A direct approach using the shifted Legendre series expansion for near optimum control of linear time-varying systems with multiple state and control delays. (English) Zbl 0586.93048

A computationally efficient method, which uses finite orthogonal expansions to approximate state and control variables, is developed to obtain suboptimal control for time-varying systems with multiple state and control delays and with quadratic cost. Shifted Legendre polynomials form the basis of expansions as a consequence of their useful properties. The Galerkin method is used to reduce the problem to one of minimizing a quadratic form with linear algebraic constraints. The solution of a coupled two-point boundary-value problem with both delayed and advanced terms, which is always required in applying the Pontryagin’s maximum principle to the optimization of delay systems, is thus avoided.


93C99 Model systems in control theory
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
44A45 Classical operational calculus
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
93B40 Computational methods in systems theory (MSC2010)
93C05 Linear systems in control theory
Full Text: DOI


[1] DOI: 10.1109/TAC.1970.1099605 · doi:10.1109/TAC.1970.1099605
[2] DOI: 10.1080/00207728208926416 · Zbl 0486.93021 · doi:10.1080/00207728208926416
[3] DOI: 10.1109/TAC.1969.1099301 · doi:10.1109/TAC.1969.1099301
[4] FINLAYSON B. A., The Method of Weighted Residuals and Variational Principles (1972) · Zbl 0319.49020
[5] GRADSHTEYN I.S., Tables of Integrals, Series and Products (1979)
[6] DOI: 10.1080/0020718508961135 · Zbl 0555.93014 · doi:10.1080/0020718508961135
[7] DOI: 10.1080/00207178408933207 · Zbl 0531.93022 · doi:10.1080/00207178408933207
[8] DOI: 10.1007/BF00940816 · Zbl 0541.93031 · doi:10.1007/BF00940816
[9] DOI: 10.1016/0005-1098(71)90005-7 · doi:10.1016/0005-1098(71)90005-7
[10] DOI: 10.1049/piee.1972.0352 · doi:10.1049/piee.1972.0352
[11] DOI: 10.1007/BF00934590 · Zbl 0394.49017 · doi:10.1007/BF00934590
[12] PONTRYAGIN L. S., The Mathematical Theory of Optimal Processes (1962) · Zbl 0112.05502
[13] DOI: 10.1080/00207728308926462 · Zbl 0502.93039 · doi:10.1080/00207728308926462
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