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Perturbation-Tolerant Structural Controllability for Linear Systems

Yuan Zhang, Yuanqing Xia, Gang Wang, Jinhui Zhang

Abstract

This paper proposes a notion termed perturbation-tolerant structural controllability (PTSC) to study the generic property of controllability preservation/resilience of structured linear systems under structured perturbations. A structured system is said to be PTSC with respect to a perturbation structure if for almost all controllable realizations of this system, there are no complex-valued perturbations obeying the zero/nonzero pattern prescribed by the perturbation structure that can make the perturbed systems uncontrollable. We prove a generic property in this notion, that for almost all controllable realizations of a structured system, either there exist such structured perturbations rendering the systems uncontrollable, or there are no such perturbations. We present a decomposition-based necessary and sufficient condition for the PTSC of single-input linear systems, whose verification has polynomial time complexity. We then discuss some intuitive graph-theoretic conditions for PTSC. As an application, our results can serve as some feasibility conditions for the conventional structured controllability radius problems from a generic viewpoint.

Perturbation-Tolerant Structural Controllability for Linear Systems

Abstract

This paper proposes a notion termed perturbation-tolerant structural controllability (PTSC) to study the generic property of controllability preservation/resilience of structured linear systems under structured perturbations. A structured system is said to be PTSC with respect to a perturbation structure if for almost all controllable realizations of this system, there are no complex-valued perturbations obeying the zero/nonzero pattern prescribed by the perturbation structure that can make the perturbed systems uncontrollable. We prove a generic property in this notion, that for almost all controllable realizations of a structured system, either there exist such structured perturbations rendering the systems uncontrollable, or there are no such perturbations. We present a decomposition-based necessary and sufficient condition for the PTSC of single-input linear systems, whose verification has polynomial time complexity. We then discuss some intuitive graph-theoretic conditions for PTSC. As an application, our results can serve as some feasibility conditions for the conventional structured controllability radius problems from a generic viewpoint.

Paper Structure

This paper contains 18 sections, 19 theorems, 33 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Structured Matrix
  4. Notion of PTSC
  5. Relations with SSC
  6. Preliminaries and Terminologies
  7. Graph Theory
  8. Structural Controllability
  9. Genericity of PTC
  10. Necessary and Sufficient Condition For Single-Input Systems
  11. One-edge Preservation Principle
  12. Conditions for Zero/Nonzero Uncontrollable Modes
  13. Necessary and Sufficient Condition
  14. Graph-Theoretic Criteria for Single-Input Case
  15. Application to SCRP
  16. ...and 3 more sections

Key Result

Lemma 1

Murota_Book For a bipartite graph ${\cal B}(M)=({\cal V}^+ ,{\cal V}^- ,{\cal E})$ associated with a generic square matrix $M$, the following conditions are equivalent: 1) ${\cal B}(M)$ is DM-irreducible; 2) ${\rm mt}({\cal B}(M)-\{v_1,v_2\})= {\rm mt}({\cal B}(M))-1$ for any $v_1\in {\cal V}^+$ and

Figures (3)

  • Figure 1: ${\cal G}(\bar{A},\bar{b})\cup {\cal G}(\bar{F})$ (a) and its cactus (b) in Example \ref{['cactus-not-necessary']}. Dotted edges represent perturbed edges.
  • Figure 2: The graphs for Example \ref{['example6']} ($i=j=3$). In (a), bold edges represent a path-cycle family; In (b), bold edges represent a path family, while the red edge stands for a self-loop; In (c), the dotted edges are reverse edges of the path family in (b); In (d), bold edges represent a path family.
  • Figure 3: Illustration of proof for property 3) of Lemma \ref{['matrix_pencil']}

Theorems & Definitions (36)

  • Definition 1: PTC
  • Definition 2: Structural controllability
  • Definition 3: PTSC
  • Example 1
  • Remark 1
  • Remark 2: Alternative definition of PTSC
  • Remark 3
  • Definition 4: SSC,mayeda1979strong
  • Example 2
  • Definition 5: DM-decomposition, Murota_Book
  • ...and 26 more