Coefficient Matrix Decomposition Method and BIBO Stabilization of Stochastic Systems with Time Delays

Abstract

The mean square BIBO stabilization is investigated for the stochastic control systems with time delays and nonlinear perturbations. A class of suitable Lyapunov functional is constructed, combined with the descriptor model transformation and the decomposition technique of coefficient matrix; thus some novel delay-dependent mean square BIBO stabilization conditions are derived. These conditions are expressed in the forms of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Finally, three numerical examples are given to demonstrate that the derived conditions are effective and much less conservative than those given in the literature.

1. Introduction

Because of the finite switching speed, memory effects, and so on, time delay is unavoidable in technology and nature, which commonly exists in various mechanical, chemical engineering, physical, biological, and economic systems. It can make the concerned control systems become of poor performance and unstable, which leads to the difficulty of hardware implementation of the control system. Thus, the stability of time-delay systems has been widely investigated. See [1–7] and some references therein. For systems with small delay, a model transformation technique is often used to transform the system with discrete delay into a system with distributed delay; the advantage of this transformation is to transform the original system to an equivalent descriptor from representation and additional dynamics in the systems will not be introduced. The delay-dependent stabilization criteria obtained by the coefficient matrix decomposition method are usually less conservative than some existing ones. Please refer to [3]. In recent years, bounded-input bounded-output (BIBO) stabilization has been investigated by many researchers in order to track out the reference input signal in real world; see [5–22] and some references therein. In [15, 16], the sufficient condition for BIBO stabilization of control systems with no delays is proposed by the Bihari-type inequality. In [6, 7], the BIBO stabilization of the systems without distributed time delays was investigated by employing the parameters technique and the Gronwall inequality. In [17–19], some BIBO stabilization criteria for a class of delayed control systems with nonlinear perturbations were established, based on Riccati equations, by constructing appropriate Lyapunov functions. In [20], the BIBO stabilization problem of a class of piecewise switched linear systems was further investigated.

However, up to now, these previous results have been assumed to be in deterministic systems, including continuous time deterministic systems and discrete time deterministic systems, but seldom in stochastic systems (see [21, 22]; in [21] Fu and Liao got several mean square BIBO stabilization criteria in terms of Razumikhin technique and comparison principle. In [22], Zhou and Zhong discussed the mean square BIBO stabilization of the stochastic delay system with nonlinear perturbations by auxiliary algebraic Riccati matrix equations). In practice, stochastic control systems are more applicable to problems that are environmentally noisy in nature or related to biological realities. Thus, the BIBO stabilization analysis problems for stochastic control case are necessary.

Motivated by the previous discussions, this paper mainly aims to study the BIBO stabilization in mean square for the stochastic control systems with time delays and nonlinear perturbations. Based on the descriptor model transformation and the decomposition technique of coefficient matrix, some sufficient conditions guaranteeing BIBO stabilization in mean square are obtained. Finally, three numerical examples provided to demonstrate the derived conditions are valid and much less conservative than those given in the literature.

Notations 1. The notations are quite standard. Throughout this letter, 𝑅^𝑛 and 𝑅^𝑛×𝑚 denote, respectively, the 𝑛-dimensioned Euclidean space and the set of all 𝑛×𝑚 real matrices. The superscript “𝑇” denotes the transpose and the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that 𝑋 and 𝑌 are symmetric matrices and that 𝑋−𝑌 is positive semidefinitive (respective positive definite). ‖⋅‖ is the Euclidean norm in 𝑅^𝑛. 𝐼 is the identity matrix with compatible dimension. 𝐴 is a matrix, denoted by ‖𝐴‖ as its operator norm; that is, ‖𝐴‖=sup{‖𝐴𝑥‖∶‖𝑥‖=1}=𝜆_max(𝐴^𝑇𝐴), where 𝜆_max(𝐴) (resp., 𝜆_min(𝐴)) means the largest (resp., smallest) eigenvalue of 𝐴. Moreover, let (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃) be a complete probability space with a filtration {𝐹_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., the filtration contains all 𝑃-null sets and is right continuous). 𝔼{⋅} stands for the mathematical expectation operator with respect to the given probability measure 𝑃. The asterisk ∗ in a matrix is used to denote term that is induced by symmetry. Matrices, if not explicitly specified, are assumed to have compatible dimensions. Sometimes, the arguments of function will be omitted in the analysis when no confusion can arise.

2. Problem Formulation and Preliminaries

Consider the stochastic control system described by the following equation: 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥𝑡−𝜏₁+𝑓𝑡,𝑥(𝑡),𝑥𝑡−𝜏₁(𝑡+(𝑡)+𝐶𝑢)𝑑𝑡𝐺𝑥+𝐻𝑥𝑡−𝜏₂𝑑𝐰(𝑡),𝑡≥𝑡₀𝑦(𝑡)(𝑡),𝑥(𝜃)(𝜃)≥0,=𝐷𝑥=𝜑∈𝐶^𝑏_ℱ0𝑡₀−𝜏,𝑡₀;𝑅^𝑛,𝜃∈𝑡₀−𝜏,𝑡₀, (1) where 𝑥(𝑡),𝑢(𝑡), and 𝑦(𝑡) are the state vector, control input, and control output of the system, respectively. 𝜏₁>0,𝜏₂>0 are discrete time delays, and 𝜏=max{𝜏₁,𝜏₂}. 𝐴,𝐵,𝐶,𝐷,𝐺, and 𝐻 are constant matrices with appropriate dimensional, 𝐰(𝑡)=(𝐰₁(𝑡),𝐰₂(𝑡),…,𝐰_𝑛(𝑡))^𝑇 is an 𝑛-dimensional standard Brownian motion defined on a complete probability space (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃) with a natural filtration {𝐹_𝑡}_𝑡≥0, and 𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏₁))∈𝐶(𝑅⁺×𝑅^𝑛×𝑅^𝑛,𝑅^𝑛) is the nonlinear vector-valued perturbation bounded in magnitude as 𝑓𝑡,𝑥(𝑡),𝑥𝑡−𝜏₁²≤𝛼₁‖𝑥(𝑡)‖²+𝛼₂𝑥𝑡−𝜏₁², (2) where 𝛼₁, 𝛼₂ are known positive constants.

