Qualitative behavior of output for sampled-data feedback control systems

MATHEMATICAL AND COMPUTER MODELLING PERGAMON

Mathematical and Computer Modelling 37 (2003) 109-133 www.eisevier.corn/locate/recto

Qualitative Behavior of Output for Sampled-Data Feedback Control Systems RONG YUAN Department of Mathematics, Beijing Normal University Beijing 100875, P.R. China ZHUJUN JING Department of Mathematics, Htman Normal University Changsha 410000, P.R. China and Academy of Mathematics and Systems Sciences Chinese Academy of Sciences Beijing 100080, P.R. China

(Received December 2000; revised and accepted July 2002) Abstract--The present paper is concerned with the qualitative behavior of output for sampleddata feedback control systems. Some sufficient conditions are established for the existence of almost periodic output, quasi-periodic output, periodic output, respectively. A special case of sampled-data feedback control systems is detailed. © 2003 Elsevier Science Ltd. All rights reserved.

K e y w o r d s - - S a m p l e d - d a t a feedback control system, Hybrid dynamical system, Plant output, Controller, Almost periodicity.

1. I N T R O D U C T I O N This paper deals with the qualitative behavior of output for sampled-data feedback control systems p

~(t)=F(t,x(t),{x([t-j])}~o,e)+Ebj(e)u(n-j),

n<_t
j=O p

N

(1.1)

u(n + 1) = E cj(e)u(n - j) + E dj(e)x(n - j) + g(n, c), j=O

j=0

This work was completed when the first author visited the Academy of Mathematics and Systems Sciences, Academy Sinica. The first author thanks the Academy of Mathematics and Systems Sciences, Academy Sinica for their hospitality. This work was partially supported by the National Natural Science Foundation of China, the National Key Basic Research Special Foundation of China (No. G. 1998020307), the Foundation for University Key Teacher by the Ministry of Education, and EYTP. The authors are grateful to the referees for their valuable suggestions that helped to improve the original manuscript. 0895-7177/03/$ - see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. PII: S0895-7177(02)00327-8

Typeset by ~4,~b~-TEX

Ii0

R. Y U A N AND Z. JING

in the case where F is almost periodic for t uniformly on R N+2 and g : Z ~ R is an almost periodic sequence for every e. The sampled-data feedback control system consists of an interconnection of a nonlinear plant (described by a system of first-order differential equations) and a linear digital controller (described by a system of higher-order linear difference equations). The interface between the plant and the controller is an A / D converter, and the interface between the controller and the plant is a D / A converter. The qualitative behavior of sampled-data feedback control systems has been under continual investigation for many years, with an emphasis on linear systems (see [1-3]). The sampled-data feedback control system is a class of hybrid dynamical systems which are in emphasis recently (see [1-6] and references therein). Many mathematical models in practice, e.g., automatic control, power systems, etc., could be written as the sampled-data feedback control systems or other classes of hybrid dynamical systems. Periodic digital controllers were discussed in [1]. The time almost periodic dependence in (1.1) reflects the effects of certain "seasonal" variations which are roughly but not exactly periodic. In this paper, we pay attention to the sampled-data feedback control system (1.1) depending on piecewise constant delay and discuss quasi-periodic and almost periodic digital controllers. We would like to ask whether or not we can choose a suitable initial position and change the coefficients such that the output of the plant is periodic, quasi-periodic, and almost periodic in time for periodic, quasi-periodic, and almost periodic digital controllers, respectively. The problem on stability of digital controller and plant output reduces to the study of stability of steady position for sampled-data feedback control systems so that the obtained results in [7] can be applied. It is assumed that the system p he(t) = F ( t , x ( t ) , { z ( [ t - - j ] ) } g , o ) + ~ ' ~ b j ( O ) u ( n - j ) , n <_ t < n + 1, j=O

N

(1.2)

u(n + 1) = ~ cj (O)u(n - j) + ~ dj (O)x(n - j) + g(n, O) j=0 j=0 has almost periodic "outer" controller and plant output which we take to be the zero solution, that is, we suppose f ( t , 0 , . . . , 0 ) -- 0, g(n, 0) -- 0, so that (x(t), u(n)) -- (0, 0) satisfies (1.2). Our aim is to seek an almost periodic controller and plant of system (1.1) near the "outer" controller and plant. Expanding (1.1) about the zero solution gives

N p X'(t) = a($, e)x(t) + ~ aj($, e)2:([t -- j]) + ~ b j ( e ) u ( n - j ) j=o

j=o

+f(t,x(t),{x([t-j])}oN, p

u(n + 1) = ~ " cj (e)u(n - j) + ~ j =0

e),

n<_t
(1.3)

N

dj (e)x(n - j) + g(n, e).

j=O

One can think of, e.g., a(t, e) as 0Err oz ~ , 0, "'" ,0, e), but, in fact, it is really (1.3) which we study in this paper. In what follows, we denote by [. [ the Euclidean norm and by [.] the greatest integer function. We say that (z(t), u(n)), n = It], is a solution of equation (1.3) (or equation (1.1)), where x : R ~ R, u : Z --* R, if the following conditions are satisfied: (i) x is continuous on R, (ii) the derivative x t of x exists on R except possibly at the point t = n, n E Z = { . . . , - 1 , 0, 1 . . . . } where one-sided derivative exists, (iii) (x(t), u(n)) satisfies equation (1.3) (or equation (1.1)) in the intervals In, n + 1], n • Z.

Qualitative Behavior

111

The following hypotheses are assumed to hold throughout the paper. (H1) a(t, ~), aj(t,e), j = O , . . . , N , are almost periodic functions in t. T h e y are continuous in e, uniformly in t E R. Let M denote a common bound for these functions on (t, e) E

R x [0, co]. (H2) a(t,O) = a °, ai(t,O) = a °, bj(O) = b°, cj(0) = ¢ , di(0) = d o , i = 0 , 1 , . . . , N , 0, 1 , . . . ,p, are constants. (H3) The module of all roots Ao,..., AN of algebraic equation AN+I

--

Ao AN .....

AN-1A

j =

- A N = O.

is not equal to one, where

Ao = ea° q- a ° - l a ° (ea° - l) , Aj = a

o-1o( _1), aj

e a°

(H4) The module of all roots # o , . . . , #p of algebraic equation ]Ap + I

- - CO/Ap . . . . .

C p _ l ] ~ - - Cp ---- 0

is not equal to one. (H5) f is almost periodic in t uniformly on (X0,...,XN+I). Furthermore, there exist nondecreasing functions M(e) and ~(p, e), 0 < e < eo, 0 < p _< P0 satisfying lim¢_~0 M(e) = 0, lim(p,~)-~(0,0) ~(p, e) = 0, such that [f(t,0,...,0, e)l
Ig(n,e)l<_M(e),

tER,

nEZ,

0
and N+I

If (t, x o , . . . , XN+I, e) -- f (t, GO,..., eN+l, eli < r/(p, e / ~

Ix~ - ~ l

i=O

hold for all t E R, Ix~I,l~il _< P0, (i = 0 , . . . , N

+ 1), 0 < e < e0.

T h e present paper is organized as follows. 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction. Some function/sequence spaces. Some lemmas. Statement of main theorems. Sampled-data feedback control system. Sampled-data feedback control system with parameter. Almost periodic quasi-linear sampled-data feedback control system with parameter. Quasi-periodic quasi-linear sampled-data feedback control system with parameter. An example.

2. S O M E

FUNCTION/SEQUENCE

SPACES

Assume t h a t Wl, w2, • • •, wr E R. Wl, w2,. • •, wr E R are called rationally independent if

k1~1 + k2w2 + . . ' + krcor ~ 0 ,

Vkl,...kr E Q \ {o}.

For example, if w is irrational, 1 and w are rationally independent.

112

R. YUAN AND Z. JING

DEFINITION 1. A s s u m e that wl,w2, . . . ,wr E R are rationally independent. A function x : R --. R p, t ---* x(t), is said to be quasiperiodic with frequencies (Wl,... ,w~) if there exists a periodic function F = F ( 0 1 , . . . , Or) in 81, 8 2 , . . . , and Or with period 1, such that x(t) = F ( w l t , w 2 t , . . . , w r t ) ,

Vt C R.

