Analytical results on the use of reduced models in the control of linear dynamical systems

Two sets of first-order differential equations denoted by X and Xr are assumed to exist, where X and Xr govern the behaviour of the plant and the reduced model, respectively. The performance of the plant when it is controlled on the false assumption that its behaviour is governed by Xr is investigated. There optimal-control problems are considered: (a) fixed-target fixed-terminal-point minimum-energy problem (b) fixed-target free-terminal-point minimum-energy tracking problem (c) output-regulation problem. Explicit relations are obtained for the deterioration in performance in model control of the plant. In the first two problems, the interconnection between the error functional commonly adopted for the model-reduction problem and the deterioration in performance is demonstrated; the results are applied to reduced models obtained by the method of optimal projection along invariant subspaces.