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Linear Optimal Control Problems and Quadratic Cost Functions Estimation

Inverse optimal control is a classical problem of control theory. It was first posed by Kalman in the early sixties. The problem, as addressed in literature, answers to the following two questions: (a) Given system matrices A,B and a gain matrix K, find necessary and sufficient conditions for K to be the optimal of an infinite time LQ problem. (b) Determine all weight matrices Q, R and S which yield the given gain matrix K. In this paper, we tackle a related, but different problem. Starting from the state trajectories of an LTI system, identify the matrices Q, R and S that have generated those trajectories. Both infinite and finite time optimal control problems are considered. The motivation lies in the characterization of the trajectories of LTI systems in terms of the control task.