Speaker
Description
In this talk, we present an efficient strategy to approximate the solutions of large-scale generalized Lyapunov equations (GLEs) while providing rigorous error guarantees. The motivation for this study stems from the use of GLEs in model order reduction (MOR) of switched linear systems (SLS) in control form. Specifically, we analyze how inaccuracies in the computed GLE solution influence the performance and reliability of the resulting MOR procedure. Furthermore, since the classical balanced-truncation error estimate for MOR of SLS is neither theoretically nor practically viable—because it relies on restrictive assumptions that require several linear matrix inequalities (LMI) to be satisfied by numerically computed solutions of the GLEs—we propose a new MOR framework for SLS. Our method is based on solving multiple GLEs and constructing projection matrices that are piecewise constant in time to appropriately balance and subsequently reduce the SLS; we therefore refer to it as piecewise balancing reduction (PBR) for SLS. We extend the standard balance-truncation error bounds to incorporate the effects of inexact LMI. We show how 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. We conclude by showing numerical experiments to corroborate our theoretical results.