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Piecewise Polynomial Control Parameterization in the Direct Solution of Optimal Control Problems

[+] Author and Article Information
Brian C. Fabien

Professor
Department of Mechanical Engineering,
University of Washington,Seattle, WA 98195
e-mail: fabien@uw.edu

Manuscript received December 27, 2011; final manuscript received December 7, 2012; published online March 28, 2013. Assoc. Editor: Alexander Leonessa.

J. Dyn. Sys., Meas., Control 135(3), 034506 (Mar 28, 2013) (5 pages) Paper No: DS-11-1407; doi: 10.1115/1.4023401 History: Received December 27, 2011; Revised December 07, 2012

This paper describes an algorithm for the direct solution of a class of optimal control problems. The algorithm is based on approximating the unknown control inputs via a finite dimensional parameterization. Specific control approximations that are implemented include (i) piecewise constant, (ii) piecewise linear, or (iii) piecewise cubic polynomials. The cubic approximation presented here is believed to be new. Another novel feature of the algorithm is that the state equations are approximated using single step Runge–Kutta methods on a fixed mesh. The parameterized optimal control problem is solved using a sequential quadratic programming technique. The paper presents examples to illustrate the convergence behavior of the various control parameterization schemes.

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References

Fabien, B. C., 2011, “OCP: An Optimal Control Problem Solver,” http://abs-5.me.washington.edu/ocp/
Agrawal, S. K., and Fabien, B. C., 1999, Optimization of Dynamic Systems, Kluwer Academic Publishers, New York.
Betts, J. T., 1998, “Survey of Numerical Methods for Trajectory Optimization,” AIAA J. Guid., Control, Dyn., 21, pp. 193–207. [CrossRef]
Bock, H. G., 1978, “Numerical Solution of Nonlinear Multipoint Boundary Value Problems With Application to Optimal Control,” ZAMM, 58, pp. T407–T409. [CrossRef]
Fabien, B. C., 1996, “Indirect Numerical Solution of Constrained Optimal Control Problems With Parameters,” Appl. Math. Comput., 80, pp. 43–62. [CrossRef]
Cuthrell, J. E., and Biegler, L. T., 1987, “On the Optimization of Differential-Algebraic Process Systems,” AIChE J., 33, pp. 1257–1270. [CrossRef]
Hargraves, C. R., and Paris, S. W., 1987, “Direct Trajectory Optimization Using Nonlinear Programming and Collocation,” AIAA J. Guid., Control, Dyn., 10, pp. 338–342. [CrossRef]
Stryk, O., and Bulirsch, R., 1992, “Direct and Indirect Methods for Trajectory Optimization,” Ann. Operat. Res., 37, pp. 357–373. [CrossRef]
Fabien, B. C., 1998, “Some Tools for the Direct Solution of Optimal Control Problems,” Adv. Eng. Softw., 29, pp. 45–61. [CrossRef]
Goh, C. J., and Teo, K. L., 1988, “MISER: A FORTRAN Program for Solving Optimal Control Problems,” Adv. Eng. Softw., 10, pp. 90–99. [CrossRef]
Kraft, D., 1994, “Algorithm 733: TOMP-Fortran Modules for Optimal Control Calculations,” ACM Trans. Math. Softw., 20, pp. 262–307. [CrossRef]
Bock, H. G., and Plitt, K. J., 1984, “A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems,” IFAC 9th Triennial World Congress, pp. 1603–1608.
Fabien, B. C., 2008, “Direct Optimization of Dynamic Systems Described by Differential-Algebraic Equations,” Opt. Control Appl. Methods, 29, pp. 445–466. [CrossRef]
Fabien, B. C., 2008, “Parameter Optimization Using the L Exact Penalty Function and Strictly Convex Quadratic Programming Problems,” Appl. Math. Comput., 198, pp. 834–848. [CrossRef]
Fabien, B. C., 2008, “Implementation of a Robust SQP Algorithm,” Opt. Methods Softw., 23, pp. 827–846. [CrossRef]
Walter, A., 2007, “Automatic Differentiation of Explicit Runge-Kutta Methods for Optimal Control,” Comput. Opt. Appl., 36, pp. 83–108. [CrossRef]
Teo, K. L., Goh, C. J., and Wong, K. H., 1991, A Unified Computational Approach to Optimal Control Problems, Longman Scientific, New York.
Conte, S. D., and de Boor, C., 1980, Elementary Numerical Analysis, 3rd ed., McGraw-Hill, New York.
Swartz, B., and Varga, R., 1972, “Error Bounds for Spline and L-Spline Interpolation,” J. Approx. Theory, 6, pp. 6–49. [CrossRef]
Hairer, E., Norsett, S. P., and Wanner, G., 2008, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised ed., Springer, New York.
Lawden, D. F., 2003, Analytical Methods of Optimization, Dover Publications, New York.

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