# American Institute of Mathematical Sciences

## Multiobjective model predictive control for stabilizing cost criteria

 Chair of Applied Mathematics, Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany

Received  April 2018 Revised  September 2018 Published  January 2019

Fund Project: The authors are supported by DFG Grant Gr 1569/13-1

In this paper we demonstrate how multiobjective optimal control problems can be solved by means of model predictive control. For our analysis we restrict ourselves to finite-dimensional control systems in discrete time. We show that convergence of the MPC closed-loop trajectory as well as upper bounds on the closed-loop performance for all objectives can be established if the ‘right’ Pareto-optimal control sequence is chosen in the iterations. It turns out that approximating the whole Pareto front is not necessary for that choice. Moreover, we provide statements on the relation of the MPC performance to the values of Pareto-optimal solutions on the infinite horizon, i.e. we investigate on the inifinite-horizon optimality of our MPC controller.

Citation: Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018336
##### References:
 [1] A. Bemporad and D. Muñoz de la Peña, Multiobjective model predictive control, Automatica, 45 (2009), 2823-2830. doi: 10.1016/j.automatica.2009.09.032. [2] D. P. Bertsekas, Dynamic Programming and Optimal Control, vol. 1, 2nd edition, Athena Scientific, 2000. [3] J. Doležal, Existence of optimal solutions in general discrete systems, Kybernetika, 11 (1975), 301-312. [4] M. Ehrgott, Multicriteria Optimization, 2nd edition, Springer, 2005. [5] J. J. V. García, V. G. Garay, E. I. Gordo, F. A. Fano and M. L. Sukia, Intelligent multi-objective nonlinear model predictive control (imo-nmpc): Towards the "on-line" optimization of highly complex control problems, Expert systems with applications, 39 (2012), 6527-6540. [6] P. Giselsson and A. Rantzer, Distributed Model Predictive Control with Suboptimality and Stability Guarantees, in 49th IEEE Conference on Decision and Control (CDC), IEEE, 2010, 7272-7277. [7] L. Grüne and J. Pannek, Nonlinear Model Predictive Control: Theory and Algorithms, 2nd edition, Communications and Control Engineering, Springer, 2017. doi: 10.1007/978-3-319-46024-6. [8] L. Grüne and A. Rantzer, On the infinite horizon performance of receding horizon controllers, IEEE Transactions on Automatic Control, 53 (2008), 2100-2111. doi: 10.1109/TAC.2008.927799. [9] L. Grüne and M. Stieler, Performance guarantees for multiobjective Model Predictive Control, in Proceedings of the IEEE 56th Annual Conference on Decision and Control (CDC) Held in Melbourne, Australia, 2017, Melbourne, Australia, 2017, 5545-5550. [10] C. M. Hackl, F. Larcher, A. Dötlinger and R. M. Kennel, Is multiple-objective model-predictive control "optimal"?, in 2013 IEEE International Symposium on Sensorless Control for Electrical Drives and Predictive Control of Electrical Drives and Power Electronics (SLED/PRECEDE), 2013. [11] A. Hajiloo, W. Xie and X. Ren, Multi-objective robust model predictive control using game theory, in Proceedings of the 2015 IEEE International Conference on Information and Automation, IEEE, 2015, 2026-2030. [12] N. Hayek, Infinite horizon multiobjective optimal control problems in the discrete time case, Optimization, 60 (2011), 509-529. doi: 10.1080/02331930903480352. [13] D. He, L. Wang and J. Sun, On stability of multiobjective NMPC with objective prioritization, Automatica, 57 (2015), 189-198. doi: 10.1016/j.automatica.2015.04.024. [14] C. M. Kellett, A compendium of comparison function results, Mathematics of Control, Signals, and Systems, 26 (2014), 339-374. doi: 10.1007/s00498-014-0128-8. [15] K. Laabidi, F. Bouani and M. Ksouri, Multi-criteria optimization in nonlinear predictive control, Mathematics and Computers in Simulation, 76 (2008), 363-374. doi: 10.1016/j.matcom.2007.04.002. [16] J. Lee and D. Angeli, Cooperative distributed model predictive control for linear plants subject to convex economic objectives, in Proceeding of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011, 3434-3439. [17] F. Logist, B. Houska, M. Diehl and J. F. Van Impe, Robust multi-objective optimal control of uncertain (bio)chemical processes, Chemical Engineering Science, 66 (2011), 4670-4682. [18] 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. [19] M. A. Müller, M. Reble and F. Allgöwer, Cooperative control of dynamically decoupled systems via distributed model predictive control, International Journal of Robust and Nonlinear Control, 22 (2012), 1376-1397. doi: 10.1002/rnc.2826. [20] J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design, Nob Hill Publishing, 2009. [21] Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Elsevier, 1985. [22] B. T. Stewart, A. N. Venkat, J. B. Rawlings, S. J. Wright and G. Pannocchia, Cooperative distributed model predictive control, Control Letters, 59 (2010), 460-469. doi: 10.1016/j.sysconle.2010.06.005. [23] S. E. Tuna, M. J. Messina and A. R. Teel, Shorter horizons for model predictive control, in Proceedings of the 2006 American Control Conference, IEEE, Minneapolis, Minnesota, USA, 2006,863-868. [24] V. M. Zavala and A. Flores-Tlacuahuac, Stability of multiobjective predictive control: A utopia-tracking approach, Automatica, 48 (2012), 2627-2632. doi: 10.1016/j.automatica.2012.06.066.