To obtain the control law described by (1) and to track out the reference input of the system, we let the controller be in the form of 𝑢(𝑡)=𝐾𝑥(𝑡)+𝑟(𝑡), (3) where 𝐾 is the feedback gain matrix and 𝑟(𝑡) is the reference inputs.

At the end of this section, let us introduce some important definitions and lemmas which will be used in the sequel.

Definition 1 (see [21]). A vector function 𝑟(𝑡)=(𝑟₁(𝑡),𝑟₂(𝑡),…,𝑟_𝑛(𝑡))^𝑇 is said to be an element of 𝐿^𝑛_∞, if ‖𝑟‖_∞=sup_{𝑡∈[𝑡0,+∞)}‖𝑟(𝑡)‖<+∞, where ‖⋅‖ denotes the Euclid norm in 𝑅^𝑛, or the norm of a matrix.

Definition 2 (see [21]). The nonlinear stochastic control system (1) is mean square BIBO stabilization, if one can construct controller (3) such that the output 𝑦(𝑡) satisfies 𝔼𝑦(𝑡)²≤𝑁₁+𝑁₂‖𝑟‖²_∞, (4) where 𝑁₁,𝑁₂ are positive constants.

Definition 3 (see [21], ℒ operator). Let Lyapunov functionals 𝑉∶𝐶([−𝜏,0];𝑅^𝑛)×𝑅₊→𝑅; its infinitesimal operator ℒ, acting on functional 𝑉, is defined by ℒ𝑉𝑥_𝑡,𝑡=lim_Δ→0+1supΔ𝔼𝑉𝑥_𝑡+Δ,𝑡+Δ−𝑉𝑥_𝑡,𝑡. (5)

Lemma 4 (see [23]). For any constant symmetric matrix 𝑀∈𝑅^𝑛×𝑛, 𝑀=𝑀^𝑇>0, scalar 𝑟>0, and vector function 𝑔∶[0,𝑟]→𝑅^𝑛, such that the integrations in the following are well defined, and then 𝑟^𝑟₀𝑔^𝑇(𝑠)𝑀𝑔(𝑠)𝑑𝑠≥^𝑟₀𝑔(𝑠)𝑑𝑠^𝑇𝑀^𝑟₀𝑔(𝑠)𝑑𝑠. (6)

Lemma 5 (see [24]). Let 𝑥,𝑦∈𝑅^𝑛 and any 𝑛×𝑛 positive-definite matrix 𝑄>0. Then, one has 2𝑥^𝑇𝑦≤𝑥^𝑇𝑄⁻¹𝑥+𝑦^𝑇𝑄𝑦. (7)

3. BIBO Stabilization for Nonlinear Stochastic Systems

To derive delay-dependent mean square stabilization conditions, which include the information of the time delay 𝜏₁, one usually uses the fact 𝑥𝑡−𝜏₁=𝑥(𝑡)−^𝑡_𝑡−𝜏1𝑥̇(𝑠)𝑑𝑠 (8) to transform the original system to a system with distributed delays. Let us decompose the coefficient matrix 𝐵 as 𝐵=𝐵₁+𝐵₂, where 𝐵₁,𝐵₂ are constant matrices. Then the original system (1) can be represented in the form of the descriptor system with discrete and distributed delays: 𝑑𝑥(𝑡)+𝐵₂^𝑡_𝑡−𝜏1𝑥(𝑠)=𝑑𝑠𝐴+𝐵₂+𝐶𝐾𝑥(𝑡)+𝐵₁𝑥𝑡−𝜏₁+𝐶𝑟(𝑡)+𝑓𝑡,𝑥(𝑡),𝑥𝑡−𝜏₁+(𝑡)𝑑𝑡𝐺𝑥+𝐻𝑥𝑡−𝜏₂𝑑𝐰(𝑡),𝑡≥𝑡₀𝑦(𝑡)(𝑡),𝑥(𝜃)(𝜃)≥0,=𝐷𝑥=𝜑∈𝐶^𝑏_ℱ0𝑡₀−𝜏,𝑡₀;𝑅^𝑛,𝜃∈𝑡₀−𝜏,𝑡₀. (9)

Letting 𝐷(𝑥_𝑡) be a new operator, we have 𝐷𝑥_𝑡=𝑥(𝑡)+𝐵₂^𝑡_𝑡−𝜏1𝑥(𝑠)𝑑𝑠. (10) To guarantee that the difference operator 𝐷(𝑥_𝑡)∶𝐶[−𝜏,0]→𝑅^𝑛 given by (10) is stable, we assume the following [25]:A1:let 𝜏₁‖𝐵₂‖<1, where ‖⋅‖ is any matrix norm.

For the mean square BIBO stabilization of the system described by (9) and (3), we have the following results.

