H-infinity optimal repetitive control: Continuous-time and sampled-data formulations

We consider robust performance and nominal performance (with robust stability) continuous-time formulations and a sampled data formulation of the repetitive control problem. The robust performance problem yields a generalized repetitive controller structure and is related to a SIMO H-infinity problem with a scalar delay. This vector H-infinity problem is solved, by an extension of the existing theory, using the commutant lifting theorem. Formulas are given for calculating the optimal performance and the optimal controller. A numerical example is presented. The nominal performance formulation is based on a two step design process. The first step is initial stabilization and approximate inversion of the nominal plant. The second step is the infinite dimensional H-infinity repetitive control design for the resulting inner equivalent plant. This formulation yields classical repetitive controllers for properly selected weighting functions under appropriate approximations. The formulation also yields a modified repetitive controller structure for non-minimum phase plants. A cascade repetitive design structure with implications for classical repetitive control design is also obtained. The practicality of this approach is demonstrated through numerical examples. The sampled-data formulation is designed to satisfy the natural definition of sampled-data repetitive control: discrete-time controllers with a digital repetitive structure designed directly to satisfy continuous-time requirements. The solution requires calculation of a discrete-time equivalent problem from the sampled-data problem. Toward this end, we extend the theory of Bamieh and Pearson to the general case. A novel approach is detailed for obtaining controllers with a digital repetitive structure by searching the solution space using genetic algorithms. While the approach appears very promising, evaluation awaits resolution of numerical difficulties in the calculation of the discrete-time equivalent problem.