Robust 𝐻_∞ Control for Discrete-Time Stochastic Interval System with Time Delay

Abstract

The robust 𝐻_∞ control problem for discrete-time stochastic interval system (DTSIS) with time delay is investigated in this paper. The stochastic interval system is equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing the appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robust 𝐻_∞ controller for DTSIS are obtained in terms of linear matrix inequality (LMI) form, and the robust 𝐻_∞ controller is designed. Finally, a numerical example with simulation is given to show the effectiveness and correctness of the designed robust 𝐻_∞ controller.

1. Introduction

In the past few years, much research effort has paid to the robust control problems for stochastic systems which have come to play an important role in many fields including communication network, image processes, and mobile robot localization. So far, plenty of significant results also have been published; see, for example, [1–3] and the references therein. In the meantime, we all know that time delay arises naturally in many mathematical (non) linear models of real phenomena, such as communication, circuits theory, biology, mechanics, electronics, hydraulic, rolling mill, chemical systems, and computer controlled systems, and which is frequently one of the main sources of instability, oscillation and poor performance of control systems. So the stability analysis and robust control for dynamic time-delay systems have attracted a number of researchers; see, for example, [4–8] and the references therein.

Meanwhile, from Hinrichsen who presented the 𝐻_∞ control problems for stochastic systems in 1998 [9], more and more experts begin to study the stochastic 𝐻_∞ control problems [10–17]. In the view of dissipation, Berman develops a 𝐻_∞-type theory for a large class of time-continuous stochastic nonlinear systems. In particular, it introduces the notion of stochastic dissipative systems by analogy with the familiar notion of dissipation associated with the deterministic systems. The problem of 𝐻_∞ output feedback control for uncertain stochastic systems with time-varying delay is discussed in [12], and the parameter uncertainties are assumed to be time-varying norm-bounded. Xu et al. [13] investigate the problems of robust stochastic stabilization and robust 𝐻_∞ control for uncertain neutral stochastic time-delay systems.

On the other hand, when modeling real-time plants, the parameter uncertainties are unavoidable, which are very often the cause of instability and poor performance, such as modeling error, external perturbation, and parameter fluctuation during the physical implementation. As a result, the parameters of a system matrix are estimated only within certain closed intervals. We call this kind of system interval system or stochastic interval system (SIS) with the following form: 𝑥̇(𝑡)=𝐴𝑥(𝑡), (1) or 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)𝑑𝑡+𝐻𝑥(𝑡)𝑑𝜔(𝑡), (2) where 𝐴∈[𝐴,𝐴], 𝐻∈[𝐻,𝐻],and⋅,⋅ denote the lower and upper bounds of the interval for the coefficients, respectively. Interval system and stochastic interval system have been well known for their importance in practice applications. And in recent years, the stability analysis and stabilization problems of various interval systems have received plenty of research attention; see, for example, [18–24] and the references therein. Of course, there are a lot of research works on stability analysis and stabilization problems for stochastic interval system that have been reported; see, for example, [25–29] and the references therein. However, to the best of authors’ knowledge, so far little results on robust 𝐻_∞ control problem for stochastic interval system with time-delay are available in the existing literature.

Inspired by the above discussions, in this paper we study the robust 𝐻_∞ control problem for discrete-time stochastic interval system (DTSIS) with time delay. The stochastic interval system will be equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robust 𝐻_∞ controller for DTSIS are obtained in terms of linear matrix inequality (LMI) form, and the robust 𝐻_∞ controller can be designed by MATLAB LMI control toolbox.

