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Direct approach for the optimal control of linear time-delay systems via shifted Legendre polynomials. (English) Zbl 0586.93026

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 Legendre- coefficients vector of controls. In the second step, the relation between Legendre-coefficients vectors of states and controls, which satisfy 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.

MSC:

93C05 Linear systems in control theory
34K35 Control problems for functional-differential equations
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
44A45 Classical operational calculus
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