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

Abstract

The problems of robust stability analysis and synthesis for a class of uncertain switched time-delay 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.

1. 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], 𝐿₂-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 𝐿₂-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^𝑛denotes 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.

2. Problem Formulation

Given a class of linear discrete time switched systems with time delay 𝑥(𝑘+1)=𝐴_𝜎𝑥(𝑘)+𝐴_𝑑𝜎𝑥(𝑘−𝑑)+𝐵_𝜎𝑢(𝑘)+Γ_𝜎𝑤(𝑘),𝑧(𝑘)=𝐶_𝜎𝑥(𝑘)+𝐶_𝑑𝜎𝑥(𝑘−𝑑)+𝐷_𝜎𝑢(𝑘)+Φ_𝜎𝑤(𝑘),𝑦(𝑘)=𝐿_𝜎𝑥(𝑘),𝜎∈{1,…,𝑁}. (2.1)(2.2)(2.3)𝑥(𝑘)∈𝑅^𝑛, 𝑢(𝑘)∈𝑅^𝑚, 𝑤(𝑘)∈𝑅^𝑞, 𝑦(𝑘)∈𝑅^𝑝, 𝑧(𝑘)∈𝑅^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: 𝜎(𝑘+1)=𝛿(𝜎(𝑘),𝑥(𝑘)). (2.4) 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 𝐴_𝑗𝐴_𝑑𝑗𝐵_𝑗Γ_𝑗𝐿_𝑗𝐶_𝑗𝐶_𝑑𝑗𝐷_𝑗Φ_𝑗0≜{^𝑀𝑗_𝑝=1𝜆_𝑖𝑝𝐴_𝑗𝑝𝐴_𝑑𝑗𝑝𝐵_𝑗𝑝Γ_𝑗𝑝𝐿_𝑗𝑝𝐶_𝑗𝑝𝐶_𝑑𝑗𝑝𝐷_𝑗𝑝Φ_𝑗𝑝0}, (2.5) where 𝜆_𝑗=(𝜆_𝑗1,𝜆_𝑗2,…,𝜆_𝑗𝑀𝑗)∈Λ_𝑗 belongs to the unit simplex of 𝑀_𝑗 vertices Λ_𝑗≜{𝜆_𝑗∶^𝑀𝑗_𝑝=1𝜆_𝑗𝑝=1,𝜆_𝑗𝑝≥0}, (2.6) 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 𝑥(𝑘+1)=𝐴_𝜎𝑥(𝑘)+𝐴_𝑑𝜎𝑥(𝑘−𝑑)+Γ_𝜎𝑤(𝑘),𝑧(𝑘)=𝐶_𝜎𝑥(𝑘)+𝐶_𝑑𝜎𝑥(𝑘−𝑑)+Φ_𝜎𝑤(𝑘), (2.7)(2.8)𝜎∈{1,…,𝑁}.

We have the following definitions.

Definition 2.1. Switched system (2.7)-(2.8) is Uniform Quadratic Stable (UQS), if there exist a Lyapunov functional 𝑉(𝑥,𝑘)>0, a constant 𝜀>0, such that for all admissible uncertainties satisfying (2.5)-(2.6) and arbitrary switching rule 𝜎(⋅) activating subsystem 𝑗∈{1,…,𝑁} at instant 𝑘+1 and subsystem 𝑖∈{1,…,𝑁} at instant 𝑘, the Lyapunov functional difference Δ𝑉(𝑥(𝑘),𝑘) satisfies Δ𝑉(𝑥(𝑘),𝑘)≜𝑉(𝑥(𝑘+1),𝑘+1)−𝑉(𝑥(𝑘),𝑘)≤−𝜀𝑥^𝑇(𝑘)𝑥(𝑘),∀𝑥(𝑘)≠0. (2.9)

Definition 2.2. Given a scalar 𝛾≥0, the 𝐿₂-gain Υ of switched system (2.7)-(2.8) over 𝑆 is Υ≜inf{𝛾≥0∶‖𝑧(𝑘)‖₂<𝛾²‖𝑤(𝑘)‖₂,∀𝜎∈𝑆,∀𝜆_𝑗∈Λ_𝑗}. (2.10)

