Using spectral discretisation for the optimalℋ 2 design of time-delay systems

Abstract

The stabilisation and robustification of a time-delay system is the topic of this article. More precisely, we want to minimise the ℋ2 norm of the transfer function corresponding to the class of linear time-invariant input–output systems with fixed time delays in the states. Due to the presence of the delays, the transfer function is a nonrational, nonlinear function, and the classical procedure which involves solving Lyapunov equations is no longer applicable. We therefore propose an approach based on a spectral discretisation applied to a reformulation of the time-delay system as an infinite-dimensional standard linear system. In this way, we obtain a large delay-free system, which serves as an approximation to the original time-delay system, and which allows the application of standard ℋ2 norm optimisation techniques. We give an interpretation of this approach in the frequency domain and relate it to the approximation of the nonlinear terms in the time-delay transfer function by means of a rational function. Using this property, we can provide some insight into the convergence behaviour of the approximation, justifying its use for the purpose of ℋ2 norm computation. Along with this, the easy availability of derivatives with respect to the original matrices allows for an efficient integration into any standard optimisation framework. A few numerical examples finally illustrate how the presented method can be employed to perform optimal ℋ2 norm design using smooth optimisation techniques.