ( (( x(2) x(1) , ( * u(0) . u(0) = u(2)(0) , u(2)(0) { ( , 9(

*. D ( x(1) * x(1) = 'Mx(0) u(2)(0)] . * * , * * ' * J2 u(2)(0) = u(2)(0) , ! u(2)(1) (, * J200 ' * . u(2)(1) , J200 , ' # (( (( u(2)(0) 9( ) 9 % (7.2.12), . . u(2)(1) = r(1)Mx(1)]:

(7.2.16)

< 00 = 1Mx(1)] = 1M'(0) u(2)(0)]:

min J u(2) (1) 2

(7.2.17)

) * 9 (# (7.2.15) u(2)(0) = u(2)(0) , u(2)(1) = u(2)(1) = r(1)Mx(1)]

J2 = qMx(1)] + Mu(2)(0)] + 1Mx(1)]: 0 < u(2)(0) , * 233

(7.2.18)

2Mx(0)] = u(2)min f q M'Mx(0) u(2)(0)] + Mu(2)(0)] + 1M'Mx(0) u(2)(0)]g: (0)2U

(7.2.19)

; 9 ! ((! 9 (# % %

% u(2)(0) . @ # 1, * 9 (# u(2)(0) = r(2)Mx(0)], ( % < ' * . ,( , !' #

u(2)(0) = r(2)Mx(0)]I u(2)(1) = r(1)Mx(1)]: < <( * 9 (# (7.2.15) 2Mx(0)] = qM'Mx(0) r(2)Mx(0)]]] + Mr(2)Mx(0)]] + 1M'Mx(0) r(2)Mx(0)]]]:

(7.2.20)

(7.2.21)

( N = 3 . 7 < * x(1) = 'Mx(0) u(0)]I x(2) = 'Mx(1) u(1)]I x(3) = 'Mx(2) u(1)]I

(7.2.22)

J3 = qMx(1)] + Mu(0)] + qMx(2)] + Mu(1)] + qMx(3)] + Mu(2)]: (7.2.23) J3 = J30 + J300 , J30 = qMx(1)] + Mu(0)] . u(3)(0) = u(3)(0) , u(3)(0) - , 9(

*. 7 < !' # , * ' 9( , ! u(3)(1) u(3)(2) (, * J300 = qMx(2)] + Mu(1)] + qMx(3)] + Mu(2)] ' * . 7 u(3)(1) u(3)(2) , *

!' # * x(1) = 'Mx(0) u(3)(0)] . 4 (7.2.20) * u(3)(1) = r(2)Mx(1)]I u(2)(2) = r(1)Mx(2)]: (7.2.24) ) * 9 (# (7.2.23) (7.2.24) u(3)(0) = u(3)(0) J3 = qMx(1)] + Mu(3)(0)] + 2(x(1)]: 0 u(3)(0) , * 3Mx(0)] = u(3)min fqM'Mx(0) u(3)(0)]] + Mu(3)(0)] + 2M'Mx(0) u(3)(0)]]g: (0)2U 234

(7.2.25)

; 9 ! ((! 9 (# % %

% u(3)(0) . @ # 1, !

u(3)(0) = r(3)Mx(0)]: (7.2.26) ,( , !' # (7.2.24), (7.2.26). < N = 4 5 : : : * ' N Mx(0)] = ( min f qM'Mx(0) u(N )(0)]] + Mu(N )(0)] + N ;1M'Mx(0) u(N )(0)]]g (7.2.27) ) u (0)2U ( 9 (# ( ! . ; < , ( % 9 (: %, (( * ( ) ' ( ( (! ! '!), ! '! , (' ' . 5( ' * 7.2.1 , !,

% # ? = % # 9 (# N

! (7.2.6). 1 (# (7.2.27) * 9 (# N

! ( * % * # 9 (#% %

%. 7 % , (7.2.27) N = 1 2 : : : * 9 (#: 9 u(1)(0) = r(1)Mx(0)] 1 Mx(0)] (N = 1)I u(2)(0) = r(2)Mx(0)]I > > > = 2Mx(0)] (N = 2)I : : : (7.2.28) > uN ;1(0) = r(N ;1)Mx(0)]I N ;1Mx(0)] (N = N ; 1)I > > u(N )(0) = r(N )Mx(0)]I N Mx(0)] (N = N ): , * 9 (# u(i)(0) = r(i)Mx(0)] (i = 1 N ) * # 9 (# fqM'Mx(0) u(i)(0)]] + Mu(i)(0)] + Vi;1M'Mx(0) u(i)(0)]] %

% u(i)(0) . ; * .3.1 % (7.2.4) N

u(0) = r0Mx(0)]I u(1) = r1Mx(1)]I u(2) = r2Mx(2)] : : : : * , * ( *

' u(0) = r(N )Mx(0)]I u(1) = rN ;1 Mx(1)]I u(2) = r(N ;2)Mx(2)] : : : u(N ; 2) = r(2)Mx(N ; 2)]I u(N ; 1) = r(1)Mx(N ; 1)]: (7.2.29) 235

,( , ( 9 (#

rk Mx(k)] = r(N ;k) Mx(k)] (k = 0 N ; 1):

(7.2.30) ' & . %,

! # , ! (7.2.29). <! % * M(N ; 1) N ] , * x(N ; 1) . & # , u(N ; 1) < ** , < :

J (N ;1) = qMx(N )] + Mu(N ; 1)]: * , * x(N ) = 'Mx(N ; 1) u(N ; 1)] , *

(7.2.31)

J (N ;1) = qM'Mx(N ; 1)] u(N ; 1)]] + Mu(N ; 1)]: (7.2.32) D * * % (7.2.11) <, # 1, * *( : u(N ; 1)] = r(1)Mx(N ; 1)]: 0 * (7.2.32)

(7.2.33)

(N ;1)Mx(N ; 1)] = qM'Mx(N ; 1)] r(1)Mx(N ; 1)]]] + Mr(1)Mx(N ; 1)]]: (7.2.34) $ M(N ; 2) N ] , % . D ** J (N ;2) = qMx(N ; 1)] + Mu(N ; 2)] + qMx(N )] + Mu(N ; 1)] = (7.2.35) = qMx(N ; 1)] + Mu(N ; 2)] + J (N ;1): & x(N ; 2) . @ # , * ' x(N ; 2) # ( # J (N ;2)) M(N ; 2) N ] . 4% J (N ;2) u(N ; 1) u(N ; 2) . * < min J (N ;1) = (N ;1)Mx(N ; 1)] = (N ;1)M'Mx(N ; 2)] u(N ; 2)]]:

u(N ;1)2U

! (7.2.35) u(N ; 1) <

236

(N ;2) = min fq M'Mx(N ; 2) u(N ; 2)]]+ (N ;2) = u(Nmin ;2)2U J u(N ;2)2U + Mu(N ; 2)] + u(Nmin f q M ' Mx(N ; 1) u(N ; 1)]] + Mu(N ; 1)]gg = ; 1) 2 U n ; 2) u(N ; 2)]] + Mu(N ; 2)] + (N ;1)M'Mx(N ; 2) u(N ; 2)]]o : q M ' M x ( N = u(Nmin ;2)2U (7.2.36) ( 9 ! ((!, * * % (7.2.19), # 1, * M(N ; 2) (N ; 1)] . <

u(N ; 2)] = r(2)Mx(N ; 2)] (N ; 2) = (N ;2)Mx(N ; 2)]: (7.2.37) % ( M(N ; 3) (N )] , ! ! . D ** J (N ;3) = qMx(N ; 2)] + Mu(N ; 3)] + J (N ;2): (7.2.38) x(N ; 3) , u(N ; 3) , u(N ; 2) , u(N ; 1) , (7.2.38). ; # , < ! ! . * , * ! (7.2.38) u(N ; 2) , u(N ; 1), ' N ;3 = min fq M'Mx(N ; 3) u(N ; 3)]]+ (N ;3) = u(Nmin ;1)2U J u(N ;3)2U u(N ;2)2U u(N ;3)2U (N ;2) g = min fq M'Mx(N ; 3) u(N ; 3)]]+ + Mu(N ; 3)] + u(Nmin J ;2)2U u(N ;3)2U u(N ;1)2U + Mu(N ; 3)] + (N ;2)M'Mx(N ; 3) uM(N ; 3)]]g: (7.2.39) # 1, * M(N ; 3) (N ; 2)]

u(N ; 3)] = r(3)Mx(N ; 3)] (N ; 3) = (N ;3)Mx(N ; 3)]: < # , * (

9 (N ;k) = min fq M'Mx(N ; k ) u(N ; k )]]+ (N ;k) = u(Nmin ;k)2U J u(N ;k)2U u(N ;1)2U + Mu(N ; k)] + (N ;k;1)M'Mx(N ; k) u(N ; k)]]g (k = 1 N ):

