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Input to State Stability: Basic Concepts and Results

  • Chapter
Nonlinear and Optimal Control Theory

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1932))

The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decomposition approaches can result in effective and easily computable control laws. These two aspects are complementary, since the latter approaches are, typically, based upon the inductive verification of the validity of the former system properties under compositions (in the terminology used in [62], the “activation” of theoretical concepts leads to “constructive” control).

This expository presentation addresses the first of these aspects, and in particular the precise formulation of questions of robustness with respect to disturbances, formulated in the paradigm of input to state stability. We provide an intuitive and informal presentation of the main concepts. More precise statements, especially about older results, are given in the cited papers, as well as in several previous surveys such as [103, 105] (of which the present paper represents an update), but we provide a little more detail about relatively recent work. Regarding applications and extensions of the basic framework, we give some pointers to the literature, but we do not focus on feedback design and specific engineering problems; for the latter we refer the reader to textbooks such as [27, 43, 44, 58, 60, 66, 96].

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References

  1. Aamo, O.M., A. Balogh, M. Krstić, “Optimal mixing by feedback in pipe flow,” in Proceedings of the 15th IFAC World Congress on Automatic Con-trol, Barcelona, Spain, 2002.

    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., “Input-to-state stability of PD-controlled robotic systems,” Auto-matica, 35 (1999): 1285-1290.

    MATH  MathSciNet  Google Scholar 

  4. Angeli, D., “A Lyapunov approach to incremental stability properties” IEEE Transactions on Automatic Control 47 (2002): 410-422.

    Article  MathSciNet  Google Scholar 

  5. Angeli, D., “An almost global notion of input-to-state stability” IEEE Trans-actions on Automatic Control 49 (2004): 866-874.

    Article  MathSciNet  Google Scholar 

  6. Angeli, D., B. Ingalls, E.D. Sontag, Y. Wang,“Separation principles for input-output and integral-input to state stability,” SIAM J. Control and Opt. 43 (2004): 256-276.

    Article  MATH  MathSciNet  Google Scholar 

  7. Angeli, D., B. Ingalls, E.D. Sontag, Y. Wang, “Uniform global asymptotic stability of differential inclusions,” Journal of Dynamical and Control Systems 10 (2004): 391-412.

    Article  MATH  MathSciNet  Google Scholar 

  8. Angeli, D., D. Nesic, “Power formulations of input to state stability notions,” in Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2001, IEEE Publi-cations, 2001, pp. 894-898.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Angeli, D., E.D. Sontag, Y. Wang, “Input-to-state stability with respect to in-puts and their derivatives,” Intern. J. Robust and Nonlinear Control 13 (2003): 1035-1056.

    Article  MATH  MathSciNet  Google Scholar 

  11. Angeli, D., E.D. Sontag, Y. Wang, “A characterization of integral input to state stability,” IEEE Trans. Autom. Control 45 (2000): 1082-1097.

    Article  MATH  MathSciNet  Google Scholar 

  12. Angeli, D., E.D. Sontag, Y. Wang, “Further equivalences and semiglobal ver-sions of integral input to state stability,” Dynamics and Control 10 (2000): 127-149.

    Article  MATH  MathSciNet  Google Scholar 

  13. Arcak, M., D. Angeli, E.D. Sontag, “A unifying integral ISS framework for sta-bility of nonlinear cascades,” SIAM J. Control and Opt. 40 (2002): 1888-1904.

    Article  MATH  MathSciNet  Google Scholar 

  14. Arcak, M., A. Teel, “Input-to-state stability for a class of Lurie systems,” Automatica 38 (2002): 1945-1949.

    Article  MATH  MathSciNet  Google Scholar 

  15. Arslan, G., T. Basar, “Disturbance attenuating controller design for strict-feedback systems with structurally unknown dynamics,” Automatica 37 (2001): 1175-1188.

    Article  MATH  MathSciNet  Google Scholar 

  16. Astolfi, A., L. Praly, “Global complete observability and output-to-state sta-bility imply the existence of a globally convergent observer,” in Proc. IEEE Conf. Decision and Control, Maui, Dec. 2003, IEEE Publications, 2003, pp. 1562-1567.

    Google Scholar 

  17. Besanon, G., S. Battilotti, L. Lanari, “A new separation result for a class of quadratic-like systems with application to Euler-Lagrange models,” Automat-ica 39 (2003): 1085-1093.