Theorem 6. For any given positive constants 𝛽_𝑖>0,𝑖=1,2,3,4,5, the nonlinear stochastic control system (1) with the controller (3) is mean square BIBO stabilization, if there exist symmetric positive-definite matrices 𝑃, 𝑅₁, 𝑅₂, 𝑅₃, and 𝑋, such that 𝛽₅𝜆_min(𝑃)−𝛽₁𝜆_max𝑅₁−𝛽₂𝜆_max𝑅₂−𝛽₄𝛼₁+𝛼₂−𝜏²₁𝜆_max𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂>0, (11) and the linear matrix inequality ()Ξ=1,1𝑃𝐵₁𝐺^𝑇()𝑃∗()()𝑃𝐻1,42,2000∗∗3,3−𝐻^𝑇𝑃𝐺𝐵₂0∗∗∗(4,4)0∗∗∗∗−𝛽₄𝐼≤0, (12) with (1,1)=𝑃𝐴+𝐴^𝑇𝑃+𝑃𝐵₂+𝐵^𝑇₂𝑃+𝐺^𝑇𝑃𝐺−𝑋−𝑋^𝑇+𝛽₅()𝑃+𝑃,1,4=−𝑃𝐴𝐵₂−𝑃𝐵²₂−𝐺^𝑇𝑃𝐺𝐵₂+𝑋𝐵₂−𝛽₅𝑃𝐵₂,(2,2)=−𝛽₁𝑅₁,(3,3)=𝐻^𝑇𝑃𝐻−𝛽₂𝑅₂,(4,4)=𝐵^𝑇₂𝐺^𝑇𝑃𝐺𝐵₂−𝛽₃𝑅₃. (13)

