On asymptotic behaviour of certain discrete delayed-control systems Текст научной статьи по специальности «Математика»

UDC 681.511.42 Vestnik of St. Petersburg University. Serie 10. 2014. Issue 4

A. V. Stepanov

ON ASYMPTOTIC BEHAVIOUR OF CERTIAN DISCRETE DELAYED-CONTROL SYSTEMS

D. I. Mendeleyev Institute for Metrology (VNIIM), 19, Moskovsky pr., St. Petersburg, 190005, Russian Federation

Sufficient conditions are obtained for the existence of stable periodic or almost periodic solutions of some discrete delayed control systems based on solution roughness concept. Namely, by analogy with elemental case it is demonstrated that any rough solution of discrete delayed system with stable controlled object and piecewise constant control statement having finite codomain, tends in time to a stable periodic solution of that system (or almost periodic, if e.g. the delay sequence is almost periodic). Both cases when delay presents itself in controlled object and in control action are considered. A case is also considered when the control delay depends on previous system states. Bibliogr. 11.

Keywords: discrete control systems, delay, stability, periodic solutions, almost periodic solutions.

А. В. Степанов

ОБ АСИМПТОТИЧЕСКОМ ПОВЕДЕНИИ РЕШЕНИЙ НЕКОТОРЫХ ДИСКРЕТНЫХ СИСТЕМ УПРАВЛЕНИЯ С ЗАПАЗДЫВАНИЕМ

ВНИИМ им. Д. И. Менделеева, Российская Федерация, 190005, Санкт-Петербург, Московский пр., 19

Опираясь на понятие грубости решений дискретных систем управления, получены достаточные условия существования устойчивых решений для некоторых классов таких систем с запаздыванием. А именно, по аналогии с простейшим случаем, показано, что любое грубое решение дискретной запаздывающей системы с собственно устойчивым управляемым объектом и кусочно-постоянным управлением, принимающим не более чем конечное число значений, сходится с течением времени к устойчивому периодическому решению данной системы (или почти периодическому, если, например, запаздывание носит почти периодический характер). Отдельно рассмотрены случаи, когда запаздывание имеет место в объекте управления и в управляющем воздействии. Также изучен случай, когда запаздывание зависит от состояния системы в каждый дискретный момент времени. Библиогр. 11 назв.

Ключевые слова: дискретные системы управления, запаздывание, устойчивость, периодические решения, почти периодические решения.

Introduction. The aim of this work is to extend some results concerning stability and asymptotic behaviour of one class of discrete-time control systems. Namely, in the work [1] authors considered a system

xk+i = Axk + cuk, k e N, uk = п(ак), = YX,

where xk,c,y e 1", A e 1nxn, uk e 1, u is a simple relay nonlinearity (e.g., u(a) = sign(a)); and stated that if all eigenvalues of A are located strictly inside of a unit circle, then any rough solution (the definition will be given below) tends to an asymptotically stable (in Lyapunov sense) periodic solution when k ^ж.

Stepanov Alekscmdr Vlo,d,imirovich — research fellow; e-mail: [email protected]

Степанов Александр Владимирович — старший научный сотрудник; e-mail: [email protected]

Consider a generalized discrete-time delayed system

Xk+1 =$(k,xk, Xk-1, Xk-m) , k > ko. (1)

A sequence {xk} c 1" to be called a solution of (1) if it satisfies that system identically for some given initial conditions xko-m,..., xko.

Definitions of uniformly asymptotic stability (UAS) and exponential stability (ES) given in [2, 3] should be used for solutions of (1). For any solution {xk} of (1) let us define a set

Q (xk) = \ w G 1" I 3{km} c N, lim km = : lim xkm = w \ .

I 1 m—m—m J

Let us specify the considered system a bit. Suppose that

xk+1 = $ (k, xk , xk-1, . .., xk-m) = F (k, xk , xk-1, . .., xk-m) + Uk, (2)

m

Uk = u(k,a), a = £x'k-il(i), Y(i) G 1";

i=0

here F describes a controlled object and U is a control action (the prime designates transposition operation). Let any component of U hereafter be a bounded piecewise constant function. Suppose also that a codomain for any component of U is a finite set (i.e. uk G U c 1", where U is finite). Let F be a continuous function with respect to the state variables (e. g., linear or bilinear).

Denote by Du a set of discontinuity points of function u.

Definition 1 [1]. Solution {xk} of (2) is said to be rough (with degree of roughness S > 0) if p (Q(xk), Du) > S (here p is a distance between sets).

