Abstract
A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates of the optimal control problem. More precisely, it is shown that the dual multipliers for the discrete scheme correspond to a pseudospectral approximation of the adjoint equation using polynomials one degree smaller than that used for the state equation. The relationship between the coefficients of the pseudospectral scheme for the state equation and for the adjoint equation is established. Also, it is shown that the inverse of the pseudospectral LGR differentiation matrix is precisely the matrix associated with an implicit LGR integration scheme. Hence, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Numerical results show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions for both finite and infinite-horizon optimal control problems.
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References
Elnagar, G., Kazemi, M., Razzaghi, M.: The pseudospectral Legendre method for discretizing optimal control problems. IEEE Trans. Autom. Control 40(10), 1793–1796 (1995)
Elnagar, G., Kazemi, M.: Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems. Comput. Optim. Appl. 11(2), 195–217 (1998)
Fahroo, F., Ross, I.M.: A spectral patching method for direct trajectory optimization. J. Astronaut. Sci. 48(2–3), 269–286 (2000)
Fahroo, F., Ross, I.M.: Costate estimation by a Legendre pseudospectral method. J. Guid. Control Dyn. 24(2), 270–277 (2001)
Ross, I.M., Fahroo, F.: Legendre Pseudospectral Approximations of Optimal Control Problems. Lecture Notes in Control and Information Sciences. Springer, Berlin (2003)
Rao, A.V.: Extension of a pseudospectral Legendre method for solving non-sequential multiple-phase optimal control problems. In: AIAA Guidance, Navigation, and Control Conference, Austin, Texas, August 11–14, 2003. AIAA Paper 2003-5634
Williams, P.: Jacobi pseudospectral method for solving optimal control problems. J. Guid. Control Dyn. 27(2), 293–297 (2004)
Williams, P.: Application of pseudospectral methods for receding horizon control. J. Guid. Control Dyn. 27(2), 310–314 (2004)
Williams, P.: Hermite-Legendre-Gauss-Lobatto direct transcription methods in trajectory optimization. In: American Astronautical Society, Spaceflight Mechanics Meeting, August 2005
Ross, I.M., Fahroo, F.: Pseudospectral knotting methods for solving optimal control problems. J. Guid. Control Dyn. 27(3), 397–405 (2004)
Fahroo, F., Ross, I.M.: On discrete-time optimality conditions for pseudospectral methods. In: AIAA Guidance, Navigation, and Control Conference, Keystone, Colorado, August 2006. AIAA Paper 2006-6304
Fahroo, F., Ross, I.M.: Pseudospectral methods for infinite-horizon nonlinear optimal control problems. J. Guid. Control Dyn. 31(4), 927–936 (2008)
Benson, D.A.: A Gauss pseudospectral transcription for optimal control. Ph.D. thesis, MIT (2004)
Benson, D.A., Huntington, G.T., Thorvaldsen, T.P., Rao, A.V.: Direct trajectory optimization and costate estimation via an orthogonal collocation method. J. Guid. Control Dyn. 29(6), 1435–1440 (2006)
Huntington, G.T.: Advancement and analysis of a Gauss pseudospectral transcription for optimal control. Ph.D. thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology (2007)
Kameswaran, S., Biegler, L.T.: Convergence rates for direct transcription of optimal control problems using collocation at Radau points. Comput. Optim. Appl. 41(1), 81–126 (2008)
Reddien, G.W.: Collocation at Gauss points as a discretization in optimal control. SIAM J. Control Optim. 17(2) (1979)
Cuthrell, J.E., Biegler, L.T.: On the optimization of differential-algebraic processes. AIChe J. 33(8), 1257–1270 (1987)
Cuthrell, J.E., Biegler, L.T.: Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13(1/2), 49–62 (1989)
Ross, I.M., Fahroo, F.: Advances in pseudospectral methods for optimal control. In: AIAA Guidance, Navigation, and Control Conference, Honolulu, Hawaii, August 2008. AIAA Paper 2008-7309
Ross, I.M., Fahroo, F.: Convergence of the costates do not imply convergence of the controls. J. Guid. Control Dyn. 31(4), 1492–1497 (2008)
Huntington, G.T., Benson, D.A., Rao, A.V.: Optimal configuration of tetrahedral spacecraft formations. J. Astronaut. Sci. 55(2), 141–169 (2007)
Huntington, G.T., Rao, A.V.: Optimal reconfiguration of spacecraft formations using the Gauss pseudospectral method. J. Guid. Control Dyn. 31(3), 689–698 (2008)
Hager, W.W.: Numerical analysis in optimal control. In: Hoffmann, K.-H., Lasiecka, I., Leugering, G., Sprekels, J., Troeltzsch, F. (eds.) International Series of Numerical Mathematics, vol. 139, pp. 83–93. Birkhäuser, Basel (2001)
Hager, W.W., Dontchev, A., Poore, A., Yang, B.: Optimality, stability, and convergence in nonlinear control. Appl. Math. Optim. 31, 297–326 (1995)
Hager, W.W.: Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87, 247–282 (2000)
Patterson, M.A.: OptimalPrime: A MATLAB software for solving non-sequential multiple-phase optimal control problems using pseudospectral methods. Dept. of Mechanical and Aerospace Engineering, University of Florida (August 2008)
Gill, P.E., Murray, W., Saunders, M.A.: User’s Guide for SNOPT Version 7: Software for Large Scale Nonlinear Programming (February 2006)
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Garg, D., Patterson, M.A., Francolin, C. et al. Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method. Comput Optim Appl 49, 335–358 (2011). https://doi.org/10.1007/s10589-009-9291-0
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DOI: https://doi.org/10.1007/s10589-009-9291-0