Novel Stability Criteria of Nonlinear Uncertain Systems with Time-Varying Delay

Abstract

The problem of robust exponential stabilization for dynamical nonlinear systems with uncertainties and time-varying delay is considered in the paper. By constructing the proposed Lyapunov-Krasovskii functional approach, continuous state feedback controllers are put forward, and the criteria which guarantee the exponential stabilization of the nonlinear systems with uncertainties and time-varying delay are established in terms of solutions to the standard Riccati differential equations. Furthermore, based on the Lyapunov method and the linear matrix inequality approach, the sufficient conditions of exponential stability for a class of uncertain systems with time-varying delays and nonlinear perturbations are derived. Finally, two numerical examples are given to demonstrate the validity of the results.

1. Introduction

The stability problem and stabilization problem of time-delay systems are important problems not yet completely solved and continuously investigated by many people. For control systems with delayed state, existing stability criteria can be classified into two categories, that is, delay-independent ones and delay-dependent ones, and delay-independent ones are usually more conservative than the delay-dependent ones. On the other hand, it is unavoidable to include uncertain parameters and perturbations in practical control systems due to modeling errors, measurement errors, approximations, and so on. Therefore, the robust stabilization problem of uncertain dynamical systems with time-varying delays has attracted considerable attention of many researchers in recent years, and most of these papers have always adopted linear matrix equalities to guarantee the exponential stability of dynamical systems by employing Leibniz-Newton formula or different transformations. For instance, by employing a descriptor model transformation and a decomposition technique of the delay term matrix, the robust stability of uncertain linear systems with a single time-varying delay and nonlinear perturbations is investigated in [1]. Without introducing any free weighting matrices, in [2], new delay-dependent stability criteria for time-delay systems have been derived by introducing a new type of Lyapunov functional which contains some triple-integral terms and fully uses the information on the lower bound of the delay. In [3], the exponential stability of linear distributed parameter systems with time-varying delays is studied through Linear Operator Inequalities which are reduced to standard Linear Matrix Inequalities. Based on the Lyapunov-Krasovskii functional approach, in [4], a stability criterion of uncertain linear systems with interval time-varying delay is derived by introducing some relaxation matrices that can be used to reduce the conservatism of the criteria. In terms of linear matrix inequality, [5] proposes a new delay-dependent stability criterion for dynamic systems with time-varying delays and nonlinear perturbations. In [6], a class of linear systems with unknown norm-bounded uncertainties and time-varying delays is investigated with Leibniz-Newton formula and linear matrix inequalities, which allow computing simultaneously the two bounds that characterize the exponential stability rate of the solution. In [7], the system is transformed into another one by the transformation of 𝑧(𝑡)=𝑒^𝛼𝑡𝑥(𝑡), and then delay-dependent stability criteria have been derived in terms of a matrix inequality (LMI) which can be easily solved using efficient convex optimization algorithms. In [8], by constructing a suitable augmented Lyapunov’s functional and utilizing free weight matrices, the criterion for stabilization of uncertain dynamic systems with time-varying delays is established in terms of linear matrix inequalities. Using the method of the matrix equality, [9] has presented some improved stability criteria to guarantee that the uncertain systems with two successive delay components are robustly, asymptotically stable. In [10, 11], the stabilization of uncertain dynamic systems is considered. In [12], the stability analysis problem of linear neutral delay differential systems with multiple time delays is investigated. Using the Lyapunov method, some sufficient conditions are presented for the asymptotic stability of systems. In [13], the problem of robust stabilization of a class of nonlinear dynamical systems with delayed perturbations is considered. Based on the stability of the nominal systems, a new stabilizing control law for exponential stability of the system is designed using Lyapunov stability theory.

Being different from them, this paper chooses the Riccati differential equation to solve the stabilization of uncertain time-delay systems. The sufficient conditions which guarantee that the uncertain systems with time-varying delay and nonlinear perturbations are exponentially stabilizable are presented by employing Lyapunov-Krasovskii functional, and the controllers are constructed.

The rest of this paper is organized as follows. Section 2 presents notations, assumptions, and lemmas which can be used in the proof of theorems. In Section 3, the main results on the robust exponential stabilization for the uncertain systems with time-varying delay and nonlinear perturbations are given. Two numerical examples are provided to illustrate the use of our results in Section 4. Finally, the conclusion follows in Section 5.

2. Preliminaries

For convenience, we now introduce the following notations that will be employed throughout the paper. The notation 𝑅^𝑛 is used to denote the 𝑛-dimensional space. 𝑅⁺ denotes the set of all real nonnegative numbers. 𝑅^𝑛×𝑟 denotes the space of all (𝑛×𝑟)  matrices. ⟨𝑥,𝑦⟩ or 𝑥^𝑇𝑦 denotes the scalar product of two vectors 𝑥 and 𝑦. ‖𝑥‖ denotes the Euclidean vector norm of 𝑥. 𝜆(𝐴) denotes the set of all eigenvalues of 𝐴. 𝜆_max(𝐴)is defined by 𝜆_max(𝐴)=max{Re𝜆∶𝜆∈𝜆(𝐴)}. 𝜆_min(𝐴)  is defined by 𝜆_min(𝐴)=min{Re𝜆∶𝜆∈𝜆(𝐴)}. For ℎ≥0, 𝐶([−ℎ,0],𝑅^𝑛) denotes the set of all 𝑅^𝑛-valued continuous functions mapping [−ℎ,0] into  𝑅^𝑛. A matrix 𝐴 is semipositive definite (𝐴≥0) if ⟨𝐴𝑥,𝑥⟩≥0, for all 𝑥∈𝑅^𝑛. 𝐴 is positive definite (𝐴>0) if ⟨𝐴𝑥,𝑥⟩>0 for all 𝑥≠0. 𝐴≥𝐵 means 𝐴−𝐵≥0. Also, 𝑥_𝑡  is defined by 𝑥_𝑡={𝑥(𝑡+𝑠),𝑠∈[−ℎ,0]},  and  ‖𝑥_𝑡‖ is defined by ‖𝑥_𝑡‖=sup_{𝑠∈[−ℎ,0]}‖𝑥(𝑡+𝑠)‖.

