Solving Generalized Lyapunov Equations with guarantees: application to the Model Reduction of Switched Linear Systems

TL;DR

The paper tackles the challenge of solving large-scale generalized Lyapunov equations (GLEs) with rigorous error guarantees to enable reliable model order reduction (MOR) for switched linear systems (SLS). It develops a certified stationary iteration for the GLE, derives computable a posteriori error bounds, and analyzes how GLE solution accuracy affects the MOR, including ROM stability. To overcome limitations of exact LMIs in balancing-based MOR, it introduces piecewise balancing reduction (PBR), which uses mode-specific GLEs and piecewise constant projection bases, together with extended BT-type bounds that account for LMI inexactness and time-varying projections. The framework is validated through synthetic and parametric PDE-driven examples, showing accurate reduced models and demonstrable gains in robustness and scalability for SLS MOR.

Abstract

We present an efficient strategy to approximate the solutions of large-scale generalized Lyapunov equations (GLEs) with rigorous, computable error guarantees. This work is motivated by applications in model order reduction (MOR) of switched linear systems (SLS) in control form, where GLEs play a central role. We analyze how inaccuracies in the numerical solution of GLEs propagate through the MOR procedure and affect the accuracy and reliability of the reduced order model. Furthermore, the classical balanced-truncation error estimate for SLS is neither theoretically nor practically viable, as they rely on restrictive assumptions requiring several requiring several linear matrix inequalities (LMI) to be satisfied exactly by numerically computed solutions of the GLEs. To overcome these limitation, we propose a new MOR framework for SLS, called piecewise balanced reduction (PBR). The method is based on solving multiple GLEs and the construction of projection matrices that are piecewise constant in time to appropriately balance and subsequently reduce the SLS. We extend the standard balanced-truncation error bounds and demonstrate that the PBR formulation allows us to control the error arising from the inexact LMI. In addition, our new error bound accounts for the influence of the piecewise constant time-varying projection matrices. Altogether, this renders the PBR approach for SLS applicable to a broad and flexible class of SLS. Numerical experiments are provided to corroborate our theoretical results.

Figures (4)

• Figure 1: Synthetic example \ref{['eqn:synt:ex']}.
• Figure 2: Synthetic example \ref{['eqn:synt:ex']} for $n=200$. MOR for SLS through GLEs solution.
• Figure 3: Example derived from the Black-Scholes model for $n=1000$. The GLEs are solved with tolerance $\texttt{tol}=10^{-8}$.
• Figure 4: Example derived from the Black-Scholes model for $n=1000$. GLE solved with $\texttt{tol}=10^{-8}$, input function $\bm{u}_2$ defined in \ref{['eqn:inp:BS']}, and $K=10$ switches times.

Theorems & Definitions (27)

• Remark 1.1
• Definition 2.1
• Definition 2.2
• Lemma 2.3: PetWL13
• Theorem 2.4: PetWL13
• proof
• Remark 3.2
• Proposition 3.3: Error bound for approximate solution of Lyapunov equations
• proof
• Proposition 3.4: Stopping criterion for \ref{['alg:statIterGLE']}