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Lyapunov and Riccati Equations from a Positive System Perspective

Dongjun Wu, Yankai Lin

TL;DR

The paper addresses the constructive analysis of Lyapunov and algebraic Riccati equations by recasting them as positive dynamical systems and leveraging Hilbert metric convergence to Perron–Frobenius vectors. By homogenizing the equations with a trace-based term and imposing observability/controllability, the authors prove existence and uniqueness (up to scaling) of solutions and establish convergence properties for the associated positive dynamics in both continuous and discrete time. Key results include exponential Hilbert-metric convergence for Lyapunov dynamics and asymptotic convergence for Riccati dynamics, with a tunable parameter to enforce the fixed-point condition. This positive-systems perspective unifies classical matrix equations, offers constructive solution pathways, and opens avenues for extensions to stochastic, time-varying, and more general Riccati-type equations.

Abstract

This paper presents a new interpretation of the Lyapunov and Riccati equations from the perspective of positive system theory. We show it is possible to construct positive systems related to these equations, and then certain conclusions -- such as the existence and uniqueness of solutions -- can be drawn from positive systems theory. Specifically, under standard observability assumptions, a strictly positive linear system can be constructed for Lyapunov equations, leading to exponential convergence in Hilbert metric to the Perron-Frobenius vector -- closely related to the solution of the Lyapunov equation. For algebraic Riccati equations, homogeneous strictly positive systems can be constructed, which exhibit more complex dynamical behaviors. While the existence and uniqueness of the solution can still be proven, only asymptotic convergence can be obtained.

Lyapunov and Riccati Equations from a Positive System Perspective

TL;DR

The paper addresses the constructive analysis of Lyapunov and algebraic Riccati equations by recasting them as positive dynamical systems and leveraging Hilbert metric convergence to Perron–Frobenius vectors. By homogenizing the equations with a trace-based term and imposing observability/controllability, the authors prove existence and uniqueness (up to scaling) of solutions and establish convergence properties for the associated positive dynamics in both continuous and discrete time. Key results include exponential Hilbert-metric convergence for Lyapunov dynamics and asymptotic convergence for Riccati dynamics, with a tunable parameter to enforce the fixed-point condition. This positive-systems perspective unifies classical matrix equations, offers constructive solution pathways, and opens avenues for extensions to stochastic, time-varying, and more general Riccati-type equations.

Abstract

This paper presents a new interpretation of the Lyapunov and Riccati equations from the perspective of positive system theory. We show it is possible to construct positive systems related to these equations, and then certain conclusions -- such as the existence and uniqueness of solutions -- can be drawn from positive systems theory. Specifically, under standard observability assumptions, a strictly positive linear system can be constructed for Lyapunov equations, leading to exponential convergence in Hilbert metric to the Perron-Frobenius vector -- closely related to the solution of the Lyapunov equation. For algebraic Riccati equations, homogeneous strictly positive systems can be constructed, which exhibit more complex dynamical behaviors. While the existence and uniqueness of the solution can still be proven, only asymptotic convergence can be obtained.

Paper Structure

This paper contains 12 sections, 7 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Positive Systems Theory
  3. Lyapunov equations
  4. Continuous-time Lyapunov equations
  5. Discrete-time Lyapunov equation
  6. Discussion
  7. Algebraic Riccati equations
  8. Discrete-time Algebraic Riccati Equation
  9. Continuous-time Algebraic Riccati Equation
  10. Riccati equations in positive control systems
  11. Conclusion
  12. Acknowledgement

Key Result

Corollary 1

Let $\mathcal{K}$ be a normal cone in a finite dimensional vector space $\mathcal{V}$. Consider the dynamics sys:ODE-Homo. The following hold.

Figures (2)

  • Figure 1: Eigenvalue corresponding to the Perron-Frobenius vector as a function of $\alpha$ for the matrix $A+\alpha c \mathds{1}^\top$.
  • Figure 2: Eigenvalue corresponding to the Perron-Frobenius vector as a function of $\alpha$ for the operator $L(X) = A^\top X A+ \alpha {\rm tr}(X) C^\top C$ with $A$ Schur stable and $(A,C)$ observable.

Theorems & Definitions (16)

  • Definition 1: Cone
  • Definition 2: Hilbert metric
  • Definition 3: Order-preserving maps
  • Remark 1
  • Definition 4: Positive systems
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Remark 2
  • Remark 3
  • ...and 6 more