Proof. We define a Lyapunov functional 𝑉(𝑡,𝑥_𝑡) as 𝑉𝑡,𝑥_𝑡=𝑉₁𝑡,𝑥_𝑡+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡), (14) where 𝑉₁𝑡,𝑥_𝑡=𝐷^𝑇𝑥_𝑡𝑃𝐷𝑥_𝑡,𝑉₂(𝑡)=^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝛽₁𝑅₁+𝛽₄𝛼₂𝑥(𝑠)𝑉𝑑𝑠,₃(𝑡)=𝛽₂^𝑡_𝑡−𝜏2𝑥^𝑇(𝑠)𝑅₂𝑥(𝑠)𝑉𝑑𝑠,₄(𝑡)=𝜏₁^𝑡_𝑡−𝜏1𝑠−𝑡+𝜏₁𝑥^𝑇(𝑠)𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂𝑥(𝑠)𝑑𝑠. (15) Taking the operator ℒ of 𝑉₁(𝑡,𝑥_𝑡) along the trajectory of system (1), we have ℒ𝑉₁𝑡,𝑥_𝑡=𝐷^𝑇𝑥_𝑡𝑃𝐴+𝐵₂+𝐶𝐾𝑥(𝑡)+𝐵₁𝑥𝑡−𝜏₁+𝑓𝑡,𝑥(𝑡),𝑥𝑡−𝜏₁(𝑡++𝐶𝑟)𝐴+𝐵₂+𝐶𝐾𝑥(𝑡)+𝐵₁𝑥𝑡−𝜏₁(𝑡)(𝑡)+𝐶𝑟+𝑓𝑡,𝑥,𝑥𝑡−𝜏₁^𝑇𝑃𝐷𝑥_𝑡+12trace𝐺𝑥(𝑡)+𝐻𝑥𝑡−𝜏₂^𝑇×2𝑃𝐺𝑥(𝑡)+𝐻𝑥𝑡−𝜏₂≤𝜉(𝑡)^𝑇Ξ𝜉(𝑡)+𝛽₁𝑥^𝑇𝑡−𝜏₁𝑅₁𝑥𝑡−𝜏₁+𝐷^𝑇𝑥_𝑡𝑃𝐶𝑟(𝑡)+𝛽₂𝑥^𝑇𝑡−𝜏₂𝑅₂𝑥𝑡−𝜏₂−𝛽₅𝐷^𝑇𝑥_𝑡𝑃𝐷𝑥_𝑡+𝛽₃^𝑡_𝑡−𝜏1𝑥(𝑠)𝑑𝑠^𝑇𝑅₃^𝑡_𝑡−𝜏1𝑥(𝑠)𝑑𝑠+𝑟^𝑇(𝑡)𝐶^𝑇𝑃𝐷𝑥_𝑡+𝛽₄𝑓^𝑇𝑡,𝑥(𝑡),𝑥𝑡−𝜏₁×𝑓𝑡,𝑥(𝑡),𝑥𝑡−𝜏₁, (16) where 𝜉(𝑡)^𝑇=𝐷^𝑇𝑥_𝑡,𝑥^𝑇𝑡−𝜏₁,𝑥^𝑇𝑡−𝜏₂,_{𝑡𝑡−𝜏1}𝑥(𝑠)𝑑𝑠^𝑇,𝑓^𝑇𝑥,𝑥(𝑡),𝑥𝑡−𝜏₁,1Ξ=1,𝑃𝐵₁𝐺^𝑇𝑃𝐻1,4𝑃∗−𝛽₁𝑅₁000∗∗𝐻^𝑇𝑃𝐻−𝛽₂𝑅₂−𝐻^𝑇𝑃𝐺𝐵₂0∗∗∗𝐵^𝑇₂𝐺^𝑇𝑃𝐺𝐵₂−𝛽₃𝑅₃0∗∗∗∗−𝛽₄𝐼≤0, (17)(18) with 1,1=𝑃𝐴+𝐴^𝑇𝑃+𝑃𝐵₂+𝐵^𝑇₂𝑃+𝐺^𝑇𝑃𝐺−𝑋−𝑋^𝑇+𝛽₅4𝑃,1,=−𝑃𝐴𝐵₂−𝑃𝐵²₂−𝐺^𝑇𝑃𝐺𝐵₂+𝑋𝐵₂. (19)
By Lemmas 4 and 5, (2), and Definition 1 we conclude that ℒ𝑉₁𝑡,𝑥_𝑡≤𝜉(𝑡)^𝑇Ξ𝜉(𝑡)+𝑥^𝑇𝑡−𝜏₁𝛽₁𝑅₁+𝛽₄𝛼₂𝑥𝑡−𝜏₁+𝛽₂𝑥^𝑇𝑡−𝜏₂𝑅₂𝑥𝑡−𝜏₂+𝛽₄𝛼₁‖𝑥(𝑡)‖²−𝛽₅𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝜏₁^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂𝑥(𝑠)𝑑𝑠+𝐷^𝑇𝑥_𝑡𝑃𝐷𝑥_𝑡−𝛽₅𝐷^𝑇𝑥_𝑡𝑃𝐵₂^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝑑𝑠−𝛽₅^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇₂𝑃𝐷𝑥_𝑡+𝐶^𝑇𝑃𝐶‖𝑟‖²_∞. (20) Taking the operator ℒ of 𝑉_𝑖(𝑡), 𝑖=2,3,4, along the trajectory of system (1), we get ℒ𝑉₂(𝑡)=𝑥^𝑇(𝑡)𝛽₁𝑅₁+𝛽₄𝛼₂𝑥(𝑡)−𝑥^𝑇𝑡−𝜏₁𝛽₁𝑅₁+𝛽₄𝛼₂𝑥𝑡−𝜏₁,ℒ𝑉₃(𝑡)=𝛽₂𝑥^𝑇(𝑡)𝑅₂𝑥(𝑡)−𝛽₂𝑥^𝑇𝑡−𝜏₂𝑅₂𝑥𝑡−𝜏₂,ℒ𝑉₄(𝑡)=𝜏²₁𝑥^𝑇(𝑡)𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂𝑥(𝑡)−𝜏₁^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂𝑥(𝑠)𝑑𝑠. (21)(22) Combining (20) with (22), we have ℒ𝑉𝑡,𝑥_𝑡≤𝜉(𝑡)^𝑇Ξ𝜉(𝑡)−𝛽₅𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝛽₁𝑅₁+𝛽₂𝑅₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝛽₄𝛼₁+𝛽₄𝛼₂𝑥(𝑡)+𝐶^𝑇𝑃𝐶‖𝑟‖²_∞+𝜏²₁𝑥^𝑇(𝑡)𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂𝑥(𝑡)≤𝜉(𝑡)^𝑇(𝑡)−𝛽Ξ𝜉₅𝜆_min(𝑃)−𝛽₁𝜆_max𝑅₁−𝛽₂𝜆_max𝑅₂−𝛽₄𝛼₁+𝛼₂−𝜏²₁𝜆_max𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂‖𝑥(𝑡)‖²+𝐶^𝑇𝑃𝐶‖𝑟‖²_∞. (23) Let 𝑎=𝛽₅𝜆_min(𝑃)−𝛽₁𝜆_max(𝑅₁)−𝛽₂𝜆_max(𝑅₂)−𝛽₄(𝛼₁+𝛼₂)−𝜏²₁𝜆_max(𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂). If (11) and LMI (12) hold, we have ℒ𝑉𝑡,𝑥_𝑡≤−𝑎‖𝑥(𝑡)‖²+𝐶^𝑇𝑃𝐶‖𝑟‖²_∞. (24) Under an assumption that 𝑉(𝑡,𝑥_𝑡)≤𝑉(𝑡₀,𝑥_𝑡0) for all 𝑡≥𝑡₀, then 𝜆_min(𝑃)𝔼𝑥(𝑡)+𝐵₂^𝑡_𝑡−𝜏1𝑥(𝑠)𝑑𝑠²≤𝑉𝑡,𝑥_𝑡≤𝑉𝑡₀,𝑥_𝑡0≤𝜌𝔼𝜑(𝜃)², (25) where 𝜌=𝜆_max(𝑃) + 𝜏₁(‖𝐵^𝑇₂𝑃‖+‖𝑃𝐵₂‖)+𝜏²₁‖𝐵^𝑇₂𝑃𝐵₂‖² + 𝜏₁𝛽₁𝜆_max(𝑅₁) + 𝜏₁𝛽₄𝛼₂+𝜏₂𝛽₂𝜆_max(𝑅₂) + (𝜏³₁/2)𝜆_max(𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂).
Thus, according to Theorem 1.3 in page 331 of [26], it can easily be gotten that 𝔼‖𝑥(𝑡)‖²≤1+𝜏₁𝐵₂1−𝜏₁𝐵₂²𝜌𝔼𝜑(𝜃)²𝜆_min(𝑃). (26) If not, there exist 𝑡>𝑡₀, such that 𝑉(𝑡,𝑥(𝑡))≥𝑉(𝑠,𝑥(𝑠)) for all 𝑠∈[𝑡₀,𝑡), and we get 𝐷⁺𝔼𝑉(𝑡,𝑥(𝑡))≥0. (27) In view of Ito’s formula, we obtain 𝐷⁺𝔼𝑉(𝑡,𝑥(𝑡))=𝔼ℒ𝑉(𝑡,𝑥(𝑡)). (28) By (24), (27), and (28), we get 0≤𝐷⁺((𝑡((𝑡‖𝑥(𝑡𝔼𝑉𝑡,𝑥))=𝔼ℒ𝑉𝑡,𝑥))≤−𝑎𝔼)‖²+𝐶^𝑇𝑃𝐶‖𝑟‖²_∞. (29) So, 𝔼‖𝑥(𝑡)‖²≤𝐶^𝑇𝑃𝐶𝑎‖𝑟‖²_∞. (30) By (26) and (30), we get 𝔼‖𝑥(𝑡)‖²≤1+𝜏₁𝐵₂1−𝜏₁𝐵₂²𝜌𝔼𝜑(𝜃)²𝜆_min(𝑃)+𝐶^𝑇𝑃𝐶𝑎‖𝑟‖²_∞. (31) Thus 𝔼𝑦(𝑡)²≤‖𝐷‖²𝔼‖𝑥(𝑡)‖²≤1+𝜏₁𝐵₂1−𝜏₁𝐵₂²‖𝐷‖²𝜌𝔼𝜑(𝜃)²𝜆_min(𝑃)+‖𝐷‖²𝐶^𝑇𝑃𝐶𝑎‖𝑟‖²_∞=𝑁₁+𝑁₂‖𝑟‖²_∞, (32) where 𝑁₁=1+𝜏₁𝐵₂1−𝜏₁𝐵₂²‖𝐷‖²𝜌𝔼𝜑(𝜃)²𝜆_min(𝑃),𝑁₂=‖𝐷‖²𝐶^𝑇𝑃𝐶𝑎. (33) By Definition 2, the nonlinear stochastic control system (1) is mean square BIBO stabilization. This completes the proof.

