Quasi‐finite‐rank approximation of compression operators on L ∞ [0, h ) with application to stability analysis of time‐delay systems
Jung Hoon Kim, Tomomichi Hagiwara · IET Control Theory and Applications · 2013
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.