Definition 2.3. Switched system (2.7)-(2.8) is UQS with an 𝐿₂-gainΥ<𝛾, if for all switching signal 𝜎∈𝑆 and for all admissible uncertainties satisfying (2.5)-(2.6), it is UQS and ∀𝑤(𝑘)≠0,‖𝑧(𝑘)‖₂<𝛾²‖𝑤(𝑘)‖₂. (2.11) We consider the following switching quadratic Lyapunov function: 𝑉_𝜎(𝑥(𝑘),𝑘)≜𝑉_1𝜎(𝑥(𝑘),𝑘)+𝑉₂(𝑥(𝑘),𝑘), (2.12) where 𝑉_1𝜎(𝑥(𝑘),𝑘)=𝑥^𝑇(𝑘)𝑃_𝜎𝑥(𝑘),𝑉₂(𝑥(𝑘),𝑘)=^𝑑_𝑙=1𝑥^𝑇(𝑘−𝑙)𝑄_𝜎𝑥(𝑘−𝑙), (2.13)𝑃_𝜎=𝑃^𝑇_𝜎>0,𝑄_𝜎=𝑄^𝑇_𝜎>0.

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 𝐿₂-gainΥ<𝛾.(2)There exist matrices 𝑃_𝑖=𝑃_𝑖^𝑇>0, 𝑄_𝑖=𝑄^𝑇_𝑖>0, 𝑋_𝑗=𝑋^𝑇_𝑗>0, and a scalar 𝛾>0 satisfying the LMIs −𝑃_𝑖+𝑄_𝑖00𝐴^𝑇_𝑖𝑝𝐶^𝑇_𝑖𝑝∗−𝑄_𝑖0𝐴^𝑇_𝑑𝑖𝑝𝐶^𝑇_𝑑𝑖𝑝∗∗−𝛾²𝐼Γ^𝑇_𝑖𝑝Φ^𝑇_𝑖𝑝∗∗∗−𝑋_𝑗0∗∗∗∗−𝐼<0, (2.14)(𝑖,𝑗)∈{1,…,𝑁}×{1,…,𝑁}, 𝑝∈{1,…,𝑀_𝑖}.

