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.

在本文中，我们表明，如果存在所谓的小性条件，则可以通过具有与原始系统相同的状态维的线性无延迟系统来近似具有分布式状态延迟的线性系统。最小条件是系统矩阵的最大滞后和范数必须满足的不等式。近似系统的特征值对应于具有分布延迟的原始系统的主要特征值。我们提供了一种数值稳定的迭代算法来计算近似系统的状态矩阵。此外，我们表明，基于提出的近似值，可以使用为无延迟系统开发的方法来执行所解决的分布式延迟系统类别的稳定性，极点放置和设定点跟踪控制问题。