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Exponential Stability and Robust [H.sub.[infinity]] Control for Discrete- Time Time-Delay Infinite Markov Jump Systems.

1. Introduction

During the past decades, Markov jump systems have been the subject of a great deal of research since they have been used extensively both in theory and in applications. Markov jump systems are hybrid dynamical systems composed of subsystems with the transitions determined by a Markov chain. A number of results that focused on Markov jump systems have been published ranging from filtering, stability, observability, and control to engineering application; see, for example, [1-15] and the references therein.

Note that most of the theoretical works related to Markov jump systems in the literatures concentrated on the case where the state space of the Markov chain is finite. However, it may be more appropriate to characterize abrupt changes in many real plants via an infinite-state Markov chain. As far as applications are concerned, infinite Markov jump systems are critical in some physics plants, such as solar thermal receiver, aircraft, and robotic manipulator systems. Theoretically, finite Markov jump systems are fundamentally different from those governed by infinite-state space. The work in [14] studied exponential almost sure stability of random jump systems. The work in [16] considered the definition and computation of an [H.sub.2]-type norm for stochastic systems with infinite Markov jump and periodic coefficients. LQ-optimal control problem has been dealt with for discrete-time infinite Markov jump systems in [17]. The work in [18] demonstrated the inequivalence between stochastic stability and mean square exponential stability in discrete-time case. With this motivation, infinite Markov jump systems have stirred widespread research interests.

Time-delay is one of the inherent features of many practical systems and also is the big source of instability and poor performances in systems [19]. Moreover, stochastic modeling has had extensive applications. Hence, dynamical time-delay stochastic systems deserve our consideration. Stability analysis and controller design of time-delay Markov jump systems have been investigated by many authors [15,20, 21]. Unfortunately, the literature about these issues for infinite Markov jump case is less developed. And, to the best of our knowledge, only a few results have been presented so far [18, 22, 23], let alone the problem involving time-delay. Actually, [18, 23] investigated the exponential stability and infinite horizon [H.sub.2]/[H.sub.[infinity]] control problem for discrete-time infinite Markov jump systems with multiplicative noises, respectively, but they neglected the effects of time-delay. Meanwhile, the authors in [22] considered time-delay, when discussing the stabilization problem for linear stochastic delay differential equations with infinite Markovian switching, but it was hard for the obtained stability results to deal with control problem. As mentioned above, stability and control for time-delay stochastic systems with infinite Markov jump and multiplicative noises have not received enough attention despite their importance in practical applications, which motivates us for the present research.

We aim to address the exponential stability and [H.sub.[infinity]] control problem for a class of discrete-time time-delay stochastic systems with infinite Markov jumps and multiplicative noises in this paper. The main contributions of this paper are as follows: First of all, we investigate exponential stability of the equilibrium point for the considered systems by employing a novel Lyapunov-Krasovskii functional. Further, a sufficient condition is established to ensure exponential stability with a given [H.sub.[infinity]] performance index of the closed-loop system. And we introduce the slack matrix to decouple the Lyapunov matrices, which makes the [H.sub.[infinity]] controller design feasible. Moreover, some numerical examples are provided to show the effectiveness of the proposed design approaches.

The remaining part of this paper is constructed as follows. In Section 2, we formulate the system model and recall some definitions and lemmas. In Section 3, we present our main results, where we derive some sufficient conditions for exponential stability with a given [H.sub.[infinity]] performance index. Two numerical examples and their simulations are given to illustrate the effectiveness of the obtained results in Section 4. Conclusions are made in Section 5.

For convenience, we fix some notations that will be used throughout this paper. The n-dimensional real Euclidean space is denoted by [R.sup.n] x [R.sup.mxn] stands for the linear space of all m by n real matrices. Let [parallel] * [parallel] be the Euclidean norm of [R.sup.n] or the operator norm of [R.sup.mxn]. By [S.sub.n] and I(0) we denote the set of all n x n symmetric matrices and the identity (zero) matrix, respectively. A' denotes the transpose of a matrix (or vector) A. We say that A is positive (semipositive) definite if A > 0([greater than or equal to] 0). [[lambda].sub.max](A)([[lambda].sub.min](A)) represent the maximum (minimum) eigenvalue of A. [[delta].sub.(x)] is called the Kronecker function. [Z.sub.+] = {0, 1, ...}. D = {1, 2, ...}. [l.sup.2]([Z.sub.+]; [R.sup.m]) = {[??] [member of] [R.sup.m] | [??] is [F.sub.t]- measurable, and [([[summation].sup.[infinity].sub.t=0] E[[parallel]y(t)[parallel].sup.2]).sup.1/2] < [infinity]}.

