Bellman pseudospectral method
The Bellman pseudospectral method is a pseudospectral method for optimal control based on Bellman's principle of optimality. The method is named after Richard E. Bellman. It was introduced by Ross et al[1] [2] first as a means to solve multiscale optimal control problems, and later expanded to obtain suboptimal solutions for general optimal control problems.
Theoretical foundations
The multiscale version of the Bellman pseudospectral mehod is based on the spectral convergence property of the Ross–Fahroo pseudospectral methods. That is, because the Ross–Fahroo pseudospectral method converges at an exponentiallly fast rate, pointwise convergence to a solution is obtained at very low number of nodes even when the solution has high-frequency components. This aliasing phenomenon in optimal control was first discovered by Ross et al.[1] Rather than use signal processing techniques to anti-alias the solution, Ross et al proposed that Bellman's principle of optimality can be appllied to the converged solution to extract information between the nodes. Because the Gauss–Lobatto nodes cluser at the boundary points, Ross et al suggested that if the node density around the initial conditions satisfy the Nyquist–Shannon sampling theorem, then the complete solution can be recovered by solving the optimal control problem in a recursive fashion over piecewise segments known as Bellman segments.[1]
In an expanded version of the method, Ross et al,[2] proposed that method could also be used to generate feasible solutions that were not necessarily optimal. In this version, one can apply the Bellman pseudospectral method at even lower number of nodes even under the knowledge that the solution may not have converged to the optimal one. In this situation, one obtains a feasible solution.
A remarkable feature of the Bellman pseudospectral method is that it automatically determines several measures of suboptimality based on the original pseudospectral cost and the cost generated by the sum of the Bellman segments.[1][2]
Applications
The Bellman pseudospectral method was first applied by Ross et al[1] to solve the challenging problem of very low thrust trajectory optimization. It has been successfully applied to solve a practical problem of generating very high accuracy solutions to a trans-Earth-injection problem of bringing a space capsule from a lunar orbit to a pin-pointed Earth-interface condition for successful reentry.[3] [4]
The Bellman pseudospectral method is most commonly used as an additional check on the optimality of a pseudospectral solution generated by the Ross–Fahroo pseudospectral methods. That is, in addition to the use of Pontryagin's minimum principle in conjunction with the solutions obtained by the Ross–Fahroo pseudospectral methods, the Bellman pseudospectral method is used as a primal-only test on the optimality of the computed solution. [5] [6]
See also
References
- ^ a b c d e I. M. Ross, Q. Gong and P. Sekhavat, P. Low-Thrust, High-Accuracy Trajectory Optimization, Journal of Guidance, Control and Dynamics, Vol. 30, No. 4, July–Aug 2007, pp.921–933.
- ^ a b c I. M. Ross, Q. Gong and P. Sekhavat, The Bellman pseudospectral method, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, AIAA-2008-6448, August 18-21, 2008.
- ^ H. Yan, Q. Gong, C. Park, I. M. Ross, and C. N. D’Souza, High Accuracy Trajectory Optimization for a Trans-Earth Lunar Mission, Journal of Guidance, Control and Dynamics, Vol. 34, No. 4, 2011, pp. 1219–1227.
- ^ H. Yan, Q. Gong, C. D. Park, I. M. Ross and C. N. D'Souza, High-Accuracy Moon to Earth trajectory optimization, AIAA Guidance, Navigation, and Control Conference, 2010.
- ^ A. Fleming, P. Sekhavat, and I. M. Ross, Minimum-Time Reorientation of a Rigid Body, Journal of Guidance, Control and Dynamics, Vol.33, No. 1, Jan–Feb 2010, pp. 160–170.
- ^ I. M. Ross, P. Sekhavat, A. Fleming and Q. Gong, Optimal feedback control: foundations, examples, and experimental results for a new approach, Journal of Guidance, Control, and Dynamics, vol. 31 no. 2, pp. 307–321, 2008.