Skip to main content

The ISS philosophy as a unifying framework for stability-like behavior

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 259))

Abstract

The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography is also included, listing extensions, applications, and other current work.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Angeli, D., “Input-to-state stability of PD-controlled robotic systems,” Automatica 35 (1999): 1285–1290.

    Article  MATH  MathSciNet  Google Scholar 

  2. Angeli, D., “Intrinsic robustness of global asymptotic stability,” Systems & Control Letters 38(1999): 297–307

    Article  MATH  MathSciNet  Google Scholar 

  3. Angeli, D., “A Lyapunov approach to incremental stability properties,” submitted for publication, January 2000.

    Google Scholar 

  4. Angeli, D., and E.D. Sontag, “Forward completeness, unboundedness observability, and their Lyapunov characterizations,” Systems and Control Letters 38(1999): 209–217.

    Article  MATH  MathSciNet  Google Scholar 

  5. Angeli, D., E.D. Sontag, and Y. Wang, “A characterization of integral input to state stability,” IEEE Trans. Autom. Control, to appear.

    Google Scholar 

  6. Angeli, D., E.D. Sontag, and Y. Wang, “Further equivalences and semiglobal versions of integral input to state stability,” Dynamics and Control, to appear.

    Google Scholar 

  7. Christofides, P.D., and A.R. Teel, “Singular perturbations and input-to-state stability,” IEEE Trans. Automat. Control 41(1996): 1645–1650.

    Article  MATH  MathSciNet  Google Scholar 

  8. Coron, J.M., L. Praly, and A. Teel, “Feedback stabilization of nonlinear systems: sufficient conditions and Lyapunov and input-output techniques,” in Trends in Control, A. Isidori, ed., Springer-Verlag, London, 1995.

    Google Scholar 

  9. Deng, H., and M. Krstić, “Output-feedback stablization of stochastic nonlinear systems driven by noise of unknown covariance,” Systems Control Lett. 39(2000): 173–182.

    Article  MATH  MathSciNet  Google Scholar 

  10. Edwards, H., Y. Lin, and Y. Wang, “On input-to-state stability for time varying nonlinear systems,” submitted, February 2000.

    Google Scholar 

  11. Fah, N.C.S., “Input-to-state stability with respect to measurement disturbances for one-dimensional systems,” Control, Optimisation and Calculus of Variations 4(1999): 99–121.

    Article  MATH  MathSciNet  Google Scholar 

  12. Freeman, R.A., “Global internal stabilizability does not imply global external stabilizability for small sensor disturbances,” IEEE Trans. Automat. Control 40(1996): 2119–2122.

    Article  MathSciNet  Google Scholar 

  13. Freeman, R.A., and P.V. Kokotovi'c, Robust Nonlinear Control Design, State-Space and Lyapunov Techniques, Birkhauser, Boston, 1996.

    MATH  Google Scholar 

  14. Fujimoto, K., and T. Sugie, “State-space characterization of Youla parametrization for nonlinear systems based on input-to-state stability”, Proc. 37th IEEE Conf. Decision and Control, Tampa, Dec. 1998, pp. 2479–2484.

    Google Scholar 

  15. Grüne, L., “Input-to-state stability of exponentially stabilized semilinear control systems with inhomogeneous perturbations,” System & Control Letters 38(1999): 27–35.

    Article  MATH  Google Scholar 

  16. Grune, L., F.R. Wirth, and E.D. Sontag, “Asymptotic stability equals exponential stability, and ISS equals finite energy gain — if you twist your eyes,” Systems and Control Letters 38 (1999): 127–134.

    Article  MathSciNet  Google Scholar 

  17. Hespanha, J.P, and A.S. Morse, “Certainty equivalence implies detectability,” Systems and Control Letters 36(1999): 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  18. Hespanha, J.P, and A.S. Morse, “Supervisory control of integral-input-to-state stabilizing controllers,” Proc. of the 5 th European Control Conference, Karlsruhe, September 1999.

