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Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics
Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability
Systec–ISR, Faculdade de Engenharia, Universidade do Porto, 4200-465, Porto, Portugal |
This article addresses the problem of controlling a constrained, continuous–time, nonlinear system through Model Predictive Control (MPC). In particular, we focus on methods to efficiently and accurately solve the underlying optimal control problem (OCP). In the numerical solution of a nonlinear OCP, some form of discretization must be used at some stage. There are, however, benefits in postponing the discretization process and maintain a continuous-time model until a later stage. This is because that way we can exploit additional freedom to select the number and the location of the discretization node points.We propose an adaptive time–mesh refinement (AMR) algorithm that iteratively finds an adequate time–mesh satisfying a pre–defined bound on the local error estimate of the obtained trajectories. The algorithm provides a time–dependent stopping criterion, enabling us to impose higher accuracy in the initial parts of the receding horizon, which are more relevant to MPC. Additionally, we analyze the conditions to guarantee closed–loop stability of the MPC framework using the AMR algorithm. The numerical results show that the proposed AMR strategy can obtain solutions as fast as methods using a coarse equidistant–spaced mesh and, on the other hand, as accurate as methods using a fine equidistant–spaced mesh. Therefore, the OCP can be solved, and the MPC law obtained, faster and/or more accurately than with discrete-time MPC schemes using equidistant–spaced meshes.
References:
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ICLOCS2: A MATLAB toolbox for optimization based control, URL http://www.ee.ic.ac.uk/ICLOCS/. Google Scholar |
[2] |
J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings and M. Diehl, CasADi – A software framework for nonlinear optimization and optimal control, Mathematical Programming Computation, (2018), 1–36.
doi: 10.1007/s12532-018-0139-4. |
[3] |
J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, 2001. |
[4] |
J. T. Betts and W. P. Huffman,
Mesh refinement in direct transcription methods for optimal control, Optimal Control Applications and Methods, 19 (1998), 1-21.
doi: 10.1002/(SICI)1099-1514(199801/02)19:1<1::AID-OCA616>3.0.CO;2-Q. |
[5] |
J. Frederic Bonnans, D. Giorgi, V. Grelard, B. Heymann, S. Maindrault, P. Martinon, O. Tissot and J. Liu, Bocop – A Collection of Examples, Technical report, INRIA, 2017, URL http://www.bocop.org. Google Scholar |
[6] |
R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. R. W. Brockett, R. S. Millman and H. S. Sussmann), Birkhouser, Boston, 27 (1983), 181–191. |
[7] |
A. Caldeira and F. Fontes, Model predictive control for path-following of nonholonomic systems, in Proceedings of the 10th Portuguese Conference in Automatic Control (ed. IFAC), 2010, 374–379. Google Scholar |
[8] |
H. Chen and F. Allgöwer,
A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability, Automatica, 34 (1998), 1205-1217.
doi: 10.1016/S0005-1098(98)00073-9. |
[9] |
F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Transactions on Automatic Control, 42 (1997), 1394–1407.
doi: 10.1109/9.633828. |
[10] |
D. M. de la Peña and D. Limón (eds.), IFAC-PapersOnLine | 5th IFAC Conference on Nonlinear Model Predictive Control NMPC 2015 - Seville, Spain, 17–20 September 2015 | ScienceDirect.com, vol. 48, 2015, URL http://www.sciencedirect.com/journal/ifac-papersonline/vol/48/issue/23. Google Scholar |
[11] |
M. Diehl, H. G. Bock, J. P. Schlöder, R. Findeisen, Z. Nagy and F. Allgöwer,
Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations, Journal of Process Control, 12 (2002), 577-585.
doi: 10.1016/S0959-1524(01)00023-3. |
[12] |
D. Dochain, D. Henrion and D. Peaucelle (eds.), IFAC-PapersOnLine | 20th IFAC World Congress | ScienceDirect.com, vol. 50, 2017, URL https://www.sciencedirect.com/journal/ifac-papersonline/vol/50/issue/1. Google Scholar |
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P. Falugi, E. Kerrigan and E. van Wyk, Imperial college london optimal control software: User guide, 2010, URL http://www.ee.ic.ac.uk/ICLOCS/user_guide.pdf, Imperial College London, London, England. Google Scholar |
[14] |
T. Faulwasser and R. Findeisen, Nonlinear model predictive path-following control, in Nonlinear Model Predictive Control (eds. L. Magni, D. M. Raimondo and F. Allgöwer), no. 384 in Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, 2009,335–343. Google Scholar |
[15] |
R. Findeisen and F. Allgöwer, An introduction to nonlinear model predictive control, in Control, 21st Benelux Meeting on Systems and Control, Veidhoven, 2003, 1–23. Google Scholar |
[16] |
F. A. C. C. Fontes, Discontinuous feedback stabilization using nonlinear model predictive controllers, in Proceedings of CDC 2000 – 39th IEEE Conference on Decision and Control,, vol. 5, IEEE, Sydney, Australia, 2000, 4969–4971.
