Research ArticleRobust Stability, Stabilization, and𝐻

∞Control of a Class of

Nonlinear Discrete Time Stochastic Systems

Tianliang Zhang,1 Yu-Hong Wang,1 Xiushan Jiang,1 and Weihai Zhang2

1College of Information and Control Engineering, China University of Petroleum (East China), Qingdao, Shandong 266510, China2College of Electrical Engineering andAutomation, ShandongUniversity of Science andTechnology, Qingdao, Shandong 266590, China

Correspondence should be addressed to Weihai Zhang; w hzhang@163.com

Received 14 November 2015; Revised 20 March 2016; Accepted 31 March 2016

Academic Editor: Mingcong Deng

Copyright © 2016 Tianliang Zhang et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper studies robust stability, stabilization, and𝐻∞control for a class of nonlinear discrete time stochastic systems. Firstly, the

easily testing criteria for stochastic stability and stochastic stabilizability are obtained via linear matrix inequalities (LMIs). Then arobust𝐻

∞state feedback controller is designed such that the concerned system not only is internally stochastically stabilizable but

also satisfies robust 𝐻∞

performance. Moreover, the previous results of the nonlinearly perturbed discrete stochastic system aregeneralized to the system with state, control, and external disturbance dependent noise simultaneously. Two numerical examplesare given to illustrate the effectiveness of the proposed results.

1. Introduction

Stochastic control has been one of the most importantresearch topics in modern control theory. The study ofstochastic stability can be traced back to the 1960s; see [1]and the recently well-known monographs [2, 3]. Stabilityis the first considered problem in system analysis andsynthesis, while stabilization is to look for a controller tostabilize an unstable system. 𝐻

∞control is one of the most

important robust control approaches, which aims to designthe controller to restrain the external disturbance below agiven level. We refer the reader to [4–9] for stability andstabilization of Itô-type stochastic systems and [10–14] forstability and stabilization of discrete time stochastic systems.Stochastic 𝐻

∞control of Itô-type systems starts from [15],

which has been extensively studied in recent years; see [16–20] and the references therein. Discrete time𝐻

∞control with

multiplicative noise can be found in [21–25].Along with the development of computer technology,

discrete time difference systems have attracted a great deal ofattention, which have been studied extensively; see [26, 27].The reason is twofold: Firstly, discrete time systems are idealmathematical models in the study of satellite attitude control[28], mathematical finance [29], single degree of freedom

inverted pendulums [21], and gene regulator networks [30].Secondly, as said in [27], the study for discrete time systemshas the advantage over continuous time differential systemsfrom the perspective of computation; moreover, it presentsa very good approach to study differential equations andfunctional differential equations.

From the existing works on stability, stabilization, and𝐻∞

control of discrete time stochastic systems with multi-plicative noise, we can find that, except for linear stochasticsystemswhere perfect results have been obtained [22–25], fewworks are on the stability of the general nonlinear discretetime stochastic system [12]

𝑥 (𝑡 + 1) = 𝑓 (𝑥 (𝑡) , 𝑤 (𝑡) , 𝑡) , 𝑥 (0) = 𝑥0

(1)

or the𝐻∞control of affine nonlinear discrete time stochastic

system [21]

𝑥 (𝑡 + 1)

= 𝑓 (𝑥 (𝑡)) + 𝑔 (𝑥 (𝑡)) 𝑢 (𝑡) + ℎ (𝑥 (𝑡)) ] (𝑡)

+ [𝑓1(𝑥 (𝑡)) + 𝑔

1(𝑥 (𝑡)) 𝑢 (𝑡) + ℎ

1(𝑥 (𝑡))] 𝑤 (𝑡) ,

𝑧 (𝑡) = 𝐿 (𝑥 (𝑡)) .

(2)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 5185784, 11 pageshttp://dx.doi.org/10.1155/2016/5185784

2 Mathematical Problems in Engineering

Up to now, the results of the deterministic discrete timenonlinear 𝐻

∞control [31] have not been perfectly general-

ized to the above nonlinear stochastic systems. For example,although some stability results in continuous time Itô systems[3] can be extended to nonlinear discrete stochastic systems[12], the corresponding criteria are not easily applied inpractice; this is because the mathematical expectation of thetrajectory is involved in the preconditions. In addition, [21]tried to discuss a general nonlinear 𝐻

∞control of discrete

time stochastic systems, but only the 𝐻∞

control of a classof norm bounded systems was perfectly solved based onlinear matrix inequality (LMI) approach. As said in [32],the general𝐻

∞control of nonlinear discrete time stochastic

multiplicative noise systems remains unsolved. We have toadmit such a fact that some research issues of discrete systemsare more difficult to solve than those of continuous timesystems. For instance, a stochastic maximum principle for Itôsystems was obtained in 1990 [33], but a nonlinear discretetime maximum principle has just been presented in [34].

Recently, [7, 13] investigated the robust quadratic stabilityand feedback stabilization of a class of nonlinear continuetime and discrete time systems, respectively, where thenonlinear terms are quadratically bounded. Such a nonlinearconstraint possesses great practical importance and has beenwidely used in many types of systems, such as singularly per-turbed systems with uncertainties [35, 36], neutral systemswith nonlinear perturbations [37], impulsive Takagi-Sugenofuzzy systems [38], and some time-delay systems [18]. Itshould be pointed out that the small gain theorem can also beused to examine the robustness as done in [39] for the study ofthe simple adaptive control system within the framework ofthe small gain theorem. In addition, the robustness of a classof nonlinear feedback systems with unknown perturbationswas discussed based on the robust right coprime factorizationand passivity property [40]. All these methods are expectedto play important roles in stochastic uncertain𝐻

∞control.

This paper deals with a class of nonlinear uncertaindiscrete time stochastic systems, for which the system state,control input, and external disturbance depend on noisesimultaneously, which was often called (𝑥, 𝑢, V)-dependentnoise for short [24] and which mean that not only thesystem state as in [21] but also the control input and externaldisturbance are subject to random noise. Hence, our con-cerned models have more wide applications. The considerednonlinear dynamic term is priorly unknown but belongsto a class of functions with a bounded energy level, whichrepresent a kind of very important nonlinear functions, andhas been studied by many researchers; see, for example,[41]. For such a class of nonlinear discrete time stochasticsystems, the stochastic stability, stabilization, and𝐻

∞control

have been discussed, respectively, and easily testing criteriahave also been obtained. What we have obtained extends theprevious works to more general models.