237

(7.2.40)

(7.2.41)

@ # 1, * M(N ; k) (N ; k ; 1)] : (7.2.42) u(N ; k)] = r(k)x(N ; k) (k = 1 N ): D (7.2.29), 9 (# (7.2.41) k = N (7.2.27).

;$ 3 #) # ' '"# #(+ #% . (

x = M(k + 1)]T = R(kT )x(kT ) + R(kT )u(kT ) (k = 1 n):

, # C 0(kT ) (<99#

(7.3.1)

u(kT ) = C 0(kT )x(kT ) (k = 1 N )

(7.3.2) (, * ! (7.3.1), (7.3.2),

!

* , 9 (#

J=

N X x0(kT )Q(kT )x(kT ) + u0M(k ; 1)T ]uM(k ; 1)T ]:

k=1

(7.3.3)

) R(kT ) , R(kT ) (k = 0 1 : : :) {

# n n , n m

I Q(kT ) { #

# n n. 7 ' <% * # . , * M(N ; 1)T NT ] . 4 < * (7.3.1) **

J N ;1 = x0(NT )Q(NT )x(NT ) + u0M(N ; 1)T ]uM(N ; 1)T ] = = fRM(N ; 1)T ]xM(N ; 1)T ] + RM(N ; 1)T ]uM(N ; 1)T ]g0 Q(NT ) f0RM(N ; 1)T ]xM(N ; 1)T ] + RM(N ; 1)T ]uM(N ; 1)T ]g + +u M(N ; 1)T ]uM(N ; 1)T ]: ( , ' ' *

(7.3.4)

(N ;1) = u(min Mx0RQx + 2x0RQRu + u0(R0QR + E )u]: (7.3.5) N ;1) u ( ! ((!: 2x0R0QRu + 2u0(R0QR + E ) = 0: 238

(7.3.6)

$ ' < , * *( uM(N ; 1)T ] = ;fR0M(N ; 1)T ]Q(NT )RM(N ; 1)T ] + E g;1 (7.3.7) R0M(N ; 1)T ]Q(NT )RM(N ; 1)T ]xM(N ; 1)T ]: ,( ,

uM(N ; 1)T ] = C 0M(N ; 1)T ]xM(N ; 1)T ]

(7.3.8)

C 0M(N ; 1)T ] = ;fR0M(N ; 1)T ]QMNT ]RM(N ; 1)T ] + E g;1 (7.3.9) R0M(N ; 1)T ]Q(NT )RM(N ; 1)T ]: (7.3.7) (7.3.5), * * (7.3.4): (N ;1) = x0RQx ; 2x0R0QR MR0QR + E ];1 R0QRx+ +x0RQR (R0QR + E );1 R0 QRx = x0M(N ; 1)T ]AM(N ; 1)T ]xM(N ; 1)T ]

AM(N ; 1)T ] = R0M(N ; 1)T ] fQ(NT ) ; Q(NT )RM(N ; 1)T ] o R0M(N ; 1)T ]Q(NT )RM(N ; 1)T ] + E ];1 R0M(N ; 1)T ]Q(NT ) (7.3.10) RM(N ; 1)T ]: ,( , * (N ;1) ( * % 9 % xM(N ; 1)T ] . ' ' (7.2.36), ( * N ;2 = u(min fx0M(N ; 1)T ]QM(N ;o 1)T ])xM(N ; 1)T ]+ N ;2)T ] + u0M(N ; 2)T ]uM(N ; 2)T ] + (N ;1) = = u(min fx0M(N ; 1)T ] MQM(N ; 1)T ]) + AM(N ; 1)T ]xM(N ; 1)T ]+ N ;2)T ] +u0M(N ; 2)T ]uM(N ; 2)T ]g = u(min ffRM(N ; 2)T ]xM(N ; 2)T ]+ N ;2)T ] +RM(N ; 2)T ]uM(N ; 2)T ]g0 MQM(N ; 1)T ]) + AM(N ; 1)T ] fRM( N ; 2)T ]xM(N ; 2)T ]g + RM(N ; 2)T ])uM(N ; 2)T ]g+ 0 +u M(N ; 2)T ]uM(N ; 2)T ]gg = u(min fx0R0MQ + A]Rx+ N ;2)T ] +2x0R0MQ + A]Ru + u0MR0MQ + A]R + E ]ug:

(7.3.11)

9 ! ((!, *

uM(N ; 2)T ] = C 0M(N ; 2)T ]xM(N ; 2)T ] 239

(7.3.12)

C 0M(N ; 2)T ] = ;fR0M(N ; 2)T ]MQM(N ; 1)T ]+ (7.3.13) +AM(N ; 1)T ]RM(N ; 2)T ]] + E g;1R0M(N ; 2)T ] fQM(N ; 1)T ] + A(N ; 1)T ]gRM(N ; 2)T ]: , * (7.3.11) *

Q (Q + A) % * (7.3.5), < (7.3.13) (7.3.9), Q Q + A , N ; 1 N ; 2 . (7.3.12), (7.3.13) (7.3.11), * * ** % ! ! !

(N ;2) = x0M(N ; 2)T ]AM(N ; 2)T ]xM(N ; 2)T ]

(7.3.14)

AM(N ; 2)T ] = R0M(N ; 2)T ]fQM(N ; 2)T ] + AM(N ; 1)T ]; ;MQM(N ; 1)T ] + AM(N ; 1)T ]] RM(N ; 2)T ]f;R10M(N ; 2)T ] (7.3.15) MQ0 M(N ; 1)T ] + AM(N ; 1)T ]]R(N ; 2)T ] + E R M(N ; 2)T ]MQM(N ; 1)T ] + AM(N ; 1)T ]]gRM(N ; 2)T ]: < # , M(N ;j )T (N ;j +1)T ] % * ' # (N ;j) = u(min fx0M(N ; j + 1)T ]QM(oN ; j + 1)T ]xM(N ; j + 1)T ]+ N ;j )T ] + u0M(N ; j )T ]uM(N ; j )T ] + (N ;j+1) ( ((

(7.3.16)

0M(N ; j + 1)T ]QM(N ; j + 1)T ])+ (N ;j) = u(min f x N ;j )T ] +AM(N ; j + 1)T ]]xM(N ; j + 1)T ] + u0M(N ; j )T ]uM(N ; j )T ]g = = u(min ffRM(N ; j)T ]xM(N ; j )T ] + RM(N ; j)T ]uM(N ; j)T ]g0 (7.3.17) N ;j )T ] MQM(N ; j + 1)T ] + AM(N ; j +0 1)T ]]fRM(N ; j )T ]xM(N ; j)T ]+ +RM(N ; j )T ]uM(N ; j )T ]g + u M(N ; j )T ]uM(N ; j )T ]g: u 9 ! ((!, *

uM(N ; j )T ] = C 0M(N ; j )T ]gxM(N ; j )T ] C 0M(N ; j )T ] = ;fR0M(N ; j )T ]MQM(N ; j + 1)T ]+ +AM(N ; j + 1)T ]]RM(N ; j )T ] + E g;1R0M(N ; j )T ] fQM(N ; j + 1)T ] + AM(N ; j + 1)T ]gRM(N ; j)T ]: 240

