# American Institute of Mathematical Sciences

May  2019, 24(5): 2219-2235. doi: 10.3934/dcdsb.2019092

## 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

* Corresponding author: Z. Foroozandeh

Received  January 2018 Revised  January 2019 Published  March 2019

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 [7]. This algorithm consists on converting the optimal control problem to a Mixed Binary Optimal Control (MBOC) problem which is then transcribed to a mixed binary non-linear programming problem ($\text{MBNLP}$) problem using Legendre-Radau pseudospectral method. Our case study shows that, in contrast with the first three approaches we test (all relying on $\text{IPOPT}$ or other numerical optimization software packages like $\text{KNITRO}$), the $\text{MBOC}$ approach detects the structure of the AUV's problem without a priori information of optimal control and computes the switching times accurately.

Citation: Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092
##### 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.

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##### 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.
Controls computed by the method of section 3.2 with $s = 5$ and $n = 20$
The control functions using implicit Euler method computed with AMPL interfaced with IPOPTS with $N = 10000$
The control function computed by ICLOCS with $N = 10000$
The control function computed by GPOPS with $N = 40$
Control computed by the method of section 3.2 with $s = 4$ and $n = 14$
States computed by the method of section 3.2 with $s = 4$ and $n = 20$
AUV's Problem: computed values of switching times and performance index for $s = 4$ and various values of $n$
 $n$ $t_1$ $t_2$ $t_3$ $t_4$ $t_f$ 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
 $n$ $t_1$ $t_2$ $t_3$ $t_4$ $t_f$ 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
Comparing results for AUV's problem with no initial guess
 $\text{Methods}$ $N$ $t_i$ $\mathcal{J}$ $\text{Iter}$ $\text{CPU Time}$ $\text{Euler}$ 1000 $\text{Not available explicitly}$ 14.941407 180 1809.557 $\text{ICLOCS}$ 1000 $\text{Not available explicitly}$ 14.943225 1075 20.265 $\text{GPOPS}$ 80 $\text{Not available explicitly}$ 14.949946 20682 1027.5 Section 3.2 14 Explicitly available 14.950161 6 4.472
 $\text{Methods}$ $N$ $t_i$ $\mathcal{J}$ $\text{Iter}$ $\text{CPU Time}$ $\text{Euler}$ 1000 $\text{Not available explicitly}$ 14.941407 180 1809.557 $\text{ICLOCS}$ 1000 $\text{Not available explicitly}$ 14.943225 1075 20.265 $\text{GPOPS}$ 80 $\text{Not available explicitly}$ 14.949946 20682 1027.5 Section 3.2 14 Explicitly available 14.950161 6 4.472
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