The paper is organized as follows: in Section 2, we givea description of the considered nonlinear stochastic sys-tems and define robust stochastic stability and stabilization.Section 3 contains our main results. Section 3.1 presents arobust stability criterion which extends the result of [13] tomore general stochastic systems. Section 3.2 gives a sufficient

condition for robust stabilization criterion. Section 3.3 isabout 𝐻

∞control, where an LMI-based sufficient condition

for the existence of a static state feedback 𝐻∞

controller isestablished. All our main results are expressed in terms ofLMIs. In Section 4, two examples are constructed to show theeffectiveness of our obtained results.

For convenience, the notations adopted in this paper areas follows.𝑀 is the transpose of the matrix𝑀 or vector𝑀, 𝑀 ≥ 0

(𝑀 > 0): 𝑀 is a positive semidefinite (positive definite)symmetric matrix; 𝐼 is the identity matrix; 𝑅𝑛 is the 𝑛-dimensional Euclidean space; 𝑅𝑛×𝑚 is the space of all 𝑛 ×𝑚 matrices with entries in 𝑅; 𝑁 is the natural numberset; that is, 𝑁 represents {0, 1, 2, . . .}; 𝑁

𝑡denotes the set of

{0, 1, . . . , 𝑡}; 𝑙2𝑤(𝑁, 𝑅𝑛

) denotes the set of all nonanticipativesquare summable stochastic processes

𝑦 = {𝑦𝑛: 𝑦𝑛

∈ 𝐿2

(Ω, 𝑅𝑛

) , 𝑦𝑛is F𝑛−1

measurable}𝑛∈𝑁

.

(3)

The 𝑙2-norm of 𝑦 ∈ 𝑙2𝑤(𝑁, 𝑅𝑛

) is defined by

𝑦𝑙2

𝑤(𝑁,𝑅𝑛)= (

∞

∑

𝑛=0

𝐸𝑦𝑛

2

)

1/2

. (4)

Similarly, 𝑙2𝑤(𝑁𝑇, 𝑅𝑛

) and ‖𝑦‖𝑙2

𝑤(𝑁,𝑅𝑛)can be defined.

2. System Descriptions and Definitions

Consider the discrete stochastic iterative system described bythe following equation:

𝑥 (𝑡 + 1) = 𝐴𝑥 (𝑡) + ℎ1(𝑡, 𝑥 (𝑡)) + 𝐵𝑢 (𝑡)

+ (𝐶𝑥 (𝑡) + ℎ2(𝑡, 𝑥 (𝑡)) + 𝐷𝑢 (𝑡)) 𝑤 (𝑡) ,

𝑥 (0) = 𝑥0∈ 𝑅𝑛

, 𝑡 ∈ 𝑁,

(5)

where 𝑥(𝑡) ∈ 𝑅𝑛 is the 𝑛-dimensional state vector and 𝑢(𝑡) ∈𝑅𝑚 is the 𝑚-dimensional control input. {𝑤(𝑡)}

𝑡≥0is a se-

quence of one-dimensional independent white noise proc-esses defined on the complete filtered probability space (Ω,F, {F

𝑡}𝑡≥0,P), whereF

𝑡= 𝜎{𝑤(0), 𝑤(1), . . . , 𝑤(𝑡)}.Assume

that 𝐸[𝑤(𝑡)] = 0, 𝐸[𝑤(𝑡)𝑤(𝑗)] = 𝛿𝑡𝑗, where 𝐸 stands for the

expectation operation and 𝛿𝑡𝑗is a Kronecker function defined

by 𝛿𝑡𝑗= 0 for 𝑡 ̸= 𝑗 while 𝛿

𝑡𝑗= 1 for 𝑡 = 𝑗. Without loss of

generality, 𝑥0is assumed to be determined. The following is

assumed to hold throughout this paper.

Assumption 1. The nonlinear functions ℎ1(𝑡, 𝑥(𝑡)) and

ℎ2(𝑡, 𝑥(𝑡)) describe parameter uncertainty of the system and

satisfy the following quadratic inequalities:

ℎ

1(𝑡, 𝑥 (𝑡)) ℎ

1(𝑡, 𝑥 (𝑡)) ≤ 𝛼

2

1𝑥

(𝑡)𝐻

1𝐻1𝑥 (𝑡) , (6)

ℎ

2(𝑡, 𝑥 (𝑡)) ℎ

2(𝑡, 𝑥 (𝑡)) ≤ 𝛼

2

2𝑥

(𝑡)𝐻

2𝐻2𝑥 (𝑡) , (7)

for all 𝑡 ∈ 𝑁, where 𝛼𝑖is a constant related to the function ℎ

𝑖

for 𝑖 = 1, 2.𝐻𝑖is a constant matrix reflecting structure of ℎ

𝑖.

Mathematical Problems in Engineering 3

We note that inequalities (6) and (7) can be written as amatrix form:

[

[

[

𝑥

ℎ1

ℎ2

]

]

]

[

[

[

−𝛼2

1𝐻

1𝐻1− 𝛼2

2𝐻

2𝐻20 0

0 𝐼 0

0 0 𝐼

]

]

]

[

[

[

𝑥

ℎ1

ℎ2

]

]

]

≤ 0. (8)

System (5) is regarded as the generalized version of thesystem in [13, 42]. We note that, in system (5), the systemstate, control input, and uncertain terms depend on noisesimultaneously, which makes (1) more useful in describingmany practical phenomena.

Definition 2. The unforced system (5) with 𝑢 = 0 is said to berobustly stochastically stable with margins 𝛼

1> 0 and 𝛼

2> 0

if there exists a constant 𝛿(𝑥0, 𝛼1, 𝛼2) such that

𝐸[

∞

∑

𝑡=0

𝑥

(𝑡) 𝑥 (𝑡)] ≤ 𝛿 (𝑥0, 𝛼1, 𝛼2) . (9)

Definition 2 implies 𝐸{‖𝑥(𝑡)‖2} → 0.

