An Iterative method for Solving the Container Crane Constrained Optimal Control Problem Using Chebyshev Polynomials

In this paper, a computational method for solving constrained nonlinear optimal control problems is presented with an application to the container crane. The method is based on Banks' et al. iterative approach, in which the nonlinear system state equations are replaced by a sequence of time-varying linear systems. Therefore, The constrained nonlinear optimal control problem can be converted into sequence of constrained time varying linear quadratic optimal control problems. Combining this iterative approach with parameterization of the state variables using Chebyshev polynomials will result in converting the hard constrained nonlinear optimal control problem into sequence of quadratic programming problems. To show the performance and the behavior of this method compared with other known approaches, we apply it on a practical problem namely the container crane problem and the simulation results are presented and compared with other methods.

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

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

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

[4]  Yoshiyuki Sakawa,et al.  Optimal control of container cranes , 1981, Autom..

[5]  Anil V. Rao,et al.  Practical Methods for Optimal Control Using Nonlinear Programming , 1987 .

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

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

[8]  F. Breitenecker,et al.  COMPUTING OPTIMAL CONTROLS FOR SYSTEMS WITH STATE AND CONTROL CONSTRAINTS , 1989 .

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

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

[11]  J. Betts Issues in the direct transcription of optimal control problems to sparse nonlinear programs , 1994 .

[12]  Hussein Mohammad Hussein Jaddu Numerical methods for solving optimal control problems using chebyshev polynomials , 1998 .

[13]  Hussein Jaddu Computational method based on state parametrization for solving constrained nonlinear optimal control problems , 1999, Int. J. Syst. Sci..

[14]  H. Maurer,et al.  SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control , 2000 .

[15]  Stephen P. Banks,et al.  Approximate Optimal Control and Stability of Nonlinear Finite- and Infinite-Dimensional Systems , 2000, Ann. Oper. Res..

[16]  Hussein Jaddu,et al.  Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials , 2002, J. Frankl. Inst..

[17]  Milan Vlach,et al.  Successive approximation method for non‐linear optimal control problems with application to a container crane problem , 2002 .

[18]  Stephen P. Banks,et al.  Observer design for nonlinear systems using linear approximations , 2003, IMA J. Math. Control. Inf..

[19]  Stephen P. Banks,et al.  Linear approximations to nonlinear dynamical systems with applications to stability and spectral theory , 2003, IMA J. Math. Control. Inf..

[20]  Stephen P. Banks,et al.  An Iterative Approach to Eigenvalue Assignment for Nonlinear Systems. , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[21]  M. Tomas-Rodriguez,et al.  Sliding Mode Control for Nonlinear Systems: An Iterative Approach. , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[22]  H. Jaddu,et al.  Legendre Polynomials Iterative Technique for Solving a Class of Nonlinear Optimal Control Problems , 2014 .

[23]  Hussein Jaddu,et al.  An Iterative Technique for Solving a Class of Nonlinear Quadratic Optimal Control Problems Using Chebyshev Polynomials , 2014 .