Robust limit cycle control in a class of nonlinear discrete-time systems

ABSTRACT

This paper studies generation of robust periodic solutions in a class of nonlinear discrete-time system. The sustained oscillations, with the desired frequency and amplitude, are achieved through the creation of the appropriate elliptic limit cycle in the phase plane of the uncertain closed-loop discrete-time system. In the first step, the nominal control law is designed to enforce the trajectories of the nominal closed-loop system to converge to the desired limit cycle. Next, considering uncertain terms, an additional robustifying term is designed. This term is added to the nominal controller to sustain the desirable stable oscillations in the presence of uncertain terms. The resulted robust controller brings the trajectories of the uncertain closed-loop discrete-time system to a boundary layer (with adjustable width) around the desired limit cycle. Moreover, the domain of attraction of the limit cycle and also the ultimate boundary layer around it are calculated via the Lyapunov analysis. Additionally, in order to verify the applicability of the proposed method, it is implemented on the discretised model of a spring–damper system. Computer simulations confirm the theoretical results in generating robust stable oscillations.

KEYWORDS:

• Nonlinear discrete-time systems
• sustained oscillations
• elliptic limit cycles
• robust limit cycle control
• $\kappa$-periodic solutions

Additional information

Notes on contributors

Ali Reza Hakimi

Ali Reza Hakimi received his B.Sc. degree in electronic engineering from Islamic Azad University-Yazd Branch and M.Sc. degree in control engineering from Shiraz University of Technology. He is currently a Ph.D. student in the Department of Electrical and Electronic Engineering, Shiraz University of Technology. His research interests include nonlinear and robust control.

Tahereh Binazadeh

Tahereh Binazadeh received her B.Sc. and M.Sc. degrees from Shiraz University, and Ph.D. degree from University of Tehran all in control Engineering. She is now Associate Professor in the Department of Electrical and Electronic Engineering, Shiraz University of Technology. Her research interests includes nonlinear and robust control, multi agent systems and complex systems.