Control system design for robust stability and robust performance.

A central problem in control system design is how to design a controller to guarantee that the closed-loop system is robustly stable and that performance requirements are satisfied despite the presence of model uncertainties and exogenous disturbance signals. The analysis problem, that is the assessment of control systems with respect to robust stability and robust performance, can be adequately solved using the structured singular value u as introduced by Doyle. The corresponding design problem (how to choose a controller K to minimize u) is still largely unsolved, but an approximate solution can be found using Doyle's D - K iteration. In this thesis we present an alternative algorithm, called u - K iteration, which works by flattening the structured singular value u over frequency. As a prelude to this a classical loop shaping approach to robust performance is presented for SISO systems, and is also based on flattening u. In u-synthesis it is often the case that real uncertainties are modelled as complex perturbations but the conservatism so introduced can be severe. On the other hand, if real uncertainties are modelled as real perturbations then D - K iteration is not relevant. It is shown that u - K iteration still works for real perturbations. In addition, a geometric approach for computing the structured singular value for a scalar problem with respect to real and/or complex uncertainty is described. This provides insight into the relationship between real u and complex u. A robust performance problem is considered for a 2-input 2-output high purity distillation column which is an ill-conditioned plant. Analysis reveals the potentially damaging effects on robustness of ill-conditioning. A design is carried out using u - K iteration and the "optimum" u compared with that obtained by Doyle and by Freudenberg for the same problem.