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+1and subsystem 𝑖 at instant 𝑘. Thus, Δ𝑉_𝜎(𝑥(𝑘),𝑘)=𝑉_𝜎(𝑘+1)(𝑥(𝑘+1),𝑘+1)−𝑉_𝜎(𝑘)(𝑥(𝑘),𝑘)=𝑥^𝑇(𝑘+1)𝑃_𝑗𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+^𝑑_𝑙=1𝑥^𝑇(𝑘+1−𝑙)𝑄_𝑗−𝑥(𝑘+1−𝑙)_𝑑𝑙=1𝑥^𝑇(𝑘−𝑙)𝑄_𝑖𝑥(𝑘−𝑙), (2.15) Then, it can be shown that Δ𝑉_𝜎(𝑥(𝑘),𝑘)+𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)=𝑥^𝑇(𝑘+1)𝑃_𝑗𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝑄_𝑗𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑)𝑄_𝑖𝑥(𝑘−𝑑)+𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)=[𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑)+Γ_𝑖𝑤(𝑘)]^𝑇𝑃_𝑗[𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑)+Γ_𝑖𝑤(𝑘)]−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝑄_𝑗𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑)𝑄_𝑖×𝑥(𝑘−𝑑)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)+[𝐶_𝑖𝑥(𝑘)+𝐶_𝑑𝑖𝑥(𝑘−𝑑)+Φ_𝑖𝑤(𝑘)]^𝑇[𝐶_𝑖𝑥(𝑘)+𝐶_𝑑𝑖𝑥(𝑘−𝑑)+Φ_𝑖=𝑤(𝑘)]𝑥(𝑘)𝑥(𝑘−𝑑)𝑤(𝑘)^𝑇Θ𝑥(𝑘)𝑥(𝑘−𝑑)𝑤(𝑘)<0, (2.16) where Θ=−𝑃_𝑖+𝑄_𝑗+𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑖+𝐶^𝑇_𝑖𝐶_𝑖𝐴^𝑇_𝑖𝑃_𝑗𝐴_𝑑𝑖+𝐶^𝑇_𝑖𝐶_𝑑𝑖𝐴^𝑇_𝑖𝑃_𝑗Γ_𝑖+𝐶^𝑇_𝑖Φ_𝑖∗−𝑄_𝑖+𝐴^𝑇_𝑑𝑖𝑃_𝑗𝐴_𝑑𝑖+𝐶^𝑇_𝑑𝑖𝐶_𝑑𝑖𝐴^𝑇_𝑑𝑖𝑃_𝑗Γ_𝑖+𝐶^𝑇_𝑑𝑖Φ_𝑖∗∗−𝛾²𝐼+Γ^𝑇_𝑖𝑃_𝑗Γ_𝑖+Φ^𝑇_𝑖Φ_𝑖. (2.17) Since ‖𝑥(𝑘)𝑥(𝑘−𝑑)𝑤(𝑘)‖≠0, then Θ<0.
By the Schur complement, Θ<0 is equivalent to −𝑃_𝑖+𝑄_𝑗00𝐴^𝑇_𝑖𝑃_𝑗𝐶^𝑇_𝑖∗−𝑄_𝑖0𝐴^𝑇_𝑑𝑖𝑃_𝑗𝐶^𝑇_𝑑𝑖∗∗−𝛾²𝐼Γ^𝑇_𝑖𝑃_𝑗Φ^𝑇_𝑖∗∗∗−𝑃_𝑗0∗∗∗∗−𝐼<0. (2.18) Applying the congruent transformation diag{𝐼,𝐼,𝐼,𝑋_𝑗,𝐼},𝑋_𝑗=𝑃_𝑗⁻¹, (2.19) we obtain −𝑃_𝑖+𝑄_𝑗00𝐴^𝑇_𝑖𝐶^𝑇_𝑖∗−𝑄_𝑖0𝐴^𝑇_𝑑𝑖𝐶^𝑇_𝑑𝑖∗∗−𝛾²𝐼Γ^𝑇_𝑖Φ^𝑇_𝑖∗∗∗−𝑋_𝑗0∗∗∗∗−𝐼<0. (2.20) 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 𝑃_𝑖=𝑃_𝑖^𝑇>0,𝑄_𝑖=𝑄^𝑇_𝑖>0, and 𝑋_𝑗=𝑋^𝑇_𝑗>0 satisfying the LMIs −𝑃_𝑖+𝑄_𝑗0𝐴^𝑇_𝑖𝑝𝐶^𝑇_𝑖𝑝∗−𝑄_𝑖𝐴^𝑇_𝑑𝑖𝑝𝐶^𝑇_𝑑𝑖𝑝∗∗−𝑋_𝑗0∗∗∗−𝐼<0, (2.21)(𝑖,𝑗)∈{1,…,𝑁}×{1,…,𝑁}, 𝑝∈{1,…,𝑀_𝑖}.

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).

3. 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.

3.1. 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_𝑖𝑥(𝑘),𝑖∈{1,…,𝑁} such that the closed-loop system 𝑥(𝑘+1)=(𝐴_𝑖+𝐵_𝑖𝐾_𝑖)𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑)+Γ_𝑖=𝑤(𝑘)𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑)+Γ_𝑖𝑤(𝑘),𝑧(𝑘)=(𝐶_𝑖+𝐷_𝑖𝐾_𝑖)𝑥(𝑘)+𝐶_𝑑𝑖𝑥(𝑘−𝑑)+Φ_𝑖=𝑤(𝑘)𝐶_𝑖𝑥(𝑘)+𝐶_𝑑𝑖𝑥(𝑘−𝑑)+Φ_𝑖𝑤(𝑘) (3.1)(3.2) is UQS with an 𝐿₂-gain Υ<𝛾.

Theorem 3.1. Switched system (3.1)-(3.2) is UQS with an 𝐿₂-gainΥ<𝛾, if there exist matrices 𝑃_𝑖=𝑃^𝑇_𝑖>0,𝑄_𝑖=𝑄^𝑇_𝑖>0,𝑋_𝑗=𝑋^𝑇_𝑖>0,𝑍_𝑖,𝑇_𝑖=𝑋_𝑖𝑄𝑋_𝑖, and a scalar 𝛾>0, such that the following LMIs hold: −𝑋_𝑖+𝑇_𝑗00𝑋_𝑖𝐴^𝑇_𝑖𝑝+𝑍^𝑇_𝑖𝐵^𝑇_𝑖𝑝𝑋_𝑖𝐶^𝑇_𝑖𝑝+𝑍^𝑇_𝑖𝐷^𝑇_𝑖𝑝∗−𝑇_𝑖0𝑋_𝑖𝐴^𝑇_𝑑𝑖𝑝𝑋_𝑖𝐶^𝑇_𝑑𝑖𝑝∗∗−𝛾²𝐼Γ^𝑇_𝑖𝑝Φ^𝑇_𝑖𝑝∗∗∗−𝑋_𝑗0∗∗∗∗−𝐼<0. (3.3) Moreover, the gain matrix is given by𝐾_𝑖=𝑍_𝑖𝑋_𝑖⁻¹.

