Dissipative Delay-Feedback Control for Nonlinear Stochastic Systems with Time-Varying Delay

Abstract

The dissipative delay-feedback control problems for nonlinear stochastic delay systems (NSDSs) based on dissipativity analysis are studied in this paper. Based on the Lyapunov stability theory and stochastic analysis technique, both delay-independent and delay-dependent dissipativity criteria are established as linear matrix inequalities- (LMIs-) based feasibility tests. The obtained results in this paper for the nominal systems include the available results on 𝐻_∞ approach and passivity for stochastic delay systems as special cases. The delay-dependent feedback controller is designed by considering the relationship among the time-varying delay, its lower and upper bound, and its differential without ignoring any terms, which effectively reduces the conservative. A numerical example is given to illustrate the theoretical developments.

1. Introduction

The stochastic differential systems appear as a natural description of many observed phenomena of real world, which have been come to play an important role in many fields including population dynamics, macroeconomics, chemical reactor control, communication network, image processes, and mobile robot localization. Therefore, the stability and stabilizability of nonlinear stochastic differential systems affine in the control have been studied in the past years by means of the stochastic Lyapunov theory [1–4]. As we all know, time delay in a control loop is one of the main sources of instability, oscillation, and poor performance and naturally encountered in a number of engineering control problems and physical systems. Therefore, time-delay systems [5, 6] and stochastic time-delay systems [7–12] have attracted many researchers’ attention and have been extensively studied. The problems include stability analysis [5, 6], stabilization problems [7, 8], and robust controller design [9–12].

On the other hand, studying the dissipativity analysis and synthesis problems has a strong motivation due to their simplicity and effectiveness in dealing with robust and nonlinear systems. Since the notation of dissipative dynamical system was introduced by Willems [13], dissipative systems have been of particular interest to researchers in areas of systems, circuits, networks and control, and so forth. Moreover, passivity of a certain system in a feedback interconnected system will ensure the overall stability of that feedback system if uncertainties or nonlinearities can be characterized by a strict passive system. Hence, dissipative theory has wide ranging implications and applications in control theory. For instance, dissipativity was crucially used in the stability analysis of nonlinear system [14]; the theory of dissipative systems generalizes basic tools including the passivity theory, bounded real lemma, Kalman Yakubovich lemma, and the circle criterion [15]. Among the relevant topics are the dissipativity analysis and synthesis for time-delay systems [16, 17]. These results show that the dissipativity-based methods are highly effective in design the robust controller.

Due to what is above mentioned, we believe that time delay is often harmful factor of systems. However, time delay is also surprising since plenty of studies have shown that time delay can also benefit the control, such as time-delay control. As mentioned previously, many results have been published about the control of systems with state delays but without input delays, which is called memoryless controllers, or to more general, memoryless controllers only includes an instantaneous feedback term. The time-delay control is an approach which gives a small delay ℎ in the controller design, so as to reduce the effect of instability factor and exogenous disturbance. See, for example [18–20], and the references therein. Rather than adjusting control gains or identifying model parameters, its essential idea is to use past observations regarding both the control input and system response, which is an open problem now. In this paper, dissipative delay-feedback control problems for nonlinear stochastic systems with time-varying delay are studied based on dissipativity. The delay-dependent feedback controller is designed by considering the relationship among the time-varying delay, its lower and upper bound, and its differential without ignoring any terms, which effectively reduces the conservative.

2. Problems Statement and Preliminaries

In this paper, we consider the following nonlinear stochastic delayed systems (NSDSs) defined on a probability space (Ω,ℱ,𝒫): (𝑡)={𝐹((𝑡)((𝑡((𝑡(𝑡((𝑡)((𝑡(𝑡),𝑧(𝑡)(𝑡)d𝑥𝑡,𝑥,𝑥𝑡−𝜏)))+𝐵𝑢𝑡,𝑡−𝜏))+𝐷𝑣)}d𝑡+𝐺𝑡,𝑥,𝑥𝑡−𝜏)))d𝜔=𝐶𝑥+𝐶_𝑑𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡−𝜏))=𝜑,𝑡∈−𝜏₂,0, (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector; 𝑢(𝑡,𝑡−𝜏(𝑡))∈ℝ^𝑚 is the control input, which depends on not only the real time but also the delay; we call the controller memory controller; 𝑧(𝑡)∈ℝ^𝑞 is the control output; 𝑣(𝑡)∈ℝ^𝑝 is the exogenous disturbance input, which satisfies 𝑣(𝑡)∈𝐿₂([0,∞),ℝ^𝑝), where 𝐿₂([0,∞),ℝ^𝑝) is the space of nonanticipatory square-summable stochastic process with respect to (ℱ_𝑡)_𝑡>0 with the following norm: ‖𝑣(𝑡)‖²₂=𝔼∫^∞₀‖𝑣(𝑡)‖²d𝑡. 𝜔(𝑡)∈ℝ^𝑙 is a scalar Brownian motion defined on a complete probability space (Ω,ℱ,𝑃) with 𝔼[d𝜔(𝑡)]=0, 𝔼[d²𝜔(𝑡)]=d𝑡.

In the sequel, we seek to study the problems of dissipative analysis and delay-feedback control for the two cases of time delay.

Case 1. Time delay is a constant 𝜏.

Case 2. 𝜏(𝑡) is the time-varying delay, which is a differential function satisfying 0≤𝜏₁≤𝜏(𝑡)≤𝜏₂,𝜏̇(𝑡)≤𝜏_𝑑≤1, (2) where 𝜏₁, 𝜏₂, and 𝜏_𝑑 are nonnegative constants.

Remark 1. Obviously, when 𝜏_𝑑=0, 𝜏(𝑡)=𝜏₁=𝜏₂, that means the time delay is a constant; this case has been extensively studied. On the other hand, the time-varying delay 𝜏(𝑡)≥𝜏₁; here 𝜏₁ is equal to or greater than 0, which has less conservativeness than 𝜏(𝑡)>0.

Assumption 2. 𝐹(⋅,⋅,⋅) is a nonlinear vector function which can be decomposed as follows: 𝐹(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))=𝐴₀𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+𝑓_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡))), (3) where 𝑓, 𝑓_𝑑 are vector-valued functions; we assume 𝑓(𝑡,𝑥(𝑡))≤𝛽‖𝑥(𝑡)‖,𝑓_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))≤𝛽_𝑑‖𝑥(𝑡−𝜏(𝑡))‖, (4)(5) where 𝛽, 𝛽_𝑑 are known real positive constants.

