September  2015, 35(9): 4385-4414. doi: 10.3934/dcds.2015.35.4385

Robustness of performance and stability for multistep and updated multistep MPC schemes

1. 

University of Bayreuth, Chair of Applied Mathematics, Universitätsstraße 30, 95440 Bayreuth, Germany, Germany

Received  May 2014 Revised  August 2014 Published  April 2015

We consider a model predictive control approach to approximate the solution of infinite horizon optimal control problems for perturbed nonlinear discrete time systems. By reducing the number of re-optimizations, the computational load can be lowered considerably at the expense of reduced robustness of the closed-loop solution against perturbations. In this paper, we propose and analyze an update strategy based on re-optimizations on shrinking horizons which is computationally less expensive than that based on full horizon re-optimization, and at the same time allowing for rigorously quantifiable robust performance estimates.
Citation: Lars Grüne, Vryan Gil Palma. Robustness of performance and stability for multistep and updated multistep MPC schemes. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4385-4414. doi: 10.3934/dcds.2015.35.4385
References:
[1]

D. P. Bertsekas, Dynamic Programming and Optimal Control. Vol. 1 and 2,, Athena Scientific, (1995).   Google Scholar

[2]

H. G. Bock, M. Diehl, E. A. Kostina and J. P. Schlöder, Constrained optimal feedback control of systems governed by large differential algebraic equations,, In L. Biegler, 3 (2007), 3.  doi: 10.1137/1.9780898718935.ch1.  Google Scholar

[3]

C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems,, in M. Grötschel, (2001), 3.   Google Scholar

[4]

L. Grüne, Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems,, SIAM Journal on Control and Optimization, 48 (2009), 1206.  doi: 10.1137/070707853.  Google Scholar

[5]

L. Grüne, Economic receding horizon control without terminal constraints,, Automatica, 49 (2013), 725.  doi: 10.1016/j.automatica.2012.12.003.  Google Scholar

[6]

L. Grüne and V. G. Palma, On the Benefit of Re-optimization in Optimal Control under Perturbations,, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems - MTNS, (2014), 439.   Google Scholar

[7]

L. Grüne and J. Pannek, Practical NMPC suboptimality estimates along trajectories,, Systems & Control Letters, 58 (2009), 161.  doi: 10.1016/j.sysconle.2008.10.012.  Google Scholar

[8]

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

[9]

L. Grüne, J. Pannek, M. Seehafer and K. Worthmann, Analysis of unconstrained nonlinear MPC schemes with varying control horizon,, SIAM Journal on Control and Optimization, 48 (2010), 4938.  doi: 10.1137/090758696.  Google Scholar

[10]

L. Grüne and A. Rantzer, On the infinite horizon performance of receding horizon controllers,, IEEE Trans. Automat. Control, 53 (2008), 2100.  doi: 10.1109/TAC.2008.927799.  Google Scholar

[11]

C. M. Kellett, H. Shim and A. R. Teel, Further results on robustness of (possibly discontinuous) sample and hold feedback,, IEEE Trans. Automat. Control, 49 (2004), 1081.  doi: 10.1109/TAC.2004.831184.  Google Scholar

[12]

H. K. Khalil, Nonlinear Systems,, Prentice Hall PTR, (2002).   Google Scholar

[13]

H. Maurer and H. J. Pesch, Solution Differentiability for Parametric Nonlinear Control Problems with Control-State Constraints,, SIAM Journal on Control and Optimization, 86 (1995), 285.  doi: 10.1007/BF02192081.  Google Scholar

[14]

V. Palma and L. Grüne, Stability, performance and robustness of sensitivity-based multistep feedback NMPC,, Extended Abstract in: Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems - MTNS 2012, (2012).   Google Scholar

[15]

J. Pannek, J. Michael and M. Gerdts, A general framework for nonlinear model predictive control with abstract updates,, arXiv preprint, ().   Google Scholar

[16]

H. J. Pesch, Numerical computation of neighboring optimum feedback control schemes in real-time,, Applied Mathematics and Optimization, 5 (1979), 231.  doi: 10.1007/BF01442556.  Google Scholar

[17]

J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design., Nob Hill Publishing, (2009).   Google Scholar

[18]

