Constrained Control: Polytopic Techniques

  • D. Q. Mayne
Part of the The International Series on Discrete Event Dynamic Systems book series (DEDS, volume 14)

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

Keywords

Manifold Hull Lution Dition Harness 

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References

  1. 1.
    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.Google Scholar
  2. 2.
    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.Google Scholar
  3. 3.
    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.Google Scholar
  4. 4.
    D. P. Bertsekas and I. B. Rhodes. On the minimax reachability of target sets and target tubes. Automatica, 7:233–247, 1971.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    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.MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. R. Bitmead, M. Gevers, and V. Wertz. Adaptive Optimal Control—The Thinking Man’s GPC. Prentice Hall Int., 1990.Google Scholar
  7. 7.
    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.Google Scholar
  8. 8.
    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.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    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.Google Scholar
  10. 10.
    D. Chmielewski and V. Manousiouthakis. On constrained infinite-time linear quadratic optimal control. Systems éf Control Letters, 29:121–129, 1996.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    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.Google Scholar
  12. 12.
    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.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    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.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    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.Google Scholar
  15. 15.
    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.MATHCrossRefGoogle Scholar
  16. 16.
    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.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Eric. C. Kerrigan. Robust constraint satisfaction: invariant sets and predictive control. PhD thesis, University of Cambridge, 2000.Google Scholar
  18. 18.
    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.Google Scholar
  19. 19.
    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.Google Scholar
  20. 20.
    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.MATHCrossRefGoogle Scholar
  21. 21.
    L. Magni and R. Sepulchre. Stability margins of nonlinear receding-horizon control via inverse optimality. Systems F.4 Control Letters, 32:241–245, 1997.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    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.CrossRefGoogle Scholar
  23. 23.
    D. Q. Mayne. Control of constrained dynamic systems. European Journal of Control, 23, 2001.Google Scholar
  24. 24.
    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.MathSciNetMATHCrossRefGoogle Scholar
  25. [25.
    D. Q. Mayne and W. R. Schroeder. Robust time-optimal control of constrained linear systems. Automatica, 33(12):2103–2118, 1997.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    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.Google Scholar
  27. 27.
    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.Google Scholar
  28. 28.
    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.Google Scholar
  29. 29.
    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.MathSciNetCrossRefGoogle Scholar
  30. 30.
    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.CrossRefGoogle Scholar
  31. 31.
    J. B. Rawlings and K. R. Muske. Stability of constrained receding horizon control. IEEE Transactions on Automatic Control, AC-38(10):1512–1516, 1993.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    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.Google Scholar
  33. 33.
    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.Google Scholar
  34. 34.
    Jeff S. Shamma and Dapeng Xiong. Linear non quadratic optimal control. IEEE Transactions on Automatic Control, 42:875–879, 1997.MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    S. M. Veres. Numerical error control in polytope computations. Journal of Optimization Theory and Applications, 2001. Accepted.Google Scholar
  36. 36.
    S. M. Veres and J. P. Norton. Predictive self-tuning control by parameter bounding and worst-case design. Automatica, 29(4):911–928, 1993.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • D. Q. Mayne
    • 1
  1. 1.Imperial College of ScienceTechnology and MedicineLondon

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