2. Problems Formulation and Preliminaries

Consider the following discrete-time stochastic interval system (DTSIS) with time delay: 𝑥(𝑘+1)=𝐴^𝑙𝑥(𝑘)+𝐴^𝑙_𝑑𝑥()(𝑘)(𝑘)+𝐻𝑘−𝑑+𝐵𝑢+𝐷𝑣_𝑙𝑥(𝑘)+𝐻^𝑙_𝑑𝑥(𝜔(𝑘),𝑧(𝑘)𝑘−𝑑)=𝐶^𝑙𝑥(𝑘)+𝐶^𝑙_𝑑𝑥(),𝑥(𝑘)(𝑘)[],𝑘−𝑑=𝜑,𝑘∈−𝑑,0 (3) where𝑥(𝑘)∈ℝ^𝑛is the state vector;𝑑>0 is the time delay; 𝑢(𝑘)∈ℝ^𝑚 is the control input;𝑧(𝑘)∈ℝ^𝑞is the control output;𝑣(𝑘)∈ℝ^𝑝 is the exogenous disturbance input, which satisfies𝑣(𝑘)∈𝐿₂([0,∞),ℝ^𝑝), where 𝐿₂([0,∞),ℝ^𝑝) is the space of nonanticipatory square-summable stochastic process with respect to(ℱ_𝑘)_𝑘∈𝐼⁺ with the following norm: ‖𝑣(𝑘)‖²₂=𝔼^∞_𝑘=0‖𝑣(𝑘)‖²=^∞_𝑘=0𝔼‖𝑣(𝑘)‖². (4)

𝜔(𝑘)∈ℝ^𝑙is a scalar Brownian motion defined on a complete probability space (Θ,ℱ,𝑃) with 𝔼[𝜔(𝑘)]=0,𝔼𝜔²(𝑘)=1. (5)

In the system (3), 𝐴^𝑙is an interval matrix with appropriate dimension, which means 𝐴^𝑙=𝐴,𝐴=𝑎_𝑖𝑗∈ℝ^𝑛×𝑛∣𝑎_𝑖𝑗≤𝑎_𝑖𝑗≤𝑎_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛, (6) where 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 are determinate matrices.

Remark 1. It is not difficult to see that the matrices𝐴^𝑙,𝐴^𝑙_𝑑,𝐻^𝑙, 𝐻^𝑙_𝑑,𝐶^𝑙,𝐶^𝑙_𝑑can also be time-dependent as long as the mappings𝐴^𝑙∶𝑅₊→[𝐴,𝐴], 𝐴^𝑙_𝑑∶𝑅₊→[𝐴_𝑑,𝐴_𝑑], 𝐻^𝑙∶𝑅₊→[𝐻,𝐻], 𝐻^𝑙_𝑑∶𝑅₊→[𝐻_𝑑,𝐻_𝑑], 𝐶^𝑙∶𝑅₊→[𝐶,𝐶], and 𝐶^𝑙_𝑑∶𝑅₊→[𝐶_𝑑,𝐶_𝑑]are continuous.

Set 1𝐴=2𝐴+𝐴,1Δ=2𝐴−𝐴=𝛿_{𝑖𝑗𝑛×𝑛},𝐷₁=𝛿₁₁𝑒₁,…,𝛿_1𝑛𝑒₁,…,𝛿_𝑛1𝑒_𝑛,…,𝛿_𝑛𝑛𝑒_𝑛,𝐺₁=𝛿₁₁𝑒₁,…,𝛿_1𝑛𝑒₁,…,𝛿_𝑛1𝑒_𝑛,…,𝛿_𝑛𝑛𝑒_𝑛^𝑇, (7) where𝑒_𝑖(𝑖=1,2,…,𝑛)is the𝑖th column of𝑛×𝑛identification matrix, setting 𝐹=𝑓∣𝑓=diag𝛼₁₁,…,𝛼_1𝑛,…,𝛼_𝑛1…,𝛼_𝑛𝑛, (8) where ‖𝛼_𝑖𝑗‖≤1. Obviously, we have 𝐹₁𝐹^𝑇₁≤𝐼_𝑛2; here𝐼_𝑛2is the𝑛²×𝑛²identification matrix. So the interval matrix𝐴^𝑙can equivalently be written as 𝐴^𝑙=𝐴,𝐴=𝐴+Δ𝐴=𝐴+𝐷₁𝐹₁𝐺₁,𝐹₁∈𝐹. (9)

