On Dimension-Varying Control Systems: A Universal State Space Approach

Abstract

This paper develops a unified framework for the analysis and design of dimension-varying control systems by constructing an intrinsic quotient state space, $Ω$. A significant challenge in non-fixed-dimensional systems is the lack of a common metric space that enables comparison of states across dimensions without relying on arbitrary external embeddings. To address this, we propose a cross-dimensional pseudo-metric $d_{\mathcal{V}}$ on $\mathbb{R}^{\infty}$ and derive $Ω$ by identifying zero-distance representatives. We demonstrate that $Ω$ preserves the essential topological and metric geometry of Euclidean space, providing the necessary foundation to extend fundamental control notions to the dimension-varying case. Specifically, we establish conditions for controllability, observability and stabilizability, and we address the complexities of Lipschitz switching and disturbance decoupling within this common space. The framework is further extended to hierarchical dimension-varying networks. The practical utility of the results is illustrated through a generator-removal-and-reconnection scenario in a three-machine power system. This case study demonstrates the use of translated representatives and projection/lift benchmarks, quantifies event-wise $d_{\mathcal{V}}$-gaps, and provides a finite-schedule dwell-time consistency check to validate the system's structural transitions.