2. Preliminaries

Consider the following discrete-time time-delay stochastic system with infinite Markov jump parameter and multiplicative noises:

[mathematical expression not reproducible], (1)

where x(t) [member of] [R.sup.n] represents the system state, [mathematical expression not reproducible] is the control input, [mathematical expression not reproducible] denotes the disturbance, and [mathematical expression not reproducible] is the system output. w(t) = {w(t) | w(t) = ([w.sub.1](t), [w.sub.2](t), ..., [w.sub.r](t))', t [member of] [Z.sub.+]} is a sequence of independent random vectors defined on a given complete probability space ([OMEGA], F, P), which satisfies E(w(t)) = 0 and E(w(t)w(s)') = [I.sub.r][[delta].sub.(t-s)]. [phi]([t.sub.0]) is a vector- valued initial condition. d is the bounded constant delay with 0 [less than or equal to] d [less than or equal to] d. Markov chain [mathematical expression not reproducible] takes values in a countably infinite set D with transition probability matrix P = [p(i,j)], where p(i, j) = P([s.sub.t+1] = j | [s.sub.t] = i), and P is nondegenerate, P([s.sub.0] = i) > 0 for all i [member of] D. Assume [mathematical expression not reproducible] are mutually independent, and [F.sub.t] = {[s.sub.k], [w.sub.s] | 0 [less than or equal to] k [less than or equal to] t, 0 [less than or equal to] s [less than or equal to] t - 1}, [F.sub.0] = [sigma]([s.sub.0]). Assume v(t) belongs to [mathematical expression not reproducible].

We introduce the Banach spaces [mathematical expression not reproducible]. The notations [A.sup.mxn.sub.1] ([A.sup.mxn.sub.[infinity]]) will be written as [A.sup.n.sub.1] (resp., [A.sub.n.sub.[infinity]]) and [A.sup.n+.sub.1] (resp., [A.sup.n+.sub.[infinity]] if and only if m = n and A(i) [member of] [S.sub.n], A(i) [greater than or equal to] 0, i [member of] D, respectively. When Y, Z [member of] [A.sup.n+.sub.1], Y [less than or equal to] Z means that Y(i) [less than or equal to] Z(i), i [member of] D. Therefore, we have [[parallel]Y[parallel].sub.1] [less than or equal to] [[parallel]Z[parallel].sub.1]. For all coefficients of the considered systems, we suppose they have a finite norm [[parallel]x[parallel].sub.[infinity]].

Definition 1 (see [10, 18]). System (1) with u(t) = 0 and v(t) = 0 is called mean square exponential stability if there exist [lambda] [greater than or equal to] 1 and [tau] [member of] (0, 1) such that

[mathematical expression not reproducible] (2)

for all t [member of] [Z.sub.+], i [member of] D and [x.sub.0] [member of] [R.sup.n]. Further, system (1) with v(t) = 0 is called exponential stabilizable if there exists a sequence [mathematical expression not reproducible] such that the closed-loop system

[mathematical expression not reproducible], (3)

with v(t) = 0 has mean square exponential stability, where u(t) = K([s.sub.t])x(t) is called exponentially stabilizing feedback.

Definition 2. Closed-loop system (3) is said to have an [H.sub.[infinity]] noise disturbance attenuation level [gamma] > 0, if under zero initial value the following condition is satisfied:

[mathematical expression not reproducible] (4)

for any [mathematical expression not reproducible].

Lemma 3 (see [22]). We denote [[??].sup.n+.sub.[infinity]] = [A | A [member of] [A.sup.n+.sub.[infinity]], their exists [epsilon] > 0 not depengding upon i such that A(i) [greater than or equal to] [epsilon][I.sub.n] for all i [member of] D}. Let

[mathematical expression not reproducible]. (5)

Assume that [mathematical expression not reproducible] for all i [member of] D for some [epsilon] > 0. Then, B [member of] [[??].sup.n+.sub.[infinity]] if and only if [mathematical expression not reproducible], where n = [n.sub.1] + [n.sub.2] and B | [B.sub.22] = [{[B.sub.B11](i) - [B.sub.12](i)[B.sub.22][(i).sup.- 1][B.sub.12](i)'}.sub.i[member of]D] is calld the Schur complement of [B.sub.22] in B.

