May  2019, 24(5): 2335-2364. doi: 10.3934/dcdsb.2019098

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

Received  January 2018 Revised  January 2019 Published  March 2019

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.

Citation: Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098
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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.

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J. Frederic Bonnans, D. Giorgi, V. Grelard, B. Heymann, S. Maindrault, P. Martinon, O. Tissot and J. Liu, BocopA Collection of Examples, Technical report, INRIA, 2017, URL http://www.bocop.org.

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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.

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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.

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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.

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F. A. C. C. Fontes, Discontinuous feedback stabilization using nonlinear model predictive controllers, in Proceedings of CDC 200039th IEEE Conference on Decision and Control,, vol. 5, IEEE, Sydney, Australia, 2000, 4969–4971. doi: 10.1109/CDC.2001.914720.

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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.

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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.

[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. HouskaH. 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.

[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.

[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. MayneJ. B. RawlingsC. 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. PannocchiaJ. RawlingsD. 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. PattersonW. 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.

[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.

[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.

[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/.

[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, BocopA Collection of Examples, Technical report, INRIA, 2017, URL http://www.bocop.org.

[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.

[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, 1720 September 2015 | ScienceDirect.com, vol. 48, 2015, URL http://www.sciencedirect.com/journal/ifac-papersonline/vol/48/issue/23.

[11]

M. DiehlH. G. BockJ. P. SchlöderR. FindeisenZ. 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.

[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.

[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.

[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.

[16]

F. A. C. C. Fontes, Discontinuous feedback stabilization using nonlinear model predictive controllers, in Proceedings of CDC 200039th 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.

[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. HouskaH. 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.

[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.

[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. MayneJ. B. RawlingsC. 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. PannocchiaJ. RawlingsD. 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. PattersonW. 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.

[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.

[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.

[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.

