Construction of optimal feedback control for nonlinear systems via Chebyshev polynomials

A method is proposed to determine the optimal feedback control law of a class of nonlinear optimal control problems. The method is based on two steps. The first step is to determine the open-hop optimal control and trajectories, by using the quasilinearization and the state variables parametrization via Chebyshev polynomials of the first type. Therefore the nonlinear optimal control problem is replaced by a sequence of small quadratic programming problems which can easily be solved. The second step is to use the results of the last quasilinearization iteration, when an acceptable convergence error is achieved, to obtain the optimal feedback control law. To this end, the matrix Riccati equation and another n linear differential equations are solved using the Chebyshev polynomials of the first type. Moreover, the differentiation operational matrix of Chebyshev polynomials is introduced. To show the effectiveness of the proposed method, the simulation results of a nonlinear optimal control problem are shown.

[1]  P. N. Paraskevopoulos,et al.  Chebyshev series approach to system identification, analysis and optimal control , 1983 .

[2]  Sanjoy K. Mitter,et al.  Successive approximation methods for the solution of optimal control problems , 1966, Autom..

[3]  R. V. Dooren,et al.  A Chebyshev technique for solving nonlinear optimal control problems , 1988 .

[4]  Computation of optimal control for time-delay systems , 1970 .

[5]  Application of Chebyshev polynomials to the optimal control of time-varying linear systems , 1985 .

[6]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[7]  William L. Garrard,et al.  Design of nonlinear automatic flight control systems , 1977, Autom..

[8]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[9]  S. Torkel Glad.  Robustness of nonlinear state feedback - A survey , 1987, Autom..

[10]  Jacques Vlassenbroeck,et al.  A chebyshev polynomial method for optimal control with state constraints , 1988, Autom..

[11]  H. J. Pesch,et al.  New general guidance method in constrained optimal control, part 1: Numerical method , 1990 .

[12]  Gamal N. Elnagar,et al.  The pseudospectral Legendre method for discretizing optimal control problems , 1995, IEEE Trans. Autom. Control..

[13]  Mark Enns,et al.  Computation of optimal controls by a method combining quasi-linearization and quadratic programming† , 1972 .

[14]  Y. Nishikawa,et al.  A method for suboptimal design of nonlinear feedback systems , 1971 .

[15]  Designing Linear Optimal Regulators Via Chebyshev Polynomials , 1993, 1993 American Control Conference.

[16]  H. J. Pesch.  Real‐time computation of feedback controls for constrained optimal control problems. part 1: Neighbouring extremals , 1989 .

[17]  Chih-Fan Chen,et al.  Design of piecewise constant gains for optimal control via Walsh functions , 1975 .

[18]  Y. Shih,et al.  Analysis and optimal control of time-varying systems via Chebyshev polynomials , 1983 .

[19]  G. Franklin,et al.  A second-order feedback method for optimal control computations , 1967, IEEE Transactions on Automatic Control.

[20]  Kenneth A. Loparo,et al.  Quadratic regulatory theory for analytic non-linear systems with additive controls , 1989, Autom..

[21]  P. Frick,et al.  Solution of optimal control problems on a parallel machine using the epsilon method , 1995 .

[22]  V. Rao,et al.  Optimal feedback control via block-pulse functions , 1979 .

[23]  L. Fox,et al.  Chebyshev polynomials in numerical analysis , 1970 .

[24]  M. Urabe,et al.  Numerical solution of boundary value problems in Chebyshev series — A method of computation and error estimation , 1969 .

[25]  R. Tibshirani,et al.  An introduction to the bootstrap , 1993 .

[26]  D. Lukes.  Optimal Regulation of Nonlinear Dynamical Systems , 1969 .