Proof. It follows from Theorem 2.4 that switched system (3.1)-(3.2) is UQS with an 𝐿₂-gainΥ<𝛾, if, for ∀(𝑖,𝑗)∈{1,…,𝑁}×{1,…,𝑁}, there exist matrices 𝑃_𝑖=𝑃^𝑇_𝑖>0, 𝑄=𝑄^𝑇>0, and𝑋_𝑗=𝑋^𝑇_𝑗>0 holding the following LMIs: −𝑃_𝑖+𝑄_𝑗00𝐴^𝑇_𝑖𝐶^𝑇_𝑖∗−𝑄_𝑖0𝐴^𝑇_𝑑𝑖𝐶^𝑇_𝑑𝑖∗∗−𝛾²𝐼Γ^𝑇_𝑖Φ^𝑇_𝑖∗∗∗−𝑋_𝑗0=∗∗∗∗−𝐼−𝑃_𝑖+𝑄_𝑗00𝐴^𝑇_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝐶^𝑇_𝑖+𝐾^𝑇_𝑖𝐷^𝑇_𝑖∗−𝑄_𝑖0𝐴^𝑇_𝑑𝑖𝐶^𝑇_𝑑𝑖∗∗−𝛾²𝐼Γ^𝑇_𝑖Φ^𝑇_𝑖∗∗∗−𝑋_𝑗0∗∗∗∗−𝐼<0. (3.4)
Applying the congruent transformation diag{𝑋_𝑖,𝑋_𝑖,𝐼,𝐼,𝐼} to LMIs (3.4), and let 𝑋_𝑖=𝑃_𝑖⁻¹,𝑍_𝑖=𝐾_𝑖𝑋_𝑖, we obtain −𝑋_𝑖+𝑇_𝑗00𝑋_𝑖𝐴^𝑇_𝑖+𝑍^𝑇_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐶^𝑇_𝑖+𝑍^𝑇_𝑖𝐷^𝑇_𝑖∗−𝑇_𝑖0𝑋_𝑖𝐴^𝑇_𝑑𝑖𝑋_𝑖𝐶^𝑇_𝑑𝑖∗∗−𝛾²𝐼Γ^𝑇_𝑖Φ^𝑇_𝑖∗∗∗−𝑋_𝑗0∗∗∗∗−𝐼<0. (3.5) Upon using vertex representation (2.5)-(2.6), we get (3.3) from (3.5).

3.2. 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 𝐴₁₁=0.30.1−0.40.2,𝐴₁₂=0.20.10.60.3,𝐵₁₁=,𝐵210.61₁₂=20.90.72,𝐴_𝑑11=00.10.20,𝐴_𝑑12=,Γ0−0.1−0.10₁₁=0.20.3,Γ₁₂=0.20.8,𝐶₁₁=0.10.3,𝐶₁₂=,𝐷0.70.3₁₁=0.10.3,𝐷₁₂=0.90.3,𝐶_𝑑11=,𝐶02_𝑑12=0−1,Φ₁₁=0.6,Φ₁₂=0.1. (3.6)

Mode 2: consider 𝐴₂₁=0.10.20.30.4,𝐴₂₂=0.20.10.60.3,𝐵₂₁=,𝐵1431₂₂=20.90.72,𝐴_𝑑21=00.10.20,𝐴_𝑑22=,Γ0−0.1−0.10₂₁=0.10.5,Γ₂₂=0.20.8,𝐶₂₁=0.60.2,𝐶₂₂=,𝐷0.90.3₂₁=0.80.3,𝐷₂₂=0.90.3,𝐶_𝑑21=,𝐶−0.10_𝑑22=−10,Φ₂₁=0.3,Φ₂₂=0.1. (3.7) We set time delay 𝑑=1, exogenous disturbance 𝑤(𝑘)=0.02sin(0.02𝜋𝑘), then the feasible solution of LMIs (3.3) is given by 𝛾=2.1, and 𝑋₁=0.20310.01760.01760.1763,𝑋₂=,𝑍0.2056−0.3407−0.34071.6231₁=0.0024−0.0065−0.0473−0.0334,𝑍₂=,𝑇0.0079−0.0993−0.0017−0.0844₁=0.06610.01990.01990.1558,𝑇₂=,𝑄0.1276−0.2793−0.2793−0.9434₁=1.5700−0.0322−0.03225.0029,𝑄₂=.2.88440.23200.23200.3284 (3.8) Since 𝐾_𝑖=𝑍_𝑖𝑋_𝑖⁻¹, the control gains are 𝐾₁=0.0154−0.0386−0.2182−0.1678,𝐾₂=−0.0964−0.0814−0.1450−0.0825. (3.9) 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.

