Exponential output tracking via a saturated RLC for a class of nonlinear systems under PSD of periodic references

This paper considers the same class of zero-relative-degree nonlinear systems for which a saturated Repetitive Learning Control (RLC) has been recently shown to ensure exponential – and not just asymptotical – convergence to zero of both the output and the input tracking errors in the case of periodic output reference signals with a known single period. The explicit role of the nonlinear unstructured uncertainties is here investigated within the more general scenario in which the output reference signal is multi-periodic. Special emphasis is provided to the generation of the input reference whose effect has to be exponentially nullified by the RLC. In particular, the following question is answered. Restrict the design to the sub-class of periodic output reference signals that admit a Periodic Signal Decomposition (PSD) (namely, the ones that can be written as a finite sum of periodic functions) with pairwise-commensurable periods (so that the knowledge of the multiple periods characterizing the input reference is preserved once the common multiple among the single periods is additionally included). Then, which robustness and convergence properties can be still achieved by the same output-feedback (definitely saturated) RLC, once it is intuitively generalized to transiently include an additional bank of learning estimation schemes corresponding to the single periods? Interestingly, the theoretical tools of this paper can be also used to successfully address identifiability and convergence issues regarding the identification of periodic components of general multi-periodic signals.