doi: 10.1109/CDC.2001.914720. |
[17] |
F. A. C. C. Fontes,
A general framework to design stabilizing nonlinear model predictive controllers, Systems & Control Letters, 42 (2001), 127-143.
doi: 10.1016/S0167-6911(00)00084-0. |
[18] |
F. A. C. C. Fontes,
Discontinuous feedbacks, discontinuous optimal controls, and continuous-time model predictive control, International Journal of Robust and Nonlinear Control, 13 (2003), 191-209.
doi: 10.1002/rnc.813. |
[19] |
F. A. C. C. Fontes and L. Magni,
Min-max model predictive control of nonlinear systems using discontinuous feedbacks, IEEE Transactions on Automatic Control, 48 (2003), 1750-1755.
doi: 10.1109/TAC.2003.817915. |
[20] |
F. Fontes and L. Magni, A generalization of Barbalat's lemma with applications to robust model predictive control, in Proceedings of Sixteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS2004), Leuven, Belgium July 5-9, vol. 4, 2004. Google Scholar |
[21] |
F. A. C. C. Fontes and H. Frankowska,
Normality and nondegeneracy for optimal control problems with state constraints, Journal of Optimization Theory and Applications, 166 (2015), 115-136.
doi: 10.1007/s10957-015-0704-1. |
[22] |
F. A. C. C. Fontes, L. Magni and E. Gyurkovics, Sampled-Data Model Predictive Control for Nonlinear Time-Varying Systems: Stability and Robustness, in Assessment and Future Directions of Nonlinear Model Predictive Control (eds. D.-I. R. Findeisen, P. D. F. Allgöwer and P. D. L. T. Biegler), no. 358 in Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, 2007,115–129.
doi: 10.1007/978-3-540-72699-9_9. |
[23] |
F. A. C. C. Fontes and F. L. Pereira, Model predictive control of impulsive dynamical systems, in Nonlinear Model Predictive Control, 45 (2012), 305–310.
doi: 10.3182/20120823-5-NL-3013.00086. |
[24] |
F. A. Fontes and L. T. Paiva,
Guaranteed constraint satisfaction in continuous-time control problems, IEEE Control Systems Letters, 3 (2019), 13-18.
doi: 10.1109/LCSYS.2018.2849853. |
[25] |
M. Gerdts, Optimal Control of ODEs and DAEs, De Gruyter, Berlin, Boston, 2012, URL https://www.degruyter.com/view/product/119403.
doi: 10.1515/9783110249996. |
[26] |
L. Grüne and V. G. Palma,
Robustness of performance and stability for multistep and updated multistep MPC schemes, Discrete and Continuous Dynamical Systems, 35 (2015), 4385-4414.
doi: 10.3934/dcds.2015.35.4385. |
[27] |
L. Grüne, D. Nesic and J. Pannek, Model predictive control for nonlinear sampled-data systems, in Assessment and Future Directions of Nonlinear Model Predictive Control (NMPC05) (ed. R. F. e. F. Allgöwer L. Biegler), vol. 358 of Lecture Notes in Control and Information Sciences, Springer Verlag, Heidelberg, 358 (2007), 105–113.
doi: 10.1007/978-3-540-72699-9_8. |
[28] |
L. Grüne and J. Pannek, Nonlinear Model Predictive Control, Springer, 2011.
doi: 10.1007/978-0-85729-501-9. |
[29] |
B. Houska, H. J. Ferreau and M. Diehl,
ACADO toolkit–An open-source framework for automatic control and dynamic optimization, Optimal Control Applications and Methods, 32 (2011), 298-312.
doi: 10.1002/oca.939. |
[30] |
I. Kolmanovsky and N. McClamroch, Developments in nonholonomic control problems, IEEE Control Systems, 15 (1995), 20-36. Google Scholar |
[31] |
M. Lazar, F. Allgower, P. M. Van den Hof and B. Cott (eds.), 4th IFAC Conference on Nonlinear Model Predictive Control, NMPC 12, IFAC Proceedings Volumes (IFAC-PapersOnline), IFAC, Noordwijkerhout; Netherlands, 2012. Google Scholar |
[32] |
L. Magni and R. Scattolini, Model predictive control of continuous-time nonlinear systems with piecewise constant control, IEEE Transactions on Automatic Control, 49 (2004), 900–906.