Now, let us consider a class of uncertain systems with time-varying delay and nonlinear perturbations of the form𝑥𝑥̇((𝑡𝑡))==+𝜙[𝐴[(𝐵𝑡()𝑡)(,𝑡+)+𝑡Δ∈Δ𝐴𝐵[(−𝜐(,ℎ𝜍𝑡),,]0𝑡)𝑥]],(𝑢𝑡)ℎ(𝑡+)≥+0𝐴𝑓,₁((𝑡𝑡,)𝑥+(𝑡Δ),𝐴𝑥₁((𝑡𝜉−,𝑡)ℎ(𝑥𝑡)()𝑡,−𝑢(ℎ𝑡)()𝑡),)(2.1) where 𝑥(𝑡)∈𝑅^𝑛 is the state, 𝑢(𝑡)∈𝑅^𝑚 is the control function, 𝐴(𝑡),𝐴₁(𝑡),𝐵(𝑡) are continuous matrixes of appropriate dimensions on 𝑅⁺, and Δ𝐴(𝜐,𝑡), Δ𝐴₁(𝜉,𝑡), Δ𝐵(𝜍,𝑡) represent the system uncertainties and are assumed to be continuous in all their arguments. Moreover, (𝜐,𝜉,𝜍)∈𝜓 is the uncertain vector, and 𝜓⊂𝑅^𝐿 is a compact set. In addition, perturbation 𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))∶𝑅⁺×𝑅^𝑛×𝑅^𝑛×𝑅^𝑚→𝑅^𝑛 is a given nonlinear function satisfying 𝑓(𝑡,0,0,0)=0 for all 𝑡≥0, and there exist scalars 𝑎,𝑏,𝑑>0 such that‖𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))‖≤𝑎‖𝑥(𝑡)‖+𝑏‖𝑥(𝑡−ℎ(𝑡))‖+𝑑‖𝑢(𝑡)‖,(2.2) for all (𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))∈𝑅⁺×𝑅^𝑛×𝑅^𝑛×𝑅^𝑚. The initial function 𝜙(𝑡)∈𝐶([−ℎ,0],𝑅^𝑛), ℎ>0, has its norm ‖𝜙‖=sup_{𝑠∈[−ℎ,0]}‖𝜙(𝑠)‖. The delay ℎ(𝑡) is a continuous function satisfying (H1)0≤ℎ(𝑡)≤ℎ, ℎ̇(𝑡)≤𝛿<1, for all 𝑡≥0.

The purpose of this paper is to design a state feedback controller 𝑢(𝑡)=𝐾(𝑡)𝑥(𝑡) such that the closed-loop system of  (2.1)𝑥̇(𝑡)=+[𝐴𝐴(𝑡)+Δ𝐴(𝜐,𝑡)+𝐵(𝑡)𝐾(𝑡)+Δ𝐵(𝜍,𝑡)𝐾(𝑡)]𝑥(𝑡)₁𝑥(𝑡)=𝜙(𝑡),(𝑡)𝑡+∈Δ[𝐴−₁ℎ(,𝜉0,]𝑡,)ℎ𝑥≥(𝑡−0,ℎ(𝑡))+𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡)),(2.3) is robustly 𝛼-exponentially stable, that is, every solution 𝑥(𝑡,𝜙) of the system satisfies∃𝑁>0,𝛼>0,‖𝑥(𝑡,𝜙)‖≤𝑁‖𝜙‖𝑒^−𝛼𝑡,∀𝑡∈𝑅⁺,(2.4) for all the uncertainties Δ𝐴(𝜐,𝑡), Δ𝐴₁(𝜉,𝑡), Δ𝐵(𝜍,𝑡).

Before proposing our theorems, we introduce for (2.1) the following standard assumptions and lemmas that will be needed for deriving the main results.

Assumption 1. For all (𝑥,𝑡)∈𝑅^𝑛×𝑅 there exist continuous matrix functions 𝐻(𝜐,𝑡), 𝐻₁(𝜉,𝑡),  𝐸(𝜍,𝑡) of appropriate dimensions such that Δ𝐴(𝜐,𝑡)=𝐵(𝑡)𝐻(𝜐,𝑡),Δ𝐴₁(𝜉,𝑡)=𝐵(𝑡)𝐻₁(𝜉,𝑡),Δ𝐵(𝜍,𝑡)=𝐵(𝑡)𝐸(𝜍,𝑡).(2.5)

Remark 2.1. Assumption 1 defines the matching condition about the uncertainties and is a rather standard assumption for robust control problem (see, e.g., [10, 14–16]).

Lemma 2.2 (see [15]). For any real vectors 𝑎, 𝑏 and any matrix 𝑄>0 with appropriate dimensions, it follows that 2𝑎^𝑇𝑏≤𝑎^𝑇𝑄𝑎+𝑏^𝑇𝑄⁻¹𝑏.(2.6)

Lemma 2.3 (see [11]). Given constant symmetric matrices 𝑆₁, 𝑆₂, 𝑆₃, and 𝑆₁=𝑆^𝑇1<0, 𝑆₃=𝑆^𝑇3>0, then 𝑆₁+𝑆₂𝑆³⁻¹𝑆^𝑇2<0 if and only if 𝑆¹𝑆₂𝑆^𝑇2−𝑆₃<0.(2.7)

3. Main Results

In this section, we will present our main results on the robust exponential stabilization of system (2.1).

