Robust Exponential Stabilization of Uncertain Impulsive Bilinear Time-Delay Systems with Saturating Actuators

Abstract

This paper investigates the problem of robust exponential stabilization for uncertain impulsive bilinear time-delay systems with saturating actuators. By using the Lyapunov function and Razumikhin-type techniques, two classes of impulsive systems are considered: the systems with unstable discrete-time dynamics and the ones with stable discrete-time dynamics. Sufficient conditions for robust stabilization are obtained in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the theoretical results.

1. Introduction

In practical control systems, impulsive dynamical systems are very important and have attracted considerable interest in science and engineering during the past decades. Two classical monographs are Lakshmikantham et al. [1] and Bainov and Simeonov [2]. In general, as reported in [1], impulsive systems provide a natural framework for mathematical modeling of many real-world evolutionary processes where the states undergo abrupt changes at certain instants or at variable instants, or autonomous systems with impulsive effects. Stability properties of impulsive systems have been extensively studied in the literatures. We refer to Bainov and Simeonov [2], Li, Soh and Xu [3, 4], Li, Wen and Soh [5], Yang [6], Chen and Zheng [7], Barreiro and Baños [8], and the references therein. The impulsive control method based on stability theory of impulsive dynamical systems has been widely used, see [7], Xu and Teo [9], and Liu, Eberhard and Teo [10].

Saturating nonlinear actuators is one of the common nonlinearities and exists in many practical systems. If a control system is designed to be stable without considering the saturation, the stability of the closed-loop-controlled system cannot be guaranteed, see Ma and Zhang [11]. Two important papers on saturation control are Full [12] and Sontag and Sussmann [13]. However, as we have known, no papers have considered the saturation control in impulsive systems.

This work is inspired by Wu and Wei [7], in which the authors considered the problems of robust stability and stabilization for uncertain impulsive time-delay systems. Unfortunately, they need all the impulsive time sequences to satisfy some strict conditions. That is, the length of the intervals between two adjacent time instants must have upper bound or lower bound. But in practical systems, it is always impossible or difficult to obtain it. In this paper, robust stabilization for uncertain impulsive time-delay systems with saturating actuators is considered. With saturation control, new conditions based on Lapunov-Razumikhin function and LMIs are established which can easily be used for systems with any time sequences.

2. Problem Formulation and Preliminaries

Throughout this paper, if not explicitly given, matrices are assumed to have compatible dimensions. For symmetric matrices 𝐴 and 𝐵, the notation 𝐴≥𝐵 (𝐴>𝐵, 𝐴≤𝐵, 𝐴<𝐵) means 𝐴−𝐵 is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix. 𝜆_max(⋅) and 𝜆_min(⋅) represent the maximum and minimum eigenvalues of the corresponding matrix, respectively. ‖⋅‖ denotes Euclidean norm for vectors or the spectral norm of matrices. 𝑃𝐶([−𝜏,0],ℝ^𝑛) denotes the set of piecewise right continuous function 𝜙∶[−𝜏,0]→ℝ^𝑛 with the norm defined by ‖𝜙‖_𝜏=sup_{−𝜏≤𝜃≤0}‖𝜙(𝜃)‖, where 𝜏 is a known positive constant.