Theorem 7. For any given positive integer 𝛿_𝑖>0, 𝑖=1,2,3,4, the nonlinear stochastic control system (1) with controller (3) is mean square BIBO stabilization, if there exist symmetric positive-definite matrices 𝑃, 𝑄 and some positive constants 𝜂, 𝛿₅, and 𝛿₆ such that 𝐵₂𝑃=𝑃𝐵₂,𝛿₁−𝛿₂−𝛿₃𝛿>0,₁−𝛿₂−𝛿₃𝜆_min(𝑃)−𝜏²₁𝜆_max𝛿₄𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂−𝜆_max𝐺^𝑇𝑃𝐺−𝜆_max𝑄⁻¹−𝜆_max𝐻^𝑇𝑃𝐺𝑄𝐺^𝑇𝑃𝐻−𝜆_max𝐻^𝑇𝑃𝐻−𝛿₅𝛼²>0, (34)(35) and the linear matrix inequality 1Ω=^∗,1_∗𝐵₁𝑆01_∗,4_∗𝐶𝐼∗−𝛿₂𝑆0000∗∗−𝛿₃𝑆000∗∗∗−𝛿₄𝑆00∗∗∗∗−𝛿₅𝐼0∗∗∗∗∗−𝛿₆𝐼≤0 (36) holds, where 𝑆=𝑃⁻¹,1_∗,1_∗=𝐴𝑆+𝑆𝐴^𝑇+𝐵₂𝑆+𝑆𝐵^𝑇₂−2𝜂𝐶𝐶^𝑇+𝛿₁1𝑆,_∗,4_∗=−𝐴𝐵₂𝑆−𝐵₂𝐵₂𝑆+𝜂𝐶𝐶^𝑇𝐵₂−𝛿₁𝐵₂𝑆. (37)