Figure 1 

The switching signal of the switched systems.

Figure 2 

State response of the closed-loop system.

Figure 3 

Control input of the switched discrete-time systems.

Figure 4 

Controlled output signal of the closed-loop switched systems.

Figure 5 

State response of the closed-loop switched discrete-time systems.

Figure 6 

Control input of the switched discrete time systems.

Figure 7 

Output signal of the closed-loop switched discrete time systems.

From the simulation results, it can be clearly seen that the proposed control law guarantees the asymptotic stability of the closed-loop system.

3.3. Switched Static Output Feedback

The objective of this subsection is to design switched output feedback 𝑢(𝑘)=𝐺_𝑖𝑦(𝑘), 𝑖∈{1,…,𝑁}, such that the closed-loop system 𝑥(𝑘+1)=(𝐴_𝑖+𝐵_𝑖𝐺_𝑖𝐿_𝑖)𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑)+Γ_𝑖𝑤(𝑘)=𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑)+Γ_𝑖𝑤(𝑘),𝑧(𝑘)=(𝐶_𝑖+𝐷_𝑖𝐺_𝑖𝐿_𝑖)𝑥(𝑘)+𝐶_𝑑𝑖𝑥(𝑘−𝑑)+Φ_𝑖𝑤(𝑘)=𝐶_𝑖𝑥(𝑘)+𝐶_𝑑𝑖𝑥(𝑘−𝑑)+Φ_𝑖𝑤(𝑘) (3.10) is UQS with an 𝐿₂-gain Υ<𝛾.

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 𝐿_𝑖=𝑈_𝑖Λ_𝑝𝑖0𝑉^𝑇_𝑖, (3.11) where 𝑈_𝑖∈𝑅^𝑝×𝑝,𝑉_𝑖∈𝑅^𝑛×𝑛 are unitary matrices and Λ_𝑝𝑖∈𝑅^𝑝×𝑝 is a diagonal positive matrix. Let 𝑋_𝑖=𝑋^𝑇_𝑖>0,𝑋_𝑖∈𝑅^𝑛×𝑛, then there exists a matrix 0<𝐸_𝑖∈𝑅^𝑝×𝑝, such that 𝐿_𝑖𝑋_𝑖=𝐸_𝑖𝐿_𝑖 (3.12) if and only if 𝑋_𝑖=𝑉_𝑖𝑋_𝑖𝑢0∗𝑋_𝑖𝑣𝑉^𝑇_𝑖,𝑋_𝑖𝑢∈𝑅^𝑝×𝑝,𝑋_𝑖𝑣∈𝑅^{(𝑛−𝑝)×(𝑛−𝑝)}. (3.13) Thus, we get 𝐸_𝑖⁻¹=𝑈_𝑖Λ_𝑝𝑖𝑋⁻¹_𝑖𝑢Λ⁻¹_𝑝𝑖𝑈_𝑖⁻¹.