Definition 3. System (5) is said to be robustly stochasticallystabilizable if there exists a state feedback control law 𝑢(𝑡) =𝐾𝑥(𝑡), such that the closed-loop system

𝑥 (𝑡 + 1) = (𝐴 + 𝐵𝐾) 𝑥 (𝑡) + ℎ1(𝑡, 𝑥 (𝑡))

+ ((𝐶 + 𝐷𝐾) 𝑥 (𝑡) + ℎ2(𝑡, 𝑥 (𝑡))) 𝑤 (𝑡) ,

𝑥 (0) = 𝑥0∈ 𝑅𝑛

, 𝑡 ∈ 𝑁

(10)

is robustly stochastically stable for all nonlinear functionsℎ𝑖(𝑡, 𝑥(𝑡)) (𝑖 = 1, 2) satisfying (6) and (7).

When there is the external disturbance ](⋅) in system (5),we consider the following nonlinear perturbed system:

𝑥 (𝑡 + 1)

= 𝐴𝑥 (𝑡) + 𝐴0] (𝑡) + 𝐵𝑢 (𝑡) + ℎ

1(𝑡, 𝑥 (𝑡))

+ (𝐶𝑥 (𝑡) + 𝐶0] (𝑡) + 𝐷𝑢 (𝑡) + ℎ

2(𝑡, 𝑥 (𝑡))) 𝑤 (𝑡) ,

𝑧 (𝑡) = [

𝐿𝑥 (𝑡)

𝑀] (𝑡)]

𝑥 (0) = 𝑥0∈ 𝑅𝑛

, 𝑡 ∈ 𝑁,

(11)

where ](𝑡) ∈ 𝑅𝑞 and 𝑧(𝑡) ∈ 𝑅𝑝 are, respectively, thedisturbance signal and the controlled output. ](𝑡) is assumedto belong to 𝑙2

𝑤(𝑁, 𝑅𝑞

), so ](𝑡) is independent of 𝑤(𝑡).

Definition 4 (𝐻∞

control). For a given disturbance attenua-tion level 𝛾 > 0, 𝑢(𝑡) = 𝐾𝑥(𝑡) is an𝐻

∞control of system (11),

if

(i) system (11) is internally stochastically stabilizable for𝑢(𝑡) = 𝐾𝑥(𝑡) in the absence of ](𝑡); that is,

𝑥 (𝑡 + 1) = (𝐴 + 𝐵𝐾) 𝑥 (𝑡) + ℎ1(𝑡, 𝑥 (𝑡))

+ [(𝐶 + 𝐷𝐾) 𝑥 (𝑡) + ℎ2(𝑡, 𝑥 (𝑡))] 𝑤 (𝑡)

(12)

is robustly stochastically stable;

(ii) The𝐻∞norm of system (11) is less than 𝛾 > 0; that is,

‖𝑇‖ = sup]∈𝑙2𝑤(𝑁,𝑅𝑞),𝑢(𝑡)=𝐾𝑥(𝑡),] ̸=0,𝑥

0=0

‖𝑧 (𝑡)‖𝑙2

𝑤(𝑁,𝑅𝑝)

‖] (𝑡)‖𝑙2

𝑤(𝑁,𝑅𝑞)

= sup]∈𝑙2𝑤(𝑁,𝑅𝑞),𝑢(𝑡)=𝐾𝑥(𝑡),] ̸=0,𝑥

0=0

(∑∞

𝑡=0𝐸 ‖𝑧 (𝑡)‖

2

)

1/2

(∑∞

𝑡=0𝐸 ‖] (𝑡)‖2)

1/2

< 𝛾.

(13)

3. Main Results

In this section, we give ourmain results on stochastic stability,stochastic stabilization, and robust 𝐻

∞control via LMI-

based approach. Firstly, we introduce the following twolemmas which will be used in the proof of our main results.

Lemma 5 (Schur’s lemma). For a real symmetric matrix 𝑆 =[

𝑆11𝑆12

𝑆𝑇

12𝑆22

], the following three conditions are equivalent:

(i) 𝑆 < 0.

(ii) 𝑆11< 0, 𝑆

22− 𝑆𝑇

12𝑆−1

11𝑆12< 0.

(iii) 𝑆22< 0, 𝑆

11− 𝑆12𝑆−1

22𝑆𝑇

12< 0.

Lemma 6. For any real matrices 𝑈, 𝑁 = 𝑁 > 0 and𝑊 withappropriate dimensions, we have

𝑈

𝑁𝑊+𝑊

𝑁𝑈 ≤ 𝑈

𝑁𝑈 +𝑊

𝑁𝑊. (14)

Proof. Because 𝑁 = 𝑁 > 0, 𝑈𝑁𝑊 + 𝑊𝑁𝑈 =(𝑈

𝑁1/2

)(𝑁1/2

𝑊) + (𝑊

𝑁1/2

)(𝑁1/2

𝑊). Inequality (14) is animmediate corollary of the well-known inequality

𝑋

𝑌 + 𝑌

𝑋 ≤ 𝑋

𝑋 + 𝑌

𝑌. (15)

3.1. Robust Stability Criteria. Consider the following un-forced stochastic discrete time system:

𝑥 (𝑡 + 1) = 𝐴𝑥 (𝑡) + ℎ1(𝑡, 𝑥 (𝑡))

+ (𝐶𝑥 (𝑡) + ℎ2(𝑡, 𝑥 (𝑡))) 𝑤 (𝑡) ,

𝑥 (0) = 𝑥0∈ 𝑅𝑛

, 𝑡 ∈ 𝑁,

(16)

where {ℎ1(𝑡, 𝑥(𝑡))}

𝑡≥0and {ℎ

2(𝑡, 𝑥(𝑡))}

𝑡≥0satisfy (8). The

following theorem gives a sufficient condition of robuststochastic stability for system (16).

