Control of Dynamic Systems with Time-Periodic Coefficients via the Lyapunov-Floquet Transformation and Backstepping Technique

Abstract

We address the problem of designing controllers that guarantee asymptotic stability of a class of linear as well as nonlinear dynamical systems with time-periodic coefficients. Using a repeated procedure consisting of the Lyapunov-Floquet transformation, the backstepping technique, and Floquet theory, the asymptotic stability of the closed-loop linearized system is guaranteed. Further, a Lyapunov matrix for the closed-loop asymptotically stable linearized system is constructed. This Lyapunov function is then used to design a combination of linear and nonlinear controllers in order to guarantee the asymptotic stability of the nonlinear system. The methodology is illustrated by designing linear and nonlinear control laws for a system consisting of two statically coupled pendula, each subjected to a time-periodic force acting in the axial direction.