For convenience, we denote w = (Wl, ... ,w~), m = ( m l , . . . , m r ) E Z ~, and (re, w) = rnlwl + ... + m r w r. Set QP(w)= If(t)= k

EfmC

2~(m'~)t

m

E[fm m

, < +C¢~, f _ m = f m / . )

(2.1)

It is easy to see that every function in QTa(w) is quasiperiodic with frequencies w. Setting [[/[[ = ~-~-m ]fm[, it is easily shown that (QP(w), I[" ][) is a Banach space. If r = 1, that is, m = m l and w = ~1, then (m,w> = m l w l = mw, and f ( t ) = ~-:~,~free i2~'~t is a periodic function with period 1/w. We set

DEFINITION 2. (See [8,9].) A function f is called an almost periodic function, if for any e > 0, the e-translation set of f , r(f,

~) = {~- e R; If(t + r) - f(t)l < e, t ~ R}

is a relative dense set on R (i.e., there exists I > 0, such that for any a c R, [a, a + l] n T ( f , e) ¢ 0). Let {Aj} denote the set of all real numbers such that lim T---*oc

1 f0 T f ( t ) exp ( - i A j t ) dt ¢ O. -T

It is well known that the set of numbers {Aj} in the above formula is countable. The set { ~ N n j A j } for all integers N and integers nj is called the module of f ( t ) , denoted by m o d ( f ) . If we denote by P, Q7v, .A:P the set of periodic functions, quasi-periodic functions, almost periodic functions, respectively, it is well known that the following relation holds:

P c QP c A P . We define ( G Q P ( 1 , w ) = { ¢ ( t ) e C(R)

¢(t) = ECk({t})ei2~(k'~)[t]; ¢-k(S) = Ck(s), k

vk, ¢,(s) c c[0,1];

sup tCk(s)l <

k 0
(2.2)

)

where C(R) denotes the set of continuous functions defined on R. It is easy to see that 6Q7'(1, ,;) is a linear space on R (or C). Clearly, if ¢(t) ~ QP(w), then ¢(t) ~ gIQT'(1,w), since we can write ¢(t) = ~ eke ~2~/~'~t = E Ck({t}lei2~(k'~)It]' k k

where Ck(s) = eke ~2~(k'~)s, which implies that GQP(1, w) is not empty. We define II¢IIG = Y~ sup [¢k(s)[, V¢ e GQP(1,w). --k'- 0 5 s ~ 1

Clearly, I[" [[a is a norm of GQ~(1,w).

Q u a l i t a t i v e Behavior

113

LEMMA 1. (gQP(1,w), I1 IIc) is a Banach space. PROOF. It is well known that (C[0,1],I. [) is a Banach space, where [¢1 = suP0 0, there exists a N > 0 such that when n, m > N, it follows [[¢,~ - Cm[[a < e. It is easily shown that {¢,~} is a bounded sequence in (gQTa(1,w), ][. IIc). By definition, {¢~} can be written in the form ¢~(t)=A.~

k

L JJ

t•R.

k

Since k

it follows sup

¢(n)(s) - ¢~m)(s) < e,

n, m > N,

uniformly for k.

O
This implies that there exists ¢~(s) • C[0, 1] such that

¢~n~(s) -~ ¢~(~),

as ~ -~ ~ ,

uniformly for k.

Since {¢n} is a bounded sequence, it follows that k~

sup I¢~(s)[ < +oo.

O
Define

¢*(t) = ~

¢~({t})~'~<~,~/I~l,

t•R.

k

Then ¢*(t) • gQP(1,w). Clearly, ¢,~ ~ ¢* in (gQP, [[. Ila). For w = (wl . . . . , wr), we define a sequence space QT'(w, Z) as

QT)(w,Z) =

u(n)= Eu~ k

. o,=~.[ E [ u k [ < + o o ,

u_k=~k

}.

(2.3)

k

Setting Ilull = ~ k lual, it is easily shown that ( Q P ( w , Z ) , t1" II) is a Banach space. So, every element in Q79(w, Z) can be seen as the restriction of one element in QP(w) on Z. If 1 and w are rational independent, then every element in Q~o(w, Z) can be seen as a quasi-periodic sequence with frequencies (1,w). If m is an integer, set

Then every element in P ( m , Z) is a m-periodic sequence. If r = 1 and ~o = 1/m with some integer m, then every element in QP(w, Z) is a periodic sequence with period m. DEFINITION 3. (See [8,9].) A sequence x : Z ~ R p is called an almost periodic sequence if the e-translation set of x,

T(x,e) -- (-r 6 Z ] ]x(n + r) - x(n)] < e, V n 6 Z} is a relatively dense set in Z for all e > 0. ~- is ca/led an e-a/most period for x. L E M M A 2.

(See [9].)

(i) If f (t) is an almost periodic function, then {f(n) }nez is an almost periodic sequence. (ii) If {x(n) }n6z is an almost periodic sequence, then there exists an almost periodic function f(t) such that x(n) = f(n) for all n • Z. If we denote by P ( Z ) , QT~(Z), A79(Z) the set of periodic sequences, quasi-periodic sequences, and almost periodic sequences, respectively, it is well known t h a t the following relations hold:

p ( z ) c Qp(z) ¢ A p ( z ) .

114

R. YUANAND Z, JING

3. S O M E

LEMMAS

We will list some lemmas, which will be useful. LEMMA 3. (See [9].) Suppose that {x(n)}n~z is an almost periodic sequence and f(t) is an Mmost periodic function. Then the sets T ( f , e) (3 Z and T(x, e) N T ( f , e) are relatively dense. LEMMA 4. If f(t) is an almost periodic function, then the sequence

{h(n)}nez = {~n T M e"(n+l-s)f(s) dS}neZ

(3.1)

is an almost periodic sequence. Furthermore, if f(t) is m-periodic, m E Z +, then the sequence {h(n) },~ez is a m-periodic sequence. Here a is a constant. PROOF. The conclusion follows easily from h ( n + T) - h(n) =

n+l

f

e a(n+l-s) [f(s + T) - - / ( S ) ] ds.

Jn

LEMMA 5. If f E GQP(1,w), then {h(n)} defined by (3.1) is in Q'P(w,Z). PROOF. Since f E 6QP(1,w), we have

f(t) = E h({t})e~2~(k'~)[t]" k

Clearly,

{°+'

ea(n+l-s) f(s) as =

Jn

lJn °+' ea(n+l-s) Ek fk({s})e i27r(k'w)nds

=

e~f~(1 -- S) ds. ei2~r(k'~)n.

LEMMA 6. K f(t), g(t) • GQP(1,w), then f(t)g(t) • ~QP(1,~o). PROOF. If

f(t) : ~ f~({t})e '~'(~,~/'J,

g(t) = ~ a~({t})e '2"~,~)t'l,

k

k

it follows from Cauchy's theorem that we can write

h(t) = f ( t l g ( t ) = y ~ h({t}le '2~k'~[tj • ~ gj({t}le '2~'~)c'j k j

|

and E m Ihml < +oc. LEMMA 7. I l l ( t )

e ~Qp(1,o~), then

~

t ea(t-s)f(s) ds

Ek ]k({t})ei2'r(~'~)n,

"

n < t < n + I,

where ]k(~-) is continuous for each k. PROOF. Since f(t) E GQP(1,w), we have f ( t ) = E fk({t})ei2*t(k'~)[t]' k

sup Jf~(r)l < +o¢. k rE[O,l]

(3.2)

115

Qualitative Behavior

/)

e~(t-~)f(s) ds =

/)

e~(t-~) E fk({s})ei2~(k'~)[s] ds k

= E ea(t-n) ~0 t - n e-aSfk(s ) ds. e i21r(k'~)n k

=E

fk({t})ei2~(k'~>n'

n < t < n + 1,

k

where

]k('r) = e ar

f

e-aSfk(s ) ds.

Clearly, ~ k sup~e[0,1] ]/k(T)] < +OO. LEMMA 8. Consider the difference equation N

x(n + 1) = E

(3.3)

A j x ( n - j) + h(n).

j=O

Assume that (H3) holds. Then for any a/most periodic sequence {h(n)}, system (3.3) possesses a unique almost periodic sequence solution {x* (n)}. Fhrthermore, there exists a constant MA such that Ix*(n)l <_ MA sup Ih(n)l. (3.4) nEZ

PROOF. Suppose that the different roots of algebraic equation in (H3) are denoted by A1,A2,. ,As $ with multiplicities n l , n 2 , . . . , n s , ~ j = l n j = N + 1. Set L = {l I IAzl < 1, 1 < I < s}, L' = {l I IAll > 1, 1 < l < s}. Then L N L' = 0, L U L' = { 1 , . . . , s } . We define a sequence {z(n)} by st

-

1

• (n) : E E k,,, E [~ -('~ + l)V;C-(m+l)h(m) IEL j=O

m<_R-i

~,-1

(3.5)

+ ~] Z k~,j~ [ ~ - (~+ 1)]J~?-(~+i)h(~), IEL' j = 0

m>_n

where the unknown constants k~j, 0 < j _< nl - 1, 1 < l < s, are determined later. Inserting (3.5) into (3.3) (similar to [10]), it is easy to see that we can choose the k~,j, denoted by kl*,j (the coefficients of linear algebraic equations are the Casorati matrix and its determinant is different from zero (see [11])), such that the sequence tel -

• *(n) = ~

1

] L k5

IEL j=O ~t - 1

Z

[~ - ('~ + 1)]J~?-(m+l)h(~)

re
+ Z Z k,*jZ [~- (~ + 1)]Jx?-(~+~)h(~), IEL' j=O

(3.6)

