The closed-loop expression of the minimum energy control for multi-input linear digital control systems

The minimum energy control problem for the multi-input linear time-invariant discrete-time system is discussed. Though the open-loop expression of the optimal inputs has been obtained, it is difficult to solve the two-point boundary value problem with the zero terminal state and to obtain the closed-loop expression with bounded gains. First, the general inputs which satisfy the zero terminal state condition are obtained. By using the general inputs, the system is transformed into a new time-variant system with the variable structure of the new inputs, and the zero terminal state problem is transformed into a new free terminal state problem. The performance index is the quadratic form with time-variant weighting matrices and cross terms. The optimal inputs are given by variable gain state feedback in the first phase, by linear combinations of a special deadbeat control and variable gain state feedback in the second phase, and by the special deadbeat control in the third phase. The variable gains are obtained by using the special nonstationary Riccati equations