    Article  Google Scholar 

  18. Brockett, R.W., “Asymptotic stability and feedback stabilization,” in Dif-ferential Geometric Control theory (R.W. Brockett, R.S. Millman, and H.J. Sussmann, eds.), Birkhauser, Boston, 1983, pp. 181-191.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Coron, J.M., L. Praly, 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 

  21. Deng, H., and M. Krstić, “Output-feedback stabilization of stochastic nonlin- ear systems driven by noise of unknown covariance,” Systems Control Lett. 39 (2000): 173-182.

    Article  MATH  MathSciNet  Google Scholar 

  22. Deng, H., M. Krstic, R.J. Williams, “Stabilization of stochastic nonlinear sys- tems driven by noise of unknown covariance,” IEEE Trans. Automat. Control 46 (2001): 1237-1253.

    Article  MATH  MathSciNet  Google Scholar 

  23. Doyle, J., B. Francis, A. Tannenbaum, Feedback Control Systems, MacMil- lan Publishing Co, 1992. Also available on the web at: http://www.control. utoronto.capeople/profs/francis/dft.html.

  24. Edwards, H., Y. Lin, Y. Wang, “On input-to-state stability for time varying nonlinear systems,” in Proc. 39th IEEE Conf. Decision and Control, Sydney, Australia, 2000, pp.3501-3506.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. 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 

  27. Freeman, R.A., P.V. Kokotović, Robust Nonlinear Control Design, State-Space and Lyapunov Techniques, Birkhauser, Boston, 1996.

    MATH  Google Scholar 

  28. Fujimoto, K., 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 

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

    Article  MATH  Google Scholar 

  30. Grüne, L., Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization. Lecture Notes in Mathematics, Vol. 1783. Springer-Verlag, Heidelberg, 2002.

    Google Scholar 

  31. Grüne, L., “Input-to-state dynamical stability and its Lyapunov function char-acterization,” IEEE Trans. Autom. Control 47 (2002): 1499-1504.

    Article  Google Scholar 

  32. Grüne, L., “Attraction rates, robustness and discretization of attractors,” SIAM J. Numer. Anal. 41 (2003): 2096-2113.

    Article  MATH  Google Scholar 

  33. Grune, L., E.D. Sontag, F.R. Wirth, “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 

  34. Hespanha, J.P., D. Liberzon, D. Angeli, E.D. Sontag, “Nonlinear observability notions and stability of switched systems,” IEEE Trans. Autom. Control, 2005, to appear.

    Google Scholar 

  35. Hespanha, J.P., D. Liberzon, A.S. Morse, “Supervision of integral-input-to-state stabilizing controllers,” Automatica 38 (2002): 1327-1335.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  37. Hu, X.M., “On state observers for nonlinear systems,” Systems & Control Let- ters 17 (1991), pp. 645-473.

    Google Scholar 

  38. Ingalls, B., E.D. Sontag, Y. Wang, “Measurement to error stability: a notion of partial detectability for nonlinear systems,” in Proc. IEEE Conf. Decision and Control, Las Vegas, Dec. 2002, IEEE Publications, 2002, pp. 3946-3951.

    Google Scholar 

  39. Ingalls, B., E.D. Sontag, Y. Wang, “A relaxation theorem for differential inclu-sions with applications to stability properties,” in Proc. 15th Int. Symp. Math-ematical Theory of Networks and Systems (MTNS 2002), CD-ROM, FM5.3.

    Google Scholar 

  40. Ingalls, B., E.D. Sontag, “A small-gain lemma with applications to input/ output systems, incremental stability, detectability, and interconnections,” J. Franklin Institute 339 (2002): 211-229.

    Article  MATH  MathSciNet  Google Scholar 

  41. Ingalls, B., E.D. Sontag, Y. Wang, “An infinite-time relaxation theorem for differential inclusions,” Proc. Amer. Math. Soc. 131 (2003): 487-499.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  44. Isidori, A., L. Marconi, A. Serrani, Robust Autonomous Guidance: An Internal Model-Based Approach, Springer-Verlag, London, 2003.

    Google Scholar 

  45. Ito, H., “New characterization and solution of input-to-state stabilization: a state-dependent scaling approach,” Automatica 37 (2001): 1663-1670.

