Incremental Stabilization of Cascade Nonlinear Systems and Harmonic Regulation: A Forwarding-Based Design

In this work, we address the problem of designing a control law for a system in feedforward form to be globally incrementally exponentially stable. To do that, we develop an incremental version of the so-called forwarding mod
$\lbrace L_{g}V\rbrace$
approach. Then, we apply such a control design to the problem of compensating matched disturbances assumed to be given by the superimposition of a finite number of harmonics with unknown amplitude. For this, we propose a dynamic controller made of
$L$
linear oscillators processing the regulation error and a stabilizer making the closed-loop system incrementally globally exponentially stable, uniformly with respect to the external signals. This guarantees that the closed-loop system asymptotically converges to a periodic trajectory having the first
$L$
-Fourier coefficients of the error to be zero. Then, we specialize our design for the class of linear systems with a scalar nonlinearity and of minimum-phase systems possessing contractive zero dynamics.