In other words, the solution {xk} is rough if there exists a positive S and a natural index Kg such that

p (xk, Du) > S Vk > Kg.

In more general case, the control vector uk can also depend on its previous values. In such a case the definition of solution roughness should be specified (see, e. g., such a definition for simple hysteresis nonlinearity in [4]).

Hereafter for any x G 1" ||x|| is the Euclidean vector norm, for any matrix A norm || Ay is spectral. Definitions for almost periodic vector and matrix sequences could be found in [2].

Delay in Controlled Object. Consider a delayed system in 1":

m

xk+1 =53 A(i)xk-i + Uk, k G N, A(i) G 1"x", (3)

i=0

i. e. the controlled object is described by linear function F.

By analogy with [1] the following statement can be proved.

Theorem 1. If trivial (zero) solution of corresponding homogeneous system

m

yk+1 = Y, A(i)yk-i (4)

i=0

is exponentially stable, then any rough solution of (3) tends to an exponentially stable periodic solution of this system when k

Proof. Denote by {xk} a rough solution of (3) having a roughness degree S. Let us use a variation-of-constants formula for (3) (see [5]): If Y(k, k0) is a fundamental system of solutions for (4) such that

Y (k,ko) =

0, k < ko,

1, k = ko,

(here 0,I e Rnxn are null and identity matrices, correspondingly) and {yk} is a solution of (4) satisfying the given initial conditions, then

k-1 k-k0-1 Xk = yk + 53 y(k,j + l)uj = yk + 53 Y(k,k — j — 1) Uk-j-1.

j=ko j=0

As lim \\yk\\ = 0, then for steady-state solution (k — k0 ^ <x>) one has k

Xk = 53 Y (k, k — j — 1) Uk-j-i. j=o

This series converges absolutely. By definition,

3M > 0, a e (0,1) : \\Y(k, k — j — 1)\\ < Maj+1,

and for any Kg e N

R(k, Kg)

j=Ks + 1

Y (k, k — j — 1) Uk-j-1

aK* +2

1 - a ' veu 11 1

Let us fix an index

Kg = logc

As U is a finite set, then

(1 -a)6' 2 Mû

, then R(k, Kg) < - VA; G N.

3k1, k2 e N, k1 < k2 : Uk1-j = Uk2-j Vj = 1,Kg.

(5)

So

\ Xk 1 — Xk2

j=Ks + 1

Y(k1, k1 — j — 1) Uki-j-1 —

— 53 Y(k2,k2 — j — 1) Uk2-j-1

j=Ks + 1

< \\R(h,Kg)\\ + \\R(k2,Kg)\\ = S.

Thus, as the solution under consideration is rough, then

Uki = Uk2 .

Repeating the constructions above one can verify that ukl+j = uk2+j f°r any natural j. So {xk} tends to periodic solution in time. ■

Thus if system (3) is rough, then any solution of (3) tends to one of stable limit cycles (in particular case, it may be a single point). Note that roughness is a "typical" property for contractive systems: set of rough systems is everywhere dense in a special metric space of contractive systems (for more details please see [1]).

Note also that number of rough steady-state periodic solutions of (3) is finite [1]. This fact holds true for all the systems considered below.

The fact of exponential stability of the zero solution of (4) can be checked by means of various criteria [3, 6]. For example (see [6]), if for any given a > 1

E<

j=0

j+1

< 1,

then (4) is exponentially stable. Another criterion mentioned in the same work is based upon Lyapunov approach.

Moreover, as (4) can be transformed into an augmented system without delay and exponential stability of such a system follows from its UAS [2], thus it is enough to use some criteria for asymptotic stability of the augmented system (or, what is the same, for (3)) which may be less restrictive (see, e.g., [7]).

By analogy with (3) we can consider a bit more complex system with non-stationary controlled object:

m

xk+i =^2 Akxk-i + uk, (6)

i=0

here j A^i^ is a periodic sequence of n x n matrices for any i. The previous statement can be extrapolated on system (6) if a trivial solution of a corresponding homogeneous system

m

yk+i = ^2 A<i)Vk-i i=0

is exponentially stable (to check the exponential stability, various criteria can be used; see, e. g., [5, 8]). This statement can be proved in the same way as the previous, the only difference is that the value k2 — k1 should be multiple to a period common for all the matrix sequences {a^}, 0 < i < m.