Given positive numbers 𝛼, 𝛽, 𝜀_𝑖, 𝑖=1,2,3,4, we set𝜔𝑃=𝜀₁+𝜀₃+𝜀₄+𝜀₂ℎ,𝐴(𝑡)=𝐴(𝑡)+𝛼𝐼,_𝛽𝜌(𝑡)=𝑃(𝑡)+𝛽𝐼,𝜌_𝜐(𝑡)=max_𝜐‖𝐻(𝜐,𝑡)‖,_𝜉(𝑡)=max_𝜉‖‖𝐻₁(𝜉,𝑡)‖‖,𝜇(𝑡)=min_𝜍12𝜆^min𝐸(𝜍,𝑡)+𝐸^𝑇𝑅(𝑡)=−4(𝜍,𝑡),𝑝=sup_𝑡∈𝑅+‖𝑃(𝑡)‖,𝜀₁𝑒^−2𝛼ℎ(1−𝛿)𝐴¹(𝑡)𝐴^𝑇1(𝑡)−14(1+𝜇(𝑡))²+4𝜌^2𝜉(𝑡)𝜀₁𝑒^−2𝛼ℎ(1−𝛿)+𝜀³⁻¹−𝜀𝜌^2𝜐(𝑡)−1𝐵(𝑡)𝐵^𝑇(𝑡)₄₋₁𝑎²+2𝑏²𝜀₁𝑒^−2𝛼ℎ(1−𝛿)+𝑑²𝐼.(3.1)

We need the following assumption.

Assumption 2. For any 𝑡>0, 𝜇(𝑡)>−1.

Theorem 3.1. Suppose that condition (H1) and Assumptions 1-2 hold. If there exist positive numbers 𝛼, 𝛽, 𝜀_𝑖, 𝑖=1,2,3,4, and a symmetric positive semidefinite matrix function 𝑃(𝑡) satisfying the following Riccati differential equation: 𝑃̇(𝑡)+A^𝑇(𝑡)𝑃_𝛽(𝑡)+𝑃_𝛽(𝑡)A(𝑡)−𝑃_𝛽(𝑡)𝑅(𝑡)𝑃_𝛽(𝑡)+𝜔𝐼=0,(3.2) then system (2.1) is robustly 𝛼-exponentially stabilizable with feedback control 𝑢(𝑡)=−12(1+𝜇(𝑡))𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡).(3.3) Moreover, the solution 𝑥(𝑡,𝜙) satisfies the condition ‖𝑥(𝑡,𝜙)‖≤𝑝+𝛽+ℎ𝜀₁+(1/2)ℎ²𝜀₂𝛽‖𝜙‖𝑒^−𝛼𝑡,𝑡≥0.(3.4)