Consider uncertain bilinear impulsive systems with time-delay and saturating actuators𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐴₁(𝑡)𝑥(𝑡−𝜏(𝑡))+𝐵(𝑡)sat(𝑢(𝑡)),𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=(𝐶(𝑡)−𝐼)𝑥(𝑡⁻)+^𝑚_𝑖=1sat𝑢_𝑖(𝑡)𝐵_𝑖(𝑡)𝑥(𝑡⁻),𝑡=𝑡_𝑘,𝑥𝑡₀+𝜃=𝜙(𝜃),𝑡₀=0,𝜃∈[−𝜏,0], (1) where 𝑥(𝑡)∈ℝ^𝑛 is the system state, 𝑢(𝑡)∈ℝ^𝑚 is the continuous control input, 0≤𝜏(𝑡)≤𝜏, Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘) describes the state jumping at impulsive time instant 𝑡=𝑡_𝑘 with 𝑥(𝑡⁺_𝑘)=𝑥(𝑡_𝑘)=lim_ℎ→0+𝑥(𝑡_𝑘+ℎ), 𝑥(𝑡⁻_𝑘)=lim_ℎ→0−𝑥(𝑡_𝑘+ℎ), 𝑘∈ℕ={1,2,…} and 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯ (𝑡_𝑘→∞ as 𝑘→∞).  𝐴(𝑡),𝐴₁(𝑡),𝐵(𝑡),𝐵_𝑖(𝑡)(𝑖=1,2,…,𝑚), and 𝐶(𝑡) are matrix functions with time-varying uncertainties Δ𝐴(𝑡), Δ𝐴₁(𝑡), Δ𝐵(𝑡), Δ𝐵_𝑖(𝑡) (𝑖=1,2,…,𝑚) and Δ𝐶(𝑡). In this paper, we impose the following hypothesis on the above uncertainties:Δ𝐴(𝑡),Δ𝐴₁(𝑡),Δ𝐵(𝑡)=𝐷₁𝐹₁(𝑡)𝑁_𝐴,𝑁_𝐴1,𝑁_𝐵,Δ𝐶(𝑡),Δ𝐵_𝑖(𝑡)=𝐷₂𝐹₂(𝑡)𝑁_𝐶,𝑁_𝐵𝑖,(𝑖=1,2,…,𝑚), (2) where 𝐷₁,𝐷₂,𝑁_𝐴,𝑁_𝐴1,𝑁_𝐵,𝑁_𝐶,𝑁_𝐵𝑖(𝑖=1,2,…,𝑚) are known real constant matrices and 𝐹_𝑖(𝑡) are unknown matrix functions satisfying ‖𝐹_𝑖(𝑡)‖≤1 with the elements of 𝐹_𝑖(𝑡) is lebesgue measurable, (𝑖=1,2). The saturating function Sat(𝑢(𝑡)) is defined asSat(𝑢(𝑡))=sat𝑢₁(𝑡),…,sat𝑢_𝑚(𝑡)^𝑇,sat𝑢_𝑗(𝑡)=−𝑈_𝑗,𝑢_𝑗(𝑡)≤−𝑈_𝑗,𝑢_𝑗(𝑡),−𝑈_𝑗<𝑢_𝑗(𝑡)<𝑈_𝑗,𝑈_𝑗,𝑢_𝑗(𝑡)≥𝑈_𝑗, (3)(4) where 𝑈_𝑗>0(𝑗=1,2,…,𝑚) are known real constants representing the upper bound of the control.

In the following, we will divide two cases to consider the robust stabilization of system (1). We denote by 𝒩_min(𝛽) the class of impulsive time sequences that satisfy inf_𝑘{𝑡_𝑘−𝑡_𝑘−1}≥𝛽 and denote by 𝒩_max(𝛽) the class of impulsive time sequences that satisfy sup_𝑘{𝑡_𝑘−𝑡_𝑘−1}≤𝛽.

We need the following lemmas.

Lemma 1 (see Su, Chu, Lu and Ji [14]). For any vectors 𝑥,𝑦∈ℝ^𝑛 and real matrices 𝐴,𝐷,𝐸,𝐹 and 𝑃>0 with 𝐹^𝑇𝐹≤𝐼, the following inequalities hold for any scalar 𝜀>0:(1)2𝑥^𝑇𝐷𝐹𝐸𝑦≤(1/𝜀)𝑥^𝑇𝐷𝐷^𝑇𝑥+𝜀𝑦^𝑇𝐸^𝑇𝐸𝑦,(2)2𝑥^𝑇𝑦≤𝑥^𝑇𝑃⁻¹𝑥+𝑦^𝑇𝑃𝑦 and for 𝑃−𝜀𝐷𝐷^𝑇>0,(3)(𝐴+𝐷𝐹𝐸)^𝑇𝑃⁻¹(𝐴+𝐷𝐹𝐸)≤(1/𝜀)𝐸^𝑇𝐸+𝐴^𝑇(𝑃−𝜀𝐷𝐷^𝑇)⁻¹𝐴.

Lemma 2 (Schur complement [14]). For given symmetric matrix 𝑆=(^𝑆11𝑆12_𝑆21𝑆22) with 𝑆₁₁∈ℝ^𝑟×𝑟, the following three conditions are equivalent:(1)𝑆<0,(2)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0,(3)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

3. Robust Stabilization

In this section, we consider the robust stabilization of system (1).

Definition 3. System (1) with 𝐹_𝑖(𝑡)=0 (𝑖=1,2) and 𝑢(𝑡)=0 is said to be exponentially stable over a given class 𝒩 if there are scalars 𝜀>0,𝛿>0 and 𝑣>0 such that 𝜙∈𝑃𝐶([−𝜏,0],ℝ^𝑛) with ‖𝜙‖_𝜏≤𝛿 implying ‖𝑥(𝑡;𝑡₀,𝜙)‖<𝜀exp(−𝑣(𝑡−𝑡₀)) for 𝑡≥𝑡₀.

Definition 4. System (1) is said to be robustly exponentially stabilizable over a given class 𝒩 with respect to static state feedback if there exists a memoryless linear state feedback law, such that the corresponding closed-loop system is robustly exponentially stable for all uncertainties satisfying (2).

For any 𝜙∈𝑃𝐶([−𝜏,0],ℝ^𝑛), we denote 𝑥(𝑡;𝑡₀,𝜙) by 𝑥(𝑡). Let us now consider designing a memoryless state feedback control law of the form𝑢(𝑡)=2𝐾𝑥(𝑡) (5) to stabilize system (1), where 2𝐾 is a constant gain to be designed.

