Costate Estimation by a Legendre Pseudospectral Method
Ross, I. Michael
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We present a Legendre pseudospectral method for directly estimating the costate of the Bolza problem encountered in optimal control theory. The method is based on calculating the state and control variables at the Legendre'Gauss'Lobatto (LGL) points. An Nth degree Lagrange polynomial approximation of these variables allows a conversion of the optimal control problem into a standard nonlinear programming (NLP) problem with the state and control values at the LGL points as optimization parameters. By applying the Karush'Kuhn'Tucker (KKT) theorem to the NLP problem, we show that the KKT multipliers satisfy a discrete analog of the costate dynamics including the transversality conditions. Indeed, we prove that the costates at the LGL points are equal to the KKT multipliers divided by the LGL weights. Hence, the direct solution by this method also automatically yields the costate by way of the Lagrange multipliers that can be extracted from an NLP solver. One important advantage of this technique is that it allows a very simple way to check the optimality of the direct solution. Numerical examples are included to demonstrate the method.
Journal of Guidance, Control, and Dynamics Vol. 24, No. 2, March-April 2001
RightsThis publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.
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Fahroo, Fariba; Ross, I. Michael (The American Institute of Aeronautics and Astronautics (AIAA), 2008-08-18);Recently, the Legendre Pseudospectral (PS) method migrated from theory to fight application onboard the International Space Station for performing a finite-horizon, zero- propellant maneuver. A small technical modification ...
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