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_𝑖𝑦(𝑘), and output matrix 𝐿_𝑖 having the SVD form 𝐿_𝑖=𝑈_𝑖[Λ_𝑝𝑖0]𝑉^𝑇_𝑖,Λ_𝑝𝑖∈𝑅^𝑝×𝑝. The resulting closed-loop system is UQS with an 𝐿₂-gain Υ<𝛾, if there exist matrices 0<𝑋_𝑖𝑢=𝑋^𝑇_𝑖𝑢∈𝑅^𝑝×𝑝, 0<𝑋_𝑖𝑣=𝑋^𝑇_𝑖𝑣∈𝑅^{(𝑛−𝑝)×(𝑛−𝑝)}, 0<𝑋_𝑗=𝑋^𝑇_𝑗, 0<𝑇_𝑖=𝑇^𝑇_𝑖, 𝐸_𝑖>0, 𝑅_𝑖, 𝐿_𝑖𝑋_𝑖=𝐸_𝑖𝐿_𝑖, and a scalar 𝛾>0, such that the LMIs −𝑋_𝑖+𝑇_𝑗00𝑋_𝑖𝐴^𝑇_𝑖𝑝+𝐿^𝑇_𝑖𝑅^𝑇_𝑖𝐵^𝑇_𝑖𝑝𝑋_𝑖𝐶^𝑇_𝑖𝑝+𝐿^𝑇_𝑖𝑅^𝑇_𝑖𝐷^𝑇_𝑖𝑝∗−𝑇_𝑖0𝑋_𝑖𝐴^𝑇_𝑑𝑖𝑝𝑋_𝑖𝐶^𝑇_𝑑𝑖𝑝∗∗−𝛾²𝐼Γ^𝑇_𝑖𝑝Φ^𝑇_𝑖𝑝∗∗∗−𝑋_𝑗0∗∗∗∗−𝐼<0 (3.14) have a feasible solution. Moreover, the feedback gain is given by 𝐺_𝑖=𝑅_𝑖𝑈_𝑖Λ_𝑝𝑖𝑋⁻¹_𝑖𝑢Λ⁻¹_𝑝𝑖𝑈_𝑖⁻¹, 𝑖∈{1,…,𝑁}.

Proof. In view of Theorem 2.4, we have −𝑋_𝑖+𝑇_𝑗00𝑋_𝑖𝐴^𝑇_𝑖+𝑋_𝑖𝐿^𝑇_𝑖𝐺^𝑇_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐶^𝑇_𝑖+𝑋_𝑖𝐿^𝑇_𝑖𝐺^𝑇_𝑖𝐷^𝑇_𝑖∗−𝑇_𝑖0𝑋_𝑖𝐴^𝑇_𝑑𝑖𝑋_𝑖𝐶^𝑇_𝑑𝑖∗∗−𝛾²𝐼Γ^𝑇_𝑖Φ^𝑇_𝑖∗∗∗−𝑋_𝑗0∗∗∗∗−𝐼<0. (3.15) Let 𝑅_𝑖=𝐺_𝑖𝐸_𝑖, then (3.15) can be rewritten as follows: −𝑋_𝑖+𝑇_𝑗00𝑋_𝑖𝐴^𝑇_𝑖+𝐿^𝑇_𝑖𝑅^𝑇_𝑖𝐵^𝑇_𝑖𝑋_𝑖𝐶^𝑇_𝑖+𝐿^𝑇_𝑖𝑅^𝑇_𝑖𝐷^𝑇_𝑖∗−𝑇_𝑖0𝑋_𝑖𝐴^𝑇_𝑑𝑖𝑋_𝑖𝐶^𝑇_𝑑𝑖∗∗−𝛾²𝐼Γ^𝑇_𝑖Φ^𝑇_𝑖∗∗∗−𝑋_𝑗0∗∗∗∗−𝐼<0. (3.16) Upon using vertex representation (2.5)-(2.6), we get (3.14) from (3.16). According to Lemma 3.2, we obtain that 𝐺_𝑖=𝑅_𝑖𝑈_𝑖Λ_𝑝𝑖𝑋⁻¹_𝑖𝑢Λ⁻¹_𝑝𝑖𝑈_𝑖⁻¹,𝑖∈{1,…,𝑁}.

3.4. Example 2

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.

Mode 1: consider 𝐿₁₁=0.5000.5,𝐿₁₂=0.6000.4. (3.17)

