Design of reduced-order linear quadratic optimal regulators

It is perhaps possible to distinguish between two main approaches to problem of model reduction [1]. The first uses open-loop criteria (not necessarily optimal) for producing low-order models [2,3], while in the second, criteria which are optimal in some sense are employed. The latter approach can too be divided into two subgroups. In the first, error criteria are used to optimally approximate the open-loop behaviour of the original system [4,5]. In the second, either a given low-order model is optimally controlled and the resultant controller is used to (sub-optimally) control the original system [6]; or, a low-order model, whose dimension equals the number of outputs of the original system, is designed such that the error between the optimally controlled model and original system is minimized [7]. A completely different approach is considered in this paper. For a desired model order, the model is designed such that, controlling it by a constant state-variable feedback and using the resulting input to control the original system, the performance index of the original system, is minimized. Therefore, we are not contrained a priori by the model order (as in [7] for example, where a single-output original system necessarily results in a first order model). Furthermore, the performance index of the original system itself is being minimized under the constraint of the desired order model. Notice, however, that the proposed solution could not be considered open-loop in some sense and thus, could not be used for designing low-order models for unstable systems.