Quasi‐finite‐rank approximation of compression operators on L ∞ [0, h ) with application to stability analysis of time‐delay systems

Abstract

This study discusses a new method for approximating compression operators, which play important roles in the operator-theoretic approach to sampled-data systems and time-delay systems. Stimulated by the success in the application of quasi-finite-rank approximation of compression operators defined on the Hilbert space L 2[0, h), the authors study a parallel problem for compression operators defined on the Banach space L ∞[0, h). In spite of similarity between these problems, they are led to applying a completely different approach because of essential differences in the underlying spaces. More precisely, they apply the idea of the conventional fast-sample/fast-hold (FSFH) approximation technique, and show that the approximation problem can be transformed into such a linear programming problem that asymptotically leads to optimal approximation as the FSFH approximation parameter M tends to infinity. Finally, they demonstrate the effectiveness of the L ∞[0, h)-based approximation technique through numerical examples, with particular application to stability analysis of time-delay systems.