Similar to the above derivation, we can have 𝐴^𝑙_𝑑=𝐴_𝑑,𝐴_𝑑=𝐴_𝑑+Δ𝐴_𝑑=𝐴_𝑑+𝐷₂𝐹₂𝐺₂,𝐻^𝑙=𝐻,𝐻=𝐻+Δ𝐻=𝐻+𝐷₃𝐹₃𝐺₃,𝐻^𝑙_d=𝐻_𝑑,𝐻_𝑑=𝐻_𝑑+Δ𝐻_𝑑=𝐻_𝑑+𝐷₄𝐹₄𝐺₄,𝐶^𝑙=𝐶,𝐶=𝐶+Δ𝐶=𝐶+𝐷₅𝐹₅𝐺₅,𝐶^𝑙_𝑑=𝐶_𝑑,𝐶_𝑑=𝐶_𝑑+Δ𝐶_𝑑=𝐶_𝑑+𝐷₆𝐹₆𝐺₆; (10)(11) where𝐹_𝑖∈𝐹,𝑖=1,2,…,6; let 𝐹=diag𝐹₁,𝐹₂,…,𝐹₆,𝐸₁=𝐷₁,𝐷₂,𝐸,0,0,0,0₂=0,0,𝐷₃,𝐷₄,𝐸,0,0₃=0,0,0,0,𝐷₅,𝐷₆,𝑁^𝑇₁=𝐺^𝑇₁,0,𝐺^𝑇₃,0,𝐺^𝑇₅,𝑁,0_𝑇2=0,𝐺^𝑇₂,0,𝐺^𝑇₄,0,𝐺^𝑇₆. (12)

Then (9)–(11) can be rewritten as Δ𝐴Δ𝐴_𝑑Δ𝐻Δ𝐻_𝑑Δ𝐶Δ𝐶_𝑑=𝐸₁𝐸₂𝐸₃𝐹𝑁₁𝑁₂. (13)

So the system (3) can be readily derived as 𝑥(𝑘+1)=(𝐴+Δ𝐴)𝑥(𝑘)+𝐴_𝑑+Δ𝐴_𝑑𝑥()(𝑘)(𝑘)+()𝑥(𝑘)𝜔(𝑘)+𝐻𝑘−𝑑+𝐵𝑢+𝐷𝑣𝐻+Δ𝐻_𝑑+Δ𝐻_𝑑𝑥()𝜔(𝑘),𝑧(𝑘)=()𝑥(𝑘)+𝐶𝑘−𝑑𝐶+Δ𝐶_𝑑+Δ𝐶_𝑑𝑥(),𝑥(𝑘)(𝑘)[].𝑘−𝑑=𝜑,𝑘∈−𝑑,0 (14) For system (14), we design the following state feedback controller: 𝑢(𝑘)=𝐾𝑥(𝑘), (15) where the matrix 𝐾 is the controller gain, which is to be designed. Then the closed-loop stochastic control system can be rewritten as 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴_𝑑𝑥()(𝑘)+(𝑘)+𝐻𝑘−𝑑+𝐷𝑣𝐻𝑥_𝑑𝑥(𝜔(𝑘),𝑧(𝑘)=(𝑘)+𝐶𝑘−𝑑)𝐶𝑥_𝑑𝑥(𝑘−𝑑), (16) where𝐴=𝐴+𝐵𝐾+Δ𝐴,𝐴_𝑑𝐴_𝑑+Δ𝐴_𝑑, 𝐻=𝐻+Δ𝐻, 𝐻_𝑑=𝐻_𝑑+Δ𝐻_𝑑, 𝐶=𝐶+Δ𝐶, 𝐶_𝑑=𝐶_𝑑+Δ𝐶_𝑑.

Definition 2. Discrete-time stochastic interval system (DTSIS) with time delay (3) is said to be stochastic exponentially stable in mean square, if there exist scalars𝑐>0,𝜇>1, such that 𝔼|𝑥(𝑁)|²≤𝑐𝜇^−𝑁max_{−𝑑≤𝑖≤0}𝔼𝜑(𝑖)². (17)

In this paper, we aim to design the controller gain matrix𝐾in (16) such that the requirements are simultaneously satisfied. (a)The zero-solution of the closed-loop stochastic control system (16) with𝑣(𝑘)=0is stochastic exponentially stable in mean square. (b)Under the zero-initial condition, the control output𝑧(𝑘)satisfies ^∞_𝑘=0𝔼‖𝑧(𝑘)‖²≤𝛾^2∞_𝑘=0𝔼‖𝑣(𝑘)‖² (18) for all nonzero𝑣(𝑘).

Which is also said the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation level𝛾>0.

Lemma 3 (see [30]). For given symmetric matrixΣ, and matrices𝐻,𝐸with appropriate dimension,Σ+𝐻𝐹𝐸+𝐸^𝑇𝐹^𝑇𝐻^𝑇<0with𝐹satisfying 𝐹^𝑇𝐹≤𝐼 holds if and only if Σ+𝑎⁻¹𝐻𝐻^𝑇+𝑎𝐸^𝑇𝐸<0 holds for any𝑎>0.

