Robust Quadratic Stabilizability and H ∞ Control of Uncertain Linear Discrete-Time Stochastic Systems with State Delay

This papermainly discusses the robust quadratic stability and stabilization of linear discrete-time stochastic systemswith state delay and uncertain parameters. By means of the linear matrix inequality (LMI) method, a sufficient condition is, respectively, obtained for the stability and stabilizability of the considered system. Moreover, we design the robust H∞ state feedback controllers such that the systemwith admissible uncertainties is not only quadratically internally stable but also robustH∞ controllable. A sufficient condition for the existence of the desired robustH∞ controller is obtained. Finally, an example with simulations is given to verify the effectiveness of our theoretical results.


Introduction
It is well known that stability and stabilization are very important concepts in linear system theory.Due to a great number of applications of stochastic systems in the realistic world, the studies of stability and stabilization for stochastic systems attract lots of researchers' attention in recent years; we refer the reader to the classic book [1] and the followup books [2,3], together with references [4][5][6][7][8][9][10][11] and the references therein, which include robust stochastic stability [4], exponential stabilization [6], mean-square stability, and D-stability and D  -stability [8].The stabilization of various systems, including impulsive Markovian jump delay systems [4], stochastic singular systems [10,12,13], uncertain stochastic T-S fuzzy systems [14], and time-delay systems [6,11,[15][16][17], has been studied extensively. ∞ control is one of the most important robust control approaches when the system is subject to the influence of external disturbance, which has been shown to be effective in attenuating the disturbance.The objective of standard  ∞ control requires designing a controller to attenuate  2 -gain from the external disturbance to controlled output below a given level  > 0; see [18].The study of  ∞ control of general linear discrete-time stochastic systems with multiplicative noise seems to be first initiated by [19].Then, stochastic  ∞ control and its applications have been investigated extensively; see [14,16,[20][21][22][23][24].
Because time-delay exists widely in practice and affects the system stability, there have been many works concerning the study in stability or  ∞ control of stochastic systems [4, 6, 9, 11, 14-16, 22, 25].Due to limitations of measurement technique and tools, it is not easy to construct exact mathematical models.Compared with the nominal stochastic systems without uncertain terms investigated in [2,5,24], our considered system allows the coefficient matrix to vary in a certain range.
Discrete-time stochastic difference systems have attracted a great deal of attention with the development of computer technology in recent years.In our viewpoint, there are at least two motivations to study discrete-time stochastic systems, Firstly, discrete-time stochastic systems are ideal mathematical models in practical modeling such as genetic regulatory networks [23].Secondly, discrete-time stochastic systems provide a better approach to understand extensively continuous-time stochastic Itô systems [2,3,26].Therefore, it is of significance to study the stabilization and  ∞ control of discrete-time stochastic time-delay uncertain systems.

Mathematical Problems in Engineering
This paper will study quadratic stability, stabilization, and robust state feedback  ∞ control for uncertain discretetime stochastic systems with state delay.The parameter uncertainties are time varying and norm bounded.It can be found that, up to now, many criteria for testing quadratic stabilization and  ∞ control have been given in terms of LMIs and algebraic Riccati equations by applying Lyapunov function approach.One of our main contributions is to study quadratic stability and stabilization via LMIs instead of algebraic Riccati equations which is hardly solved.What we have obtained extended the work of [15] about the quadratic stability and stabilization of deterministic uncertain systems.Another contribution is to solve the state feedback  ∞ control and present a state feedback  ∞ controller design.
The paper is organized as follows.In Section 2 we give some adequate preliminaries and useful definitions.In Section 3, sufficient conditions for quadratic stability and stabilization are given in terms of LMIs which is convenient to compute by the MATLAB LMI toolbox.Section 4 designs a state feedback  ∞ controller.Two numerical examples with simulations are given in Section 5 to verify the efficiency of the proposed results.Finally, we end this paper in Section 6 with a brief conclusion.

Preliminaries
Consider a class of uncertain linear discrete-time stochastic systems with state delay described by where () ∈ R  is the system state and () ∈ R  is the control input, and {()} ≥0 are independent white noise process satisfying the following assumptions: where   is a Kronecker function defined by   = 0 for  ̸ =  while   = 1 for  = .

