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

and Applied Analysis 3 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 ofA. λmax A is defined by λmax A max{Re λ : λ ∈ λ A }. λmin A is defined by λmin A min{Re λ : λ ∈ λ A }. For h ≥ 0, C −h, 0 , R denotes the set of allR-valued continuous functions mapping −h, 0 into R. A matrixA is semipositive definite A ≥ 0 if 〈Ax, x〉 ≥ 0, for all x ∈ R. A is positive definite A > 0 if 〈Ax, x〉 > 0 for all x/ 0.A ≥ B means A − B ≥ 0. Also, xt is defined by xt {x t s , s ∈ −h, 0 }, and ‖xt‖ is defined by ‖xt‖ sups∈ −h,0 ‖x t s ‖. Now, let us consider a class of uncertain systems with time-varying delay and nonlinear perturbations of the form ẋ t A t ΔA υ, t x t A1 t ΔA1 ξ, t x t − h t B t ΔB ς, t u t f t, x t , x t − h t , u t , x t φ t , t ∈ −h, 0 , h ≥ 0, 2.1 where x t ∈ R is the state, u t ∈ R 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 is a compact set. In addition, perturbation f t, x t , x t − h t , u t : R × R × R × R → R 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 × R × R. The initial function φ t ∈ C −h, 0 , R , h > 0, has its norm ‖φ‖ sups∈ −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 t ΔA υ, t B t K t ΔB ς, t K t x t A1 t ΔA1 ξ, t x t − h t f t, x t , x t − h t , u t , x t φ t , t ∈ −h, 0 , h ≥ 0, 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. 4 Abstract and Applied Analysis Assumption 1. For all x, t ∈ R × 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 matrixQ > 0 with appropriate dimensions, it follows that 2ab ≤ aQa bTQ−1b. 2.6 Lemma 2.3 see 11 . Given constant symmetric matrices S1, S2, S3, and S1 S1 < 0, S3 S T 3 > 0, then S1 S2S−1 3 S T 2 < 0 if and only if [ S1 S2 S2 −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, εi, i 1, 2, 3, 4, we set ω ε1 ε3 ε4 ε2h , A t A t αI, Pβ t P t βI, ρυ t max υ ‖H υ, t ‖, ρξ t max ξ ‖H1 ξ, t ‖, μ t min ς [ 1 2 λmin ( E ς, t E ς, t )] , p sup t∈R ‖P t ‖, R t − 4 ε1e−2αh 1 − δ A1 t A1 t − [ 1 4 ( 1 μ t )2 4ρ2 ξ t ε1e−2αh 1 − δ ε−1 3 ρ 2 υ t − 1 ] B t B t − ( ε−1 4 a 2 2b2 ε1e−2αh 1 − δ d2 ) I. 3.1 We need the following assumption. Assumption 2. For any t > 0, μ t > −1. Abstract and Applied Analysis 5 Theorem 3.1. Suppose that condition (H1) and Assumptions 1-2 hold. If there exist positive numbers α, β, εi, i 1, 2, 3, 4, and a symmetric positive semidefinite matrix function P t satisfying the following Riccati differential equation:and Applied Analysis 5 Theorem 3.1. Suppose that condition (H1) and Assumptions 1-2 hold. If there exist positive numbers α, β, εi, 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 Pβ t Pβ t A t − Pβ t R t Pβ t ωI 0, 3.2 then system 2.1 is robustly α-exponentially stabilizable with feedback control u t − 1 2 ( 1 μ t )B t Pβ t x t . 