Approximation of ℋ∞ Controllers for Unstable Time Delay Systems

We examine approximations of the ${\mathcal{H}_\infty }$-optimal controller arising from the mixed sensitivity minimization for a first order unstable time delay system, with first order weights. The optimal controller is known to have a special structure where a PI controller and an FIR filter form the finite dimensional and the infinite dimensional parts, respectively. We approximate the infinite dimensional part by fixed order rational transfer functions using various methods from the literature. We compare the ${\mathcal{H}_\infty }$ approximation errors obtained with the following rational approximation methods for the FIR part of the controller: systune and ssest commands of Matlab, Hankel norm approximation (HNA), moment matching (MM), balanced reduction (balred), iterative rational Krylov algorithm (IRKA) and Pade approximation.