Obviously, we know that 𝑓(0,0)=0,𝑓_𝑑(0,0)=0. (6)

Equivalently stated, condition (4) implies that there exists a scalar 𝜅>0 such that 𝜅𝛽²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇(𝑡,𝑥(𝑡))𝑓(𝑡,𝑥(𝑡))≥0. (7)

Similarly, condition (5) implies that there exists a scalar 𝜅_𝑑>0 such that 𝜅_𝑑𝛽²_𝑑𝑥^𝑇(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏(𝑡))−𝑓^𝑇_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))𝑓_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))≥0. (8)

Assumption 3. 𝐺(⋅,⋅,⋅) is a nonlinear vector function which satisfies Trace𝐺^𝑇(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝐺(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))≤𝑥^𝑇(𝑡)Θ^𝑇₁Θ₁𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡))Θ^𝑇₂Θ₂𝑥(𝑡−𝜏(𝑡)), (9) where Θ₁, Θ₂ are known real matrices.

Hence, nonlinear stochastic delay systems (NSDSs) (1) can be rewritten as d𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+𝑓_𝑑(((𝑡((𝑡(𝑡((𝑡)((𝑡(𝑡),𝑧(𝑡)(𝑡)𝑡,𝑥𝑡−𝜏)))+𝐵𝑢𝑡,𝑡−𝜏))+𝐷𝑣)d𝑡+𝐺𝑡,𝑥,𝑥𝑡−𝜏)))d𝜔=𝐶𝑥+𝐶_𝑑𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡−𝜏))=𝜑,𝑡∈−𝜏₂,0. (10)

Definition 4 (see [21]). Given matrices 𝑄^𝑇=𝑄≤0, 𝑅^𝑇=𝑅≥0, and 𝑆, nonlinear stochastic delay systems (NSDSs) (10) are called (𝑄,𝑆,𝑅)-dissipative if, for some real function 𝜂(⋅), 𝜂(0)=0, 𝔼^𝑇₀𝑧^𝑇(𝑠)𝑄𝑧(𝑠)+2𝑣^𝑇(𝑠)𝑆𝑧(𝑠)+𝑣^𝑇(𝑠)(𝑠𝑥𝑅𝑣)d𝑠+𝜂₀≥0,∀𝑇≥0. (11)
Furthermore, if, for a scalar 𝛼>0, 𝔼^𝑇₀𝑧^𝑇(𝑠)𝑄𝑧(𝑠)+2𝑣^𝑇(𝑠)𝑆𝑧(𝑠)+𝑣^𝑇(𝑠)(𝑠𝑥𝑅𝑣)d𝑠+𝜂₀≥𝔼^𝑇₀𝛼𝑣^𝑇(𝑠)𝑣(𝑠)d𝑠,∀𝑇≥0, (12)

NSDSs (10) are called strictly (𝑄,𝑆,𝑅)-dissipative.

Lemma 5 (see [22]). Given three constant matrices 𝑆₁, 𝑆₂, and 𝑆₃, where 𝑆₃=𝑆^𝑇₃<0 and 𝑆₁=𝑆^𝑇₁<0, then 𝑆₁−𝑆^𝑇₂𝑆₃⁻¹𝑆₂<0 holds if and only if ^𝑆1𝑆2_{𝑆𝑇2𝑆3}<0 or ^𝑆3𝑆2_{𝑆𝑇2𝑆1}<0.

Lemma 6 (see [23]). For given positive symmetric matrix 𝑀=𝑀^𝑇>0, two scalars 𝑎 and 𝑏 satisfying 𝑎<𝑏, and vector function 𝑥(𝑡)∶[𝑎,𝑏]→ℝ^𝑛, then ^𝑏_𝑎𝑥(𝑠)𝑑𝑠^𝑇𝑀^𝑏_𝑎𝑥(𝑠)𝑑𝑠≤(𝑏−𝑎)^𝑏_𝑎𝑥^𝑇(𝑠)𝑀𝑥(𝑠)𝑑𝑠. (13)

3. Dissipativity Analysis for NSDSs

In this section, our primary purpose is to develope delay-independent and delay-dependent stochastically stability and dissipativity criteria for NSDSs (10) based on Definition 4.

3.1. Delay-Independent Dissipativity

In this sequel, we consider the time delay as unknown constant pertaining to Case 1 and hence the results developed hereinafter will be independent of the size of delay. Without regard to the control input, setting 𝑢(𝑡,𝑡−𝜏(𝑡))=0, then (10) can be rewritten as d𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏)+𝑓_𝑑(((𝑡((𝑡)((𝑡),𝑧(𝑡)(𝑡)𝑡,𝑥𝑡−𝜏))+𝐷𝑣)d𝑡+𝐺𝑡,𝑥,𝑥𝑡−𝜏))d𝜔=𝐶𝑥+𝐶_𝑑𝑥(),𝑥(𝑡)(𝑡)[).𝑡−𝜏=𝜑,𝑡∈−𝜏,0 (14)

Theorem 7. Consider the NSDSs (14). Given some scalars 𝛼>0, 𝛽>0, and 𝛽_𝑑>0 and matrices 𝑄=𝑄^𝑇≤0, 𝑅=𝑅^𝑇>0, and 𝑆, suppose there exist matrices 𝑃=𝑃^𝑇>0, 𝑊=𝑊^𝑇>0 and positive scalars 𝜅>0, 𝜅_𝑑>0 such that the following LMI holds: Σ₁𝑃𝐴_𝑑Σ₂𝑃𝑃𝐶^𝑇𝑄∗Σ₃−𝐶^𝑇_𝑑𝑆00𝐶^𝑇_𝑑𝑄∗∗−𝑅_𝛼000∗∗∗−𝜅𝐼00∗∗∗∗−𝜅_𝑑𝐼0∗∗∗∗∗𝑄<0; (15) then the NSDSs (14) are strictly (𝑄,𝑆,𝑅)-dissipative independent of delay, where Σ₁=𝐴^𝑇𝑃+𝑃𝐴+𝑊+Θ^𝑇₁𝑃Θ₁+𝜅𝛽²𝐼, Σ₂=𝑃𝐷−𝐶^𝑇𝑆, Σ₃=Θ^𝑇₂𝑃Θ₂−𝑊+𝜅_𝑑𝛽²_𝑑𝐼, and 𝑅_𝛼=(𝑅−𝛼𝐼).