A quasilinear system can also be considered. Namely, let

xk+i =V A(i)xk-i + Y] r(i) (xk-i )+Uk, (7)

k-i

i=0 i=0

where

r(i) (0) = 0, 0 < i < m,

and any r(i) is a Lipschitz function (denote by Li corresponding Lipschitz constants). Theorem 2. If

(m \ 2 / m \ 2

mj^a2^ ||A(i),A(i)||J + i (m + 1)^ a2(j+1) L?J < 1 (8)

for some real a > 1, then any rough solution of (7) tends to a stable periodic solution of this system when k ^m.

Proof. The proof of this statement is similar to the proof of the previous theorem. Note that condition (8) guarantees exponential stability for uncontrolled system

yk+1 A(i)yk-i + 53 r(i) (yk_i)

i=0 i=0

(see [9]). So we obtain an analogue for (5) for steady-state solution and all the above constructions can be repeated. ■

Consideration of some control systems with a piecewise linear control law can lead to more exotic cases than (3). For example, consider a system

Xk+1 = Axk + CUk-m, Uk = u(ffk ), Ok = Y'xk, O, \o\ < 1,

u(a) \ sign(o), \a\ > 1,

i. e. u is a saturation nonlinearity. Obviously, the latter system can be rewritten as a bilinear controlled system

xk+1 = Axk + Bxk-muk-m + c(uh-m + m) , B = CY', (9)

u^ = u(i)(xk) is an indicator function of a set Qi,

Qi = {x : \a\ < 1}, = {x : a< -1}, Q3 = {x : a> 1}

(each u(i) is evidently bounded and piecewise constant). It can be shown easily that any rough solution of (9) tends to an asymptotically stable periodic solution if ||A|| < 1 and a trivial solution of system

xk+1 = Axk + Bxk-m

is asymptotically stable (the latter condition may set some limitations onto components of c and/or y vectors).

Almost Periodic Delay. Linear Object. Let us suppose now that the delay is implanted into control and the controlled object does not contain it. Consider a delayed system

xk+1 = Axk + uk-mk, k > ko, uk = u(xk) e U C R", (10)

where xk e R", A e R"x" is square matrix, integer-valued sequence {mk} is nonnegative, almost periodic and (consequently) bounded, i.e.

3M e N : mk < M Vk e Z+.

Suppose uko, uko-1, ..., uk0-M to be known.

By the analogy with the previous theorems let us formulate following statement. Theorem 3. If ||A|| < 1, then any rough solution of (10) tends to an asymptotically stable almost periodic solution of this system when k ^m.

Proof. To prove the statement let us consider an auxiliary system

xk+1 = Axk + ^ (k, uk, ...,uk-m) , (11)

where m G Z+; the function p G 1" is almost periodic with respect to k and bounded:

3 p > 0: \\p (k, Uk ,...,Uk-m )|| < p yk G N, Uk ,...,Uk-m G U.

Statement. If \\A\\ < 1, then any rough solution of (11) tends to an asymptotically stable almost periodic solution of this system and the respective sequence {uk} is periodic. Let {xk} be a rough steady-state solution of (11) with degree of roughness S:

Xk Aj p (k - j - 1, Uk-j-1, . .., Uk-j-m-1)

j=o

(the series above is absolutely convergent). Denote by a = \\A\\ < 1; let

(1 - a)S "

Kg =

logQ

8p

then ajp < (12)

j=Ks + 1

Let U = {v1, v2, ..., vN}, m = Kg+m; there exists N = Nm different sets of elements of U of size m:

Si = {vi ,v2 ,...,vf}, 1 < i < N, vj G u.

Consider N different m-periodic sequences {Ui^j=l

U\+i■ m+j = hm-j, 0 < j < m - 1, l G Z, 1 < i < N, (13)

and N responding sequences

Xk = ^ A P (k - j - 1, Uk-j-1, - . Uk-m-j-l) . j=0

It is obvious that any {X\} is almost periodic.

As it known from [2], for any pair of almost periodic sequences {ak}, {bk} and any real e > 0 exists a number T(e) g N such that between any pair lT(e), (l + 1)T(e) (l G N) there exists at least one e-almost period which is common for both {ak} and {bk}. Moreover, it is easy to see that this almost period can be selected to be multiple of any predefined natural p. This statement in an obvious manner can be extended onto a case of arbitrary finite set of almost periodic sequences. Let us use this fact for the sequences {xlk}. Suppose

and select an e-almost period h = lKg to be common for all the sequences {X\} , ...,{X?}. Denote by

Sk = {Uk-1, Uk-2, ..., Uk-m-1} G {S1, S2, ..., S^ j (14)

and consider N +1 of sets Sk+i f, 1 < i < N +1. At least one of these sets is encountered two (or more) times:

3mo G 1, N, 3i 12 G 1, N +1: Sk+h f = S\+i2 f = Sm0.