Figure 1.  Illustration of the multi–level adaptive time–mesh refinement strategy
Figure 2.  Illustration of the extended (time–dependent) time–mesh refinement strategy with different refinement thresholds
Figure 3.  Illustration of the extended time–mesh refinement algorithm for MPC
Figure 4.  Construction of the (extended) admissible control $ {\bf{\tilde u}} $ with $ \Pi = \{t_k\}_{k \in \mathbb{N}} $, $ t_k = k \delta $, and with $ \pi_r = \{s_i\}_{i \in 0, 1, \ldots N_r} $, $ s_i = i \delta/2 $
Figure 5.  Car–like system geometry
Figure 6.  Pathwise state constraints (13) for (PCP)
Figure 7.  Optimal path computed in the initial coarse mesh
Figure 8.  Discretization error estimate in the initial coarse mesh
Figure 9.  Optimal path computed in the final mesh $ \pi_{\rm{AMR}} $
Figure 10.  Optimal trajectory and control
Figure 11.  Discretization error in the coarse mesh and the MPC refining levels
Figure 12.  Path resulting from the AMR–MPC scheme
Figure 13.  Trajectory and control resulting from the AMR–MPC scheme
Table 1.  Results for problem (PCP) solved in each time-mesh
$ \pi_j $ $ N_j $ $ \Delta t_j $ $ I_j $ $ \left|\left|\varepsilon_{\bf{x}}^{(j)}\right|\right|_\infty $ CPU time (s)
Solver $ \varepsilon_{\bf{x}} $
$ \pi_0 $ 21 $ 0.5 $ 42 $ 1.0016{\rm{E}}^{-4} $ $ 0.9816 $ $ 0.0563 $
$ \pi_1 $ 82 $ 1/54 $ 42 $ 3.3801{\rm{E}}^{-7} $ $ 0.7061 $ $ 0.0642 $
$ \pi_{\rm{AMR}} $ 82 $ 1/54 $ 84 $ 3.3801{\rm{E}}^{-7} $ $ 1.6877 $ $ 0.1205 $
$ \pi_{\rm{F}} $ 541 $ 1/54 $ 403 $ 4.0358{\rm{E}}^{-7} $ $ 13.2473 $ $ 0.4675 $
$ \pi_j $ $ N_j $ $ \Delta t_j $ $ I_j $ $ \left|\left|\varepsilon_{\bf{x}}^{(j)}\right|\right|_\infty $ CPU time (s)
Solver $ \varepsilon_{\bf{x}} $
$ \pi_0 $ 21 $ 0.5 $ 42 $ 1.0016{\rm{E}}^{-4} $ $ 0.9816 $ $ 0.0563 $
$ \pi_1 $ 82 $ 1/54 $ 42 $ 3.3801{\rm{E}}^{-7} $ $ 0.7061 $ $ 0.0642 $
$ \pi_{\rm{AMR}} $ 82 $ 1/54 $ 84 $ 3.3801{\rm{E}}^{-7} $ $ 1.6877 $ $ 0.1205 $
$ \pi_{\rm{F}} $ 541 $ 1/54 $ 403 $ 4.0358{\rm{E}}^{-7} $ $ 13.2473 $ $ 0.4675 $
Table 2.  Results for each MPC and AMR iterations
MPC Iter AMR Iter $ N_j $ $ \Delta t_j $ $ I_j $ $ \left|\left|\varepsilon_{\bf{x}}^{(j)}\right|\right|_\infty $ CPU time (s)
Solver $ \varepsilon_{\bf{x}} $
$ \pi_{0} $ 21 0.5 $ 42 $ $ 1.002{\rm{E}}^{-4} $ $ 0.982 $ $ 0.0563 $
1 $ \pi_{1} $ 21 0.5 $ 8 $ $ 1.002{\rm{E}}^{-4} $ $ 0.105 $ $ 0.0156 $
$ \pi_{2} $ 52 0.0625 $ 22 $ $ 3.525{\rm{E}}^{-6} $ $ 0.344 $ $ 0.0374 $
$ \pi_{\rm{AMR}} $ 52 0.0625 $ 30 $ $ 3.525{\rm{E}}^{-6} $ $ 0.449 $ $ 0.0530 $
2 $ \pi_{1}=\pi_{\rm{AMR}} $ 31 0.0625 $ 11 $ $ 3.525{\rm{E}}^{-6} $ $ 0.1564 $ $ 0.0230 $
3 $ \pi_{1}=\pi_{\rm{AMR}} $ 21 0.5 $ 11 $ $ 2.042{\rm{E}}^{-7} $ $ 0.1639 $ $ 0.0139 $
4 $ \pi_{1}=\pi_{\rm{AMR}} $ 21 0.5 $ 7 $ $ 4.321{\rm{E}}^{-7} $ $ 0.0936 $ $ 0.0126 $
5 $ \pi_{1}=\pi_{\rm{AMR}} $ 21 0.5 $ 7 $ $ 4.515{\rm{E}}^{-7} $ $ 0.0912 $ $ 0.0123 $
MPC Iter AMR Iter $ N_j $ $ \Delta t_j $ $ I_j $ $ \left|\left|\varepsilon_{\bf{x}}^{(j)}\right|\right|_\infty $ CPU time (s)
Solver $ \varepsilon_{\bf{x}} $
$ \pi_{0} $ 21 0.5 $ 42 $ $ 1.002{\rm{E}}^{-4} $ $ 0.982 $ $ 0.0563 $
1 $ \pi_{1} $ 21 0.5 $ 8 $ $ 1.002{\rm{E}}^{-4} $ $ 0.105 $ $ 0.0156 $
$ \pi_{2} $ 52 0.0625 $ 22 $ $ 3.525{\rm{E}}^{-6} $ $ 0.344 $ $ 0.0374 $
$ \pi_{\rm{AMR}} $ 52 0.0625 $ 30 $ $ 3.525{\rm{E}}^{-6} $ $ 0.449 $ $ 0.0530 $
2 $ \pi_{1}=\pi_{\rm{AMR}} $ 31 0.0625 $ 11 $ $ 3.525{\rm{E}}^{-6} $ $ 0.1564 $ $ 0.0230 $
3 $ \pi_{1}=\pi_{\rm{AMR}} $ 21 0.5 $ 11 $ $ 2.042{\rm{E}}^{-7} $ $ 0.1639 $ $ 0.0139 $
4 $ \pi_{1}=\pi_{\rm{AMR}} $ 21 0.5 $ 7 $ $ 4.321{\rm{E}}^{-7} $ $ 0.0936 $ $ 0.0126 $
5 $ \pi_{1}=\pi_{\rm{AMR}} $ 21 0.5 $ 7 $ $ 4.515{\rm{E}}^{-7} $ $ 0.0912 $ $ 0.0123 $
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