doi: 10.1109/TAC.2004.829595. |
[33] |
D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert,
Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814.
doi: 10.1016/S0005-1098(99)00214-9. |
[34] |
D. Mayne and H. Michalska,
Receding horizon control of nonlinear systems, IEEE Transactions on Automatic Control, 35 (1990), 814-824.
doi: 10.1109/9.57020. |
[35] |
H. Michalska and D. Mayne,
Robust receding horizon control of constrained nonlinear systems, IEEE Transactions on Automatic Control, 38 (1993), 1623-1633.
doi: 10.1109/9.262032. |
[36] |
L. T. Paiva and F. A. C. C. Fontes, A sufficient condition for stability of sampled–data model predictive control using adaptive time–mesh refinement, in Proceedings of NMPC 2018- 6th IFAC International Conference on Nonlinear Model Predictive Control, Madison, WI, USA, August 2018 (ed. IFAC), 51 (2018), 104–109.
doi: 10.1016/j.ifacol.2018.10.182. |
[37] |
L. T. Paiva and F. A. Fontes, Sampled-data model predictive control using adaptive time-mesh refinement algorithms, in CONTROLO 2016: Proceedings of the 12th Portuguese Conference on Automatic Control, 402 (2016), 143-153.
doi: 10.1007/978-3-319-43671-5_13. |
[38] |
L. T. Paiva and F. A. C. C. Fontes,
Adaptive time-mesh refinement in optimal control problems with state constraints, Discrete and Continuous Dynamical Systems, 35 (2015), 4553-4572.
doi: 10.3934/dcds.2015.35.4553. |
[39] |
G. Pannocchia, J. Rawlings, D. Mayne and G. Mancuso,
Whither Discrete Time Model Predictive Control?, IEEE Transactions on Automatic Control, 60 (2015), 246-252.
doi: 10.1109/TAC.2014.2324131. |
[40] |
M. A. Patterson, W. W. Hager and A. V. Rao,
A ph mesh refinement method for optimal control, Optimal Control Applications and Methods, 36 (2015), 398-421.
doi: 10.1002/oca.2114. |
[41] |
I. Prodan, S. Olaru, F. A. C. C. Fontes, F. L. Pereira, J. B. d. Sousa, C. S. Maniu and S.-I. Niculescu, Predictive control for path-following. from trajectory generation to the parametrization of the discrete tracking sequences, in Developments in Model-Based Optimization and Control, Lecture Notes in Control and Information Sciences, Springer, Cham, 2015,161–181. Google Scholar |
[42] |
I. Prodan, S. Olaru, F. A. Fontes, C. Stoica and S.-I. Niculescu, A predictive control-based algorithm for path following of autonomous aerial vehicles, in Control Applications (CCA), 2013 IEEE International Conference on, IEEE, 2013, 1042–1047.
doi: 10.1109/CCA.2013.6662889. |
[43] |
J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design, Nob Hill Pub., 2009. Google Scholar |
[44] |
A. Rucco, A. P. Aguiar, F. A. Fontes, F. L. Pereira and J. B. de Sousa, A model predictive control-based architecture for cooperative path-following of multiple unmanned aerial vehicles, in Developments in Model-Based Optimization and Control, Springer, 464 (2015), 141–160.
doi: 10.1007/978-3-319-26687-9_7. |
[45] |
J.-J. E. Slotine and W. Li, Applied nonlinear control, Prentice Hall, New York, 1991. Google Scholar |
[46] |
R. B. Vinter, Optimal Control, Birkhauser, Boston, 2000. |
[47] |
A. Wächter and L. T. Biegler,
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.
doi: 10.1007/s10107-004-0559-y. |
[48] |
Y. Zhao and P. Tsiotras,
Density functions for mesh refinement in numerical optimal control, Journal of Guidance, Control, and Dynamics, 34 (2011), 271-277.
doi: 10.2514/1.45852. |
show all references
References:
[1] |
ICLOCS2: A MATLAB toolbox for optimization based control, URL http://www.ee.ic.ac.uk/ICLOCS/. Google Scholar |
[2] |
J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings and M. Diehl, CasADi – A software framework for nonlinear optimization and optimal control, Mathematical Programming Computation, (2018), 1–36.