Proof. Let 𝑢(𝑡)=𝐾(𝑡)𝑥(𝑡), where 𝐾(𝑡)=−12(1+𝜇(𝑡))𝐵^𝑇(𝑡)𝑃_𝛽(𝑡),𝑡≥0.(3.5) For the closed-loop system (2.3) of system (2.1), we consider the following Lyapunov-Krasovskii functional: 𝑉𝑡,𝑥_𝑡=𝑉₁+𝑉₂+𝑉₃,(3.6) where 𝑉₁=𝑉⟨𝑃(𝑡)𝑥(𝑡),𝑥(𝑡)⟩+𝛽⟨𝑥(𝑡),𝑥(𝑡)⟩,₂=𝜀₁^{𝑡𝑡−ℎ(𝑡)}𝑉𝑒^{2𝛼(𝑠−𝑡)}‖𝑥(𝑠)‖²𝑑𝑠,₃=𝜀₂^0−ℎ^{𝑡𝑡+𝜏−ℎ(𝑡+𝜏)}𝑒^{2𝛼(𝑠−𝑡)}‖𝑥(𝑠)‖²𝑑𝑠𝑑𝜏.(3.7) The time derivative of 𝑉 along the trajectory of (2.3) is given by 𝑉̇𝑡,𝑥_𝑡=𝑉₁̇+𝑉₂̇+𝑉₃̇,(3.8) where 𝑉₁̇==𝑃̇𝑃(̇𝑡()𝑡𝑥)+(𝑡)𝐴,𝑥_𝑇(𝑡)+2𝑃_𝛽(𝑡)𝑥̇(𝑡),𝑥(𝑡)+2𝑃_𝛽(𝑡)𝑃_𝛽(𝑡)+𝑃_𝛽(𝑡)𝐴(𝑡)𝑥(𝑡),𝑥(𝑡)+2𝑃_𝛽(𝑡)𝐵(𝑡)𝐻(𝜐,𝑡)𝑥(𝑡),𝑥(𝑡)+2𝑃_𝛽(𝑡)𝐴₁(𝑡)𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡)+2𝑃_𝛽(𝑡)𝐵(𝑡)𝐻₁(𝜉,𝑡)𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡)+2𝑃_𝛽(𝑡)𝐵(𝑡)𝐾(𝑡)𝑥(𝑡),𝑥(𝑡)+2𝑃_𝛽(𝑡)𝐵(𝑡)𝐸(𝜍,𝑡)𝐾(𝑡)𝑥(𝑡),𝑥(𝑡)𝑉̇(𝑡)𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡)),𝑥(𝑡)₂,=≤−−22𝛼𝛼𝑉𝑉₂₂+𝜀₁‖𝑥(𝑡)‖²−𝜀₁𝑒^{−2𝛼ℎ(𝑡)}‖𝑥(𝑡−ℎ(𝑡))‖²1−ℎ̇(𝑡)𝑉̇+𝜀₁‖𝑥(𝑡)‖²−𝜀₁𝑒^−2𝛼ℎ‖𝑥(𝑡−ℎ(𝑡))‖²(1−𝛿),₃=−2𝛼𝑉₃+𝜀₂^0−ℎ‖𝑥(𝑡)‖²≤−2𝛼𝑉₃−𝑒^{−2𝛼(𝜏−ℎ(𝑡+𝜏))}‖𝑥(𝑡+𝜏−ℎ(𝑡+𝜏))‖²1−ℎ̇(𝑡+𝜏)𝑑𝜏+𝜀₂ℎ‖𝑥(𝑡)‖²−𝜀₂^0−ℎ≤−2𝛼𝑉₃𝑒^{−2𝛼(𝜏−ℎ(𝑡+𝜏))}‖𝑥(𝑡+𝜏−ℎ(𝑡+𝜏))‖²(1−𝛿)𝑑𝜏+𝜀₂ℎ‖𝑥(𝑡)‖².(3.9) Applying Lemma 2.2 gives 2≤𝑃_𝛽(𝑡)𝐴₁(4𝑡)𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡)𝜀₁(1−𝛿)𝑒^−2𝛼ℎ𝑃_𝛽(𝑡)𝐴₁(𝑡)𝐴^𝑇1(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡),𝑥(𝑡)+𝜀¹(1−𝛿)𝑒^−2𝛼ℎ2𝑃4⟨𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡−ℎ(𝑡))⟩,_𝛽≤𝜀(𝑡)𝐵(𝑡)𝐻₁(𝜉,𝑡)𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡)₁(1−𝛿)𝑒^−2𝛼ℎ+44⟨𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡−ℎ(𝑡))⟩𝜀₁(1−𝛿)𝑒^−2𝛼ℎ≤𝜀𝑃_𝛽(𝑡)𝐵(𝑡)𝐻₁(𝜉,𝑡)𝐻^𝑇1(𝜉,𝑡)𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡),𝑥(𝑡)₁(1−𝛿)𝑒^−2𝛼ℎ4⟨𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡−ℎ(𝑡))⟩+4𝜌^2𝜉(𝑡)𝜀₁(1−𝛿)𝑒^−2𝛼ℎ2𝑃𝑃_𝛽(𝑡)𝐵(𝑡)𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡),𝑥(𝑡)_𝛽,≤𝜀(₃𝑡)𝐵(𝑡)𝐻(𝜐,𝑡)𝑥(𝑡),𝑥(𝑡)≤𝜀₃⟨𝑥(𝑡),𝑥(𝑡)⟩+𝜀³⁻¹𝑃_𝛽(𝑡)𝐵(𝑡)𝐻(𝜐,𝑡)𝐻^𝑇(𝜐,𝑡)𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡),𝑥(𝑡)⟨𝑥(𝑡),𝑥(𝑡)⟩+𝜀³⁻¹𝜌^2𝜐(𝑡)𝑃_𝛽(𝑡)𝐵(𝑡)𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡),𝑥(𝑡).