Theorem 7. System (16) with margins 𝛼1> 0 and 𝛼

2>

0 is said to be robustly stochastically stable, if there exists

4 Mathematical Problems in Engineering

a symmetric positive definite matrix 𝑄 > 0 and a scalar 𝛼 > 0such that

[

[

[

[

[

[

[

[

−𝑄 + 2𝛼2

1𝛼𝐻

𝐻 + 2𝛼2

2𝛼𝐻

𝐻 𝐴

𝑄 𝐶

𝑄 0

∗ −

1

2

𝑄 0 0

∗ ∗ −

1

2

𝑄 0

∗ ∗ ∗ 𝑄 − 𝛼𝐼

]

]

]

]

]

]

]

]

< 0.

(17)

Proof. If (17) holds, we set 𝑉(𝑥(𝑡)) = 𝑥(𝑡)𝑄𝑥(𝑡) as aLyapunov function candidate of system (16), where 0 < 𝑄 <𝛼𝐼 by (17). Note that 𝑥(𝑡) and 𝑤(𝑡) are independent, so thedifference generator is

𝐸Δ𝑉 (𝑥 (𝑡)) = 𝐸 [𝑉 (𝑥 (𝑡 + 1)) − 𝑉 (𝑥 (𝑡))]

= 𝐸 {𝑥

(𝑡) (𝐴

𝑄𝐴 + 𝐶

𝑄𝐶 − 𝑄)𝑥 (𝑡)

+ 𝑥

(𝑡) 𝐴

𝑄ℎ1(𝑡, 𝑥 (𝑡)) + ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄𝐴𝑥 (𝑡)

+ ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄ℎ

1(𝑡, 𝑥 (𝑡)) + 𝑥

(𝑡) 𝐶

𝑄ℎ2(𝑡, 𝑥 (𝑡))

+ ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄𝐶𝑥 (𝑡) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄ℎ

2(𝑡, 𝑥 (𝑡))} .

(18)

Applying Lemma 6 and inequalities (6)-(7), by 0 < 𝑄 < 𝛼𝐼,we have

𝑥

(𝑡) 𝐴

𝑄ℎ1(𝑡, 𝑥 (𝑡)) + ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄𝐴𝑥 (𝑡)

≤ 𝑥

(𝑡) 𝐴

𝑄𝐴𝑥 (𝑡) + ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄ℎ

1(𝑡, 𝑥 (𝑡))

≤ 𝑥

(𝑡) (𝐴

𝑄𝐴 + 𝛼2

1𝛼𝐻

𝐻)𝑥 (𝑡) ,

𝑥

(𝑡) 𝐶

𝑄ℎ2(𝑡, 𝑥 (𝑡)) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄𝐶𝑥 (𝑡)

≤ 𝑥

(𝑡) (𝐶

𝑄𝐶 + 𝛼2

2𝛼𝐻

𝐻)𝑥 (𝑡) ,

ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄ℎ

1(𝑡, 𝑥 (𝑡)) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄ℎ

2(𝑡, 𝑥 (𝑡))

≤ 𝑥

(𝑡) (𝛼2

1𝛼𝐻

𝐻 + 𝛼2

2𝛼𝐻

𝐻)𝑥 (𝑡) .

(19)

Substituting (19) into (18), we achieve that

𝐸Δ𝑉 (𝑥 (𝑡)) ≤ 𝐸 {𝑥

(𝑡) (2𝐴

𝑄𝐴 + 2𝐶

𝑄𝐶 − 𝑄

+ 2𝛼2

1𝛼𝐻

𝐻 + 2𝛼2

2𝛼𝐻

𝐻)𝑥 (𝑡)} .

(20)

By Schur’s complement,

Ω fl 2𝐴𝑄𝐴 + 2𝐶𝑄𝐶 − 𝑄 + 2𝛼21𝛼𝐻

𝐻 + 2𝛼2

2𝛼𝐻

𝐻

< 0

(21)

is equivalent to

[

[

[

[

[

[

−𝑄 + 2𝛼2

1𝛼𝐻

𝐻 + 2𝛼2

2𝛼𝐻

𝐻 𝐴

𝑄 𝐶

𝑄

∗ −

1

2

𝑄 0

∗ ∗ −

1

2

𝑄

]

]

]

]

]

]

< 0, (22)

which holds by (17). We denote 𝜆max(Ω) and 𝜆min(Ω) to bethe largest and the minimum eigenvalues of the matrix Ω,respectively; then (20) yields

𝐸Δ𝑉 (𝑥 (𝑡)) ≤ 𝜆max (Ω) 𝐸 ‖𝑥 (𝑡)‖2

. (23)

Taking summation on both sides of the above inequality from𝑡 = 0 to 𝑡 = 𝑇 ≥ 0, we get

𝐸 [𝑉 (𝑥 (𝑇))] − 𝑉 (𝑥0) = 𝐸[

𝑇

∑

𝑡=0

Δ𝑉 (𝑥 (𝑡))]

≤ 𝜆max (Ω) 𝐸[𝑇

∑

𝑡=0

𝑥

(𝑡) 𝑥 (𝑡)] .

(24)

Therefore,

𝜆min (−Ω) 𝐸[𝑇

∑

𝑡=0

𝑥

(𝑡) 𝑥 (𝑡)] ≤ 𝑉 (𝑥0) , (25)

which leads to

𝐸[

𝑇

∑

𝑡=0

𝑥

(𝑡) 𝑥 (𝑡)] ≤ 𝛿 (𝑥0, 𝛼1, 𝛼2) fl

𝑉 (𝑥0)

𝜆min (−Ω). (26)

Hence, the robust stochastic stability of system (16) isobtained by (26) via letting 𝑇 → ∞.

Remark 8. FromTheorem7, if LMI (17) has feasible solutions,then, for any bounded parameters �̂�

1and �̂�

2on the uncertain

perturbation satisfying �̂�1≤ 𝛼1and �̂�

2≤ 𝛼2, system (16) is

robustly stochastically stable with margins �̂�1and �̂�

2.

3.2. Robust Stabilization Criteria. In this subsection, a suffi-cient condition about robust stochastic stabilization via LMIwill be given.

