Numerical solution to optimal control problems of oscillatory processes

Kamil Aida-Zade; Jamila Asadova

doi:10.3934/jimo.2021166

Article Contents

2022, Volume 18, Issue 6: 4433-4445. Doi: 10.3934/jimo.2021166

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Numerical solution to optimal control problems of oscillatory processes

Kamil Aida-Zade and
Jamila Asadova^,

The Institute of Control Systems, Azerbaijan National Academy of Sciences, 9, B. Vahabzadeh str., AZ1141, Baku, Azerbaijan

^* Corresponding author: asadova.jamilya64@gmail.com
^* Corresponding author: asadova.jamilya64@gmail.com

Received: September 30, 2020

Revised: June 30, 2021

Early access: September 2021

Published: September 2022

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Abstract

A numerical approach to the investigation of optimal control problems of oscillatory processes with boundary and intermediate concentrated (lumped) control actions has been proposed in this paper. The corresponding analytical formulas for the components of the target functional gradient with respect to control actions considered on the class of piecewise continuous functions are obtained. The results of numerical experiments on the examples of solving model problems of optimal control of oscillatory processes with boundary and intermediate concentrated controls are provided. These results illustrate the dependence of the minimum settling time of oscillatory processes on the number and locations of concentrated control actions, on the process parameters, on the resistance coefficient of the medium (dissipation factor), and other factors.

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Mathematics Subject Classification: Primary: 49J15, 49J35.

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Figure 1. Plots of optimal piecewise continuous control of the oscillatory process for Problem $ III $: a) control at the left end; b) control at the right end; c) intermediate pointwise control located at the point $ \bar{x}_{1} = 0,5 $

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Figure 2. Plots of optimal piecewise-continuous control actions by intermediate concentrated sources in Problem $ VI $ under the following arrangement: a) $ \bar{x}_{1} = 0,4 $; b) $ \bar{x}_{2} = 0,8 $

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Figure 3. Plots of optimal control actions of four ILS for problem $ III $ at the following arrangement of ILS: a) $ \bar{x}_{1} = 0,2 $; b) $ \bar{x}_{2} = 0,4 $; c) $ \bar{x}_{3} = 0,6 $; d) $ \bar{x}_{4} = 0,8 $

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Table 1. Parameters of six model problems of oscillatory processes control with boundary and intermediate lumped controls

Problem No.	$ \alpha $	$ \hat{{\varphi }}_{10} \, $	$ \, \hat{{\varphi }}_{1T} \, $	$ \eta _{0} (0) $	$ \eta _{T} (0) $
$ I $	2,112	1	1.5	3, 72	5, 34
$ II $	3,168	1	1.5	3, 72	5, 34
$ III $	2,112	1	2	3, 1	5, 17
$ IV $	2,112	1	3	3, 87	7, 05
$ V $	1,408	1	2	3, 1	5, 17
$ VI $	2,112	1	4	3, 23	9, 38

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Table 2. Dependence of the optimal settling time for problems $ I $, $ III $, and $ VI $ on the arrangement of the single intermediate lumped control action

		$ T_{opt}^{} $
$ \bar{x}_{1} $	$ f_{0}^{1} (0) $	$ I $ problem	$ III $ problem	$ VI $ problem
0, 1	0, 2112	0, 95	1	1, 05
0.2	0, 4224	0, 85	0, 85	0, 9
0, 3	0, 6336	0, 75	0, 75	0, 8
0, 4	0, 8448	0, 85	0, 85	0, 95
0, 5	1,056	1, 05	1, 05	1, 1
0, 6	1, 2672	0, 85	0, 85	0, 95
0, 7	1, 4784	0, 75	0, 75	0, 8
0, 8	1, 6896	0.85	0.85	0, 9
0, 9	1, 9008	0.95	0, 95	1, 05

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Table 3. The optimal settling time of oscillatory processes in the problems $ I $, $ III $ and $ VI $ in the presence of two ILS

