Abstract
In this paper, we present a technique for approximate robust dynamic programming that allows to generate feedback controllers with guaranteed stability, even for worst case disturbances. Our approach is closely related to robust variants of the Model Predictive Control (MPC), and is suitable for linearly constrained polytopic systems with piecewise affine cost functions. The approximation method uses polyhedral representations of the cost-to-go function and feasible set, and can considerably reduce the computational burden compared to recently proposed methods for exact dynamic programming for robust MPC [1, 8]. In this paper, we derive novel conditions for guaranteeing closed loop stability that are based on the concept of a “uroborus”. We finish by applying the method to a state constrained tutorial example, a parking car with uncertain mass.
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References
A. Bemporad, F. Borrelli, and M. Morari. Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, vol. 48, no. 9, 1600-1606, 2003.
A. Bemporad and M. Morari. Robust model predictive control: A survey. In A. Garulli, A. Tesi, and A. Vicino, editors, Robustness in Identification and Control, number 245 in Lecture Notes in Control and Information Sciences, 207–226, Springer-Verlag, 1999.
D. P. Bertsekas. Dynamic Programming and Optimal Control, volume 1 and 2. Athena Scientific, Belmont, MA, 1995.
J. Björnberg and M. Diehl. Approximate robust dynamic programming and robustly stable MPC. Automatica, 42(5):777–782, May 2006.
J. Björnberg and M. Diehl. Approximate robust dynamic programming and robustly stable MPC. Technical Report 2004-41, SFB 359, University of Heidelberg, 2004.
J. Björnberg and M. Diehl. The software package RDP for robust dynamic programming. http://www.iwr.uni-heidelberg.de/Moritz.Diehl/RDP/, January 2005.
H. Chen and F. Allgöwer. A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica, 34(10):1205–1218, 1998.
M. Diehl and J. Björnberg. Robust dynamic programming for min-max model predictive control of constrained uncertain systems. IEEE Transactions on Automatic Control, 49(12):2253–2257, December 2004.
R. Findeisen and F. Allgöwer. Robustness properties and output feedback of optimization based sampled-data open-loop feedback. In Proc. of the joint 45th IEEE Conf. Decision Contr., CDC’05/9th European Control Conference, ECC’05, 7756–7761, 2005.
R. Findeisen, L. Imsland, F. Allgöwer, and B.A. Foss. Towards a sampled-data theory for nonlinear model predictive control. In W. Kang, C. Borges, and M. Xiao, editors, New Trends in Nonlinear Dynamics and Control, volume 295 of Lecture Notes in Control and Information Sciences, 295–313, New York, 2003. Springer-Verlag.
L. Grüne and A. Rantzer. On the infinite horizon performance of receding horizon controllers. Technical report, University of Bayreuth, February 2006.
E. C. Kerrigan and J. M. Maciejowski. Feedback min-max model predictive control using a single linear program: Robust stability and the explicit solution. International Journal on Robust and Nonlinear Control, 14:395–413, 2004.
W. Langson, S.V. Rakovic I. Chryssochoos, and D.Q. Mayne. Robust model predictive control using tubes. Automatica, 40(1):125–133, 2004.
J. H. Lee and Z. Yu. Worst-case formulations of model predictive control for systems with bounded parameters. Automatica, 33(5):763–781, 1997.
B. Lincoln and A. Rantzer. Suboptimal dynamic programming with error bounds. In Proceedings of the 41st Conference on Decision and Control, 2002.
D. Q. Mayne. Nonlinear model predictive control: Challenges and opportunities. In F. Allgöwer and A. Zheng, editors, Nonlinear Predictive Control, volume 26 of Progress in Systems Theory, 23–44, Basel Boston Berlin, 2000. Birkhäuser.
D. Q. Mayne. Control of constrained dynamic systems. European Journal of Control, 7:87–99, 2001.
D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: stability and optimality. Automatica, 26(6):789–814, 2000.
S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. Control Engineering Practice, 11:733–764, 2003.
P. O. M. Scokaert and D. Q. Mayne. Min-max feedback model predictive control for constrained linear systems. IEEE Transactions on Automatic Control, 43:1136–1142, 1998.
H. S. Witsenhausen. A minimax control problem for sampled linear systems. IEEE Transactions on Automatic Control, 13(1):5–21, 1968.
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Björnberg, J., Diehl, M. (2008). Approximate Dynamic Programming for Generation of Robustly Stable Feedback Controllers. In: Bock, H.G., Kostina, E., Phu, H.X., Rannacher, R. (eds) Modeling, Simulation and Optimization of Complex Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79409-7_6
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DOI: https://doi.org/10.1007/978-3-540-79409-7_6
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