Robust Finite-Time H ∞ Control for Impulsive Switched Nonlinear Systems with State Delay

This paper investigates robust finite-time H∞ control for a class of impulsive switched nonlinear systems with time-delay. Firstly, using piecewise Lyapunov function, sufficient conditions ensuring finite-time boundedness of the impulsive switched system are derived. Then, finite-time H∞ performance analysis for impulsive switched systems is developed, and a robust finite-time H∞ state feedback controller is proposed to guarantee that the resulting closed-loop system is finite-time bounded with H∞ disturbance attenuation. All the results are given in terms of linear matrix inequalities LMIs . Finally, two numerical examples are provided to show the effectiveness of the proposed method.


Introduction
A switched system is a hybrid dynamical system consisting of a family of continuous-time or discrete-time subsystems and a switching law that orchestrates the switching between them 1 .In the last decades, in the stability analysis and stabilization for switched systems, lots of valuable results are established see 2-5 .Most recently, on the basis of Lyapunov functions and other analysis tools, the stability problem of linear and nonlinear switched systems with time-delay has been further investigated see 6-15 , and lots of valuable results are established for H ∞ control problems see 16-22 .It is well known that impulsive dynamical behaviors inevitably exist in some practical systems like physical, biological, engineering, and information science systems due to abrupt changes at certain instants during the dynamical process.Although hybrid system and switched system are important models for dealing with complex real systems, there is little work concerned with the above impulsive phenomena.Such a phenomenon can be modeled as an impulsive switched system, it is characteristic that their states change during the switching because of the occurrence of impulses 23 .In recent years, the impulsive switched systems have drawn more and more attention and many useful conclusions have been obtained.Multiple Krasovskii-Lyapunov function approach is employed to study the problem of ISS stability of a class of impulsive switched systems with time-delay in 24 .By the Lyapunov-Razumikhin technique, a delayindependent criterion of the exponential stability is established on the minimum dwell time in 25 .The problem of robust H ∞ stabilization of nonlinear impulsive switched system with time-delays is studied in 23 .Usually, the stability of a system is defined over an infinite-time interval.But in many practical systems, we focus on the dynamical behavior of a system over a fixed finite-time interval.Based on this, finite-time stability is first proposed by Dorato in 1961 26 .Compared with the classical Lyapunov stability, finite-time stability is proposed for the study of the transient performance of the system, which is a totally different concept.The so-called finitetime stability means the boundedness of the state of a system over a fixed finite-time interval.Finite-time stability problems can be found in 27-32 .The finite-time stability of linear impulsive systems is analyzed in 33 , the finite-time stability and stabilization of impulsive dynamic systems are carried out in 34-36 .The finite-time stability and stabilization of switched systems are investigated in 37 .Recently, robust finite-time control of switched systems is studied in 38, 39 .However, to the best of our knowledge, there are very few results on finite-time boundedness and robust H ∞ control of the impulsive switched systems, which motivates the present study.The paper is organized as follows.In Section 2, problem formulation and some necessary lemmas are given.In Section 3, based on the dwell time approach, finite-time boundedness and finite-time H ∞ performance for switched impulsive systems are addressed, and sufficient conditions for the existence of a robust finite-time H ∞ state feedback controller are proposed in terms of a set of matrix inequalities.Numerical examples are provided to show the effectiveness of the proposed approach in Section 4. Concluding remarks are given in Section 5. Notations.The notations used in this paper are standard.The notation P > 0 means that P is a real positive definite matrix; diag{• • • } stands for a block-diagonal matrix; λ max P and λ min P denote the maximum and minimum eigenvalues of matrix P , respectively; x t x T t x t and x t 2
are uncertain real-valued matrices with appropriate dimensions.We assume that the uncertainties are of the form where T 2 − T 1 /τ a holds for given N 0 ≥ 0, τ a > 0, then the constant τ a is called the average dwell time.In this paper we let N 0 0. Definition 2.9.For a given time constant T f , impulsive switched system 2.1a -2.1d with u 1 t ≡ 0, u 2 t ≡ 0 is said to have finite-time H ∞ performance with respect to 0, c 2 2 , T f , d 2 , γ, R, σ t if the system is finite-time bounded and the following inequality holds: where c 2 > 0, γ > 0, R is a positive definite matrix and σ t is a switching signal.
Definition 2.10.For a given time constant T f , impulsive switched system 2.1a -2.1d is said to be robust finite-time stabilization with H ∞ disturbance attenuation level γ, if there exists a switched controller u 1 t K σ t x t , t / t k and an impulsive controller u 2 t k K σ t x t k , t t k , where t ∈ t 0 , T f such that i the corresponding closed-loop system is finite-time bounded with respect to 0, c 2 2 , T f , d 2 , R, σ t ; ii under zero initial condition, inequality 2.7 holds for any w t satisfying 2.5 .
Lemma 2.11.Let U, V , W, and X be real matrices of appropriate dimensions with X satisfying X X T , then for all if and only if there exists a scalar ε > 0 such that 2.9

