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).

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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).

[15]

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

[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.

[17]

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

[18]

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

[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.

show all references

References:
[1]

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

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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).

[15]

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

[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.

[17]

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

[18]

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

[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.

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