Positive real control for descriptor systems with uncertainties in the derivative matrix via a proportional plus derivative feedback

Abstract

This article deals with the problem of normalisation and positive real control synthesis for a class of descriptor systems with uncertainties in the derivative matrix. Attention is focused on the design of a proportional plus derivative (PD) state feedback, which guarantees that the closed-loop system is quadratically normal and quadratically stable (QNQS) and the closed-loop transfer function is extended strictly positive real (ESPR). We first present a new linear matrix inequality (LMI) characterisation of positive realness for normal descriptor systems. This new characterisation provides a kind of decoupling among the Lyapunov matrix, the system matrix and the derivative matrix. Based on this, a necessary and sufficient condition for the desired PD controllers is given and a PD controller is constructed by solving LMIs. Finally, a numerical example is given to demonstrate that the proposed method is effective.

Keywords:

• descriptor systems
• positive real control
• proportional plus derivative feedback
• parameter uncertainty
• singular systems

Acknowledgements

The authors are grateful to the Associate Editor and the anonymous referees for their constructive comments and suggestions which helped improving the presentation of this article. This work is partially supported by the Natural Science Foundation of China under Grant nos 61040014, 60974004 and Applied Basic Research Fund of Shenyang Municipal Science & Technology Project under Grant no. F10-205-1-50.