Abstract
There is a resurgence of interest in the control of dynamic systems with hard constraints on states and controls and many significant advances have been made. A major reason for the success of model predictive control, which, with over 2000 applications, is the most widely used modern control technique, is precisely its ability to handle effectively hard constraints. But there are also other important developments. The solution of the constrained linear quadratic regulator problem has been characterized permitting, at least in principle, explicit determination of the value function and the optimal state feedback controller. Maximal output admissible sets have been effectively harnessed to provide easily implementable regulation and control of constrained linear systems. The solution of the robust, constrained time-optimal control problem has also been characterized. A common feature of all these advances is their reliance on polytopic techniques. Knowledge of robust controllability sets is required in model predictive control of constrained dynamic systems; these sets are polytopes when the system being controlled is linear and the constraints polytopic. In other problems, such as robust time optimal control and (unconstrained) $1 optimal control, the value function itself is polytopic. Partitioning of the state space into polytopes is required for the characterization of the solution of the constrained linear quadratic regulator problem for which the value function is piecewise quadratic and the optimal control piecewise affine. It is possible that polytopic computation may become as useful a tool for the
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Frank Allgöwer, Thomas A. Badgwell, Joe S. Qin, James B. Rawlings, and Stephen J. Wright. Nonlinear predictive control and mov-ing horizon estimation - an introductory overview. In Paul M. Frank, editor, Advances in Control: highlights of ECC’99, pages 391–449, London, 1999. Springer.
A. Bemporad, F. Borelli, and M. Morari. The explicit solution of constrained LP-based receding horizon control. In Proceedings of the 39th IEEE Conference on Decision and Control, page 632, Sydney, 2000.
A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Technical Report AUT99–16, Automatic Control Laboratory, ETH-Swiss Federal Institute of Technology, 1999.
D. P. Bertsekas and I. B. Rhodes. On the minimax reachability of target sets and target tubes. Automatica, 7:233–247, 1971.
D. P. Bertsekas and I. B. Rhodes. Recursive state estimation for a set-membership description of uncertainty. IEEE Transactions on Automatic Control, 16:117–128, 1971.
R. R. Bitmead, M. Gevers, and V. Wertz. Adaptive Optimal Control—The Thinking Man’s GPC. Prentice Hall Int., 1990.
F. Blanchini. Control synthesis for discrete-time linear systems with control and state bounds in the presence of disturbances. In Proceedings 30th IEEE Conference on Decision and Control, pages 3464–3467, 1990.
Franco Blanchini and Stefano Miani. Any domain of attraction for a linear constrained system is a tracking domain of attraction. SIAM Journal of Control and Optimization, 38(3):971–994, 2000.
Franco Blanchini and Stefano Miani. Any domain of attraction for a linear constrained system is a tracking domain of attraction. In Proceedings 39th IEEE Conference on Decision and Control, Sydney, 2000.
D. Chmielewski and V. Manousiouthakis. On constrained infinite-time linear quadratic optimal control. Systems éf Control Letters, 29:121–129, 1996.
G. De Nicolao, L. Magni, and R. Scattolini. Stability and robustness of nonlinear model predictive control. In Frank Allgöwer and Alex Zheng, editors, Nonlinear Model Predictive Control, pages 322. Birkhäuser Verlag, Basle, 2000.
E. C. Gilbert, I. Kolmanovsky, and K. T. Tan. Discrete-time reference governors and the nonlinear control of systems with state and control constraints. Journal of Robust and Nonlinear Control, 5:487–504, 1995.
E. G. Gilbert and K. T. Tan. Linear systems with state and control constraints: the theory and application of maximal output admissible sets. IEEE Transactions on Automatic Control, AC-36:1008–1020, 1991.
Ali Jadbabaie, Jie Yu, and John Hauser. Stabilizing receding horizon control of nonlinear systems: a control Lyapunov function approach. In Proceedings of American Control Conference, 1999.
S. S. Keerthi and E. G. Gilbert. Computation of minimum-time feedback control laws for systems with state-control constraints. IEEE Transactions on Automatic Control, AC-32:432–435, 1987.