Theorem 9. System (5) with margins 𝛼1and 𝛼

2is robustly

stochastically stabilizable if there exist real matrices 𝑌 and𝑋 >0 and a real scalar 𝛽 > 0 such that

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

−𝑋 𝛼1𝑋𝐻

𝛼2𝑋𝐻

𝐽

1𝐽

20

∗ −

1

2

𝛽𝐼 0 0 0 0

∗ ∗ −

1

2

𝛽𝐼 0 0 0

∗ ∗ ∗ −

1

2

𝑋 0 0

∗ ∗ ∗ ∗ −

1

2

𝑋 0

∗ ∗ ∗ ∗ ∗ 𝛽𝐼 − 𝑋

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

< 0 (27)

holds, where

𝐽1= 𝐴𝑋 + 𝐵𝑌,

𝐽2= 𝐶𝑋 + 𝐷𝑌.

(28)

In this case, 𝑢(𝑡) = 𝐾𝑥(𝑡) = 𝑌𝑋−1𝑥(𝑡) is a robustlystochastically stabilizing control law.

Mathematical Problems in Engineering 5

Proof. We consider synthesizing a state feedback controller𝑢(𝑡) = 𝐾𝑥(𝑡) to stabilize system (5). Substituting 𝑢(𝑡) = 𝐾𝑥(𝑡)into system (5) yields the closed-loop system described by

𝑥 (𝑡 + 1) = 𝐴𝑥 (𝑡) + ℎ1(𝑡, 𝑥 (𝑡))

+ (𝐶𝑥 (𝑡) + ℎ2(𝑡, 𝑥 (𝑡))) 𝑤 (𝑡) ,

𝑥 (0) = 𝑥0∈ 𝑅𝑛

, 𝑡 ∈ 𝑁,

(29)

where 𝐴 = 𝐴 + 𝐵𝐾 and 𝐶 = 𝐶 + 𝐷𝐾. By Theorem 7, system(29) is robustly stochastically stable if there exists a matrix𝑄, 0 < 𝑄 < 𝛼𝐼, such that the following LMI

Ω̃ fl[

[

[

[

[

[

−𝑄 + 2𝛼2

1𝛼𝐻

𝐻 + 2𝛼2

2𝛼𝐻

𝐻 𝐴

𝑄 𝐶

𝑄

∗ −

1

2

𝑄 0

∗ ∗ −

1

2

𝑄

]

]

]

]

]

]

< 0

(30)

holds. Let 𝑄−1 = 𝑋 and pre- and postmultiply

diag [𝑋,𝑋,𝑋] (31)

on both sides of inequality (30), and it yields

[

[

[

[

[

[

−𝑋 + 2𝛼2

1𝛼𝑋𝐻

𝐻𝑋 + 2𝛼2

2𝛼𝑋𝐻

𝐻𝑋 𝑋𝐴

𝑋𝐶

∗ −

1

2

𝑋 0

∗ ∗ −

1

2

𝑋

]

]

]

]

]

]

< 0.

(32)

In order to transform (32) into a suitable LMI form, we set𝛽 = 1/𝛼; then 0 < 𝑄 < 𝛼𝐼 is equivalent to

𝛽𝐼 − 𝑋 < 0, 𝛽 > 0. (33)

Combining (33) with (32) and setting the gain matrix 𝐾 =𝑌𝑋−1, LMI (27) is obtained. The proof is completed.

3.3. 𝐻∞Control. In this subsection,main result about robust

𝐻∞

control will be given via LMI approach.

Theorem 10. Consider system (11) with margins 𝛼1> 0 and

𝛼2> 0. For the given 𝛾 > 0, if there exist real matrices 𝑋 > 0

and 𝑌 and scalar 𝛽 > 0 satisfying the following LMI,

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

−𝑋 𝐿

𝛼2

1𝑋𝐻

1𝛼2

2𝑋𝐻

2𝐽

1𝐽

20 0 0 0 𝐽

1𝐽

20

∗ −𝐼 0 0 0 0 0 0 0 0 0 0 0

∗ ∗ −

1

3

𝛽𝐼 0 0 0 0 0 0 0 0 0 0

∗ ∗ ∗ −

1

3

𝛽𝐼 0 0 0 0 0 0 0 0 0

∗ ∗ ∗ ∗ −𝑋 0 0 0 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ −𝑋 0 0 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ −𝛾2

𝐼 𝐴

0𝐶

0𝑀

𝐴0𝐶0

0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝑋 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝑋 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝐼 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝑋 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝑋 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 𝛽𝐼 − 𝑋

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

< 0, (34)

where 𝐽1= 𝐴𝑋 + 𝐵𝑌, 𝐽

2= 𝐶𝑋 + 𝐷𝑌, then system (11) is𝐻

∞

controllable, and the robust𝐻∞

control law is 𝑢(𝑡) = 𝐾𝑥(𝑡) =𝑌𝑋−1

𝑥(𝑡) for 𝑡 ∈ 𝑁.

Proof. When ](𝑡) = 0, byTheorem 9, system (11) is internallystabilizable via 𝑢(𝑡) = 𝐾𝑥(𝑡) = 𝑌𝑋−1𝑥(𝑡), because LMI (34)implies LMI (27). Next, we only need to show ‖𝑇‖ < 𝛾.

Take 𝑢(𝑡) = 𝐾𝑥(𝑡) and choose the Lyapunov function𝑉(𝑥(𝑡)) = 𝑥

(𝑡)𝑄𝑥(𝑡), where

0 < 𝑄 < 𝛼𝐼, 𝛼 > 0 (35)

for some 𝛼 > 0 to be determined; then for the system

𝑥 (𝑡 + 1) = 𝐴𝑥 (𝑡) + 𝐴0] (𝑡) + ℎ

1(𝑡, 𝑥 (𝑡))

+ (𝐶𝑥 (𝑡) + 𝐶0] (𝑡) + ℎ

2(𝑡, 𝑥 (𝑡))) 𝑤 (𝑡) ,

𝑧 (𝑡) = [

𝐿𝑥 (𝑡)

𝑀] (𝑡)] ,

𝑥 (0) = 𝑥0∈ 𝑅𝑛

, 𝑡 ∈ 𝑁

(36)