(7.3.18)

(7.3.19)

(7.3.18), (7.3.19) (7.3.17), *

(N ;j) = x0M(N ; j )T ]AM(N ; j )T ]xM(N ; j )T ] AM(N ; j )T ] = R0M(N ; j )T ]fQM(N ; j + 1)T ]+ +AM(N ; j + 1)T ]gRM(N ; j )T ] ; R0M(N ; j )T ]fQM(N ; j + 1)T ]+ +AM(N ; j + 1)T ]gRM(N ; j )T ]fR0M(N ; j )T ]MQM(N ; j + 1)T ]+ +AM(N ; j + 1)T ]]RM(N ; j )T ] + E g;1R0M(N ; j )T ] MQM(N ; j + 1)T ] + AM(N ; j + 1)T ]]RM(N ; j)T ]:

(7.3.20)

(7.3.21)

; (7.3.18) ... (7.3.21) ! i = 1 N (A(N ) = 0) , ( , (7.3.20), (7.3.19) (

' * (% #

C 0MkT ] = C 0M(N ; j )T ] (j = 1 N ):

(7.3.22)

;$ 4

& '$ % . (, %

x_ = Ax + B u

(7.4.1)

u_ = C 0x

(7.4.2)

, % # C

(, * ! ( * (%

D(s) = det(Es ; A ; BC 0) (7.4.3)

( ( ) 1 : : : n . D * * % . ' ' ( . ; < * (7.4.1) (7.4.2) B = b , C = c , b c { n - ( , # ! #%. 1. (7.4.1) ( 9 1

x]_ = A]x] + b] u 241

(7.4.4)

::: 0 0 ::: 0 0 . I b] = ... I (7.4.5) En;1 .. ::: 1 0 ;d0 ;d1 : : : ;dn;1 1 En;1 { * # (n ; 1) (n ; 1)I d0 , d1 , ... dn;1 { (<99# ! ( * ( . ( (7.4.1)I 0 0 A] = ... 0

p p

p

p p p p p p p p p p p p p p p p p p p

p

1 0 ... 0 p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p p p p p p p p p p p p p p p p p p p p

p p

p p

p p

p p

p p

p

p

p

p

p

p

p

p

p

p

p

p

p

pp

D(s) = sn + dn;1 sn;1 + : : : + d1s + d0: (7.4.6) ! (3.2.19) ( (7.4.4)

x = Q;y 1x] ;1 Qy = ]b A]]b : : : A]n;1]b b Ab : : : An;1b : 4 , * . ( (7.4.1)

(7.4.7) (7.4.8)

det Qy 6= 0: (7.4.9) 2. @ ( # A] , * (7.4.4), '

% x]1 , -

D(s)]x1 = u:

(7.4.10) n Y & <

% D(s) = (s;i ) = sn +dn;1 sn;1+: : : i=1 +d1s + d0 * u(s) = ; c]nsn;1 + : : : + c]2s + c]1 x]1 (7.4.11)

c](i+1) = di ; di (i = 0 n ; 1): , * sx]i = x]i+1 (i = 1 n ; 1) n X u = ; c]ix]i = ;c]0x] i=1 0 ( ]c { n - % ( * ). 3. ; (

, * ( % ( 242

(7.4.12) (7.4.13)

c]0 = ;]c0Qy

(7.4.14) *%

( ! ( * ( (3.2.19), (3.2.20). $ *%, ( . ( ,

y.; < * . ( 9 "! - !": n s + dn;1 sn;1 + + d1s + d0 y = (kmsm + + kp1 s + kp0 ) u: (7.4.15) > ( (<99# n dn s + + d 1 s + d 0 u = kpm sm + + k 1 s + k 0 y (7.4.16) (, * ! ( * (% (7.4.15), (7.4.16)

( . 7 , * * <! ( % nc n . &9 % : n Y D(s) = (s ; i ) = sn + dn ;1sn ;1 + + d1s + d0: (7.4.17) p

c

i=1

c

c

c

b ( * (% %

D(s) = d(s)d (s) ; k(s)k (s): &9 4 8 :

(7.4.18)

d(s)d (s) ; k(s)k (s) = D(s): (7.4.19) & (<99# ( ! ! s % % *! < , * % ! * (! % N (d k) = d (7.4.20) iT h = d 0 : : : d n k 0 : : : k m { n +m +r - % ( ( ! (<99#h iT (7.4.16) d = d0 d1 : : : dn { ( (<99# , N (d k) { ( #,

! (<99# . ( (7.4.15). @ ,M6.5] * c

det N (d k) 6= 0 (7.4.21) . ( , ! ! % 243

np < m mp < n: . $ *% n = 2 , m = 1 , nc = 3 . ; < * . ( 2 s + d1s + d0 y = (k1s + k0) u % D(s) = s3 + d2s2 + d1 s + d0: > (

(7.4.22)

(7.4.23) (7.4.24)

(d 1 s + d 0 )u = (k 1 s + k 0 )y: , > (s2 + d1s + d0)(d 1 s + d 0 ) ; (k1s + k0)(k 1 s + k 0 ) = s3 + d2s2 + d1s + d0: & (<99# ( ! ! s , * % d0d 0 ; k0k 0 = d0 d0dp1 + d1dp0 ; k0kp1 ; k1kp0 = d1 : d1d 1 + d2d 0 ; k0k 1 = d2 d2d 1 = d3 4 , * * 2 3 2 3 2 3 d 0 ; k 0 d 0 66 d0 77 66 0 77 66 0 77 N (d k) = 666 d1 d0 ;k1 ;k0 777 I = 666 d 1 777 I d = 666 d1 777 : 4 d2 5 4 d2 d1 0 ;k1 5 4 k 0 5 1 0 d2 0 0 k 1

244

;7#&# 1 ( #=" $ 5$ %%'% ( # ( * * ( (m = 1) u(t) ( % ( # ( , . . , * * xi1 (i = 1 n) . % u0(t) < ( x00(t) , x01(t) , : : : , x0n(t) , * *( (2.2.3). 4 . .2.1 ( , ( ( * * * ( .

$. .2.1 $ ( * (

; " < t < (7.1.1) " { ( * * , 0 < < t1 . @ < ( ( % * uO 6= u0 , < uO 2 U . 4 ! ! (0 ; ") ( t1)

u0(t). ,( # <( u0(t) " )" . @ * # # * % (* ( #

* , # <( %

! * 99 # . , * * uO ; u0 M ; " ] %, ' * uO ! ! . 4 , ;u u u , uO ; u0 ! 0 2u . 7 *% # ! ( , *, ( * * uO ; u0 , <% # . ( ( * , ( (( (Ou ; u0)" ( * . <. ; ( * ; " < t < %' x0(t) , x1(t) , : : : , xn(t) t > * ( . .2.2).