doi: 10.1007/s12532-018-0139-4. |
[3] |
J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, 2001. |
[4] |
J. T. Betts and W. P. Huffman,
Mesh refinement in direct transcription methods for optimal control, Optimal Control Applications and Methods, 19 (1998), 1-21.
doi: 10.1002/(SICI)1099-1514(199801/02)19:1<1::AID-OCA616>3.0.CO;2-Q. |
[5] |
J. Frederic Bonnans, D. Giorgi, V. Grelard, B. Heymann, S. Maindrault, P. Martinon, O. Tissot and J. Liu, Bocop – A Collection of Examples, Technical report, INRIA, 2017, URL http://www.bocop.org. Google Scholar |
[6] |
R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. R. W. Brockett, R. S. Millman and H. S. Sussmann), Birkhouser, Boston, 27 (1983), 181–191. |
[7] |
A. Caldeira and F. Fontes, Model predictive control for path-following of nonholonomic systems, in Proceedings of the 10th Portuguese Conference in Automatic Control (ed. IFAC), 2010, 374–379. Google Scholar |
[8] |
H. Chen and F. Allgöwer,
A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability, Automatica, 34 (1998), 1205-1217.
doi: 10.1016/S0005-1098(98)00073-9. |
[9] |
F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Transactions on Automatic Control, 42 (1997), 1394–1407.
doi: 10.1109/9.633828. |
[10] |
D. M. de la Peña and D. Limón (eds.), IFAC-PapersOnLine | 5th IFAC Conference on Nonlinear Model Predictive Control NMPC 2015 - Seville, Spain, 17–20 September 2015 | ScienceDirect.com, vol. 48, 2015, URL http://www.sciencedirect.com/journal/ifac-papersonline/vol/48/issue/23. Google Scholar |
[11] |
M. Diehl, H. G. Bock, J. P. Schlöder, R. Findeisen, Z. Nagy and F. Allgöwer,
Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations, Journal of Process Control, 12 (2002), 577-585.
doi: 10.1016/S0959-1524(01)00023-3. |
[12] |
D. Dochain, D. Henrion and D. Peaucelle (eds.), IFAC-PapersOnLine | 20th IFAC World Congress | ScienceDirect.com, vol. 50, 2017, URL https://www.sciencedirect.com/journal/ifac-papersonline/vol/50/issue/1. Google Scholar |
[13] |
P. Falugi, E. Kerrigan and E. van Wyk, Imperial college london optimal control software: User guide, 2010, URL http://www.ee.ic.ac.uk/ICLOCS/user_guide.pdf, Imperial College London, London, England. Google Scholar |
[14] |
T. Faulwasser and R. Findeisen, Nonlinear model predictive path-following control, in Nonlinear Model Predictive Control (eds. L. Magni, D. M. Raimondo and F. Allgöwer), no. 384 in Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, 2009,335–343. Google Scholar |
[15] |
R. Findeisen and F. Allgöwer, An introduction to nonlinear model predictive control, in Control, 21st Benelux Meeting on Systems and Control, Veidhoven, 2003, 1–23. Google Scholar |
[16] |
F. A. C. C. Fontes, Discontinuous feedback stabilization using nonlinear model predictive controllers, in Proceedings of CDC 2000 – 39th IEEE Conference on Decision and Control,, vol. 5, IEEE, Sydney, Australia, 2000, 4969–4971.
doi: 10.1109/CDC.2001.914720. |
[17] |
F. A. C. C. Fontes,
A general framework to design stabilizing nonlinear model predictive controllers, Systems & Control Letters, 42 (2001), 127-143.
doi: 10.1016/S0167-6911(00)00084-0. |
[18] |
F. A. C. C. Fontes,
Discontinuous feedbacks, discontinuous optimal controls, and continuous-time model predictive control, International Journal of Robust and Nonlinear Control, 13 (2003), 191-209.
doi: 10.1002/rnc.813. |
[19] |
F. A. C. C. Fontes and L. Magni,
Min-max model predictive control of nonlinear systems using discontinuous feedbacks, IEEE Transactions on Automatic Control, 48 (2003), 1750-1755.
doi: 10.1109/TAC.2003.817915. |
[20] |
F. Fontes and L. Magni, A generalization of Barbalat's lemma with applications to robust model predictive control, in Proceedings of Sixteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS2004), Leuven, Belgium July 5-9, vol. 4, 2004. Google Scholar |
[21] |
F. A. C. C. Fontes and H. Frankowska,
Normality and nondegeneracy for optimal control problems with state constraints, Journal of Optimization Theory and Applications, 166 (2015), 115-136.
