Periodic Nonlinear Adaptive Control of Rapidly Time-Varying Linear Systems

In adaptive control the goal is to deal with systems that have unknown and/or time-varying parameters. Most techniques are proven for the case in which any time-variation is slow, with results for systems with fast time-variations limited to those for which the time-variation is of a known form or for which the plant has stable zero dynamics. Here we propose a new adaptive controller design methodology for which the time-variation can be rapid. While the plant is allowed to have unstable-zero dynamics, it must satisfy several structural conditions which have been proven to be necessary in the literature; we also impose some mild regularity conditions. The proposed controller is nonlinear and periodic, and in each period the parameter values are estimated and an appropriate stabilizing control signal is applied. Under the technical assumptions that the plant is relative degree one and that the plant uncertainty is in terms of a single scalar variable, it is proven that the closed loop system is stable under fast parameter variations with persistent jumps