Robust Quadratic Stabilization
In this section, a sufficient condition about robust quadratic stability and robust quadratic stabilization will be presented via LMIs, respectively.First, we cite the following lemma which is essential in proving our main results.
Lemma 3 (see [27]).Suppose that  =   , () satisfies (2), and then for any real matrices , , and  of suitable dimensions we have if and only if (iff), for some  > 0, Theorem 4. Consider uncertain discrete-time stochastic delay system (1) with () = 0.This system is robustly quadratically stable if there exist positive definite matrices  > 0,  > 0 such that the following LMI holds. [ where Proof.From Definition 1, taking a Lyapunov function   as in the form of (7), if uncertain discrete time-delay stochastic system (1) is quadratically stable, then, for all admissible uncertainties of (1), there exist matrices  > 0,  > 0 and a scalar  > 0 such that E(Δ  ) associated with unforced system (8) satisfies (6).In view of the assumption (H 1 ), it is easy to compute where  0Δ ,  0Δ ,  0Δ , and  0Δ are given in (5) and Π is shown as By Definition 1, system (1) with () = 0 is robustly quadratically stable, only if which is equivalent to Mathematical Problems in Engineering Note that Π 2 can be rewritten as By Schur's complement, it is easy to derive that Π < 0 is equivalent to where Then, using the same way as in ( 16)-( 19) yields The above inequality can be rewritten as where Because Π 3 is a symmetric matrix, applying Lemma 3, (21) holds iff the following inequality holds: where Take and then by substituting ( 25) into (23), for  > 0, we get where Δ 11 , Δ 12 , Δ 22 are shown in (12).Using the same method as in ( 16)-( 20), ( 11)-( 12) follow immediately from the above inequality.
Theorem 5. System (1) is robustly quadratically stabilizable if there exist positive matrices  > 0,  > 0,  ∈ R × and a scalar  > 0 with  −  −1 < 0 such that the following LMI holds. where Moreover, a quadratically stabilizing state feedback controller is given by Proof.By Definition 2, using the same way as in the proof of Theorem 4, the following inequality which has a similar form to ( 11)-( 12) can be obtained by taking () = () where In order to eliminate the nonlinear quadratic terms pre-and postmultiplying diag (, ,  −1 , , , ) on both sides of (30) and considering  −1 > , ( 27)-( 28) can be obtained easily.This theorem is proved.
Remark 6.Compared with the results about quadratic stability and quadratic stabilizability of deterministic systems given in [14], our two theorems not only extend the results of [14] to stochastic systems, but also provide the corresponding LMI criteria which can be easily tested by MATLAB LMI toolbox.

State Feedback 𝐻 ∞ Control
In this section we consider the state feedback discretetime  ∞ control problem for the following uncertain linear stochastic system with state delay: where () ∈ R   and () ∈ R  are called the controlled output and external disturbance, respectively.In addition, the effect of the disturbance () on the controlled output () is described by a perturbation operator G  :   → , which maps any finite energy disturbance signal  into the corresponding finite energy output signal  of the closedloop system.The size of this linear operator, that is, ‖G  ‖, measures the influence of the disturbances in the worst case.We denote by  2  (N 0 , R  ) the set of all nonanticipative square summable R  -valued stochastic processes Firstly, for system (37), we define the perturbed operator G  and its norm as follows.
Definition 8.The perturbed operator of system (37), G  :  2  (N 0 , R  )  →  2  (N 0 , R   ), is defined as with its norm Next, we present the definition about stochastic robust  ∞ control.
Besides, if  * () exists, then system (37) is called  ∞ controllable in the disturbance attenuation.Furthermore, it is called strongly robust  ∞ controllable if  = 1.
So in the case of () = 0,  = 0, −1, . . ., −, we have where Obviously, it is easy to get that ‖G  ‖ <  if Ξ < 0.Then, we need to eliminate the uncertainties.Using the same method as in the proof of Theorem 4, we know that, for some  > 0, a sufficient condition for Ξ < 0 can be got from the following matrix inequality. [ where Then, by pre-and postmultiplying on both sides of (47), we have For some constant  > 0 with  −1 > , Theorem 10 is concluded; that is, an  ∞ control of system (37) is obtained by solving LMIs (42)-(43).This completes the proof.

Simulation Example
In this section, we consider two simple examples with simulations to illustrate the effectiveness of the proposed approach.
Example 11.Consider discrete-time stochastic system (1) with the following parameters: Using LMI toolbox to solve (11)- (12) in Theorem 4, we find out that  min = 0.0086 > 0 which means that there is no feasible solution and indicates that system (1) with  ≡ 0 is unstable.By Theorem 5, the system is mean-square stabilizable which is verified by Figure 2 (54) For perturbed system (42), we take the external disturbance as () =  − and the certain level as  = 0.8.In addition, according to Lemma 3, an appropriate  is given as  = 4.9.Then, by the result of Theorem 10, using LMI toolbox to solve (43) and (47), we find that  min = −0.1046,which means we have got a group of feasible solutions with This further verifies the effectiveness of Theorem 10.

Figure 1 Figure 1 :x 2 Figure 2 :
Figure 1: State trajectories of the autonomous system.

x 2 Figure 3 :z 2 Figure 4 :
Figure 3: State trajectories of the closed-loop system.