Robust Stability Analysis and Synthesis for Switched Discrete-Time Systems with Time Delay

The problems of robust stability analysis and synthesis for a class of uncertain switched timedelay systems with polytopic type uncertainties are addressed. Based on the constructive use of an appropriate switched Lyapunov function, sufficient linear matrix inequalities LMIs conditions are investigated to make such systems a uniform quadratic stability with an L2-gain smaller than a given constant level. System synthesis is to design switched feedback schemes, whether based on state, output measurements, or by using dynamic output feedback, to guarantee that the corresponding closed-loop system satisfies the LMIs conditions. Two numerical examples are provided that demonstrate the efficiency of this approach.


Introduction
The stability analysis and synthesis problem of switched systems is one of the fundamental and challenging research topics, and various approaches has been obtained so far.For arbitrary switching law, a common Lyapunov function gives stability 1-3 .Liberzon and Hespanha have studied the global uniform asymptotic stability problem from the viewpoint of Lie algebra 4, 5 .On the other hand, Branicky 6 , Johansen and Rantzer 7 , respectively, have proposed the multiple Lyapunov function method for analysis and synthesis of switched systems with prescribed switching law.Furthermore, average dwell time technique 8-12 is an effective tool of choosing such switching law.In the recent literatures 13-16 , stability analysis and synthesis of discrete time switched system were studied.A survey of switched systems problems have been proposed by Liberzon and Morse 5 .On the other hand, in many industrial hybrid systems such as power systems 17 and network control systems 13 , time delay often occurs in the transition of the discrete states and the interior running of each subsystem.Therefore, more recently, much research attention has been devoted to the study of switched systems with time delay.In 18 , L 2 -gain properties under arbitrary switching for a class of switched symmetric delay systems were studied.In 19 , sufficient conditions of asymptotical stability were established for switched linear delay systems.Based on switched Lyapunov function approach, H ∞ filtering problem of discrete-time switched systems with state delay was developed.In these papers mentioned above, only stable analysis is considered and state feedback results are proposed.In addition, output measurement and dynamic output feedback that are important synthesis methods for switched systems without delay are also expected to be effective for switched delay systems.However, no such results have been available up to now.
This paper studies the robust stability analysis and synthesis problems for uncertain switched delay systems with polytopic type uncertainties.Compared with the existing results on switched delay systems, the results of this paper have two features.Firstly, we give design of a novel switched Lyapunov function while the existing works commonly aim at the design of the common Lyapunov function.Secondly, dynamic output feedback is achieved while the existing literature usually addresses the state feedback control.
The rest of the paper is organized as follows.Section 2 briefly presents the uncertain switched time delay systems with polytopic type uncertainties and robust stability analysis with L 2 -gain, based on the switched Lyapunov function.Section 3 presents the robust synthesis with the switched state feedback or output feedback design schemes, while Section 4 reports the results for switched dynamic output feedback.Finally, the main conclusions are summarized in Section 5.

Notation
The notation used in this paper is fairly standard.The superscript T stands for the matrix transposition, the notation • refers to the Euclidea vector norm, R n denotes the n dimensional Euclidean space.In addition, in the symmetric block matrices or long matrix expressions, we use * as an ellipsis for the terms that are introduced by symmetry, and diag{• • • } stands for a block-diagonal matrix.A symmetric matrix P > 0 ≥ 0 means that P is a positive semipositive definite matrix.

Problem Formulation
Given a class of linear discrete time switched systems with time delay r are the system state vector, control input, exogenous disturbance, measured output, and controlled output, respectively, and d presents the time delay.The particular mode σ at any given time instant may be a selective procedure characterized by a switching rule of the following form: The function δ • is usually defined using a partition of the continuous state space.Let S denote the set of all selective rules.Therefore, the linear discrete time switched systems under consideration are composed of N subsystems, each of which is activated at particular switching instant.For a switching mode j ∈ {1, . . ., N}, the associated matrices A j , . . ., Φ j contain uncertainties represented by a real convex-bounded polytopic model of the type where λ j λ j1 , λ j2 , . . ., λ jM j ∈ Λ j belongs to the unit simplex of M j vertices where A jp , . . ., Φ jp , p 1, . . ., M j are known as real constant matrices of appropriate dimensions.
Distinct from 2.1 -2.3 is the free switched system Definition 2.2.Given a scalar γ ≥ 0, the L 2 -gain Υ of switched system 2.7 -2.8 over S is

2.11
We consider the following switching quadratic Lyapunov function: where 13 Theorem 2.4.The following statements are equivalent.
1 There exist a switching Lyapunov function of the type 2.12 with σ ∈ S and a scalar γ > 0 such that switched system 2.7 -2.8 with polytopic representation 2.5 -2.6 is UQS with L 2 -gain Υ < γ.
2 There exist matrices Proof. 1 ⇒ 2 Suppose that there exist a constant γ > 0 and a switching Lyapunov function of the type 2.12 .Let the switching rule σ • activates subsystem j at instant k 1 and subsystem i at instant k.Thus, Then, it can be shown that where By the Schur complement, Θ < 0 is equivalent to Applying the congruent transformation diag I, I, I, X j , I , X j P −1 j , 2.19 we obtain Upon using the vertex representation 2.5 -2.6 , we get 2.14 from 2.20 . 2 ⇒ 1 Follow by reversing the steps in the proof and applying Definitions 2.1-2.3 to the system 2.7 -2.8 for all modes i, j ∈ {1, . . ., N} × {1, . . ., N} and using 2.5 -2.6 .
The proof is complete.
Consider switched system 2.7 -2.8 with w k 0, a special case of Theorem 2.4 is provided.

