Synthesis of an Optimal Output Regulating System with a Reference Vector

The difficult problem of synthesizing an optimal regulation scheme for linear time-invariant systems, assuming an integral quadratic cost functional, when the desired output is a constant nonzero vector and when the upper time limit approaches infinity, is studied in this paper. The problem is solved by properly redefining the cost functional, so that consistence with the stability requirement is achieved. The problem of the optimal closed-loop regulation of a singleinput/single-output system has always been an interesting topic for control engineers. The most common assumption is that the cost functional to minimize be of quadratic type, with infinity as an upper limit. The classical frequency domain approach, assuming the absence of initial conditions, suggests an optimal compensator in forward loop. On the other hand, the modern time-domain approach, based on the direct application of the maximum principle, yields a closed-form solution for the case in which the desired output is zero. This paper presents results which allow one to consider from a unifying point of view the two above approaches and, in addition, suggests a stable optimal scheme which is valid for any initial state and for any desired constant output.