On Dimension Varying Control Systems: A Universal State Space Approach
Daizhan Cheng
TL;DR
This paper introduces a universal state space for dimension-varying control systems by formulating $\\mathbb{R}^{\infty}$ and the quotient $\\Omega=\\mathbb{R}^{\infty}/\\leftrightarrow$ endowed with the distance $d_{\\mathcal{V}}$. It defines a dimension-keeping semi-tensor product (DK-STP) and Lipschitz switching to fuse dynamics across different dimensions, enabling a unified stability and robustness analysis. Building a differential-geometric toolkit on $\\Omega$ (smooth functions, vector fields, distributions), the paper derives controllability, observability, stabilization, and disturbance decoupling results for dimension-varying systems and shows how to embed them into a classical switching framework. It further proposes a hierarchical-network aggregation approach for large-scale, time-varying networks, highlighting practical paths to scalable design.
Abstract
A cross-dimensional Euclidian space ($Ω$) is proposed for the state space of dimension varying (control) systems. It is shown that the topological structure of $Ω$ is consistent with all $\mathbb{R}^n$, which are the state spaces of each component modes of a dimension varying system. Using the universal metric from $Ω$, the switching laws are assumed to be Lipschitz. Some reasonable conventional switching laws are proposed. Under the topology deduced by the metric on $Ω$ some fundamental properties of dimension varying dynamic systems, such as stability and robustness, are investigated. Then some control problems of dimension varying control systems, including controllability, observability, stabilization, disturbance decoupling, etc. are investigated. Finally, an aggregation approach for large scale hierarchical dimension varying networks is proposed.