Abstract
This paper provides a review of the types of optimization problems that arise when implementing model predictive control, a feedback control method based on the on-line solution of open-loop optimal control problems. For linear, unconstrained processes, the issues are fully resolved; the controller is linear and the optimization reduces to a least squares problem that can be solved off line. For linear, constrained processes, the controller is nonlinear and the optimization is a convex quadratic program that must be solved on line. Maintaining feasibility of the optimization problem in the presence of the constraints is the remaining open issue. For nonlinear processes, the controller is nonlinear and the optimization is a nonlinear and non-convex problem. Because on-line solution of non-convex problems is difficult, alternatives to this optimization problem are presented. The available theory to establish robustness of the nonlinear controller to perturbations is summarized. Throughout the paper, we highlight the interplay between optimization theory and model predictive control theory.
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References
D. Chimielewski and V. Manousiouthakis. On constrained infinite-time linear quadratic optimal control. Technical Report ENG 95–150, Department of Chemical Engineering, UCLA, Los Angeles, CA, 1995.
S.L. De Oliveira, V. Nevistic, and M. Morari. Control of nonlinear systems subject to input constraints. In IFAC Symposium on Nonlinear Control System Design, Tahoe City, California, pages 15–20, 1995.
E.G. Gilbert. Linear control systems with pointwise-in-time constraints: What do we do about them. In Proceedings of the 1992 American Control Conference, page 2565, Chicago, Illinois, June 1992. American Automatic Control Council.
E.G. Gilbert and K.T. Tan. Linear systems with state and control constraints: The theory and application of maximal output admissible sets. IEEE Trans. Auto. Cont., 36 (9): 1008–1020, September 1991.
W.W.Hager. Lipschitz continuity for constrained processes. SIAM J. Cont. Opt., 17 (3): 321–338, 1979.
R.E. Kalman. A new approach to linear filtering and prediction problems. Trans. ASME, J. Basic Engineering, pages 35–45, March 1960.
S.S. Keerthi and E.G. Gilbert. Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations. J. Optim. Theory Appi., 57 (2): 265–293, May 1988.
D.L. Kleinman. An easy way to stabilize a linear constant system. IEEE Trans. Auto. Cont., 15 (12): 692, December 1970.
M.J. Kurtz and M. Henson. Input-output linearizing control of constrained nonlinear systems. Accepted for publication in J. Process Control, 1996.
W.H. Kwon and A. E. Pearson. On feedback stabilization of time-varying discrete linear systems. IEEE Trans. Auto. Cont., 23 (3): 479–481, June 1978.
J.P. Lasalle. The stability of dynamical systems. In Regional Conference Series in Applied Mathematics #25. SIAM, 1976.
E.B. Lee and L. Markus. Foundations of Optimal Control Theory. John Wiley and Sons, New York, 1967.
D.Q. Mayne. Optimization in model based control. In Proceedings of the IFAC Symposium Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes, pages 229–242, Helsingor, Denmark, June 1995.
E.S. Meadows, M.A. Henson, J.W. Eaton, and J.B. Rawlings. Receding horizon control and discontinuous state feedback stabilization. Int. J. Control, 62 (5): 1217–1229, 1995.
E.S. Meadows, K.R. Muske, and J.B. Rawlings. Implementable model predictive control in the state space. In Proceedings of the 1995 American Control Conference, pages 3699–3703, 1995.
E.S. Meadows and J.B. Rawlings. Model predictive control. In M.A. Henson and D.E. Seborg, editors, Nonlinear Process Control. Prentice Hall, 1996.
H. Michalska and D.Q. Mayne. Robust receding horizon control of constrained nonlinear systems. IEEE Trans. Auto. Cont., 38 (11): 1623–1633, 1993.
M. Morari. Some control problems in the process industries. In H.L. Trentelman and J.C. Willems, editors, Essays on Control: Perspectives in the Theory and its Applications. Birkhäuser, 1993.
E. Polak. On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems. Math. Prog., 62: 385–414, 1993.
A.I. Propoi. Use of linear programming methods for synthesizing sampled-data automatic systems. Automn. Remote Control, 24 (7): 837–844, July 1963.
J.B. Rawlings and K.R. Muske. Stability of constrained receding horizon control. IEEE Trans. Auto. Cont., 38 (10): 1512–1516, October 1993.
P.O. Scokaert and J.B. Rawlings. On infeasibilities in model predictive control. In J.C. Kantor, editor, Chemical Process Control-CPCV. CACHE, 1996.
P.O. Scokaert and J.B. Rawlings. Constrained linear quadratic regulation. Submitted for publication in IEEE Trans. Auto. Cont., December 1995.
P.O. Scokaert and J.B. Rawlings. Feasibility issues in model predictive control. Submitted for publication in IEEE Trans. Auto. Cont., January 1996.
P.O. Scokaert, J.B. Rawlings, and E.S. Meadows. Discrete-time stability with perturbations: Application to model predictive control. Accepted for publication in Automatica, 1996.
M. Sznaier and M.J. Damborg. Suboptimal control of linear systems with state and control inequality constraints. In Proceedings of the 26th Conference on Decision and Control, pages 761–762, 1987.
S.J. Wright. Applying new optimization algorithms to model predictive control. Fifth International Conference on Chemical Process Control, Lake Tahoe, California, January 1996.
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Scokaert, P.O.M., Rawlings, J.B. (1997). Optimization Problems in Model Predictive Control. In: Biegler, L.T., Coleman, T.F., Conn, A.R., Santosa, F.N. (eds) Large-Scale Optimization with Applications. The IMA Volumes in Mathematics and its Applications, vol 93. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1960-6_8
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DOI: https://doi.org/10.1007/978-1-4612-1960-6_8
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