On numerical methods for singular optimal control problems: An application to an AUV problem

Z. Foroozandeh; Maria do rosário de Pinho; M. Shamsi

doi:10.3934/dcdsb.2019092

Article Contents

2019, Volume 24, Issue 5: 2219-2235. Doi: 10.3934/dcdsb.2019092

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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

^* Corresponding author: Z. Foroozandeh
^* Corresponding author: Z. Foroozandeh

Received: December 31, 2017

Revised: December 31, 2018

Early access: March 2019

Published: March 2019

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Abstract

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.

Keywords:

Singular optimal control problem,
switching points,
AUV's problem,
Legendre-Gauss-Radau pseudospectral method,
mixed-binary non-linear programming,
mixed-binary optimal control,
ICLOCS,
GPOPS,
Implicit Euler Method.

Mathematics Subject Classification: Primary: 49J30, 65K05; Secondary: 49M37.

Citation:

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Figure 4. Controls computed by the method of section 3.2 with $ s = 5 $ and $ n = 20 $

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Figure 1. The control functions using implicit Euler method computed with AMPL interfaced with IPOPTS with $ N = 10000 $

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Figure 2. The control function computed by ICLOCS with $ N = 10000 $

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Figure 3. The control function computed by GPOPS with $ N = 40 $

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Figure 5. Control computed by the method of section 3.2 with $ s = 4 $ and $ n = 14 $

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Figure 6. States computed by the method of section 3.2 with $ s = 4 $ and $ n = 20 $

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Table 1. 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

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Table 2. 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

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