~e z

m>n

is a solution of the difference equation (3.3). Clearly,

x*(n + ~) - x*(n) = ~

~

IEL j=O

k~*j ~

In - (m + llF~?-
m<_n-1

+ E E kl:J E [n -- (77~+ 1)]J)~?-(m+l)[h(m + T) - h ( m ) ] , IEL' j = o m>_n

(3.7)

116

R. YUAN AND Z. JING

which easily implies that {x*(n)} is an almost periodic sequence. The uniqueness of almost periodic sequence solution follows easily from the fact that the module of all roots is not equal to one and the representation of solutions of homogeneous difference equation (e.g., see [10,11] for details). Equation (3.4) follows easily from (3.6). | REMARK 1. If we consider the following difference equation: p

~(n + 1) = ~ csu(~ - j) + g(~),

(3.s)

5=0

we can obtain the same conclusions under Condition (H4). At this time, we use M e instead of MA in Lemma 8. REMARK 2. The following hypothesis is usually needed. (H6) M c Y'~-;=0]da°l • M A Y~.5=o P [Bj[ < 1. Here B 5 = (e a° - 1)a°-lb °, j = 0 , 1 , . . . , p . REMARK 3. After submitting this paper, our attention was drawn to Corduneanu's work [12]. In [12], Corduneanu studied the existence of almost periodic sequence solution for a difference system of the form y(n + 1) = Ay(n) + h(n), where A is a d x d constant matrix, y 6 R d, h : Z ---, l~ d is an almost periodic sequence. We showed the proof's outline of Lemma 8, mainly because it is useful for later lemmas' proof. LEMMA 9. Assume that (H3) holds. Let wl = (w~1), ... ,w(p) and w2 = (w~2), • • •, ~r2'(2)\/ be rationM independent. Then for any {h(n)} 6 QP(o;1, Z) $ QP(w2, Z), system (3.3) possesses a unique sequence solution {x*(n)} E QT~(a;1,Z) @ QT~(w2,Z), where

Q*'(~Ol, Z) @ QS°(w2, Z) = / {u(n)}

u(n) = Z

/

u(x)

+Z

kl u(kli) + ~ kl

uk2(2)ei2'qk2'~)n',

k2 U(k22) < +OO,. (1) = ~(1),. (2) : ~(2)/ ~-k, ki ~-k2 k2 "

)

k2

Furthermore, there exists a constant M A such that

IIz*ll <_ MAIIhlI. PROOF• We use the same notations as in Lemma 8. From Lemma 8, it follows that there exists k~,i such that the sequence {x*(n)} defined by (3.6) is a solution of the difference equation (3.3). Since {h(n)} E QP(~I, Z) ~ QP(w2, Z), {h(n)} can be written as h(n)

X " l.(1).~i2rr(kl,wl} n J_ ~

: Z.., '°k~ ~" kl

l~(2).~i2r(k2,w2) n

" ~ '°k2 " k2

"

~± ~V" "(2) "

'

Thus, {x*(n)} can be rewritten as x * (n) = ~ " X(1) kz

k2

where XO)kl= h(1)kl

~ k~je-~2r(k~'~OP5 5=0

Aze-i2~r(k~'~°O

+ Z

~l-I ] Z ki*,'(-1)SA/tq5 (A~-le-~2~(k''~)) '

lEL' j=0

Qualitative Behavior

117

and x k2 (2) has the same form. Here,

Let

p j ( z ) = x + 2ix 2 + ... + uJx u + ... ,

Ixl < 1;

qj(x) = 1 + 2ix + . . . + uJx u-1 + ... ,

Ixl < 1.

/~l--1

/~--1

lEL j=O

IEL' j=O

We have x(kll) <- C1 h01)

,

x (2) _< C1 h(2)

,

(1) ---- 2~(1) kl,

X(2)

-:__2~(2)

The rest is the same as those in Lemma 8.

|

LEMMA 10. Under the condition of Lamina 8, then for any periodic sequence {h(n)} with period mo C Z +, system (3.3) possesses a unique periodic sequence solution {x*(n)} with period too. Furthermore, there exists a constant MA such that llx*ll

MAIIhlI.

PROOF. From the proof of Lemma 8, we know that system (3.3) possesses a sequence solution {x* (n)}. The periodicity of {x*(n)} is easily seen from (3.7). | 4.

STATEMENT

OF

MAIN

THEOREMS

We now formulate our main theorem. The proof will be stated later. THEOREM 1. A s s u m e that f ( t , x o , . . . , XN-i-1, £) iS almost periodic in t uniformly for ( x o , . . . , XN+I) and { g ( n , e ) } n e z is an almost periodic sequence. If (H1)-(H6) hold, then there exist ~2,pl,0 < ~2 ~-- £0 and 0 < Pl ~- Po, such that for each e satisfying 0 < e ~_ £2, (1.3) has a unique almost periodic solution (x*(t, e), u*(n, e)) satisfying ][x[l ~ Pl, flu][ -~ Pl, i.e., x*(t) is an almost periodic plant and { u* ( n ) } is an almost periodic digital controller, and this solution is continuous in e uniformly in t C R , n E Z and satisfies ][x*(e)[I + ][u*(e)][ = O ( M ( e ) ) as e -* O. THEOREM 2. A s s u m e that a(t,e), aj(t,e), 0 <_ j <_ N , f ( ' , x 0 , . . . , X N + l , e ) are real analytic quasi-periodic functions with frequencies Wx, {g(., e)}nez E QP(w2, Z) and 1, wl, w2 are rational independent. If (H1)-(H6) hold, then there exist e2,pl,0 < e2 <_ eo and 0 < Pl <_ Po, such that for each e satisfying 0 < e <_ e2, (1.3) has a unique quasi-periodic solution (x*(t, e), u*(n,e)) E QP(1,w) × Q P ( w , Z ) satisfying [Ix[I <_ Pl, [full <_ Pl, i.e., x*(t) is a quasi-periodic plant and {u*(n)} is a quasi-periodic digitM controller, and this solution is continuous in e uniformly in t E R, n c Z and satisfies llx*(e)ll + llu*(e)ll =

O(M(e)) as

O, where w = (Wl,W2).

THEOREM 3. A s s u m e that a(t,e), aj(t,e), 0 <_ j < N , f ( t , x o , . . . , x N + l , e ) are periodic in t with period T and {g(., e)}nez E P(m2, Z) with m2 E Z +. If (H1)-(H6) hold, then there exist e2,Pl,0 < e2 <_ e0 and 0 < Pl <_ Po, such that for each e satisfying 0 < ~ < e2, (1.3) has a unique solution (x*(t, e), u*(n, e) ) satisfying llxll _< pl, llull _< pl and this solution is continuous in e uniformly in t c R, n E Z and satisfies [[x*(e)[[ + ]]u*(e)[ I = O ( M ( e ) ) as e ~ O. Furthermore, the following conclusions hold. (1) I f T = nl E Z +, then the unique solution (x*(t,e),u*(n,e)) is mc = (nl,m.2)-periodic, where (El, m2) denotes the minimal common multiple of nl and m2. (2) If T = n l / m l , El, m l E Z +, n l and m l are mutually prime, then the unique solution (x*(t, e), u*(n, e)) is mc = (mlT, m2)-periodic. (3) If T is irrational, T = 1/w, and f 6 P(w), then x*(t,e) 6 QTv(1,w) and u*(n,e) c

ep(w, z).

118

R . YUAN AND Z. JING ,

SAMPLED-DATA

FEEDBACK

CONTROL

SYSTEM

In this section, we consider the following system: N

p

• '(t) = a0~(t) + ~ a0~({t - Jl) + ~ j=o

u ( . + 1) = ~

p

b°u(n - J) + I(t),

n
j=o N

c°u(~ - j) + ~

j=0

(~.l)

eo=(. - j) + g(~).

y=0

If (x(t), u(n)) is a solution of system (5.1), we have the following relations: N

z(t) = ea°(t-n)x(n) + (e a°(t-n) - 1)

Ea°-~a%(n- j) j=0

+ (°°o(,_.>,)

J..°o(,_.,:(.)

P

t

j=O

n < t < n + 1. In view of the continuity of x(t) at a point and (H3), we arrive at the following difference systerff: N

p

x(n + 1) = ~

Ajx(n - j) + Z

j =0 p

Bju(n - j) + h(n),

j =0

(5.2)

N

u(n + 1) = ~

c ° u ( n - j) + E 4 x(n - J) + g(n),

j=0

j=0

where Bj -'- (e a° - 1)a°-lb °, (j = O, 1 .... ,p), h(n) = fn+l eaO(n+l_s)f(s) ds. Let A P ( Z ) be the set of all almost periodic sequences {x(n)} defined on Z, which are equipped with the norm Ilxl[ = sup~ez Ix(n)l. Let A P ( R ) be the set of all almost periodic functions f(t) defined on R, which is equipped with the norm IIfH = supteR [f(t)l. It can be shown that ATa(Z) and AT)(R) are Banach spaces. THEOREM 4. Assume that (H3), (H4), and (H6) hold. Then for any almost periodic sequences {h(n)} and {g(n)} the difference system (5.2) possesses a unique almost periodic sequence solution (x* ( n ) , u *(n ) ) . Furthermore, there exist constants M1, M2, N1, N2 such that

Iz*(n)l _< Mlllhll + N~llgll, [u*(n)[ _< M211h[I + N211gll,

n e z.

