Loop shaping for robust performance in systems with structured perturbations

The authors consider the problem of controller synthesis to satisfy quantitative performance specifications for a family of single-input single-output linear systems given by y(s) = h(s,α)u(s), where α denotes a parameter vector indexing the unknown but bounded uncertainty and h(s,α) is the transfer function. The controller parameters are considered fixed. The controller synthesis is performed in the frequency domain. This is done by formulating pointwise constraints on compensators so as to satisfy ω-domain bounds on the overall closed loop system. Kharitonov's theorem is used to obtain frequency domain constraints on the family of plants h(s,α) which can be exploited to obtain a transparent loop shaping procedure for controller synthesis. The loop shaping procedure and performance specifications are illustrated on a third order example for a special disturbance rejection problem.