Finite-Time Boundedness Analysis
In this subsection, we focus on the finite-time boundedness of the following impulsive switched system: Before proceeding to Lemma 3.2, we first introduce a function v t .For given positive definite matrices Q i k , i k ∈ N, by Assumption 2.4, there exists a real number Furthermore, we define the following function Finally, a piecewise continuous function v t is as follows: , and v t is monotonically nonincreasing and bounded function, , which can be obtained by setting ρ i k 1 in 3.2 .Thus, the proposed approach may provide more relaxed conditions.

Lemma 3.2. Consider the following Lyapunov functional candidate:
V t x T t P σ t x t t t−h v s x T s e α t−s Q σ s x s ds 3.5 for system 3.1a , 3.1b , and 3.1c , where P i and Q i , i ∈ N are symmetric positive definite matrices with appropriate dimensions.
The following inequality is derived:

3.7
Mathematical Problems in Engineering 7 From 3.2 , we can obtain that
ii When t k h < t k 1 , 1 t ∈ t k , t k h , the proof is similar to the proof line in the situation i .

3.9
The proof for this situation is omitted.The proof is completed.
we have where ρ * max{ρ i k , i k ∈ N}.
Proof.Without loss of generality, let σ t k i, σ t k j.Then, we have Combining 3.13 with 3.14 , we have

3.15
where ij I E j T P i I E j e αh ρ * E T j Q i E j − e αh ρ * P j .

3.16
Using Schur complement, 3.11 is equivalent to The proof is completed.
hold, under the average dwell time scheme

3.28
Using Schur complement, we obtain from 3.19 that where

3.30
Noticing that the above inequality holds for all i, j ∈ N, then we have

3.32
Noticing that N σ t 0 , T f < T f /τ a and according to 3.21 , we have x T t Rx t .

3.34
According to the Lyapunov function that we have chosen, we have

3.35
According to 3.21 , the following inequality is derived:

3.37
Using Schur complement, 3.22 is equivalent to λ 2 and symmetric positive matrices P i , P j , Q i for all i, j ∈ N with appropriate dimensions such that hold with average dwell time
14 Mathematical Problems in Engineering

H ∞ Performance Analysis
In this subsection, H ∞ performance of the following system is investigated: Suppose that there exist positive scalars ρ i ≥ 1, i ∈ N, ρ * max{ρ i , i ∈ N}, α, γ, ε and symmetric positive matrices P i , P j , Q i for all i, j ∈ N such that hold with average dwell time

3.51
Using Schur complement, we obtain from 3.44 that where Noticing that the above inequality holds for all i, j ∈ N, then we have Thus, V x t − αV x t z T t z t − γ 2 w T t w t < 0, 3.53 Let γ 2 w T s w s − z T s z s Δ s , from 3.32 , we have V t < e α t−t 0 e αh ρ * N σ t 0 ,t V t 0 t t 0 e α t−s e αh ρ * N σ s,t Δ s ds.

3.54
Under zero initial condition, we have e α t−s e αh ρ * N σ s,t γ 2 w T s w s ds < e αt e αh ρ * N σ t 0 ,t t t 0 γ 2 w T s w s ds.