    Google Scholar 

  19. Hu, X.M., “On state observers for nonlinear systems,” Systems & Control Letters 17 (1991), pp. 645–473.

    Article  Google Scholar 

  20. Ingalls, B., and E.D. Sontag, “A purely input/output version of the IOS small gain theorem,” in preparation.

    Google Scholar 

  21. Ingalls, B., E.D. Sontag, and Y. Wang, in preparation.

    Google Scholar 

  22. Isidori, A., “Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems,” Systems & Control Letters 28(1996): 115–122.

    Article  MATH  MathSciNet  Google Scholar 

  23. Isidori, A., Nonlinear Control Systems, Third Edition, Springer-Verlag, London, 1995.

    MATH  Google Scholar 

  24. Isidori, A., Nonlinear Control Systems II, Springer-Verlag, London, 1999.

    MATH  Google Scholar 

  25. Jiang, Z.-P., and I.M. Mareels, “A small-gain control method for nonlinear cascaded systems with dynamic uncertainties,” IEEE Trans. Automat. Control 42(1997): 292–308.

    Article  MATH  MathSciNet  Google Scholar 

  26. Jiang, Z.-P., and L. Praly, “Preliminary results about robust Lagrange stability in adaptive nonlinear regulation,” Int. J. of Adaptive Control and Signal Processing 6(1992): 285–307.

    Article  MATH  Google Scholar 

  27. Jiang, Z.-P., A. Teel, and L. Praly, “Small-gain theorem for ISS systems and applications,” Mathematics of Control, Signals, and Systems 7(1994): 95–120.

    Article  MATH  MathSciNet  Google Scholar 

  28. Jiang, Z.-P., F. Khorrami, and D.J. Hill, “Decentralized output-feedback control with disturbance attenuation for large-scale nonlinear systems,” Proc. 38th IEEE Conf. Decision and Control, Phoenix, Dec. 1999, pp. 3271–3276.

    Google Scholar 

  29. Jiang, Z-P., E.D. Sontag, and Y. Wang, “Input-to-state stability for discrete-time nonlinear systems,” in Proc. 14th IFAC World Congress (Beijing), Vol E, pp. 277–282, 1999.

    Google Scholar 

  30. Jiang, Z-P., and Y. Wang, “A converse lyapunov theorem and input-to-state stability properties for discrete-time nonlinear systems,” Automatica, to appear.

    Google Scholar 

  31. Kazakos, D., and J. Tsinias, “The input-to-state stability condition and global stabilization of discrete-time systems,” IEEE Trans. Automat. Control 39(1994): 2111–13.

    Article  MATH  MathSciNet  Google Scholar 

  32. Khalil, H.K., Nonlinear Systems, Second Edition, Prentice-Hall, Upper Saddle River, NJ, 1996.

    Google Scholar 

  33. Kokotović, P. and M. Arcak, “Constructive nonlinear control: progress in the 90's,” Invited Plenary Talk, IFAC Congress, in Proc. 14th IFAC World Congress, the Plenary and Index Volume, pp. 49–77, Beijing, 1999.

    Google Scholar 

  34. Krener, A.J., “A Lyapunov theory of nonlinear observers,” in Stochastic analysis, control, optimization and applications, Birkhäuser Boston, Boston, MA, 1999, pp. 409–420.

    Google Scholar 

  35. Krichman, M., A Lyapunov approach to detectability of nonlinear systems, Ph.D. Thesis (E. Sontag, advisor), Rutgers University, Mathematics Department, January 2000.

    Google Scholar 

  36. Krichman, M., E.D. Sontag, and Y. Wang, “Input-output-to-state stability,” submitted.

    Google Scholar 

  37. Krstić, M., and H. Deng, Stabilization of Uncertain Nonlinear Systems, Springer-Verlag, London, 1998.

    Google Scholar 

  38. Krstić, M., I. Kanellakopoulos, and P.V. Kokotović, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, 1995.

    Google Scholar 

  39. Krstić, M., and Z.H. Li, “Inverse optimal design of input-to-state stabilizing nonlinear controllers,” IEEE Trans. Automat. Control 43(1998): 336–350.