6 Mathematical Problems in Engineering

with 𝐴 = 𝐴 + 𝐵𝐾 and 𝐶 = 𝐶 + 𝐷𝐾, we have, with 𝑥(𝑡) and](𝑡) independent of 𝑤(𝑡), in mind that

𝐸Δ𝑉 (𝑥 (𝑡)) = 𝐸 [𝑉 (𝑥 (𝑡 + 1)) − 𝑉 (𝑥 (𝑡))]

= 𝐸 [𝑥

(𝑡 + 1)𝑄𝑥 (𝑡 + 1) − 𝑥

(𝑡) 𝑄𝑥 (𝑡)]

= 𝐸 {𝑥

(𝑡) [𝐴

𝑄𝐴 + 𝐶

𝑄𝐶 − 𝑄]𝑥 (𝑡)

+ 𝑥

(𝑡) 𝐴

𝑄ℎ1(𝑡, 𝑥 (𝑡)) + 𝑥

(𝑡) 𝐶

𝑄ℎ2(𝑡, 𝑥 (𝑡))

+ 𝑥

(𝑡) [𝐴

𝑄𝐴0+ 𝐶

𝑄𝐶0] ] (𝑡)

+ ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄𝐴𝑥 (𝑡) + ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄ℎ

1(𝑡, 𝑥 (𝑡))

+ ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄𝐴

0] (𝑡) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄𝐶𝑥 (𝑡)

+ ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄ℎ

2(𝑡, 𝑥 (𝑡)) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄𝐶

0] (𝑡)

+ ] (𝑡) (𝐴0𝑄𝐴0+ 𝐶

0𝑄𝐶0) ] (𝑡)

+ ] (𝑡) [𝐴0𝑄𝐴 + 𝐶

0𝑄𝐶] 𝑥 (𝑡)

+ ] (𝑡) 𝐴0𝑄ℎ1(𝑡, 𝑥 (𝑡)) + ] (𝑡) 𝐶

0𝑄ℎ2(𝑡, 𝑥 (𝑡))} .

(37)

Set 𝑥0= 0, and then for any ](𝑡) ∈ 𝑙2

𝑤(𝑁, 𝑅𝑝

),

‖𝑧 (𝑡)‖2

𝑙2

𝑤(𝑁𝑇,𝑅𝑝)− 𝛾2

‖] (𝑡)‖2𝑙2

𝑤(𝑁𝑇,𝑅𝑞)

= 𝐸

𝑇

∑

𝑡=0

[𝑥

(𝑡) 𝐿

𝐿𝑥 (𝑡) + ] (𝑡)𝑀𝑀] (𝑡)

− 𝛾2] (𝑡) ] (𝑡) + Δ𝑉 (𝑡)] − 𝑥 (𝑇)𝑄𝑥 (𝑇)

≤ 𝐸

𝑇

∑

𝑡=0

{𝑥

(𝑡) 𝐿

𝐿𝑥 (𝑡) + ] (𝑡)𝑀𝑀] (𝑡)

− 𝛾2] (𝑡) ] (𝑡) + ] (𝑡) (𝐴

0𝑄𝐴0+ 𝐶

0𝑄𝐶0) ] (𝑡)

+ 𝑥

(𝑡) [𝐴

𝑄𝐴 + 𝐶

𝑄𝐶 − 𝑄]𝑥 (𝑡)

+ 𝑥

(𝑡) 𝐴

𝑄ℎ1(𝑡, 𝑥 (𝑡)) + 𝑥

(𝑡) 𝐶

𝑄ℎ2(𝑡, 𝑥 (𝑡))

+ 𝑥

(𝑡) [𝐴𝑄𝐴0+ 𝐶

𝑄𝐶0] ] (𝑡)

+ ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄𝐴𝑥 (𝑡) + ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄ℎ

1(𝑡, 𝑥 (𝑡))

+ ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄𝐴

0] (𝑡) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄𝐶𝑥 (𝑡)

+ ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄ℎ

2(𝑡, 𝑥 (𝑡)) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄𝐶

0] (𝑡)

+ ] (𝑡) 𝐴0𝑄ℎ1(𝑡, 𝑥 (𝑡)) + ] (𝑡) 𝐶

0𝑄ℎ2(𝑡, 𝑥 (𝑡))

+ ] (𝑡) [𝐴0𝑄𝐴 + 𝐶

0𝑄𝐶] 𝑥 (𝑡)} .

(38)

Using Lemma 6 and setting 𝑥(𝑡)𝐴 = 𝑈, ℎ1(𝑡, 𝑥(𝑡)) = 𝑊,

and𝑁 = 𝑄 > 0, we have

𝑥

(𝑡) 𝐴

𝑄ℎ1(𝑡, 𝑥 (𝑡)) + ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄𝐴𝑥 (𝑡)

≤ 𝑥

(𝑡) 𝐴

𝑄𝐴𝑥 (𝑡) + ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄ℎ

1(𝑡, 𝑥 (𝑡)) .

(39)

Similarly, the following inequalities can also be obtained:

𝑥

(𝑡) 𝐶

𝑄ℎ2(𝑡, 𝑥 (𝑡)) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄𝐶𝑥 (𝑡)

≤ 𝑥

(𝑡) 𝐶

𝑄𝐶𝑥 (𝑡) + ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄ℎ

2(𝑡, 𝑥 (𝑡)) ,

ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄𝐴

0] (𝑡) + ] (𝑡) 𝐴

0𝑄ℎ1(𝑡, 𝑥 (𝑡))

≤ ℎ

1(𝑡, 𝑥 (𝑡)) 𝑄ℎ

1(𝑡, 𝑥 (𝑡)) + ] (𝑡) 𝐴

0𝑄𝐴0] (𝑡) ,

ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄𝐶

0] (𝑡) + ] (𝑡) 𝐶

0𝑄ℎ2(𝑡, 𝑥 (𝑡))

≤ ℎ

2(𝑡, 𝑥 (𝑡)) 𝑄ℎ

2(𝑡, 𝑥 (𝑡)) + ] (𝑡) 𝐶

0𝑄𝐶0] (𝑡) .