3.58
Let t T f , because τ a > h/ε, we have w T s w s ds, 3.59 then w T s w s ds.

3.60
Thus, system 3.42a -3.42d is finite-time bounded and has H ∞ performance with respect to 0, The proof is completed.
Remark 3.7.When ρ * 1, Theorem 3.6 degenerates to the result of 41 , which cannot guarantee the finite-time boundedness of the addressed system if ρ * > 1.

Robust Finite-Time H ∞ Control
Consider system 2.1a -2.1d , under the switching controller u 1 t K σ t x t , t / t k and impulsive controller u 2 t k K σ t x t k , t t k , the corresponding closed-loop system is given by x t ϕ t , t ∈ t 0 − h, t 0 .

3.61d
Theorem 3.8.Consider impulsive switched system 2.1a -2.1d , let max{ρ i , i ∈ N} α, γ, ε, δ i and positive definite symmetric matrices P i , Q i , and matrices Y i , i ∈ N, with appropriate dimensions, such that the following inequalities hold

3.66
Then, under the controller K i Y i P −1 i , K i −E i , and the following average dwell time scheme the corresponding closed-loop system is finite-time bounded with H ∞ performance with respect to 0, c 2 2 , T f , d 2 , γ, R, σ t and γ 2 e 1 ε αT f ρ * εT f /h γ 2 .
Proof.According to Assumption 2.1, we have

3.68
Now replacing A i , A di , C i in the left side of 3.44 with where

3.70
From 3.69 , we know that where

3.74
From Lemma 2.11, we can obtain that

3.76
Now we choose K i −E i , and replacing E i in 3.45 with E i K i , we know that .77 by 3.64 , we know that the condition 3.45 hold.
Then, system 2.1a -2.1d is robust finite-time bounded with H ∞ performance with respect to 0, c 2 2 , T f , d 2 , γ, R, σ t , and γ 2 e 1 ε αT f ρ * εT f /h γ 2 .The proof is completed.Remark 3.9.In order to eliminate the impulsive jump, we design an impulsive feedback controller K i −E i , t t k .Then the system becomes a switched system with continuous states.

Numerical Examples
In this section, we present two examples to illustrate the effectiveness of the proposed approach.
Example 4.1.Consider system 2.1a -2.1d with the following parameters.and τ a > τ * a 3.8340.We choose τ a 4, ε 0.05, γ 2 0.9464, the initial condition x t 0, t ∈ −h, 0 , the switching signal is shown in Figure 1, and state trajectories of the closed-loop system are shown in Figure 2.

Subsystem 1
We can see from Figure 2 that the states of the system are continuous due to the feedback K i in impulsive instants.

4.7
Obviously, the above inequalities do not satisfy the conditions of 41, Theorem 3 .Thus, we cannot draw the conclusion that the closed-loop system is finite-time bounded from Theorem 3 in 41 .

Conclusions
This paper has investigated robust finite-time H ∞ control for a class of impulsive switched nonlinear systems with time-delay.Based on piecewise Lyapunov function, sufficient conditions which guarantee finite-time boundedness of the impulsive switched system are derived.Then, a feedback control scheme consisting of an impulsive feedback controller and a switching controller is proposed, and the proposed control strategy can guarantee that the closed-loop system is finite-time bounded with H ∞ disturbance attenuation level.Finally, the results are illustrated by means of two numerical examples.
and E Bi are known real-valued constant matrices with appropriate dimensions, F i t is the uncertain matrix satisfying Equation 2.6 stands for the boundedness of the state of a system over a fixed finite-time interval t 0 , T f , when the initial state is bounded.For any T 2 > T 1 > 0, let N σ t T 1 , T 2 denote the switching number of σ t on an interval T 1 , T 2 .If N σ t T 1 , T 2 ≤ N 0 ∈ t k , t k 1 , according to 3.18 and Lemma 3.2, we have Example 4.2.Consider system 2.1a -2.1d with the following parameters.Let h 0.2, T f 12, d 2 10, R I, α 0.001, C 2 2 21, ρ * 1.3, γ 2 0.9344.By solving the LMIs in 3.62 -3.66 , we can get