Stability of nonlinear hybrid dynamical systems with time delay via sum of squares decomposition

Considering an infinite number of eigenvalues for time delay systems, it is difficult to determine their stability. We have developed a new approach for the stability test of time delay nonlinear hybrid systems. Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are found, stability’s analysis of the system is straight-forward. In this paper both delay-independent and delay-dependent stability tests are proposed, based on the construction of appropriate Lyapunov-Krasovskii functionals. The methodology is based on the sum of squares decomposition of multivariate polynomials and the algorithmic construction is achieved through the use of semidefinite programming. The reduction techniques provide numerical solution of large-scale instances; otherwise they will be computationally infeasible to solve. The introduced method can be used for hybrid systems with linear or nonlinear vector fields. Finally simulation results show the correctness and validity of the designed method. Recommended by Editorial Board member Young Soo Suh under the direction of Editor Jae Weon Choi. The authors wish to express their thanks to Dr. A. Papachristodoulou and Dr. M. Peet for their helpful comments and suggestions. Mohammad Ali Badamchizadeh was born in Tabriz, Iran, in December 1975. He received the B.S. degree in Electrical Engineering from University of Tabriz in 1998 and the M.Sc. degree in Control Engineering from University of Tabriz in 2001. He received the Ph.D. degree in Control Engineering from University of Tabriz in 2007. He is now an Assistant Professor in the Faculty of Electrical and Computer Engineering at University of Tabriz. His research interests include Hybrid dynamical systems, Stability of systems, Time delay systems, Robot path planning. Sohrab Khanmohammadi received the B.S. degree in Industrial Engineering from Sharif University, Iran in 1977 and the M.Sc. degree in Automatic from University Paul Sabatie, France in 1980 and the Ph.D. degree in Automatic from National University, ENSAE, France in 1983. He is now a Professor of Electrical Engineering at University of Tabriz. His research interests are Fuzzy control, Artificial Intelligence applications in control and simulation on industrial systems and human behavior. Gasem Alizadeh was born in Tabriz, Iran in 1967. He received the B.S. degree in Electrical Engineering from Sharif University, Iran in 1990 and the M.Sc. degree from Khajeh Nasir Toosi University, Iran in 1993 and the Ph.D. degree in Electrical Engineering from Tarbiat Modarres University, Iran in 1998. From 1998, he is a Member of University of Tabriz in Iran. His research interests are robust and optimal control, guidance, navigation and adaptive control. Ali Aghagolzadeh was born in Babol, Iran. He received the B.S. degree in Electrical Engineering in 1985 from University of Tabriz, Tabriz, Iran, and the M.Sc. degree in Electrical Engineering in 1988 from the Illinois Institute of Technology, Chicago, IL. He also attended the School of Electrical Engineering at Purdue University in August 1998 where he was also employed as a part-time research assistant and received the Ph.D. degree in 1991. He is currently an Associate Professor of Electrical Engineering at University of Tabriz, Tabriz, Iran. His research interests include digital signal and image processing, image coding and communication, computer vision, and image analysis.