(5.3)

PROOF. Let (u0(n)} E A P ( Z ) . We consider the difference system N

x(n + 1) = ~

p

Ajx(n - j) + ~

j=O

j=0

p

N

Bjuo(n - j) + h(n),

(5.4)

u(n + 1) = E c°u(n - J) + Z ~ x ( n - j) + g(n). j=o j=o Note from Lemma 8 that the first equation in (5.4) is solved first. It has a unique almost periodic sequence solution {x(n)}. Then this x is substituted to the second equation in (5.4) which is then solved for {u(n)}. Therefore, the inhomogeneous difference system (5.4) has an almost periodic sequence solution (x(n), u(n)). Writing u = T(uo), then solving (5.2) is equivalent to find a fixed point of T(uo).

Qualitative

Behavior

119

If u = T(uo) and r = T(fio), the first equation in (5.4) is solved first so that it has a unique almost periodic sequence solution {x(n)} and {~(n)}, respectively. Then these x and ~ are substituted to the second equation in (5.4) which is then solved for {u(n)} and {r(n)}, respectively. Let p = u - r, X = x - ~, and #0 = u0 - rio. We find that # and X satisfy N

p

x(n + 1) = ~ Abx(n - j) + ~ Bb#o(n - j), j=0 5=o p

(5.5)

N

c°#(n - j) + ~ ~ x ( n - j).

#(n + 1) = ~ j=O

j=O

From Lemma 8 and Remark 1, it follows that N

I#(n)l <_Mc ~

Id°l sup Ix(n)[

5=0 N

p

Z 141MAZ 151 supI.o/n/I

<

j=0

5= 0

n

If (H6) holds, then the contraction mapping principle implies that T has a unique fixed point {u* (n)}. Correspondingly, the unique almost periodic sequence solution {x* (n)} can be obtained from the first equation of system (5.2). And, P

[x*(n)l <_ MA ~ 5= 0

lu*(n)l <_Mc ~ 5=o

tSj[ sup lu*(n)[ + MA sup th(n)l, n

n

la~jl MA ~ L

]

[Bst sup lu*(n)l + MA sup Ih(n)l + Mc sup I9(n)l,

5=0

~

which implies that

lu*(n)l ~ M2 sup Ih(n)l + N2 sup Ig(n)l, n

n

where

--1

M2 = (1 - M C

Id°l MA j=o

I t )_1

~ I~t McMA, j=o

IBjl

j=o

N2 = (1 - M e y ~ [a°l MA ]Bj[ j=o j=o

Me.

So, we have Ix*(n)l _< M1 sup Ih(n)[ + N1 sup Ig(n)[, n

where

P

MI=MAM2~-~IBj[+MA,

P

NI=MA~-~IBjlN2.

j=o

j=o

THEOREM 5. Assume that (H3), (H4), and (H6) hold.

(1) If Cg1 and w2 are rational independent, then for any {h(n)} e Q ~ l : ) ( O J l , Z ) and {g(n)} • QTP(w2,Z), the difference system (5.2) possesses a unique sequence solution (x* (n), u* (n) ) • Q7~(~1, z) ~ Q p ( ~ , z). (2) H m l and m2 are integer, then for any {h(n)} • 79(ml,Z) and {g(n)} • P(m2, Z), the difference system (5.2) possesses a unique sequence solution (x*(n),u*(n)) • 7~(mc, Z), where mc = (ml, m2) is the minimal common multiple ofml and m2.

120

R. YUAN AND Z, JING

Furthermore, there ex/st constants Mx , 1142,N1, N2 such that

I[x*ll <__Mxllhll + Nlllgll,

nEZ.

Ilu*ll -< M211hl[ + N211gll, PROOF.

(1) Let {u0(n)} • QT~(wl, Z) @ QT~(w2, Z). We consider the difference system N

p

x(n + 1) = E A j x ( n - j) + E B j u ° ( n - j) + h(n), j=0

j=0

p

N

u(n + 1) -- ~

(5.6)

c~u(n - j) + E ~ x ( n - j) + g(n).

5=o

j=o

Note from Lemma 9 that the first equation in (5.6) is solved first. It has a unique sequence solution {x(n)} • QT~(wl,Z) @ QP(w2, Z). Then this x is substituted to the second equation in (5.6) which is then solved for {u(n)} • Q'P(Wl, Z)@QP(w2, Z). Therefore, the inhomogeneous difference equation (5.6) has a sequence solution (x(n), u(n)) • QP(wx, Z) @ QP(w2, Z). Writing u = T(uo), then solving (5.2) is equivalent to finding a fixed point of T(uo). The rest of the proof is similar to that of Theorem 4. (2) Let {u0(n)} E "P(mc, Z). The rest is similar to Part (1), where Lemma 10 is employed in this case. II THEOREM 6. Assume that (H3), (H4), and (H6) hold. Then for any almost periodic f(t) and (g(n) }, the sampled-data feedback control system (5.1) possesses a unique almost periodic solution (x(t), u(n)), i.e., x(t) is an a/most periodic plant and {u(n)} is an almost periodic digital controller. Furthermore, there exist constants Mx, M2, N1, N2 such that Ix(t)l _< M1 sup If(t)l +/V1 sup ]g(n)l,

t ~ lu(n)l _< M2 sup If(t)l + 192 sup Ig(n)[. t

(5.7)

n

PROOF. From Theorem 4, we know that the difference system (5.2) possesses a unique almost periodic sequence solution (x*(n), u*(n)). We define %

2

•

P

+ (e a°(t-n) - 1) E a ° - l b ° u * ( n - j ) +

N

1) Ea0-1aox,(n

f'

J)

5=0 ea°(t-s)f(s)ds,

(5.8) n <_t < n + 1.

5=o It can be easily shown that x(t) is almost periodic (e.g., see [10] for details) and (x(t),u(n)) satisfies system (5.1), where u(n) = u*(n). From Theorem 4, it is known that there exist constants M1, M2, N1, N2 such that (5.3) holds. It is easily seen that there exist constants 2t5/1,A:/2, 5/1, hr2 such that (5.7) holds. | THEOREM 7. Assume that (H3), (H4), and (H6) hold and wl and w2 are rationM independent. Then for any f (t ) C QP (wi') , {g(n)} e Q'P (w2 , Z ) , the sampled-data feedback control system (5.1) possesses a unique solution (x(t), u(n) ), where x(t) e GQT~(1, (Wl,W2)) and {u(n)} E Q~O(oJ1,Z)~ QP(w2, Z), i.e., x(t) is a quasi-periodic plant and {u(n)} is a quasi-periodic digital controller. Furthermore, there ex/st constants Iffl, 2fiR, fll, lq2 such that (5.7) holds. PROOF. Since f(t) • Q'P(wl), f can be denoted as f(t) = Y]~kl fk, e~2~
h(n) = J n

ea°(n+l-s)f(s) ds = ~A-, h-~,e i2~r(ka ,Wl)n , kl

Qualitative

where

121

Behavior

{ ca° [e~2~(kl,wl)-a° _ 1]

hkl =

i~-~-a-g

¢0, -'g~]L \ if $~7rk~1,°31] -- a 0 : 0.

fkl,

if i2~r(kl,Wl) - a °

a o

e Jkl,

From Theorem 5, we know that the difference system (5.2) possesses a unique sequence solution (x*(n), u*(n)) and

z*(n)

= ~x~,k

e ~ 2 ~ < k ~ ' ~ / n -b - v2.~ --

k~ US(n) -~--E U*],klei~Tr(kl'L~)l>n

, ei2~(k~,~)~ X2,k2

k2 _L ~

~,* ei27r(k2,o)2> n ~ t~2,k2 k2

kl

We define x(t) by (5.8). From the constructions of x*(n) and u*(n), we know that x(t) is continuous on R and (x(t), u(n)) satisfies system (5.1), where u(n) = u*(n), x(t) can be rewritten in the form

x(t) = e ~°{t} E

"~1, ~* k l ~~i2n{kl,wl}t~-i2~(kl,wl}{t}

k~ N

E a 0- aj0 E x*l,kl e-i2~r{kl'a~Oj~i2~r(kl'wt}t~-i27r{kl'~l}{t}~

+

j=0 P

--

kl

a

Oj ~. Ul,kl e-i2~(kl'wl)Jei27r(kl'wl}te -12~(kl'wl}{t}

j=0

_}_ea°{t} ~'~ ~*

kl

~i2~r(k2,w2}t~-i27r{k2,w2}{t}

k2 N --

j=0 P

-~ (e a°{t} --

1) E

a0 1 O aj E

~'* e -i2x
~2,k2

~

k~ a ° - l "~---~oj ° ~ ?A2,k2*e-i2~r(k2'w2)Je*27r(k2'w2}te-i2~r(k2'w2}{t}

j=0

k2

+ E f~" ~i2~r{k~'wl)t~(a°-i2~(k~'w~)) {t} f kl

{t} e( i2~r(k''"~O-a°)s ds.

