Approximation and control of a class of distributed delay systems

Abstract

Abstract In this paper, we show that a linear system with distributed state delay can be approximated by a linear delay-free system that has the same state dimension as the original system if a so-called smallness condition holds. The smallness condition is an inequality which the maximum lag and norms of the system matrices have to satisfy. The eigenvalues of the approximate system correspond to the dominant eigenvalues of the original system with distributed delay. We provide a numerically stable, iterative algorithm to compute the state matrix of the approximate system. Furthermore, we show that, based on the proposed approximation, the stabilisation, pole placement and setpoint tracking control problems of the addressed class of distributed delay systems can be performed using methods developed for delay-free systems. Simulation results are provided to show the applicability of the proposed approximation and control design method.