(40)

Substituting (39)-(40) into inequality (38) and considering(35), it yields

‖𝑧 (𝑡)‖2

𝑙2

𝑤(𝑁𝑇,𝑅𝑝)− 𝛾2

‖] (𝑡)‖2𝑙2

𝑤(𝑁𝑇,𝑅𝑞)≤ 𝐸

𝑇

∑

𝑡=0

{𝑥

(𝑡) [−𝑄

+ 2𝐴

𝑄𝐴 + 2𝐶

𝑄𝐶 + 𝐿

𝐿 + 3𝛼2

1𝛼𝐻

1𝐻1

+ 3𝛼2

2𝛼𝐻

2𝐻2] 𝑥 (𝑡) + 𝑥

(𝑡) [𝐴

𝑄𝐴0+ 𝐶

𝑄𝐶0] ] (𝑡)

+ ] (𝑡) [𝐴0𝑄𝐴 + 𝐶

0𝑄𝐶] 𝑥

(𝑡) + ] (𝑡) (2𝐴0𝑄𝐴0

+ 2𝐶

0𝑄𝐶0− 𝛾2

𝐼 +𝑀

𝑀) ] (𝑡)} = 𝐸𝑇

∑

𝑡=0

[

𝑥 (𝑡)

] (𝑡)]

⋅ Ξ [

𝑥 (𝑡)

] (𝑡)] ,

(41)

where

Ξ fl [Ξ11Ξ12

∗ Ξ22

] (42)

with

Ξ11= −𝑄 + 2𝐴

𝑄𝐴 + 2𝐶

𝑄𝐶 + 𝐿

𝐿 + 3𝛼2

1𝛼𝐻

1𝐻1

+ 3𝛼2

2𝛼𝐻

2𝐻2,

Ξ12= 𝐴

𝑄𝐴0+ 𝐶

𝑄𝐶0,

Ξ22= 2𝐴

0𝑄𝐴0+ 2𝐶

0𝑄𝐶0− 𝛾2

𝐼 +𝑀

𝑀.

(43)

Let 𝑇 → ∞ in (41); then we have

‖𝑧 (𝑡)‖2

𝑙2

𝑤(𝑁,𝑅𝑝)− 𝛾2

‖] (𝑡)‖2𝑙2

𝑤(𝑁,𝑅𝑞)

≤ 𝐸

∞

∑

𝑡=0

[

𝑥 (𝑡)

] (𝑡)]

Ξ[

𝑥 (𝑡)

] (𝑡)] .

(44)

Mathematical Problems in Engineering 7

It is easy to see that if Ξ < 0, then ‖𝑇‖ < 𝛾 for system (11).Next, we give an LMI sufficient condition for Ξ < 0. Noticethat

Ξ = [

Ξ11− 𝐴

𝑄𝐴 𝐶

𝑄𝐶0

∗ Ξ22− 𝐴

0𝑄𝐴0

]

+ [

𝐴

𝐴

0

]𝑄 [𝐴 + 𝐵𝐾 𝐴0] < 0

⇐⇒

[

[

[

[

Ξ11− 𝐴

𝑄𝐴 𝐶

𝑄𝐶0

𝐴

∗ Ξ22− 𝐴

0𝑄𝐴0𝐴0

∗ ∗ −𝑄−1

]

]

]

]

< 0

⇐⇒

[

[

[

[

[

[

[

Ξ̃110 𝐴

𝐶

∗ Ξ̃22𝐴0𝐶0

∗ ∗ −𝑄−1

0

∗ ∗ ∗ −𝑄−1

]

]

]

]

]

]

]

< 0,

(45)

where

Ξ̃11= Ξ11− 𝐴

𝑄𝐴 − 𝐶

𝑄𝐶,

Ξ̃22= Ξ22− 𝐴

0𝑄𝐴0− 𝐶

0𝑄𝐶0.

(46)

Using Lemma 5, Ξ < 0 is equivalent to

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

−𝑄 𝐿

𝛼2

1𝐻

1𝛼2

2𝐻

2𝐴

𝑄 𝐶

𝑄 0 0 0 0 𝐴

𝐶

∗ −𝐼 0 0 0 0 0 0 0 0 0 0

∗ ∗ −

1

3𝛼

𝐼 0 0 0 0 0 0 0 0 0

∗ ∗ ∗ −

1

3𝛼

𝐼 0 0 0 0 0 0 0 0

∗ ∗ ∗ ∗ −𝑄 0 0 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ −𝑄 0 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ −𝛾2

𝐼 𝐴0𝑄 𝐶0𝑄 𝑀

𝐴0𝐶0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝑄 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝑄 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝐼 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝑄−1

0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝑄−1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

< 0. (47)

It is obvious that seeking 𝐻∞

gain matrix 𝐾 needs to solveLMIs (47) and𝑄−𝛼𝐼 < 0. Setting𝑄−1 = 𝑋 and 𝛽 = 1/𝛼, andpre- and postmultiplying

diag [𝑋, 𝐼, 𝐼, 𝐼, 𝑋,𝑋, 𝐼, 𝑋,𝑋, 𝐼, 𝐼, 𝐼] (48)on both sides of (47) and considering (35), (34) is obtainedimmediately. The proof is completed.

4. Numerical Examples

This section presents two numerical examples to demonstratethe validity of our main results described above.

Example 1. Consider system (5) with parameters as

𝐴 = [

0.8 0.3

0.4 0.9

] ,

𝐶 = [

0.3 0.2

0.4 0.2

] ,

𝐻1= 𝐻2= [

1 0

0 1

] ,

𝐵 = 𝐷 = [

1

1

] ,

𝑥 (0) = [

𝑥1(0)

𝑥2(0)

] = [

50

−50

] .

(49)

For the unforced system (16) with 𝛼1= 𝛼2= 0.3, its

corresponding state locus diagram is made in Figure 1. FromFigure 1, it is easy to see that the status values are of seriousdivergences through 50 iterations.Hence the unforced systemis not stable.

