Nonlinear Periodic Adaptive Control for Linear Time-Varying Plants

Abstract

In adaptive control the goal is to deal with systems that have unknown and/or time-varying parameters. Adaptive control techniques have been developed since 1950’s and most results were proven in the cases when the time-variations were non-existent or slow. However the results pertaining to systems with fast time-variations are still limited, in particular, when it comes to plants with unstable zero dynamics.

In this work we adopt the controller design technique from the area of gain scheduling, where the time-varying parameter is assumed to be measurable. We propose the design of a nonlinear periodic controller, where in each period the state and parameter values are estimated and an appropriate stabilizing control signal is applied. It is shown that the closed loop system is stable under fast parameter variations with persistent jumps: the trajectory of the closed loop state in response to the initial condition is bounded by a decaying exponential plus a gain times the size of the noise. Our approach imposes several constraints on the plant; however, we show that there exists at least one interesting class of systems, which includes plants with unstable zero dynamics, that can be stabilized by our controller.