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

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.     

Robust control synthesis for discrete-time uncertain semi-Markov jump systems

This paper studies the control synthesis for uncertain semi-Markov jump systems in a discrete-time domain subjected to external disturbance. The switching between modes is determined by a function of the transition probability and the sojourn-time distribution between two neighbouring modes. Based on the σ-error mean square stability criterion, time-varying controllers are designed to stabilise the system. By constructing a holding time dependent Lyapunov function, time-varying state-feedback controllers are obtained that meet a set of sufficient conditions in the form of linear matrix inequalities. Two examples, including a DC motor system, are presented to show the validity of the proposed control scheme.