    Article  MATH  Google Scholar 

  46. Ito, H., “Scaling supply rates of ISS systems for stability of feedback intercon-nected nonlinear systems,” in Proc. IEEE Conf. Decision and Control, Maui, Dec. 2003, IEEE Publications, 2003, pp. 5074-5079.

    Google Scholar 

  47. Jiang, Z.-P., “A combined backstepping and small-gain approach to adaptive output feedback control,” Automatica 35 (1999): 1131-1139.

    Article  MATH  Google Scholar 

  48. Jiang, Z.-P., I.M. Mareels, D.J. Hill, Jie Huang, “A unifying framework for global regulation via nonlinear output feedback: from ISS to integral ISS,” IEEE Trans. Automat. Control 49 (2004): 549-562.

    Article  MathSciNet  Google Scholar 

  49. Jiang, Z.-P., F. Khorrami, 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 

  50. Jiang, Z.-P., L. Praly, “Preliminary results about robust Lagrange stability in adaptive nonlinear regulation.” Intern. J. Control 6 (1992): 285-307.

    MATH  Google Scholar 

  51. Jiang, Z.-P., D.W. Repperger, D.J. Hill, “Decentralized nonlinear output-feedback stabilization with disturbance attenuation,” IEEE Trans. Automat. Control 46 (2001): 1623-1629.

    Article  MATH  MathSciNet  Google Scholar 

  52. Jiang, Z.-P., E.D. Sontag, 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  54. Jiang, Z.-P., Y. Wang, “Small gain theorems on input-to-output stability”, in Proc. of the Third International DCDIS Conference on Engineering Applications and Computational Algorithms, pp. 220-224, 2003.

    Google Scholar 

  55. Jiang, Z.-P., Y. Wang, “A converse Lyapunov theorem for discrete time systems with disturbances”, Systems & Control Letters, 45 (2002): 49-58.

    Article  MATH  MathSciNet  Google Scholar 

  56. Jiang, Z.P., Y. Wang, “Input-to-state stability for discrete-time nonlinear sys-tems”, Automatica, 37 (2001): 857-869.

    Article  MATH  MathSciNet  Google Scholar 

  57. Kaliora, G., A. Astolfi, L. Praly, “Norm estimators and global output feedback stabilization of nonlinear systems with ISS inverse dynamics,” Proc. 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, Dec. 2004, paper ThC07.2, IEEE Publications, Piscataway.

    Google Scholar 

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

    Google Scholar 

  59. Krener, A.J., “A Lyapunov theory of nonlinear observers,” in Stochastic analy-sis, control, optimization and applications, Birkhäuser Boston, Boston, MA, 1999, pp. 409-420.

    Google Scholar 

  60. Krstić, M., I. Kanellakopoulos, P.V. Kokotović, Nonlinear and Adaptive Con-trol Design, John Wiley & Sons, New York, 1995.

    Google Scholar 

  61. Kazakos, D., J. Tsinias, “The input-to-state stability condition and global sta-bilization of discrete-time systems,” IEEE Trans. Automat. Control 39 (1994): 2111-2113.

    Article  MATH  MathSciNet  Google Scholar 

  62. Kokotović, P., M. Arcak, “Constructive nonlinear control: a historical perspec- tive,” Automatica 37 (2001): 637-662.

    MATH  Google Scholar 

  63. Kokotović, P., M. Arcak, “Nonlinear observers: a circle criterion design and robustness analysis,” Automatica 37 (2001): 1923-1930.

    Article  MATH  Google Scholar 

  64. Krichman, M., E.D. Sontag, “Characterizations of detectability notions in terms of discontinuous dissipation functions,” Intern. J. Control 75 (2002): 882-900.

    Article  MATH  MathSciNet  Google Scholar 

  65. Krichman, M., E.D. Sontag, Y. Wang, “Input-output-to-state stability,” SIAM J. Control and Optimization 39 (2001): 1874-1928.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  67. Laila, D.S., D. Nesic, “Changing supply rates for input-output to state sta-ble discrete-time nonlinear systems with applications,” Automatica 39 (2003): 821-835.

    Article  MATH  MathSciNet  Google Scholar 

  68. Lakshmikantham, V., S. Leela, A.A. Martyuk, Practical Stability of Nonlinear Systems, World Scientific, New Jersey, 1990.