Proof. We define a Lyapunov functional 𝑉(𝑡,𝑥_𝑡) as 𝑉𝑡,𝑥_𝑡=𝑉₁𝑡,𝑥_𝑡+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡), (38) where 𝑉₁𝑡,𝑥_𝑡=𝐷^𝑇𝑥_𝑡𝑃𝐷𝑥_𝑡,𝑉₂(𝑡)=𝛿₂^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝑅₁𝑥(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝑡_𝑡−𝜏2𝑥^𝑇(𝑠)𝛿₃𝑃+𝐻^𝑇𝑃𝐺𝑄𝐺^𝑇𝑃𝐻+𝐻^𝑇𝑥(𝑠)𝑉𝑃𝐻𝑑𝑠,₄(𝑡)=𝜏₁^𝑡_𝑡−𝜏1𝑠−𝑡+𝜏₁𝑥^𝑇(𝑠)𝛿₄𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂𝑥(𝑠)𝑑𝑠. (39) Taking the operator ℒ of 𝑉₁(𝑡,𝑥_𝑡) along the trajectory of system (1), by (2), (34), and Definition 1, we have ℒ𝑉₁𝑡,𝑥_𝑡=𝐷^𝑇𝑥_𝑡𝑃𝐴+𝐵₂+𝐶𝐾𝑥(𝑡)+𝐵₁𝑥𝑡−𝜏₁((𝑡(𝑡++𝑓𝑡,𝑥))+𝐶𝑟)𝐴+𝐵₂+𝐶𝐾𝑥(𝑡)+𝐵₁𝑥(𝑡−𝜏₁)+𝐶𝑟(𝑡)+𝑓(𝑡,𝑥(𝑡))^𝑇𝑃𝐷𝑥_𝑡+12trace𝐺𝑥(𝑡)+𝐻𝑥𝑡−𝜏₂^𝑇×2𝑃𝐺𝑥(𝑡)+𝐻𝑥𝑡−𝜏₂≤𝜉(𝑡)^𝑇Ω𝜉(𝑡)−𝛿₁𝐷^𝑇𝑥_𝑡𝑃𝐷𝑥_𝑡+𝛿₂𝑥^𝑇𝑡−𝜏₁𝑃𝑥𝑡−𝜏₁+𝛿₃𝑥^𝑇𝑡−𝜏₂𝑃𝑥𝑡−𝜏₂+𝑥𝑡−𝜏₂𝐻^𝑇𝑃𝐻𝑥𝑡−𝜏₂+𝛿₄^𝑡_𝑡−𝜏1𝑥(𝑠)𝑑𝑠^𝑇𝑃^𝑡_𝑡−𝜏1𝑥(𝑠)𝑑𝑠+𝑥𝑡−𝜏₂𝐻^𝑇𝑃𝐺𝑥(𝑡)+𝑥^𝑇(𝑡)𝐺^𝑇𝑃𝐺𝑥(𝑡)+𝑥^𝑇(𝑡)𝐺^𝑇𝑃𝐻𝑥𝑡−𝜏₂+𝛿₅𝛼²‖𝑥(𝑡)‖²+𝛽₆‖𝑟‖²_∞+𝛿₁𝐷^𝑇𝑥_𝑡𝑃𝐵₂^𝑡_𝑡−𝜏1𝑥(𝑠)𝑑𝑠+𝛿₁^𝑡_𝑡−𝜏1𝑥(𝑠)𝑑𝑠^𝑇𝐵^𝑇₂𝑃𝐷𝑥_𝑡, (40) where 𝜉(𝑡)^𝑇=𝐷^𝑇𝑥_𝑡,𝑥^𝑇𝑡−𝜏₁,𝑥^𝑇𝑡−𝜏₂,_{𝑡𝑡−𝜏1}𝑥(𝑠)𝑑𝑠^𝑇,𝑓^𝑇(𝑥,𝑥(𝑡)),𝑟^𝑇(𝑡,1)Ω=^∗,1^∗𝛿₁𝑃01^∗,4^∗𝑃𝐶𝑃∗−𝛿₂𝑃0000∗∗−𝛿₃𝑃000∗∗∗−𝛿₄𝑃00∗∗∗∗−𝛿₅𝐼0∗∗∗∗∗−𝛿₆𝐼, (41) with 1^∗,1^∗=𝑃𝐴+𝐵₂+𝐴+𝐵₂^𝑇𝑃+𝑃𝐵₁−2𝜂𝑃𝐶^𝑇1𝐶𝑃,_∗,4^∗=−𝑃𝐴𝐵₂−𝑃𝐵²₂+𝜂𝑃𝐶𝐶^𝑇𝐵₂𝑃−𝛿₁𝑃𝐵₂. (42) Pre- and postmultiplying (36) by diag[𝑃,𝑃,𝑃,𝑃,𝐼,𝐼], we can obtain Ω≤0. (43) Then, by Lemmas 4 and 5, (40), and (43), we conclude that ℒ𝑉₁𝑡,𝑥_𝑡≤−𝛿₁𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝛿₂𝑥^𝑇𝑡−𝜏₁𝑃𝑥𝑡−𝜏₁+𝛿₆‖𝑟‖²_∞+𝛿₃𝑥^𝑇𝑡−𝜏₂𝑃𝑥𝑡−𝜏₂+𝑥^𝑇(𝑡)𝐺^𝑇𝑃𝐺𝑥(𝑡)+𝜏₁^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝛿₄𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂𝑥(𝑠)𝑑𝑠+𝑥^𝑇𝑡−𝜏₂𝐻^𝑇𝑃𝐺𝑄𝐺^𝑇𝑃𝐻𝑥𝑡−𝜏₂+𝑥^𝑇(𝑡)𝑄⁻¹𝑥(𝑡)+𝑥^𝑇𝑡−𝜏₂𝐻^𝑇𝑃𝐻𝑥𝑡−𝜏₂+𝛿₅𝛼²‖𝑥(𝑡)‖². (44) Taking the operator ℒ of 𝑉_𝑖(𝑡),𝑖=2,3,4, along the trajectory of system (1), we get ℒ𝑉₂(𝑡)=𝛿₂𝑥^𝑇(𝑡)𝑃𝑥(𝑡)−𝛿₂𝑥^𝑇𝑡−𝜏₁𝑃𝑥𝑡−𝜏₁,ℒ𝑉₃(𝑡)=𝑥^𝑇(𝑡)𝛿₃𝑃+𝐻^𝑇𝑃𝐺𝑄𝐺^𝑇𝑃𝐻+𝐻^𝑇𝑃𝐻𝑥(𝑡)−𝑥^𝑇𝑡−𝜏₂𝛿₃𝑃+𝐻^𝑇𝑃𝐺𝑄𝐺^𝑇𝑃𝐻+𝐻^𝑇𝑃𝐻×𝑥𝑡−𝜏₂,ℒ𝑉₄(𝑡)=𝜏²₁𝑥^𝑇(𝑡)𝛿₄𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂𝑥(𝑡)−𝜏₁^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝛿₄𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂𝑥(𝑠)𝑑𝑠. (45)(46) Combining (44) with (46), we have ℒ𝑉(𝑡,𝑥(𝑡))≤−𝛿₁−𝛿₂−𝛿₃𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝛽₅𝛼²‖𝑥(𝑡)‖²+𝛽₆‖𝑟‖²_∞+𝜏²₁𝑥^𝑇(𝑡)𝛿₄𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝑄⁻¹𝑥(𝑡)+𝑥^𝑇(𝑡)𝐻^𝑇𝑃𝐻𝑥(𝑡)+𝑥^𝑇(𝑡)𝐺^𝑇𝑃𝐺𝑥(𝑡)+𝑥^𝑇(𝑡)𝐻^𝑇𝑃𝐺𝑄𝐺^𝑇(𝑡)𝛿𝑃𝐻𝑥≤−₁−𝛿₂−𝛿₃𝜆_min(𝑃)−𝜏²₁𝜆_max𝛿₄𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂−𝛽₅𝛼²−𝜆_max𝐻^𝑇𝑃𝐻−𝜆_max𝐺^𝑇𝑃𝐺−𝜆_max𝑄⁻¹−𝜆_max𝐻^𝑇𝑃𝐺𝑄𝐺^𝑇×‖‖𝑃𝐻𝑥(𝑡)²+𝛽₆‖𝑟‖²_∞. (47) Let 𝑏=(𝛿₁−𝛿₂−𝛿₃)𝜆_min(𝑃) − 𝜏²₁𝜆_max(𝛿₄𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂) − 𝛽₅𝛼² − 𝜆_max(𝐻^𝑇𝑃𝐻) − 𝜆_max(𝐺^𝑇𝑃𝐺) − 𝜆_max(𝑄⁻¹) − 𝜆_max(𝐻^𝑇𝑃𝐺𝑄𝐺^𝑇𝑃𝐻); we have ℒ𝑉(𝑡,𝑥(𝑡))≤−𝑏‖𝑥(𝑡)‖²+𝛽₆‖𝑟‖²_∞. (48) The rest of the proof is essentially the same as Theorem 6 and hence is omitted. This completes the proof.

