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Rank Resilience of Pattern Matrices against Structured Perturbations with Applications

Yuan Zhang, Yuanqing Xia, Gang Wang

TL;DR

This paper considers the rank resilience of pattern matrices against structured perturbations that can arbitrarily alter the values of a prescribed set of entries, corresponding to link weight variations, and establishes a generic property in this concept.

Abstract

In structured system theory, a pattern matrix is a matrix with entries either fixed to zero or free to take arbitrary numbers. The (generic) rank of a pattern matrix is the rank of almost all its realizations. The resilience of various system properties is closely tied to the rank resilience of the corresponding pattern matrices. Yet, existing literature predominantly explores the latter aspect by focusing on perturbations that change the zero-nonzero structure of pattern matrices, corresponding to link additions/deletions. In this paper, we consider the rank resilience of pattern matrices against structured perturbations that can arbitrarily alter the values of a prescribed set of entries, corresponding to link weight variations. We say a pattern matrix is structurally rank $r$ resilient against a perturbation pattern if almost all realizations of this pattern matrix have a rank not less than $r$ under arbitrary complex-valued realizations of the perturbation pattern. We establish a generic property in this concept. We then present combinatorial necessary and sufficient conditions for a rectangular pattern matrix to lose full rank under given perturbation patterns. We also generalize them to obtain a sufficient condition and a necessary one for losing an arbitrarily prescribed rank. We finally show our results can be applied to the resilience analysis of various properties of structured (descriptor) systems, including controllability and input-state observability, as well as characterizing zero structurally fixed modes.

Rank Resilience of Pattern Matrices against Structured Perturbations with Applications

TL;DR

This paper considers the rank resilience of pattern matrices against structured perturbations that can arbitrarily alter the values of a prescribed set of entries, corresponding to link weight variations, and establishes a generic property in this concept.

Abstract

In structured system theory, a pattern matrix is a matrix with entries either fixed to zero or free to take arbitrary numbers. The (generic) rank of a pattern matrix is the rank of almost all its realizations. The resilience of various system properties is closely tied to the rank resilience of the corresponding pattern matrices. Yet, existing literature predominantly explores the latter aspect by focusing on perturbations that change the zero-nonzero structure of pattern matrices, corresponding to link additions/deletions. In this paper, we consider the rank resilience of pattern matrices against structured perturbations that can arbitrarily alter the values of a prescribed set of entries, corresponding to link weight variations. We say a pattern matrix is structurally rank resilient against a perturbation pattern if almost all realizations of this pattern matrix have a rank not less than under arbitrary complex-valued realizations of the perturbation pattern. We establish a generic property in this concept. We then present combinatorial necessary and sufficient conditions for a rectangular pattern matrix to lose full rank under given perturbation patterns. We also generalize them to obtain a sufficient condition and a necessary one for losing an arbitrarily prescribed rank. We finally show our results can be applied to the resilience analysis of various properties of structured (descriptor) systems, including controllability and input-state observability, as well as characterizing zero structurally fixed modes.

Paper Structure

This paper contains 15 sections, 20 theorems, 26 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Rank resilience of pattern matrices
  3. Genericity in $k$-GRR
  4. Conditions for completability of GLRM$({\mathcal{M}},1)$
  5. Basis preservation approach
  6. Graph-theoretic form
  7. The case $k>1$
  8. Sufficient condition
  9. Necessary condition
  10. Applications to structured systems
  11. Proofs of Theorems \ref{['BasisPreservation']}, \ref{['BasisApproach']} , and \ref{['poly_verify']}
  12. Proof of Theorem \ref{['BasisPreservation']}
  13. Proof of Proposition \ref{['BasisApproach']}
  14. Proof of Theorem \ref{['poly_verify']}
  15. Conclusions

Key Result

Lemma 1

(Hilbert's Nullstellensatz and dimension of solutions for undetermined systems) Sombra1999,Hartshorne2013 The following statements are true with respect to (w.r.t.) the solutions of a system of polynomial equations $(f_1,...,f_t)$ in variables $(x_1,...,x_n)$: 1) $(f_1,...,f_t)$ is inconsistent if a 2) If $(f_1,...,f_t)$ is underdetermined, then it has either infinitely many complex solutions or i

Figures (1)

  • Figure 1: Bipartite graph representation of $\mathcal{M}$ in Example \ref{['exp3']}. Solid lines represent $*$-edges and dotted lines represent $?$-edges.

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Lemma 1
  • Lemma 2: Effective Nullstellensatz
  • Lemma 3
  • Theorem 1
  • Remark 1
  • Example 2
  • ...and 26 more