On LMI conditions to design robust static output feedback controller for continuous-time linear systems subject to norm-bounded uncertainties

Abstract

This paper addresses the problem of Static Output Feedback (SOF) stabilisation for continuous-time linear systems subject to norm-bounded parameter uncertainties using the Linear Matrix Inequality (LMI) approach. Usually, this issue leads to the feasibility of a Bilinear Matrix Inequality (BMI), which is difficult to linearise to get non-conservative LMI conditions. We present first, in this paper, some background results on the SOF controller design and that are found to be extended to the case of norm-bounded uncertainties. We show that some restrictions on the feasibility of these results should be guaranteed. Moreover, by means of some technical Lemmas, we transform the BMI into a new LMI with a line search over a scalar variable. An enhanced and less conservative LMI condition with a line search over two scalar variables is also developed. Furthermore, a simplified version of each LMI condition without a priori fixed parameters is also presented. An extensive portfolio of numerical examples is presented in order to evaluate the conservativeness and to show the superiority of the proposed design method to the background results.

Keywords:

• Uncertain linear systems
• static output feedback control
• robust control
• norm-bounded parametric uncertainty
• linear matrix inequality (LMI)

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Hassène Gritli

Hassène Gritli received the Engineer's degree in 2006 and the Master's degree in 2007 in Industrial Computing and Automation from the National Institute of Applied Sciences and Technology (INSAT), Tunisia. He received also the Ph.D. degree in Electrical Engineering from the National Engineering School of Tunis (ENIT) in 2013, Tunisia. From 2008 to 2014, he was an Assistant Professor at the Higher Institute of Applied Sciences and Technology of Mateur and at the Higher Institute of Technological Studies of Kélibia, Tunisia. Since September 2014, he was an Associate Professor in Robotics and Automation at the Higher Institute of Information and Communication Technologies (ISTIC), Tunisia. He is presently a tenured research scientist in Robotics and Automatic Control at the Laboratory of Robotics, Informatics and Complex Systems (RISC) at ENIT. His main research interests include: impulsive and hybrid systems; biped robots; manipulator robots; exoskeleton robots; chaos and bifurcations theory; linear and nonlinear control; Linear Matrix Inequality (LMI); observer design; observer-based feedback control; static output feedback control; robust control; ADRC, etc. He serves as a regular referee for many international journals in the field of Control and Robotics.

Ali Zemouche

Ali Zemouche received his Ph.D. degree in automatic control in 2007, from the University Louis Pasteur, Strasbourg, France, where he held post-doctorate degree from October 2007 to August 2008. Dr. Zemouche joined, as Associate Professor, the Centre de Recherche en Automatique de Nancy (CRAN UMR CNRS 7039) at the Université de Lorraine, since September 2008. He held an academic position at Inria – Saclay (EPI DISCO) from September 2016 to August 2017. His research activities include nonlinear systems, state observers, observer-based control, time-delay systems, robust control, and application to real-world models. Dr. Zemouche is currently associate editor in international journals: SIAM Journal of Control and Optimization, European Journal of Control, Cogent Engineering, IEEE Access, DESIGN – MDPI, International Journal of Aerospace System Science and Engineering. He is also a member of the Conference Editorial Board of IEEE Control Systems Society and IFAC TC2.3 (Non-Linear Control Systems).

Safya Belghith

Safya Belghith received his engineering diploma in ‘Electricity’ (1981) and his D.E.A (1982) and the PhD degree (1985) in ‘Automatic and Signal Processing’ from the High School of Electricity, Lab Signals & Systems, University Paris XI Orsay. In 1997, she received her Status Doctorate in ‘Physical Sciences’ from the Faculty of Sciences Tunis (FST) with collaboration of the Laboratory Signals & Systems, High School of Electricity University Paris XI Orsay. She is now a Professor at the National Engineering School of Tunis (ENIT), Tunisia and director of the Laboratory of Robotics, Informatics and Complex Systems (RISC) at ENIT. His research domain includes nonlinear systems, chaos, communication and cryptography, robotics, control and estimation, etc.