Linear-quadratic regulator with output feedback and optimal observer

It is shown that by means of an appropriate choice of the coefficients of the regulator of the order smaller by one from that of the plant, it is possible to obtain the freely prescribed poles of the closed loop (CL) system. Then, the algorithm of design of the regulator-observer for the CL system with output feedback is described. The resulting CL system retains all the poles of the CL system with linear-quadratic regulator and state feedback and has additionally freely chosen poles coming from the observer. Next, it is shown that the designed regulator contains the Luenberger observer, which is optimal for adequate initial conditions. For the plants with a zero pole, under stepwise excitations, the deterministic equivalence principle is proved to be true