245

$. .2.2 * , * x ; x ; ") = "' Mx0( ) u0( )] I xO ( ) ; xO( ; ") = "' MOx0( ) uO 0( )] . , * x( ; ") = x0( ; ") , * n h io xO ( ) ; x0( ) = " ' MOx( ) uO( )] ; ' x0( ) u0( ) : (7.1.2) D ( * , * , < t > ! ( xO (t) x0(t) . ( < ! ( ( * . ; ( # x(t) ( xi(t) (i = 0 n) % (( 0( )

0(

x(t) = xO (t) ; xO 0(t):

(7.1.3) < (7.1.2) * * <% # n h io

x( ) = " ' MOx( ) uO( )] ; ' x0( ) u0( ) : (7.1.4) ( x0(t) " %" ( %,

(2.1.1). D n @' (x u) 0 d(xi) = X i xj (i = 0 n): (7.1.5) dt @xj j =0 $ ' <! % * ! ! (7.1.4) ! ( % x0i (t) xOi(t) (i = 0 n) t > . ; * x0(t1) % J ' ( ) * 9 (# (2.1.5),

*% # %. ,( (( u(t) * ' * J , x0(t1) ' * , <

J = x0(t1) 0: ' < '

;J = ;x0(t1) = Mx(t1)]0 (t1) 0

(7.1.6)

(7.1.7) (t1) { ( ,

% (, * Mx(t1)]0 (t1) x0(t1) . 246

* , *

(t1) = f;1 0 0 : : : 0g : (7.1.8) uO( ) = u0( ) (;J ) ' * ,

, ' (7.1.6) * , * "! ", * : '% <99 (, * . ) * uO( ) (, * * (;J ) ' %, ! % < *

;J = Mx(t1)]0 (t1):

(7.1.9) ) : * uO( ) # , ' ' , < * (, * ( * (;J ) , (7.1.9). @ (7.1.9) uO( ) , * Mx(t1)]0 (t1) ! (7.1.5) * ! ! (7.1.4) % x(t1) <! * ! % u0( ) . ; < ( , * ( (t), % ' (2.1.9), ' Mx(t)]0 (t) = Mx(t1)]0 (t1) t t1: , t = *

(7.1.10)

Mx( )]0 ( ) = Mx(t1)]0 (t1) t t1: (7.1.11) , * J % * < , ( u0( ) . @ (7.1.10) , *

Mx(t)]0 (t) = const ( t t1)

(7.1.12)

d = nMx(t)]0 (t)o = 0 ( t t ) 1 dt

(7.1.13)

d Mx(t)0] (t) + Mx(t)0] d(t) = 0: dt dt )' < 247

n d n X X d M x ( t )] ( t ) + x i i i i (t) = 0: dt i=0 dt i=0 @ (2.1.9) (7.1.5), * 3 2n 3 2n n X n X X X @' @' j 5 i 5 4 4 @x xj i(t) ; @x j xi 0: i=0 j =0

j

i=0 j =0

i

,( , ' (7.1.10) ( . , * ;J = Mx( )]0 ( ) 0 . < ' * (7.1.4) n h io0 " MOx( ) uO( )] ; x0( ) u0( ) ( ) 0: (7.1.14) @ (7.1.2) , * xO ( ) ; x0( ) < ( , ( ( * ( , * " , < ( xO ( ) (7.1.14) x0( ) . &( (7.1.14) " > 0 * (2.1.7), (* , * H x0 uO H x0 u0 : (7.1.15) D * , * 9 (# H ( uO = u0 , ( , (2.1.13) ( .

;7#&# 2 ( - ) ' " #

)' (3.1.1), (3.1.2) 9 (# (3.1.3) % 9 : n m X X (7.2.1) x_ i = aij xj + bik uk I (i = 1 n)I j =1

k=1

n X uk = cik xi (k = 1 m)I i=1 1 0 Z1 X n m X J = @ qij xixj + u2k A dt 0

n X

i j =1

(7.2.2) (7.2.3)

k=1

qij xixj 0 ! * % xi (i = 1 n) . * ! ! (2.3.11), (2.3.12) * 2n 3 n n @ X m m X X @ ; @t = @x 4 aij xj + bik uk5 + X qij xixj + X u2k : (7.2.4)

i j =1

i=1

i j =1

k=1

248

i j =1

k=1

n @ n @ X 1X b + 2 u = 0 u = ; (7.2.5) ik k k 2 i=1 @xi bik (k = 1 m): i=1 @xi @(* uk (k = 1 m) (7.2.4) (7.2.5), * % * ! !

0n 1 m n !2 X n n X X X X @ 1 @ @ ; @t = i=1 @xi @j=1 aij xj A ; 4 i=1 @xi bik + i j=1 qij xi xj = 0: k=1 $ ' < ( ( * % 9 n X = qij xixj : i j =1

(7.2.6)

(7.2.7)

(7.2.7) (7.2.6), *

0n n X X @

i=1 j =1

10 n 1 m 2n 0n 1 32 n X X X X X 1 A @ A 4 @ A 5 pij xj aij xj ; pij xj bik 4 + qij xixj = 0: j =1

k=1 i=1 j =1

i j =1

(7.2.8)

D 6n n 0n 1 n ! X ! 3 n X n 6 X X X X 6 1 @ pj xj A pibk xj 77 aj xi ; 4 4 7 =1 j =1 =1 i=1 k=1 j =1 "X ! # X n X n n pibk xi + qij xixj = 0: i=1 =1

i j =1

( (<99# ! xixj , (i j = 1 n) * , * xixj = xj xi , * n(n2+ 1) * (! % (<99# pij (i j = 1 n) : ! X ! n m X n n X X (7.2.9) (piaj + aj pi) ; pibk pj bk + qij = 0 =1 =1 k=1 =1 < pij = pji . ( * 9 (7.2.7) (7.2.5), * ! n X n X uk = ; pibk xi (k = 1 m): (7.2.10) i=1 =1

P * (! % (7.2.9) ' ! * pij (i j = 1 n) , ( 9 (7.2.7)

%, n X pij xixj > 0 i j =1

249

! xi (i = 1 n) , ( (7.2.2), ' *, (7.2.10) , , ( (<99# cik (i = 1 nI k = 1 m)

n X (7.2.11) cik = ; pibk (i = 1 nI k = 1 m) =1

) (7.2.9) (7.2.10) * % 9 , * (3.1.15), (3.1.16).

;7#&# 3 ( ' 1 $#%& -2 " 1. 4% * '( (e = x ; x^ ) .

7 < * . ( (4.2.1) (4.2.6), *

e_ = MA(t) ; K (t)D(t)] e ; K (t) + Q(t)f I e(t0) = e(0):

(7.3.1) 2. , * * ( (4.2.7) ' ! (7.3.1). ; < *

eO(t) = M fe(t)g I

(7.3.2)

P~e (t) = M fMe(t) ; eO(t)]Me(t) ; eO(t)]0g (7.3.3) Oe(t) - * ( I P~e (t) - # % '( . 4 , * P~e (t) = M fMe(t) ; eO(t)]Me(t) ; eO(t)]0g = = M fe(t)e0(t)g ; M fe(t)g Oe0(t) ; eO(t)M fe0(t)g + fOe(t)Oe0(t)g = M fe(t)e0(t)g ; eO(t)Oe0(t): (7.3.4) , *

M fe(t)e0(t)g = P~e (t) + Oe(t)Oe0(t): (7.3.5) 0 # M fe(t)e0(t)g ) . D # Ge(t) = M fe(t)e0(t)g gij(e)(t) = M fei(t)ej (t)g : ,( , (7.3.5)

(7.3.6)

Ge (t) = P~e (t) + Oe(t)Oe0(t):

(7.3.7)

250

5 6 A. & * ( '( (4.2.7) -

* * ( # % '( :

M fe0(t)q(t)e(t)g = tr MP~e (t) + q(t)] + eO0(t)q(t)Oe(t):

(7.3.8)

7( < ' * , * (7.3.6), (7.3.7) '

