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On numerical methods for singular optimal control problems: An application to an AUV problem
1. | Faculdade de Engenharia Universidade do Porto, DEEC, Porto, Portugal |
2. | Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran |
We discuss and compare numerical methods to solve singular optimal control problems by the direct method. Our discussion is illustrated by an Autonomous Underwater Vehicle (AUV) problem with state constraints. For this problem, we test four different approaches to solve numerically our problem via the direct method. After discretizing the optimal control problem we solve the resulting optimization problem with (ⅰ) A Mathematical Programming Language ($ \text{AMPL} $), (ⅱ) the Imperial College London Optimal Control Software ($ \text{ICLOCS} $), (ⅲ) the Gauss Pseudospectral Optimization Software ($ \text{GPOPS} $) as well as with (ⅳ) a new algorithm based on mixed-binary non-linear programming reported in [
References:
[1] |
R. Baltensperger and M. R. Trummer,
Spectral differencing with a twist., SIAM J. Sci. Comput., 24 (2003), 1465-1487.
doi: 10.1137/S1064827501388182. |
[2] |
J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, vol. 19 of Advances in Design and Control, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010.
doi: 10.1137/1.9780898718577. |
[3] |
R. Byrd, J. Nocedal and R. Waltz, Knitro: An integrated package for nonlinear optimization, in Large-Scale Nonlinear Optimization (eds. G. Di Pillo and M. Roma), vol. 83 of Nonconvex Optimization and Its Applications, Springer US, 2006, 35–59.
doi: 10.1007/0-387-30065-1_4. |
[4] |
C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-3-642-84108-8. |
[5] |
M. d. R. de Pinho, Z. Foroozandeh and A. Matos, Optimal control problems for path planing of auv using simplified models, in 2016 IEEE 55th Conference on Decision and Control (CDC), 2016,210–215.
doi: 10.1109/CDC.2016.7798271. |
[6] |
B. Fornberg, A Practical Guide to Pseudospectral Methods, vol. 1 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511626357. |
[7] |
Z. Foroozandeh, M. Shamsi and M. d. R. de Pinho, A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems, International Journal of Control, 1–16. |
[8] |
Z. Foroozandeh, M. Shamsi and M. d. R. De Pinho,
A hybrid direct–indirect approach for solving the singular optimal control problems of finite and infinite order, Iranian Journal of Science and Technology, Transactions A: Science, 42 (2018), 1545-1554.
doi: 10.1007/s40995-017-0176-2. |
[9] |
Z. Foroozandeh, M. Shamsi, V. Azhmyakov and M. Shafiee,
A modified pseudospectral method for solving trajectory optimization problems with singular arc, Mathematical Methods in the Applied Sciences, 40 (2017), 1783-1793.
doi: 10.1002/mma.4097. |
[10] |
R. Fourer, D. M. Gay and B. Kernighan, Algorithms and model formulations in mathematical programming, Springer-Verlag New York, Inc., New York, NY, USA, 1989, chapter AMPL: A Mathematical Programming Language, 150–151. |
[11] |
D. Garg, Advances in Global Pseudospectral Methods for Optimal Control, PhD thesis, 2011, Thesis (Ph.D.)–University of Florida. |
[12] |
D. Garg, M. Patterson, W. W. Hager, A. V. Rao, D. A. Benson and G. T. Huntington,
A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 46 (2010), 1843-1851.
doi: 10.1016/j.automatica.2010.06.048. |
[13] |
H. Maurer,
Numerical solution of singular control problems using multiple shooting techniques, J. Optimization Theory Appl., 18 (1976), 235-257.
doi: 10.1007/BF00935706. |
[14] |
H. Maurer,
On optimal control problems with bounded state variables and control appearing linearly, SIAM J. Control Optim., 15 (1977), 345-362.
doi: 10.1137/0315023. |
[15] |
M. A. Patterson and A. V. Rao, GPOPS-Ⅱ: A matlab software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Softw., 41 (2014), Art. 1, 37 pp.
doi: 10.1145/2558904. |
[16] |
P. D. Pinto da Silva, Planeamento Otimizado de Movimento de Robot Submarino, (universidade do porto, faculdade de engenharia, msc thesis), 2014. |
[17] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962. |
[18] |
S. Takriti, R. Fourer, M. Gay and B. Kernighan, Ampl: A Modeling Language for Mathematical Programming, 1994. |
[19] |
E. J. Van Wyk, P. Falugi and E. C. Kerrigan, Iclocs, 2010, URL http://www.ee.ic.ac.uk/ICLOCS. |
[20] |
R. Vinter, Optimal Control, Birkhäuser Basel, 2010.
doi: 10.1007/978-0-8176-8086-2. |
[21] |
A. Wächter and L. T. Biegler,
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.
doi: 10.1007/s10107-004-0559-y. |
[22] |
J. A. C. Weideman and S. C. Reddy,
A MATLAB differentiation matrix suite, ACM Trans. Math. Software, 26 (2000), 465-519.
doi: 10.1145/365723.365727. |
[23] |
B. D. Welfert,
Generation of pseudospectral differentiation matrices. I., SIAM J. Numer. Anal., 34 (1997), 1640-1657.
doi: 10.1137/S0036142993295545. |
show all references
References:
[1] |
R. Baltensperger and M. R. Trummer,
Spectral differencing with a twist., SIAM J. Sci. Comput., 24 (2003), 1465-1487.
doi: 10.1137/S1064827501388182. |
[2] |
J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, vol. 19 of Advances in Design and Control, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010.
doi: 10.1137/1.9780898718577. |
[3] |
R. Byrd, J. Nocedal and R. Waltz, Knitro: An integrated package for nonlinear optimization, in Large-Scale Nonlinear Optimization (eds. G. Di Pillo and M. Roma), vol. 83 of Nonconvex Optimization and Its Applications, Springer US, 2006, 35–59.