Mode 2: consider 𝐿₂₁=0.4000.6,𝐿₂₂=0.6000.4. (3.18) In this example, we set time delay 𝑑=1, the exogenous disturbance 𝑤(𝑘)=0.02sin(0.02𝜋𝑘), then the feasible solution of LMIs (3.14) is given by 𝛾=2 and 𝑋₁=0.19550.02540.02540.1803,𝑋₂=,𝑅0.1655−0.1082−0.10820.5388₁=0.0115−0.0072−0.0927−0.0927,𝑅₂=,𝐸−0.0071−0.0789−0.0425−0.0504₁₁=0.19550.02540.02540.1803,𝐸₁₂=,𝐸0.19550.03810.01690.1803₁=0.1955−0.0343−0.01950.1803,𝐸₂₁=,𝐸0.1655−0.0721−0.16230.5388₂₂=0.1655−0.1623−0.07210.5388,𝐸₂=,𝑇0.1655−0.1352−0.09920.5388₁=0.06880.02180.02180.1611,𝑇₂=,𝑄0.0920−0.1057−0.10570.1611₁=0.06880.00780.03690.1631,𝑄₂=.0.1176−0.2333−0.06650.2616 (3.19) Since 𝐺_𝑖=𝑅_𝑖𝐸_𝑖⁻¹, the control gains are 𝐺₁=0.0642−0.0521−0.4312−0.4320,𝐺₂=−0.1537−0.1849−0.3685−0.1860. (3.20) With the initial state 𝑥(0)=[−0.54−0.56]^𝑇, and the switching signal as Example 1, the system state, control, and output signal are shown as follows.

4. 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: 𝜉(𝑘+1)=𝐴_𝑐𝑖𝜉(𝑘)+𝐵_𝑐𝑖𝑦(𝑘),𝑢(𝑘)=𝐶_𝑐𝑖𝜉(𝑘). (4.1) 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 𝜂(𝑘+1)=𝐴_𝑖𝜂(𝑘)+𝐴_𝑑𝑖𝜂(𝑘−𝑑)+Γ_𝑖𝐶𝑤(𝑘),𝑧(𝑘)=_𝑖𝜂(𝑘)+𝐶_𝑑𝑖𝜂(𝑘−𝑑)+Φ_𝑖𝑤(𝑘), (4.2) where 𝐴_𝑖=𝐴_𝑖𝐵_𝑖𝐶_𝑐𝑖𝐵_𝑐𝑖𝐿_𝑖𝐴_𝑐𝑖,𝐴_𝑑𝑖=𝐴_𝑑𝑖0,Γ00^𝑖=Γ_𝑖0,𝐶_𝑖=𝐶_𝑖𝐷_𝑖𝐶_𝑐𝑖,𝐶_𝑑𝑖=𝐶_𝑑𝑖0. (4.3)

Theorem 4.1. Consider switched system (4.2) with polytopic representation (2.5)-(2.6) and output matrix 𝐿_𝑖 having the SVD form 𝐿_𝑖=𝑈_𝑖[Λ_𝑝𝑖0]𝑉^𝑇_𝑖,Λ_𝑝𝑖∈𝑅^𝑝×𝑝, 𝐿_𝑖𝑋_𝑖=𝐸_𝑖𝐿_𝑖. This system is UQS with an 𝐿₂-gain Υ<𝛾, if there exist 𝑃_𝑖=𝑃_𝑠𝑖0∗𝑃_𝑐𝑖,𝑇_𝑖=𝑇_𝑠𝑖0∗𝑇_𝑐𝑖>0,𝑋_𝑖≜𝑃_𝑖⁻¹=𝑋_𝑠𝑖0∗𝑋_𝑐𝑖,0<𝑋_𝑠𝑖𝑢=𝑋^𝑇_𝑠𝑖𝑢,0<𝑋_𝑠𝑖𝑣=𝑋^𝑇_𝑠𝑖𝑣,𝑌_𝑗=𝑌_𝑠𝑗0∗𝑌_𝑐𝑗,0<𝑌_𝑠𝑗=𝑌^𝑇_𝑠𝑗,0<𝑌_𝑐𝑗=𝑌^𝑇_𝑐𝑗,Ω_𝑐𝑖,Π_𝑐𝑖,Ψ_𝑐𝑖,(𝑖,𝑗)∈{1,…,𝑁}×{1,…,𝑁}, (4.4) and a scalar 𝛾>0 satisfying the systems LMIs −𝑋_𝑠𝑖+𝑇_𝑠𝑖0000𝑋_𝑠𝑖𝐴^𝑇_𝑖𝑝𝐿^𝑇_𝑖𝑝Ω^𝑇_𝑐𝑖𝑋_𝑠𝑖𝐶^𝑇_𝑖𝑝∗−𝑋_𝑐𝑖+𝑇_𝑐𝑖000Π^𝑇_𝑐𝑖𝐵^𝑇_𝑖𝑝Ψ^𝑇_𝑐𝑖Π^𝑇_𝑐𝑖𝐷^𝑇_𝑖𝑝∗∗−𝑇_𝑠𝑖00𝑋_𝑠𝑖𝐴^𝑇_𝑑𝑖𝑝0𝑋_𝑠𝑖𝐶^𝑇_𝑑𝑖𝑝∗∗∗−𝑇_𝑐𝑖0000∗∗∗∗−𝛾²𝐼Γ^𝑇_𝑖𝑝0Φ^𝑇_𝑖𝑝∗∗∗∗∗−𝑌_𝑠𝑗𝑌00∗∗∗∗∗∗−_𝑐𝑗0∗∗∗∗∗∗∗−𝐼<0. (4.5) Moreover, the gain matrix is given by 𝐴_𝑐𝑖=Ψ_𝑐𝑖𝑋⁻¹_𝑐𝑖,𝐶_𝑐𝑖=Π_𝑐𝑖𝑋⁻¹_𝑐𝑖, and 𝐵_𝑐𝑖=Ω_𝑐𝑖𝐸_𝑖⁻¹=Ω_𝑐𝑖𝑈_𝑖×Λ_𝑝𝑖𝑋⁻¹_𝑠𝑖𝑢Λ⁻¹_𝑝𝑖𝑈_𝑖⁻¹.