Proof. At first we introduce the following Lyapunov-Krasovskii functional (LKF): 𝑉𝑥_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑊𝑥(𝑠)d𝑠. (16) Evaluating the Itô derivative of 𝑉(𝑥_𝑡) along the solution of NSDSs (14), we have ℒ𝑉𝑥_𝑡=2𝑥^𝑇𝑃𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏)+𝑓_𝑑(𝑡,𝑥(𝑡−𝜏))+𝐷𝑣(𝑡)+𝑥^𝑇𝑊𝑥−𝑥^𝑇(𝑡−𝜏)𝑊𝑥(𝑡−𝜏)+𝐺^𝑇(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑃𝐺(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏)). (17)
Noting (7)–(9), we obtain ℒ𝑉𝑥_𝑡≤ℒ𝑉𝑥_𝑡+𝜅𝛽²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇(𝑡,𝑥(𝑡))𝑓(𝑡,𝑥(𝑡))+𝜅_𝑑𝛽²_𝑑𝑥^𝑇(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏(𝑡))−𝑓^𝑇_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))×𝑓_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))≤𝜉^𝑇Σ𝜉, (18) where 𝜉=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏),𝑣^𝑇(𝑡),𝑓^𝑇((𝑡,𝑓𝑡,𝑥))_𝑇𝑑(𝑡,𝑥(𝑡−𝜏))^𝑇,ΣΣ=¹𝑃𝐴_𝑑𝑃𝐷𝑃𝑃∗Σ₃000∗∗000∗∗∗−𝜅𝐼0∗∗∗∗−𝑘_𝑑𝐼. (19)
Hence, we have ℒ𝑉𝑥_𝑡−𝑧^𝑇(𝑠)𝑄𝑧(𝑠)−2𝑣^𝑇(𝑠)𝑆𝑧(𝑠)−𝑣^𝑇(𝑠)(𝑅−𝛼𝐼)𝑣(𝑠)≤𝜉^𝑇Σ𝜉, (20) where 𝐶Σ=Σ−^𝑇𝑄𝐶𝐶^𝑇𝑄𝐶^𝑇_𝑑𝐶^𝑇𝑆00∗𝐶^𝑇_𝑑𝑄𝐶_𝑑𝐶^𝑇_𝑑().𝑆00∗∗𝑅−𝛼𝐼00∗∗∗00∗∗∗∗0 (21)
According to Lemma 5 and applying the congruent transformation, we know that ℒ𝑉𝑥_𝑡−𝑧^𝑇(𝑠)𝑄𝑧(𝑠)−2𝑣^𝑇(𝑠)𝑆𝑧(𝑠)−𝑣^𝑇(𝑠)(𝑅−𝛼𝐼)𝑣(𝑠)≤0. (22) Then, integrating (22) from 0 to 𝑇 and taking mathematical expectation, we obtain that (12) holds which completes the proof.

Remark 8. When 𝑄=−𝐼, 𝑆=0, and 𝑅_𝛼=𝛾²𝐼, strictly (𝑄,𝑆,𝑅)-dissipativity reduces to the 𝐻_∞ performance level. When 𝑄=0, 𝑆=𝐼, and 𝑅_𝛼=𝛾𝐼, strictly (𝑄,𝑆,𝑅)-dissipativity reduces to the strictly passivity.

So the following corollaries stand out as special cases.

Corollary 9. Consider the NSDSs (14). Given some scalars 𝛾>0, 𝛽>0, and 𝛽_𝑑>0, suppose there exist matrices 𝑃=𝑃^𝑇>0, 𝑊=𝑊^𝑇>0 and positive scalars 𝜅>0, 𝜅_𝑑>0 such that the following LMI holds: Σ₁𝑃𝐴𝑃𝐷𝑃𝑃−𝐶^𝑇∗Σ₃000−𝐶^𝑇_𝑑∗∗−𝛾²𝐼000∗∗∗−𝜅𝐼00∗∗∗∗−𝜅_𝑑𝐼0∗∗∗∗∗−𝐼<0; (23) then the NSDSs (14) are stochastically asymptotically stable and independent of delay with disturbance level 𝛾.

Corollary 10. Consider the NSDSs (14). Given some scalars 𝛾>0, 𝛽>0, and 𝛽_𝑑>0, suppose there exist matrices 𝑃=𝑃^𝑇>0, 𝑊=𝑊^𝑇>0 and positive scalars 𝜅>0, 𝜅_𝑑>0 such that the following LMI holds: Σ₁𝑃𝐴𝑃𝐷−𝐶^𝑇𝑃𝑃∗Σ₃−𝐶^𝑇_𝑑00∗∗−𝛾𝐼00∗∗∗−𝜅𝐼0∗∗∗∗−𝜅_𝑑𝐼<0; (24) then the NSDSs (14) are strictly passive independent of delay.

3.2. Delay-Dependent Dissipativity

We now direct attention to the type of dissipativity which depends on the time-varying delay, which pertains to Case 2. Without consideration of the control input, and defining a new state variable 𝑦(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+𝑓_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))+𝐷𝑣(𝑡), (25) then, the NSDSs (10) can be rewritten as (𝑡)(𝑡)((𝑡)((𝑡(𝑡),𝑧(𝑡)(𝑡)d𝑥=𝑦d𝑡+𝐺𝑡,𝑥,𝑥𝑡−𝜏)))d𝜔=𝐶𝑥+𝐶_𝑑𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡−𝜏))=𝜑,𝑡∈−𝜏₂,0. (26)

