Nonlinear robust periodic output regulation of minimum phase systems

Abstract

In linear system theory, it is a well-known fact that a regulator given by the cascade of an oscillatory dynamics, driven by some regulated variables, and of a stabiliser stabilising the cascade of the plant and of the oscillators has the ability of blocking on the steady state of the regulated variables any harmonics matched with the ones of the oscillators. This is the well-celebrated internal model principle. In this paper, we are interested to follow the same design route for a controlled plant that is a nonlinear and periodic system with period T: we add a bunch of linear oscillators, embedding \(n_o\) harmonics that are multiple of \(2 \pi /T\), driven by a “regulated variable” of the nonlinear system, we look for a stabiliser for the nonlinear cascade of the plant and the oscillators, and we study the asymptotic properties of the resulting closed-loop regulated variable. In this framework, the contributions of the paper are multiple: for specific class of minimum-phase systems we present a systematic way of designing a stabiliser, which is uniform with respect to \(n_o\), by using a mix of high-gain and forwarding techniques; we prove that the resulting closed-loop system has a periodic steady state with period T with a domain of attraction not shrinking with \(n_o\); similarly to the linear case, we also show that the spectrum of the steady-state closed-loop regulated variable does not contain the n harmonics embedded in the bunch of oscillators and that the \(L_2\) norm of the regulated variable is a monotonically decreasing function of \(n_o\). The results are robust, namely the asymptotic properties on the regulated variable hold also in the presence of any uncertainties in the controlled plant not destroying closed-loop stability.