Analysis Tools and Control Algorithms for Nonlinear Dynamical Systems

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

9988813 Teel While the field of control design for linear, time-invariant, finite dimensional systems is very mature, there is still a lot remaining to be discovered about control of nonlinear, time-varying, (even finite dimensional) systems. Since nonlinearities are ubiquitous in engineering systems, it is not uncommon to see the lack of knowledge about this topic slowing down technological advances. It is noteworthy that, for example, modern anti-windup control synthesis developed in the last three years was recently needed to solve an industrial, high-performance vibrational attenuation control problem where actuator saturation was non-negligible. The objective of this research is to contribute additional stability analysis tools and control algorithms for nonlinear continuous-time dynamical systems. These tools and algorithms will be keyed to problems of engineering significance. For the sake of generality, systems that are modeled as differential and forward-shift inclusions will be stressed. This class of models is able to address systems that rely on discontinuous controls and those that implement control algorithms with discrete or hybrid dynamics. In this work Lyapunov-type characterizations of set asymptotic stability will be emphasized. Some of the specific goals of the project are to 1) provide important extensions to the most recent converse Lyapunov theorems on strong asymptotic stability of sets; 2) modify these converse Lyapunov function constructions so that they apply to weak asymptotic stability of sets and generate locally Lipschitz weak Lyapunov functions; 3) show how these locally Lipschitz weak Lyapunov functions produce locally Lipschitz control Lyapunov functions for asymptotically controllable nonlinear systems; 4) show how the optimization problem associated with the converse Lyapunov function construction yields a meaningful receding horizon control strategy; 5) analyze the efficacy of these receding horizon control strategies when using discrete-time model approximations; 6) provide new, integral characterizations of uniform global asymptotic stability; 7) use these new characterizations to generate new nonlinear control algorithms. These objectives complement other work of the principal investigator that is aimed at transitioning recently developed nonlinear control theories to industry. Moreover, the objectives are consistent with the broader goal of making the tools of nonlinear analysis and control design more powerful, more concise, better understood and more widely applicable to today's control engineering problems. ***