Theorem 11. Consider the NSDSs (26). For the given scalars 𝛼>0, 𝜏₂>𝜏₁≥0, 𝜏_𝑑>0, 𝛽>0, and 𝛽_𝑑>0, the NSDSs (26) are strictly (𝑄,𝑆,𝑅)-dissipative for all time-varying delays if there exist symmetric positive-definite matrices 𝑃, 𝑉₁, 𝑉₂, 𝑉₃, 𝑊₁, 𝑊₂, 𝑍₁, and 𝑍₂; any appropriately dimensioned matrices 𝑁, 𝑀, 𝐹, and 𝐻; and two positive scalars 𝜅>0, 𝜅_𝑑>0 such that the following LMIs hold: Ξ−𝜏₂𝑀∗−𝜏₂𝑊₁𝜏<0,Ξ−₂−𝜏₁𝑁∗−𝜏₂−𝜏₁𝑊₁+𝑊₂𝜏<0,Ξ−₂−𝜏₁𝐹∗−𝜏₂−𝜏₁𝑊₂<0, (27) where ΞΞ=¹¹Ξ₁₂Ξ₁₃Ξ₁₄𝐴^𝑇𝐻Ξ₁₆∗Ξ₂₂−𝐶^𝑇_𝑑𝑆0𝐴^𝑇_𝑑𝐻Ξ₂₆∗∗−𝑅_𝛼0𝐷^𝑇𝐻0∗∗∗Ξ₄₄Ξ₄₅0∗∗∗∗Ξ₅₅0∗∗∗∗∗Ξ₆₆,Ξ₁₁=𝑃𝐴+𝐴^𝑇𝑃+𝑉₁+𝑉₂+𝑉₃+𝜏₂𝑍₁+𝜏₂−𝜏₁𝑍₂+𝑀+𝑀^𝑇+𝜅𝛽²Ξ𝐼,₁₂=𝑁−𝐹−𝑀+𝑃𝐴_𝑑,Ξ₁₃=𝑃𝐷−𝐶^𝑇Ξ𝑆,₁₄=,Ξ𝐹−𝑁𝑃𝑃₁₆=𝐶^𝑇𝑄Θ₁𝑃0,Ξ₂₂=−1−𝜏_𝑑𝑉₂+𝜅_𝑑𝛽²_𝑑Ξ𝐼,₂₆=𝐶^𝑇_𝑑𝑄0Θ₂𝑃,Ξ₄₄=diag−𝑉₁,−𝑉₃,−𝜅𝐼,−𝜅_𝑑𝐼,Ξ₅₅=𝜏₂𝑊₁+𝜏₂−𝜏₁𝑊₂−𝐻−𝐻^𝑇,Ξ₆₆=diag{𝑄,−𝑃,−𝑃},Ξ₄₅=00𝐻^𝑇𝐻^𝑇𝑇,𝑁=𝑁^𝑇,0_1×10^𝑇,𝑀=𝑀^𝑇,0_1×10^𝑇,𝐹=𝐹^𝑇,0_1×10^𝑇. (28)

