On Dimension Varying Control Systems: A Universal State Space Approach

A cross-dimensional Euclidian space ($\mathrm{\Omega }$) is proposed for the state space of dimension varying (control) systems. It is shown that the topological structure of $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$, the switching laws are assumed to be Lipschitz. Some reasonable conventional switching laws are proposed. Under the topology deduced by the metric on $\mathrm{\Omega }$ 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.