Structured controller synthesis for mechanical servo-systems: Algorithms, relaxations and optimality certificates

In many application areas of mechanical servo-systems the high demands on the performance often imply a tightly tuned feedback controller, that takes dynamical interaction into account. Model-based H-optimal controller synthesis is a well-suited technique for this purpose. However, the state-of-the-art synthesis approach yields controllers with high McMillan degree that can not be implemented in real-time at high sampling-rates, because of the limited computational capacity. This motivates to constrain the McMillan degree of the controller. The aim of this thesis is to provide numerical tools for H-optimal degree constrained (or otherwise structured) controller synthesis. For this problem we have developed relaxations that are based on Sum-Of-Squares polynomials. Their optimal values are lower bounds on the globally optimal structured controller synthesis problem and can be computed by solving LMI problems. It is guaranteed, that the bounds converge to best achievable performance as we improve our relaxations. To make this technique feasible for plants with high McMillan degree, we proposed a computationally less demanding scheme based on partial dualization. The Sum-Of-Squares relaxations have also been applied to robust polynomial Semi-Definite Programs (SDPs). Also for this case a sequence of relaxations has been developed, whose optimal values converge from below to the optimal value of the robust SDP. Furthermore for the structured controller synthesis problem an Interior Point algorithm has been developed. It is shown how this algorithm can be made more efficient, by exploiting the control-theoretic characteristics of the problem. Conditions have been derived to verify local optimality of the optimized controller. Finally, it has been illustrated by real-time experiments that the algorithms described in this thesis can be used to synthesize high-performing fixed-order controllers for a new prototype of a wafer stage