Proof. We construct a LKF as follows: 𝑉𝑥_𝑡=𝑉₁𝑥_𝑡+𝑉₂𝑥_𝑡+𝑉₃𝑥_𝑡+𝑉₄𝑥_𝑡, (29) where 𝑉₁𝑥_𝑡=𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂𝑥_𝑡=^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝑉₁𝑥(𝑠)d𝑠+^𝑡_{𝑡−𝜏(𝑡)}𝑥^𝑇(𝑠)𝑉₂𝑥(𝑠)+d𝑠_{𝑡𝑡−𝜏2}𝑥^𝑇(𝑠)𝑉₃𝑥(𝑠)𝑉d𝑠,₃𝑥_𝑡=⁰_−𝜏2^𝑡_𝑡+𝜃𝑦^𝑇(𝑠)𝑊₁𝑦(𝑠)+d𝑠_{−𝜏1−𝜏2𝑡𝑡+𝜃}𝑦^𝑇(𝑠)𝑊₂𝑦(𝑠)𝑉d𝑠,₄𝑥_𝑡=⁰_−𝜏2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑍₁𝑥(𝑠)+d𝑠_{−𝜏1−𝜏2𝑡𝑡+𝜃}𝑥^𝑇(𝑠)𝑍₂𝑥(𝑠)d𝑠. (30)
Then, the weak infinitesimal operator ℒ of the stochastic process 𝑥_𝑡 along the evolution of 𝑉(𝑥_𝑡) is given by ℒ𝑉𝑥_𝑡=ℒ𝑉₁𝑥_𝑡+ℒ𝑉₂𝑥_𝑡+ℒ𝑉₃𝑥_𝑡+ℒ𝑉₄𝑥_𝑡, (31) where ℒ𝑉₁𝑥_𝑡≤2𝑥^𝑇(𝑡)𝑃𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+𝑓_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))+𝐷𝑣(𝑡)+𝑥^𝑇(𝑡)Θ^𝑇₁𝑃Θ₁𝑥(𝑡)+𝑥(𝑡−𝜏(𝑡))Θ^𝑇₂𝑃Θ₂𝑥(𝑡−𝜏(𝑡)),ℒ𝑉₂𝑥_𝑡≤𝑥^𝑇(𝑡)𝑉₁+𝑉₂+𝑉₃𝑥(𝑡)−𝑥^𝑇𝑡−𝜏₁𝑉₁𝑥𝑡−𝜏₁−𝑥^𝑇𝑡−𝜏₂𝑉₃𝑥𝑡−𝜏₂−𝑥^𝑇(𝑡−𝜏(𝑡))1−𝜏_𝑑𝑉₂𝑥(𝑡−𝜏(𝑡)),ℒ𝑉₃𝑥_𝑡=𝜏₂𝑦^𝑇(𝑡)𝑊₁𝑦(𝑡)−^𝑡_𝑡−𝜏2𝑦^𝑇(𝑠)𝑊₁𝑦(𝑠)+𝜏d𝑠₂−𝜏₁𝑦^𝑇(𝑡)𝑊₂𝑦(𝑡)−^𝑡−𝜏1_𝑡−𝜏2𝑦^𝑇(𝑠)𝑊₂𝑦(𝑠)d𝑠. (32)(33)(34)
By using Lemma 6, we get the following inequality: ℒ𝑉₄𝑥_𝑡≤𝜏₂𝑥^𝑇(𝑡)𝑍₁𝑥(𝑡)+𝜏₂−𝜏₁𝑥^𝑇(𝑡)𝑍₂𝑥(𝑡)−1𝜏₂^𝑡_{𝑡−𝜏(𝑡)}𝑥(𝑠)d𝑠^𝑇𝑍₁^𝑡_{𝑡−𝜏(𝑡)}𝑥(𝑠)−1d𝑠𝜏₂−𝜏₁^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝑥(𝑠)d𝑠^𝑇𝑍₂^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝑥(𝑠)−1d𝑠𝜏₂−𝜏₁^𝑡−𝜏1_{𝑡−𝜏(𝑡)}𝑥(𝑠)d𝑠^𝑇𝑍₂^𝑡−𝜏1_{𝑡−𝜏(𝑡)}𝑥(𝑠)d𝑠, (35)
setting 𝜉(𝑡)=(𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏(𝑡)),𝑣^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏₁),𝑥^𝑇(𝑡−𝜏₂),𝑓^𝑇(𝑥(𝑡)),𝑓^𝑇_𝑑(𝑥(𝑡−𝜏(𝑡))),𝑦^𝑇(𝑡),(∫^𝑡_{𝑡−𝜏(𝑡)}𝑥(𝑠)d𝑠)^𝑇,(∫^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝑥(𝑠)d𝑠)^𝑇,(∫^𝑡−𝜏1_{𝑡−𝜏(𝑡)}𝑥(𝑠)d𝑠)^𝑇)^𝑇. And we introduce the following four zero equations: 2𝑥^𝑇(𝑡)𝑀𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))−^𝑡_{𝑡−𝜏(𝑡)}d𝑥(𝑠)=0,2𝑥^𝑇(𝑡)𝑁𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏₂−^{𝑡−𝜏(𝑡)}_𝑡−𝜏2d𝑥(𝑠)=0,2𝑥^𝑇(𝑡)𝐹𝑥𝑡−𝜏₁−𝑥(𝑡−𝜏(𝑡))−^𝑡−𝜏1_{𝑡−𝜏(𝑡)}d𝑥(𝑠)=0,2𝑦^𝑇(𝑡)𝐻𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+𝑓_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))+𝐷𝑣(𝑡)−𝑦(𝑡)=0. (36)(37)(38)(39)
Summing up (31)–(39), we obtain ℒ𝑉𝑥_𝑡≤ℒ𝑉𝑥_𝑡+𝜅𝛽²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇(𝑡,𝑥(𝑡))𝑓(𝑡,𝑥(𝑡))+𝜅_𝑑𝛽²_𝑑𝑥^𝑇(𝑡−𝜏(𝑡))𝑥(𝑡−𝜏(𝑡))−𝑓^𝑇_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))𝑓_𝑑(𝑡,𝑥(𝑡−𝜏(𝑡)))+2𝑥^𝑇(𝑡)𝑀𝑥(𝑡)((𝑡−−𝑥𝑡−𝜏))_{𝑡(𝑡)𝑡−𝜏}𝑦(𝑠)d𝑠−2𝑥^𝑇(𝑡)𝑀^𝑡_{𝑡−𝜏(𝑡)}𝐺𝑥_𝑠d𝜔(𝑠)+2𝑥^𝑇(𝑡)𝑁𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−𝜏₂−^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝑦(𝑠)d𝑠−2𝑥^𝑇(𝑡)𝑁^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝐺𝑥_𝑠d𝜔(𝑠)+2𝑥^𝑇(𝑡)𝐹𝑥𝑡−𝜏₁((𝑡−−𝑥𝑡−𝜏))_{𝑡−𝜏1(𝑡)𝑡−𝜏}𝑦(𝑠)d𝑠−2𝑥^𝑇(𝑡)𝐹^𝑡−𝜏1_{𝑡−𝜏(𝑡)}𝐺𝑥_𝑠d𝜔(𝑠)+2𝑦^𝑇(𝑡)𝐻𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+𝑓_𝑑(((𝑡(𝑡)(𝑡,𝑡,𝑥𝑡−𝜏)))+𝐷𝑣−𝑦) (40) where 𝑀, 𝑁, 𝐹, and 𝐻 are matrices with appropriate dimensions. Hence, ℒ𝑉𝑥_𝑡−𝑧^𝑇(𝑠)𝑄𝑧(𝑠)−2𝑣^𝑇(𝑠)𝑆𝑧(𝑠)−𝑣^𝑇(𝑠)(𝑅−𝛼𝐼)𝑣(𝑠)≤𝜉^𝑇(𝑡)Φ𝜉(𝑡)−2𝑥^𝑇(𝑡)𝑀×^𝑡_{𝑡−𝜏(𝑡)}𝑦(𝑠)d𝑠+^𝑡_{𝑡−𝜏(𝑡)}𝐺𝑥_𝑠d𝜔(𝑠)−2𝑥^𝑇(𝑡)𝑁^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝑦(𝑠)d𝑠+^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝐺𝑥_𝑠d𝜔(𝑠)−2𝑥^𝑇(𝑡)𝐹^𝑡−𝜏1_{𝑡−𝜏(𝑡)}𝑦(𝑠)d𝑠+^𝑡−𝜏1_{𝑡−𝜏(𝑡)}𝐺𝑥_𝑠(𝑠−d𝜔)_{𝑡𝑡−𝜏2}𝑦^𝑇(𝑠)𝑊₁𝑦(𝑠)d𝑠−^𝑡−𝜏1_𝑡−𝜏2𝑦^𝑇(𝑠)𝑊₂𝑦(𝑠)d𝑠, (41) where ΦΦ=¹¹Φ₁₂Φ₁₃Φ₁₄𝐴^𝑇𝐻0∗Φ₂₂−𝐶^𝑇_𝑑𝑆0𝐴^𝑇_𝑑𝐻0∗∗−𝑅_𝛼0𝐷^𝑇𝐻0∗∗∗Φ₄₄Φ₄₅0∗∗∗∗Φ₅₅0∗∗∗∗∗Φ₆₆,Φ₁₁=𝑃𝐴+𝐴^𝑇𝑃+Θ^𝑇₁𝑃Θ₁+𝑉₁+𝑉₂+𝑉₃+𝜏₂𝑍₁+𝜏₂−𝜏₁𝑍₂+𝑀+𝑀^𝑇−𝐶^𝑇𝑄𝐶+𝜅𝛽²Φ𝐼,₁₂=𝑁−𝐹−𝑀+𝑃𝐴_𝑑−𝐶^𝑇𝑄𝐶_𝑑,Φ₁₃=𝑃𝐷−𝐶^𝑇Φ𝑆,₁₄=,Φ𝐹−𝑁𝑃𝑃₂₂=Θ^𝑇₂𝑃Θ₂−𝐶^𝑇_𝑑𝑄𝐶_𝑑−1−𝜏_𝑑𝑉₂+𝜅_𝑑𝛽²_𝑑Φ𝐼,₄₄=diag−𝑉₁,−𝑉₃,−𝜅𝐼,−𝜅_𝑑𝐼,Φ₄₅=00𝐻^𝑇𝐻^𝑇𝑇,Φ₅₅=𝜏₂𝑊₁+𝜏₂−𝜏₁𝑊₂−𝐻−𝐻^𝑇,Φ₆₆−1=diag𝜏₂𝑍₁1,−𝜏₂−𝜏₁𝑍₂1,−𝜏₂−𝜏₁𝑍₂. (42) So it easy to obtain that ℒ𝑉𝑥_𝑡−𝑧^𝑇(𝑠)𝑄𝑧(𝑠)−2𝑣^𝑇(𝑠)𝑆𝑧(𝑠)−𝑣^𝑇(𝑠)𝑣(𝑠)≤1)(𝑅−𝛼𝐼𝜏₂^𝑡_{(𝑡)𝑡−𝜏}𝜂^𝑇(𝑡,𝑠)Φ−𝜏₂𝑀∗−𝜏₂𝑊₁𝜂()+1𝑡,𝑠d𝑠𝜏₂−𝜏₁^𝑡_{(𝑡)𝑡−𝜏}𝜂^𝑇()×𝜏𝑡,𝑠Φ−₂−𝜏₁𝑁∗−𝜏₂−𝜏₁𝑊₁+𝑊₂𝜂()+1𝑡,𝑠d𝑠𝜏₂−𝜏₁^𝑡_{(𝑡)𝑡−𝜏}𝜂^𝑇()𝜏𝑡,𝑠Φ−₂−𝜏₁𝐹∗−𝜏₂−𝜏₁𝑊₂𝜂(𝑡,𝑠)d𝑠−2𝑥^𝑇(𝑡)𝑀^𝑡_{𝑡−𝜏(𝑡)}𝐺𝑥_𝑠d𝜔(𝑠)−2𝑥^𝑇(𝑡)𝑁^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝐺𝑥_𝑠d𝜔(𝑠)−2𝑥^𝑇(𝑡)𝐹^𝑡−𝜏1_{𝑡−𝜏(𝑡)}𝐺𝑥_𝑠d𝜔(𝑠), (43) where 𝜂^𝑇(𝑡,𝑠)=[𝜉^𝑇(𝑡),𝑦^𝑇(𝑠)].
By Lemma 5 and applying the congruent transformation to (27), it follows that ℒ𝑉𝑥_𝑡−𝑧^𝑇(𝑠)𝑄𝑧(𝑠)−2𝑣^𝑇(𝑠)𝑆𝑧(𝑠)−𝑣^𝑇(𝑠)(𝑅−𝛼𝐼)𝑣(𝑠)≤−2𝑥^𝑇(𝑡)𝑀^𝑡_{𝑡−𝜏(𝑡)}𝐺𝑥_𝑠d𝜔(𝑠)−2𝑥^𝑇(𝑡)𝑁^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝐺𝑥_𝑠d𝜔(𝑠)−2𝑥^𝑇(𝑡)𝐹^𝑡−𝜏1_{(𝑡)𝑡−𝜏}𝐺𝑥_𝑠d𝜔(𝑠). (44)
Then, integrating both sides of (44) from 0 to 𝑇 and taking mathematical expectation, we obtain that (12) holds, which completes the proof.

