Using spectral discretization for the optimal H₂ design of time-delay systems

The stabilization and robustification of a time-delay system is the topic of this paper. More precisely, we want to minimize the H₂ norm of the transfer function corresponding to this 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function is a nonrational, nonlinear function, and the classical procedure which involves solving Lyapunov equations is no longer applicable.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 discretization 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 optimization 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 in the convergence behaviour of the approximation, justifying its use for the purpose of H₂ norm computation. Along with this, the easy availibility of derivatives with respect to the original matrices allows for an efficient integration into any standard optimization framework. A numerical example finally illustrates how the presented method can be employed to perform optimal H₂ norm design using smooth optimization techniques.nrpages: 24status: publishe