(3.10) Using (2.2) and (3.3), we get 2𝑃_𝛽≤(𝑡2)‖𝑓𝑓(𝑡(,𝑡𝑥,𝑥(𝑡()𝑡),𝑥,𝑥(𝑡(𝑡−−ℎℎ(𝑡()𝑡))),𝑢,𝑢(𝑡()𝑡))),‖𝑥‖‖(𝑡𝑃)_𝛽≤2𝑎‖𝑥(𝑡)‖‖‖𝑃_𝛽(𝑡)𝑥(𝑡)‖‖≤𝜀₄₋₁(𝑡)𝑥(𝑡)‖‖+2𝑏‖𝑥(𝑡−ℎ(𝑡))‖‖‖𝑃_𝛽(𝑡)𝑥(𝑡)‖‖+2𝑑‖𝑢(𝑡)‖‖‖𝑃_𝛽(𝑡)𝑥(𝑡)‖‖𝑎²‖‖𝑃_𝛽(𝑡)𝑥(𝑡)‖‖²+𝜀₄‖𝑥(𝑡)‖²+2𝑏²𝜀₁𝑒^−2𝛼ℎ(1−𝛿)‖‖𝑃^𝛽(𝑡)𝑥(𝑡)‖‖²+𝜀¹𝑒^−2𝛼ℎ(1−𝛿)2‖𝑥(𝑡−ℎ(𝑡))‖²+𝑑²‖‖𝑃_𝛽(𝑡)𝑥(𝑡)‖‖²+‖𝑢(𝑡)‖²=𝜀⁴⁻¹𝑎²+2𝑏²𝜀₁𝑒^−2𝛼ℎ(1−𝛿)+𝑑²𝑃^2𝛽+1(𝑡)𝑥(𝑡),𝑥(𝑡)+𝜀₄⟨𝑥(𝑡),𝑥(𝑡)⟩4(1+𝜇(𝑡))²𝑃_𝛽(𝑡)𝐵(𝑡)𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡),𝑥(𝑡)+𝜀¹𝑒^−2𝛼ℎ(1−𝛿)2‖𝑥(𝑡−ℎ(𝑡))‖².(3.11) In addition, it is easy to obtain the following: 2𝑃_𝛽=(𝑡−)𝐵(𝑡1)𝐾(𝑡)𝑥(𝑡),𝑥(𝑡)+2𝑃_𝛽(𝑡)𝐵(𝑡)𝐸(𝜍,𝑡)𝐾(𝑡)𝑥(𝑡),𝑥(𝑡)1+𝜇(𝑡)𝑥^𝑇(𝑡)𝑃_𝛽(𝑡)𝐵(𝑡)𝐼+12𝐸(𝜍,𝑡)+𝐸^𝑇≤−1(𝜍,𝑡)𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡)1+𝜇(𝑡)𝜆^min𝐼+12𝐸(𝜍,𝑡)+𝐸^𝑇(𝜍,𝑡)‖‖𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡)‖‖²=−𝑃_𝛽(𝑡)𝐵(𝑡)𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)𝑥(𝑡),𝑥(𝑡).(3.12) The last equality is got because of 𝜇(𝑡)=min_𝜍[(1/2)𝜆_min(𝐸(𝜍,𝑡)+𝐸^𝑇(𝜍,𝑡))].
Therefore, we get 𝑉̇𝑡,≤𝑥_𝑡+2𝑃𝛼̇𝑉(𝑡)+𝑡,𝑥_𝑡𝐴^𝑇(𝑡)𝑃_𝛽(𝑡)+𝑃_𝛽(𝑡)𝐴(𝑡)+4𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+1𝑃_𝛽(𝑡)𝐴₁(𝑡)𝐴^𝑇1(𝑡)𝑃_𝛽(𝑡)4(1+𝜇(𝑡))²+4𝜌^2𝜉(𝑡)𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+𝜀+𝜀³⁻¹𝜌^2𝜐(𝑡)−1𝑃_𝛽(𝑡)𝐵(𝑡)𝐵^𝑇(𝑡)𝑃_𝛽(𝑡)₄₋₁𝑎²+2𝑏²𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+𝑑²𝑃^2𝛽(𝑡)+𝜀₁+𝜀₃+𝜀₄+𝜀₂ℎ𝐼𝑥(𝑡),𝑥(𝑡).(3.13) Then introducing (3.2) into (3.13) we can get 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤0,∀𝑡≥0.(3.14) It is obvious that 𝑒^2𝛼𝑡𝑉̇𝑡,𝑥_𝑡+2𝛼𝑒^2𝛼𝑡𝑉𝑡,𝑥_𝑡≤0.(3.15) Integrating both sides of (3.15) from 0 to 𝑡, we get 𝑉𝑡,𝑥_𝑡≤𝑉0,𝑥₀𝑒^−2𝛼𝑡,∀𝑡≥0.(3.16) On the other hand, from the expression of 𝑉(𝑡,𝑥_𝑡), it is easy to see that 𝛽‖𝑥(𝑡)‖²≤𝑉𝑡,𝑥_𝑡,∀𝑡≥0.(3.17) In addition, since 𝑉₁0,𝑥₀≤(𝑝+𝛽)‖𝜙‖²,𝑉₂0,𝑥₀=𝜀₁^0−ℎ(0)𝑒^2𝛼𝑠‖𝑥(𝑠)‖²𝑑𝑠≤𝜀₁^0−ℎ𝑉₃‖𝜙‖²𝑑𝑠≤𝜀₁ℎ‖𝜙‖²,0,𝑥₀=𝜀₂^0−ℎ^{0𝜏−ℎ(𝜏)}𝑒^2𝛼𝑠‖𝑥(𝑠)‖²𝑑𝑠𝑑𝜏≤𝜀₂^0−ℎ^0𝜏−ℎ‖𝜙‖²𝑑𝑠𝑑𝜏≤21𝜀²ℎ²‖𝜙‖²,(3.18) we get 𝑉(0,𝑥₀)≤(𝑝+𝛽+ℎ𝜀₁+(1/2)ℎ²𝜀₂)‖𝜙‖².
Hence we have ‖𝑥(𝑡,𝜙)‖≤𝑝+𝛽+ℎ𝜀₁+(1/2)ℎ²𝜀₂𝛽‖𝜙‖𝑒^−𝛼𝑡,𝑡≥0.(3.19) So the closed-loop system (2.3) is exponentially stable. This completes the proof.