3. Main Results

Theorem 4. If there exist two positive definite matrices𝑃and𝑄, and for a given positive constant𝛾>0, such that the following matrix inequality holds: 𝐴Σ=𝑄−𝑃00^𝑇𝐻^𝑇𝐶^𝑇∗−𝑄0𝐴^𝑇_𝑑𝐻^𝑇_𝑑𝐶^𝑇_𝑑∗∗−𝛾²𝐼𝐷^𝑇00∗∗∗−𝑃⁻¹00∗∗∗∗−𝑃⁻¹0∗∗∗∗∗−𝐼<0, (19) then the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation level𝛾.

Proof. Let𝑣(𝑘)=0; we construct the Lyapunov-Krasovskii functional as 𝑉(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+^𝑑_𝑖=1𝑥^𝑇(𝑘−𝑖)𝑄𝑥(𝑘−𝑖). (20)
Calculating the difference of 𝑉(𝑘)along with the system (16) and taking the mathematical expectation, we have 𝔼{Δ𝑉(𝑘)}=𝐸{𝑉(𝑘+1)−𝑉(𝑘)}=𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−𝑥^𝑇()()=𝑘−𝑑𝑄𝑥𝑘−𝑑𝑥(𝑘)𝑥(𝑘−𝑑)^𝑇Π₁𝑥(𝑘)𝑥(),𝑘−𝑑 (21) where Π₁=+𝑄−𝑃0𝐴0−𝑄𝐴^𝑇𝑇_𝑑𝑃𝐴^𝑇𝐴^𝑇_𝑑^𝑇+𝐻^𝑇𝐻^𝑇_𝑑𝑃𝐻^𝑇𝐻^𝑇_𝑑^𝑇. (22)
It is clear formΣ<0that there exists a sufficient scalar𝑎>0, such that Σ+𝑎diag𝐼_𝑛×𝑛,0<0. (23)
Therefore, we have 𝔼{Δ𝑉(𝑘)}≤−𝑎𝔼|𝑥(𝑘)|². (24)
We now in a position to analyze of the exponential stable in mean square for the stochastic interval delay system (16). According to the definition of 𝑉(𝑘), we have 𝔼{𝑉(𝑘)}≤𝜌₁𝔼|𝑥(𝑘)|²+𝜌₂^𝑑_𝑘=1𝔼|𝑥(𝑘−𝑖)|², (25) where 𝜌₁=𝜆_max(𝑃) and 𝜌₂=𝜆_max(𝑄). For any scalar 𝜇>1, the earlier inequality, together with (23), implies that 𝜇^𝑘+1𝔼{𝑉(𝑘+1)}−𝜇^𝑘𝔼{𝑉(𝑘)‖=𝜇^𝑘+1𝔼‖Δ𝑉(𝑘)‖+𝜇^𝑘𝜇−1𝔼{𝑉(𝑘)}≤𝛽₁𝜇𝜇^𝑘𝔼|𝑥(𝑘)|²+𝛽₂𝜇𝜇^𝑘𝑑_𝑖=1𝔼|𝑥(𝑘−𝑖)|², (26) where 𝛽₁(𝜇)=−𝑎𝜇+𝜌₁(𝜇−1),𝛽₂(𝜇)=(𝜇−1)𝜌₂.
Furthermore, summing up both sides of the earlier inequality from0to𝑁−1with respect to𝑘, we get 𝜇^𝑁𝔼{𝑉(𝑁)}−𝔼{𝑉(0)}≤𝛽₁𝜇^𝑁−1_𝑘=0𝜇^𝑘𝔼|𝑥(𝑘)|²+𝛽₂𝜇^𝑁−1𝑑_𝑘=0𝑖=1𝜇^𝑘𝔼|𝑥(𝑘−𝑖)|². (27)
For 𝑑>1, ^𝑁−1𝑑_𝑘=0𝑖=1𝜇^𝑘𝔼|𝑥(𝑘−𝑖)|²≤⁻¹_𝑖=−𝑑^𝑖+𝑑_𝑘=0+^{𝑁−1−𝑑}_𝑖=0^𝑖+𝑑_𝑘=𝑖+1+^𝑁−1_{𝑖=𝑁−1−𝑑}^𝑁−1_𝑘=𝑖+1𝜇^𝑘𝔼|𝑥(𝑖)|²≤⁻¹_𝑖=−𝑑𝜇^𝑖+𝑑𝔼|𝑥(𝑖)|²+𝑑^{𝑁−1−𝑑}_𝑖=0𝜇^𝑖+𝑑𝔼|𝑥(𝑖)|²+𝑑^𝑁−1_{𝑖=𝑁−1−𝑑}𝜇^𝑖+𝑑𝔼|𝑥(𝑖)|²≤𝑑𝜇^𝑑max_{−𝑑≤𝑖≤0}𝔼𝜑(𝑖)²+𝑑𝜇^𝑑𝑁−1_𝑖=0𝔼|𝑥(𝑖)|². (28)
