The preview control problem is one in which a controller can use future as well as present and past information to determine the control. In the optimal discrete finite preview problem studied in this paper, the controller is assumed to make use of preview information with respect to command signal (desired trajectory) from the present time i up to Nla time units in the future. Nla is called the preview time, and is usually shorter than N, the problem duration. To make the solution applicable to a broad class of problems, measurement time delay, observation noise and driving noise are included in formulating the problem. The solution shows how to utilize the local future information obtained by finite preview (Nla) to minimize an optimality criterion evaluated over the problem duration (N). An illustrative example is given to show that preview drastically improves the performance of a system relative to non-preview optimal performance, and that there is a critical preview length essential for such an improvement.

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