Similarly, as the special case, we can easily obtain the following corollaries.

Corollary 12. Consider the NSDSs (26). For the given scalars 𝛾>0, 𝜏₂>𝜏₁≥0, 𝜏_𝑑>0, 𝛽>0, and 𝛽_𝑑>0, the NSDSs (26) are stochastically asymptotically stable with 𝐻_∞ performance level for all time-varying delays if there exist symmetric positive-definite matrices 𝑃, 𝑉₁, 𝑉₂, 𝑉₃, 𝑊₁, 𝑊₂, 𝑍₁, and 𝑍₂; any appropriately dimensioned matrices 𝑁, 𝑀, 𝐹, and 𝐻; and two positive scalars 𝜅>0, 𝜅_𝑑>0 such that the following LMIs hold: Ξ−𝜏₂𝑀∗−𝜏₂𝑊₁𝜏<0,Ξ−₂−𝜏₁𝑁∗−𝜏₂−𝜏₁𝑊₁+𝑊₂𝜏<0,Ξ−₂−𝜏₁𝐹∗−𝜏₂−𝜏₁𝑊₂<0, (45) where ΞΞ=¹¹Ξ₁₂𝑃𝐷Ξ₁₄𝐴^𝑇𝐻Ξ₁₆∗Ξ₂₂00𝐴^𝑇_𝑑𝐻Ξ₂₆∗∗−𝛾²𝐼0𝐷^𝑇𝐻0∗∗∗Ξ₄₄Ξ₄₅0∗∗∗∗Ξ₅₅0∗∗∗∗∗Ξ₆₆,Ξ₁₆=−𝐶^𝑇Θ₁𝑃0,Ξ₂₆=−𝐶^𝑇_𝑑0Θ₂𝑃,Ξ₆₆{},Ξ=diag−𝐼,−𝑃,−𝑃₁₁,Ξ₁₂,Ξ₁₄,Ξ₂₂,Ξ₄₄,Ξ₄₅,Ξ₅₅, (46) defined in Theorem 11.

Corollary 13. Consider the NSDSs (26). For the given scalars 𝛾>0, 𝜏₂>𝜏₁≥0, 𝜏_𝑑>0, 𝛽>0, and 𝛽_𝑑>0, the NSDSs (26) are strictly (𝑄,𝑆,𝑅)-passive for all time-varying delays if there exist symmetric positive-definite matrices 𝑃, 𝑉₁, 𝑉₂, 𝑉₃, 𝑊₁, 𝑊₂, 𝑍₁, 𝑍₂; any appropriately dimensioned matrices 𝑁, 𝑀, 𝐹, and 𝐻; and two positive scalars 𝜅>0, 𝜅_𝑑>0 such that the following LMIs hold: Ξ−𝜏₂𝑀∗−𝜏₂𝑊₁𝜏<0,Ξ−₂−𝜏₁𝑁∗−𝜏₂−𝜏₁𝑊₁+𝑊₂𝜏<0,Ξ−₂−𝜏₁𝐹∗−𝜏₂−𝜏₁𝑊₂<0, (47) where ΞΞ=¹¹Ξ₁₂𝑃𝐷−𝐶^𝑇Ξ₁₄𝐴^𝑇𝐻Ξ₁₆∗Ξ₂₂−𝐶^𝑇_𝑑0𝐴^𝑇_𝑑𝐻Ξ₂₆∗∗−𝛾𝐼0𝐷^𝑇𝐻0∗∗∗Ξ₄₄Ξ₄₅0∗∗∗∗Ξ₅₅0∗∗∗∗∗Ξ₆₆,Ξ₁₆=Θ₁𝑃0,Ξ₂₆=0Θ₂𝑃,Ξ₆₆=diag{−𝑃,−𝑃},𝑁=𝑁^𝑇,0_1×9^𝑇,𝑀=𝑀^𝑇,0_1×9^𝑇,𝐹=𝐹^𝑇,0_1×9^𝑇,Ξ₁₁,Ξ₁₂,Ξ₁₄,Ξ₂₂,Ξ₄₄,Ξ₄₅,Ξ₅₅ (48) defined in Theorem 11.

