Optimization and Engineering

, Volume 16, Issue 4, pp 695–711 | Cite as

Ellipsoidal bounds on state trajectories for discrete-time systems with linear fractional uncertainties

  • Masako KishidaEmail author
  • Richard D. Braatz


Computation of exact ellipsoidal bounds on the state trajectories of discrete-time linear systems that have time-varying or time-invariant linear fractional parameter uncertainties and ellipsoidal uncertainty in the initial state is known to be NP-hard. This paper proposes three algorithms to compute ellipsoidal bounds on such a state trajectory set and discusses the tradeoffs between computational complexity and conservatism of the algorithms. The approach employs linear matrix inequalities to determine an initial estimate of the ellipsoid that is refined by the subsequent application of the skewed structured singular value \(\nu \). Numerical examples are used to illustrate the application of the proposed algorithms and to compare the differences between them, where small conservatism for the tightest bounds is observed.


Uncertain dynamical systems Discrete-time systems Bounding method Structured singular value Linear matrix inequalities 



The authors acknowledge support from the Institute for Advanced Computing Applications and Technologies.


  1. Berger M (1979) Géométrie. CEDIC/Nathan, ParisGoogle Scholar
  2. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  3. Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaGoogle Scholar
  4. Braatz RD, Young PM, Doyle JC, Morari M (1994) Computational complexity of \(\mu \) calculation. IEEE Trans Autom Control 39:1000–1002zbMATHMathSciNetCrossRefGoogle Scholar
  5. Chernousko FL (2010) Optimal ellipsoidal estimates of uncertain systems: an overview and new results. In: Marti K, Ermoliev Y, Makowski M (eds) Coping with uncertainty, Springer, BerlinGoogle Scholar
  6. Durieu C, Walter E, Polyak B (2001) Multi-input multi-output ellipsoidal state bounding. JOTA 111:273–303zbMATHMathSciNetGoogle Scholar
  7. El Ghaoui L, Calafiore G (1999) Worst-case simulation of uncertain systems. In: Garulli A, Tesi A, Vicino A (eds) Robustness in identification and control. Springer, LondonGoogle Scholar
  8. Fan MKH, Tits A (1992) A measure of worst-case \(H_{\infty }\) performance and of largest acceptable uncertainty. Syst Control Lett 18:409–421zbMATHMathSciNetCrossRefGoogle Scholar
  9. Ferreres G (1999) A practical approach to robustness analysis with aeronautical applications. Springer, New YorkzbMATHGoogle Scholar
  10. Ferreres G, Biannic JM (2009) Skew Mu Toolbox (SMT). Accessed 14 March 2014
  11. Horak DT (1988) Failure detection in dynamic systems with modeling errors. AIAA JGCD 11:508–516Google Scholar
  12. Huang H, Adjiman C, Shah N (2002) Quantitative framework for reliable safety analysis. AIChE J 48:78–96CrossRefGoogle Scholar
  13. Kishida M, Braatz RD (2011) Ellipsoid bounds on state trajectories for discrete-time systems with time-invariant and time-varying linear fractional uncertainties. In: Proceedings of IEEE Conference on Decision and Control and European Control Conference, Orlando, pp 216–221Google Scholar
  14. Kishida M, Rumschinski P, Findeisen R, Braatz RD (2011) Efficient polynomial-time outer bounds on state trajectories for uncertain polynomial systems using skewed structured singular values. In: Proc. of IEEE International Symposium on Computer-Aided Control System Design, Denver, CO, pp 216–221Google Scholar
  15. Löfberg J (2004) YALMIP: A toolbox for modeling and optimization in MATLAB. In: Proceedings of IEEE International Symposium on Computer-Aided Control System Design, Taipei, pp 284–289.
  16. Moore R (1996) Interval analysis. Prentice-Hall, Englewood CliffsGoogle Scholar
  17. Polyak BT, Nazin SA, Durieu C, Walter E (2004) Ellipsoidal parameter or state estimation under model uncertainty. Automatica 40:1171–1179zbMATHMathSciNetCrossRefGoogle Scholar
  18. Pronzato L, Walter E (1994) Minimum-volume ellipsoids containing compact sets: application to parameter bounding. Automatica 30:1731–1739zbMATHMathSciNetCrossRefGoogle Scholar
  19. Puig V, Stancu A, Quevedo J (2005) Simulation of uncertain dynamic systems described by interval models: A survey. In: Proceedings of of the 16th IFAC World Congress, Prague, paper Fr-A13-TO/6Google Scholar
  20. Rokityanski DY, Veres SM (2005) Application of ellipsoidal estimation to satellite control. Math Comput Model Dyn Syst 11:239–249MathSciNetCrossRefGoogle Scholar
  21. Russell EL, Braatz RD (1998) Model reduction for the robustness margin computation of large scale uncertain systems. Comput Chem Eng 22:913–926CrossRefGoogle Scholar
  22. Russell EL, Power CPH, Braatz RD (1997) Multidimensional realizations of large scale uncertain systems for multivariable stability margin computation. Int J Robust Nonlinear Control 7:113–125zbMATHMathSciNetCrossRefGoogle Scholar
  23. Sanyal AK, Lee T, Leok M, McClamroch NH (2008) Global optimal attitude estimation using uncertainty ellipsoids. Syst Control Lett 57:236–245zbMATHMathSciNetCrossRefGoogle Scholar
  24. Schweppe FC (1968) Recursive state estimation: unknown but bounded errors and system inputs. IEEE Trans Automatic Control 13:22–28CrossRefGoogle Scholar
  25. Schweppe FC (1973) Uncertain dynamic systems: modelling, estimation, hypothesis testing identification and control. Prentice-Hall, Englewood CliffsGoogle Scholar
  26. Smith RSR (1990) Model validation for uncertain systems. PhD thesis, California Institute of Technology, PasadenaGoogle Scholar
  27. Tibken B, Hofer EP (1995) Simulation of controlled uncertain nonlinear systems. Appl Math Comp 70:329–338zbMATHMathSciNetCrossRefGoogle Scholar
  28. Young PM, Newlin MP, Doyle JC (1995) Computing bounds for the mixed \(\mu \) problem. Int J Robust Nonlinear Control 5:573–590zbMATHMathSciNetCrossRefGoogle Scholar
  29. Zhou K, Doyle JC, Glover K (1995) Robust and optimal control. Prentice Hall, Upper Saddle RiverGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Electrical and Computer EngineeringUniversity of CanterburyChristchurchNew Zealand
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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