So

||xk+ii f — xk+i2 f || ^ ll^+i, f — xk+i2 t II + ||x^fe+0ii f — xk+ii T || +

^ s

+ ||i™+°i2f-xk+i2f\\ < (i2-¿i)e + 4 aj<f< (_N+l)e + -<5

j=Ks + 1

(see (12)), and

uk+ii f uk+i2f

(by definition of rough solution).

Repeating the above constructions, one can state that

3m1 e 1,N: Sk+iif+1 = Sk+i2t+1 = Smi,

implying

|xk+ii T+1 — xk+i2 T+1 y ^ llxk+h f+1 — xk + i2 f+11 + i ^^mo || I II ~mo || ^ e

+ \\xk+ii f+1 — xk+ii f+11| + |\xk+i2 f+1 — xk+i2 f+MI < S

(here we again used (12) and also the fact that S is a common e-almost period for all xik). So, k

uk+ii f+1 = uk+i2T+1,

etc. Therefore

uk+ii f+i = uk+i2 f+i Vl e N,

and {uk} is periodic (with period (i2 — n)S), implying almost periodicity of the solution {xk}. Stability of the solution follows in an obvious way from its roughness and inequality | A| < 1 .

Now let us return to the system (10). Let vlk = Simk (where S is Kronecker delta function), then (10) may be rewritten as (11), where

uk-mk = ^ (k, uk, . .., uk-M) = vkuk + vluk-1 + . .. + VMuk-M.

The sequence {^k} is almost periodic, so we can apply the auxiliary statement just proved to system (10). ■

If the sequence {mk} is periodic, then the steady-state rough solution under consideration should obviously be periodic.

As it follows from the proof of the statement, the similar result holds true in case if any almost periodic external action is present in the right-side part of (10). The statement also holds true in case of non-stationary object (see [4]), i.e.:

xk+1 = Akxk + uk-mk ,

where {Ak } is almost periodic sequence of matrices, if the trivial solution of a responding homogeneous system xk+1 = Akxk is uniformly asymptotically stable.

We can also return here to (6) and consider a case when every matrix sequence j A^ j is almost periodic.

Bilinear Object. Suppose uk to be scalar and consider bilinear system

xk+1 = Axk + (Bxk + c) uk-mk , (15)

A,B e R"x", C e R".

Theorem 4. If zero solution of homogenous system

Vk+i = (A + Buk) yk (16)

is UAS for any sequence {uk} C U, then the responding rough solution {xk} of (15) tends to an asymptotically stable almost periodic solution of this system.

Proof. The proof in general is similar to the proof of the previous theorem. System (15) can be rewritten as

Xk+l = (A + Buk-mk ) Xk + CUk-mk = Ak Xk + OUk-mk,

where Ak = A + Buk-mk. Denote by

Xi,j = A— A- x ... x Aj,

then

k — ko — l

Xk = Xk,ko Xko + Xk,k-j+i cvk-j-i, vk = uk-

/ J ^^k, k-]+i^uk-j-U uk — u'k-mk j=0

and for steady-sate solution:

oo

Xk = ^^Xk,k-j-i cvk-j-i. j=0

As zero solution of (16) is UAS, then (see [2]) there exists a pair of constants L > 0 and a G (0; 1) such that

WXi,jy < Lai-j.

Let S > 0 be a roughness degree for the solution considered. Suppose

(1 - a)S'

Ks = 1 +

uk are given by (13) and

log0

8L llcll v

Ak = A + BUUk, Xj,l = Aj-1 Aj-2 X ... X Ai, XXk = 53 Xk,k-j-1 cUk-j-1.

j=o

Let e = jS lj and f = I Kg, I e Z, is e-almost period common for all sequences

{.Xk} , {X"k} ,...,1x^1 . Consider N +1 sets Sk+if, given by (14). At least one of these sets encountered two (or more) times:

3mo, 1 < mo < N, h, i2, 1 < i1 <'h < N +1 : Sk+i1 f = Sk+i2 f = Sm0.

Then estimate \\Xk+i1 f - Xk+i2 f\\:

W^+n f - Xk + i2 f \\ ^ 11 Xfe+°i1 f+1 - X k + i2 f+1 II +

i II ~mo |I . || ~mo || . e

+ llXk+i1 f+1 - Xk+ii f+1 || + ||Xk+i2 f+1 - Xk+i2 f+11| < S.

Thus, as the solution under consideration is rough, then

uki = uk2 .