3.3 Moreover, the solution x t, φ satisfies the condition ∥ ∥x ( t, φ )∥ ∥ ≤ √ p β hε1 1/2 hε2 β ∥ ∥φ ∥ ∥e−αt, t ≥ 0. 3.4 Proof. Let u t K t x t , where K t − 1 2 ( 1 μ t )B t Pβ 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, xt V1 V2 V3, 3.6 where V1 〈P t x t , x t 〉 β〈x t , x t 〉, V2 ε1 ∫ t t−h t e2α s−t ‖x s ‖ds, V3 ε2 ∫0 −h ∫ t t τ−h t τ e2α s−t ‖x s ‖ds dτ. 3.7 The time derivative of V along the trajectory of 2.3 is given by V̇ t, xt V̇1 V̇2 V̇3, 3.8 6 Abstract and Applied Analysis where V̇1 〈 Ṗ t x t , x t 〉 2 〈 Pβ t ẋ t , x t 〉 〈( Ṗ t A t Pβ t Pβ t A t ) x t , x t 〉 2 〈 Pβ t B t H υ, t x t , x t 〉 2 〈 Pβ t A1 t x t − h t , x t 〉 2 〈 Pβ t B t H1 ξ, t x t − h t , x t 〉 2 〈 Pβ t B t K t x t , x t 〉 2 〈 Pβ t B t E ς, t K t x t , x t 〉 2 〈 Pβ t f t, x t , x t − h t , u t , x t 〉 , V̇2 −2αV2 ε1‖x t ‖ − ε1e−2αh t ‖x t − h t ‖ ( 1 − ḣ t ) ≤ −2αV2 ε1‖x t ‖ − ε1e−2αh‖x t − h t ‖ 1 − δ , V̇3 −2αV3 ε2 ∫0 −h [ ‖x t ‖ − e−2α τ−h t τ ‖x t τ − h t τ ‖2(1 − ḣ t τ ) ] dτ ≤ −2αV3 ε2h‖x t ‖ − ε2 ∫0 −h e−2α τ−h t τ ‖x t τ − h t τ ‖ 1 − δ dτ ≤ −2αV3 ε2h‖x t ‖. 3.9 Applying Lemma 2.2 gives 2 〈 Pβ t A1 t x t − h t , x t 〉 ≤ 4 ε1 1 − δ e−2αh 〈 Pβ t A1 t A1 t Pβ t x t , x t 〉 ε1 1 − δ e−2αh 4 〈x t − h t , x t − h t 〉, 2 〈 Pβ t B t H1 ξ, t x t − h t , x t 〉 ≤ ε1 1 − δ e −2αh 4 〈x t − h t , x t − h t 〉 4 ε1 1 − δ e−2αh 〈 Pβ t B t H1 ξ, t H 1 ξ, t B T t Pβ t x t , x t 〉 ≤ ε1 1 − δ e −2αh 4 〈x t − h t , x t − h t 〉 4ρ2 ξ t ε1 1 − δ e−2αh 〈 Pβ t B t B t Pβ t x t , x t 〉 , 2 〈 Pβ t B t H υ, t x t , x t 〉 ≤ ε3〈x t , x t 〉 ε−1 3 〈 Pβ t B t H υ, t H υ, t B t Pβ t x t , x t 〉 ≤ ε3〈x t , x t 〉 ε−1 3 ρ2 υ t 〈 Pβ t B t B t Pβ t x t , x t 〉 . 3.10 Abstract and Applied Analysis 7 Using 2.2 and 3.3 , we getand Applied Analysis 7 Using 2.2 and 3.3 , we get 2 〈 Pβ t f t, x t , x t − h t , u t , x t 〉 ≤ 2∥∥f t, x t , x t − h t , u t ∥∥∥∥Pβ t x t ∥ ∥ ≤ 2a‖x t ‖ ∥ Pβ t x t ∥ ∥ 2b‖x t − h t ‖ ∥ Pβ t x t ∥ ∥ 2d‖u t ‖ ∥ Pβ t x t ∥ ∥ ≤ ε−1 4 a2 ∥ Pβ t x t ∥ ∥ 2 ε4‖x t ‖ 2b 2 ε1e−2αh 1 − δ ∥ Pβ t x t ∥ ∥ 2 ε1e−2αh 1 − δ 2 ‖x t − h t ‖ d2∥∥Pβ t x t ∥ ∥ 2 ‖u t ‖ ( ε−1 4 a 2 2b2 ε1e−2αh 1 − δ d2 ) 〈 P 2 β t x t , x t 〉 ε4〈x t , x t 〉 1 4 ( 1 μ t )2 〈 Pβ t B t B t Pβ t x t , x t 〉 ε1e−2αh 1 − δ 2 ‖x t − h t ‖. 