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An instrumental matrix approach to design output feedback passive controller for switched singular systems is proposed in this paper. The nonlinear inequality condition including Lyapunov inverse matrix and controller gain matrix is decoupled by introducing additional instrumental matrix variable. Combined with multiple Lyapunov function method, the nonlinear inequality is transformed into linear matrix inequality (LMI). An LMI condition is presented for switched singular system to be stable and passive via static output feedback under designed switching signal. Moreover, the conditions proposed do not require the decomposition of Lyapunov matrix and its inverse matrix or fixing to a special structure. The theoretical results are verified by means of an example. The method introduced in the paper can be effectively extended to a single singular system and normal switched system.

The switched singular systems arise from, for instance, power systems, economic systems, and complex networks. As pointed out in [

It has been shown that passivity is a suitable design approach in power systems [

Up to now, little attention has been paid to passive control problem for switched singular systems. This motivates us to investigate this problem. Furthermore, considering the operational cost and the reliability of systems and the simplicity of implementation, output feedback is always adopted to stabilize a system. Thus, we study passive control of switched singular systems through output feedback.

In this paper, by introducing instrumental matrix variable, the nonlinear inequality including Lyapunov inverse matrix and controller gain matrix is decoupled, which makes the design of output feedback passive controllers for continuous-time switched singular systems easy. Based on multiple Lyapunov functions and variable substitution techniques, a new and simple sufficient condition is presented in terms of LMI, by solving which static output feedback passive controller can be designed. The novelty of the conditions proposed in this paper lies in the following aspect. Decomposition of Lyapunov matrix and its inverse matrix is not required. Moreover, the Lyapunov inverse matrix is not fixed to a special structure.

The rest of this paper is organized as follows. Problem statement and preliminaries are given in Section

Consider the following switched singular system:

Let us consider the following static output feedback controller:

Then, the resulting closed-loop system can be described as

We are now considering the output feedback passive control problem for system (

System (

For a give scalar

To obtain the main results of this paper, the following transformation is introduced.

Since

The following theorem provides a sufficient condition under which system (

If there exist simultaneously nonnegative real number

When

Construct

Design switching signal as

When

Suppose that (

Next, we prove the stability of system (

Condition (

For any

Let

The instrumental matrix variable

Based on the above lemma, an LMI condition is presented, under which system (

If there exist simultaneously nonnegative real number

Since

By making use of the existent methods (e.g., [

If

When switched singular system reduces to a single singular system (i.e., no switching), Theorem

If there exist a real number

If there exist simultaneously nonnegative real number

Consider the switched singular system composed of two subsystems

The corresponding switching signal is chosen as

State response of the corresponding closed-loop system.

In this paper, the output feedback passive control problem for a class of switched singular systems is investigated. A novel method is proposed to solve static output feedback passive controllers. Sufficient linear matrix inequality condition is presented by means of introducing instrumental matrix variable

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China under Grants nos. 61203001, 61104066, and 61473140, Liaoning Educational Committee Foundation under Contract L2014525, and the Natural Science Foundation of Liaoning Province under Grant no. 2014020106.

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