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Advanced Stability Methods and Applications

  • Iasson Karafyllis
  • Zhong-Ping Jiang
Part of the Communications and Control Engineering book series (CCE)

Abstract

Chapter 5 is devoted to the description of advanced analysis methods for checking the internal and external global stability properties introduced previously, for various important classes of nonlinear dynamic systems. A focus will be placed on small-gain techniques and vector Lyapunov functions.

Keywords

Gain Function Strategic Game Nash Equilibrium Point Vector Lyapunov Function Robust Asymptotic Stability 
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.

References

  1. 1.
    Angeli, D., Astolfi, A.: A tight small-gain theorem for not necessarily ISS systems. Systems and Control Letters 56, 87–91 (2007) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Angeli, D., De Leenheer, P., Sontag, E.D.: A small-gain theorem for almost global convergence of monotone systems. Systems and Control Letters 52(5), 407–414 (2004) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Burton, T.A.: Stability by Fixed Point Theory for Functional Differential Equations. Dover, Mineola (2006) MATHGoogle Scholar
  4. 4.
    Chen, Z., Huang, J.: A simplified small gain theorem for time-varying nonlinear systems. IEEE Transactions on Automatic Control 50(11), 1904–1908 (2005) CrossRefGoogle Scholar
  5. 5.
    Dashkovskiy, S., Rüffer, B.S., Wirth, F.R.: A small-gain type stability criterion for large scale networks of ISS systems. In: Proceedings of the Joint 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, pp. 5633–5638 (2005) CrossRefGoogle Scholar
  6. 6.
    Dashkovskiy, S., Rüffer, B.S., Wirth, F.R.: Discrete time monotone systems: Criteria for global asymptotic stability and applications. In: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Kyoto, Japan, pp. 89–97 (2006) Google Scholar
  7. 7.
    Dashkovskiy, S., Rüffer, B.S., Wirth, F.R.: An ISS small-gain theorem for general networks. Mathematics of Control, Signals and Systems 19, 93–122 (2007) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dashkovskiy, S., Rüffer, B.S., Wirth, F.R.: Small-gain theorems for large scale systems and construction of ISS Lyapunov functions. SIAM Journal on Control and Optimization 48(6), 4089–4118 (2010) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Desoer, C., Vidyasagar, M.: Feedback Systems: Input–Output Properties. Academic Press, New York (1975) MATHGoogle Scholar
  10. 10.
    Enciso, G.A., Sontag, E.D.: Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete and Continuous Dynamical Systems 14(3), 549–578 (2006) MathSciNetMATHGoogle Scholar
  11. 11.
    Grüne, L.: Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Transactions on Automatic Control 47(9), 1499–1504 (2002) CrossRefGoogle Scholar
  12. 12.
    Haddad, W.M., Chellaboina, V.: Nonlinear Dynamical Systems and Control. A Lyapunov-Based Approach. Princeton University Press, Princeton (2008) MATHGoogle Scholar
  13. 13.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) MATHGoogle Scholar
  14. 14.
    Hill, D.J.: A generalization of the small-gain theorem for nonlinear feedback systems. Automatica 27(6), 1043–1045 (1991) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ito, H.: Stability Criteria for interconnected iISS and ISS systems using scaling of supply rates. Proceedings of the American Control Conference 2, 1055–1060 (2004) Google Scholar
  16. 16.
    Ito, H.: State-dependent scaling problems and stability of interconnected iISS and ISS systems. IEEE Transactions on Automatic Control 51(10), 1626–1643 (2006) CrossRefGoogle Scholar
  17. 17.
    Ito, H., Jiang, Z.-P.: Nonlinear small-gain condition covering iISS systems: Necessity and sufficiency from a Lyapunov perspective. In: Proceedings of the Joint 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, pp. 355–360 (2005) Google Scholar
  18. 18.
    Ito, H., Jiang, Z.-P.: Necessary and sufficient small-gain conditions for integral input-to-state stable systems: A Lyapunov perspective. IEEE Transactions on Automatic Control 54(10), 2389–2404 (2009) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jiang, Z.P., Mareels, I.M.Y.: A small-gain control method for nonlinear cascaded systems with dynamic uncertainties. IEEE Transactions on Automatic Control 42, 292–308 (1997) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Jiang, Z.P., Wang, Y.: A generalization of the nonlinear small-gain theorem for large-scale complex systems. In: Proceedings of the 7th World Congress on Intelligent Control and Automation, Chongqing, China, pp. 1188–1193 (2008) Google Scholar
  21. 21.
    Jiang, Z.P., Teel, A., Praly, L.: Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals and Systems 7, 95–120 (1994) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Jiang, Z.P., Mareels, I.M.Y., Wang, Y.: A Lyapunov formulation of the nonlinear small-gain theorem for interconnected systems. Automatica 32, 1211–1214 (1996) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Jiang, Z.P., Lin, Y., Wang, Y.: A local nonlinear small-gain theorem for discrete-time feedback systems and its applications. In: Proceedings of 3rd Asian Control Conference, Shanghai, pp. 1227–1232 (2000) Google Scholar
  24. 24.
    Jiang, Z.P., Lin, Y., Wang, Y.: Nonlinear small-gain theorems for discrete-time feedback systems and applications. Automatica 40(12), 2129–2136 (2004) MathSciNetMATHGoogle Scholar
  25. 25.
    Karafyllis, I.: The non-uniform in time small-gain theorem for a wide class of control systems with outputs. European Journal of Control 10(4), 307–323 (2006) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Karafyllis, I., Jiang, Z.-P.: A small-gain theorem for a wide class of feedback systems with control applications. SIAM Journal Control and Optimization 46(4), 1483–1517 (2007) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Karafyllis, I., Jiang, Z.-P.: A vector small-gain theorem for general nonlinear control systems. Accepted for publication in IMA Journal of Mathematical Control and Information. A short version was published in the Proceedings of the 48th IEEE Conference on Decision and Control 2009, Shanghai, China, pp. 7996–8001 (2009). A preliminary version is also available at arXiv:0904.0755