JO

It is easy to see that x(t) can be written in the form

x(t) = F (t, Wlt, w2t), which is continuous on R, where F(81,02,03) is periodic in 0~, 02, and 03 with period 1. It follows that x(t) is quasiperiodic with frequencies (1, w~, w~). We can also write the quasi-periodic solution x(t) as follows:

x(t)

~ X(1){lt ~i2~r(k~'~)[t] + ~'~ x(2)(It~e i2~(k~'w~)[t] , k~ k2

where N j=0 P j=0

+ "Jkl e a%

/o s e(i2~(kl'~l)-a°) ° dO

122

R. Y U A N A N D Z. JING

and N

x(~ ) (s) = ea%x*2,k~ + ( ea% - 1) ~',_..,a°-la°x~,k~e -i2~(k2,~2)j j=0 P

+ (ea%- 1) Ea°-lb°u~,k2e -i2r(k2'w2>j j=O

are continuous on [0, 1], x 0), (s) = 2(11) (s), x(2) 2 (s) = 2(22) (s). It follows from Theorem 5 that (5.7) holds. This implies x(t) E ~QP(1, (wl,w2)). | REMARK 4. The function x(t) in the above-mentioned theorem is a new class of quasi-periodic functions with frequencies (1, w) and is called a generalized quasi-periodic function with frequencies (1, w). THEOREM 8. Assume that (H3), (H4), and (H6) hold. Then for any T-periodic function f(t) and any m2-periodic sequence {g(n) }, m2 6 Z*, the sampled-data feedback control system (5.1) possesses a unique solution (x*(t), u*(n)). The unique solution (x*(t), u*(n)) has the following properties. (i) I f T = ml 6 Z +, then (x*(t), u*(n)) is mc = (ml, m2)-periodic, where me is the minimal

common multiple of ml and m2. (ii) I f T = mz/nl, m l , n l • Z +, then (x*(t),u*(n)) is mc = ( n l T , m2)-periodic, where mc is

the minimal common multiple of n i t and m 2. (iii) If f(t) • P(w), w is irrational, {g(n)} • P(1/m2), then x*(t) is quasiperiodic with frequencies (w , l / m 2 ) and {u(n)} • QP ( (w , l / m 2 ) , Z ) . Furthermore, there exist constants 1VI1,iVI2, N1, N2 such that (5.7) holds. PROOF.

(i) If T = m l • Z +, then it follows from L e m m a 4 that h(n) = fn+l eaO(n+l_s) f(s) ds is also ml-periodic. It follows from Theorem 5 that the difference system (5.2) possesses a unique sequence solution (x*(n), u*(n)), and (x*(n), u*(n)) is mc = (ml, m2) periodic. We define x(t) by (5.8). It is easy to prove that x(t) is mc-periodic. (ii) If T = m l / n l , then it follows from L e m m a 4 t h a t h(n) = f~+l eaO(n+l_s) f(s)ds is niT-periodic. It follows from Theorem 5 that the difference system (5.2) possesses a unique sequence solution (x*(n),u*(n)), and (x*(n), u*(n)) is mc = (niT, m2)-periodic. It is easily proved that x(t) defined by (5.8) is m~-periodic. (iii) The result follows from Theorem 7 directly. |

6. S A M P L E D - D A T A F E E D B A C K CONTROL SYSTEM WITH PARAMETER In this section, we now consider the following system: N

p

x'(t) = a(t, e)x(t) + E aj(t, e)x([t - j]) + E j=0 p

u(n + 1) = E j=o

bj(e)u(n - j) + f(t),

j=0 N

(~.1)

cj(e)u(n - j) + E dj(e)x(n - j) + g(n), ./=o

where f : R -* R is an almost periodic function and g : Z --~ R is an almost periodic sequence for each c.

Qualitative

Behavior

123

THEOREM 9. Under Assumptions (H1)-(H6), there exists an el > 0 such that for each (f, g) c ATe(R) x ATe(Z), 0 < e _
Ix*(t)l <_IVI1sup If(t)l + N1 sup Ig(n)l, t

n

lu*(n)l
n

The map (f, g) --+ (x*(., f, g, e), u*(., f, g, e)) defines a bounded linear operator £(e), and e --+/:(e) is continuous for 0 < e < el. PROOF. Suppose (xo(t), uo(n)) e AP(R) x AP(Z). We consider the following system: N

p

x'(t) = a°x(t) + ~ a°x([t - j]) + ~-~ b~u(n - j) + (a(t,e) - a °) xo(t) j =o j =o

N

p

+ ~ (aj(t,e) - a~) x o ( [ t - j]) + ~ (bj(e) - b°) u o ( n - j) + f(t), j=o j=o

~(~+ 1) = ~

p

(6.2)

N

c°u(n - j) + Z d°~( ~ - 5)

j=o

j=o p

N

+ ~ ( c j ( e ) - c o) u o ( n - j ) + ~ ( d , ( e ) - ~ ) x o ( n - j ) + g(n), j=o j=o n < t < n + 1. From the first equation and the continuity of x(t), we have N

p

x(r~ + 1) = ~ Ajx(n - j) + ~ Bju(n - j) + h(n, e), a=o j=o where

~n+l

h(n, e) = L

+[

ea°(n+l-*) (a(s, e) - a 0) Xo(S) ds

n+l

N

e°°(~+l-s>~(aj(s,~)-a j=O

,]?2

°) d = x o ( n - j )

P

+ (e " ° - I ) a ° - 1 ~

n+l

(b,(e)-b~)uo(n-j)+

L

(6.a)

0

e" ("+'-*)f(s)ds.

j=O From Theorem 4, it follows that the difference system N

x(n + 1) = ~ A j x ( n j=0 p

u(n + 1) = ~ c ° u ( n j=o p

p

j) + ~

Bju(n - j) + h(n,e),

j=0 N

j) + ~ ~ z ( n j=o

j)

(6.4)

N

+ E (c,(~) - c o) ~o(~ - J) + E (d,(~) - e o) xo(n - 5) + g(n) j=o j=o

has a unique almost periodic sequence solution (x(n, e), u(n, e)). We define N

X(t,g) : ea°(t-n>x(n,g)-}- (e ao(t-n) - 1) ~ a ° - l a l x ( n - j=o P

+ @oo(,_,,)_ I) E a ° - ~ b % ( n j=O

- y,~)+ h(t,~),

j,£) (6.5)

124

R. YUANAND Z. JING

where

h(t,,) = ~nt ~oo(,-,) (~(~, ~) _ ~o) ~o(4 ~ +

~n t

o

N

e a (t-s) E

(aj(s,e) - a °) ds. x o ( n - j)

(6.6)

j=O P

-F (e a°(t-n) -- 1)a°-iE

t

(bj(c) - bO) uo(n - j) 21-~n ea°(t-s) f(s) ds' j=o

n < t < n + 1. It can be easily shown t h a t x(t, e) is an a l m o s t p e r i o d i c function (e.g., see [10] for details). We write x(t) = x(t,~) and u(n) = u(n, e) for short. W r i t i n g ( x , u ) = T(xo,uo), t h e n solving (6.1) is equivalent to finding a fixed p o i n t of 7"(., .). If (x, u) = T(xo, uo), (2, ~2) = T ( 2 o , t o ) and let X = x - ~, # = u - fi, Xo = xo - 20, and #o = uo - ~2o, we find t h a t X a n d # satisfy N

p

x' (t) = a°x(t) + E a°x([t - J]) + E j=o

b° #(n - j) + (aCt' ~) - a°) xo(t)

j=o

N

p

+ E (a.(t,.) - ~o) ~o([t - jJ) + E (b.(.) - b~) ~o(~ - J), j=o

j=o

p

(6.7)

N

c°#(n - J) + E d°x(n - J)

tt(n + 1) = E j=0

j=0 p

N

+ ~ (~j(~)-¢).o(n- j) + ~ (d,(~)-e o) xo(~- J). j=o

j=o

F r o m T h e o r e m 6, it follows t h a t t h e r e exist c o n s t a n t s A7/1,-/17/2,fi/'l,/92 such t h a t

Ix(t)l < ~

sup (~(t, ~) - ~o) xo(t) t~n

N

+E

p

(aj(t, e) - a °) Xo([t - j]) + ~

j=o

t,n j = 0

I#(n)l _
N

-

J)

j=o

+ ~ s u p ~p ( ~ j ( 4 - 4 ) . o ( n -

+E

(b~(e) - b°) # o ( n

J) + ~N (~j(~) - ~°) ~o(n - J ) j=0

(a(t, e) - a °) Xo(t)

(aj(t, e) - a °) Xo([t - j]) +

j =o

~

(bj (e) - b°) # o ( n -

J)

j =o

p

N

+ / 9 2 sup E (cj(e) - c°) #o(n - j) + E (dj(e) - d °) xo(n t,n j = 0 j=O

-J)

