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

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 [

On the other hand, in many industrial hybrid systems such as power systems [

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

The notation used in this paper is fairly standard. The superscript

Given a class of linear discrete time switched systems with time delay

For a switching mode

Distinct from (

We have the following definitions.

Switched system (

Given a scalar

Switched system (

The following statements are equivalent.

There exist a switching Lyapunov function of the type (

There exist matrices

(1)

By the Schur complement,

(2)

The proof is complete.

Consider switched system (

The following statements are equivalent.

There exists a switching Lyapunov function of type (

There exist matrices

Since the exogenous disturbance

Extending on Section

With reference to system (

Switched system (

It follows from Theorem

Applying the congruent transformation

In this example, we design a switched state feedback control for this system based on Theorem

Mode 1: consider

Mode 2: consider

The switching signal of the switched systems.

State response of the closed-loop system.

Control input of the switched discrete-time systems.

Controlled output signal of the closed-loop switched systems.

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

Control input of the switched discrete time systems.

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.

The objective of this subsection is to design switched output feedback

Before developing our main results, we give the following lemma.

Given a matrix

Consider switched system (

In view of Theorem

Similar to Example

Mode 1: consider

Mode 2: consider

We consider the more general case and employ at every mode

Consider switched system (

In view of Theorem

We have examined

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).