Then, from (27) and (28), it follows that 𝜇^𝑁𝔼{𝑉(𝑁)}≤𝔼{𝑉(0)}+𝛽₁𝜇+𝑑𝜇^𝑑𝛽₂𝜇×^𝑁−1_𝑘=0𝜇^𝑘𝔼|𝑥(𝑘)|²+𝑑𝜇^𝑑𝛽₂𝜇max_{−𝑑≤𝑖≤0}𝔼𝜑(𝑖)². (29)
Let 𝜌₀=𝜆_min(𝑃) and 𝜌=max{𝜌₁,𝜌₂}; we have 𝔼{𝑉(𝑁)}≥𝜌₀𝔼|𝑥(𝑁)|². (30)
It also follows that 𝔼{𝑉(0)}≤𝜌max_{−𝑑≤𝑖≤0}𝔼𝜑(𝑖)². (31)
We can choose appropriate scalar𝜎>0such that 𝛽₁(𝜎)+𝑑𝜎^𝑑𝛽₂(𝜎)=0, (32)
which implies 𝔼|𝑥(𝑁)|²≤𝑐𝜇^−𝑁max_{−𝑑≤𝑖≤0}𝔼𝜑(𝑖)², (33) where 𝑐=(1/𝜌₀)(𝜌+𝑑𝜎^𝑑𝛽₂(𝜎)). Hence the closed-loop stochastic control system (16) with𝑣(𝑘)=0is stochastic exponentially stable in mean square according to Definition 2.
Under the zero-initial condition, let 𝑣(𝑘)≠0, setting 𝐽(𝑁)=𝔼^𝑁_𝑘=0𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑣^𝑇(𝑘)𝑣(𝑘)=𝔼^𝑁_𝑘=0𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑣^𝑇(𝑘)𝑣(𝑘)()(𝑘()+𝑉𝑘+1−𝑉)−𝑉𝑁+1≤𝔼^𝑁_𝑘=0𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑣^𝑇(𝑘)𝑣(𝑘)(𝑘=+Δ𝑉)_𝑁𝑘=0𝑥(𝑘)𝑥(𝑘−𝑑)𝑣(𝑘)^𝑇Π₂𝑥(𝑘)𝑥()𝑣(𝑘)𝑘−𝑑, (34) where Π₂=Q−𝑃000−𝑄000−𝛾²𝐼+𝐴^𝑇𝐴^𝑇_𝑑𝐷^𝑇𝑃𝐴^𝑇𝐴^𝑇_𝑑𝐷^𝑇𝑇+𝐻^𝑇𝐻^𝑇_𝑑0𝑃𝐻^𝑇𝐻^𝑇_𝑑0^𝑇+𝐶^𝑇𝐶^𝑇_𝑑0𝐶^𝑇𝐶^𝑇_𝑑0^𝑇. (35)
So (19) implies that Π₂<0; hence,𝐽(𝑁)<0. Thus𝑁→∞, 𝐽(∞)<0, such that (18) holds. The proof is completed.

To design the controller (15), matrix inequality (15) must be solvable and matrix inequality (15) in Theorem 4 must be inverted into LMI form. In the following section, we will find a way for the solution of (15).

Remark 5. The results obtained in the paper are based on the Lyapunov-Krasovskii functional and the corresponding technique used in the proof of Theorem 4, and it is easy to extend the results to stochastic system with time-varying delay or multiple delays. Details are omitted here due to page length consideration.

