State-space Realization of Linear Time-invariant Systems with Maximum Measure of Quality

In this paper, we give a state-space realization that maximizes the measure of quality (MoQ). MoQ of a system is a performance index that determines how easily the system can be controlled, while the standard notion of reachability/controllability just gives a binary information, e.g. reachable or unreachable. In this paper, we formulate the realization problem as maximization of the MoQ by a coordinate transformation matrix. We show that the optimal coordinate transformation is given in a closed form. The proposed method is effective in particular for preconditioning of an ill-conditioned grammian, the inverse of which is used to solve the reachability problem. We show a numerical example to illustrate this merit of the proposed method.