Generalized Cactus and Structural Controllability of Switched Linear Continuous-Time Systems

TL;DR

The paper addresses structural controllability of switched continuous-time systems by identifying a gap in the sufficiency proof of a key criterion and introduces a novel graph-theoretic framework—multi-layer dynamic graphs, generalized stems/buds, and generalized cactus configurations—to provide a rigorous proof and a generalized cactus-based criterion. This approach links algebraic rank conditions to combinatorial linkings in a dynamic, layered graph, yielding a lower bound on the generic dimension of controllable subspaces via the generic rank $\mathrm{grank}$ and extending Lin's cactus criterion to switched systems for the first time. The methodology delivers a polynomial-time verification pathway using DFS and maximum matching, and clarifies the structural role of cactus-like configurations in switched networks. Overall, the work advances structural controllability analysis for switched systems and lays the groundwork for further characterization of the generic controllable subspace and extensions to discrete-time settings.

Abstract

This paper explores the structural controllability of switched linear continuous-time systems. It first identifies a gap in the proof for a pivotal criterion for the structural controllability of switched linear systems in the literature. To address this void, we develop novel graph-theoretic concepts, such as multi-layer dynamic graphs, generalized stems/buds, and generalized cacti, and based on them, provide a comprehensive proof for this criterion. Our approach also induces a new, generalized cactus based graph-theoretic criterion for structural controllability. This not only extends Lin's cactus-based graph-theoretic condition to switched systems for the first time, but also provides a lower bound for the generic dimension of controllable subspaces.

Figures (4)

• Figure 1: (a): a boost converter (borrowed from LiuStructural). (b): the corresponding structured system model of (a), where $*$ represents nonzero entries and $0$ represents zero entries.
• Figure 2: Illustration of graph representations of a switched system with two subsystems (hereafter, edges with different color indices are depicted with different styles of lines). The subsystem digraphs ${\cal G}_1$ and ${\cal G}_2$ correspond to respectively $(A_1,B_1)$ and $(A_2,B_2)$ given in Example \ref{['example1']}. Next to ${\cal G}_1$ and ${\cal G}_2$ are $W_0,W_1$, and $W_2$. Graph $\hat{\cal G}_2$ is the MDG with $3$ layers. Bold red lines represent a $\hat{U}_{}-\hat{X}_{0}$ linking $L$ with size $3$, which corresponds to $3$ nonzero product terms in $W_1=[B_1,A_1B_1,A_2B_1]$ that are in different rows and columns. The permutation $\pi$ associated with $L$ is $\pi(x_{10}^{00})=u_{110}^{00}$, $\pi(x_{20}^{00})=u^{11}_{111}$, and $\pi(x_{30}^{00})=u_{111}^{21}$. According to (\ref{['determinant-2']}), the determinant of $W_2({\rm head}(L), {\rm tail}(L))$ is $w(L)=a_{21}a_{31}b_1^3$.
• Figure 3: (a): ${\cal G}_c$ of a switched system with $N=2$ and $n=7$. No multiple edges exist in ${\cal G}_c$. (b) and (d) are two generalized stems, and (c) is a generalized bud contained in ${\cal G}_c$.
• Figure 4: (a): ${\cal G}_c$ (no multiple edges) of a switched system with $N=2$ and $n=10$. (b): a generalized cactus configuration in ${\cal G}_c$ (depicted in bold red lines). (c): a conventional cactus configuration in ${\cal G}_c$ (bold red lines).

Theorems & Definitions (23)

• Definition 1: Z.S2002Controllability
• Lemma 1: Z.S2002Controllability
• Definition 2: LiuStructural
• Definition 3
• Definition 4: LiuStructural
• Lemma 2: LiuStructural
• Example 1
• Example 2
• Proposition 1
• Remark 1