doi: 10.1007/s10957-015-0704-1. |
[22] |
F. A. C. C. Fontes, L. Magni and E. Gyurkovics, Sampled-Data Model Predictive Control for Nonlinear Time-Varying Systems: Stability and Robustness, in Assessment and Future Directions of Nonlinear Model Predictive Control (eds. D.-I. R. Findeisen, P. D. F. Allgöwer and P. D. L. T. Biegler), no. 358 in Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, 2007,115–129.
doi: 10.1007/978-3-540-72699-9_9. |
[23] |
F. A. C. C. Fontes and F. L. Pereira, Model predictive control of impulsive dynamical systems, in Nonlinear Model Predictive Control, 45 (2012), 305–310.
doi: 10.3182/20120823-5-NL-3013.00086. |
[24] |
F. A. Fontes and L. T. Paiva,
Guaranteed constraint satisfaction in continuous-time control problems, IEEE Control Systems Letters, 3 (2019), 13-18.
doi: 10.1109/LCSYS.2018.2849853. |
[25] |
M. Gerdts, Optimal Control of ODEs and DAEs, De Gruyter, Berlin, Boston, 2012, URL https://www.degruyter.com/view/product/119403.
doi: 10.1515/9783110249996. |
[26] |
L. Grüne and V. G. Palma,
Robustness of performance and stability for multistep and updated multistep MPC schemes, Discrete and Continuous Dynamical Systems, 35 (2015), 4385-4414.
doi: 10.3934/dcds.2015.35.4385. |
[27] |
L. Grüne, D. Nesic and J. Pannek, Model predictive control for nonlinear sampled-data systems, in Assessment and Future Directions of Nonlinear Model Predictive Control (NMPC05) (ed. R. F. e. F. Allgöwer L. Biegler), vol. 358 of Lecture Notes in Control and Information Sciences, Springer Verlag, Heidelberg, 358 (2007), 105–113.
doi: 10.1007/978-3-540-72699-9_8. |
[28] |
L. Grüne and J. Pannek, Nonlinear Model Predictive Control, Springer, 2011.
doi: 10.1007/978-0-85729-501-9. |
[29] |
B. Houska, H. J. Ferreau and M. Diehl,
ACADO toolkit–An open-source framework for automatic control and dynamic optimization, Optimal Control Applications and Methods, 32 (2011), 298-312.
doi: 10.1002/oca.939. |
[30] |
I. Kolmanovsky and N. McClamroch, Developments in nonholonomic control problems, IEEE Control Systems, 15 (1995), 20-36. Google Scholar |
[31] |
M. Lazar, F. Allgower, P. M. Van den Hof and B. Cott (eds.), 4th IFAC Conference on Nonlinear Model Predictive Control, NMPC 12, IFAC Proceedings Volumes (IFAC-PapersOnline), IFAC, Noordwijkerhout; Netherlands, 2012. Google Scholar |
[32] |
L. Magni and R. Scattolini, Model predictive control of continuous-time nonlinear systems with piecewise constant control, IEEE Transactions on Automatic Control, 49 (2004), 900–906.
doi: 10.1109/TAC.2004.829595. |
[33] |
D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert,
Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814.
doi: 10.1016/S0005-1098(99)00214-9. |
[34] |
D. Mayne and H. Michalska,
Receding horizon control of nonlinear systems, IEEE Transactions on Automatic Control, 35 (1990), 814-824.
doi: 10.1109/9.57020. |
[35] |
H. Michalska and D. Mayne,
Robust receding horizon control of constrained nonlinear systems, IEEE Transactions on Automatic Control, 38 (1993), 1623-1633.
doi: 10.1109/9.262032. |
[36] |
L. T. Paiva and F. A. C. C. Fontes, A sufficient condition for stability of sampled–data model predictive control using adaptive time–mesh refinement, in Proceedings of NMPC 2018- 6th IFAC International Conference on Nonlinear Model Predictive Control, Madison, WI, USA, August 2018 (ed. IFAC), 51 (2018), 104–109.
doi: 10.1016/j.ifacol.2018.10.182. |
[37] |
L. T. Paiva and F. A. Fontes, Sampled-data model predictive control using adaptive time-mesh refinement algorithms, in CONTROLO 2016: Proceedings of the 12th Portuguese Conference on Automatic Control, 402 (2016), 143-153.
