Output regulation of nonlinear systems evolving in the neighborhood of a periodic orbit (Q1890999)

The author considers the output regulation problem for a nonlinear system of the form \[ \begin{cases} \dot z & = f(z)+ g(z) u+ p(z)w\\ \dot w & = s(w)\\ y & = h(z)+ q(w),\end{cases} \] where \(z\) denotes the plant state and \(w\) is an exogenous variable. Similar problems have been previously studied in the literature, under the assumption that the plant state variable \(z\) evolves in a neighborhood of an equilibrium position. In this paper, the author assumes that the vector field \(f\) has a periodic orbit, and that the plant state evolves in a neighborhood of this orbit. The problem consists in finding a feedback law \(u= \alpha(z, w)\) for which the output variable \(y(t)\) tends to zero when \(t\to \infty\). At the same time, it is required that the feedback law stabilizes exponentially the plant state evolution to the periodic orbit. A necessary and sufficient condition is given in the form of a set of partial differential equations. The result is illustrated by means of a worked example.