Now we decompose the saturation nonlinear function Sat(𝑢(𝑡)) as the sum of a linear part and another nonlinear part (dead zone nonlinearity) Δ𝑢(𝑡) as Sat(𝑢(𝑡))=𝑢(𝑡)+Δ𝑢(𝑡), then by [14], (5) implies thatΔ𝑢(𝑡)^𝑇Δ𝑢(𝑡)≤𝑥^𝑇(𝑡)𝐾^𝑇𝐾𝑥(𝑡). (6)

Substituting (5) into system (1), we have𝑥̇(𝑡)=(𝐴(𝑡)+2𝐵(𝑡)𝐾)𝑥(𝑡)+𝐴₁(𝑡)𝑥(𝑡−𝜏(𝑡))+𝐵(𝑡)Δ𝑢(𝑡),𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=(𝐶(𝑡)−𝐼)𝑥(𝑡⁻)+^𝑚_𝑖=1sat𝑢_𝑖(𝑡)𝐵_𝑖(𝑡)𝑥(𝑡⁻),𝑡=𝑡_𝑘,𝑥𝑡₀+𝜃=𝜙(𝜃),𝑡₀=0,𝜃∈[−𝜏,0], (7)

Lemma 5. Assume that there exist matrix 𝑋>0 and positive scalars 𝜀₁, 𝜀₂, μ such that−𝜇𝑋𝑋𝑁^𝑇_𝐶𝑋𝐶^𝑇𝑋0⋯0∗−𝜀₁𝐼00𝑈₁𝑁_𝐵1⋯𝑈_𝑚𝑁_𝐵𝑚∗∗𝜀₁𝐷₂𝐷^𝑇₂−𝑋0𝑈₁𝐵₁⋯𝑈_𝑚𝐵_𝑚∗∗∗−𝜀₂^𝑚_𝑖=1𝑈_𝑖⁻¹𝐼0⋯0∗∗∗∗−𝜀₂𝐼⋯0⋯⋯⋱∗∗∗∗∗⋯−𝜀₂𝐼≤0, (8)and set 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡), where 𝑃=𝑋⁻¹. Then 𝑉𝑡_𝑘≤𝜇𝑉𝑡⁻_𝑘. (9)

Proof. By the second equation of (1), we have 𝑥𝑡_𝑘=𝐶𝑡_𝑘+^𝑚_𝑖=1sat𝑢_𝑖𝑡_𝑘𝐵_𝑖𝑡_𝑘𝑥𝑡⁻_𝑘𝑉𝑡_𝑘−𝜇𝑉𝑡⁻_𝑘=𝑥^𝑇𝑡⁻_𝑘Ξ₁𝑥𝑡⁻_𝑘, (10) where Ξ₁=(𝐶(𝑡)+∑^𝑚_𝑖=1sat(𝑢_𝑖(𝑡))𝐵_𝑖(𝑡))^𝑇𝑃(𝐶(𝑡)+∑^𝑚_𝑖=1sat(𝑢_𝑖(𝑡))𝐵_𝑖(𝑡)). By Lemmas 1 and 2, (2), (4), and (8) imply that Ξ₁≤0, hence (9) holds.

Theorem 6. Assume that there exist matrices 𝑋>0, 𝑌 and positive scalars 𝛼, 𝑑, 𝜀₁, 𝜀₂, 𝜀₃ and 0<𝜇<1 such that (8) and Ψ₁𝐴₁𝑋𝐵𝑌^𝑇2𝑌^𝑇𝑁^𝑇_𝐵1+𝑋𝑁^𝑇_𝐴𝐷₁∗−𝛼𝑋00𝑋𝑁^𝑇_𝐴10∗∗−𝑑⁻¹𝐼0𝑁^𝑇_𝐵0∗∗∗−𝑑𝐼00∗∗∗∗−𝜀₃𝐼0∗∗∗∗∗−𝜀₃⁻¹𝐼<0 (11) hold, then for any bounded time-delay 𝜏(𝑡)≤𝜏, system (1) can be robustly stabilized by the control law (5) with 𝐾=𝑌𝑋⁻¹ for any impulsive time sequence {𝑡_𝑘}, where Ψ₁=𝛼𝜇⁻¹𝑋+𝐴𝑋+𝑋𝐴^𝑇+2𝐵𝑌+2𝑌^𝑇𝐵^𝑇.

