Reachability of Dimension-Bounded Linear Systems

TL;DR

This paper investigates the reachability problem in dimension-bounded linear systems with time-varying state dimensions, providing methods to determine reachability, properties of reachable sets, and their relationships over time.

Contribution

It introduces a new approach to analyze reachability in systems with changing state dimensions, including a rank condition and the use of annihilator polynomials.

Findings

t-step reachable subset forms a linear space

Rank condition for verifying reachability

Relationship between invariant space and reachable subset

Abstract

In this paper, the reachability of dimension-bounded linear systems is investigated.Since state dimensions of dimension-bounded linear systems vary with time, the expression of state dimension at each time is provided.A method for judging the reachability of a given vector space is proposed. In addition, this paper proves that the t-step reachable subset is a linear space, and gives a computing method. The t-step reachability of a given state is verified via a rank condition. Furthermore, annihilator polynomials are discussed and used to illustrate the relationship between the invariant space and the reachable subset after the invariant time point t*. The inclusion relation between reachable subsets at times t*+i and t*+j is shown via an example.