Repeating the constructions above it is easy to verify that uki+j = uk2+j for any natural j. So, {xk} tends to periodic solution in time. ■

Some Corollaries. Control Delay Depending on Previous States or Delays.

Consider some more exotic cases. Let

xk+1 = Axk + uk-mk, (17)

where, e.g., uk = u(xk) (u satisfies all the conditions mentioned above) and delay depends on current state

'mk=g(xk), g : 1" —>■ 0, to, or previous control value (or values)

TOfc = h(umk-1), h\U Q,m.

Suppose that functions g, h are bounded and piecewise constant integer-valued functions:

g : R" ^ Z+, 0 < g(x) < M Vx e R"; h : U ^ Z+, 0 < h(v) < M Vv e U;

here M is some natural constant.

In that cases the roughness definition should be specified:

Definition 2. Consider any solution {xk} of the system (17). Denote by Dg and Dh sets of discontinuity points for the functions g and superposition h(u), correspondingly. The solution {xk } is said to be rough (with degree of roughness S > 0)ifp (Q(xk), Du U Dg) > S (or, correspondingly, p (Q(xk),Du U Dh) > S).

Theorem 5. Let ||A|| < 1, then any rough solution of (17) tends to an asymptotically stable periodic solution of this system when k

Let us skip the proof: it is similar to [1] and the above proofs; the main idea is that delay values (and therefore control values) should repeat due to finiteness of possible delay and control range.

Delay in both controlled object and control. Of course the controlled object and control action may contain delay simultaneously, e. g.,

xk+1 = Axk + Bxk-mk + uk, uk = u (xk, xk-mk ) . (18)

As 0 < mk < m, then this system can be rewritten as

m

xk+1 = 53 Ak^ xk-j + uk,

j=0

where

Aj = A + BI (mk ,j), I (mk ,j) = Smk j I. Theorem 6. Consider homogeneous system

m

yk+1 =53 Akj) yk-j. (19)

j=0

Assume that Cauchy function Y(k, k0) of the system (19) satisfies the following condition [10]:

^ \\Y(k,k q)||

fr VUTTMII < M° (20)

J = k0

for k ^ k0 and some positive M0 and any sequence {mk}. Then any rough solution of (18) tends in time to an asymptotically stable almost periodic solution of this system.

To proof this statement it is enough to note that condition (20) implies existence of a pair of constants M > 0 and a e (0; 1) such that

HY (k,ko)H < Mak-k0;

so one can easily repeat all the constructions for the system (6).

Connection to Continuous Control Systems. Note that all the above discrete-time systems were considered irrelatively to any continuous-time control systems. But use of the foregoing approach can provide us with similar results in case if some additional assumptions are put over the control law.

Consider a bilinear system of differential equations in R":

x = Px + p(t) + (Qx + q) u(t - t), x e R", t > t0, (21)

P,Q e R"x", let function p be continuous and periodic or almost periodic. Let control statement u is being formed at discrete moments of time tk = t0 + kh, k e Z+ (switching moments) and is constant between them. Let T/h = m e N. Using the Cauchy formula for (21) (by analogy with [11]) one can obtain

xk+1 = A(uk-m) xk + p(k, uk-m) + c(uk-m)uk-m, (22)

where

tk + i

A(uk-m) = eP+Quk-m, c(uk-m)= J e(P+Quk-m)(tk+i-s)qds,

tk

tk+i

p(k,u-m)= J e(P+Quk-m)(tk+i-s)p(s) ds. tk

Sequence {p(k, uk)} should be periodic (if p is T-periodic and T/h e Q) or almost periodic with respect to k.

Consider an auxiliary homogeneous system

xk+1 = A(vk )xk, {vk }™0 C U. (23)

Theorem 7. If zero solution of (23) is UAS for any sequence {vk}, then any rough solution of (22) tends in time to an asymptotically stable almost periodic solution of this system.

It also means that the initial continuous-time control system (21) has almost periodic asymptotically stable solutions (due to its continuous dependence from initial conditions).

Conclusion. Thus the approach proposed in [1] to prove the existence of stable periodic solutions of relatively simple stationary systems is applicable to more complex systems with delay in control or in the controlled object. Moreover, applying the roughness concept, one can also obtain sufficient conditions for the existence of stable almost periodic solutions of such delayed systems.

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The article is recommended for publication by prof. A. M. Kamachkin. The article is received by the editorial office on June 26, 2014. Статья рекомендована к печати проф. А. М. Камачкиным. Статья поступила в редакцию 26 июня 2014 г.