A subspace framework for L ∞ model reduction

Abstract

We propose an approach for the L infinity ${\mathcal{L}}_{\infty }$ model reduction of descriptor systems based on the minimization of the L infinity ${\mathcal{L}}_{\infty }$ objective by means of smooth optimization techniques. Direct applications of smooth optimization techniques are not feasible even for systems of modest order, since the optimization techniques converge at best at a linear rate requiring too many evaluations of the costly L infinity ${\mathcal{L}}_{\infty }$ -norm objective to be practical. We replace the original system with a system of smaller order interpolating the original system at points on the imaginary axis, minimize the L infinity ${\mathcal{L}}_{\infty }$ objective after this replacement, and refine the smaller system based on the minimization. We also describe how asymptotic stability constraints on the reduced system sought can be incorporated into our approach. The numerical experiments illustrate that the approach leads to locally optimal solutions to the L infinity ${\mathcal{L}}_{\infty }$ model reduction problem, and its capability to deal with systems of order a few ten thousands.