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. By augmenting the classical immersion viewpoint, we propose the trajectory-matching system immersion framework. It reformulates the regulator equation as a forced system, and demonstrates that finding an internal model is equivalent to reproducing the non-decaying output trajectories of this forced system by constructing an unforced one. This perspective yields an exact algebraic boundary for finite-dimensional internal models, termed finite linear parameterization. It further reveals a distinctive obstruction in time-varying systems: even highly structured, finite-dimensional affine parametric uncertainties can generate infinite-dimensional families of non-decaying error-zeroing signals, thereby precluding exact robust regulation via linear finite-dimensional internal models in general. Hence, we develop 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 recovers exact regulation when the uncertainty influences the system in some specified ways. Overall, these results clarify the intrinsic limitation of exact finite-dimensional robust regulation for uncertain LTV systems, and provide a general, executable framework for constructing an internal model-based design.