Proof. Let 𝑃=𝑋⁻¹, 𝜆₀=𝜆_min(𝑃), 𝜆₁=𝜆_max(𝑃). Substituting 𝑌=𝐾𝑋⁻¹ in (11) and pre- and postmultiplying it by diag{𝑃,𝑃,𝐼,𝐼,𝐼,𝐼} yield Ψ^′₁𝑃𝐴₁𝑃𝐵𝐾^𝑇𝑁_𝐴+2𝑁_𝐵𝐾^𝑇𝑃𝐷₁∗−𝛼𝑃00𝑁^𝑇_𝐴10∗∗−𝑑⁻¹𝐼0𝑁^𝑇_𝐵0∗∗∗−𝑑𝐼00∗∗∗∗−𝜀₃𝐼0∗∗∗∗∗−𝜀₃⁻¹𝐼<0, (12) where Ψ^′₁=𝛼𝜇⁻¹𝑃+(𝐴+2𝐵𝐾)^𝑇𝑃+𝑃(𝐴+2𝐵𝐾). Hence, there exists small positive scalar 𝜇, such thatΨ₁^′′𝑃𝐴₁𝑃𝐵𝐾^𝑇𝑁_𝐴+2𝑁_𝐵𝐾^𝑇𝑃𝐷₁∗−𝛼exp−𝜇𝜏𝑃00𝑁^𝑇_𝐴10∗∗−𝑑⁻¹𝐼0𝑁^𝑇_𝐵0∗∗∗−𝑑𝐼00∗∗∗∗−𝜀₃𝐼0∗∗∗∗∗−𝜀₃⁻¹𝐼<0, (13)where Ψ₁^′′=(𝛼𝜇⁻¹+𝜇)𝑃+(𝐴+2𝐵𝐾)^𝑇𝑃+𝑃(𝐴+2𝐵𝐾).

For given 𝜌>0, choose 0<𝛿≤𝜇𝜆₀/𝜆₁𝜌. We assume that the initial function 𝜙∈𝑃𝐶([−𝜏,0],ℝ^𝑛) satisfying ‖𝜙‖_𝜏≤𝛿. Set 𝑊(𝑡)=exp(𝜇𝑡)𝑉(𝑡). In the following, we will prove that for 𝑡≥𝑡₀−𝜏,𝑊(𝑡)<𝜆₀𝜌². (14)

First, we prove that (14) holds for 𝑡∈[𝑡₀−𝜏,𝑡₁). It is noticed that for 𝜃∈[−𝜏,0], 𝑊(𝑡₀+𝜃)≤𝜆₁‖𝜙‖²_𝜏≤𝜇𝜆₀𝜌²<𝜆₀𝜌². So it needs only to prove that, for 𝑡∈(𝑡₀,𝑡₁), (14) holds. Otherwise, there exists 𝑡∈(𝑡₀,𝑡₁), such that 𝑊(𝑡)≥𝜆₀𝜌². Set 𝑡^∗=inf{𝑡∈[𝑡₀,𝑡₁)∶𝑊(𝑡)≥𝜆₀𝜌²}. Then 𝑡^∗∈(𝑡₀,𝑡₁), 𝑊(𝑡^∗)=𝜆₀𝜌². Set 𝑡=sup{𝑡∈[𝑡₀,𝑡^∗)∶𝑊(𝑡)≤𝜇𝜆₀𝜌²}. Then 𝑡∈[𝑡₀,𝑡^∗) and 𝑊(𝑡)=𝜇𝜆₀𝜌². Hence, for 𝑡∈[𝑡,𝑡^∗] and 𝜃∈[−𝜏,0], we have𝜇𝑊(𝑡+𝜃)≤𝜇𝜆₀𝜌²≤𝑊(𝑡)≤𝜆₀𝜌². (15) It follows with (6) that𝐷⁺𝑊(𝑡)≤exp𝜇𝑡𝜇𝑉(𝑡)+𝐷⁺𝑉(𝑡)+𝛼𝜇⁻¹𝑊(𝑡)−𝑊(𝑡−𝜏(𝑡))+𝑑⁻¹𝑥^𝑇(𝑡)𝐾^𝑇𝐾𝑥(𝑡)−Δ𝑢(𝑡)^𝑇(𝑡)𝜂Δ𝑢exp𝜇𝑡≤exp𝜇𝑡^𝑇(𝑡)Ξ₂𝜂(𝑡), (16) where 𝜂^𝑇(𝑡)=𝑥^𝑇(𝑡),𝑥(𝑡−𝜏(𝑡))^𝑇,Δ𝑢(𝑡)^𝑇,Ξ₂=Ψ^′₂𝑃𝐴₁−𝑃𝐵∗−𝛼exp𝜇𝜏𝑃0∗∗−𝑑⁻¹𝐼+𝑑⁻¹𝐾^𝑇00𝐾00+𝜀₃⁻¹𝑁_𝐴+2𝑁_𝐵𝐾^𝑇𝑁^𝑇_𝐴1𝑁^𝑇_𝐵𝑁_𝐴+2𝑁_𝐵𝐾𝑁_𝐴1𝑁_𝐵+𝜀₃𝑃𝐷₁00𝐷^𝑇₁𝑃00, (17) By Lemma 2, (13) implies that Ξ₂<0, which yields by (16) that for 𝑡∈[𝑡,𝑡^∗],𝐷⁺𝑊(𝑡)<0. (18) It follows that 𝑊(𝑡^∗)≤𝑊(𝑡). This is a contradiction with the fact 𝑊(𝑡^∗)=𝜆₀𝜌²>𝑊(𝑡)=𝜇𝜆₀𝜌². Thus, (14) holds for 𝑡∈[𝑡₀−𝜏,𝑡₁).