Therefore, t h e r e exists el > 0 such t h a t if 0 < e < el, the c o n t r a c t i o n m a p p i n g p r i n c i p l e implies t h a t T has a u n i q u e fixed p o i n t so t h a t s y s t e m (6.1) possesses a unique a l m o s t p e r i o d i c solution

Qualitative Behavior

125

(x*(t), u*(n)). From Theorem 6, it follows t h a t there exist constants 217/1,M2, N1, N2 such that

[x*(t)t <_ h:/lsup (a(t,e)

-

a °) x*(t)

t,r~

N P f(t) + E (aj(t,e) - a °) x * ( [ t - j]) + E (bj(e) - b°) u * ( n - j) + j=o -b

j=o

/~1 sup

~(

cj(e) - c °) 3 ~'(n - j) + ~N (dj(~) - d °) x*(n - j) +

~(~1

j=O

t,n

lu*(n)l <_ 117/2su p (a(t,e)

-

a °) x*(t)

t,r~

+ E N( a j ( t , e ) - a ° ) x * ( [ t - j ] ) +

~

j=o

(bj(e)-b°)u*(n-j)+

f(t)

j=o

t,n

j=O

j=O

From these, it follows that there exist constants el > 0, /V/l,/V/2, N1, fi/2 such that when 0 < e _< q , we have Ix*(t)[ ~ ~/1 sup If(t)[ + IV1 sup [g(n)], t

lu*(n)[ <

n

,Q2 sup If(t)[

+ N2 sup Ig(n)l,

t

n

here we would not like to use more notations so that we use the same q . If we set (z(f, 9, e), u(f, g, e)) =/:(e)(f, g), and X = x(f, g, e) - x(f, g, e'), # = u(f, g, e) - u(f, g, e'), then we have N

P

x'(t) = a(t, e)x(t) + E aj (t, e)x([t - j ] ) + E bj(e)#([t - j]) j=0

j=0 N

+ (a(t, ~) - a (t, ~')) z (t, ~') + ~

(aj(t, ~) - aj (t, ~')) x (It - j],

~')

j=0 P

+ ~ (bj(~) - bj (¢1) u (n - j, ~'), j=O p

/z(n + 1) - E

N

cj(e)l~(n - j) + E dj(e)x(n - j)

j=O

j=O p

N

+ E (cj(e) - cj (e')) u (n - j, e') + E (dj(e) - dj (e')) x (n - j, e'), j =o j =0 here we write x(e') = x(f,g, e'), u(e') = u(f,g, e') for short. It follows that e ~ E(e) is continuous. U THEOREM 10. Assume that wl and w2 are rational independent, a(t, e), aj(t, e) E Q'P(wl). Un-

der Assumptions (H1)-(H6), there exists an el > 0 such that for any f(t) C QP(wl), g(n) E Q'~(c~32, Z ) , 0 < e ~ el, system (6.1)possesses a unique solution (x*(t,f,g,e),u*(n,f,g,e)),

126

R. YUAN AND Z, JING

where x * ( t , f , g , e ) • QQP(1,(Wl,W2)), u*(n,f,g,e) • Q79((wl, ca2),Z), i.e., x*(t) is a quasiperiodic plant and {u*(n)} is a quasi-periodic digital controller. Furthermore, there exist constants 1~/I1,IVI2, 91,1~ 2 such that Ix*(t, e)l _< / ~ 1

sup t

If(t)l + N1 sup ]g(n)l, n

[u*(n, e)l < A~/2sup [f(t)l + -~2 sup Ig(n)l. t

n

The map (f,g) ~ (x*(f,g,e), u*(f,g,e)) defines a bounded linear operator £(e), and ~ ~ £(e) is continuous for 0 < e < Q. PROOF. Let w = (wl,w2). For any zo(t) • GQP(1,w) and {u0(n)} • QP(w, Z), we still consider systems (6.2) and (6.4). In this case, we know from L e m m a 5 that {h(n,e)} • Q P ( w , Z ) . It follows from Theorem 5, that system (6.4) has a sequence solution (x(n, c), u(n, e)) • QP(w, Z), i.e., we have the following formula:

x(n, ,)

=

,) = k

Ixk(

k

)l <

k

lu ( )l <

=

=

k

for each c > 0. We define x(t, e) by (6.5). rewritten as follows: x(t, ,) =

From Lemmas 5-7, we know that x(t, ~) can be

k •

sup [xk(s,e)[<+c~, 8~[0,11

x k ( ( t } , ,)e k

for each e, which is continuous on R so that x(t, e) c GQP(1,w). The rest is similar to these in Theorem 9 and follows easily from the proof of Theorem 9. | THEOREM 11. Assume that a(t, e), aj(t, e) are T-periodic. Under Assumptions (H1)-(H6), there exists an el > 0 such that for any T-periodic function f ( t ) and any m2-periodic sequence {g(n)},

m2 integer, 0 < e < el, system (6.1)possesses a unique solution (x*(t, f, g, e), u*(n, f, g, e)). The unique solution (x* (t, f, g, e), u* (n, f, g, e) ) has the following properties. (i) I f T --- ml e Z +, ( x * ( t , f , g , e ) , u * ( n , f , g , e ) ) is mc = (ml,m2)-periodic, where mc is the

minimal common multiple of ml and m2. (ii) I f T = m l / n l , m l , n l E Z +, m l and nl are mutually prime, ( x * ( t , f , g , e ) , u * ( n , f , g , e)) is mc = (niT, rn2)-periodic, where me is the minimal common multiple of n i T and m2. (iii) I f a(t, e), aj(t, e), f(t) • P(w), w = 1 / T is irrational, {g(n)} • P ( ( 1 / m 2 ) , Z), x*(t, f, g, ~) is quasiperiodic with frequencies (w, 1/m2) and {u*(n, f ,g, e)} • QP((a;, 1/m2), Z). Fur-

thermore, there exist constants 1~1,1~2, ~[1,1~2 such that ]x*(t)[ _< /~/1

sup t

If(t)]

+/V1

sup [g(n)[, n

lu*(t)l _< ~r2 sup ]f(t)l + N2 sup [g(n)r. t

n

The map (f,g) -~ (x*(f,g,e),u*(f,g,c)) defines a bounded linear operator £(e), and c --* £(e) is continuous [or 0 < e < ~1. PROOF. (i) If T = m l • Z + and f ( t ) is T-periodic, it follows from L e m m a 4 that hi(n) = f~+l ea°(n+l-s)f(s)ds is ml-periodic. Set me = (ml,m2). For any me-periodic xo(t) and {u0(n)}, it follows from Theorem 5 that system (6.4) has a unique me-periodic solution

Qualitative Behavior

127

(x(n, e), u(n, e)) for each e. We define x(t, e) by (6.5). It is easily proved that x(t, e) is me-periodic (e.g., see [10] for details). The rest is similar to those in Theorem 9. (ii) If T = m l / n l , m l , n l E Z + and f(t) is T-periodic, it follows from Lemma 4 that hi(n) = fn+l eaO(,~+l_,)f(s)ds is niT-periodic. Set mc = (niT, m2) to be the minimal common multiple of niT and m2. For any mc-periodic xo(t) and {u0(n)}, it follows from Theorem 5 that system (6.4) possesses a unique mc-periodic solution (x(n, e), u(n, e)) for each ~. We define x(t, e) by (6.5). It is easily proved that x(t, e) is rnc-periodic for each e. The rest is similar to those in Theorem 9. (iii) Follows from Theorem 10 directly. |

7. A L M O S T P E R I O D I C Q U A S I - L I N E A R S A M P L E D - D A T A FEEDBACK CONTROL SYSTEMS WITH PARAMETER We now consider the following system: N

p

x'(t) = a(t, e)x(t) + E aj (t, e)x([t - j]) + E bj (e)u(n - j) j=0

j=0

+f t,x(t),{x([t--jl)}N,e), p

~(~ + 1) = ~

(7.1)

n<_t
N

cj(~)u(~ - i) + ~

j=0

dj(~)z(~ - j) + g(~, ~),

j=0

where f : R x R n+2 --, R is almost periodic in t uniformly for ( x 0 , x l , . . . ,x/v+1) c R N+2 and g : Z --~ R is an almost periodic sequence. LEMMA 11. (See [9,10].) Suppose that {x(n)} is an almost periodic sequence and g is an almost periodic function for t uniformly on R/v+2. Then T(x, e) N T(g, e, W) is relatively dense, where W C R N+2 is a compact subset. LEMMA 12. (See [10].) Suppose that x(t) is almost periodic and g(t,... ) is an almost periodic function for t uniformly on R N+2. Then the function g(t,x(t), {x([t + / ] ) } oN) is also almost periodic.

