Abstract
This chapter considers two approaches to explicit min-max NMPC of general constrained nonlinear discrete-time systems in the presence of bounded disturbances and/or parameter uncertainties. The approach in Section 6.2 is based on an open-loop min-max NMPC formulation and constructs a piecewise linear (PWL) approximation of the optimal solution. An explicit open-loop min-max NMPC controller is designed for a continuous stirred tank reactor, whose heat transfer coefficient is an uncertain parameter. The approach in Section 6.3 adopts a closed-loop (also referred to as feedback) min-max NMPC formulation and builds a piecewise nonlinear (PWNL) approximation of the optimal sequence of feedback control policies. The approach is applied to design an explicit feedback min-max NMPC controller for a cart and spring system in the presence of bounded disturbances.
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Keywords
- Control Input
- Uncertain Parameter
- Model Predictive Control
- State Trajectory
- Nonlinear Model Predictive Control
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Non-linear programming: theory and algorithms. Wiley, New York (1993)
Bemporad, A., Borrelli, F., Morari, M.: Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control 48, 1600–1606 (2003)
Bentley, J.L.: Multidimensional binary search trees used for associative searching. Communications of the ACM 18, 509–517 (1975)
Björnberg, J., Diehl, M.: Approximate robust dynamic programming and robustly stable MPC. Automatica 42, 777–782 (2006)
Campo, P.J., Morari, M.: Robust model predictive control. In: Proceedings of the American Control Conference, Minneapolis, Minn., vol. 2, pp. 1021–1026 (1987)
Cychowski, M., O’Mahony, T.: Efficient off-line solutions to robust model predictive control using orthogonal partitioning. In: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005), www.IFAC-PapersOnLine.net
Fiacco, A.V.: Introduction to sensitivity and stability analysis in nonlinear programming. Academic Press, Orlando (1983)
Grancharova, A., Johansen, T.A.: Explicit min-max model predictive control of constrained nonlinear systems with model uncertainty. In: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005), www.IFAC-PapersOnLine.net
Grancharova, A., Johansen, T.A.: Explicit approximate approach to feedback min-max model predictive control of constrained nonlinear systems. In: Proceedings of the IEEE Conference on Decision and Control, San Diego, USA, pp. 4848–4853 (2006)
Grancharova, A., Johansen, T.A., Tøndel, P.: Computational aspects of approximate explicit nonlinear model predictive control. In: Findeisen, R., Allgöwer, F., Biegler, L. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control. LNCIS, vol. 358, pp. 181–192. Springer, Heidelberg (2007)
Grancharova, A., Johansen, T.A.: Computation, approximation and stability of explicit feedback min-max nonlinear model predictive control. Automatica 45, 1134–1143 (2009)
Grossmann, I.E., Halemane, K.P., Swaney, R.E.: Optimization strategies for flexible chemical processes. Computers and Chemical Engineering 7, 439–462 (1983)
Hicks, G., Ray, W.: Approximation methods for optimal control synthesis. The Canadian Journal of Chemical Engineering 49, 522–528 (1971)
Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40, 293–300 (2004)
Kerrigan, E.C., Maciejowski, J.M.: Feedback min-max model predictive control using a single linear program: Robust stability and the explicit solution. International Journal of Robust and Nonlinear Control 14, 395–413 (2004)
Kerrigan, E.C., Mayne, D.Q.: Optimal control of constrained piecewise affine systems with bounded disturbances. In: Proceedings of the IEEE Conference on Decision and Control, Las Vegas, NV, pp. 1552–1557 (2002)
Khalil, H.K.: Nonlinear systems, 3rd edn. Prentice Hall, USA (2002)
Lazar, M., Heemels, W.P.M.H., Bemporad, A., Weiland, S.: Discrete-time non-smooth nonlinear MPC: Stability and robustness. In: Findeisen, R., Allgöwer, F., Biegler, L. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control: Towards New Challenging Applications. LNCIS, vol. 358, pp. 93–103. Springer, Heidelberg (2007)
Limon, D., Alamo, T., Camacho, E.F.: Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties. In: Proceedings of the IEEE Conference on Decision and Control, Las Vegas, NV, pp. 4619–4624 (2002)
Limon, D., Alamo, T., Salas, F., Camacho, E.F.: Input-to-state stability of min-max MPC controllers for nonlinear systems with bounded uncertainties. Automatica 42, 797–803 (2006)
Magni, L., de Nicolao, G., Scattolini, R., Allgöwer, F.: Robust model predictive control for nonlinear discrete-time systems. International Journal of Robust and Nonlinear Control 13, 229–246 (2003)
Magni, L., Scattolini, R.: Robustness and robust design of MPC for nonlinear discrete-time systems. In: Findeisen, R., Allgöwer, F., Biegler, L. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control. LNCIS, vol. 358, pp. 239–254. Springer, Heidelberg (2007)
Mayne, D.Q., Rawlings, J.B., Rao, C.V., Scokaert, P.O.M.: Constrained model predictive control: Stability and optimality. Automatica 36, 789–814 (2000)
Mayne, D.Q., Raković, S.V., Vinter, R.B., Kerrigan, E.C.: Characterization of the solution to a constrained H ∞ optimal control problem. Automatica 42, 371–382 (2006)
Mhaskar, P.: Robust model predictive control design for fault-tolerant control of process systems. Industrial Engineering & Chemistry Research 45, 8565–8574 (2006)
Michalska, H., Mayne, D.Q.: Robust receding horizon control of constrained nonlinear systems. IEEE Transactions on Automatic Control 38, 1623–1633 (1993)
Mukai, M., Azuma, T., Kojima, A., Fujita, M.: Approximate robust receding horizon control for piecewise linear systems via orthogonal partitioning. In: Proceedings of the European Control Conference, Cambridge, U.K. (2003)
Muñoz de la Peña, D., Alamo, T., Ramirez, D.R., Camacho, E.F.: Min-max model predictive control as a quadratic program. IET Control Theory and Applications 1, 328–333 (2007)
Muñoz de la Peña, D., Bemporad, A., Filippi, C.: Robust explicit MPC based on approximate multi-parametric convex programming. IEEE Transactions on Automatic Control 51, 1399–1403 (2006)
Raković, S.V., Kerrigan, E.C., Mayne, D.Q.: Optimal control of constrained piecewise affine systems with state- and input-dependent disturbances. In: Proceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems, Leuven, Belgium (2004)
Wan, Z., Kothare, M.: An efficient off-line formulation of robust model predictive control using linear matrix inequalities. Automatica 39, 837–846 (2003)
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Grancharova, A., Johansen, T.A. (2012). Explicit Min-Max MPC of Constrained Nonlinear Systems with Bounded Uncertainties. In: Explicit Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol 429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28780-0_6
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