Robust Output Regulation of Uncertain Linear Time-Varying Systems

Abstract

Robust output regulation for linear time-varying systems has remained an open problem for decades. To address this, we propose the trajectory-matching system immersion framework, by reformulating the regulator equation into a more insightful form. This perspective demonstrates that finding an internal model is equivalent to reproducing the steady-state output trajectories of a given forced system by constructing an unforced one. This reveals the fundamental influence of parametric uncertainties, yielding the precise algebraic boundary for robust regulation, termed finite linear parameterization. With this, we further demonstrate that uncertainties in time-varying systems can easily excite infinite-dimensional families of functions, making it impossible for a finite-dimensional regulator to achieve exact robust regulation. Hence, we establish a comprehensive approximate robust design, which yields a bounded tracking error that can be arbitrarily small, and avoids explicitly solving the regulator equation. Additionally, it can ensure exact regulation when the uncertainty influences the system in some specified ways. Overall, these results provide a general, executable framework for constructing an internal model-based design, and simplify the robust control implementation process.