Remark 4. Lemma 3 is the infinite-dimensional version of Schur complements (see [24]).

3. Main Results

Firstly, stability will be analyzed, and a sufficient condition is obtained for system (1) with u(t) = 0 and v(t) = 0 to have mean square exponential stability.

Theorem 5. System (1) with u(t) = 0 and v(t) = 0 is exponentially mean square stable, if we can find matrices P [member of] [[??].sup.n+.sub.[infinity]], Q [member of] [[??].sup.n+.sub.[infinity]] such that the following matrix inequality holds:

[mathematical expression not reproducible] (6)

uniformly with respect to (i, q) [member of] D x D, where

[mathematical expression not reproducible]. (7)

Proof. Construct the following Lyapunov-Krasovskii functional:

[mathematical expression not reproducible]. (8)

By the assumption that w(t) is independent of the Markov chain [mathematical expression not reproducible], besides [F.sub.t-[??]] [subset] [F.sub.t], we have

[mathematical expression not reproducible], (9)

and

[mathematical expression not reproducible], (10)

and

[mathematical expression not reproducible]. (11)

Thus, combining (8) with (9)-(11), we get

E[V(x(t + 1), [s.sub.t+1]) - V(x(t), [s.sub.t]) | [F.sub.t], [s.sub.t] = i] [less than or equal to] a(t)' [R.sub.iq](P)a(t), (12)

where

[mathematical expression not reproducible] (13)

with q = [s.sub.t-d] and a(t) is defined as a(t) = [x(t)' x(t - d)']'.

Applying Lemma 3 to (6) leads to

diag {-P(i) + ([??] + 1)Q(i), -Q(q)} + M(i)' [P.sup.-1]M(i) < 0, (14)

where M(i) = [[C.sub.x](i) [D.sub.d](i)]. Further, we have [R.sub.iq](P) < 0. It is clear from [R.sub.iq](P) < 0 that there exists a sufficiently small scalar [epsilon] > 0 such that [R.sub.iq](P) < -[epsilon][I.sub.n]. Therefore, it follows that

E[V(x(t + 1), [s.sub.t+1]) - V(x(t), [s.sub.t])] < -[epsilon]E[[parallel]x(t)[parallel].sup.2]]. (15)

On the other hand, by using (8), we deduce that

[mathematical expression not reproducible], (16)

where

[mathematical expression not reproducible]. (17)

Noting (15) and (16), for any constant [kappa] > 1, we obtain that

[mathematical expression not reproducible], (18)

where [[theta].sub.3] = ([??] +1)[[theta].sub.2]. By taking summation from 0 to T - 1 on both sides of (18), for T [greater than or equal to] [??] + 1, it implies that

[mathematical expression not reproducible]. (19)

Recalling (8) and (16), denoting [[theta].sub.0] = [min.sub.l[member of]D][[lambda].sub.min](P(l)) and [theta] = max[[[theta].sub.1], ([??] + 1)[[theta].sub.2]}, we have

E[V (x(T), [s.sub.T])] [greater than or equal to] [[theta].sub.0]E[[[parallel]x(T)[parallel].sup.2]], (20)

and

[mathematical expression not reproducible], (21)

respectively. Furthermore, it suffices to show that there exists a constant [[kappa].sub.0] > 1 such that

[-[[kappa].sub.0][epsilon] + ([[kappa].sub.0] - 1)[[theta].sub.1]] + ([[kappa].sub.0] - 1)[[theta].sub.3][??][[kappa].sup.[??].sub.0] = 0. (22)

Actually, letting f([kappa]) = [-[kappa][epsilon] + ([kappa] - 1)[[theta].sub.1]] + ([kappa] - 1)[[theta].sub.3][??][[kappa].sup.[??].sub.0], then we have f'(k) > 0 and f(1) < 0. Therefore, (22) has a unique solution [[kappa].sub.0] > 1. By substituting (20)-(22) into (19), we obtain

[mathematical expression not reproducible], (23)

where [[lambda].sub.0] = ([theta] + ([[kappa].sub.0] - 1)[[theta].sub.3][??][[kappa].sup.[??].sub.0])/[[theta].sub.0]. This indicates that system (1) with u(t) = 0 and v(t) = 0 has mean square exponential stability. The proof is completed.

