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Minimal Input Structural Modifications for Strongly Structural Controllability

Geethu Joseph, Shana Moothedath, Jiabin Lin

TL;DR

The paper addresses how to enforce strong structural controllability in generalized zero/nonzero/arbitrary systems with minimal input-pattern changes. It develops a formal framework based on pattern matrices, zero-forcing sets, and color-change rules to characterize SSC and feasibility, and it reformulates the problem into a cost-minimization over feasible pattern matrices. Two algorithms are proposed: a greedy method that is fast but can be arbitrarily suboptimal, and a randomized Monte Carlo Markov Chain (MCMC) strategy with convergence guarantees and probabilistic error bounds. Numerical experiments on random graphs illustrate the trade-offs: greedy is usually fast and effective, while MCMC provides robustness and often near-optimal solutions, with costs corresponding to the number of changes needed to achieve SSC. The work advances practical design tools for guaranteeing SSC under uncertainty in large networks and suggests directions like restricted modifications and time-varying extensions.

Abstract

This paper studies the problem of modifying the input matrix of a structured system to make the system strongly structurally controllable. We focus on the generalized structured systems that rely on zero/nonzero/arbitrary structure, i.e., some entries of system matrices are zeros, some are nonzero, and the remaining entries can be zero or nonzero (arbitrary). We analyze the feasibility of the problem, and if it is feasible, we reformulate it into another equivalent problem. This new formulation leads to a greedy heuristic algorithm. However, we also show that the greedy algorithm can give arbitrarily poor solutions for some special systems. Our alternative approach is a randomized Markov chain Monte Carlo-based algorithm. Unlike the greedy algorithm, this algorithm is guaranteed to converge to an optimal solution with high probability. Finally, we numerically evaluate the algorithms on random graphs to show that the algorithms perform well.

Minimal Input Structural Modifications for Strongly Structural Controllability

TL;DR

The paper addresses how to enforce strong structural controllability in generalized zero/nonzero/arbitrary systems with minimal input-pattern changes. It develops a formal framework based on pattern matrices, zero-forcing sets, and color-change rules to characterize SSC and feasibility, and it reformulates the problem into a cost-minimization over feasible pattern matrices. Two algorithms are proposed: a greedy method that is fast but can be arbitrarily suboptimal, and a randomized Monte Carlo Markov Chain (MCMC) strategy with convergence guarantees and probabilistic error bounds. Numerical experiments on random graphs illustrate the trade-offs: greedy is usually fast and effective, while MCMC provides robustness and often near-optimal solutions, with costs corresponding to the number of changes needed to achieve SSC. The work advances practical design tools for guaranteeing SSC under uncertainty in large networks and suggests directions like restricted modifications and time-varying extensions.

Abstract

This paper studies the problem of modifying the input matrix of a structured system to make the system strongly structurally controllable. We focus on the generalized structured systems that rely on zero/nonzero/arbitrary structure, i.e., some entries of system matrices are zeros, some are nonzero, and the remaining entries can be zero or nonzero (arbitrary). We analyze the feasibility of the problem, and if it is feasible, we reformulate it into another equivalent problem. This new formulation leads to a greedy heuristic algorithm. However, we also show that the greedy algorithm can give arbitrarily poor solutions for some special systems. Our alternative approach is a randomized Markov chain Monte Carlo-based algorithm. Unlike the greedy algorithm, this algorithm is guaranteed to converge to an optimal solution with high probability. Finally, we numerically evaluate the algorithms on random graphs to show that the algorithms perform well.
Paper Structure (12 sections, 7 theorems, 30 equations, 1 figure, 3 algorithms)

This paper contains 12 sections, 7 theorems, 30 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Minimal Structural Modification Problem
  4. Structural Modification Algorithms
  5. Greedy Algorithm
  6. Monte Carlo Markov Chain Algorithm
  7. Numerical Results
  8. Conclusion
  9. Proof of \ref{['thm:feasibility']}
  10. Proof of \ref{['thm:eq_opt']}
  11. Proof of \ref{['prop:greedyopt']}
  12. Proof of \ref{['prop:prob_error']}

Key Result

Theorem 1

The system $(\mathcal{A},\mathcal{B})$ is strong structural controllable if and only if the following two conditions hold:

Figures (1)

  • Figure 1: Variation of the number of changes required by the greedy and MCMC algorithms with the state and control input dimensions. For the special case discussed in the proof of \ref{['prop:greedyopt']}, the greedy algorithm performed poorly while the MCMC algorithm returned the optimal cost in most cases. However, for Erdős Rényi graphs, the greedy algorithm has a performance comparable to that of the MCMC algorithm.

Theorems & Definitions (13)

  • Theorem 1: jia2020unifying
  • Theorem 2: jia2020unifying
  • Definition 1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Proposition 1
  • proof
  • Theorem 5
  • ...and 3 more