Figure 1.
Optimal control $ t\longmapsto u(t) $ for the problem of Example 1
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Solving nonlinear differential equations using hybrid method between Lyapunov's artificial small parameter and continuous particle swarm optimization
December
2021, 11(4): 645-663.
doi: 10.3934/naco.2021002
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 |
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:
Linear-quadratic optimal control,
Cauchy discretization technique,
quadratic optimization problem,
active-set method,
numerical simulation.
Mathematics Subject Classification:
Primary: 93C05; Secondary: 53C35.
Citation:
Mohamed Aliane, Mohand Bentobache, Nacima Moussouni, Philippe Marthon. Direct method to solve linear-quadratic optimal control problems. Numerical Algebra, Control and Optimization,
2021, 11
(4)
: 645-663.
doi: 10.3934/naco.2021002
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References:
| [1] |
J. Awerjcewicz, Modeling, Simulation and Control of Non-Linear Engineering Dynamical Systems, State-of-the-Art, Presperctives and Applications, Springer, Heidelberg, Germany, 2008.
doi: 10.1007/978-3-319-08266-0. |
| [2] |
M. Aliane, N. Moussouni and M. Bentobache,
Optimal control of a rectilinear motion of a rocket, Statistics, Optimization & Information Computing, 8 (2020), 281-295.
doi: 10.19139/soic-2310-5070-741. |
| [3] |
M. Aliane, N. Moussouni and M. Bentobache, Non-linear optimal control of the heel angle of a rocket, The 6th International Conference on Control, Decision and Information Technologies, CODIT'19, Paris, France, (2019), 756–760.
doi: 10.19139/soic-2310-5070-741. |
| [4] |
M. Bentobache, M. Telli and A. Mokhtari, A sequential linear programming algorithm for continuous and mixed-integer non-convex quadratic programming, Optimization of Complex Systems: Theory, Models, Algorithms and Applications, WCGO 2019, Advances in Intelligent Systems and Computing, Springer, Cham, 991 (2020), 26–36. |
| [5] |
M. O. Bibi and M. Bentobache,
A hybrid direction algorithm for solving linear programs, International Journal of Computer Mathematics, 92 (2015), 201-216.
doi: 10.1080/00207160.2014.890188. |
| [6] |
L. D. Duncan, Basic Considerations in the Development of an Unguided Rocket Trajectory Simulation Model, Technical report $N^0$ 5076, Atmospheric Sciences Laboratory, United States Army Electronics Command, 1966. |
| [7] |
A. Chinchuluun, R. Enkhbat and P. M. Pardalos,
A novel approach for non-convex optimal control problems, Optimization, 58 (2009), 781-789.
doi: 10.1080/02331930902943962. |
| [8] |
R. Gabasov, F. M. Kirillova and S. V. Prischepova, Optimal Feedback Control, Springer-Verlag, London, 1995. |
| [9] |
G. R. Rose, Numerical Methods for Solving Optimal Control Problems, Master's Thesis, University of Tennessee, Knoxville, 2015. |
| [10] |
P. Howlett,
The optimal control of a train, Annals of Operations Research, 98 (2000), 65-87.
doi: 10.1023/A:1019235819716. |
| [11] |
N. Khimoum and M. O. Bibi,
Primal-dual method for solving a linear-quadratic multi-input optimal control problem, Optimization Letters, 14 (2019), 653-669.
doi: 10.1007/s11590-018-1375-2. |
| [12] |
K. Louadj, P. Spiteri, F. Demim, M. Aidene, A. Nemra and F. Messine,
Application optimal control for a problem aircraft flight, Engineering Science and Technology Review, 11 (2018), 156-164.
|
| [13] |
N. Moussouni and M. Aidene,
An algorithm for optimization of cereal output, Acta Applicandae Mathematicae, 11 (2011), 113-127.
doi: 10.1007/s10440-011-9664-0. |
| [14] |
N. Moussouni and M. Aidene,
Optimization of cereal output in presence of locusts, An International Journal of Optimization and Control: Theories & Applications, 6 (2016), 1-10.
doi: 10.11121/ijocta.01.2016.00254. |
| [15] |
O. Oukacha, Direct Method for the Optimization of Control Problems, PhD Dissertation, University of Tizi-Ouzou, Algeria, 2016 (in French). |
| [16] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Intersciences Publisher, New York, 1962. |
| [17] |
E. Trélat, Optimal Control: Theory and Applications, Vuibert, Paris, France, 2005 (in French). |
| [18] |
J. Zhu and R. Zeng,
A mathematical formulation for optimal control of air pollution, Science in China, 46 (2003), 994-1002.
|
| [19] |
M. A. Zaitri, M. O. Bibi and M. Bentobache,
A hybrid direction algorithm for solving optimal control problems, Cogent Mathematics & Statistics, 6 (2019), 1-12.