Remark 6. Due to the consideration of an infinite-state Markov chain, the infinite dimension Banach spaces have been introduced. Furthermore, it should be pointed out that a novel Lyapunov-Krasovskii functional (8) has been constructed to analyze the mean square exponential stability for system (1) with u(t) = 0 and v(t) = 0.

Next, we prove that system (1) with u(t) = 0 verifies the Hot performance disturbance attenuation [gamma].

Theorem 7. System (1) has mean square exponential stability for u(t) = 0 and v(t) = 0 with a prescribed [H.sub.[infinity]] performance [gamma] for u(t) = 0, if we can find matrices P [member of] [[??].sup.n+.sub.[infinity]] Q [member of] [[??].sup.n+.sub.[infinity]] such that the following matrix inequality holds:

[mathematical expression not reproducible], (24)

uniformly with respect to (i, q) [member of] D x D, where

[H.sub.v](i) = [[H.sub.0](i)', [H.sub.1](i)', ..., [H.sub.r](i)']'. (25)

Proof. It is well established that (24) implies (6). Applying Theorem 5 one obtains that system (1) has mean square exponential stability for u(t) = 0 and v(t) = 0.

Let us now show that system (1) with u(t) = 0 satisfies a prescribed [H.sub.[infinity]] performance level. To this end, constructing the same Lyapunov-Krasovskii functional V(x(t), [s.sub.t]) as in Theorem 5 and under the zero initial condition, the following index is introduced:

[mathematical expression not reproducible], (26)

where

[mathematical expression not reproducible], (27)

and b(t) is defined as b(t) = [x(t)' x(t - d)' v(t)'] . The last '[less than or equal to]' in (26) holds as a result of the similar line with (12). Then, by using Lemma 3 in (24), we obtain that A([s.sub.t], [s.sub.t-d]) - B([s.sub.t])'C[([s.sub.t]).sup.- 1]B([s.sub.t]) < 0. Thus, [J.sup.T] < 0. Taking the limit T [right arrow] [infinity] in (26), we have

[mathematical expression not reproducible]. (28)

This ends the proof.

Combining Theorem 5 with Theorem 7, the following corollary can be easily derived for closed-loop system (3).

Corollary 8. Let the feedback control gain K(i), i [member of] D, be given. Then closed-loop system (3) has mean square exponential stability for v(t) = 0 with a prescribed [H.sub.[infinity]] performance [gamma] if there exist two matrices P [member of] [[??].sup.n+.sub.[infinity]] and Q [member of] [[??].sup.n+.sub.[infinity]], such that

[mathematical expression not reproducible], (29)

uniformly with respect to (i, q) [member of] D x D, where

[mathematical expression not reproducible]. (30)

Below, based on Corollary 8, we are ready to present the [H.sub.[infinity]] controller design for system (1).

Theorem 9. For system (1), a state feedback controller can be designed such that closed-loop system (3) has mean square exponential stability for v(t) = 0 and a given [H.sub.[infinity]] performance [gamma] can be ensured if there exist matrices [mathematical expression not reproducible] such that

[mathematical expression not reproducible], (31)

uniformly with respect to (i, q) [member of] D x D, where

[mathematical expression not reproducible]. (32)

Moreover, if matrix inequalities (31) are feasible, then an exponentially stabilizing feedback gain can be given by

K(i) = [??](i)[F.sup.-1]. (33)

Proof. Via Lemma 3, we conclude that (29) is equivalent to the following matrix inequality:

[mathematical expression not reproducible]. (34)

Premultiply diag{F', F', I, I, I} and postmultiply diag{F, F, I, I, I} with (34), and let

[mathematical expression not reproducible]. (35)

By a tedious calculation, one can rewrite (34) as

[mathematical expression not reproducible] (36)

where

[mathematical expression not reproducible]. (37)

According to Corollary 8 and the fact that

[mathematical expression not reproducible], (38)

namely,

-F'[??][(i).sup.-1] F [less than or equal to] [??](i) - F - F', (39)

the desired result is derived.

