On the Solvability of Quasi-Regulator Equations in Non-smooth Output Regulation
This work provides a theoretical foundation for output regulation in a challenging class of systems, but the results are incremental as they extend existing regulator theory to non-smooth signals.
The paper addresses the solvability of quasi-regulator equations for output regulation of linear systems with non-smooth, non-periodic exogenous signals. It provides a necessary and sufficient condition for solvability based on a non-smooth non-resonance condition and relative degree.
Motivated by the prevalence of non-smooth, possibly non-periodic signals in real-world applications, the output regulation of linear systems subject to non-smooth non-periodic exogenous signals has emerged as a challenging problem. A fundamental prerequisite for solving this problem is the existence of solutions to the so-called ``quasi-regulator equations''. In this paper, we investigate the solvability of these equations. To this end, we reformulate the quasi-regulator equations as differential-algebraic equations and highlight the critical role played by the system's relative degree. We finally propose a ``non-smooth non-resonance condition'' that, under specific relative degree requirements, provides a necessary and sufficient characterization of the solvability of the quasi-regulator equations.