Matthias Voigt | Department of Mathematics, University of Hamburg
Numerical Methods in H-infinity-Control
In engineering problems it is often desired to attenuate the influence of external noise and to deal with uncertainties in a dynamical system.
In classical control theory, this amounts to the design of so-called H-infinity-controllers which requires the solution of a certain nonlinear optimization problem. In this talk, numerical techniques for its solution will be presented.
After reviewing classical methods known in the literature, we will focus on a more direct design procedure that is based on nonsmooth constrained optimization techniques. These techniques allow a greater flexibility in the controller structure and the treatment of systems with a high state-space dimension. The numerical optimization methods require multiple evaluations of the H-infinity-norm and its gradient with respect to the controller variables. Therefore, efficient techniques for their evaluation are necessary, especially in the high-dimensional setting.
In this talk we address new efficient methods for the computation of the H-infinity-norm that are tailored to high-dimensional dynamical systems with possibly irrational transfer functions. We discuss a subspace projection approach for solving this problem using interpolatory techniques that are well-known in model reduction. More precisely, after performing the reduction, we compute the H-infinity-norm of the reduced transfer function and choose the point at which the H-infinity-norm is attained as a new interpolation point. In this manner, the projection matrices and the reduced transfer function are updated iteratively until convergence. We further discuss how to better deal with irrational transfer functions.
These can also be reduced by rational approximation techniques and thus, we get a sequence of rational reduced transfer functions whose H-infinity norm can be evaluated more efficiently. Finally, we show how to apply this approach in the context of controller design.