To design a feedback controller such that the closed-loopsystem is stochastically stable, usingMatlab LMI Toolbox, we

8 Mathematical Problems in Engineeringx1andx2

5 10 15 20 25 30 35 40 45 500

t (sec)

−2000

0

2000

4000

6000

8000

10000

Figure 1: State trajectories of the autonomous system with 𝛼 = 0.3.

x1andx2

−50

−40

−30

−20

−10

0

10

20

30

40

50

2 4 6 8 10 12 14 16 18 200

t (sec)

Figure 2: State trajectories of the closed-loop system with 𝛼 = 0.3.

find that a symmetric, positive definitematrix𝑋, a realmatrix𝑌, and a scalar 𝛽 given by

𝑋 = [

3.318 0.177

0.177 1.920

] ,

𝑌 = [−1.587 −0.930] ,

𝛽 = 1.868

(50)

solve LMI (27). So we get the feedback gain 𝐾 =[−0.455 −0.443]. Submitting

𝑢 (𝑡) = [−0.455 −0.443] [

𝑥1(𝑡)

𝑥2(𝑡)

] (51)

into system (5), the state trajectories of the closed-loop systemare shown as in Figure 2.

From Figure 2, one can find that the controlled systemachieves stability using the proposed controller. Meanwhile,

x1andx2

5 10 15 20 25 30 35 40 45 500

t (sec)

−6000

−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Figure 3: State trajectories of the autonomous system with 𝛼 = 0.1.

in the case 𝛼 ≤ 0.3, the controlled system maintainsstabilization.

In order to show the robustness, we use different valuesof 𝛼 in Example 1 below. We reset 𝛼 = 0.1 and 𝛼 =1 in system (5) with the corresponding trajectories shownin Figures 3 and 4, respectively. By comparing Figures 1,3, and 4, intuitively speaking, the more value 𝛼 takes,the more divergence the autonomous system (5) has. Byusing Theorem 9, we can get that, under the conditionof 𝛼 = 0.1 and 𝛼 = 1, the corresponding controllersare 𝑢𝛼=0.1(𝑡) = [−0.4750 −0.4000] [

𝑥1(𝑡)

𝑥2(𝑡)] and 𝑢

𝛼=1(𝑡) =

[−0.4742 −0.4016] [𝑥1(𝑡)

𝑥2(𝑡)], respectively. Substitute 𝑢

𝛼=0.1(𝑡)

and 𝑢𝛼=1(𝑡) into system (5) in turn, and the corresponding

closed-loop system is shown in Figures 5 and 6, respectively.From the simulation results, we can see that the larger

the value 𝛼 takes, the slower the system converges. Thisobservation is reasonable, because the larger uncertainty thesystem has, the stronger the robustness of controller requires.

Example 2. Consider system (11) with parameters as

𝐶0= [

0.2 0

0 0.1

] ,

𝐴0= [

0.3 0

0 0.3

] ,

𝐿 = [

0.2 0

0 0.2

] ,

𝑀 = [

0.2 0

0 0.2

] ,

𝐻1= 𝐻2= [

0.8 0

0 0.8

] .

(52)

Mathematical Problems in Engineering 9

×104

−8

−6

−4

−2

0

2

4

6

8

10

x1andx2

5 10 15 20 25 30 35 40 45 500

t (sec)

Figure 4: State trajectories of the autonomous system with 𝛼 = 1.

−60

−40

−20

0

20

40

60

x1andx2

2 4 6 8 10 12 14 16 18 200

t (sec)

Figure 5: State trajectories of the closed-loop system with 𝛼 = 0.1.

x1andx2

t (sec)50454035302520151050

−500

−400

−300

−200

−100

0

100

200

300

400

500

Figure 6: State trajectories of the closed-loop system with 𝛼 = 1.

t (sec)302520151050

x1andx2

−50

−40

−30

−20

−10

0

10

20

30

40

50

Figure 7: State trajectories of the closed-loop system with 𝛼 = 0.3.

𝐴, 𝐵, 𝐶,𝐷 have the same values as in Example 1. Setting 𝐻∞

norm bound 𝛾 = 0.6 and 𝛼1= 𝛼2= 0.3, a group of the

solutions for LMI (34) are

𝑋 = [

55.257 3.206

3.206 31.482

] ,

𝑌 = [−26.412 −15.475] ,

𝛽 = 30.596,

(53)

and the𝐻∞

controller is

𝑢 (𝑡) = 𝐾𝑥 (𝑡) = 𝑌𝑋−1

𝑥 (𝑡)

= [−0.452 −0.446] [

𝑥1(𝑡)

𝑥2(𝑡)

] .

(54)

The simulation result about Theorem 10 is described inFigure 7.This further verifies the effectiveness ofTheorem 10.

In order to give a comparison with 𝛼 = 0.3, we set 𝛼 = 0.1and 𝛼 = 1. By using Theorem 10, the 𝐻

∞controller for 𝛼 =

0.1 is 𝑢(𝑡) = [−0.4737 −0.4076] [ 𝑥1(𝑡)𝑥2(𝑡)] with the simulation

result given in Figure 8. When 𝛼 = 1, LMI (34) is infeasible;that is, in this case, system (11) is not𝐻

∞controllable.

5. Conclusion

This paper has discussed robust stability, stabilization, and𝐻∞

control of a class of nonlinear discrete time stochasticsystems with system state, control input, and external distur-bance dependent noise. Sufficient conditions for stochasticstability, stabilization, and robust𝐻

∞control law have been,

respectively, given in terms of LMIs. Two examples have alsobeen supplied to show the effectiveness of our main results.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

10 Mathematical Problems in Engineering

5 10 15 20 25 300

t (sec)

−50

−40

−30

−20

−10

0

10

20

30

40

50

x1andx2

Figure 8: State trajectories of the closed-loop system with 𝛼 = 0.1.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (no. 61573227), the Research Fundfor the Taishan Scholar Project of Shandong Province ofChina, theNatural Science Foundation of Shandong Provinceof China (no. 2013ZRE28089), SDUST Research Fund (no.2015TDJH105), State Key Laboratory of Alternate Electri-cal Power System with Renewable Energy Sources (no.LAPS16011), and the Postgraduate Innovation Funds of ChinaUniversity of Petroleum (East China) (nos. YCX2015048 andYCX2015051).

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