Remark 3.2. Note that from the proof of Theorem 3.1, condition (3.2) can be relaxed via the following matrix inequality: 𝑃̇(𝑡)+𝐴^𝑇(𝑡)𝑃_𝛽(𝑡)+𝑃_𝛽(𝑡)𝐴(𝑡)−𝑃_𝛽(𝑡)𝑅(𝑡)𝑃_𝛽(𝑡)+𝜔𝐼≤0.(3.20)

In addition, we consider a class of uncertain systems with time-varying delays and simple nonlinear perturbations as follows:𝑥𝑥̇((𝑡𝑡))==+𝜙[𝐴((𝐵𝑡+)+,ΔΔ𝐴𝑡𝐵∈(𝜐(,𝜍[𝑡−,)]𝑡ℎ)𝑥),(𝑢0𝑡])+,+𝑓ℎ(𝐴≥𝑡,₁𝑥0+,(𝑡Δ),𝐴𝑥₁((𝑡𝜉−,𝑡)ℎ(𝑥𝑡)()𝑡)−,ℎ(𝑡))(3.21) where 𝐴, 𝐴₁, 𝐵 are constant matrices of appropriate dimensions and Δ𝐴(𝜐,𝑡), Δ𝐴₁(𝜉,𝑡), Δ𝐵(𝜍,𝑡) are unknown time-varying uncertain matrices, and there exist scalars 𝑎,𝑏,𝑑,𝑙₁,𝑙₂,𝑙₃>0 such that‖𝑓(𝑡,Δ𝑥𝐴(𝑡)(,𝜐𝑥,𝑡()𝑡Δ−𝐴ℎ_𝑇(𝑡)))‖≤𝑎‖𝑥(𝑡)‖+𝑏‖𝑥(𝑡−ℎ(𝑡))‖,∀(𝑡,𝑥(𝑡))∈𝑅⁺×𝑅^𝑛,(𝜐,𝑡)≤𝑙₁𝐼,Δ𝐴₁(𝜉,𝑡)Δ𝐴^𝑇1(𝜉,𝑡)≤𝑙₂𝐼,‖Δ𝐵‖≤𝑙₃.(3.22)

Given positive numbers 𝛼, 𝛽, 𝜀_𝑖, 𝑖=1,2,3,4, we set 𝐴^⌢=𝐴+𝛼𝐼,𝑃_𝛽=𝑃+𝛽𝐼,𝜔=𝜀₁+𝜀₃+𝜀₄+𝜀₂ℎ,𝑅=−4𝜀₁𝑒^−2𝛼ℎ(1−𝛿)𝐴¹𝐴^𝑇1+𝐵𝐵^𝑇−𝜀³⁻¹𝑙₁+𝜀⁴⁻¹𝑎²+4𝑙²𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+2𝑏²𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+𝑙₃‖‖𝐵^𝑇‖‖𝐼.(3.23)

We have the following theorem.

