Minimization of the input energy of regulator problems with the fixed terminal state. The case of multi-input linear time-invariant discrete-time systems.

Abstract

This paper discusses the regulator problem of minimizing the input energy for the multi-input linear time-invariant discrete-time system with the zero terminal state. The performance index is the sum of the quadratic forms of the input vectors. By using the deadbeat principle, the system is transformed into the new time-variant system with the time-variant structure of the new input. The performance index is transformed into the sum of the quadratic forms of the new state vectors and the new input vectors with cross terms, and the new terminal state is free. The optimal inputs are given by the variable gain state feedback in the first phase, by linear combinations of the constant gain state feedback and the variable gain state feedback in the second phase, and by the constant gain state feedback in the third phase. The variable gains are obtained by using the special Riccati equation. Both gains are independent of the initial state.