Skip to main content
SpringerLink
Log in

Robust MPC for Multiplicative and Mixed Uncertainty

  • Chapter
  • First Online:
Model Predictive Control

Part of the book series: Advanced Textbooks in Control and Signal Processing ((C&SP))

Abstract

In this chapter, we consider constrained linear systems with imprecisely known parameters, namely systems that are subject to multiplicative uncertainty.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For this example, the time required to solve (5.50) using the Newton–Raphson method is between one and two orders of magnitude less than the computation time for (5.18) using the Mosek.

  2. 2.

    In general, it is not possible to compute \(\rho \) exactly. However, upper bounds on \(\rho \) can be computed to any desired accuracy, for example, using sum of squares programming [28].

  3. 3.

    With \(H_c\) and \(H^{(j)}\), \(j=1,\ldots ,m\) chosen so as to minimize sum of elements in each of their rows, the conditions (5.92) and (5.93) include the corresponding conditions that were derived in Sect. 5.4.2 for low-complexity polytopes as a special case. Thus, expressing the low-complexity polytopic set \(\{x: \underline{\alpha } \le V_0 x \le \overline{\alpha }\}\) equivalently as \(\{x: Vx\le \alpha \}\) with \(V = [V_0^T \ {-V_0^T}]^T\) and \(\alpha = [\overline{\alpha }^T \ {-\underline{\alpha }}^T]^T\), the solutions of (5.94) and (5.95) can be obtained in closed form as

    $$ H_c = [(\tilde{F} + G\tilde{K})^+ \ (\tilde{F} + G\tilde{K})^-] \text { and } H^{(j)} = \begin{bmatrix} (\tilde{\varPhi }^{(j)})^+&(\tilde{\varPhi }^{(j)})^- \\ (\tilde{\varPhi }^{(j)})^-&(\tilde{\varPhi }^{(j)})^+ \end{bmatrix}, \ j= 1,\ldots ,m. $$

    Therefore conditions (5.78) and (5.79) are identical to (5.92) and (5.93) for this case.

References

  1. R.W. Liu, Convergent systems. IEEE Trans. Autom. Control 13(4), 384–391 (1968)

    Article  Google Scholar 

  2. J. Richalet, A. Rault, J.L. Testud, J. Papon, Model predictive heuristic control: applications to industrial processes. Automatica 14(5), 413–428 (1978)

    Article  Google Scholar 

  3. P.J. Campo, M. Morari, Robust model predictive control, in Proceedings of the 1987 American Control Conference, Minneapolis, USA. vol. 2 (1987), pp. 1021–1026

    Google Scholar 

  4. J.C. Allwright, G.C. Papavasiliou, On linear programming and robust model-predictive control using impulse-responses. Syst. Control Lett. 18(2), 159–164 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Zheng, M. Morari, Robust stability of constrained model predictive control, in Proceedings of the 1993 American Control Conference, San Francisco, USA (1993), pp. 379–383

    Google Scholar 

  6. H. Genceli, M. Nikolaou, Robust stability analysis of constrained \(l_1\)-norm model predictive control. AIChE J. 39(12), 1954–1965 (1993)

    Article  MathSciNet  Google Scholar 

  7. B. Kouvaritakis, J.A. Rossiter, Use of bicausal weighting sequences in least squares identification of open-loop unstable dynamic systems. Control Theory Appl. IEE Proc. D 139(3), 328–336 (1992)

    Article  MATH  Google Scholar 

  8. M.V. Kothare, V. Balakrishnan, M. Morari, Robust constrained model predictive control using linear matrix inequalities. Automatica 32(10), 136–1379 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Schuurmans, J.A. Rossiter, Robust predictive control using tight sets of predicted states. Control Theory Appl., IEE Proc. 147(1), 13–18 (2000)

    Article  Google Scholar 

  10. B. Kouvaritakis, J.A. Rossiter, J. Schuurmans, Efficient robust predictive control. IEEE Trans. Autom. Control 45(8), 1545–1549 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Kouvaritakis, M. Cannon, J.A. Rossiter, Who needs QP for linear MPC anyway? Automatica 38(5), 879–884 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Cannon, B. Kouvaritakis, Optimizing prediction dynamics for robust MPC. IEEE Trans. Autom. Control 50(11), 1892–1897 (2005)

