| 摘要(英) |
This thesis aims to study a full-space Lagrange-Newton algorithm for solving nonlinear optimal control problems. These problems represent a wide range of applications in computational science and engineering, such as trajectory optimization problems, industrial robots problems that can be also mathematically formulated as equality-constrained optimization problems.
In this method, the first step is to introduce the Lagrange multiplier into the objective function and then solve the optimization problem by finding the critical solution of the first-order necessary optimality condition (also known as the KKT condition) by the Newton-type method. One of the advantages of the Newton-type method is fast convergence provided that an initial guess is close to the solution. However, such a good initial guess is not easy to obtain. And Newton′s method suffers from the convergence issue when the nonlinearity of the system is not well balanced even some globalization technique is used. One of the drawbacks of the Lagrange-Newton method is that the KKT matrix needs to construct. The computation of the Hessian matrix of the KKT system could be expensive for example using finite-difference approximation. To improve the robustness of Newton′s method, we propose a new three-stage decoupling preconditioner. The key point of the new proposed algorithm is that before performing the global Newton update, in the three-stage decoupling preconditioning phase, we correct the Lagrange multiplier, control variables, state variables in order.
Our numerical results based on several benchmark test problems show that the three-stage decoupling preconditioner helpful for the convergence of the Lagrange-Newton algorithm and can reduce the number of iterations. Besides, we report a series of comparative study to investigate the full-space method with different approaches for constructing the Hessian matrix, including analytical approach, finite differences, automatic differentiation, and the low-rank update based method. We also show numerically that the full-space method is about a few hundred times faster than an optimizer in Matlab toolbox, which is implemented using the reduced-space Lagrange-Newton method. |
| 參考文獻 |
[1] J.T. Betts. Survey of numerical methods for trajectory optimization. J. Guid. Contr.
Dynam., 21:193–207, 1998.
[2] F. L. Chernousko. Optimization in control of robots. Computational Optimal Control,
1994.
[3] D. Gaur and M. S. Prasad. Orbit transfer using optimal control theory. In 2019 3rd
International Conference on Recent Developments in Control, Automation Power
Engineering (RDCAPE), 27–31, 2019.
[4] D. L. Black and S. V. Gunn. Space nuclear propulsion. In Robert A. Meyers,
editor, Encyclopedia of Physical Science and Technology (Third Edition), 555 – 575.
Academic Press, New York, third edition edition, 2003.
[5] F. S. Forbes and P. A. Van Splinter. Liquid rocket propellants. In Robert A. Meyers,
editor, Encyclopedia of Physical Science and Technology (Third Edition), 741 – 777.
Academic Press, New York, third edition edition, 2003.
[6] R. G. Jahn and E. Y. Choueiri. Electric propulsion. In Robert A. Meyers, editor, En-
cyclopedia of Physical Science and Technology (Third Edition), 125 – 141. Academic
Press, New York, third edition edition, 2003.
[7] D. Altman. Rocket motors, hybrid. In Robert A. Meyers, editor, Encyclopedia of
Physical Science and Technology (Third Edition), 303 – 321. Academic Press, New
York, third edition edition, 2003.
[8] P. W. Keaton. Low-thrust rocket trajectories. Technical report, Los Alamos National
Lab., NM (USA), 1987.
[9] A.V. Rao. Trajectory optimization: A survey. In Optimization and Optimal Control
in Automotive Systems, 3–21. Springer, 2014.
[10] O. von Stryk and R. Bulirsch. Direct and indirect methods for trajectory optimiza-
tion. Annals Oper. Res., 37:357–373, 1992.
[11] J.T. Betts. Practical Methods for Optimal Control and Estimation Using Nonlinear
Programming. SIAM, Philadelphia, 2nd edition, 2010.
[12] F. Fahroo and I.M. Ross. Costate estimation by a Legendre pseudospectral method.
J. Guid., Contr. Dynam, 24:270–277, 2001.
[13] D.G. Hull. Optimal Control Theory for Applications. Springer-Verlag, New York,
2003.
[14] P. Williams. Jacobi pseudospectral method for solving optimal control problems. J.
Guid. Contr. Dynam., 27:293–297, 2004.
[15] J. Nocedal and S.J. Wright. Numerical Optimization. Springer-Verlag, New York,
2006.
[16] M.R. Sentinella and L. Casalino. Cooperative evolutionary algorithm for space tra-
jectory optimization. Celestial Mech. Dyn. Astron., 105:211–227, 2009.
[17] S. Subchan and R. Żbikowski. Computational Optimal Control: Tools and Practice.
John Wiley & Sons, 2009.
[18] A. Wuerl, T. Crain, and E. Braden. Genetic algorithm and calculus of variations-
based trajectory optimization technique. J. Spacecraft Rockets, 40:882–888, 2003.
[19] M. Pontani. Particle swarm optimization of ascent trajectories of multistage launch
vehicles. Acta Astronaut., 94:852–864, 2014.
[20] P. Lu and M.A. Khan. Nonsmooth trajectory optimization-an approach using con-
tinuous simulated annealing. J. Guid. Contr, Dynam., 17:685–691, 1994.
[21] Y.-S. Lo. A full space lagrange-newton-krylov algorithm for minimum time trajectory
optimization. Master’s thesis, Natianal Central University, 2012.
[22] J.E. Dennis Jr. and R.B. Schnabel. Numerical Methods for Unconstrained Optimiza-
tion and Nonlinear Equations. SIAM, Philadelphia, 1996.
[23] Stanley C Eisenstat and Homer F Walker. Globally convergent inexact newton meth-
ods. SIAM Journal on Optimization, 4(2):393–422, 1994.
[24] H. Yang, F.-N. Hwang, and X-.C. Cai. Nonlinear preconditioning techniques for full-
space Lagrange-Newton solution of PDE-constrained optimization problems. SIAM
J. Sci. Comput., 2016. |
[Endnote RIS 格式]
[相關文章]
[文章引用]
[完整記錄]
[館藏目錄]
[檢視]
plurk
funp
live
udn
HD
myshare
netvibes
friend
youpush
delicious
baidu
Google bookmarks
del.icio.us
hemidemi
facebook
twitter
google
reddit