doi: 10.1080/25742558.2019.1612614. |
show all references
References:
| [1] |
J. Awerjcewicz, Modeling, Simulation and Control of Non-Linear Engineering Dynamical Systems, State-of-the-Art, Presperctives and Applications, Springer, Heidelberg, Germany, 2008.
doi: 10.1007/978-3-319-08266-0. |
| [2] |
M. Aliane, N. Moussouni and M. Bentobache,
Optimal control of a rectilinear motion of a rocket, Statistics, Optimization & Information Computing, 8 (2020), 281-295.
doi: 10.19139/soic-2310-5070-741. |
| [3] |
M. Aliane, N. Moussouni and M. Bentobache, Non-linear optimal control of the heel angle of a rocket, The 6th International Conference on Control, Decision and Information Technologies, CODIT'19, Paris, France, (2019), 756–760.
doi: 10.19139/soic-2310-5070-741. |
| [4] |
M. Bentobache, M. Telli and A. Mokhtari, A sequential linear programming algorithm for continuous and mixed-integer non-convex quadratic programming, Optimization of Complex Systems: Theory, Models, Algorithms and Applications, WCGO 2019, Advances in Intelligent Systems and Computing, Springer, Cham, 991 (2020), 26–36. |
| [5] |
M. O. Bibi and M. Bentobache,
A hybrid direction algorithm for solving linear programs, International Journal of Computer Mathematics, 92 (2015), 201-216.
doi: 10.1080/00207160.2014.890188. |
| [6] |
L. D. Duncan, Basic Considerations in the Development of an Unguided Rocket Trajectory Simulation Model, Technical report $N^0$ 5076, Atmospheric Sciences Laboratory, United States Army Electronics Command, 1966. |
| [7] |
A. Chinchuluun, R. Enkhbat and P. M. Pardalos,
A novel approach for non-convex optimal control problems, Optimization, 58 (2009), 781-789.
doi: 10.1080/02331930902943962. |
| [8] |
R. Gabasov, F. M. Kirillova and S. V. Prischepova, Optimal Feedback Control, Springer-Verlag, London, 1995. |
| [9] |
G. R. Rose, Numerical Methods for Solving Optimal Control Problems, Master's Thesis, University of Tennessee, Knoxville, 2015. |
| [10] |
P. Howlett,
The optimal control of a train, Annals of Operations Research, 98 (2000), 65-87.
doi: 10.1023/A:1019235819716. |
| [11] |
N. Khimoum and M. O. Bibi,
Primal-dual method for solving a linear-quadratic multi-input optimal control problem, Optimization Letters, 14 (2019), 653-669.
doi: 10.1007/s11590-018-1375-2. |
| [12] |
K. Louadj, P. Spiteri, F. Demim, M. Aidene, A. Nemra and F. Messine,
Application optimal control for a problem aircraft flight, Engineering Science and Technology Review, 11 (2018), 156-164.
|
| [13] |
N. Moussouni and M. Aidene,
An algorithm for optimization of cereal output, Acta Applicandae Mathematicae, 11 (2011), 113-127.
doi: 10.1007/s10440-011-9664-0. |
| [14] |
N. Moussouni and M. Aidene,
Optimization of cereal output in presence of locusts, An International Journal of Optimization and Control: Theories & Applications, 6 (2016), 1-10.
doi: 10.11121/ijocta.01.2016.00254. |
| [15] |
O. Oukacha, Direct Method for the Optimization of Control Problems, PhD Dissertation, University of Tizi-Ouzou, Algeria, 2016 (in French). |
| [16] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Intersciences Publisher, New York, 1962. |
| [17] |
E. Trélat, Optimal Control: Theory and Applications, Vuibert, Paris, France, 2005 (in French). |
| [18] |
J. Zhu and R. Zeng,
A mathematical formulation for optimal control of air pollution, Science in China, 46 (2003), 994-1002.
|
| [19] |
M. A. Zaitri, M. O. Bibi and M. Bentobache,
A hybrid direction algorithm for solving optimal control problems, Cogent Mathematics & Statistics, 6 (2019), 1-12.
doi: 10.1080/25742558.2019.1612614. |
Figure 2.
CPU time and Error in terms of $ N $ for Example 1
Figure 3.
CPU time and Error in terms of $ N $ for Example 2
Figure 4.
CPU time and Error in terms of $ N $ for Example 3
Figure 5.
CPU time and Error in terms of $ N $ for Example 4
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 | ||
| 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 | ||
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 | ||
| 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 | ||
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 | ||
| 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 | ||
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 | ||
| 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 | ||
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 | |
| 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 | |
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 | ||
| 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 | ||
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 | ||
| 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 | ||
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 | ||
| 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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