9 8 n n = < X X M fe0(t)q(t)e(t)g = M : ij (t)ei(t)ej (t)g = ij (t)M fei(t)ej (t) = i j =1 i j =1 n n o X ij (t)gij(e)(t) = tr Mq(t)Ge(t)] = tr q(t)MPOe(t) + eO(t)Oe0(t)] =

i j =1

= tr MP~e (t)q(t)] + eO0(t)q(t)Oe(t):

7 q = diag jj1 : : : n jj i 0 i = 1 n . 3. ! ( # ( (4.2.7), , * (7.3.8) , ( eO(t) = 0 . 4% , ( ! < . ; 9 % 5' (1.3.18) ' (7.3.1)

Zt 0 ] e(t) = H (t t0)e(t0) + H] (t )MQ( )f ( ) ; K ( )( )] d t0

(7.3.9)

H] (t t0) {

9 # ' %

e_ = MA(t) ; K (t)D(t)]e(t): (7.3.10) , * M f(t)g = M ff (t)g = 0 , * (7.3.9) M fe(t)g = H] (t t0)M fe(t0)g eO(t) = H] (t t0)Oe(t0): D * , * eO(t) '

eO_ = MA(t) ; K (t)D(t)]Oe:

P *

x^ (t0) = xO (0)

eO(t0) , , ' (7.3.11) 251

(7.3.11)

eO(t) = 0:

(7.3.12)

,( , ( ' (4.2.11). 4. 4% , ( ! (7.3.8). 7 < * , ( # % '( P~e (t) . ,( , M3.7].

5 6 D. P x(t) - ' x_ = A(t)x + Q(t)f (t) x(t0) = x(0) (7.3.13) ( f (t) { " % '" R(1)(t)I x(0) { !* (% ( , % f (t) xO (0) # % % R(0) = M fMx(0) ; xO (0)] MOx(0) ; x(0)]0g, # % Px (t) = M fx(t) ; xO (t)]Mx(t) ; xO (t)]0g xO (t) = M fx(t)g * 99 # P_x(t) = A(t)Px(t) + Px A0(t) + Q(t)R(1)(t)Q0(t) Px(t0) = R(0):

(7.3.14)

7 ( < ' * (7.3.5)

Px(t) = M fx(t)x0(t)g ; xO (t)Ox0(t): (7.3.15) 7 * * # M fx(t)x0(t)g

' (7.3.13): Zt x(t) = H (t t0)x(t0) + H (t )Q( )f ( )d: (7.3.16) ,

t0

252

M fx(t)x0(t)g = M fH (t t0)x(0)x(0)0 H 0(t t0)g+ 8 2 Zt 309 > > < = (0) 4 5 +M >H (t t0)x H (t )Q( )f ( )d > + : t0

82 Z t 3 < +M :4 H (t )Q( )f ( )d 5 x(0)0 H 0(t t0) ] g + t

(7.3.17)

0

82 t 3 2 Zt 309 > > < Z = +M >4 H (t )Q( )f ( )d 5 4 H (t )Q( )f ( )d 5 > : : t0 t0

; < , ( (( ( x(0) f (t) M ff (t)g = 0 . * (4.1.3) 82 t 3 2 Zt 309 > > < Z = 4 5 4 5 M > H (t )Q( )f ( )d H (t )Q()f ()d > = : t0 t0 =

Zt Zt t0 t0

H (t )Q( )M ff ( ) f 0()gQ0( )H 0(t )jdd =

(7.3.18)

Zt

= H (t )Q( )R(1)( )Q0( )H 0(t )d: t0

(7.3.17) (7.3.15), * * (7.3.18)

Zt 0 Px(t) = H (t t0)Gx(0) H (t t0) + H 0(t )Q( )R(1)( )Q0( )H 0(t )d ; t0

;H (t t0) xO (0)xO(0) H (t t0) 0

Gx(0) = M Mx(0)x(0)0 ]: B * (7.3.7) *

Px (t) = H (t t0

)R(0)H 0(t t

0) +

Zt t0

H (t )Q( )R(1)( )Q0( )H 0(t )d:

(7.3.19)

(4 , * Px(t0) = R(0) ). 799 # < * ' 253

dH (t ) = A(t)H (t ) d

(7.3.20)

H (t t) = En (7.3.21) * (7.3.14), ( , > ( . @ < , ' # % '( P~e(t) = MA(t) ; K (t)D(t)]P~e (t) + P~e (t)MA(t) ; K (t)D(t)]0+ (7.3.22) (2) 0 (1) 0 +K (t)R (t)K (t) + Q(t)R (t)Q (t)I Pe (t0) = R(0): (7.3.23) 5. &9 , ( % ' % * 99 # $(( (3.1.37). * , * (2.3.14) 9 (# v(t) = x0(t)P (t)x(t) (7.3.24) ' * ( , (' * % 9 (# (3.1.30) ! (3.1.29), * * 9 (# (3.1.30) ! ( !. D * min J = v(t0) = x0(t0)P (t0)x(t0): (7.3.25) ; * * 9 (# (3.1.30)

u = C~ 0(t)x

(7.3.26) C~ (t) { , 9(

# 9 (#% ,

! Mt0 t1] . . (, ( % (7.3.26),

x_ = A~(t)xI x(t0) = x(0)

A~(t) = A(t) + B (t)C~ 0(t) 9 (# (3.1.30) Zt1 J = xQ~ xdt + x0(t1)P (1)x(t1) t0

254

(7.3.27) (7.3.28) (7.3.29)

Q~ = Q(t) + C~ (t)C~ 0(t): (7.3.30) @(, ! * * 9 (# (7.3.29) ' ! (7.3.27). 4 9( ( t0 , ( * 9 (# (7.3.29) (( Zt1 0 ~ x (t)P (t)x(t) = x0(t)Q~ (t)x(t)dt + x0(t1)P (1)x(t1): (7.3.31) t

@ < , * , *

P~ (t1) = P (1): 799 # * (7.3.31) * (7.3.27), *

(7.3.32)

x0(t)MA~0(t)P~ (t) + P~_ + P~ (t)A~(t)]x(t) = ;x0(t)Q~ (t)x(t):

;P~_ (t) = P~ (t)A~(t) + A~0(t)P~ (t) + Q~ (t):

D * (7.3.28), (7.3.30)

;P~_ (t) = P~(t)MA(t) + B (t)C~ 0(t)] + MA(t) + B(t)C~ 0(t)]0P~(t) + Q(t) + C~ (t)C~ 0(t):

(7.3.33)

@ (7.3.31) * * 9 (# (7.3.29)

v~(t0) = x0(t0)P~ (t0)x(t0):

(7.3.34)

* , *

v~(t0) v(t0) (7.3.35) ( (( v(t0) { * < 9 (# . @ (7.3.34) , * P~ (t0) P (t0) , ( ( ( x(t0) %. 5 , t0 , (* , * MP~ (t) ; P (t)] 0: ,( , ( .

