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. 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. Google Scholar

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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.Google Scholar

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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. Google Scholar

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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. Google Scholar

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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

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

L. Grüne and J. Pannek, Nonlinear Model Predictive Control, Springer, 2011. doi: 10.1007/978-0-85729-501-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[3]

J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, 2001. Google Scholar

[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. Google Scholar

[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.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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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 200039th IEEE Conference on Decision and Control,, vol. 5, IEEE, Sydney, Australia, 2000, 4969–4971. doi: 10.1109/CDC.2001.914720. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

L. Grüne and J. Pannek, Nonlinear Model Predictive Control, Springer, 2011. doi: 10.1007/978-0-85729-501-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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