Now we assume that (14) holds for 𝑡∈[𝑡₀−𝜏,𝑡_𝑚); we will prove that (14) still holds for 𝑡∈[𝑡_𝑚,𝑡_𝑚+1). On the contrary, there exists some 𝑡∈[𝑡_𝑚,𝑡_𝑚+1), such that 𝑊(𝑡)>𝜆₀𝜌². Set 𝑡^∗=inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1)∶𝑊(𝑡)≥𝜆₀𝜌²}. Then because of (9), we have 𝑊(𝑡_𝑚)≤𝜇𝜆₀𝜌²<𝜆₀𝜌², which implies that 𝑡^∗∈(𝑡_𝑚,𝑡_𝑚+1) and 𝑊(𝑡^∗)=𝜆₀𝜌². Set 𝑡=sup{𝑡∈[𝑡_𝑚,𝑡_∗)∶𝑊(𝑡)≤𝜇𝜆₀𝜌²}. Then 𝑡∈[𝑡_𝑚,𝑡^∗) and 𝑊(𝑡)=𝜇𝜆₀𝜌². Hence, for 𝑡∈[𝑡,𝑡^∗], we obtain that (15) holds, which leads that (16) and (18) hold. It follows that 𝑊(𝑡^∗)≤𝑊(𝑡), which contradicts the fact 𝑊(𝑡^∗)=𝜆₀𝜌²>𝑊(𝑡)=𝜇𝜆₀𝜌². Thus, (14) holds for 𝑡∈[𝑡₀−𝜏,𝑡_𝑚+1). By mathematical induction, (14) holds for 𝑡≥𝑡₀−𝜏. That is, system (1) can be robustly stabilized.

Remark 7. By Theorem 6, in the case 𝜇<1, different from Chen and Zheng [7], we can stabilize impulsive systems with any time sequences by saturation control.

Remark 8. In Chen and Zheng [7], the authors need continuous control input 𝑢_𝑐(𝑡) and impulsive control input 𝑢_𝑑(𝑡). But in this paper, we only need continuous control input 𝑢(𝑡).

For 𝜇=1, we have the following result.

Theorem 9. Assume that there exist matrices 𝑋>0, 𝑌 and positive scalars 𝛼, 𝑑, 𝜀₁, 𝜀₂, and 𝜀₃ such that (8) and (11) hold, then for any bounded time-delay 𝜏(𝑡)≤𝜏, system (1) can be robustly stabilized by the control law (5) with 𝐾=𝑌𝑋⁻¹ for any impulsive time sequence {𝑡_𝑘}, where Ψ₂=𝛼𝑋+𝐴𝑋+𝑋𝐴^𝑇+2𝐵𝑌+2𝑌^𝑇𝐵^𝑇.

Proof. Choose a sufficiently small scalar ℎ>0, and set 𝜇=1−ℎ. Then for ‖𝜙‖_𝜏≤𝛿≤𝜇𝜆₀/𝜆₁𝜌 with 𝜙∈𝑃𝐶([−𝜏,0],ℝ^𝑛), similar to the proof of Theorem 6, we can obtain the result of Theorem 9. The proof is complete.

For 𝜇>1, we have the following result.

Theorem 10. Assume that for a prescribed scalar 𝛽>0, there exist matrices 𝑋>0, 𝑌 and positive scalars 𝛼, 𝑑, 𝜀₁, 𝜀₂, 𝜀₃, and 𝜇>1 such that (8) and Ψ₃𝐴₁𝑋𝐵𝑌^𝑇2𝑌^𝑇𝑁^𝑇_𝐵+𝑋𝑁^𝑇_𝐴𝐷₁∗−𝛼𝑋00𝑋𝑁^𝑇_𝐴10∗∗−𝑑⁻¹𝐼0𝑁^𝑇_𝐵0∗∗∗−𝑑𝐼00∗∗∗∗−𝜀₃𝐼0∗∗∗∗∗−𝜀₃⁻¹𝐼<0 (19) hold, then for any bounded time-delay 𝜏(𝑡)≤𝜏, there exists a positive scalar ℎ, such that system (1) can be robustly stabilized by the control law (5) with 𝐾=𝑌𝑋⁻¹ over time sequence 𝒩_min(𝛽−ℎ), where Ψ₃=(𝛽⁻¹ln𝜇+𝛼𝜇)𝑋+𝐴𝑋+𝑋𝐴^𝑇+2𝐵𝑌+2𝑌^𝑇𝐵^𝑇.