E. D. Sontag, Clocks and Insensitivity to Small Measurement Errors,, ESAIM Control Optim. Calc. Var, 4 (1999), 537.  doi: 10.1051/cocv:1999121.  Google Scholar

[19]

V. Zavala and L. Biegler, The advanced-step NMPC controller: Optimality, stability and robustness,, Automatica, 45 (2009), 86.  doi: 10.1016/j.automatica.2008.06.011.  Google Scholar

show all references

References:
[1]

D. P. Bertsekas, Dynamic Programming and Optimal Control. Vol. 1 and 2,, Athena Scientific, (1995).   Google Scholar

[2]

H. G. Bock, M. Diehl, E. A. Kostina and J. P. Schlöder, Constrained optimal feedback control of systems governed by large differential algebraic equations,, In L. Biegler, 3 (2007), 3.  doi: 10.1137/1.9780898718935.ch1.  Google Scholar

[3]

C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems,, in M. Grötschel, (2001), 3.   Google Scholar

[4]

L. Grüne, Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems,, SIAM Journal on Control and Optimization, 48 (2009), 1206.  doi: 10.1137/070707853.  Google Scholar

[5]

L. Grüne, Economic receding horizon control without terminal constraints,, Automatica, 49 (2013), 725.  doi: 10.1016/j.automatica.2012.12.003.  Google Scholar

[6]

L. Grüne and V. G. Palma, On the Benefit of Re-optimization in Optimal Control under Perturbations,, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems - MTNS, (2014), 439.   Google Scholar

[7]

L. Grüne and J. Pannek, Practical NMPC suboptimality estimates along trajectories,, Systems & Control Letters, 58 (2009), 161.  doi: 10.1016/j.sysconle.2008.10.012.  Google Scholar

[8]

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

[9]

L. Grüne, J. Pannek, M. Seehafer and K. Worthmann, Analysis of unconstrained nonlinear MPC schemes with varying control horizon,, SIAM Journal on Control and Optimization, 48 (2010), 4938.  doi: 10.1137/090758696.  Google Scholar

[10]

L. Grüne and A. Rantzer, On the infinite horizon performance of receding horizon controllers,, IEEE Trans. Automat. Control, 53 (2008), 2100.  doi: 10.1109/TAC.2008.927799.  Google Scholar

[11]

C. M. Kellett, H. Shim and A. R. Teel, Further results on robustness of (possibly discontinuous) sample and hold feedback,, IEEE Trans. Automat. Control, 49 (2004), 1081.  doi: 10.1109/TAC.2004.831184.  Google Scholar

[12]

H. K. Khalil, Nonlinear Systems,, Prentice Hall PTR, (2002).   Google Scholar

[13]

H. Maurer and H. J. Pesch, Solution Differentiability for Parametric Nonlinear Control Problems with Control-State Constraints,, SIAM Journal on Control and Optimization, 86 (1995), 285.  doi: 10.1007/BF02192081.  Google Scholar

[14]

V. Palma and L. Grüne, Stability, performance and robustness of sensitivity-based multistep feedback NMPC,, Extended Abstract in: Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems - MTNS 2012, (2012).   Google Scholar

[15]

J. Pannek, J. Michael and M. Gerdts, A general framework for nonlinear model predictive control with abstract updates,, arXiv preprint, ().   Google Scholar

[16]

H. J. Pesch, Numerical computation of neighboring optimum feedback control schemes in real-time,, Applied Mathematics and Optimization, 5 (1979), 231.  doi: 10.1007/BF01442556.  Google Scholar

[17]

J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design., Nob Hill Publishing, (2009).   Google Scholar

[18]

E. D. Sontag, Clocks and Insensitivity to Small Measurement Errors,, ESAIM Control Optim. Calc. Var, 4 (1999), 537.  doi: 10.1051/cocv:1999121.  Google Scholar

[19]

V. Zavala and L. Biegler, The advanced-step NMPC controller: Optimality, stability and robustness,, Automatica, 45 (2009), 86.  doi: 10.1016/j.automatica.2008.06.011.  Google Scholar

[1]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099

[2]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[3]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[4]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[5]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[6]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[7]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[8]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[9]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[10]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179

[11]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[12]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[13]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[14]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051

[15]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032

[16]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026

[17]

Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020053

[18]

Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021011

[19]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[20]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]