255

(7.3.36)

5 6 &. 0 * (7.3.33) ( (7.3.32)

' , (7.3.36). D ,

C~ (t) = ;P~ (t)B (t):

(7.3.37)

P 9 (# (3.1.30) (3.1.42), (7.3.33)

;P~_ (t) = MA(t) + B (t)C~ 0(t)]0P~ (t(1)) + P~ (t)MA(t) + B (t)C~ 0(t)]+ #

+Q + C~ (t)Q (t)C~j (t)

(7.3.38)

(7.3.39) C~ (t) = ;P~ (t)B (t)Q(1);1: 6. ; ( ' t0 = 0 ) 99 # ( % # S~(t) ( n n ), ( * (7.3.22) = t1 ; t % t :

;S~_ (t) = MA0(t1 ; t) ; D0(t1 ; t)K 0(t1 ; t)]0S~(t) + S~(t)MA0(t1 ; t); ;D0(t1 ; t)K 0(t1 ; t)] + K (t1 ; t)R(2)(t1 ; t)K 0(t1 ; t) + Q(t1 ; t)R(1)(t1 ; t)Q(t1 ; t)I

(7.3.40)

S~(t1) = R(0) (7.3.41) * , * ' % (7.3.22), (7.3.40) P~e (t) = S~(t1 ; t): (7.3.42) ; ( (7.3.40). 0 # S (t) , # K (t1 ; t) = S~(t)D0(t1 ; t)R(2);1(t1 ; t): (7.3.43) 7 % , (7.3.40) (7.3.38), B (t) = D0(t1 ; t) C~ (t) = ;K (t1 ; t) . A(t) = A0(t1 ; t) Q(t) = Q(t1 ; t)R(1)(t1 ; t)Q0(t1 ; t) Q(1)(t) = R(2)(t1 ; t) 256

(7.3.37) * (7.3.43). (7.3.43) (7.3.40), * ' * # S~(t) = S (t) : ;S_ (t) = A(t1 ; t)S(t) + S (t)A0(t1 ; t);

;S(t)D0(t1 ; t)R(2);1(t1 ; t)D(t1 ; t)S (t) + Q(t1 ; t) R(1)(t1 ; t)Q0(t1 ; t)I S(t) = R(0):

(7.3.44)

$ ' % (7.3.40) (7.3.44) MS~(t) ; S (t)] 0: (7.3.44), (* , * MP~e(t) ; Pe (t)] 0: (7.3.45) Pe(t) { ' (4.2.9), ( (7.3.44) . @ (7.3.45) tr MP~e(t)q(t)] tr MPe(t)q(t)] < # (7.3.43), S~(t) = S (t) = Pe (t) # % (4.2.8), , ( , 4.2.2 ( . , * *

% ( ( t , ( % # ( (4.2.7) # q(t) , < # (4.2.8) (4.2.7)

! t > t0 ! (

! # q(t) .

;7#&# 4 ;7#&# # %( 7 "

)' 9 (# (4.2.5) * ( # * (

Zt1

n o J = M fx0Q(t)x + u0ug dt + M x0(t1)P (1)x(t1)

t0

n o M fxn0Q(t)xg = M Mx(t) ; x^ (t) + x^ (ot)]0 Q(t) nM^x(t) ; x^ (t) + x^ (t)] =o = M Mx(t) ; x^ (t)]0 Q(t) Mx(t) ; x^ (t)] + 2M Mx(t) ; x^ (t)]0 Q(t)^x(t) + +M fx^ 0(t)Q(t)^x(t)g 257

(7.4.1)

(7.4.2)

x^ { (

! (4.2.6), ( # K (t) ' (4.2.8) ... (4.2.9). ; (7.3.8) # % '( n o M Mx(t) ; x^ (t)]0 Q(t) Mx(t) ; x^ (t)] = tr MPe (t)Q(t)] (7.4.3) ( (( e(t) = 0 , P~e = Pe , Pe { ' (4.2.9). ; (7.4.2) n o M Mx(t) ; x^ (t)]0 Q(t)^x(t) = 0 (7.4.4) . 5 6 . ; ( e(t) = x(t) ; x^ (t) x^ (t) (

. 7( < . ,( ,

M fx0(t)Q(t)x(t)g = tr MPe (t)Q(t)] + M fx^ 0(t)Q(t)^x(t)g : < t = t1 Q(t1) P (1) , * n o h i n o M x0(t1)P (1)(t)x(t1) = tr Pe (t1)P (1) + M x^ 0(t1)P (1)x^ (t1) : @ < , ' 9 (# (4.2.5)

(7.4.5) (7.4.6)

8 Zt1 9 8 Zt1 9 < 0 = < = J = M : M^x Q(t)^x + u0u] dt + x^ 0(t1)P (1)(t)^x(t1) + tr : Pe (t)Q(t) dt + Pe (t1)P (1) : t0 t0 (7.4.7) ) , * ! ! < . )' (4.2.6)

x^_ = A(t)^x + B (t)u + K (t)My ; D(t)^x]:

(7.4.8)

5 6 D. $

(t) = y ; D(t)^x

(7.4.9) *% # " % '" R(2)(t). < (4.2.2), ( (( = y ; D(t)x . B > * 9 (# (4.2.5) % 9 # ! . ( (4.2.1)I (4.2.2) ( * 9 (# 258

8 Zt1 9 < 0 = J = M : M^x Q(t)^x + u0u] dt + x^ 0(t1)P (1)x^ (t1) t

(7.4.10)

x^_ = A(t)^x + B (t)u + K (t)(t)

(7.4.11)

0

!* ( ". ("

*% # (t) , " '". $ ' <% *

u = C 0(t)^x

(7.4.12) # C 0(t) (4.1.7), (4.1.8), ( ,

( . 7 ( B '

:

x_ = A(t)x + B (t)u + Q(t)f I y = D(t)x + (t)I

(7.4.13)

x^_ = A(t)^x + B (t)u + K (t)My ; D(t)^x]:

(7.4.14) ; * (7.4.13) (7.4.14), *

e_ = MA(t) ; K (t)D(t)]e + Q(t)f ; K (t):

(7.4.15)

(7.4.14) (7.4.12), (*

x^_ = MA(t) + B (t)C 0(t)]^x + K (t)D(t)e + K (t): (7.4.16) $ '

% ( col ke(t) ^x(t)k , ( % 99 -

#

0 e + Q(t) ;K (t) f (7.4.17) e_ = A(t) ; K (t)D(t) x^_ K (t)D(t) A(t) + B (t)C 0(t) x^ 0 K (t) * e(t0) = x(t0) ;xO (0) : (7.4.18) x^ (t0) xO (0) * # % '

( *

259

( ) e ( t ) ; M f e ( t ) g 0 0 M ^ x(t) ; M fxO(t)g Me(t) ; M fe(t)g] M^x(t) ; M fx^ (t)g] = (7.4.19) = PP110 ((tt))I PP12((tt)) : 22 12 799 # # P11(t) , P12(t) , P22(t) * , 7(.3.>. ,(, # (7.4.17) (7.3.14), *, * , # P11(t) P12(t) : P_11(t) = MA(t) ; K (t)D(t)] P11(t) + P11(t) MA(t) ; K (t)D(t)]0 + (7.4.20) +Q(t)R(1)(t)Q0(t) + K (t)R(2)(t)K 0(t)I P_12(t) = MA(t) ; K (t)D(t)] P12(t) + P11(t)D0(t)K 0(t)+ (7.4.21) +P12(t) MA(t) + B (t)C 0(t)] ; K (t)R(2)(t)K 0(t)I * P11(t0) P12(t0) = P120 (t0) P22(t0) ( ) h i 0 e ( t 0) ; Oe(t0 ) 0 = M x^ (t ) ; xO^ (t )g Me(t0) ; eO(t0)g] x^ (t0) ; xO^ (t0) = 0 0 ( i0 0 R(0) 0 O x ( t ( t 0) ; x 0 ) (0) = M x(t0) ; xO 0 = 0 0 : 0

(7.4.22)

,( , P11(t0) = R(0) , P12(t0) = 0 . 4 , * (7.4.20) (4.2.9), K (t) (4.2.8). & ,

P11(t) = Pe (t): (7.4.23) < (7.4.21) (4.2.8), (* , * P11(t)D0 (t)K 0(t) ;K (t)R(2)(t)K 0(t) (7.4.21) *. ' * < 99 # * P12(t0) = 0 , ( ' P12(t) = 0: (7.4.19) * (7.3.12) *

(7.4.24)

n o P12(t) = M M^x(t) ; M fx^ (t)g] Me(t) ; eO(t)]0 = M fM^x(t) ; M fx^ (t)g] e0(t)g = = M fx^ (t)e0(t)g ; M fx^ (t)g eO0 = M fx^ (t)e0(t)g = 0 260

( , B ( . 7( > * ,

s_ = y ; D(t)^x = D(t)^e + I s(t0) = 0

(7.4.25) (7.4.15). &9 '

% ( col ks(t) e(t)k , ' # % '

( . B < ( >.

