Design of discrete-time linear-quadratic regulator with 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 discrete-time (DT), closed loop (CL) system. The algorithm of design of the regulator-observer for the DT 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 the freely chosen poles coming from the observer. It is also shown that the designed regulator is the same as that based on the Luenberger observer; the latter is optimal for adequate initial conditions. For the systems with stepwise excitations of the plant output, the deterministic equivalence principle has been proved true.