4. Dissipative Delay-Feedback Control for NSDSs

Extending on the results of the foregoing section, our aim is to develope an LMIs-based solution to the problem of designing a delay-feedback controller as 𝑢(𝑡,𝑡−𝜏(𝑡))=𝐾₀𝑥(𝑡)+𝐾₁𝑥(𝑡−𝜏(𝑡)), (49) which will render the NSDSs (10) strictly (𝑄,𝑆,𝑅)-dissipative. The closed-loop systems is now described by d𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+𝑓_𝑑(((𝑡(𝑡((𝑡)((𝑡(𝑡),𝑧(𝑡)(𝑡)𝑡,𝑥𝑡−𝜏)))+𝐷𝑣)d𝑡+𝐺𝑡,𝑥,𝑥𝑡−𝜏)))d𝜔=𝐶𝑥+𝐶_𝑑𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡−𝜏))=𝜑,𝑡∈−𝜏₂,0, (50) where 𝐴=𝐴+𝐵𝐾₀, 𝐴_𝑑=𝐴_𝑑+𝐵𝐾₁.

Applying Theorem 7, together with Lemma 5 and congruent transformation, we can get the following theorem without detailed proofs.

Theorem 14. Consider the NSDSs (50). Given some scalars 𝛼>0, 𝛽>0, 𝛽_𝑑>0, 𝜅>0, and 𝜅_𝑑>0 and matrices 𝑄=𝑄^𝑇≤0, 𝑅=𝑅^𝑇>0, and 𝑆, suppose there exist matrices 𝑋=𝑋^𝑇>0, 𝑊=𝑊^𝑇>0, 𝑌, and 𝑌_𝑑 such that the following LMI holds: Γ₁𝐴_𝑑𝑋+𝐵𝑌_𝑑𝐷−𝑋𝐶^𝑇𝑆Γ₂∗−𝑊−𝑋𝐶^𝑇_𝑑𝑆Γ₃∗∗−𝑅_𝛼0∗∗∗Γ₄<0; (51) then the NSDSs (50) are strictly (𝑄,𝑆,𝑅)-dissipative independent of delay; the feedback gain is 𝐾₀=𝑌𝑋⁻¹, 𝐾₁=𝑌_𝑑𝑋⁻¹, where Γ₁=𝑋𝐴^𝑇+𝐴𝑋+𝑌^𝑇𝐵^𝑇+𝐵𝑌+𝑊, Γ₂=(𝐼𝐼𝑋𝐶^𝑇𝑄𝑋Θ₁0𝜅𝛽𝑋0), Γ₃=(00𝑋𝐶^𝑇_𝑑𝑄0𝑋Θ₂0𝜅_𝑑𝛽_𝑑𝑋), and Γ₄=diag(−𝜅𝐼,−𝜅_𝑑𝐼,𝑄,−𝑋,−𝑋,−𝜅𝐼,−𝜅_𝑑𝐼).

Similarly, applying Theorem 11, together with Lemma 5 and congruent transformation, we can obtain the following theorem without detailed proofs.

Theorem 15. Consider the NSDSs (50). For the given scalars 𝛼>0, 𝜏₂>𝜏₁≥0, 𝜏_𝑑>0, 𝜅>0, 𝜅_𝑑>0, 𝛽>0, and 𝛽_𝑑>0, the NSDSs (50) are strictly (𝑄,𝑆,𝑅)-dissipative for all time-varying delays and the feedback gain is 𝐾₀=𝑌𝑋⁻¹, 𝐾₁=𝑌_𝑑𝑋⁻¹, if there exist symmetric positive-definite matrices 𝑋, 𝑉₁, 𝑉₂, 𝑉₃, 𝑊₁, 𝑊₂, 𝑍₁, and 𝑍₂ and any appropriately dimensioned matrices 𝑁, 𝑀, 𝐹, 𝑌, and 𝑌_𝑑 such that the following LMIs hold: Δ−𝜏₂𝑀∗−𝜏₂𝑊₁𝜏<0,Δ−₂−𝜏₁𝑁∗−𝜏₂−𝜏₁𝑊₁+𝑊₂𝜏<0,Δ−₂−𝜏₁𝐹∗−𝜏₂−𝜏₁𝑊₂<0, (52) where ΔΔ=¹¹Δ₁₂Δ₁₃Δ₁₄Δ₁₅Δ₁₆∗Δ₂₂−𝑋𝐶^𝑇_𝑑𝑆0Δ₂₅Δ₂₆∗∗−𝑅_𝛼0𝐷^𝑇0∗∗∗Δ₄₄Δ₄₅0∗∗∗∗Δ₅₅0∗∗∗∗∗Δ₆₆,Δ₁₁=𝐴𝑋+𝐵𝑌+𝑋𝐴^𝑇+𝑌^𝑇𝐵^𝑇+𝑉₁+𝑉₂+𝑉₃+𝜏₂𝑍₁+𝜏₂−𝜏₁𝑍₂+𝑀+𝑀^𝑇,Δ₁₂=𝑁−𝐹−𝑀+𝐴_𝑑𝑋+𝐵𝑌_𝑑,Δ₁₃=𝐷−𝑋𝐶^𝑇Δ𝑆,₁₄=𝐹−𝑁𝐼𝐼,Δ₁₅=𝑌^𝑇𝐵^𝑇+𝑋𝐴^𝑇,Δ₁₆=𝑋𝐶^𝑇𝑄𝑋Θ₁,Δ0𝜅𝛽𝑋0₂₂=−1−𝜏_𝑑𝑉₂,Δ₂₅=𝑌^𝑇_𝑑𝐵^𝑇+𝑋𝐴^𝑇_𝑑,Δ₂₆=𝑋𝐶^𝑇_𝑑𝑄0𝑋Θ₂0𝜅_𝑑𝛽_𝑑𝑋,Δ₄₄=diag−𝑉₁,−𝑉₃,−𝜅𝐼,−𝜅_𝑑𝐼,Δ₄₅=00𝐼𝐼^𝑇,Δ₅₅=𝜏₂𝑊₁+𝜏₂−𝜏₁𝑊₂−𝑋−𝑋^𝑇,Δ₆₆=diag𝑄,−𝑋,−𝑋,−𝜅𝐼,−𝜅_𝑑𝐼,𝑁=𝑁^𝑇,0_1×12^𝑇,𝑀=𝑀^𝑇,0_1×12^𝑇,𝐹=𝐹^𝑇,0_1×12^𝑇. (53)

