A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.

[1]  Mohsen Razzaghi,et al.  A legendre technique for solving time-varying linear quadratic optimal control problems , 1993 .

[2]  Gamal N. Elnagar,et al.  Short communication: A collocation-type method for linear quadratic optimal control problems , 1997 .

[3]  Gamal N. Elnagar,et al.  Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems , 1998 .

[4]  Anil V. Rao,et al.  Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method , 2006 .

[5]  Anil V. Rao,et al.  Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method , 2010, TOMS.

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

[7]  William W. Hager,et al.  Runge-Kutta methods in optimal control and the transformed adjoint system , 2000, Numerische Mathematik.

[8]  Fariba Fahroo.  On Discrete-Time Optimality Conditions for Pseudospectral Methods, AIAA (2006; Keystone, Colorado) , 2006 .

[9]  P. Williams.  Jacobi pseudospectral method for solving optimal control problems , 2004 .

[10]  M. Grassi,et al.  AIAA Guidance, Navigation, and Control Conference , 2008 .

[11]  I. Michael Ross,et al.  Costate Estimation by a Legendre Pseudospectral Method , 1998 .

[12]  I. Michael Ross,et al.  Convergence of the Costates Does Not Imply Convergence of the Control , 2008, Journal of Guidance, Control, and Dynamics.

[13]  G. Reddien.  Collocation at Gauss Points as a Discretization in Optimal Control , 1979 .

[14]  Qi Gong,et al.  Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control , 2008, Comput. Optim. Appl..

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

[16]  Gamal N. Elnagar,et al.  Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems , 1998, Comput. Optim. Appl..

[17]  J. E. Cuthrell,et al.  Simultaneous optimization and solution methods for batch reactor control profiles , 1989 .

[18]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[19]  I. Michael Ross,et al.  Pseudospectral Methods for Infinite-Horizon Nonlinear Optimal Control Problems , 2005 .

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

[21]  W. Kang.  On the Rate of Convergence for the Pseudospectral Optimal Control of Feedback Linearizable Systems , 2009, 0904.0833.

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

[23]  Anil V. Rao,et al.  Optimal configuration of tetrahedral spacecraft formations , 2007 .

[24]  P. Williams.  Hermite-Legendre-Gauss-Lobatto Direct Transcription in Trajectory Optimization , 2009 .

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

[26]  Lorenz T. Biegler,et al.  Convergence rates for direct transcription of optimal control problems using collocation at Radau points , 2008, Comput. Optim. Appl..

[27]  Owe Axelsson,et al.  Global integration of differential equations through Lobatto quadrature , 1964 .

[28]  William W. Hager,et al.  Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method , 2011, Comput. Optim. Appl..

[29]  Anil V. Rao,et al.  Optimal Reconfiguration of Spacecraft Formations Using the Gauss Pseudospectral Method , 2008 .

[30]  Paul Williams,et al.  Application of Pseudospectral Methods for Receding Horizon Control , 2004 .