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Stability and Performance Analysis on Self-dual Cones

Emil Vladu

TL;DR

This work extends stability and performance analysis to cone-preserving, possibly nonsymmetric systems on self-dual cones, moving beyond classical positive-system results. It develops a cone-preserving Sylvester criterion and a set of equivalent conditions for $H_ ty$ performance in monotone systems, showing that the norm bound can be certified via a stabilizing cone-preserving Riccati solution and interior conic inequalities, with the maximal gain attained at zero frequency. Key contributions include a necessary-and-sufficient condition $\\mu(A) + \\mu(D) < 0$ for cone-preserving Sylvester solvability, and a suite of equivalent tests for the bounded real lemma in the cone setting, plus illustrative examples such as a closed-loop $H_ ty$ controller and a positive Sylvester solution in the nonnegative orthant. The results broaden the toolkit for controller synthesis and stability verification in large-scale, cone-preserving systems, enabling performance guarantees beyond the traditional symmetric-cone framework.

Abstract

In this paper, we consider nonsymmetric solutions to certain Lyapunov and Riccati equations and inequalities with coefficient matrices corresponding to cone-preserving dynamical systems. Most results presented here appear to be novel even in the special case of positive systems. First, we provide a simple eigenvalue criterion for a Sylvester equation to admit a cone-preserving solution. For a single system preserving a self-dual cone, this reduces to stability. Further, we provide a set of conditions equivalent to testing a given H-infinity norm bound, as in the bounded real lemma. These feature the stability of a coefficient matrix similar to the Hamiltonian, a solution to two conic inequalities, and a stabilizing cone-preserving solution to a nonsymmetric Riccati equation. Finally, we show that the H-infinity norm is attained at zero frequency.

Stability and Performance Analysis on Self-dual Cones

TL;DR

This work extends stability and performance analysis to cone-preserving, possibly nonsymmetric systems on self-dual cones, moving beyond classical positive-system results. It develops a cone-preserving Sylvester criterion and a set of equivalent conditions for performance in monotone systems, showing that the norm bound can be certified via a stabilizing cone-preserving Riccati solution and interior conic inequalities, with the maximal gain attained at zero frequency. Key contributions include a necessary-and-sufficient condition for cone-preserving Sylvester solvability, and a suite of equivalent tests for the bounded real lemma in the cone setting, plus illustrative examples such as a closed-loop controller and a positive Sylvester solution in the nonnegative orthant. The results broaden the toolkit for controller synthesis and stability verification in large-scale, cone-preserving systems, enabling performance guarantees beyond the traditional symmetric-cone framework.

Abstract

In this paper, we consider nonsymmetric solutions to certain Lyapunov and Riccati equations and inequalities with coefficient matrices corresponding to cone-preserving dynamical systems. Most results presented here appear to be novel even in the special case of positive systems. First, we provide a simple eigenvalue criterion for a Sylvester equation to admit a cone-preserving solution. For a single system preserving a self-dual cone, this reduces to stability. Further, we provide a set of conditions equivalent to testing a given H-infinity norm bound, as in the bounded real lemma. These feature the stability of a coefficient matrix similar to the Hamiltonian, a solution to two conic inequalities, and a stabilizing cone-preserving solution to a nonsymmetric Riccati equation. Finally, we show that the H-infinity norm is attained at zero frequency.

Paper Structure

This paper contains 11 sections, 12 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Stability Analysis
  5. Performance Analysis
  6. Illustrative Examples
  7. A closed-loop positive $H_\infty$ Optimal Controller
  8. A Positive Solution to the Sylvester Equation
  9. Proofs
  10. Conclusions
  11. Acknowledgements

Key Result

Lemma 1

Let $A \in \mathbb{R}^{n \times n}$ and suppose that there exists a $P \in \mathbb{R}^{n \times n}$ with $P + P^T \succ 0$ such that If $\lambda$ is a real eigenvalue of $A$, then $\lambda < 0$.

Theorems & Definitions (26)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 16 more