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##### References:
 [1] A. Bemporad and D. Muñoz de la Peña, Multiobjective model predictive control, Automatica, 45 (2009), 2823-2830. doi: 10.1016/j.automatica.2009.09.032. [2] D. P. Bertsekas, Dynamic Programming and Optimal Control, vol. 1, 2nd edition, Athena Scientific, 2000. [3] J. Doležal, Existence of optimal solutions in general discrete systems, Kybernetika, 11 (1975), 301-312. [4] M. Ehrgott, Multicriteria Optimization, 2nd edition, Springer, 2005. [5] J. J. V. García, V. G. Garay, E. I. Gordo, F. A. Fano and M. L. Sukia, Intelligent multi-objective nonlinear model predictive control (imo-nmpc): Towards the "on-line" optimization of highly complex control problems, Expert systems with applications, 39 (2012), 6527-6540. [6] P. Giselsson and A. Rantzer, Distributed Model Predictive Control with Suboptimality and Stability Guarantees, in 49th IEEE Conference on Decision and Control (CDC), IEEE, 2010, 7272-7277. [7] L. Grüne and J. Pannek, Nonlinear Model Predictive Control: Theory and Algorithms, 2nd edition, Communications and Control Engineering, Springer, 2017. doi: 10.1007/978-3-319-46024-6. [8] L. Grüne and A. Rantzer, On the infinite horizon performance of receding horizon controllers, IEEE Transactions on Automatic Control, 53 (2008), 2100-2111. doi: 10.1109/TAC.2008.927799. [9] L. Grüne and M. Stieler, Performance guarantees for multiobjective Model Predictive Control, in Proceedings of the IEEE 56th Annual Conference on Decision and Control (CDC) Held in Melbourne, Australia, 2017, Melbourne, Australia, 2017, 5545-5550. [10] C. M. Hackl, F. Larcher, A. Dötlinger and R. M. Kennel, Is multiple-objective model-predictive control "optimal"?, in 2013 IEEE International Symposium on Sensorless Control for Electrical Drives and Predictive Control of Electrical Drives and Power Electronics (SLED/PRECEDE), 2013. [11] A. Hajiloo, W. Xie and X. Ren, Multi-objective robust model predictive control using game theory, in Proceedings of the 2015 IEEE International Conference on Information and Automation, IEEE, 2015, 2026-2030. [12] N. Hayek, Infinite horizon multiobjective optimal control problems in the discrete time case, Optimization, 60 (2011), 509-529. doi: 10.1080/02331930903480352. [13] D. He, L. Wang and J. Sun, On stability of multiobjective NMPC with objective prioritization, Automatica, 57 (2015), 189-198. doi: 10.1016/j.automatica.2015.04.024. [14] C. M. Kellett, A compendium of comparison function results, Mathematics of Control, Signals, and Systems, 26 (2014), 339-374. doi: 10.1007/s00498-014-0128-8. [15] K. Laabidi, F. Bouani and M. Ksouri, Multi-criteria optimization in nonlinear predictive control, Mathematics and Computers in Simulation, 76 (2008), 363-374. doi: 10.1016/j.matcom.2007.04.002. [16] J. Lee and D. Angeli, Cooperative distributed model predictive control for linear plants subject to convex economic objectives, in Proceeding of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011, 3434-3439. [17] F. Logist, B. Houska, M. Diehl and J. F. Van Impe, Robust multi-objective optimal control of uncertain (bio)chemical processes, Chemical Engineering Science, 66 (2011), 4670-4682. [18] 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. [19] M. A. Müller, M. Reble and F. Allgöwer, Cooperative control of dynamically decoupled systems via distributed model predictive control, International Journal of Robust and Nonlinear Control, 22 (2012), 1376-1397. doi: 10.1002/rnc.2826. [20] J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design, Nob Hill Publishing, 2009. [21] Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Elsevier, 1985. [22] B. T. Stewart, A. N. Venkat, J. B. Rawlings, S. J. Wright and G. Pannocchia, Cooperative distributed model predictive control, Control Letters, 59 (2010), 460-469. doi: 10.1016/j.sysconle.2010.06.005. [23] S. E. Tuna, M. J. Messina and A. R. Teel, Shorter horizons for model predictive control, in Proceedings of the 2006 American Control Conference, IEEE, Minneapolis, Minnesota, USA, 2006,863-868. [24] V. M. Zavala and A. Flores-Tlacuahuac, Stability of multiobjective predictive control: A utopia-tracking approach, Automatica, 48 (2012), 2627-2632. doi: 10.1016/j.automatica.2012.06.066.
Schematic illustration of a Pareto front for two objectives.
Two bicriterion optimization problems with ${\mathbb{R}}^2_{\geq 0}$-compact set of admissible values. The red parts indicate the nodominated values.
Step (1) in Algorithm 2.
Accumulated performance of the six objectives (blue) compared to the value of the Pareto optimal control sequence ${\bf{u}}^{\star, N}_{x_0}$ from step (0), Algorithm 2 (red).
Trajectories of the six systems (phase plots).
Performance without the constraints in step (1), Algorithm 2.
Trajectories and accumulated performance without terminal constraints using Algorithm 3.
Trajectories and accumulated performance without terminal constraints using Algorithm 4.
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