If the stochastic term disappears, the control system (9) reduces to 𝑑𝑥(𝑡)+𝐵₂^𝑡_𝑡−𝜏1𝑥(𝑠)=𝑑𝑠𝐴+𝐵₂+𝐶𝐾𝑥(𝑡)+𝐵₁𝑥𝑡−𝜏₁(𝑡)(𝑡)+𝐶𝑟+𝑓𝑡,𝑥,𝑥𝑡−𝜏₁𝑑𝑡𝑡≥𝑡₀𝑦(𝑡)(𝑡),𝑥(𝜃)(𝜃)≥0,=𝐷𝑥=𝜑∈𝐶^𝑏_ℱ0𝑡₀−𝜏,𝑡₀;𝑅^𝑛,𝜃∈𝑡₀−𝜏,𝑡₀. (49)

We have the following stabilization results.

Corollary 8. For any given positive constants 𝛽_𝑖>0,𝑖=1,2,3,4,5, the nonlinear stochastic control system (1) with the controller (3) is BIBO stabilization, if there exist symmetric positive-definite matrices 𝑃, 𝑅₁, 𝑅₂, 𝑅₃, and 𝑋, such that 𝛽₅𝜆_min(𝑃)−𝛽₁𝜆_max𝑅₁−𝛽₂𝜆_max𝑅₂−𝛽₄𝛼₁+𝛼₂−𝜏²₁𝜆_max𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂>0, (50) and the linear matrix inequality ()Ξ=1,1𝑃𝐵₁0()𝑃∗()()()01,42,2000∗∗3,300∗∗∗4,4∗∗∗∗−𝛽₄𝐼≤0, (51) with (1,1)=𝑃𝐴+𝐴^𝑇𝑃+𝑃𝐵₂+𝐵^𝑇₂𝑃−𝑋−𝑋^𝑇+𝛽₅()𝑃+𝑃,1,4=−𝑃𝐴𝐵₂−𝑃𝐵²₂+𝑋𝐵₂−𝛽₅𝑃𝐵₂,(2,2)=−𝛽₁𝑅₁,(3,3)=−𝛽₂𝑅₂,(4,4)=−𝛽₃𝑅₃. (52)

Corollary 9. For any given positive integer 𝛿_𝑖>0,𝑖=1,2,3, the nonlinear stochastic control system (49) with the controller (3) is BIBO stabilization, if there exist symmetric positive-definite matrix 𝑃>0 and some positive constants 𝜂>0, 𝛿₄>0, and 𝛿₅>0 such that 𝐵₂𝑃=𝑃𝐵₂,𝛿₁−𝛿₂𝛿>0,₁−𝛿₂𝜆_min(𝑃)−𝜏²₁𝜆_max𝛿₃𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂−𝛿₅𝛼²>0 (53)(54) and the linear matrix inequality 1Ω=^∗,1_∗𝐵₁𝑆1_∗,3_∗𝐶𝐼∗−𝛿₂𝑆000∗∗−𝛿₃𝑆00∗∗∗−𝛿₄𝐼0∗∗∗∗−𝛿₅𝐼≤0 (55) holds, where 𝑆=𝑃⁻¹,1_∗,1_∗=𝐴𝑆+𝑆𝐴^𝑇+𝐵₂𝑆+𝑆𝐵^𝑇₂−2𝜂𝐶𝐶^𝑇+𝛿₁1𝑆,_∗,3_∗=−𝐴𝐵₂𝑆−𝐵²₂𝑆+𝜂𝐶𝐶^𝑇𝐵₂−𝛿₁𝐵₂𝑆. (56)

4. Examples

In this section, Example 1 will be presented to show that the mean BIBO stabilization conditions in Theorem 6 are valid, Example 2 will be presented to show that the BIBO stabilization conditions in Corollary 8 are valid, and Example 3 will be presented to show that the derived conditions are much less conservative than those given in the literature [20].

