Problem No. | |||||
2,112 | 1 | 1.5 | 3, 72 | 5, 34 | |
3,168 | 1 | 1.5 | 3, 72 | 5, 34 | |
2,112 | 1 | 2 | 3, 1 | 5, 17 | |
2,112 | 1 | 3 | 3, 87 | 7, 05 | |
1,408 | 1 | 2 | 3, 1 | 5, 17 | |
2,112 | 1 | 4 | 3, 23 | 9, 38 |
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.
Citation: |
Table 1. Parameters of six model problems of oscillatory processes control with boundary and intermediate lumped controls
Problem No. | |||||
2,112 | 1 | 1.5 | 3, 72 | 5, 34 | |
3,168 | 1 | 1.5 | 3, 72 | 5, 34 | |
2,112 | 1 | 2 | 3, 1 | 5, 17 | |
2,112 | 1 | 3 | 3, 87 | 7, 05 | |
1,408 | 1 | 2 | 3, 1 | 5, 17 | |
2,112 | 1 | 4 | 3, 23 | 9, 38 |
Table 2.
Dependence of the optimal settling time for problems
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 |
Table 3.
The optimal settling time of oscillatory processes in the problems
(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 |
Table 4. The optimal settling time in the presence of three ILS
(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 |
Table 5. The minimal settling time of oscillating processes in the case of four ILS
(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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