Corollary 2.5. The following statements are equivalent.
1 There exists a switching Lyapunov function of type 2.12 with σ ∈ S such that switched system 2.7 -2.8 with w k 0, and polytopic representation 2.5 -2.6 is UQS.
2 There exist matrices Since the exogenous disturbance w k 0 in the switched system 2.7 -2.8 , we could let the coefficients Γ i , Φ i be zero matrices in the inequality 2.20 .Directly deleting the third row and the third column from the matrix inequality 2.20 , that is, deleting the term relevant to w k , we get the inequality 2.21 .

Switched Control Synthesis
Extending on Section 2, we examine here the problem of switched control synthesis using either switched state feedback or output feedback design schemes.

Switched State Feedback
With reference to system 2.1 -2.2 , we consider that the arbitrary switching rule σ • activates subsystem i at an instant k.Our objective herein is to design a switched state feedback u k K i x k , i ∈ {1, . . ., N} such that the closed-loop system is UQS with an L 2 -gain Υ < γ.
Theorem 3.1.Switched system 3.1 -3.2 is UQS with an L 2 -gain Υ < γ, if there exist matrices , and a scalar γ > 0, such that the following LMIs hold: Moreover, the gain matrix is given by K i Z i X −1 i .

3.4
Applying the congruent transformation diag{X i , X i , I, I, I} to LMIs 3.4 , and let Upon using vertex representation 2.5 -2.6 , we get 3.3 from 3.5 .

Example 1
In this example, we design a switched state feedback control for this system based on Theorem 3.1, switching occurs between two modes described by the following coefficients.Mode 1: consider  With the initial state x 0 −0.02 − 0.71 T , and the following switching signal Figure 1 , the system state, control, and output signal are shown in Figures 2, 3, and 4, respectively.
From the simulation results, it can be clearly seen that the proposed control law guarantees the asymptotic stability of the closed-loop system.

Switched Static Output Feedback
The objective of this subsection is to design switched output feedback u k G i y k , i ∈ {1, . . ., N}, such that the closed-loop system Before developing our main results, we give the following lemma.
Lemma 3.2 see 14 .Given a matrix L i ∈ R p×n , rank L i p, and having the singular value decomposition form where 12 if and only if

3.13
Thus, we get Theorem 3.3.Consider switched system 3.1 -3.2 with polytopic representation 2.5 -2.6 subject to the output feedback control u k G i y k , and output matrix L i having the SVD form have a feasible solution.Moreover, the feedback gain is given by Proof.In view of Theorem 2.4, we have Let R i G i E i , then 3.15 can be rewritten as follows: Upon using vertex representation 2.5 -2.6 , we get 3.14 from 3.16 .According to Lemma 3.2, we obtain that

Switched Dynamic Output Feedback
We consider the more general case and employ at every mode i ∈ {1, . . ., N}, a switched dynamic output feedback scheme of the following form: Applying controller 4.1 to system 2.1 -2.3 and letting η T k x T k ξ T k , we get the closed-loop system where This system is UQS with an L 2 -gain Υ < γ, if there exist and a scalar γ > 0 satisfying the systems LMIs Moreover, the gain matrix is given by A ci Ψ ci X −1 ci , C ci Π ci X −1 ci , and Proof.In view of Theorem 2.4, we have By applying 4.3 -4.4 to 4.6 , we obtain Let Ψ ci A ci X ci , Ω ci B ci E i , Π ci C ci X ci , and using vertex representation 2.5 -2.6 , we get 4.5 from 4.7 .

Figure 1 :Figure 2 :
Figure 1: The switching signal of the switched systems.

Figure 3 :Figure 4 :
Figure 3: Control input of the switched discrete-time systems.

Figure 5 :Figure 6 :
Figure 5: State response of the closed-loop switched discrete-time systems.

Figure 7 :
Figure 7: Output signal of the closed-loop switched discrete time systems.
Similar to Example 1, we wish to design a switched static output feedback control law based on Theorem 3.3.Using all coefficients in Example 1 that adapt to Example 2. , and the switching signal as Example 1, the system state, control, and output signal are shown as follows.
4.1.Consider switched system 4.2 with polytopic representation 2.5 -2.6 and output matrix L i having the SVD form L