Direct approach for the optimal control of linear time-delay systems via shifted Legend re polynomials

Abstract

The present paper proposes a direct approach to the optimal control of linear time-delay systems. In the first step, both control and state are approximated by shifted Legendre polynomials, and then the Legendre-coefficients vector of states can be expressed in terms of the Legend re-coefficients vector of controls. In the second step, the relation between Legendre-coefficients vectors of states and controls, which satisfies the state equations approximately, is introduced into the performance index. Next, the variational principle is applied to find the Legendre-coefficients vector of near-optimal control that minimizes the performance index directly. In the present approach, the Legendre-coefficients vector of sub-optimal control can be obtained directly from a set of algebraic equations. Thus, both difficulties in solving simultaneous partial differential equations of large dimension, or in solving n sets of ordinary differential equations, are avoided.