Direct method to solve linear-quadratic optimal control problems

Mohamed Aliane; Mohand Bentobache; Nacima Moussouni; Philippe Marthon

doi:10.3934/naco.2021002

Article Contents

2021, Volume 11, Issue 4: 645-663. Doi: 10.3934/naco.2021002

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Direct method to solve linear-quadratic optimal control problems

1.
Laboratory of Pure and Applied Mathematics, University of Laghouat, Bp 37G, Ghardaia road, 03000, Laghouat, Algeria
2.
Laboratory L2CSP, University of Tizi-Ouzou, 15000, Tizi-Ouzou, Algeria
3.
IRIT-ENSEEIHT, Université Fédérale Toulouse-Midi-Pyrénées, France

^* Corresponding author: m.aliane@lagh-univ.dz
^* Corresponding author: m.aliane@lagh-univ.dz

Received: April 2020

Revised: December 2020

Early access: January 2021

Published: December 2021

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Abstract

In this work, we have proposed a new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex. In this approach, we transform the continuous optimal control problem into a quadratic optimization problem using the Cauchy discretization technique, then we solve it with the active-set method. In order to study the efficiency and the accuracy of the proposed approach, we developed an implementation with MATLAB, and we performed numerical experiments on several convex and non-convex linear-quadratic optimal control problems. The obtained simulation results show that our method is more accurate and more efficient than the method using the classical Euler discretization technique. Furthermore, it was shown that our method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.

Keywords:

Mathematics Subject Classification: Primary: 93C05; Secondary: 53C35.

Citation:

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Figure 1. Optimal control $ t\longmapsto u(t) $ for the problem of Example 1

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Figure 2. CPU time and Error in terms of $ N $ for Example 1

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Figure 3. CPU time and Error in terms of $ N $ for Example 2

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Figure 4. CPU time and Error in terms of $ N $ for Example 3

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Figure 5. CPU time and Error in terms of $ N $ for Example 4

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Table 1. Numerical simulation results of EDM for Example 1

	EDM(SQP)			EDM(ASM)
N	J(u)	CPU	Error	J(u)	CPU	Error
4	0.953125	0.45	3.73E+00	0.953125	1.03	3.73E+00
10	2.588281	0.09	2.09E+00	2.588281	0.12	2.09E+00
30	3.865982	0.15	8.14E-01	3.865982	0.25	8.14E-01
50	4.176861	0.43	5.03E-01	4.176861	0.54	5.03E-01
70	4.316335	0.91	3.64E-01	4.316335	1.58	3.64E-01
100	4.422900	1.99	2.57E-01	4.422900	2.48	2.57E-01
200	4.549882	10.30	1.30E-01	4.549697	12.45	1.30E-01
400	4.614494	51.99	6.54E-02	4.614199	46.91	6.57E-02
600	4.636198	161.93	4.37E-02	4.635949	144.54	4.39E-02
800	4.647087	375.59	3.28E-02	4.646814	379.60	3.31E-02
1000	4.653632	696.03	2.63E-02	4.653362	865.84	2.65E-02

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Table 2. Numerical simulation results of CDA for Example 1

	CDA1			CDA2
N	J(u)	CPU	Error	J(u)	CPU	Error
4	4.494898	0.01	1.85E-01	4.494898	0.05	1.85E-01
10	4.658000	0.01	2.19E-02	4.658000	0.07	2.19E-02
30	4.674355	0.03	5.54E-03	4.674355	0.21	5.54E-03
50	4.677464	0.09	2.43E-03	4.677464	0.45	2.43E-03
70	4.678780	0.11	1.11E-03	4.678780	0.54	1.11E-03
100	4.679761	0.25	1.30E-04	4.679761	0.81	1.30E-04
200	4.679830	0.96	6.13E-05	4.679830	8.37	6.13E-05
400	4.679865	10.73	2.67E-05	4.679865	10.08	2.67E-05
600	4.679876	31.06	1.52E-05	4.679876	29.00	1.52E-05
800	4.679882	72.20	9.43E-06	4.679882	80.56	9.43E-06
1000	4.679886	160.93	5.97E-06	4.679886	178.74	5.97E-06

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Table 3. Numerical simulation results of EDM for Example 2

	EDM(SQP)			EDM(ASM)
N	J(u)	CPU	Error	J(u)	CPU	Error
4	-807.000006	0.42	6.33E+02	-807.000000	0.78	6.33E+02
10	-6.020544	0.04	1.68E+02	-6.021052	0.07	1.68E+02
30	-180.145998	0.17	6.15E+00	-180.145998	0.24	6.15E+00
50	-187.685200	0.74	1.37E+01	-187.685200	0.66	1.37E+01
70	-187.041677	1.80	1.30E+01	-187.041677	1.85	1.30E+01
100	-184.758275	5.04	1.08E+01	-184.758275	4.92	1.08E+01
200	-180.485036	37.17	6.49E+00	-180.485036	36.75	6.49E+00
400	-177.524433	293.36	3.52E+00	-177.523183	324.39	3.52E+00
600	-176.412810	1075.55	2.41E+00	-176.411448	1284.80	2.41E+00
800	-175.833379	2692.90	1.83E+00	-175.831972	3712.96	1.83E+00
1000	-175.478114	5611.01	1.48E+00	-175.477202	8598.05	1.48E+00

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Table 4. Numerical simulation results of CDA for Example 2