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. Journal of Optimization Theory and Applications, 57:265–293, 1988.
Eric. C. Kerrigan. Robust constraint satisfaction: invariant sets and predictive control. PhD thesis, University of Cambridge, 2000.
I. Kolmanovsky and E. C. Gilbert. Maximal output admissible sets for discrete-time systems with disturbance inputs. In Proceedings of the American Control Conference, Seattle, June 1995.
I. Kolmanovsky and E. C. Gilbert. Multi-modal regulators for systems with state and control constraints and disturbance inputs. In A. S. Morse, editor, Control using logic-based switching: Lecture Notes in Control and Information Sciences, pages 118–127. Springer-Verlag, 1996.
I. Kolmanovsky and E. C. Gilbert. Theory and computation of disturbance invariant sets for discrete-time linear systems. Mathematical Problems in Engineering, 4:317–367, 1998.
L. Magni and R. Sepulchre. Stability margins of nonlinear receding-horizon control via inverse optimality. Systems F.4 Control Letters, 32:241–245, 1997.
D. Q. Mayne. Nonlinear model predictive control: challenges and opportunities. In Frank Allgöwer and Alex Zheng, editors, Nonlinear Model Predictive Control, pages 23–44. Birkhäuser Verlag, Basle, 2000.
D. Q. Mayne. Control of constrained dynamic systems. European Journal of Control, 23, 2001.
D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: stability and optimality. Automatica, 36:789–814, June 2000. Survey paper.
D. Q. Mayne and W. R. Schroeder. Robust time-optimal control of constrained linear systems. Automatica, 33(12):2103–2118, 1997.
Robert H. Miller, Ilya Kolmanovsky, Elmer C. Gilbert, and Peter D. Wahabaugh. Constrained linear systems: a case study. IEEE Control Systems Magazine, pages 23–32, February 2000.
S. H. Mo and J. P. Norton. Parameter-bounding identification algorithms for bounded-noise records. Proceedings of the IEE, 135 Part D:127–132, 1987.
M. Morari. Mathematical programming approach to hybrid systems, analysis and control. In 25 Years of Nonlinear Control at Ècole des Mines de Parishttp://www.cas.ensmp.fr/25ans 2001.
M. A. Poubelle, R. R. Bitmead, and M. Gevers. Fake algebraic Riccati techniques and stability. IEEE Transactions on Automatic Control, AC-31:379–381, 1988.
S. J. Qin and T. A. Badgwell. An overview of nonlinear model predictive control applications. In Frank Allgöwer and Alex Zheng, editors, Nonlinear Model Predictive Control, pages 369–392. Birkhäuser Verlag, Basle, 2000.
J. B. Rawlings and K. R. Muske. Stability of constrained receding horizon control. IEEE Transactions on Automatic Control, AC-38(10):1512–1516, 1993.
P. O. M. Scokaert and J. B. Rawlings. Infinite horizon linear quadratic control with constraints. In Proceedings of the 13th IFAC triennial world congress, volume M, pages 109–114, San Francisico, June 1996.
Marià M. Seron, José A. De Doná, and Graham C. Goodwin. Global analytical model predictive control with input constraints. In Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000.
Jeff S. Shamma and Dapeng Xiong. Linear non quadratic optimal control. IEEE Transactions on Automatic Control, 42:875–879, 1997.
S. M. Veres. Numerical error control in polytope computations. Journal of Optimization Theory and Applications, 2001. Accepted.
S. M. Veres and J. P. Norton. Predictive self-tuning control by parameter bounding and worst-case design. Automatica, 29(4):911–928, 1993.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Larry Ho with affection, respect, and gratitude for the inspiration he has abundantly provided to all in our field
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Mayne, D.Q. (2003). Constrained Control: Polytopic Techniques. In: Gong, W., Shi, L. (eds) Modeling, Control and Optimization of Complex Systems. The International Series on Discrete Event Dynamic Systems, vol 14. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1139-7_6
Download citation
DOI: https://doi.org/10.1007/978-1-4615-1139-7_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5411-6
Online ISBN: 978-1-4615-1139-7
eBook Packages: Springer Book Archive