5. Numerical Example with Simulation

In this section, we will give an example to show the correctness of the derived results and the effectiveness of the designed controller. Consider the following nonlinear stochastic delay systems: (𝑡)=𝑥(𝑡)+d𝑥−0.11012−12.50−1−1.5−50002−50.1𝑥₁sin𝑥₃0.4𝑥₂cos𝑥₄0.3𝑥₃sin𝑥₁𝑥₄0.5𝑥₄cos𝑥₂𝑥₃+𝑥()+−11012−32.50−1−2−300023𝑡−𝜏0.1𝑥₁(𝑡−𝜏)sin𝑥₃(𝑡−𝜏)0.2𝑥₂(𝑡−𝜏)cos𝑥₄(𝑡−𝜏)0.3𝑥₃(𝑡−𝜏)sin𝑥₁(𝑡−𝜏)𝑥₄(𝑡−𝜏)0.3𝑥₄(𝑡−𝜏)cos𝑥₂(𝑡−𝜏)𝑥₃()+𝑢()+𝑣(𝑡)+𝑡−𝜏0−1−0.50−0.200−0.2𝑡,𝑡−𝜏0.50.10.100.10.10.50.200.1010010.4d𝑡0.35𝑥₁sin𝑥₁+0.35𝑥₁()𝑥𝑡−𝜏×cos₂(𝑡−𝜏)0.71𝑡𝑥1+𝑡²+𝑥₂(𝑡−𝜏)0.35𝑥₃sin𝑥₁𝑥₃−0.35𝑥₃()𝑥𝑡−𝜏×sin²₃(𝑡−𝜏)14𝑡𝑥7+100𝑡⁴−𝑥₄()(𝑡),𝑧(𝑡)=𝑥(𝑡)+𝑥().𝑡−𝜏d𝜔120000.5000010000.21110−10.50.5−100012002−1𝑡−𝜏 (54)

For Case 1, when the initial condition 𝑥(0)=[−2,2,3,−4]^𝑇 is used and 𝑣(𝑡) is a random vector of zero mean and 0.3 standard deviation, we can see that the uncontrolled NSDSs (54) are not stable from Figure 1. Hence the design of dissipative delay-feedback controller is necessary. From (54), we can get Θ₁=Θ₂=diag(0.5,1,0.5,2), and 𝛽=0.5, 𝛽_𝑑=0.3, for given 𝑆=𝐼, 𝑅=5𝐼, and 𝑄=−0.2𝐼; applying Theorem 14 to this example, we have ,∗,𝑌𝑋=38.2649−9.3804−3.69247.8006−9.380425.989610.5274−1.9189−3.692410.52744.5464−0.73377.8006−1.9189−0.73372.2634𝑌=1.0𝑒+004−0.02541.55970.6249−0.00440.8881−0.0450−0.01750.1783_𝑑∗.=1.0𝑒+0031.19510.50870.21000.23130.03080.24030.0994−0.0020 (55)

Figure 1 

The states curves of open-loop NSDSs in (54) without the control with initial state (−2,2,3,−4)𝑇 and time delay 𝜏=1.

So the delay-feedback controller parameters can be calculated as follows: ,𝐾𝐾=149.4677780.3725−306.768927.4064258.0621100.2942−67.4539−38.6640_𝑑=.43.186333.70780.3173−17.95045.89537.65506.9022−12.4934 (56)

The states curves and the output curves of closed-loop NSDSs in (54) can be seen in Figures 2, 3, and 4; from Figure 5, we can see that (12) holds. Hence, the closed-loop NSDSs are strictly (𝑄,𝑆,𝑅)-dissipative; we can also see that the delay-feedback controller is delay-independent.

Figure 2 

The states curves of closed-loop NSDSs in (54) under the delay-feedback control with initial state (−2,2,3,−4)𝑇 and time delay 𝜏=1.

Figure 3 

The states curves of closed-loop NSDSs in (54) under the delay-feedback control with initial state (−2,2,3,−4)𝑇 and time delay 𝜏=2.

Figure 4 

The output curves of closed-loop NSDSs in (54) under the delay-feedback control with initial state (−2,2,3,−4)𝑇 and time delay 𝜏=1.

Figure 5 

The dissipativity level of closed-loop NSDSs in (54) under the delay-feedback control with initial state (−2,2,3,−4)𝑇 and time delay 𝜏=1.

6. Conclusions

The dissipative delay-feedback control problems for nonlinear stochastic delay systems (NSDSs) have been investigated based on dissipativity analysis. The systems are subjected to stochastic disturbance, nonlinear disturbance, and two cases time-delay effects, which often exist in a wide variety of industrial processes and are the main sources of instability. Based on the Lyapunov stability theory and stochastic analysis technique, both delay-independent and delay-dependent dissipativity criteria have been established in terms of linear matrix inequalities (LMIs). The available results on 𝐻_∞ approach and passivity for stochastic delay systems as special cases of the developed results also have been given in this paper. The delay-dependent feedback controller has been designed by considering the relationship among the time-varying delay, its lower and upper bound, and its difference without ignoring any terms, which effectively reduces the conservative. A numerical example also has been given to verify the effectiveness of the proposed methods.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Science Foundation of China (NSFC) under Grants nos. 61104127, 51079057, and 61134012; China Postdoctoral Science Foundation under Grant no. 2012M521428; and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) under Grant no. Y201320.