Remark 10. The work in [20] presented a necessary and sufficient condition for the existence of the mixed [H.sub.2]/[H.sub.[infinity]] control by four coupled matrix Riccati equations (CMREs). Note that CMREs are hardly solved in practice, and this motivates us to find a new sufficient condition in terms of matrix inequalities that can be easily solved to guarantee that the resulting closed-loop system has mean square exponential stability for the zero exogenous disturbance and satisfies a prescribed [H.sub.[infinity]] performance level.

Remark 11. With the introduction of a slack matrix F, a sufficient condition is obtained in Theorem 9, in which the Lyapunov matrices are not involved in any product with system matrices. This makes the [H.sub.[infinity]] controller design feasible and can be easily carried out by solving corresponding matrix inequalities.

Remark 12. It is worth noting that the obtained results can be extended to discrete-time time-delay infinite Markov jump stochastic systems with time-varying delays. Assume that the time-varying delay d(t) satisfies [d.sub.m] [less than or equal to] d(t) [less than or equal to] [d.sub.M]; then by similar procedures to the above and choosing the following Lyapunov-Krasovskii function

[mathematical expression not reproducible], (40)

the corresponding results can be derived.

4. Illustrative Example

In this section, some illustrative examples are presented to demonstrate the effectiveness of the developed method.

Example 1. Consider the following one-dimensional discrete-time time-delay stochastic system with infinite Markov jumps:

[mathematical expression not reproducible], (41)

where the transition probability is defined by p(i, i) = 1/4, p(i, i + 1) = 3/4, p(i, j) = 0, j [not equal to] i, i + 1, i, j [member of] D. Now take

[mathematical expression not reproducible]. (42)

Let P(i) = 4(i + 1)/3i, Q(i) = 1/9i(i +1),and time-delay d = 2. By direct computation, (6) holds. According to Theorem 5, we deduce that system (41) has mean square exponential stability, and Figure 1 presents the state response of system (41) with initial conditions [phi]([t.sub.0]) = 0.5 for [t.sub.0] = -2, -1,0.

Example 2. Consider the following one-dimensional discrete-time time-delay stochastic system with infinite Markov jumps:

[mathematical expression not reproducible], (43)

where the transition probability is defined by p(i, i) = 1/2, p(i, i + 1) = 1/2, p(i, j) = 0, j [not equal to] i, i + 1, i, j [member of] D. The coefficients of system (43) are reset to be

[mathematical expression not reproducible], (44)

The purpose here is to design an [H.sub.[infinity]] controller such that the closed-loop system has mean square exponential stability and with a given [H.sub.[infinity]] norm bound [gamma] = 0.5. Applying Theorem 9, the [H.sub.[infinity]] controller can be designed as

K(i) = -1/[i + 1]. (45)

With the initial conditions [phi]([t.sup.0]) = 0.1 for [t.sub.0] = -2, -1, 0 and the exogenous disturbance v(t) = 2[e.sup.-t] sin t, Figures 2 and 3 show the state and output responses, respectively.

5. Conclusions

In this paper, the issue of exponential stability and robust [H.sub.[infinity]] control for a class of discrete-time time-delay stochastic systems with infinite Markov jumps and multiplicative noises has been studied. Time-delay and infinite Markov jump are taken into consideration simultaneously. By using Lyapunov-Krasovskii functional and introducing slack matrix, an matrix inequality approach has been adopted to ensure the mean square exponential stability and satisfy a prescribed [H.sub.[infinity]] performance level. Finally, some illustrative examples are given to demonstrate the usefulness of the proposed design methods. Further research directions would include the investigation on [H.sub.2]/[H.sub.[infinity]] control problem and asynchronous control problem for discrete-time time-delay stochastic systems with infinite Markov jumps.

https://doi.org/10.1155/2018/3676083

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant 61673013, Natural Science Foundation of Shandong Province under Grant ZR2016JL022, and the SDUST Research Fund under Grant 2015TDJH105.

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Yueying Liu and Ting Hou (iD)

College of Mathematics and Systems Science, Shandong University of Science and

Technology, Qingdao 266590, China

Correspondence should be addressed to Ting Hou; ht_math@sina.com

Received 4 August 2018; Revised 7 October 2018; Accepted 16 October 2018; Published 22 October 2018

Guest Editor: Abdul Qadeer Khan

Caption: Figure 1: System state response in Example 1.

Caption: Figure 2: System state response in Example 2.

Caption: Figure 3: System output response in Example 2.
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