Theorem 3.3. Suppose that condition (H1) holds. If there exist positive numbers 𝛼, 𝛽, 𝜀_𝑖, 𝑖=1,2,3,4, and a symmetric positive define matrix 𝑋 such that the following LMIs hold: ⎛⎜⎜⎝𝑋𝐴^⌢𝑇+𝑋𝐴^⌢𝑋−𝑅−𝜔𝑋₋₁−𝑋𝑋−𝛽𝑋₋₁𝐼⎞⎟⎟⎠≤0,(3.24)𝐼<0,(3.25) then system (3.21) is robustly exponentially stabilizable with feedback control 𝑢(𝑡)=−12𝐵^𝑇𝑋⁻¹𝑥(𝑡).(3.26)

Moreover, the solution 𝑥(𝑡,𝜙) satisfies the condition ‖𝑥(𝑡,𝜙)‖≤𝜆_max(𝑃)+𝛽+ℎ𝜀₁+(1/2)ℎ²𝜀₂𝛽‖𝜙‖𝑒^−𝛼𝑡,𝑡≥0,(3.27) where 𝑃=𝑋⁻¹−𝛽𝐼 and 𝜆_max(𝑃) represents the maximum eigenvalue of 𝑃.

Proof. From (3.25), we get that 𝑃 is a symmetric positive define matrix.
For system (3.21), select a Lyapunov-Krasovskii functional as 𝑉𝑡,𝑥_𝑡=𝑉₁+𝑉₂+𝑉₃,(3.28) where 𝑉₁𝑉=⟨𝑃𝑥(𝑡),𝑥(𝑡)⟩+𝛽⟨𝑥(𝑡),𝑥(𝑡)⟩,₂=𝜀₁^{𝑡𝑡−ℎ(𝑡)}𝑉𝑒^{2𝛼(𝑠−𝑡)}‖𝑥(𝑠)‖²𝑑𝑠,₃=𝜀₂^0−ℎ^{𝑡𝑡+𝜏−ℎ(𝑡+𝜏)}𝑒^{2𝛼(𝑠−𝑡)}‖𝑥(𝑠)‖²𝑑𝑠𝑑𝜏.(3.29) The time derivative of 𝑉 along trajectory of (3.21) is given by 𝑉̇𝑡,𝑥_𝑡=𝑉₁̇+𝑉₂̇+𝑉₃̇,(3.30) where 𝑉₁̇==2𝑃𝐴_𝛽𝑇𝑥̇(𝑡),𝑥(𝑡)+2𝑃_𝛽+𝑃_𝛽𝐴𝑥(𝑡),𝑥(𝑡)+2𝑃_𝛽Δ𝐴(𝜐,𝑡)𝑥(𝑡),𝑥(𝑡)+2𝑃_𝛽𝐴₁𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡)+2𝑃_𝛽Δ𝐴₁(𝜉,𝑡)𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡)+2𝑃_𝛽𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡))),𝑥(𝑡)𝑉̇𝑃_𝛽𝐵𝐾𝑥(𝑡),𝑥(𝑡)+2𝑃_𝛽Δ𝐵𝐾𝑥(𝑡),𝑥(𝑡)₂,=≤−−22𝛼𝛼𝑉𝑉₂₂+𝜀₁‖𝑥(𝑡)‖²−𝜀₁𝑒^{−2𝛼ℎ(𝑡)}‖𝑥(𝑡−ℎ(𝑡))‖²1−ℎ̇(𝑡)𝑉̇+𝜀₁‖𝑥(𝑡)‖²−𝜀₁𝑒^−2𝛼ℎ‖𝑥(𝑡−ℎ(𝑡))‖²(1−𝛿),₃=−2𝛼𝑉₃+𝜀₂^0−ℎ‖𝑥(𝑡)‖²≤−2𝛼𝑉₃−𝑒^{−2𝛼(𝜏−ℎ(𝑡+𝜏))}‖𝑥(𝑡+𝜏−ℎ(𝑡+𝜏))‖²1−ℎ̇(𝑡+𝜏)𝑑𝜏+𝜀₂ℎ‖𝑥(𝑡)‖²−𝜀₂^0−ℎ≤−2𝛼𝑉₃𝑒^{−2𝛼(𝜏−ℎ(𝑡+𝜏))}‖𝑥(𝑡+𝜏−ℎ(𝑡+𝜏))‖²(1−𝛿)𝑑𝜏+𝜀₂ℎ‖𝑥(𝑡)‖²,𝑃_𝛽=𝑃+𝛽𝐼.(3.31) Applying Lemma 2.2 and inequality (3.22) gives 2≤𝑃_𝛽𝜀Δ₃₋₁𝐴(𝜐,𝑡)𝑥(𝑡),𝑥(𝑡)≤𝜀₃₋₁𝑃_𝛽Δ𝐴(𝜐,𝑡)Δ𝐴^𝑇(𝜐,𝑡)𝑃_𝛽𝑥(𝑡),𝑥(𝑡)+𝜀₃⟨𝑥(𝑡),𝑥(𝑡)⟩2𝑙₁𝑃^2𝛽𝑥(𝑡),𝑥(𝑡)+𝜀₃⟨𝑥(𝑡),𝑥(𝑡)⟩,≤𝑃_𝛽𝐴₁𝑥(𝑡4−ℎ(𝑡)),𝑥(𝑡)𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+𝜀𝑃_𝛽𝐴₁𝐴^𝑇1𝑃_𝛽𝑥(𝑡),𝑥(𝑡)₁(1−𝛿)𝑒^−2𝛼ℎ24⟨𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡−ℎ(𝑡))⟩,≤𝑃_𝛽Δ𝐴₁(𝜉4,𝑡)𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡)𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+𝜀𝑃_𝛽Δ𝐴₁(𝜉,𝑡)Δ𝐴^𝑇1(𝜉,𝑡)𝑃_𝛽𝑥(𝑡),𝑥(𝑡)₁(1−𝛿)𝑒^−2𝛼ℎ≤4𝑙₂4⟨𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡−ℎ(𝑡))⟩𝜀₁(1−𝛿)𝑒^−2𝛼ℎ𝑃^2𝛽𝑥(𝑡),𝑥(𝑡)+𝜀¹(1−𝛿)𝑒^−2𝛼ℎ24⟨𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡−ℎ(𝑡))⟩,≤𝑃_𝛽2𝑓‖𝑓(𝑡,(𝑥𝑡,(𝑥𝑡)(𝑡,)𝑥,(𝑥𝑡(−𝑡−ℎ(ℎ𝑡)()𝑡))),)𝑥‖(‖‖𝑡)≤2𝑎‖𝑥(𝑡)‖‖‖𝑃_𝛽(𝑡)𝑥(𝑡)‖‖≤𝜀₄₋₁𝑃_𝛽𝑥(𝑡)‖‖+2𝑏‖𝑥(𝑡−ℎ(𝑡))‖‖‖𝑃_𝛽𝑥(𝑡)‖‖𝑎²‖‖𝑃_𝛽𝑥(𝑡)‖‖²+𝜀₄‖𝑥(𝑡)‖²+2𝑏²𝜀₁𝑒^−2𝛼ℎ(1−𝛿)‖‖𝑃_𝛽𝑥(𝑡)‖‖²+𝜀¹𝑒^−2𝛼ℎ(1−𝛿)2‖𝑥(𝑡−ℎ(𝑡))‖²=𝜀⁴⁻¹𝑎²+2𝑏²𝜀₁𝑒^−2𝛼ℎ(1−𝛿)𝑃²_𝛽𝑥(𝑡),𝑥(𝑡)+𝜀₄⟨𝑥(𝑡),𝑥(𝑡)⟩+𝜀¹𝑒^−2𝛼ℎ(1−𝛿)2‖𝑥(𝑡−ℎ(𝑡))‖².(3.32)
Noticing 22𝑃_𝛽𝐵𝐾𝑥(𝑡),𝑥(𝑡)=−𝑃_𝛽𝐵𝐵^𝑇𝑃_𝛽𝑥(𝑡),𝑥(𝑡),𝑃_𝛽Δ𝐵𝐾𝑥(𝑡),𝑥(𝑡)=𝑃_𝛽Δ𝐵𝐵^𝑇𝑃_𝛽𝑥(𝑡),𝑥(𝑡)≤𝑙₃‖‖𝐵^𝑇‖‖‖‖𝑃_𝛽𝑥(𝑡)‖‖²,(3.33) we get 𝑉̇𝑡,≤𝑥_𝑡+2_⌢𝛼𝑉𝑡,𝑥_𝑡𝐴^𝑇𝑃_𝛽+𝑃𝐴_𝛽⌢+𝑃_𝛽𝜀₁(1−4𝛿)𝑒^−2𝛼ℎ𝐴₁𝐴^𝑇1−𝐵𝐵^𝑇+𝜀₃₋₁𝑃_𝛽+𝜀₁+𝜀₃+𝜀₄+𝜀₂ℎ𝐼𝑙₁+𝜀⁴⁻¹𝑎²+4𝑙²𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+2𝑏²𝜀₁(1−𝛿)𝑒^−2𝛼ℎ+𝑙₃‖‖𝐵^𝑇‖‖𝑃^2𝛽𝑥(𝑡),𝑥(𝑡).(3.34) Hence we obtain 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤𝑥^𝑇(𝑡)𝐴^⌢𝑇𝑃_𝛽+𝑃𝐴_𝛽⌢−𝑃_𝛽𝑅𝑃_𝛽+𝜔𝐼𝑥(𝑡).(3.35)
Noticing 𝑃_𝛽=𝑋⁻¹, and from (3.24) and using Lemma 2.3, we obtain 𝑃^𝛽⌢−1𝐴^𝑇+𝐴^⌢𝑃^𝛽−1−𝑅+𝜔𝑃^𝛽−1𝑃^𝛽−1≤0.(3.36) By pre-multiplying and post-multiplying the right-hand side of (3.36) with 𝑃_𝛽 it follows that 𝐴^⌢𝑇𝑃_𝛽+𝑃𝐴_𝛽⌢−𝑃_𝛽𝑅𝑃_𝛽+𝜔𝐼≤0.(3.37) So we have 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤0,∀𝑡≥0.(3.38) It is obvious that 𝑒^2𝛼𝑡𝑉̇𝑡,𝑥_𝑡+2𝛼𝑒^2𝛼𝑡𝑉𝑡,𝑥_𝑡≤0.(3.39) Integrating both sides of (3.15) from 0 to 𝑡, we get 𝑉𝑡,𝑥_𝑡≤𝑉0,𝑥₀𝑒^−2𝛼𝑡,∀𝑡≥0.(3.40) On the other hand, from the expression of 𝑉(𝑡,𝑥_𝑡), it is easy to see that 𝛽‖𝑥(𝑡)‖²≤𝑉𝑡,𝑥_𝑡,∀𝑡≥0.(3.41) In addition, since 𝑉₁0,𝑥₀≤𝜆_max(𝑃)+𝛽‖𝜙‖²,𝑉₂0,𝑥₀≤𝜀₁ℎ‖𝜙‖²,𝑉₃0,𝑥₀≤12𝜀²ℎ²‖𝜙‖²,(3.42) we get 𝑉(0,𝑥₀)≤(𝜆_max(𝑃)+𝛽+ℎ𝜀₁+(1/2)ℎ²𝜀₂)‖𝜙‖².
Hence we have ‖𝑥(𝑡,𝜙)‖≤𝜆_max(𝑃)+𝛽+ℎ𝜀₁+(1/2)ℎ²𝜀₂𝛽‖𝜙‖𝑒^−𝛼𝑡,𝑡≥0.(3.43) So the system (3.21) is robustly exponentially stabilizable. This completes the proof of Theorem 3.3.

4. Illustrative Examples

In this section, we will give numerical examples to demonstrate the effective of the proposed methods.