Example 1. As a simple application of Theorem 6, consider the stochastic control system (1) with control law (3); the parameters are given by ,,𝐴=−211−3,𝐵=−200−1𝐶=4120,𝐺=0.05000.05𝐻=0.01000.01 (57)𝑓₁=[0.1sin(𝑡),0.1cos(𝑡)]^𝑇, 𝑓₂=[√0.2sin(𝑡),√0.2cos(𝑡)]^𝑇, 𝛽₁=0.01, 𝛽₂=0.01, 𝛽₃=0.01, 𝛽₄=1, 𝛽₅=4.8, 𝛼₁=0.01, and 𝛼₂=0.02. Let us decompose matrix 𝐵=𝐵₁+𝐵₂, where 𝐵₁=−0.50.10.2−0.5,𝐵₂=−1.5−0.1−0.2−0.5. (58) By using the Matlab LMI Toolbox, we solve LMI (12) and obtain the feasible solutions as follows: 𝑃=431.2920131.9598131.9598408.9760,𝑅₁=,𝑅5500.9−492.4−492.45151.3₂=3183.90.70.73183.7,𝑅₃=,,𝛽3423.6−251−2513991.8𝑋=695.8768469.2695469.2695801.3229₅𝜆_min(𝑃)−𝛽₁𝜆_max𝑅₁−𝛽₂𝜆_max𝑅₂−𝛽₄𝛼₁+𝛼₂−𝜏²₁𝜆_max𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂=2.0586>0. (59) From the above formula, we can see that condition (11) is met. The stabilizing feedback gain matrix is given by 𝐾=−𝐶⁻¹𝑃⁻¹𝑋=−0.3477−0.8922−0.00973.0268. (60) Meanwhile, we obtain the maximum value 𝜏_max=5.315. This example shows that the mean square BIBO stabilization conditions in Theorem 6 are valid.

Example 2. Let us consider the delayed control system (49) with parameters given by ,,𝐴=−411−4,𝐵=−200−1𝐶=2110 (61)𝛽₁=0.01, 𝛽₂=0.01, 𝛽₃=0.01, 𝛽₄=1, 𝛽₅=4.8, 𝛼₁=0.01, and 𝛼₂=0.02.
Let us decompose matrix 𝐵=𝐵₁+𝐵₂, where 𝐵₁=−0.50.10.2−0.5,𝐵₂=−1.5−0.1−0.2−0.5. (62) Now we use Corollary 8 in this paper to study the problem. By using the Matlab LMI Toolbox, we obtain the feasible solutions as follows: 𝑃=1.88691.75031.75033.1049,𝑅₁=,𝑅21.08321.91401.914024.2362₂=18.62400018.6240,𝑅₃=,,𝛽21.7687−5.7602−5.760225.2583𝑋=0.59181.00131.00134.1015₅𝜆_min(𝑃)−𝛽₁𝜆_max𝑅₁−𝛽₂𝜆_max𝑅₂−𝛽₄𝛼₁+𝛼₂−𝜏²₁𝜆_max𝛽₃𝑅₃+𝛽₅𝐵^𝑇₂𝑃𝐵₂=0.0043>0. (63) From the above formula, we can see that condition (50) is met. The stabilizing feedback gain matrix is given by 𝐾=−𝐶⁻¹𝑃⁻¹𝑋=−0.3054−2.14170.58045.7392. (64) Meanwhile, we obtain the maximum value 𝜏_max=0.48.

Example 3. For the convenience of comparison, let us consider a delayed control system (49) with parameters given by 𝐴=−411−4,𝐵=−200−1,𝐶=2110, (65)𝛿₁=0.8,𝛿₂=0.4, and 𝛿₃=0.5.
Now we use Corollary 9 in this paper to study the problem; let us decompose matrix 𝐵=𝐵₁+𝐵₂, where 𝐵₁=−1.9500−1.03,𝐵₂=−0.05000.03. (66) By using the Matlab LMI Toolbox, solving LMI (55), we can get 𝑃=14.84640.87400.87406.1426, (67) and 𝛿₄=1.0428, 𝛿₅=1.0428, and 𝜂=0.0683. In order to verify condition (54), we give the parameters as follows: 𝛿₁−𝛿₂𝜆_min(𝑃)−𝜏²₁𝜆_max𝛿₃𝑃+𝛿₁𝐵^𝑇₂𝑃𝐵₂−𝛿₅𝛼²=0.0145>0. (68) From the above formula, we can see that condition (54) is met. The stabilizing feedback gain matrix is given by 𝐾=−2.0867−0.5387−1.0135−0.0597. (69) Meanwhile, solving LMI (55), the maximum value of 𝜏_max for BIBO stabilization of system (49) is 𝜏_max=1.06. In [20], its 𝜏_max is 0.2960. The maximum value of 𝜏_max in this example in our paper is 358.11% larger than this in [20]. This example shows that the BIBO stabilization conditions in this paper are less conservative than these in [20]. The essential reasons why the stability conditions we have given are less conservative are that we decompose the coefficient matrix in a proper way.

5. Conclusions

The problem of the mean square BIBO stabilization for the stochastic control systems with delays and nonlinear perturbations is investigated. A class of suitable Lyapunov functional combined with the descriptor model transformation and the decomposition technique of coefficient matrix is constructed to derive some novel delay-dependent BIBO stabilization criteria. Numerical examples have shown that the derived conditions are valid and improvements over the existing results are significant.

Acknowledgments

The authors would like to thank the editor and the reviewers for their detailed comments and valuable suggestions which have led to a much improved paper. The work of Xia Zhou is supported by the National Natural Science Foundation of China (no. 11226140), the Anhui Provincial Colleges and Universities Natural Science Foundation (no. KJ2013Z267), and Fuyang Teachers College Natural Science Foundation (no. 2012FSKJ08). The work of Shouming Zhong is supported by National Basic Research Program of China (no. 2010CB732501).