  28. 28.
    Karafyllis, I., Jiang, Z.-P.: New results in trajectory-based small-gain with application to the stabilization of a chemostat. Submitted to International Journal of Robust and Nonlinear Control Google Scholar
  29. 29.
    Karafyllis, I., Kravaris, C.: Global stability results for systems under sampled-data control. International Journal of Robust and Nonlinear Control 19, 1105–1128 (2009) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Karafyllis, I., Kravaris, C.: Robust global stabilizability by means of sampled-data control with positive sampling rate. International Journal of Control 82(4), 755–772 (2009) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Karafyllis, I., Tsinias, J.: Non-uniform in time ISS and the small-gain theorem. IEEE Transactions on Automatic Control 49(2), 196–216 (2004) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Karafyllis, I., Kravaris, C., Syrou, L., Lyberatos, G.: A vector Lyapunov function characterization of input-to-state stability with application to robust global stabilization of the chemostat. European Journal of Control 14(1), 47–61 (2008) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Karafyllis, I., Pepe, P., Jiang, Z.-P.: Input-to-output stability for systems described by retarded functional differential equations. European Journal of Control 14(6), 539–555 (2008) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Karafyllis, I., Kravaris, C., Kalogerakis, N.: Relaxed Lyapunov criteria for robust global stabilization of nonlinear systems. International Journal of Control 82(11), 2077–2094 (2009) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Karafyllis, I., Jiang, Z.-P., Athanasiou, G.: Nash equilibrium and robust stability in dynamic games: A small-gain perspective. Computers and Mathematics with Applications 60, 2936–2952 (2010) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Lakshmikantham, V., Matrosov, V.M., Sivasundaram, S.: Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems. Kluwer Academic, Dordrecht (1991) MATHCrossRefGoogle Scholar
  37. 37.
    Liberzon, D., Nesic, D.: Stability analysis of hybrid systems via small-gain theorems. In: Hespanha, J., Tiwari, A. (eds.) Proceedings of the 9th International Workshop on Hybrid Systems: Computation and Control, Santa Barbara, 2006. Lecture Notes in Computer Science, vol. 3927, pp. 421–435. Springer, Berlin (2006) CrossRefGoogle Scholar
  38. 38.
    Mareels, I.M.Y., Hill, D.J.: Monotone stability of nonlinear feedback systems. Journal of Mathematical Systems, Estimation and Control 2, 275–291 (1992) MathSciNetMATHGoogle Scholar
  39. 39.
    Mazenc, F., Niculescu, S.-I.: Lyapunov stability analysis for nonlinear delay systems. Systems and Control Letters 42, 245–251 (2001) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Nersesov, S.G., Haddad, W.M.: On the stability and control of nonlinear dynamical systems via vector Lyapunov functions. IEEE Transactions on Automatic Control 51(2), 203–215 (2006) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Nesić, D., Teel, A.R.: Sampled-data control of nonlinear systems: An overview of recent results. In: Moheimani, R.S.O. (ed.) Perspectives on Robust Control, pp. 221–239. Springer, New York (2001) Google Scholar
  42. 42.
    Nesić, D., Teel, A.R.: Stabilization of sampled-data nonlinear systems via backstepping on their Euler approximate model. Automatica 42, 1801–1808 (2006) MATHCrossRefGoogle Scholar
  43. 43.
    Niculescu, S.I.: Delay Effects on Stability, a Robust Control Approach. Springer, Heidelberg (2001) MATHGoogle Scholar
  44. 44.
    Rüffer, B.S.: Monotone dynamical systems, graphs and stability of large-scale interconnected systems. PhD Thesis, University of Bremen, Germany (2007) Google Scholar
  45. 45.
    Rüffer, B.S.: Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean n-space. Positivity 14(2), 257–283 (2010) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Smith, H.L.: Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41. AMS, Providence (1994) Google Scholar
  47. 47.
    Sontag, E.D.: Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control 34, 435–442 (1989) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Sontag, E.D., Ingalls, B.: A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections. Journal of the Franklin Institute 339, 211–229 (2002) MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Teel, A.: A nonlinear small gain theorem for the analysis of control systems with saturations. IEEE Transactions on Automatic Control 41, 1256–1270 (1996) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Teel, A.: Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Transactions on Automatic Control 43(7), 960–964 (1998) MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Teel, A.: Input-to-state stability and the nonlinear small gain theorem. Preprint (2003) Google Scholar
  52. 52.
    Zames, G.: On the input-output stability of time-varying nonlinear feedback systems. Part I: Conditions using concepts of loop gain, conicity and positivity. IEEE Transactions on Automatic Control 11, 228–238 (1966) CrossRefGoogle Scholar
  53. 53.
    Zames, G.: On the input-output stability of time-varying nonlinear feedback systems. Part II: Conditions involving circles in the frequency plane and sector nonlinearities. IEEE Transactions on Automatic Control 11, 465–476 (1966) CrossRefGoogle Scholar
  54. 54.
    Zhou, K., Doyle, J.C., Glover, K.: Robust Optimal Control. Prentice-Hall, New Jersey (1996) MATHGoogle Scholar

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