    Article  MathSciNet  Google Scholar 

  13. Q. Cheng, M. Cannon, B. Kouvaritakis, The design of dynamics in the prediction structure of robust MPC. Int. J. Control 86(11), 2096–2103 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Y.I. Lee, B. Kouvaritakis, Stabilizable regions of receding horizon predictive control with input constraints. Syst. Control Lett. 38(1), 13–20 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Y.I. Lee, B. Kouvaritakis, Robust receding horizon predictive control for systems with uncertain dynamics and input saturation. Automatica 36(10), 1497–1504 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Y.I. Lee, B. Kouvaritakis, A linear programming approach to constrained robust predictive control. IEEE Trans. Autom. Control 45(9), 1765–1770 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Y.I. Lee, B. Kouvaritakis, Linear matrix inequalities and polyhedral invariant sets in constrained robust predictive control. Int. J. Robust Nonlinear Control 10(13), 1079–1090 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Evans, M. Cannon, B. Kouvaritakis, Robust MPC for linear systems with bounded multiplicative uncertainty, in Proceedings of the 51st IEEE Conference on Decision and Control, Maui, USA (2012), pp. 248–253

    Google Scholar 

  19. J. Fleming, B. Kouvaritakis, M. Cannon, Regions of attraction and recursive feasibility in robust MPC, in Proceedings of the 21st Mediterranean Conference on Control and Automation, Chania, Greece (2013), pp. 801–806

    Google Scholar 

  20. J. Fleming, B. Kouvaritakis, M. Cannon, Robust tube MPC for linear systems with multiplicative uncertainty. IEEE Trans. Autom. Control 60(4), 1087–1092 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Y.C. Gautam, A. Chu, Y.C. Soh, Optimized dynamic policy for receding horizon control of linear time-varying systems with bounded disturbances. IEEE Trans. Autom. Control 57(4), 973–998 (2012)

    Article  MathSciNet  Google Scholar 

  22. D. Muñoz-Carpintero, M. Cannon, B. Kouvaritakis, Recursively feasible robust MPC for linear systems with additive and multiplicative uncertainty using optimized polytopic dynamics, in Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy (2013),pp. 1101–1106

    Google Scholar 

  23. D. Muñoz-Carpintero, M. Cannon, B. Kouvaritakis, Robust MPC strategy with optimized polytopic dynamics for linear systems with additive and multiplicative uncertainty. Syst. Control Lett. (2015), in press

    Google Scholar 

  24. S.P. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, Philadelphia, 1994)

    Book  MATH  Google Scholar 

  25. B. Kouvaritakis, J.A. Rossiter, A.O.T. Chang, Stable generalised predictive control: an algorithm with guaranteed stability. Control Theory Appl., IEE Proc. D 139(4), 349–362 (1992)

    Article  MATH  Google Scholar 

  26. T. Barjas Blanco, M. Cannon, B. De Moor, On efficient computation of low-complexity controlled invariant sets for uncertain linear systems. Int. J. Control 83(7), 1339–1346 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. F. Blanchini, Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions. IEEE Trans. Autom. Control 39(2), 428–433 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. P.A. Parrilo, A. Jadbabaie, Approximation of the joint spectral radius using sum of squares. Linear Algebra Appl. 428(10), 2385–2402 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Cannon, V. Deshmukh, B. Kouvaritakis, Nonlinear model predictive control with polytopic invariant sets. Automatica 39(8), 1487–1494 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Nesterov, A. Nemirovsky, Interior-Point Polynomial Algorithms in Convex Programming (SIAM, Philadelphia, 1994)

    Book  Google Scholar 

  31. G. Bitsoris, On the positive invariance of polyhedral sets for discrete-time systems. Syst. Control Lett. 11(3), 243–248 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Benzaouia, C. Burgat, Regulator problem for linear discrete-time systems with non-symmetrical constrained control. Int. J. Control 48(6), 2441–2451 (1988)

    Article  MATH  Google Scholar 

  33. R. Fletcher, Practical Methods of Optimization, 2nd edn. (Wiley, New York, 1987)

    MATH  Google Scholar 

  34. F. Blanchini, S. Miani, Set-Theoretic Methods in Control (Birkhäuser, Boston, 2008)

    MATH  Google Scholar 

  35. B. Pluymers, J.A. Rossiter, J.A.K. Suykens, B. De Moor, The efficient computation of polyhedral invariant sets for linear systems with polytopic uncertainty, in Proceedings of the 2005 American Control Conference (2005), pp. 804–809

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Oxford, Oxford, UK

    Basil Kouvaritakis & Mark Cannon

Authors
  1. Basil Kouvaritakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark Cannon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Basil Kouvaritakis .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kouvaritakis, B., Cannon, M. (2016). Robust MPC for Multiplicative and Mixed Uncertainty. In: Model Predictive Control. Advanced Textbooks in Control and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-24853-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-24853-0_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24851-6

  • Online ISBN: 978-3-319-24853-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation