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Approximate Stability Radius Analysis and Design in Linear Systems

Ananta Kant Rai, Vaibhav Katewa

TL;DR

The paper addresses robustness of stability in linear time-invariant systems by studying the stability radius $SR$ under perturbations $A+BΔC$ with sparsity constraints. It develops two approximate SR formulations based on eigenvalue sensitivity: a linear approximation (SR_la) with a closed-form per-eigenvalue solution, and a successive-linear approximation (SR_sla) that builds small steps to improve accuracy. These approximations enable tractable system-design problems (SD_la and SD_sla) that aim to increase SR with minimal perturbation, subject to sparsity. Numerical experiments show that SR_la and SR_sla closely track the true SR, with SR_sla delivering higher accuracy at increased computation time, and the designs successfully achieve target SR values. The results offer a practical framework for making LTI systems more robust to parametric or adversarial perturbations.

Abstract

The robustness of the stability properties of dynamical systems in the presence of unknown/adversarial perturbations to system parameters is a desirable property. In this paper, we present methods to efficiently compute and improve the approximate stability radius of linear time-invariant systems. We propose two methods to derive closed-form expressions of approximate stability radius, and use these to re-design the system matrix to increase the stability radius. Our numerical studies show that the approximations work well and are able to improve the robustness of the stability of the system.

Approximate Stability Radius Analysis and Design in Linear Systems

TL;DR

The paper addresses robustness of stability in linear time-invariant systems by studying the stability radius under perturbations with sparsity constraints. It develops two approximate SR formulations based on eigenvalue sensitivity: a linear approximation (SR_la) with a closed-form per-eigenvalue solution, and a successive-linear approximation (SR_sla) that builds small steps to improve accuracy. These approximations enable tractable system-design problems (SD_la and SD_sla) that aim to increase SR with minimal perturbation, subject to sparsity. Numerical experiments show that SR_la and SR_sla closely track the true SR, with SR_sla delivering higher accuracy at increased computation time, and the designs successfully achieve target SR values. The results offer a practical framework for making LTI systems more robust to parametric or adversarial perturbations.

Abstract

The robustness of the stability properties of dynamical systems in the presence of unknown/adversarial perturbations to system parameters is a desirable property. In this paper, we present methods to efficiently compute and improve the approximate stability radius of linear time-invariant systems. We propose two methods to derive closed-form expressions of approximate stability radius, and use these to re-design the system matrix to increase the stability radius. Our numerical studies show that the approximations work well and are able to improve the robustness of the stability of the system.
Paper Structure (13 sections, 3 theorems, 35 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 13 sections, 3 theorems, 35 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Problem Formulation
  3. Approximate Stability Radius Problems
  4. Eigenvalue Approximation
  5. Stability Radius via Linear Approximation
  6. Stability Radius via Successive-Linear Approximation
  7. System Design Problems
  8. System Design via Linear Approximation
  9. System Design via Successive-Linear Approximation
  10. Numerical Simulations
  11. Approximations of Spectral Abscissa and SR
  12. System Design Problem
  13. CONCLUSION

Key Result

Lemma 1

(Eigenvalue Sensitivity hinrichsen2005mathematical) Let $\lambda_{k}$ be a simple eigenvalue of $A$ with corresponding left and right eigenvectors, $y_{k}$ and $z_{k}$, respectively, such that $y_{k}^{*}z_{k} = 1$ for $k=1,2,\cdots, n$. Then, as $A$ is perturbed to $A(\Delta)$, the sensitivity of $\ where $E_{ij}$ is a matrix with $(i,j)^{\text{th}}$ entry as $1$ and all other entries as $0$.

Figures (4)

  • Figure 1: Variation of spectral abscissas as a function of $\gamma$.
  • Figure 2: Variation of approximation errors $e_{la}$ and $e_{sla}$ as a function of normality gap $NG$ for 200 random triplets $(A,B,C)$.
  • Figure 3: Variation of (a) $\lVert \Delta_o ^{*} \rVert$, and (b) $SR$ as a function of $\epsilon$ for Case I.
  • Figure 4: Variation of (a) $\lVert \Delta_o ^{*} \rVert$, and (b) $SR$ as a function of $\epsilon$ for Case II.

Theorems & Definitions (10)

  • Remark 1: Perturbation structure
  • Definition 1: Stability Radius
  • Lemma 1
  • Remark 2: Effect of normality of $A$ on the approximations
  • Lemma 2
  • Example 1
  • Theorem 1
  • Remark 3: Implementation details of Algorithm \ref{['alg:Sr_Algo_sla']}
  • Remark 4: Comparison of LA and SLA based approaches
  • Remark 5: Approximate solutions can aid other SR algorithms