The characteristics of model-following systems as synthesized by optimal control

Synthesis through linear optimal-control theory is applied to two types of model-following systems. Although the two system concepts are fundamentally different, the optimal-control theory in both cases generates constant feedback gains which modify the plant dynamics so that the resultant system response approaches that of the model. Each synthesis method produces an optimal-control law that has a closed-form solution and minimizes an integral quadratic performance index involving both system errors and control motions. The first type of model-following system is generated by incorporating the model only in the performance index. The integral square error between the model and the system is minimized by one term in the performance index. The basic idea in this system is to alter, by the feedback, the coefficients in the equations of motion for the plant so that these coefficients approach those of the model. The feedback control obtained from this method depends on the solution of the matrix Riccati equation, and also on a product of constant matrices. For an infinite weight on the error term in the performance index, the solution of the Riccati equation approaches zero. The second type of model-following system incorporates the model as part of the control system, and places it ahead of the plant as a prefilter. The difference between the model output and the plant output is minimized in the performance index. The resulting system has not only feedback gains for the plant state variables but also feedforward gains for the model state variables. Evaluation shows that the feedback gains are independent of the model, and are exactly those which would be obtained for the normal regulator problem. The feedforward gains depend on both plant and model.