PROOF OF THEOREM 1. We choose Pl (~ Po) and e2 (_< q ) such that (N + 2)pit/(Pl, e~) + M (e2) < pl,

J~"l(N -b 2)piT] (pl, 52)-t-

(M l

-I- J~l) M @2)< pl,

(7.2)

1

/~1 (N -t- 2)V (Pl, £2) '(

Ilxoll

Suppose that zo(t) E AP(R) satisfying system:

7'

-< Pl and 0 < e _< e2. We consider the following

N

p

x'(t) = a(t, e)x(t) + E a j ( t , e)x([t - j]) + E bi(e)u(n - J) j=0

j=O

+f (t, xo(t), {Xo([t- j ] ) } ~ , e ) , p

n
+

1,

(7.3)

N

~(~ + 1) = ~ cj(~)~(~ - i) + ~ d,(~)x(n - j) + gCn, e). 5=0

j=0

Then Theorem 9 implies that system (7.3) possesses a unique almost periodic solution (x(t), u(n)). Writing x(t) = T(xo, e), then solving (7.1) is equivalent to finding a fixed point of T(., e).

128

R. YUAN AND Z. JING

It follows from (Hh) that

If (t, ~o(t), {~o([t - j]))o~, ~)1 ,7 (,Ol, ~2) (N + 2)Ilzoll + M (~2)

_<

_< (N + 2)px~(pl,e2) + M(e2),

t • R,

We estimate x(t) using Theorem 9 as

li~il

<- M, sup: (t,~o(t),

{xo([t-j]))o~,~) +.m~s~plg(n,~)l S 217/1[(N + 2)plr] (Pl, e2) + M (e2)] + N1M (e2) < Pl.

Thus, T(., e) maps the closed set f = (x c ,47); Hx[I _< Pl} into itself for each e with 0 < ~ < c2. If x0,20 • ~', we write x = T(xo,e) = E(e)(f(.,xo,e)), 5: = T(20, e) = E(e)(f(-,20, e)) and denote by (x, u) and (2, fi) the corresponding unique almost periodic solution of system (7.3), respectively. Setting X = x - 2 and # = u - fi, we find that X and # satisfy N

p

x'(t) = a(t, e)x(t) + E aj(t, e)x([t - j]) + E bj(e)#(n - j) j=0

+

f (t, xo(t),

3=0

{xo([t- j])}oN ,e) - f (t, 2o(t), {2o([t- j])}N , e ) ,

p

(7.4)

N

.(n + 1) = ~ cj(~).(n- j)+ ~ d~(~).(k -j). j=0

j=0

From Theorem 9 and (7.3), it follows that there exists a constant/t5/1 such that [x(t)l_/t7/1 sup f (t, xo(t), {xo([t- j])}N,e) -- f (t,20(t), {20(It- j])}~, e) _
< 1 s u p Ixo(t) - 2o(t)l. Hence, T is a uniform contraction. Since f is continuous in (X0,Xl,... ,XN+I) uniformly in t • R, it follows that e -o fc(t) • A P is continuous, where f~(t) = f(t, xo(t), {x0([t - j ] ) } N , e). Since e ~ E(e) is continuous, we conclude that for fixed xo • AP, the map E ~ T(x0, e) is continuous on (0, e2]. The uniform contraction principle implies the existence of a unique fixed point x*(e) • ~ which is a continuous function of e, 0 < e < e2. So, system (7.1) possesses a unique almost periodic solution (x*(t), u*(n)). Furthermore,

,l::(t)rl _< M1 sup If (t, x*(t), {x*([t - j])}o~, ~)l + N1 sup rg(n, ~) t

n

Ilu*(n)ll ~_ A5/2sup f ( t , x ' ( t ) , { x ' ( [ t - - j ] ) } N , e )

+/V2supIg(n,~)[

<_/17/1 [~ (P2, e)(N + 2)sup [ x * ( t ) i t + M(e)] +/V1M(e), which implies that

IIx*(t)l I = O(M(e)),

Ilu*(n)ll = O(M(e)).

|

Qualitative Behavior

129

PROOF OF THEOREM 3. We use the same notations as the above proof of Theorem 1. (i) If T = rnl E Z +, then for any T-periodic xo(t) satisfying Ilxoll <_ Pl, f(t,xo(t),{xo (It - j])}0N,e) is T-periodic. Thus, Theorem 11 implies that system (7.3) possesses a unique m~ = (ml,rn2)-periodic solution. Thus, T(.,e) maps the closed set $-~ = {x E P~; [Ixl[ <_ Pl} into itself for each e with 0 < ~ <_ e2, where 7v~ is the set of all w-periodic functions. The rest is similar. (ii) If T = rnl/nl, rnl,nl e Z +, then for any niT-periodic xo(t) satisfying IIx0N _< 0~,

f(t,xo(t), {x0([t- j])}oN,e) is niT-periodic. The rest is similar. (iii) If T is irrational, this conclusion follows from Theorem 2 directly. The proof of Theorem 2 will be shown in the next section. |

8. Q U A S I - P E R I O D I C Q U A S I - L I N E A R S A M P L E D - D A T A FEEDBACK CONTROL SYSTEM WITH PARAMETER In this section, we consider the following sampled-data feedback control system for simplicity: N

p

x'(t) = a(t, e)x(t) + E aj(t, e)x([t - j]) + ~ j=0

bj(e)u(n - j)

j=0

+ f(t, z(t), x([t]), e), p

u(n + 1) = ~

(8.1)

N

cj (e)u(n

-

j) + E dj (e)x(n

j =0

-

j) + g(n, e),

j=0

where f : R x R x R --, R, t --* f ( t , x , y , e ) , is quasiperiodic in t with frequencies col = (~1) . . . , w(11)). Furthermore, we suppose that f can be written in the form f(~lx, y,~) ~- ~

F k lxllyl2e i27r{k'W1)t ,

~

k

(8.2)

l

where 1 = (11,12), F-k,l = -Pk,l, and ix1 < +cx~, lYl < +ec, i.e., f is a real analytic quasi-periodic function with frequencies wz. If r = 1, we call f a real analytic periodic function with period 1/wl. The system of the form (7.1) can be easily discussed similarly. LEMMA 13. Assume that f is a real azaalytic quasi-periodic function with frequencies a;z. Then for any ¢ e ~ ° ( 1 , W l ) , {h(n)} = (fnn+l ea(n+l-s)f(s,C(s),¢([s]))ds} e Q ~ ( w , , Z ) . Here a is a constant. PROOF. It is known that f can be written as (8.2). We have

h(n) =

i

n+l

e a(~+l-~) ~

Jn

=~

Fk,l¢ 11(s)¢ 12(n)e i2~(k'wl)~ ds

k,l

e~Fk,l¢ ~2(n)e i2"(k

~ ) ~

e-USe ll (n + s)e i2~(k'~l)s ds.

k,l

Since ¢(t) E 6QP(1,~ol), it follows: 0
,rnl 1

and

nEZ. /21 ,... ~l]l 2

n~Z,

130

R. YUAN AND Z. JING

Therefore, h(n) can be written as follows:

f~,~e~¢,~ (0)... ¢~,~ (0) ~,l

ml...~l I

t/1.../212

/oe-°~¢m, (s)... ¢,~,~

(s) ~ ' ~ ' ~

• ei2rt(ml+...+mq

ds

+ u l + . . . + u t 2 +k,w~}n

Setting

al = ( m l , . . . , m l x , U l , . . . , l J l 2 ) ,

[all=ml +'"+rnh

/o

then h(n) can be rewritten in the form k,l

+ul + ' " + u l ~ ,

/21~,..,/~'/2

rO,l,...~ml I

(8.3)

where

fI~ : E E Hj3-laLl,l,cqei2~rO3'wl)n" l

cq

It is easy to see that

B

l

"

where

I1¢~1[ :

sup ICj(s)[,

vj.

|

0_
LEMMA 14. Assume that f is a real analytic quasi-periodic function with frequencies wl and ¢ C ~ Q P ( 1 , wl). Then we have the following formula:

~

t

ea
(8.4)

where/:/~(s) is continuous on 0 < s < 1 and ~-~ ]/:/~(s)[ < +oo. PROOF• It is known that f can be written as (8•2). We have

~

t

ea(t-s) f(s, ¢(s), ¢([s])) ds

=

£

ea(t -s) EFk,zCh(s)¢t=(n)ei2,~
= E Fk,lea(t-n)¢12(n)ei2r(k'wOn

e-aS¢ ll (n + s)e ~2~r(k'~l)s ds ~0 $ - n

k,l

=E E

E

k~l /21-..t]l 2

Fk'tea(t-n)¢~(O)"'¢a, (0) ~0 t - n e-aSCm'(S)'"¢m', (s)ei~È(k'"~)~ds

ml...mt 1

• ei21r(ml+'"+mq+u1+'"+ut2+k,wl)n /~

l

at

Qualitative Behavior

131

where

c~l = ( m i , . . . , m l a , u l .... ,ut2),

gk,z,~,(u)

[oq I = m l + ' - ' + m l ~ + u i + ' ' + u t 2 ,

/:

= Fk,~e°"¢,,(O)... ¢,,~(0)