Proof. In view of Theorem 2.4, we have −𝑋_𝑖+𝑇_𝑖00𝑋_𝑖𝐴^𝑇_𝑖𝑋_𝑖𝐶^𝑇_𝑖∗−𝑇_𝑖0𝑋_𝑖𝐴^𝑇_𝑑𝑖𝑋_𝑖𝐶^𝑇_𝑑𝑖∗∗−𝛾²𝐼Γ^𝑇_𝑖Φ^𝑇_𝑖∗∗∗−𝑌_𝑗0∗∗∗∗−𝐼<0. (4.6) By applying (4.3)-(4.4) to (4.6), we obtain −𝑋_𝑠𝑖+𝑇_𝑠𝑖0000𝑋_𝑠𝑖𝐴^𝑇_𝑖𝑋_𝑠𝑖𝐿^𝑇_𝑖𝐵^𝑇_𝑐𝑖𝑋_𝑠𝑖𝐶^𝑇_𝑖∗−𝑋_𝑐𝑖+𝑇_𝑐𝑖000𝑋_𝑐𝑖𝐶^𝑇_𝑐𝑖𝐵^𝑇_𝑖𝑋_𝑐𝑖𝐴^𝑇_𝑐𝑖𝑋_𝑐𝑖𝐶^𝑇_𝑐𝑖𝐷^𝑇_𝑖∗∗−𝑇_𝑠𝑖00𝑋_𝑠𝑖𝐴^𝑇_𝑑𝑖0𝑋_𝑠𝑖𝐶^𝑇_𝑑𝑖∗∗∗−𝑇_𝑐𝑖0000∗∗∗∗−𝛾²𝐼Γ^𝑇_𝑖0Φ^𝑇_𝑖∗∗∗∗∗−𝑌_𝑠𝑗𝑌00∗∗∗∗∗∗−_𝑐𝑗0∗∗∗∗∗∗∗−𝐼<0. (4.7) Let Ψ_𝑐𝑖=𝐴_𝑐𝑖𝑋_𝑐𝑖,Ω_𝑐𝑖=𝐵_𝑐𝑖𝐸_𝑖,Π_𝑐𝑖=𝐶_𝑐𝑖𝑋_𝑐𝑖, and using vertex representation (2.5)-(2.6), we get (4.5) from (4.7).

5. Conclusions

We have examined 𝐿₂-gain analysis and control synthesis for a class of switched time-delay systems with convex-bounded parameter uncertainties in the system matrices using an appropriate switched Lyapunov functional. LMI-based feasibility conditions have been developed to ensure that the linear switched time-delay system with polytopic uncertainties is UQS with an 𝐿₂-gain. Switched feedback schemes have been designed using state output measurements and by using dynamic output feedback to guarantee that the corresponding closed-loop system enjoys the uniform quadratic stability with an 𝐿₂-gain smaller than a prescribed constant level.

Acknowledgments

The authors are grateful to the comments of anonymous referees and remarks of the editors. This work was supported by the Grants from NSFC (nos. 60774037, 60904069, 10926048, and 60911120067), the Doctoral Fund of Ministry of Education of China (no. 20091103120008), and the Science and Technology Development Programs of Beijing Educational Commission (no. KM201010005031).