Proof. Let 𝑃=𝑋⁻¹, 𝜆₀=𝜆_min(𝑃), 𝜆₁=𝜆_max(𝑃), and 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡). Substituting 𝑌=𝐾𝑋⁻¹ in (19) and pre- and postmultiplying it by diag{𝑃,𝑃,𝐼,𝐼,𝐼,𝐼} yield Ψ^′₃𝑃𝐴₁𝑃𝐵𝐾^𝑇𝑁_𝐴+2𝑁_𝐵𝐾^𝑇𝑃𝐷₁∗−𝛼𝑃00𝑁^𝑇_𝐴10∗∗−𝑑⁻¹𝐼0𝑁^𝑇_𝐵0∗∗∗−𝑑𝐼00∗∗∗∗−𝜀₃𝐼0∗∗∗∗∗−𝜀₃⁻¹𝐼<0, (20) where Ψ^′₃=(𝛽⁻¹ln𝜇+𝛼𝜇)𝑃+(𝐴+2𝐵𝐾)^𝑇𝑃+𝑃(𝐴+2𝐵𝐾). Hence, there exist small positive scalars 𝜇 and ℎ, such thatΨ₃^′′𝑃𝐴₁𝑃𝐵𝐾^𝑇𝑁_𝐴+2𝑁_𝐵𝐾^𝑇𝑃𝐷₁∗−𝛼exp−𝜇𝜏𝑃00𝑁^𝑇_𝐴10∗∗−𝑑⁻¹𝐼0𝑁^𝑇_𝐵0∗∗∗−𝑑𝐼00∗∗∗∗−𝜀₃𝐼0∗∗∗∗∗−𝜀₃⁻¹𝐼<0, (21)where Ψ₃^′′=(𝜇₁+𝛼𝜇+𝜇)𝑃+(𝐴+2𝐵𝐾)^𝑇𝑃+𝑃(𝐴+2𝐵𝐾), 𝜇₁=(𝛽−ℎ)⁻¹ln𝜇.

For given 𝜌>0, choose 0<𝛿≤𝜆₀/𝜇𝜆₁𝜌. We assume that the initial function 𝜙∈𝑃𝐶([−𝜏,0],ℝ^𝑛) satisfying ‖𝜙‖_𝜏≤𝛿. Set 𝑊(𝑡)=exp(𝜇𝑡)𝑉(𝑡). In the following, we will prove that for 𝑡≥𝑡₀−𝜏, (14) holds. First, we prove that (14) holds for 𝑡∈[𝑡₀−𝜏,𝑡₁). It is noticed that for 𝜃∈[−𝜏,0], 𝑊(𝑡₀+𝜃)≤𝜆₁‖𝜙‖²_𝜏≤𝜇⁻¹𝜆₀𝜌²<𝜆₀𝜌². Similar to the proof of Theorem 6, it needs only to prove that, for 𝑡∈(𝑡₀,𝑡₁), (14) holds. Otherwise, there exists 𝑡∈(𝑡₀,𝑡₁), such that 𝑊(𝑡)≥𝜆₀𝜌². Set 𝑡^∗=inf{𝑡∈[𝑡₀,𝑡₁)∶𝑊(𝑡)≥𝜆₀𝜌²}. Then 𝑡^∗∈(𝑡₀,𝑡₁), 𝑊(𝑡^∗)=𝜆₀𝜌². Set 𝑡=sup{𝑡∈[𝑡₀,𝑡^∗)∶𝑊(𝑡)≤𝜇⁻¹𝜆₀𝜌²}. Then 𝑡∈[𝑡₀,𝑡^∗) and 𝑊(𝑡)=𝜇⁻¹𝜆₀𝜌². Hence, for 𝑡∈[𝑡,𝑡^∗] and 𝜃∈[−𝜏,0], we have𝑊(𝑡+𝜃)≤𝜆₀𝜌²≤𝜇𝑊(𝑡). (22) By (6), (7), and (22), we have 𝐷⁺𝑊(𝑡)≤exp𝜇𝑡𝜇𝑉(𝑡)+𝐷⁺𝑉(𝑡)+𝛼𝜇𝑊(𝑡)−𝑊(𝑡−𝜏(𝑡))+𝑑⁻¹𝑥^𝑇(𝑡)𝐾^𝑇𝐾𝑥(𝑡)−Δ𝑢(𝑡)^𝑇(𝑡)𝜂Δ𝑢exp𝜇𝑡≤exp𝜇𝑡^𝑇(𝑡)Ξ₃𝜂(𝑡)−𝜇₁𝑊(𝑡), (23) where Ξ₃=Ψ₃^′′𝑃𝐴₁−𝑃𝐵∗−𝛼exp𝜇𝜏𝑃0∗∗−𝑑⁻¹𝐼+𝑑⁻¹𝐾^𝑇00𝐾00+𝜀₃⁻¹𝑁_𝐴+2𝑁_𝐵𝐾^𝑇𝑁^𝑇_𝐴1𝑁^𝑇_𝐵𝑁_𝐴+2𝑁_𝐵𝐾𝑁_𝐴1𝑁_𝐵+𝜀₃𝑃𝐷₁00𝐷^𝑇₁𝑃00, (24) By Lemma 2, (21) implies that Ξ₃<0, which yields by (23) that for 𝑡∈[𝑡,𝑡^∗],𝐷⁺𝑊(𝑡)<−𝜇₁𝑊(𝑡). (25) It follows that 𝑊(𝑡^∗)≤𝑊(𝑡). This is a contradiction with the fact 𝑊(𝑡^∗)=𝜆₀𝜌²>𝑊(𝑡)=𝜇⁻¹𝜆₀𝜌². Thus, (14) holds for 𝑡∈[𝑡₀−𝜏,𝑡₁).