    Article  MathSciNet  Google Scholar 

  40. Liberzon, D., “Nonlinear stabilization by hybrid quantized feedback,” in Proc. Hybrid Systems 2000, Pittsburgh, March 2000, to appear.

    Google Scholar 

  41. Liberzon, D., A.S. Morse, and E.D. Sontag, “A new definition of the minimum-phase property for nonlinear systems, with an application to adaptive control,” submitted.

    Google Scholar 

  42. Liberzon, D., E.D. Sontag, and Y. Wang, “On integral-input-to-state stabilization,” in Proc. American Control Conf., San Diego, June 1999, pp. 1598–1602.

    Google Scholar 

  43. Lin, Y., Lyapunov Function Techniques for Stabilization, Ph.D. Thesis (E. Sontag, advisor), Rutgers University, Mathematics Department, October 1992.

    Google Scholar 

  44. Lin, Y., E.D. Sontag, and Y. Wang, “Input to state stabilizability for parameterized families of systems,” Intern. J. Robust & Nonlinear Control 5 (1995): 187–205.

    Article  MathSciNet  Google Scholar 

  45. Lin, Y., E.D. Sontag, and Y. Wang, “A smooth converse Lyapunov theorem for robust stability,” SIAM J. Control and Optimization 34 (1996): 124–160.

    Article  MATH  MathSciNet  Google Scholar 

  46. Lohmiller, W., and J-J.E. Slotine, “On contraction analysis for non-linear systems,” Automatica 34(1998): 683–696.

    Article  MATH  MathSciNet  Google Scholar 

  47. Lu, W.M., “A class of globally stabilizing controllers for nonlinear systems,” Systems & Control Letters 25 (1995), pp. 13–19.

    Article  MATH  MathSciNet  Google Scholar 

  48. Lu, W.M., “A state-space approach to parameterization of stabilizing controllers for nonlinear systems,” IEEE Trans. Automat. Control 40 (1995): 1576–1588.

    Article  MATH  MathSciNet  Google Scholar 

  49. Marino, R., G. Santosuosso, and P. Tomei, “Robust adaptive observers for nonlinear systems with bounded disturbances,” Proc. 38th IEEE Conf. Decision and Control, Phoenix, Dec. 1999, pp. 5200–5205.

    Google Scholar 

  50. Marino, R., and P. Tomei, “Nonlinear output feedback tracking with almost disturbance decoupling,” IEEE Trans. Automat. Control 44(1999): 18–28.

    Article  MATH  MathSciNet  Google Scholar 

  51. Morse, A.S., “Control using logic-based switching,” in Trends in Control: A European Perspective, A. Isidori, ed., Springer, London, 1995, 69–114.

    Google Scholar 

  52. Nešić, D., and E.D. Sontag, “Input-to-state stabilization of linear systems with positive outputs,” Systems and Control Letters 35(1998): 245–255.

    Article  MathSciNet  Google Scholar 

  53. Nešić, D., A.R. Teel, and E.D. Sontag, “Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems,” Systems and Control Letters 38 (1999): 49–60.

    Article  MathSciNet  Google Scholar 

  54. Nešić, D., and A.R. Teel, ‘Input-to-state stability for nonlinear time-varying systems via averaging”, submitted for publication, 1999.

    Google Scholar 

  55. Pan, D.J., Z.Z. Han, and Z.J. Zhang, “Bounded-input-bounded-output stabilization of nonlinear systems using state detectors”, Systems Control Lett., 21(1993): 189–198.

    Article  MATH  MathSciNet  Google Scholar 

  56. Pogromsky A.Yu., T. Glad and H. Nijmeijer “On diffusion driven oscillations in coupled dynamical systems,” Int. J. Bifurcation Chaos 9(1999): 629–644.

    Article  MATH  MathSciNet  Google Scholar 

  57. Praly, L., and Y. Wang, “Stabilization in spite of matched unmodelled dynamics and an equivalent definition of input-to-state stability,” Mathematics of Control, Signals, and Systems 9(1996): 1–33.

    Article  MATH  MathSciNet  Google Scholar 

  58. Rumyantsev, V.V., and A.S. Oziraner, Stability and Stabilization of Motion with Respect to Part of the Variables (in Russian), Nauka, Moscow, 1987.

    Google Scholar 

  59. Sanchez, E.N., and J.P. Perez, “Input-to-state stability (ISS) analysis for dynamic neural networks,” IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications, 46(1999): 1395–1398.

    Article  MATH  MathSciNet  Google Scholar 

  60. Sepulchre, R., M. Jankovic, P.V. Kokotović, Constructive Nonlinear Control, Springer, 1997.

    Google Scholar 

  61. Sepulchre, R., M. Jankovic, and P.V. Kokotović, “Integrator forwarding: a new recursive nonlinear robust design,” Automatica 33 (1997): pp. 979–984.

    Article  MATH  Google Scholar 

  62. Shiriaev, A.S., “The notion of V-detectability and stabilization of invariant sets of nonlinear systems,” Proc. 37th IEEE Conf. Decision and Control, Tampa, Dec. 1998, pp. 2509–2514.

    Google Scholar 

  63. Sontag, E.D., “Smooth stabilization implies coprime factorization,” IEEE Trans. Automatic Control 34(1989): 435–443.

    Article  MATH  MathSciNet  Google Scholar 

  64. Sontag, E.D., “Some connections between stabilization and factorization,” Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1989, IEEE Publications, 1989, pp. 990–995.

    Google Scholar 

  65. Sontag, E.D., “Remarks on stabilization and input-to-state stability,” Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1989, IEEE Publications, 1989, pp. 1376–1378.

    Google Scholar 

  66. Sontag, E.D., “Further facts about input to state stabilization”, IEEE Trans. Automatic Control 35(1990): 473–476.

    Article  MATH  MathSciNet  Google Scholar 

  67. Sontag, E.D., “Comments on integral variants of ISS,” Systems and Control Letters 34(1998): 93–100.

    Article  MATH  MathSciNet  Google Scholar 

  68. Sontag, E.D., “Stability and stabilization: Discontinuities and the effect of disturbances,” in Nonlinear Analysis, Differential Equations, and Control (Proc. NATO Advanced Study Institute, Montreal, Jul/Aug 1998; F.H. Clarke and R.J. Stern, eds.), Kluwer, Dordrecht, 1999, pp. 551–598.

    Google Scholar 

  69. Sontag, E.D., and Y. Wang, “On characterizations of the input-to-state stability property,” Systems and Control Letters 24 (1995): 351–359.

    Article  MATH  MathSciNet  Google Scholar 

  70. Sontag, E.D., and Y. Wang, “On characterizations of input-to-state stability with respect to compact sets,” in Proceedings of IFAC Non-Linear Control Systems Design Symposium, (NOLCOS’ 95), Tahoe City, CA, June 1995, pp. 226–231.

    Google Scholar 

  71. Sontag, E.D., and Y. Wang, “Various results concerning set input-to-state stability,” Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pp. 1330–1335.

    Google Scholar 

  72. Sontag, E.D., and Y. Wang, “Detectability of nonlinear systems,” in Proc. Conf. on Information Sciences and Systems (CISS 96), Princeton, NJ, 1996, pp. 1031–1036.

    Google Scholar 

  73. Sontag, E.D., and Y. Wang, “New characterizations of input to state stability,” IEEE Trans. Autom. Control 41(1996): 1283–1294.

    Article  MATH  MathSciNet  Google Scholar 

  74. Sontag, E.D., and Y. Wang, “A notion of input to output stability,” Proc. European Control Conf., Brussels, July 1997, Paper WE-E A2, CD-ROM file ECC958.pdf, 6 pages.