Example 4.1. Consider system (2.1) with ℎ(𝑡)=cos²0.25𝑡 and 𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))=⎛⎜⎜⎜⎝12cos𝑡𝑥¹(𝑡)+34sin𝑡𝑥²1(𝑡−ℎ(𝑡))2sin𝑡𝑥²(𝑡)+3𝐴(𝑡)=⎛⎜⎜⎜⎝(−6𝑒−4)cos𝑡−4−sin𝑡4cos𝑡𝑢(𝑡)⎞⎟⎟⎟⎠,(−4𝑒−2)2ccooss𝑡𝑡−+84𝑒−−412𝑒−127(−6𝑒−4()−c4o𝑒s−𝑡−2)c4o−s𝑡s−in8𝑡𝑒−4𝐴₁2cos𝑡+4−12𝑒−127⎞⎟⎟⎟⎠,(𝑡)=12𝑒^−0.50𝑒0_−0.5Δ𝐴(𝜐,𝑡)=𝜐𝜐((𝑡𝑡))00,Δ𝐴₁,𝐵(𝑡)=11,(𝜉,𝑡)=0022𝜉𝜉((𝑡𝑡)),Δ𝐵(𝜍,𝑡)=𝜍𝜍((𝑡𝑡)),(4.1) where 𝜐(𝑡)=𝜂₁, 𝜉(𝑡)=𝜂₂, 𝜍(𝑡)=𝜂₃ with |𝜂₁|≤2, |𝜂₂|≤0.5, |𝜂₃|≤0.5. It is easy to obtain that ‖𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑢(𝑡))‖≤‖𝑥(𝑡)‖+‖𝑥(𝑡−ℎ(𝑡))‖+‖𝑢(𝑡)‖,(4.2) that is, 𝑎=𝑏=𝑑=1. We can get ℎ=1, 𝛿=0.5. 𝜌_𝜐(𝑡)=2, 𝜌_𝜉(𝑡)=1, 𝜇(𝑡)=−0.5, 𝜔=4. For the positive numbers 𝜀₁=𝜀₂=𝜀₃=𝜀₄=1, 𝛽=1, 𝛼=0.5, we can verify that all the conditions of Theorem 3.1 are satisfied, and the solution of the Riccati differential equation (3.2) is given by 𝑃(𝑡)=cos0𝑡+1cos0𝑡+1.(4.3) Therefore, the system is robustly 𝛼-exponentially stabilizable with feedback control 𝑢(𝑡)=−(cos𝑡+1)𝑥₁(𝑡)−(cos𝑡+1)𝑥₂(𝑡).(4.4) Noting 𝑝=sup_𝑡∈𝑅⁺‖𝑃(𝑡)‖=2, the solution of the system satisfies ‖𝑥(𝑡,𝜙)‖≤2.1213‖𝜙‖𝑒^−0.5𝑡,𝑡∈𝑅⁺.(4.5)

Remark 4.2. The systems in the examples that are dealt by [17] do not contain uncertainty in the linear part on state and control, and those systems are robustly stabilizable. But, our Example 4.1 contains uncertainty in the linear part on state and control, and the system in Example 4.1 is robustly  𝛼-exponentially stabilizable. Using the similar approach in [17], our results can be extended to the systems with multiple delays.

Example 4.3. Consider system (3.21) with ℎ(𝑡)=0.5 and 𝐴=−215.5−205.5,𝐴₁=12𝑒^−0.50𝑒0_−0.5𝑙,𝐵=₁11,=1,𝑙₂=1,𝑙₃=√2,𝑎=1,𝑏=1.(4.6)

Taking 𝜀₁=𝜀₂=𝜀₃=𝜀₄=1, 𝛽=1, 𝛼=0.5, we can get ℎ=1,𝛿=0.5,𝐴^⌢=𝐴+𝛼𝐼=−125−025,𝜔=4,𝑅=−2𝐼+𝐵𝐵^𝑇−(1+1+8𝑒+4𝑒+2)𝐼=−(6+12𝑒)𝐼+1111.(4.7)

We can verify that all the conditions of Theorem 3.3 and LMI (3.24) and (3.25) are satisfied with𝑃=0.1101110.110111.(4.8)

By Theorem 3.3, the system is exponentially stabilizable with feedback control𝑢(𝑡)=−0.555556𝑥₁(𝑡)−0.555556𝑥₂(𝑡),(4.9) and the solution of the system satisfies‖𝑥(𝑡,𝜙)‖≤1.61589‖𝜙‖𝑒^−0.5𝑡,𝑡≥0.(4.10)

For𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)))=⎛⎜⎜⎜⎝12cos𝑡𝑥¹(𝑡)+34sin𝑡𝑥²1(𝑡−0.5)2sin𝑡𝑥²Δ𝐴(𝜐,𝑡)=0.5𝐼,Δ𝐴₁(𝑡)⎞⎟⎟⎟⎠,(𝜉,𝑡)=0.5𝐼,Δ𝐵(𝜍,𝑡)=00..55,(4.11) and the initial condition 𝜙(𝑡)=−0.50.5^𝑇,∀−0.5≤𝑡<0,(4.12) the simulation result is shown in Figure 1. It is seen from Figure 1 that the closed-loop system is exponentially stable.

Figure 1 

The state 𝑥1 and 𝑥2 of the closed-loop system in Example 4.3.

5. Conclusions

The problem of robust stabilization for a class of dynamical nonlinear systems with uncertainties and time-varying delays has been considered. On condition that the derivative of time-varying delays has restriction, a novel stability criterion which can guarantee the exponential stabilization of the uncertain systems with time-varying delay and nonlinear perturbations has been established by using the Riccati differential equation. The continuous state feedback controller has been proposed. Furthermore, based on the Lyapunov method, a linear matrix inequality approach to robust exponentially stabilization for a class of uncertain systems with time-varying delays and nonlinear perturbations via linear memoryless state feedback has been proposed. Finally, two illustrative examples have been given to demonstrate the utilization of the results.

Acknowledgments

This work was supported by the National Nature Science Foundation of China under Grant no. 50977047 and the Natural Science Foundation of Tianjin under Grant no. 11JCYBJC06800.