Now we assume that (14) holds for 𝑡∈[𝑡₀−𝜏,𝑡_𝑚); we will prove that (14) still holds for 𝑡∈[𝑡_𝑚,𝑡_𝑚+1). First, we claim that 𝑊(𝑡⁻_𝑚)≤𝜇⁻¹𝜆₀𝜌². Otherwise, we have 𝑊(𝑡⁻_𝑚)>𝜇⁻¹𝜆₀𝜌². Here we must consider two cases.

Case 1. 𝑊(𝑡)>𝜇⁻¹𝜆₀𝜌² for all 𝑡∈[𝑡_𝑚−1,𝑡_𝑚). It follows that (22) holds according to the fact that (14) holds for 𝑡∈[𝑡₀−𝜏,𝑡_𝑚). Hence (23) and (25) hold for 𝑡∈[𝑡_𝑚−1,𝑡_𝑚). They lead that 𝑊𝑡⁻_𝑚≤𝑊𝑡_𝑚−1exp−𝜇₁𝑡_𝑚−𝑡_𝑚−1≤𝜆₀𝜌²exp−𝜇₁𝛽−ℎ=𝜇⁻¹𝜆₀𝜌². (26) This is also a contradiction.

Case 2. There exists some 𝑡∈[𝑡_𝑚−1,𝑡_𝑚), such that 𝑊(𝑡)≤𝜇⁻¹𝜆₀𝜌². Set 𝑡=sup{𝑡∈[𝑡_𝑚−1,𝑡_𝑚)∶𝑊(𝑡)≤𝜇⁻¹𝜆₀𝜌²}. Then 𝑡∈[𝑡_𝑚−1,𝑡_𝑚), 𝑊(𝑡)=𝜇⁻¹𝜆₀𝜌² and (22) for 𝑡∈[𝑡,𝑡_𝑚]. It follows that (23) and (25) hold, which yields that 𝑊(𝑡⁻_𝑚)≤𝑊(𝑡)=𝜇⁻¹𝜆₀𝜌². This is a contradiction which implies that 𝑊(𝑡⁻_𝑚)≤𝜇⁻¹𝜆₀𝜌², and hence by (9), we obtain that 𝑊(𝑡_𝑚)≤𝜇𝑊(𝑡⁻_𝑚)≤𝜆₀𝜌².

Now we assume that (14) is not true for 𝑡∈[𝑡_𝑚,𝑡_𝑚+1). Set 𝑡^∗=inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1)∶𝑊(𝑡)≥𝜆₀𝜌²}. Then 𝑡^∗∈[𝑡_𝑚,𝑡_𝑚+1) and 𝑊(𝑡^∗)=𝜆₀𝜌². If there exists some 𝑡∈[𝑡_𝑚,𝑡^∗), such that 𝑊(𝑡)≤𝜇⁻¹𝜆₀𝜌², then set 𝑡=sup{𝑡∈[𝑡_𝑚,𝑡^∗)∶𝑊(𝑡)≤𝜇⁻¹𝜆₀𝜌²}, otherwise, we set 𝑡=𝑡_𝑚. Then 𝑡∈[𝑡_𝑚,𝑡^∗). 𝑊(𝑡)<𝜆₀𝜌². It follows that (22), (23), and (25) hold, which yields that 𝑊(𝑡^∗)≤𝑊(𝑡)<𝜆₀𝜌². This is a contradiction, which implies that (14) is true for 𝑡∈[𝑡_𝑚,𝑡_𝑚+1). By mathematical induction, (14) holds for 𝑡≥𝑡₀−𝜏. That is, system (1) can be robustly stabilized over time sequence 𝒩_min(𝛽−ℎ).

Remark 11. On the conditions in Theorem 10, with the prescribed scalar 𝛽>0, we can stabilize not only the impulsive systems over time sequence 𝒩_min(𝛽), but some ones over other time sequences.

4. Numerical Examples

In this section, two illustrative examples will be presented to show the effectiveness of the results obtained.