Theorem 6. If there exist two positive definite matrices 𝑋,𝑄, a matrix 𝑌 with appropriate dimension, and a positive constant 𝜀, for a given positive constant 𝛾>0, such that the following linear matrix inequality (LMI) holds 𝑄−𝑋00Ω𝑋𝐻^𝑇𝑋𝐶^𝑇𝑋𝑁^𝑇₁0∗−𝑄0𝑋𝐴^𝑇_𝑑𝑋𝐻^𝑇_𝑑𝑋𝐶^𝑇_𝑑𝑋𝑁^𝑇₂0∗∗−𝛾²𝐼𝐷^𝑇0000∗∗∗−𝑋000𝜀𝐸₁∗∗∗∗−𝑋00𝜀𝐸₂∗∗∗∗∗−𝐼0𝜀𝐸₃∗∗∗∗∗∗−𝜀𝐼0∗∗∗∗∗∗∗−𝜀𝐼<0, (36) where Ω=𝑌^𝑇𝐵^𝑇+𝑋𝐴^𝑇, then the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation level 𝛾 with the controller designed as 𝐾=𝑌𝑋⁻¹.

Proof. By (13) and (19), we have Π+Θ₁𝐹^𝑇Θ^𝑇₂+Θ₂𝐹Θ^𝑇₁<0, (37) where ()Π=𝑄−𝑃00𝐴+𝐵𝐾^𝑇𝐻^𝑇𝐶^𝑇∗−𝑄0𝐴^𝑇_𝑑𝐻^𝑇_𝑑𝐶^𝑇_𝑑∗∗−𝛾²𝐼𝐷^𝑇00∗∗∗−𝑃⁻¹00∗∗∗∗−𝑃⁻¹0Θ∗∗∗∗∗−𝐼_𝑇1=𝑁₁,𝑁₂,0_1×4,Θ^𝑇₂=0_1×3,𝐸^𝑇₁,𝐸^𝑇₂,𝐸^𝑇₃. (38)
By Lemma 3, we know that Π+𝜀⁻¹Θ₁Θ^𝑇₁+𝜀Θ₂Θ^𝑇₂<0. (39)
Using Schur complement lemma and contragradient transformation, we can get the LMI (36) holds. The proof is completed.

Without considering of interval matrices, the system (3) changes into the following discrete-time stochastic time-delay system: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴_𝑑𝑥()(𝑘)(𝑘)+(𝑘)𝑘−𝑑+𝐵𝑢+𝐷𝑣𝐻𝑥+𝐻_𝑑𝑥(𝜔(𝑘),𝑧(𝑘)(𝑘)𝑘−𝑑)=𝐶𝑥+𝐶_𝑑𝑥(),𝑥(𝑘)(𝑘)[].𝑘−𝑑=𝜑,𝑘∈−𝑑,0 (40)

We can design the controller (15) by the following the corollary without proof.

Corollary 7. If there exist two positive definite matrices 𝑋,𝑄 and a matrix 𝑌 with appropriate dimension, for a given positive constant 𝛾>0, such that the following linear matrix inequality (LMI) holds 𝑄−𝑋00Ω𝑋𝐻^𝑇𝑋𝐶^𝑇∗−𝑄0𝑋𝐴^𝑇_𝑑𝑋𝐻^𝑇_𝑑𝑋𝐶^𝑇_𝑑∗∗−𝛾²𝐼𝐷^𝑇00∗∗∗−𝑋00∗∗∗∗−𝑋0∗∗∗∗∗−𝐼<0, (41) where Ω is defined in (36), then the closed-loop stochastic control system (40) is stochastic exponentially stable in mean square with disturbance attenuation level𝛾with the controller designed as𝐾=𝑌𝑋⁻¹.

If we do not consider the stochastic disturbance in system (3), the system (3) can be written as 𝑥(𝑘+1)=𝐴^𝑙𝑥(𝑘)+𝐴^𝑙_𝑑𝑥()(𝑘)(𝑘),𝑧(𝑘)𝑘−𝑑+𝐵𝑢+𝐷𝑣=𝐶^𝑙𝑥(𝑘)+𝐶^𝑙_𝑑𝑥(),𝑥(𝑘)(𝑘)[].𝑘−𝑑=𝜑,𝑘∈−𝑑,0 (42)

Then the robust 𝐻_∞ controller can be designed by the following corollary.