doi: 10.1007/978-3-319-43671-5_13. |
[38] |
L. T. Paiva and F. A. C. C. Fontes,
Adaptive time-mesh refinement in optimal control problems with state constraints, Discrete and Continuous Dynamical Systems, 35 (2015), 4553-4572.
doi: 10.3934/dcds.2015.35.4553. |
[39] |
G. Pannocchia, J. Rawlings, D. Mayne and G. Mancuso,
Whither Discrete Time Model Predictive Control?, IEEE Transactions on Automatic Control, 60 (2015), 246-252.
doi: 10.1109/TAC.2014.2324131. |
[40] |
M. A. Patterson, W. W. Hager and A. V. Rao,
A ph mesh refinement method for optimal control, Optimal Control Applications and Methods, 36 (2015), 398-421.
doi: 10.1002/oca.2114. |
[41] |
I. Prodan, S. Olaru, F. A. C. C. Fontes, F. L. Pereira, J. B. d. Sousa, C. S. Maniu and S.-I. Niculescu, Predictive control for path-following. from trajectory generation to the parametrization of the discrete tracking sequences, in Developments in Model-Based Optimization and Control, Lecture Notes in Control and Information Sciences, Springer, Cham, 2015,161–181. Google Scholar |
[42] |
I. Prodan, S. Olaru, F. A. Fontes, C. Stoica and S.-I. Niculescu, A predictive control-based algorithm for path following of autonomous aerial vehicles, in Control Applications (CCA), 2013 IEEE International Conference on, IEEE, 2013, 1042–1047.
doi: 10.1109/CCA.2013.6662889. |
[43] |
J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design, Nob Hill Pub., 2009. Google Scholar |
[44] |
A. Rucco, A. P. Aguiar, F. A. Fontes, F. L. Pereira and J. B. de Sousa, A model predictive control-based architecture for cooperative path-following of multiple unmanned aerial vehicles, in Developments in Model-Based Optimization and Control, Springer, 464 (2015), 141–160.
doi: 10.1007/978-3-319-26687-9_7. |
[45] |
J.-J. E. Slotine and W. Li, Applied nonlinear control, Prentice Hall, New York, 1991. Google Scholar |
[46] |
R. B. Vinter, Optimal Control, Birkhauser, Boston, 2000. |
[47] |
A. Wächter and L. T. Biegler,
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.
doi: 10.1007/s10107-004-0559-y. |
[48] |
Y. Zhao and P. Tsiotras,
Density functions for mesh refinement in numerical optimal control, Journal of Guidance, Control, and Dynamics, 34 (2011), 271-277.
doi: 10.2514/1.45852. |













CPU time (s) | ||||||
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82 | 42 | |||||
82 | 84 | |||||
541 | 403 |
CPU time (s) | ||||||
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82 | 42 | |||||
82 | 84 | |||||
541 | 403 |
MPC Iter | AMR Iter | CPU time (s) | |||||
Solver | |||||||
21 | 0.5 | ||||||
1 | 21 | 0.5 | |||||
52 | 0.0625 | ||||||
52 | 0.0625 | ||||||
2 | 31 | 0.0625 | |||||
3 | 21 | 0.5 | |||||
4 | 21 | 0.5 | |||||
5 | 21 | 0.5 |
MPC Iter | AMR Iter | CPU time (s) | |||||
Solver | |||||||
21 | 0.5 | ||||||
1 | 21 | 0.5 | |||||
52 | 0.0625 | ||||||
52 | 0.0625 | ||||||
2 | 31 | 0.0625 | |||||
3 | 21 | 0.5 | |||||
4 | 21 | 0.5 | |||||
5 | 21 | 0.5 |
[1] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[2] |
Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021076 |
[3] |
Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373 |
[4] |
Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021013 |
[5] |
Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021021 |
[6] |
Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249 |
[7] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[8] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[9] |
Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021014 |
[10] |
Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021066 |
[11] |
Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $ H_{\infty} $ control design for active suspension systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021036 |
[12] |
Peng Zhang, Yongquan Zeng, Guotai Chi. Time-consistent multiperiod mean semivariance portfolio selection with the real constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1663-1680. doi: 10.3934/jimo.2020039 |
[13] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
[14] |
Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248 |
[15] |
Elimhan N. Mahmudov. Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021010 |
[16] |
Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021012 |
[17] |
Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021013 |
[18] |
Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1 |
[19] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[20] |
Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021007 |
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