3.11 In addition, it is easy to obtain the following: 2 〈 Pβ t B t K t x t , x t 〉 2 〈 Pβ t B t E ς, t K t x t , x t 〉 − 1 1 μ t x t Pβ t B t [ I 1 2 ( E ς, t E ς, t )] B t Pβ t x t ≤ − 1 1 μ t λmin [ I 1 2 ( E ς, t E ς, t )]∥ ∥ ∥B t Pβ t x t ∥ ∥ ∥ 2 − 〈 Pβ t B t B t Pβ t x t , x t 〉 . 3.12 The last equality is got because of μ t minς 1/2 λmin E ς, t E ς, t . Therefore, we get V̇ t, xt 2αV t, xt ≤ 〈{ Ṗ t A T t Pβ t Pβ t A t 4 ε1 1 − δ e−2αh Pβ t A1 t AT1 t Pβ t ( 1 4 ( 1 μ t )2 4ρ2 ξ t ε1 1 − δ e−2αh ε−1 3 ρ 2 υ t − 1 ) Pβ t B t B t Pβ t ( ε−1 4 a 2 2b2 ε1 1 − δ e−2αh d2 ) P 2 β t ε1 ε3 ε4 ε2h I } x t , x t 〉 . 3.13 8 Abstract and Applied Analysis Then introducing 3.2 into 3.13 we can get V̇ t, xt 2αV t, xt ≤ 0, ∀t ≥ 0. 3.14 It is obvious that e2αtV̇ t, xt 2αe2αtV t, xt ≤ 0. 3.15 Integrating both sides of 3.15 from 0 to t, we get V t, xt ≤ V 0, x0 e−2αt, ∀t ≥ 0. 3.16 On the other hand, from the expression of V t, xt , it is easy to see that β‖x t ‖ ≤ V t, xt , ∀t ≥ 0. 3.17 In addition, since V1 0, x0 ≤ ( p β )∥ ∥φ ∥ ∥ 2 , V2 0, x0 ε1 ∫0 −h 0 e2αs‖x s ‖ds ≤ ε1 ∫0 −h ∥ ∥φ ∥ ∥ 2 ds ≤ ε1h ∥ ∥φ ∥ ∥ 2 , V3 0, x0 ε2 ∫0 −h ∫0 τ−h τ e2αs‖x s ‖ds dτ ≤ ε2 ∫0 −h ∫0 τ−h ∥ ∥φ ∥ ∥ds dτ ≤ 1 2 ε2h 2∥∥φ ∥ ∥, 3.18 we get V 0, x0 ≤ p β hε1 1/2 hε2 ‖φ‖. Hence we have ∥ ∥x ( t, φ )∥ ∥ ≤ √ p β hε1 1/2 hε2 β ∥ ∥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 Pβ t Pβ t A t − Pβ t R t Pβ t ωI ≤ 0. 3.20 In addition, we consider a class of uncertain systems with time-varying delays and simple nonlinear perturbations as follows: ẋ t A ΔA υ, t x t A1 ΔA1 ξ, t x t − h t B ΔB ς, t u f t, x t , x t − h t , x t φ t , t ∈ −h, 0 , h ≥ 0, 3.21 Abstract and Applied Analysis 9 where A, A1, B are constant matrices of appropriate dimensions and ΔA υ, t , ΔA1 ξ, t , ΔB ς, t are unknown time-varying uncertainmatrices, and there exist scalars a, b, d, l1, l2, l3 > 0 such thatand Applied Analysis 9 where A, A1, B are constant matrices of appropriate dimensions and ΔA υ, t , ΔA1 ξ, t , ΔB ς, t are unknown time-varying uncertainmatrices, and there exist scalars a, b, d, l1, l2, l3 > 0 such that ∥ ∥f t, x t , x t − h t ∥ ≤ a‖x t ‖ b‖x t − h t ‖, ∀ t, x t ∈ R × R, ΔA υ, t ΔA υ, t ≤ l1I, ΔA1 ξ, t ΔA1 ξ, t ≤ l2I, ‖ΔB‖ ≤ l3. 3.22 Given positive numbers α, β, εi, i 1, 2, 3, 4, we set A A αI, Pβ P βI, ω ε1 ε3 ε4 ε2h, R − 4 ε1e−2αh 1 − δ A1A T 1 BB T − ( ε−1 3 l1 ε −1 4 a 2 4l2 ε1 1 − δ e−2αh 2b2 ε1 1 − δ e−2αh l3 ∥ ∥ ∥B ∥ ∥ ∥ ) I. 3.23 We have the following theorem. Theorem 3.3. Suppose that condition (H1) holds. If there exist positive numbers α, β, εi, i 1, 2, 3, 4, and a symmetric positive define matrix X such that the following LMIs hold: