Advertisement

Robust Control of Large-Scale Systems: Efficient Selection of Inputs and Outputs

  • Javad Lavaei
  • Somayeh Sojoudi
  • Amir G. Aghdam
Chapter

Abstract

There has been a growing interest in recent years in robust control of systems with parametric uncertainty [4,9,13,16,18]. The dynamic behavior of this type of systems is typically governed by a set of differential equations whose coefficients belong to fairly-known uncertainty regions. Although there are several methods to capture the uncertain nature of a real-world system (e.g., by modeling it as a structured or unstructured uncertainty [6]), it turns out that the most realistic means of describing uncertainty is to parameterize it and then specify its domain of variation.

Keywords

Robust Stability Uncertain System Matrix Polynomial Uncertainty Region Gramian Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blekherman, G.: There are significantly more nonnegative polynomials than sums of squares. Israel Journal of Mathematics 153, 355–380 (2006)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bliman, P. A., Oliveira, R. C. L. F., Montagner, V. F., Peres, P. L. D.: Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities with parameters in the simplex. Proceedings of 45th IEEE Conference on Decision and Control. San Diego, USA. 1486–1491 (2006)Google Scholar
  3. 3.
    Chesi, G.: On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems. Automatica 44, 2973–2976 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach. IEEE Transactions on Automatic Control 50, 365–370 (2005)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Davison, E. J., Chang, T. N.: Decentralized stabilization and pole assignment for general proper systems. IEEE Transactions Automatic Control 35, 652–664 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dullerud, G. E., Paganini, F.: A course in robust control theory: a convex approach. Texts in Applied Mathematics, Springer, Berlin (2005)Google Scholar
  7. 7.
    Hardy, G. H., Littlewood, J. E., Polya, G.: Inequalities. Cambridge University Press, Cambridge, UK, Second edition (1952)MATHGoogle Scholar
  8. 8.
    Hillar, C. J., Nie, J.: An elementary and constructive solution to Hilbert’s 17th Problem for matrices. Proceedings of the American Mathematical Society 136, 73–76 (2008)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kau, S., Liu, Y., Hong, L., Lee, C., Fang, C., Lee, L.: A new LMI condition for robust stability of discrete-time uncertain systems. Systems and Control Letters 54, 1195–1203 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lavaei, J., Aghdam, A. G.: Optimal periodic feedback design for continuous-time LTI systems with constrained control structure. International Journal of Control 80, 220–230 (2007)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lavaei, J., Aghdam, A. G.: A graph theoretic method to find decentralized fixed modes of LTI systems. Automatica 43, 2129–2133 (2007)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lavaei, J., Aghdam, A. G.: Control of continuous-time LTI systems by means of structurally constrained controllers. Automatica 44, 141–148 (2008)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lavaei, J., Aghdam, A. G.: Robust stability of LTI systems over semi-algebraic sets using sum-of-squares matrix polynomials. IEEE Transactions on Automatic Control 53, 417–423 (2008)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Lofberg, J.: A toolbox for modeling and optimization in MATLAB. Proc. of the CACSD Conference. Taipei, Taiwan (2004) Available via http://control.ee.ethz.ch/~joloef/yalmip.php
  15. 15.
    Prajna, S., Papachristodoulou, A., Seiler, P., Parrilo, P. A.: SOSTOOLS sum of squares optimization toolbox for MATLAB. Users guide (2004) Available via http://www.cds.caltech.edu/sostools
  16. 16.
    Oliveira, M. C., Geromel, J. C.: A class of robust stability conditions where linear parameter dependence of the Lyapunov function is a necessary condition for arbitrary parameter dependence. Systems and Control Letters 54, 1131–1134 (2005)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Oliveira, R. C. L. F., Oliveira, M. C., Peres, P. L. D.: Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions. Systems and Control Letters 57, 680–689 (2008)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Oliveira, R. C. L. F., Peres, P. L. D.: LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions. Systems and Control Letters 55, 52–61 (2006)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Parrilo, P. A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. dissertation, California Institute of Technology (2000)Google Scholar
  20. 20.
    Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42, 969–984 (1993)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sastry, S. S., Desoer, C. A.: The robustness of controllability and observability of linear time-varying systems. IEEE Transactions on Automatic Control 27, 933–939 (1982)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Savkin, A. V., Petersen, I. R.: Weak robust controllability and observability of uncertain linear systems. IEEE Transactions on Automatic Control 44, 1037–1041 (1999)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Scherer, C. W., Hol, C. W. J.: Matrix sum-of-squares relaxations for robust semi-definite programs. Mathematical Programming 107, 189–211 (2006)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Siljak, D. D.: Decentralized control of complex systems. Academic, Cambridge (1991)Google Scholar
  25. 25.
    Sojoudi, S., Aghdam, A. G.: Characterizing all classes of LTI stabilizing structurally constrained controllers by means of combinatorics. Proceedings 46th IEEE Conference on Decision and Control. New Orleans, USA, 4415–4420 (2007)Google Scholar
  26. 26.
    Sojoudi, S., Lavaei, J., Aghdam, A. G.: Robust controllability and observability degrees of polynomially uncertain systems. Automatica 45(11), 2640–2645, Nov. (2009)MATHCrossRefGoogle Scholar
  27. 27.
    Ugrinovskii, V. A.: Robust controllability of linear stochastic uncertain systems. Automatica 41, 807–813 (2005)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Wang, S. H., Davison, E. J.: On the stabilization of decentralized control systems. IEEE Transactions on Automatic Control 18, 473–478 (1973)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Javad Lavaei
    • 1
  • Somayeh Sojoudi
    • 1
  • Amir G. Aghdam
    • 2
  1. 1.Control and Dynamical SystemsCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of Electrical and Computer EngineeringConcordia UniversityMontrealCanada

Personalised recommendations