~-°~¢,~(~) .. Cm~ (~)~ ~2'~(k'~'~ as,

H~(~) : ~2 ~2 H~-,~,I,~,~, (u) l

cq

It is easy to see that

PROOF OF THEOREM 2. Set w = (wl,w~). Suppose that (xo(t), uo(n)) E G Q P ( 1 , w ) × QP(w, Z) satisfying Ux0U _< pl, Iluot[ -< pl, and 0 < e <_ e2. We consider the following system: N

P

x'(t) = a(t, e)x(t) + E aj(t, e)x([t - j]) + E bj(e)u(n - j) j=o j=o + f (t, xo(t),xo([t]),e), n < t < n + l, p

u(n + 1) = E

(8.5)

N

cj(e)u(n - j) + E

j=o

dj(~)x(n - j) + gi n, e),

j=o

where

fxo(t,e) := f (t,zo(t),xo([tl),e) = ~

Fk,tx oll (t)x oa ([t])e i2¢(k,~)t ,

~ k

l

l = (/i,/2). Because fzo(t) ~ Q ~ O ( W l ) , in general, we cannot conclude t h a t system (8.5) possesses a unique quasi-periodic solution by using the results of Theorem 10. We consider the following system: N

P

x'(t) = a °x(t) + E a°x([t - j]) + E j=0

b°u( n - J) + (a(t,e) - a °) xo(t)

j=0

N

P

+ E (aj(t, ~) - ao) x0([t - 51) + E (bj(~) - bo) uo(~ - 5) + Jxo(~, ~), j=o j=o p

u(n + 1) = E

(8.6)

N

c ° u ( n - j) + E 4 x(n - j)

j=O

j=O p

N

+ E (ej(e) - c0)3 uo(n - j) + E (dj(e) - d °) xo(n - j) + 9(n,~), j =0 j =o n < t < n + 1. From the continuity of x(t), we obtain the following difference system: N

x(n +

i) =

P

E A j x ( n - j) + E B j u ( n - j) + h(n, e), j=0

j=0

p

N

(8.7)

~ ( ~ + 1) = E & ( n - j ) + E dOx(n _ 5) j =0

j =0 p

+E j=0

N

(cj(e) - c o ) u o ( n - j) + E j=0

(dj(e) - d o ) xo(n - j) + g(n,e),

132

R. YUAN AND Z. JING

where

n+l h(n, e) =

+~

n+l

e a°(n+l-s) (a(s, e) - a °) xo(s) ds

N

ea°(n+l-s)E

~n

( a j ( s , e ) - a °) ds. x o ( n - j)

j=O

P

fn+l

+(ea°-l)a°-lE(bj(~)-b°)uo(n-j)+ j=o

ea°(n+l-S)fxo(S,~)ds.

Jn

Prom L e m m a 13, we know {h(n, e)} E Q:P(w, Z). It follows from Theorem 5 that system (8.7) has a sequence solution (x(n), u(n)) E QP(~, Z). We define

N x(t) = ea°(t-n)x(n) + (e a°(t-~) - 1) E a ° - l a ° x ( n

- j)

j=O

(8s)

P

+ (e a°(t-n) - 1)

~-~a°-~b°u(n-

j ) + h(t,~),

n ~ t < n + 1,

j=O which is continuous on R, where

h(t, ~) = ~nt e ~°(t-8) (a(s, e) - a °) Xo(S) ds t

-~~n

N

ea°(t--s)

E

(aj(s, e) - a °) dsxo(n - j)

j=O

+ (e °°(~-~) - 1)a°-~E (~j(~)-¢)uo(n- j)+

ea°(t-s)fxo(s,e)ds,

j=O

n < t < n + 1. From L e m m a 14, we know that x(t) can be rewritten as follows:

x(t) = ~ x~({t})~ ~<~, ~>I~l k and ~-~k supse[0,1] [Xk(S)I < +c~. The rest is similar to those in Theorem 9. 9. A N

EXAMPLE

We consider a special sampled-data feedback control system

x'(t) = (ao + al(t, e))x(t) + (bo + b l ( e ) ) u ( n ) + f(t), u(n + 1) : (co + cl(e))u(n) + (do + d l ( e ) ) x ( n ) + g(n),

(9.1)

where a0, b0, co, do are constants, e is a parameter. We assume that

(1) al(t, (2) al(t,

0) = 0, bl(0) = 0, c1(0) = 0, dl (0) = 0; e) and f(t) are almost periodic functions in t and continuous in e; bl(e), cl(e), dl(e) are continuous in e; {g(n)} is an almost periodic sequence;

(3) a0 ¢ o, Icol # 1. Set

E = {(x,y) l x + y < l ,

x>0}.

Qualitative Behavior

133

COROLLARY. I f

(a) (Ibodo[/lao[, min{Ic0[, [COl-l}) E E for ao < O; or (b) (]bodol/lao[e a°, min{Icol, Ic01-1}) e E for ao > O; then there exists an el > 0 such that for each 0 < e < el, the following conclusions hold. (i) For each a/most periodic (f,g), system (9.1) possesses a unique almost periodic plant output x* (t, f, g, e) and a unique almost periodic digital controller u* (n, f, g, e). (ii) If a(t,e), aj(t,e) C QP(wl), 1,wl,w2 are irrational independent, then for each quasiperiodic f(t) E QP(wl) and each {g(n)} • QP(w2, Z), system (9.1) possesses a unique quasi-periodic plant output x*(t, f , g, e) • GQP(1, (~)1,~2)) and a unique quasi-periodic digital controller u* (n, f, g, e) • QP ( (wl , w: ) , Z ) . (iii) If a(t,e), aj(t,c) are ml-periodic, then for each ml-periodic function f(t) and each m2-periodic sequence g(n), ml,m2 • Z +, system (9.1)possesses a unique me-periodic plant output x* (t, f, g, e) and a unique mc-periodic digital controller u* (n, f, g, e) , where mc = (ml, m2) is the minimal common multiple of ml and m2. (iv) Ira(t, e), aj(t, e) are T = ml/nl-periodic, then for each T-periodic function f (t) and each m2-periodic sequence g(n), nl, ml, m2 • Z +, system (9.1) possesses a unique me-periodic plant output x*(t, f, g, e) and a unique mc-periodic digital eontroller u*(n, f, g, e), where mc = (niT, m2) is the minimal common multiple of n i T and m2. (v) If a(t,e),aj(t,e) • P(w), ~ is irrational, then for each periodic f(t) • P(~) and each {g(n)} • ~0(1/m2, Z), m2 • Z +, system (9.1) possesses a unique quasi-periodic plant output x*(t, f, g, ~) with frequencies (w, l / m 2 ) and a unique quasi-periodic digital controller u*(n, f, g, e) • QT~((w, l / m 2 ) , Z). It is easy to see that under the assumptions of the corollary, we have 1 1 ifao<0, ~ l-]col' if [c0[ < 1, MA = 1 - e ~o' Me = e~° if ao > O, ~ ]Co[ en o - 1 ' [col-l' if Ic0[> 1,

(9.2)

so that (H6) hold. Therefore, we can easily see the qualitative behaviors of plant output and controller of system (9.1) when the forced terms are added. It is said that there is a digital controller such that the plant output is almost periodic, quasiperiodic, and periodic under different forced terms, respectively.

REFERENCES 1. B.A. Francis and T.T. Georgiou, Stability theory for linear time-invariant plants with periodic digital controllers, IEEE Trans. Automat. Contr. 33, 820-832, (1988). 2. T. Hagiwara and M. Araki, Design of a stable feedback controller based on the multirate sampling of the plant out, IEEE Trans. Automat. Contr. 33, 812-819, (1988). 3. P.A. Iglesias, On the stability of sampled-data linear time-varying feedback systems, In Proc. 33rd Conf. Decision and Control, Lake Buena Vista, FL, December 1994, pp. 219-224. 4. A. Back, J. Guckenheimer and M. Myers, A dynamical simulation facility for hybrid systems, In Hybmd Systems, (Edited by R. Grossman, A. Nerode, A. Ravn and H. l:tischel), New York, Springer, p p 255-267. (1993). 5. A. Gollu and P.P. Varaiya, Hybrid dynamical systems, In Proc. 28th IEEE Conf. Decision and Control, Tampa, FL, December 1989, pp. 2708-2712. 6. R. Grossman, A. Nerode, A. Ravn and H. Rischel, Editors, Hybrid Systems, New York, Springer, (1993). 7. H. Ye, A.N. Michel and L. Hou, Stability theory for hybrid dynamical systems, IEEE Trans. Automat. Contr. 43 (4), 461-474, (1998). 8. C. Corduneanu, Almost Periodic Functions, John Wiley &: Sons, New York, (1968). 9. A.M. Fink, Almost periodic differential equations, In Lecture Notes in Mathematics, Volume 377, SpringerVerlag, Berlin, (1974). 10. R. Yuan and J. Hong, Almost periodic solutions of differential equations with piecewise constant argument. Analysis 16, 171-180, (1996). 11. V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applicatzons, Academic Press, (1987). 12. C. Corduneanu, Almost periodic discrete processes, Libertas Math. 2, 159-169, (1982).