;7#&# 5 1# H1 %( 4 (5.1.14) '

+ ; =

m X a2i (!f ):

(7.5.1)

i=1

D * (5.1.11) (( m X a2i (!f ) = + ; = f sT TT (;j!f )T(j!f )f s: )9(

i=1 !f = !f ,

(7.5.2)

$ -$#

f sT TT (;j!f )T(j!f )f s max(j!f )f sT f s

(7.5.3) T f f f

max(j! ) { ( * < % # T (;j! )T (j! ) h i f f max(j!f ) = 1max im 1(j! ) : : : m (j! ) . =2 (j! f ) , * # T (j!f ) ' max(j!f ) = 1max 2 f f f ! ! T (j! )1 = sup max(j! ) , < (7.5.3) 0! 1 (7.5.4) f sT TT(;j!f )T(j!f )f s T(j!f )21 f sT f s: * (7.5.2), * ( %. f

;7#&# 6 '"# , -$) 2:1 H1 %'

3 ( , * (5.2.6) { (5.2.8) * H1 , *

, 9 , ( % ( . ; <

x~_ = A~x~ + Q~ f~ z = N~ x~ :

(7.6.1) 2 . P #

' P (P > 0) $(( 261

P A~ + A~T P + ;2P Q~ Q~ T P + N~ T N~ = 0 (7.6.2) # A~ { # , H1 * % # T~zf~ = N~ (Es ; A~);1Q~ T~zf~1 : (7.6.3) 1 . . * % * $(( sP Q~ T (;Es ; A~T ) , (Es ; A~);1Q~ . (3.3.7) ( * 2E * (

ME ; T~f~(;s)]T 2ME ; T~f~(s)] = 2E ; T~zTf~(;s)T~zf~(s)

(7.6.4)

T~f~(s) = ;2Q~ T (Es ; A~);1Q~ : (7.6.5) s = j! *, # % *, T~zTf~(;j!)T~zf~(j!) E 2 (7.6.6) ( , ! %,

! ( * 9 , * (7.6.3). = # # A~ - , (7.6.2) ~ P A~ + A~T P = ;Q (7.6.7) Q~ = ;N~ T N~ ; 2P Q~ Q~ T P { # -

# , ~ A~] { . * MN ( (5.2.12) * , ( ( . ( x(t) * . ; < * ' (5.2.13):

u = C T xI C T = ;B T P

(7.6.8)

$(( (5.2.9)

PA + AT P ; PBB T P + ;2P QQT P + N T N = 0: (7.6.9) > ( # C T (, * P (A + BC T ) + (A + BC T )T P + 12 P QQT P + N T N + CC T = 0: (7.6.10) ( 262

PA + AT P + 12 P QQT P ; PBB T P + (C + PB )(C T + B T P ) + N T N = 0:

(7.6.11)

* , *, # C T (7.6.8), K (7.6.9). P * h iT A~ = A + BC T I N~ = N T C (7.6.12) (7.6.10) (7.6.2). & % , . ( (5.1.1), ( % (7.6.8),

x_ = (A + BC T )x + Qf I

h T T iT T T T h T i T z = u = (N x ) C x = N C x (7.6.13)

( * ! (7.6.12)

x_ = A~x + Qf I z = N~ x

, kTzf k . ! ( *, ( % (5.1.1), (5.2.7), (5.2.6) h i xO = xT eT T (7.6.14) e = x ; x , ' < , % x x + , u C T x f

Kf x ,

xO_ = AOxO + QO Of z = NO xO

(7.6.15)

"

# " # " # T T A + BC BC Q 0 N 0 O O AO = QKf A + QKf ; KD I Q = ;Q K I N = C T C T : 7 % , !

! (5.2.6) * (5.1.1)

e_ = ;x_ + A(x + e) + BC T (x + e) + QKf (x + e) + K (Dx + G) ; KD(x + e) = = QKf x + (A + QKf ; KD)e ; Qf + K G: 5 , 263

h

i

z = T uT T = (N x)T C T x

T T

T T O = (Nx)T C T (x + e) = N xO :

; 5.2.2 % # C T , Kf K ( , * # PO 0 , $(( O PO AO + AOT PO + ;2PO QO QO T PO = ;NO T N: (7.6.16) > ( ' < * - % # " # P 0 O P= 0 P : (7.6.17) 1

(7.6.16) % 9

3 # 2 A + BC T T T (QKf ) P 0 64 T 75 + T T 0 P1 QKf A + QKf ; KD BC (A + QKf ; KD) " # " #" #" T T #" P 0 # P 0 P 0 Q 0 Q ; Q ; 2 0 P1 + 0 P1 ;Q K 0 P1 0 P1 = " T #" # N C N 0 = ; 0 C CT CT : (7.6.18) - % ! % ( < (7.6.10) < ( ' (5.2.7). ; ! % % ( "

#"

A + BC T

BC T

PBC T + (QKf )T + ;2P QQT P1 = ;CC T : * (5.2.7), *

(7.6.19)

Kf = ;2QT P: 4 % % ( (7.6.18)

(7.6.20)

P1(A +QKf ; KD)+(A + QKf ; KD)T + ;2P1Q QQT + KK T P1 = ;CC T : (7.6.21) * (5.2.7) (7.6.20) ' P1A + AT P1 + ;2 P1QQT P + ;2 P QQT P1 + ;2 P1(QQT + KK T )P1; ;P1KD ; DT K T P1 = ;PBBT P: & < (7.6.9) 264

(P1 + P )A + AT (P1 + P ) + ;2(P1 + P )QQT (P1 + P ) + ;2P1KK T P1; ;P1KD ; DT K T P1 = ;N T N: D (P1 + P )A + AT (P1 + P ) + ;2(P1 + P )QQT (P1 + P ) ; 2DT D+ +( DT ; ;1P1 K ) ; ( D ; ;1K T P1) = ;N T N: # (<99# (%, *

P1K = 2D: (7.6.22) (P1 + P ) * A(P1 + P );1 + (P1 + P );1 ; 2(P1 + P );1 DT D(P1 + P );1 + QQT ;2 = = (P1 + P );1N T N (P1 + P );1 :

(7.6.22) (7.6.23)

(7.6.24)

; #

Y = 2(P1 + P );1: (7.6.25) (7.6.24) (P1 + P );1 = Y ;2 2 *, Y = Pe (5.2.10). 4% , ( ! # P1

%. )' (7.6.25) (( Y (P + P1 ) = E 2 , P1 = 2Y ;1 ; P . @ 2Y ;1 ; P > 0 , PY < E 2 , ( * * Y = Pe , <( (5.2.10).