Example 12. Consider the uncertain impulsive time-delay systems (1) with the following specifications:  𝜇=0.8, 𝑑=1, 𝛼=2, 𝜀₁=0.3034, 𝜀₂=0.5, 𝜀₃=0.5, 𝑈₁=𝑈₂=1,0≤𝜏(𝑡)≤𝛾=∶const, 𝐴=(^−40.2_0.1−5),𝐴₁=(¹⁰_0.21),𝐵=(^10.1_0.21), 𝐶=(^−0.10_−0.2−0.1),𝑁_𝐴=(^0.10_10.1), 𝑁_𝐴1=(^0.10_0.10.1), 𝑁_𝐵=(^0.010_0.10.1), 𝑁_𝐶=(^0.10.1_00.1), 𝐵₁=(^0.10_−10.1), 𝐵₂=(^0.10.2_00.1), 𝑁_𝐵1=(^0.1−0.1_00.1),𝑁_𝐵2=(^0.10.1_00.1), 𝐷₁=(^10.1₀₁), 𝐷₂=(^10.1₀₁). Then by solving LMIs (8) and (11), we obtain the following results: 𝑋=(^{0.5330−0.0297}_{−0.02970.6117}),  𝑌=(^{−1.2292−0.1268}_{−0.2300−0.8236}). By Theorem 6, the controller 𝑢(𝑡)=𝑌𝑋⁻¹𝑥(𝑡)=(^{−2.3241−0.3201}_{−0.5079−1.3710})𝑥(𝑡) is obtained. It exponentially stabilizes the impulsive system over any impulsive time sequence according to Theorem 6.

Example 13. Consider the uncertain impulsive time-delay systems (1) with the following specifications: 𝜇=1,𝑑=1, 𝛼=2, 𝜀₁=0.4049, 𝜀₂=0.5, 𝜀₃=0.5, 𝑈₁=𝑈₂=1, 0≤𝜏(𝑡)≤𝛾=∶const, 𝐴=(^−30.2_0.1−5), 𝐴₁=(¹⁰_0.21), 𝐵=(^10.2_0.20.7), 𝐶=(^0.10_−0.2−0.1), 𝑁_𝐴=(^0.10.2_10.1), 𝑁_𝐴1=(^0.10_10.1), 𝑁_𝐵=(^0.10_0.20.1), 𝑁_𝐶=(^0.10.1_00.1), 𝐵₁=(^0.10_−10.1), 𝐵₂=(^0.10.2_00.1), 𝑁_𝐵1=(^0.1−0.1_00.1), 𝑁_𝐵2=(^0.10.1_00.1), 𝐷₁=(^10.1₁₁), 𝐷₂=(^10.1_0.21). Then by solving LMIs (8) and (11), we obtain the following results: 𝑋=(^{0.7144−0.0180}_{−0.01800.8380}),  𝑌=(^{−2.4545−0.8386}_{−0.6829−1.5880}). By Theorem 6, the controller 𝑢(𝑡)=𝑌𝑋⁻¹𝑥(𝑡)=(^{−3.4628−1.0751}_{−1.0042−1.9166})𝑥(𝑡) is obtained. It exponentially stabilizes the impulsive system over any impulsive time sequence according to Theorem 6.

Example 14. Consider the uncertain impulsive time-delay systems (1) with the following specifications: 𝜇=1.5, 𝑑=1, 𝛼=2, 𝜀₁=0.1100, 𝜀₂=0.5, 𝜀₃=0.5, 𝑈₁=𝑈₂=1, 0≤𝜏(𝑡)≤𝛾=∶const, 𝐴=−10𝐼, 𝐴₁=𝐵=𝐷₁=𝐼, 𝐶=−0.1𝐼, 𝑁_𝐴=𝑁_𝐴1=𝑁_𝐶=0.1𝐼, 𝑁_𝐵=𝐵₁=𝑁_𝐵1=𝑁_𝐵2=0.1𝐼, 𝐵₂=diag{1,0.1}, 𝐷₂=diag{3,1}. Then by solving LMIs (8) and (19), we obtain that 𝛽=0.4055, 𝑋=diag{1.2150,1.0491}, 𝑌=diag{0.7386,0.4180}. By Theorem 10, the controller 𝑢(𝑡)=𝑌𝑋⁻¹𝑥(𝑡)=(^0.60790_00.3984)𝑥(𝑡) is obtained. It exponentially stabilizes the impulsive system over impulsive time sequence 𝒩_min(0.4055−ℎ) according to Theorem 10, where ℎ>0 is some small scalar. In fact, we can obtain some ℎ≥0.002.

5. Conclusion

This paper studied a class of uncertain impulsive time-delay systems. Based on Lyapunov function and Razumikhin-type techniques, we obtain conditions which can guarantee the above systems robustly exponentially stabiliable for any impulsive time sequences and improves some related results.

Acknowledgments

The authors would like to thank the editor and the reviewers for their valuable comments and suggestions which helped to significantly improve the paper. This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.