	CDA1			CDA2
N	J(u)	CPU	Error	J(u)	CPU	Error
4	-174.000000	0.29	9.95E-13	-174.000000	0.56	0.00E+00
10	-161.832000	0.02	1.22E+01	-161.832000	0.11	1.22E+01
30	-172.676444	0.07	1.32E+00	-172.676444	0.33	1.32E+00
50	-173.524339	0.19	4.76E-01	-173.524339	0.61	4.76E-01
70	-173.757431	0.44	2.43E-01	-173.757431	0.98	2.43E-01
100	-174.000000	0.98	3.01E-12	-174.000000	1.81	5.00E-12
200	-174.000000	8.02	4.01E-12	-174.000000	10.29	6.00E-12
400	-174.000000	83.60	3.00E-11	-174.000000	90.35	9.00E-11
600	-174.000000	354.27	2.10E-11	-174.000000	359.11	6.40E-11
800	-174.000000	996.67	6.20E-11	-174.000000	1045.79	1.56E-10
1000	-174.000000	2354.52	3.01E-12	-174.000000	2320.88	4.01E-12

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Table 5. Numerical simulation results of EDM for Example 3

	EDM(SQP)			EDM(ASM)
N	J(u)	CPU	Error	J(u)	CPU	Error
4	32.000000	0.45	4.20E+01	32.000000	0.85	4.20E+01
10	53.422400	0.49	2.06E+01	53.422400	0.93	2.06E+01
30	66.485116	0.54	7.51E+00	66.485116	1.02	7.51E+00
50	69.407818	0.66	4.59E+00	69.407818	1.22	4.59E+00
70	70.694043	0.88	3.31E+00	70.694043	1.51	3.31E+00
100	71.672177	1.26	2.33E+00	71.672177	2.01	2.33E+00
200	72.828072	2.84	1.17E+00	72.828072	4.14	1.17E+00
400	73.412022	12.25	5.88E-01	73.412022	16.85	5.88E-01
600	73.607566	42.80	3.92E-01	73.607566	56.17	3.92E-01
800	73.705506	120.33	2.94E-01	73.705506	154.07	2.94E-01
1000	73.764324	284.73	2.36E-01	73.764324	360.15	2.36E-01

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Table 6. Numerical simulation results of CDA for Example 3

	CDA1			CDA2
N	J(u)	CPU	Error	J(u)	CPU	Error
4	74.000000	0.01	0.00E+00	74.000000	0.05	0.00E+00
10	74.000000	0.02	0.00E+00	74.000000	0.07	0.00E+00
30	74.000000	0.03	0.00E+00	74.000000	0.21	0.00E+00
50	74.000000	0.08	0.00E+00	74.000000	0.38	0.00E+00
70	74.000000	0.11	0.00E+00	74.000000	0.54	9.95E-14
100	74.000000	0.18	0.00E+00	74.000000	0.93	0.00E+00
200	74.000000	0.92	0.00E+00	74.000000	2.51	0.00E+00
400	74.000000	6.55	2.98E-13	74.000000	9.12	3.98E-13
600	74.000000	24.48	1.99E-13	74.000000	28.43	3.98E-13
800	74.000000	65.34	6.96E-13	74.000000	71.64	6.96E-13
1000	74.000000	144.16	0.00E+00	74.000000	152.09	0.00E+00

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Table 7. Numerical simulation results of EDM for Example 4

	EDM(SQP)			EDM(ASM)
N	J(u)	CPU	Error	J(u)	CPU	Error
4	0.171875	0.46	1.81E+00	0.171875	1.03	1.81E+00
10	0.922969	0.07	1.06E+00	0.922969	0.08	1.06E+00
30	1.560700	0.17	4.17E-01	1.560700	0.16	4.17E-01
50	1.719317	0.60	2.59E-01	1.719317	0.53	2.59E-01
70	1.790821	1.12	1.87E-01	1.790821	1.00	1.87E-01
100	1.845588	2.58	1.33E-01	1.844967	2.06	1.33E-01
200	1.910993	12.49	6.71E-02	1.910780	10.75	6.73E-02
400	1.944331	60.91	3.38E-02	1.944171	61.13	3.39E-02
600	1.955538	187.13	2.26E-02	1.955084	176.01	2.30E-02
800	1.961162	438.55	1.70E-02	1.960852	530.81	1.73E-02
1000	1.964543	820.38	1.36E-02	1.964187	1180.43	1.39E-02

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Table 8. Numerical simulation results of CDA for Example 4

	CDA1			CDA2
N	J(u)	CPU	Error	J(u)	CPU	Error
4	1.882653	0.01	9.55E-02	1.882653	0.06	9.55E-02
10	1.966800	0.01	1.13E-02	1.966800	0.07	1.13E-02
30	1.975251	0.03	2.86E-03	1.975251	0.21	2.86E-03
50	1.976858	0.08	1.25E-03	1.976858	0.36	1.25E-03
70	1.977538	0.11	5.74E-04	1.977538	0.58	5.74E-04
100	1.978046	0.23	6.72E-05	1.978046	0.80	6.72E-05
200	1.978081	1.05	3.17E-05	1.978081	2.92	3.17E-05
400	1.978099	10.23	1.38E-05	1.978099	10.02	1.38E-05
600	1.978105	30.87	7.86E-06	1.978105	28.47	7.86E-06
800	1.978108	73.76	4.88E-06	1.978108	79.62	4.88E-06
1000	1.978110	157.80	3.09E-06	1.978110	177.02	3.09E-06

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