    Google Scholar 

  75. Sontag, E.D., and Y. Wang, “Output-to-state stability and detectability of nonlinear systems,” Systems and Control Letters 29(1997): 279–290.

    Article  MATH  MathSciNet  Google Scholar 

  76. Sontag, E.D., and Y. Wang, “Notions of input to output stability,” Systems and Control Letters 38(1999): 235–248.

    Article  MATH  MathSciNet  Google Scholar 

  77. Sontag, E.D., and Y. Wang, “Lyapunov characterizations of input to output stability,” SIAM J. Control and Opt., to appear.

    Google Scholar 

  78. Sussmann, H.J., E.D. Sontag, and Y. Yang, “A general result on the stabilization of linear systems using bounded controls,” IEEE Trans. Autom. Control 39(1994): 2411–2425.

    Article  MATH  MathSciNet  Google Scholar 

  79. Teel, A.R., “Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem,” IEEE Trans. Automat. Control 43(1998): 960–964.

    Article  MATH  MathSciNet  Google Scholar 

  80. Teel, A.R., D. Nešić, and P.V. Kokotović, “A note on input-to-state stability of sampled-data nonlinear systems,” Proc. 37th IEEE Conf. Decision and Control, Tampa, Dec. 1998, pp. 2473–2479.

    Google Scholar 

  81. Teel, A.R., and L. Praly, “On assigning the derivative of a disturbance attenuation control,” submitted, 1999.

    Google Scholar 

  82. Teel, A.R., and L. Praly, “Results on converse Lyapunov functions from class-KL estimates,” Proc. 38th IEEE Conf. Decision and Control, Phoenix, Dec. 1999, pp. 2545–2550.

    Google Scholar 

  83. Teel, A., and E.D. Sontag, “Changing supply functions in input/state stable systems,” IEEE Trans. Autom. Control 40(1995): 1476–1478.

    Article  MATH  MathSciNet  Google Scholar 

  84. Tsinias, J., “Sontag's ‘input to state stability condition’ and global stabilization using state detection,” Systems Control Lett., 20(1993): 219–226.

    Article  MATH  MathSciNet  Google Scholar 

  85. Tsinias, J., “Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation,” ESAIM Control Optim. Calc. Var. 2(1997): 57–85.

    Article  MATH  MathSciNet  Google Scholar 

  86. Tsinias, J., “Stochastic input-to-state stability and applications to global feedback stabilization,” Int. J. Control 71(1998): 907–930.

    Article  MATH  MathSciNet  Google Scholar 

  87. Tsinias, J., and I. Karafyllis, “ISS property for time-varying systems and application to partial-static feedback stabilization and asymptotic tracking,” IEEE Trans. Automat. Control 44(1999): 2173–2184.

    Article  MathSciNet  Google Scholar 

  88. Su, W., L. Xie, and Z. Gong, “Robust input to state stabilization for minimumphase nonlinear systems,” Int J. Control 66(1997): 825–842.

    Article  MATH  MathSciNet  Google Scholar 

  89. van der Schaft, A., “L 2 Gain and Passivity Techniques in Nonlinear Control,” Springer-Verlag, London, 1999.

    Google Scholar 

  90. Vorotnikov, V.I., “Stability and stabilization of motion: Research approaches, results, distinctive characteristics,” Automation and Remote Control 54 (1993): 339–397.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Alberto Isidori Françoise Lamnabhi-Lagarrigue Witold Respondek

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag London Limited

About this paper

Cite this paper

Sontag, E.D. (2001). The ISS philosophy as a unifying framework for stability-like behavior. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds) Nonlinear control in the year 2000 volume 2. Lecture Notes in Control and Information Sciences, vol 259. Springer, London. https://doi.org/10.1007/BFb0110320

Download citation

  • DOI: https://doi.org/10.1007/BFb0110320

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-364-5

  • Online ISBN: 978-1-84628-569-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics