Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials

Abstract In this paper, a numerical method to solve nonlinear optimal control problems with terminal state constraints, control inequality constraints and simple bounds on the state variables, is presented. The method converts the optimal control problem into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem with a sequence of constrained linear-quadratic optimal control problems, then each of the state variables is approximated by a finite length Chebyshev series with unknown parameters. The method gives the information of the quadratic programming problem explicitly (The Hessian, the gradient of the cost function and the Jacobian of the constraints). To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.

[1]  M. Nagurka,et al.  Optimal control of linearly constrained linear systems via state parametrization , 1992 .

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

[3]  Hussein Jaddu,et al.  Computation of optimal control trajectories using chebyshev polynomials : Parameterization, and quadratic programming , 1999 .

[4]  Mohsen Razzaghi,et al.  Linear quadratic optimal control problems via shifted Legendre state parametrization , 1994 .

[5]  Kok Lay Teo,et al.  Control parametrization: A unified approach to optimal control problems with general constraints , 1988, Autom..

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

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

[8]  C. Neuman,et al.  A suboptimal control algorithm for constrained problems using cubic splines , 1973 .

[9]  D. Tabak,et al.  Applications of mathematical programming techniques in optimal control: A survey , 1970 .

[10]  A. Calise,et al.  Stochastic and deterministic design and control via linear and quadratic programming , 1971 .

[11]  R. Bellman,et al.  Quasilinearization and nonlinear boundary-value problems , 1966 .

[12]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[13]  Oskar von Stryk,et al.  Direct and indirect methods for trajectory optimization , 1992, Ann. Oper. Res..

[14]  J. Junkins,et al.  Optimal Spacecraft Rotational Maneuvers , 1986 .

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

[16]  Dieter Kraft,et al.  On Converting Optimal Control Problems into Nonlinear Programming Problems , 1985 .

[17]  Kok Lay Teo,et al.  A Unified Computational Approach to Optimal Control Problems , 1991 .

[18]  E. Shimemura,et al.  Computation of optimal feedback gains for time-varying LQ optimal control , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).