Corollary 8. If there exist two positive definite matrices𝑋,𝑄, a matrix𝑌with appropriate dimension, and a positive constant 𝜀, for a given positive constant𝛾>0, such that the following linear matrix inequality (LMI) holds 𝑄−𝑋00Ω𝑋𝐶^𝑇𝑋𝑁^𝑇₁0∗−𝑄0𝑋𝐴^𝑇_𝑑𝑋𝐶^𝑇_𝑑𝑋𝑁^𝑇₂0∗∗−𝛾²𝐼𝐷^𝑇000∗∗∗−𝑋00𝜀𝐸₁∗∗∗∗−𝐼0𝜀𝐸₃∗∗∗∗∗−𝜀𝐼0∗∗∗∗∗∗−𝜀𝐼<0, (43) where Ω is defined in (36), then the closed-loop stochastic control system (42) is exponentially stable with disturbance attenuation level 𝛾 with the controller designed as 𝐾=𝑌𝑋⁻¹.

4. Numerical Example with Simulation

In this section, an example is given to show the usefulness of the designed controller. Consider the following discrete-time stochastic interval system with time delay: 𝑥()=1[][]𝑥(𝑘)+𝑢(𝑘)+[]0[]𝑥()+𝑒𝑘+10.5,0.70.20.1,0.30.20.3−0.1,0.10.15−0.5,0.2𝑘−10−0.10.1−0.10.2^−0.2𝑘sin𝑘−𝑒^−0.1𝑘+[]0[]𝑥(𝑘)+[]0[]𝑥()𝜔(𝑘),𝑧(𝑘)=[]𝑥(𝑘)+[]𝑥(),𝑥(𝑘)(𝑘)[].cos2𝑘0.1,0.30.3−0.2,0.1−0.1−0.1,0.1−0.1,0.2𝑘−10−0.1,0.10.100.20,0.1][0.1,0.3−0.10𝑘−10=𝜑,𝑘∈−10,0 (44)

Without considering the state-feedback control, we see that the open-loop stochastic interval system is unstable from Figure 1.

Figure 1 

Response of open-loop DTSIS (44) to initial state(5,−3).

We aim at designing a state-feedback controller for stochastic interval system (44), such that the closed-loop stochastic interval system is stochastic exponentially stable in mean square with disturbance attenuation level𝛾.

By using the method discussed in the previous section, we can calculate the matrices 𝐴,𝐴_𝑑,𝐻,𝐻_𝑑,𝐶,𝐶_𝑑,𝐷_𝑖,𝐺_𝑖, 𝑖=1,2,…,6, setting 𝛾=0.9, solving the LMI (36) in Theorem 6 and then obtain the matrices as follows: ,,,𝑋=0.71470.07060.07061.3303𝑄=0.24420.02850.02850.9091𝑌=−2.8706−0.5642𝜀=2.2710. (45)

Note that in expression of Theorem 6, we can design the controller as 𝐾=𝑌𝑋⁻¹=−3.9954−0.2119. (46)

The response of the closed-loop stochastic interval system (44) is shown in Figures 2 and 3, which demonstrate that the designed controller for the stochastic interval system (44) is effective.

Figure 2 

Response of closed-loop DTSIS (44) to initial state(5,−3).

Figure 3 

Response of control output𝑧(𝑘)of closed-loop DTSIS (44).

5. Conclusions

In this paper, we have studied the robust𝐻_∞control problem for discrete-time stochastic interval system (DTSIS) with time delay. The stochastic interval system was equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robust𝐻_∞controller for DTSIS have been obtained in terms of linear matrix inequality (LMI) form, and the robust𝐻_∞controller has been designed easily by MATLAB LMI control toolbox. A numerical example with simulation has been given to demonstrate the feasibility and effectiveness of the proposed method.

Of course, this method proposed in this paper also can be used to the the state estimation problem and the sampled-data synchronization control problem for the complex networks, which has been studied in [31, 32] and is one of our future research topics.

Acknowledgments

The authors thank the referee and the associate editor for their very helpful comments and suggestions. This work is supported by National Science Foundation of China (NSFC) under Grant nos. 61104127, 51079057, and 61134012, China Postdoctoral Science Foundation with Grant no. 2012M521428, and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) under Grant no. Y201101.