PI-generalizing saturated repetitive learning control for a class of nonlinear uncertain systems: Robustness w.r.t. relative degree zero or one

The class of single-input, single-output, nonlinear, time-invariant systems with unknown output-dependent nonlinearities, unknown parameters and relative degree ρ∈{0,1} is considered. A priori knowledge of the system includes globally Lipschitz nature of the nonlinearities and positive sign of the high-frequency gain, besides the minimum-phase property. Single-input, single-output, observable, minimum phase, linear, time-invariant systems are included as a very special case. The aim is to track, via the output error feedback only, periodic output reference signals with known period. Here we originally show that the same saturated repetitive learning control that constitutes the most natural generalization of the PI (Proportional–Integral) control, concurrently ensures the following properties for any initial condition of the system: asymptotic convergence to zero of the output tracking error is guaranteed if ρ=1; exponential convergence to zero is achieved for both the output and the input tracking errors if ρ=0.