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

  • Somayeh SojoudiEmail author
  • Javad Lavaei
  • Amir G. Aghdam
Chapter

Abstract

There has been a growing interest in recent years in robust control of systems with parametric uncertainty [1, 2, 3, 4, 5]. 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 Control Robust Stability Uncertain System Matrix Polynomial Uncertainty Region 
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.

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References

  1. 1.
    J. Lavaei and A. G. Aghdam, “Robust stability of LTI systems over semi-algebraic sets using sum-of-squares matrix polynomials,” IEEE Transactions on Automatic Control, vol. 53, no. 1, pp. 417–423, 2008.MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. C. L. F. Oliveira and P. L. D. Peres, “LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions,” Systems & Control Letters, vol. 55, no. 1, pp. 52–61, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 365–370, 2005.MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Kau, Y. Liu, L. Hong, C. Lee, C. Fang, and L. Lee, “A new LMI condition for robust stability of discrete-time uncertain systems,” Systems & Control Letters, vol. 54, no. 12, pp. 1195–1203, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    M. C. de Oliveira and J. C. Geromel, “A class of robust stability conditions where linear parameter dependence of the Lyapunov function is a necessary condition for arbitrary parameter dependence,” Systems & Control Letters, vol. 54, no. 11, pp. 1131–1134, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    G. E. Dullerud and F. Paganini, A course in robust control theory: A convex approach, Texts in Applied Mathematics, Springer, 2005.Google Scholar
  7. 7.
    A. V. Savkin and I. R. Petersen, “Weak robust controllability and observability of uncertain linear systems,” IEEE Transactions on Automatic Control, vol. 44, no. 5, pp. 1037–1041, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    V. A. Ugrinovskii, “Robust controllability of linear stochastic uncertain systems,” Automatica, vol. 41, no. 5, pp. 807–813, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    S. S. Sastry and C. A. Desoer, “The robustness of controllability and observability of linear time-varying systems,” IEEE Transactions on Automatic Control, vol. 27, pp. 933–939, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    E. J. Davison and T. N. Chang, ”Decentralized stabilization and pole assignment for general proper systems,” IEEE Transactions on Automatic Control, vol. 35, no. 6, pp. 652–664, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    D. D. Šiljak, Decentralized control of complex systems, Cambridge: Academic Press, 1991.Google Scholar
  12. 12.
    J. Lavaei and A. G. Aghdam, “A graph theoretic method to find decentralized fixed modes of LTI systems,” Automatica, vol. 43, no. 12, pp. 2129–2133, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    J. Lavaei and A. G. Aghdam, “Control of continuous-time LTI systems by means of structurally constrained controllers,” Automatica, vol. 44, no. 1, pp. 141–148, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    S. Sojoudi and A. G. Aghdam, “Characterizing all classes of LTI stabilizing structurally constrained controllers by means of combinatorics,” in Proceedings of 46th IEEE Conference on Decision and Control, New Orleans, USA, 2007.Google Scholar
  15. 15.
    S. H. Wang and E. J. Davison, “On the stabilization of decentralized control systems,” IEEE Transactions on Automatic Control vol. 18, no. 5, pp. 473–478, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    G. Blekherman, “There are significantly more nonnegative polynomials than sums of squares,” Israel Journal of Mathematics, vol. 153, no. 1, pp. 355–380, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    P. A. Parrilo, “Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization,” Ph.D. dissertation, California Institute of Technology, 2000.Google Scholar
  18. 18.
    G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities. Cambridge University Press, Cambridge, UK, Second edition, 1952.Google Scholar
  19. 19.
    M. Putinar, “Positive polynomials on compact semi-algebraic sets,” Indiana University Mathematics Journal, vol. 42, pp. 969–984, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    J. Lavaei and A. G. Aghdam, “Optimal periodic feedback design for continuous-time LTI systems with constrained control structure,” International Journal of Control, vol. 80, no. 2, pp. 220–230, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    C. W. Scherer and C. W. J. Hol, “Matrix sum-of-squares relaxations for robust semi-definite programs,” Mathematical Programming, vol. 107, no. 1–2, pp. 189–211, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    J. L¨ofberg, “A toolbox for modeling and optimization in MATLAB,” in Proceedings of the CACSD Conference, Taipei, Taiwan, 2004 (available online at http://control.ee.ethz.ch/~joloef/yalmip.php).
  23. 23.
    S. Prajna, A. Papachristodoulou, P. Seiler and P. A. Parrilo, “SOSTOOLS sum of squares optimization toolbox for MATLAB,” Users guide, 2004 (available online at http://www.cds.caltech.edu/sostools).
  24. 24.
    C. J. Hillar and J. Nie, “An elementary and constructive solution to Hilbert’s 17th Problem for matrices,” in Proceedings of the American Mathematical Society, vol. 136, pp. 73–76, 2008.Google Scholar
  25. 25.
    P.-A. Bliman, R. C. L. F. Oliveira, V. F. Montagner, and P. L. D. Peres, “Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequal- ities with parameters in the simplex,” in Proceedings of 45th IEEE Conference on Decision and Control, San Diego, USA, pp. 1486–1491, 2006.Google Scholar
  26. 26.
    R. C. L. F. Oliveira, M. C. de Oliveira, and P. L. D. Peres, “Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions,” Systems & Control Letters, vol. 57, no. 8, pp. 680–689, 2008.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer US 2011

Authors and Affiliations

  • Somayeh Sojoudi
    • 1
    Email author
  • Javad Lavaei
    • 1
  • Amir G. Aghdam
    • 2
  1. 1.Control & Dynamical Systems Dept.California Institute of TechnologyPasadenaUSA
  2. 2.Dept. Electrical & Computer EngineeringConcordia UniversityMontreal QuébecCanada

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