doi: 10.1007/0-387-30065-1_4. |
[4] |
C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-3-642-84108-8. |
[5] |
M. d. R. de Pinho, Z. Foroozandeh and A. Matos, Optimal control problems for path planing of auv using simplified models, in 2016 IEEE 55th Conference on Decision and Control (CDC), 2016,210–215.
doi: 10.1109/CDC.2016.7798271. |
[6] |
B. Fornberg, A Practical Guide to Pseudospectral Methods, vol. 1 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511626357. |
[7] |
Z. Foroozandeh, M. Shamsi and M. d. R. de Pinho, A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems, International Journal of Control, 1–16. |
[8] |
Z. Foroozandeh, M. Shamsi and M. d. R. De Pinho,
A hybrid direct–indirect approach for solving the singular optimal control problems of finite and infinite order, Iranian Journal of Science and Technology, Transactions A: Science, 42 (2018), 1545-1554.
doi: 10.1007/s40995-017-0176-2. |
[9] |
Z. Foroozandeh, M. Shamsi, V. Azhmyakov and M. Shafiee,
A modified pseudospectral method for solving trajectory optimization problems with singular arc, Mathematical Methods in the Applied Sciences, 40 (2017), 1783-1793.
doi: 10.1002/mma.4097. |
[10] |
R. Fourer, D. M. Gay and B. Kernighan, Algorithms and model formulations in mathematical programming, Springer-Verlag New York, Inc., New York, NY, USA, 1989, chapter AMPL: A Mathematical Programming Language, 150–151. |
[11] |
D. Garg, Advances in Global Pseudospectral Methods for Optimal Control, PhD thesis, 2011, Thesis (Ph.D.)–University of Florida. |
[12] |
D. Garg, M. Patterson, W. W. Hager, A. V. Rao, D. A. Benson and G. T. Huntington,
A unified framework for the numerical solution of optimal control problems using pseudospectral methods, Automatica, 46 (2010), 1843-1851.
doi: 10.1016/j.automatica.2010.06.048. |
[13] |
H. Maurer,
Numerical solution of singular control problems using multiple shooting techniques, J. Optimization Theory Appl., 18 (1976), 235-257.
doi: 10.1007/BF00935706. |
[14] |
H. Maurer,
On optimal control problems with bounded state variables and control appearing linearly, SIAM J. Control Optim., 15 (1977), 345-362.
doi: 10.1137/0315023. |
[15] |
M. A. Patterson and A. V. Rao, GPOPS-Ⅱ: A matlab software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Softw., 41 (2014), Art. 1, 37 pp.
doi: 10.1145/2558904. |
[16] |
P. D. Pinto da Silva, Planeamento Otimizado de Movimento de Robot Submarino, (universidade do porto, faculdade de engenharia, msc thesis), 2014. |
[17] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962. |
[18] |
S. Takriti, R. Fourer, M. Gay and B. Kernighan, Ampl: A Modeling Language for Mathematical Programming, 1994. |
[19] |
E. J. Van Wyk, P. Falugi and E. C. Kerrigan, Iclocs, 2010, URL http://www.ee.ic.ac.uk/ICLOCS. |
[20] |
R. Vinter, Optimal Control, Birkhäuser Basel, 2010.
doi: 10.1007/978-0-8176-8086-2. |
[21] |
A. Wächter and L. T. Biegler,
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.
doi: 10.1007/s10107-004-0559-y. |
[22] |
J. A. C. Weideman and S. C. Reddy,
A MATLAB differentiation matrix suite, ACM Trans. Math. Software, 26 (2000), 465-519.
doi: 10.1145/365723.365727. |
[23] |
B. D. Welfert,
Generation of pseudospectral differentiation matrices. I., SIAM J. Numer. Anal., 34 (1997), 1640-1657.
doi: 10.1137/S0036142993295545. |



6 | 0.049341739 | 0.6454231790 | 14.626412022 | 14.72348234 | 14.956410213 | |||||
8 | 0.049051513 | 0.6456145668 | 14.623890223 | 14.72412511 | 14.950131513 | |||||
10 | 0.049052155 | 0.6456134127 | 14.623817123 | 14.72416201 | 14.950161798 | |||||
12 | 0.049052100 | 0.6456134114 | 14.623817653 | 14.72416210 | 14.950161782 | |||||
14 | 0.049052100 | 0.6456134114 | 14.623817653 | 14.72416210 | 14.950161782 |
6 | 0.049341739 | 0.6454231790 | 14.626412022 | 14.72348234 | 14.956410213 | |||||
8 | 0.049051513 | 0.6456145668 | 14.623890223 | 14.72412511 | 14.950131513 | |||||
10 | 0.049052155 | 0.6456134127 | 14.623817123 | 14.72416201 | 14.950161798 | |||||
12 | 0.049052100 | 0.6456134114 | 14.623817653 | 14.72416210 | 14.950161782 | |||||
14 | 0.049052100 | 0.6456134114 | 14.623817653 | 14.72416210 | 14.950161782 |
1000 | 14.941407 | 180 | 1809.557 | |||||||
1000 | 14.943225 | 1075 | 20.265 | |||||||
80 | 14.949946 | 20682 | 1027.5 | |||||||
Section 3.2 | 14 | Explicitly available | 14.950161 | 6 | 4.472 |
1000 | 14.941407 | 180 | 1809.557 | |||||||
1000 | 14.943225 | 1075 | 20.265 | |||||||
80 | 14.949946 | 20682 | 1027.5 | |||||||
Section 3.2 | 14 | Explicitly available | 14.950161 | 6 | 4.472 |
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