;7#&# 7 ( #% # 1 $ %# '"#

' * . ( . 4 , ' . (, * 9 (# (??)

x]_ = A]x] + b~ uI y = d] x] ~ b1 .. A] = 0 ;Ean;1 I ~b = ~ . bn;1 ~bn p p p p

p

p

p

p

p

p p p p

p

p

p

p

p

p

p

p

p

p

p

I

(7.7.1)

d~ = 1 0 : : : 0 I

a~ = a0 a1 : : : an;1 : 265

(7.7.2)

, . (7.7.1), (7.2.2), (7.2.10): _ ] x] A 0 0 x] b~ v_ (1) = b] d] F 0 v(1) + 0 0 : v_ (2) 0 0 F v(2) b] p

p

p p

p

p

p p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p p

p

p

p p

p p p

p

p

p

p

p p pp

p

p

p p

p p

p p

p p

p p p p

p p p p

p

p

p

p

p p

p p

p p p

p p p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

(7.7.3)

y = d] x] :

(t) = + '(t)

(7.7.4)

x_ = Ax + b Mng + '0(t) ] I y = dx

(7.7.5)

{ ( %

% ( , ' (7.7.3) (7.7.4) ((

x0 = x] 0 v(1) v(2) I 0

0

] ~ ] ~ (1)0 ~ (2)0 ~ b A + 0 bd b b ] ] A = bd I b = : (7.7.6) b] d] b] F(1)0 F + b]0(2)0 b0] 0 P '(t) 0 , = , (7.7.3) <

% . 7 % , x% { ( , p

p

p p

p p

p p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p p

p

p

p

p

p

p

p

p

p

p

p

p

p

p p

p p

p

p

p p

p p

p p

p p

p p

p

p pp p

p

p

p

p

p

p

p pp p

p

p

p

p

p

p

p

p

p

p

p

p

p

p pp p

p

p

p

p

p

p

p p

p p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

x_ % = Ax% + bngI y% = dx%

(7.7.7)

x0% = x0% v%(1) v%(2) I d~ = 1 0 : : : 0 : P

, ( ( # A ( b , <

0

0

(7.7.7), ( *

9( ( , ( * 9 (#,

y(s) , g(s) % (7.7.7) kn w (s) . k% % ; * (7.7.5) (7.7.7), *

e~_ = Ae + b M'0(t) ] I e = de~

e~(t) = x(t) ; x%(t): 266

(7.7.8)

!

( ( 7.2.2, % ( '(t) , ( # tlim !1 e = 0 . 7 < 9 (# -

= e~0P~ e + '0=' > 0

(7.7.9) P { -

# (3n ; 2)(3n ; 2) ,

' * -

A0P + PA = ;Q (7.7.10) ( Q { ( -

#. 9 (# (7.7.9) (7.7.8)

= (Ae~ + b'0 )0 + P ~e + e~0P (Ae~ + b'0 ) + '_ 0=' + '0='_ = (7.7.11) = ;e~0Qe~ + ( 0b0P ~e + '_ 0=) ' + '0 (~e0P b + ='_ ) : P # Q , (, * ' (7.7.10)

, *, *

P b = d0

(7.7.12)

'_ = ;=;1 ~e0d0 = ;=;1e

(7.7.13)

_ = ;e~0Qe~ 0:

(7.7.14) 7 ( (% # Q M6.5], ( % ( # , * 9 (# w(s) = d (Es ; A);1 b = kk w%(s) % %. ( ( w%(s) { , (7.7.9), (7.7.14) , * tlim '(t) !1 e(t) = 0 . 4 (7.7.14) , (7.7.13), ( * '(t) = (t)+

%( (7.2.16).

;7#&# 8 ( #% # 1 '"#

(7.2.25) * * (7.2.27) 267

u = 0 + '0(t) + '_ 0 = 0 + '0(_ + 0 ) + '_ 0 = 0 + (s + 0)'0:

(7.8.1)

( 7.2.2 , * u (7.8.1), *

e~_ = Ae~ + b M(s + 0)'0 ] e = de~: D ((

(7.8.2)

e~_ = Ae~ + b'0 e = de~ (7.8.3) h i d MEs ; A);1 b (s + 0) = kk w%(s + 0) { * 9 (% #. @ 9 (# - (7.7.9), (* , (( , * (7.2.28) * # tlim !1 e = 0 .

268

# #' References A *, * 1

M1.1] % ( ;.B.. ) 4.&., ' B. 0. <( . 0.,; '.'(., 1969. 296 . M1.2] : .'. . 0.,; '.'(., 1980. 288 . M1.3]

.

B. . < ( . 0.,D , 1961. 187

M1.4] = %.%. 0.1, 1963, 1966.

! * (! .

M1.5] # C. . . 0.,D , 1987. 256 . M1.6] & * ( * ( / . %.%.:. 0.,4(, 1987. 712 . M1.7] % '.D.,:) '.8.,D '..0 * ( ( .0.; '.'(,1989,488. M1.8] % %.E. .0.; '.'(,1989,263. M1.9]

:.%.,= D.'.,C D..0 ! * ( .0.@.0=, . 4.D.> ,2000.

M1.10] 0 , % - * ( ./ . 4.7.P.0.@.0=, . 4.D.> ,2002.

A 2

M2.1] E ) 7.,., : ) E.%.. 7. . 0 * ( ! . 0.,1. 1969. 512 . M2.2] % '.!., <" '.!., = ,.'. . 0.,4(, 1979. 430 . M2.3] '.%., = D.'. , ! * ( . 0.,4(, 1981. 336 . M2.4] : .. ; #

#9 . 0.,D , 1967. 440 . 269

M2.5] , <.:. #

. 0.,4(, 1977. 480 . M2.6] :) D.D. , . 0.,4(, 1968. 476 . 2.7. M2.7] 0 * ( ! # . / 7.,. . 0.,4(, 1961. 392 . M2.8] ' =. . 3

' <( ! *. 0.,4(, 1980. 518. M2.9] = (. .

' * . 0.,4(, 1978. 487 . M2.10] 8" D.,. 3

. 3.I. 0.,4(, 1973. 663 . M2.11] : D.D. 3

. 0.,4(, 1978. 512 . M2.12] ! D.D. D ! . 0.,4(, 1975. 528 . M2.13] # =.7., 8 D.'. ; #

* ! ( . 0 ,4(, 1973. 238 . M2.14] 8) '.E. ( . 0.,4(, 1973. 446 . M2.15] 8) %.E. ,

. 0.,4(, 1965. 376 . M2.16] 7 ) . 0 # ( (* ( : . . 0.,@-, 1965. 538 . M2.17] # '. ., 8 '.., ,) '.%. < ( * (

(. 0.,D , 1968. 232 . M2.18] 8) %.E., # $ %.B. < ( ! * ( %

(. 0.,D , 1972. 109 . M2.19] ") .C. 4 ( #

*,

( ( . // ( ( ! (. 1946. ,. 10. ; . 2. M2.20] 8 (. 7 * ( . 0.,@-, 1960. 400 . M2.21] (9 7.. # ( -. &. ! . //B( ! (. 1959. ,. 20. 10, .1320- 1334I 11, . 1441-1458I 12, . 1561-1578. 270

M2.22] : '.=. 0 ' #

! * * ! % I II. //B( ! (-1962. ,. 13. 12 1963. ,.14. 5. M2.23] : '.=., 8 '.3., E '.. 4 #

* ( . 0.,4(, 1969. 288 . M2.24] : '.=., E '.. 0 0.,4(, 1973. 446

* .

M2.25] F !.!. 4 ! * 9 > . //7(. B4 &&&$. ,. 242. 5. M2.26] ,

%.., ,

D.D. 5 * ( * . //@. B4 &&&$. , ! * ( ( (. 1983. 2. c. 24-32. M2.27] !) .., , .'. 77023, ( 181007 25 9 1935 . M2.28] = %.%. # ! * ( //B( ! (. 1953. 6. c. 712-728. M2-29] 8) '.E.. E (.'., 7.,. 5 ! # . //7(. B4 &&&$. 1956. ,. 110. 1. c. 7-10. M2.30] 8) '.E. 0 * ( . 0., 1966. 308 . M2.31] %.%. & % ! , ! %. 0.,4(, 1966. 390 . M2.32] ) '.%. ! * ( # 9 % % '

. -.,D , 1982. 216 . M2.33] :* %.,., : %.%. # * (! %. 0.,D , 1982. 238 .

A 3

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