Comparison Between Different Continuous-Time Realizations Based on the tau Method of a Nonlinear Repetitive Control Scheme

This paper presents a new framework to study a continuous-time finite-dimensional approximation of a repetitive control scheme for minimum-phase nonlinear systems. Based on the tau method, three state matrices are proposed to approximate the transport equation, namely tau-Fourier, tau- Legendre and tau-Chebyshev models. Then, the forwarding approach allows us to set up three finite dimensional controllers whose structure is derived from these tau models. In all three cases, we prove that a smooth steady state is reached and that the output converges to zero as the order increases. We also show that, for a fixed order, the tau-Fourier controller leads to the smallest output. Lastly, numerical comparisons are discussed and emphasize our theoretical expectations.