		$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ) $	$ (f_{0}^{1} (0),\, \, f_{0}^{2} (0)) $	$ I $ problem	$ III $ problem	$ VI $ problem
(0, 1; 0, 9)	(0, 2112; 1, 6896)	0, 85	0, 87	0, 94
(0, 2; 0, 8)	(0, 4224; 1, 2672)	0, 65	0, 69	0, 75
(0, 3; 0, 7)	(0, 6336; 0, 8448)	0, 66	0, 69	0, 74
(0,332; 0,668)	(0, 7012; 0, 7096)	0, 72	0, 75	0, 84
(0, 4; 0, 6)	(0, 8448; 0.4224)	0, 63	0, 69	0, 76
(0, 1; 0, 2)	(0.2112; 0, 2112)	0, 82	0, 85	0, 94
(0, 2; 0, 3)	(0, 4224; 0, 2112)	0, 75	0, 78	0, 85
(0, 3; 0, 4)	(0, 6336; 0, 2112)	0.65	0.68	0, 75
(0, 4; 0, 5)	(0, 8448; 0, 2112)	0.64	0, 67	0, 79
(0, 6; 0, 7)	(1, 2672; 0, 2112)	0, 66	0, 68	0, 75
(0, 2; 0, 6)	(0, 4224; 0, 8448)	0, 88	0, 9	0, 92

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Table 4. The optimal settling time in the presence of three ILS

		$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ,\bar{x}_{3} ) $	$ (f_{0}^{1} (0),\, \, f_{0}^{2} (0),\, \, f_{0}^{3} (0)) $	$ I $ prob.	$ III $ prob.	$ VI $ prob.
(0,252; 0, 5; 0,748)	(0, 5322; 0, 5237; 0, 5237)	0, 61	0, 63	0, 65
(0, 1; 0, 5; 0, 9)	(0, 2112; 0, 8448; 0, 8448)	0, 89	0, 91	0, 94
(0, 1; 0, 2; 0, 3)	(0, 2112; 0, 2112; 0, 2112)	0, 82	0, 83	0, 87
(0, 1; 0, 3; 0, 5)	(0, 2112; 0, 4224; 0, 4224)	0, 6	0, 62	0, 64
(0, 1; 0, 4; 0, 7)	(0, 2112; 0, 6336; 0, 6336)	0, 7	0, 72	0, 76
(0, 4; 0, 5; 0, 6)	(0, 8448; 0, 2112; 0, 2112)	0, 72	0, 74	0, 75
(0, 1; 0, 2; 0, 9)	(0, 2112; 0, 2112; 1, 4784)	0, 83	0, 85	0, 86
(0, 1; 0, 2; 0, 8)	(0, 2112; 0, 2112; 1, 2672)	0, 71	0, 73	0, 74
(0, 1; 0, 2; 0, 7)	(0, 2112; 0, 2112; 1,056)	0, 74	0, 74	0, 77

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Table 5. The minimal settling time of oscillating processes in the case of four ILS

		$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ,\bar{x}_{3} ,\bar{x}_{4} ) $	$ (f_{0}^{1} (0),\, \, f_{0}^{2} (0),\, \, f_{0}^{3} (0),\, \, f_{0}^{4} (0)) $	$ I $ prob.	$ III $ prob.	$ VI $ prob.
(0, 2; 0, 4; 0, 6; 0, 8)	(0, 4224; 0, 4224; 0, 4224; 0, 4224)	0, 51	0, 54	0, 57
(0, 1; 0, 2; 0, 8; 0, 9)	(0, 2112; 0, 2112; 1, 2672; 0, 2112)	0, 7	0, 72	0, 76
(0, 1; 0, 3; 0, 7; 0, 9)	(0, 2112; 0, 4224; 0, 8448; 0, 4224)	0, 51	0, 53	0, 59
(0, 1; 0, 2; 0, 3; 0, 4)	(0, 2112; 0, 2112; 0, 2112; 0, 2112)	0, 72	0, 73	0, 74
(0, 1; 0, 2; 0, 3; 0, 9)	(0, 2112; 0, 2112; 0, 2112; 1, 2672)	0, 71	0, 73	0, 77
(0, 1; 0, 2; 0, 3; 0, 8)	(0, 2112; 0, 2112; 0, 2112; 1,056)	0, 62	0, 63	0, 66
(0, 1; 0, 2; 0, 3; 0, 7)	(0, 2112; 0, 2112; 0, 2112; 0, 8448)	0, 7	0, 72	0, 75

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