    MATH  Google Scholar 

  69. Liberzon, D., “Nonlinear stabilization by hybrid quantized feedback,” in Proc. 3rd International Workshop on Hybrid Systems: Computation and Control, Lecture Notes in Computer Science vol. 1790 (N. Lynch and B. H. Krogh, Eds.), pp/ 243-257, Springer-Verlag, 2000.

    Google Scholar 

  70. Liberzon, D., “Output-input stability and feedback stabilization of multivari-able nonlinear control systems,” in Proc. IEEE Conf. Decision and Control, Maui, Dec. 2003, IEEE Publications, 2003, pp. 1550-1555.

    Google Scholar 

  71. Liberzon, D., A.S. Morse, E.D. Sontag, “Output-input stability and minimum-phase nonlinear systems,” IEEE Trans. Autom. Control 47 (2002): 422-436.

    Article  MathSciNet  Google Scholar 

  72. Liberzon, D., E.D. Sontag, Y. Wang, “Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation,” Systems and Control Letters 46 (2002): 111-127.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  74. Loria, A., D. Nesic, “Stability of time-varying discrete-time cascades,” in Proc. 15th. IFAC World Congress, (Barcelona, Spain), 2002, paper no. 652.

    Google Scholar 

  75. 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 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  78. Malisoff, M., F. Mazenc,“Further remarks on strict input-to-state stable lyapunov functions for time-varying systems,” arxiv.org/math/0411150, June 2004.

    Google Scholar 

  79. Malisoff, M., L. Rifford, E.D. Sontag,“Asymptotic controllability implies input to state stabilization,” SIAM J. Control and Optimization 42 (2004): 2221-2238.

    Article  MATH  MathSciNet  Google Scholar 

  80. ıa, “On converse Lyapunov theorems for ISS and iISS switched nonlinear systems,” Systems & Control Letters 42 (2001): 47-53.

    Google Scholar 

  81. ıa, E. Sontag, Y. Wang, “On the representa- tion of switched systems with inputs by perturbed control systems,” Nonlinear Analysis: Theory, Methods & Applications 60 (2005): 1111-1150.

    Article  Google Scholar 

  82. Marino, R., G. Santosuosso, 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 

  83. Marino, R., P. Tomei, “Nonlinear output feedback tracking with almost dis- turbance decoupling,” IEEE Trans. Automat. Control 44 (1999): 18-28.

    Article  MATH  MathSciNet  Google Scholar 

  84. Massera, J.L., “Contributions to stability theory,” Annals of Mathematics 64 (1956): 182-206.

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  86. Nesic, D., D. Angeli, “Integral versions of iss for sampled-data nonlinear sys-tems via their approximate discrete-time models,” IEEE Transactions on Au-tomatic Control 47 (2002): 2033-2037.

    Article  MathSciNet  Google Scholar 

  87. Nesic, D., D.S. Laila, “A note on input-to-state stabilization for nonlinear sampled-data systems”, IEEE Transactions on Automatic Control 47 (2002): 1153-1158.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  89. Nesic, D., A.R. Teel, “Input-to-state stability for nonlinear time-varying sys-tems via averaging,” Mathematics of Control, Signals and Systems 14 (2001): 257-280.

    Article  MATH  MathSciNet  Google Scholar 

  90. Nesic, D., A.R. Teel, E.D. Sontag,“Formulas relating KL stability esti- mates of discrete-time and sampled-data nonlinear systems,” Syst. Contr. Lett. 38 (1999): 49-60.

    Article  MATH  MathSciNet  Google Scholar 

  91. Ogren, P., N.E. Leonard, “Obstacle avoidance in formation,” in Proc. IEEE Conf. Robotics and Automation, Taipei, Taiwan, Sept. 2003, pp. 2492-2497.

    Google Scholar 

  92. Pan, D.J., Z.Z. Han, Z.J. Zhang, “Bounded-input-bounded-output stabilization of nonlinear systems using state detectors,” SCL 21 (1993): 189-198.

    MATH  MathSciNet  Google Scholar 

  93. Praly, L., Y. Wang “Stabilization in spiteof 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 

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

    Google Scholar 

  95. Sanchez, E.N., 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 

  96. Sepulchre, R., M. Jankovic, P.V. Kokotović, Constructive Nonlinear Control, Springer-Verlag, New York, 1997.

    MATH  Google Scholar 

  97. 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 

  98. Sontag, E.D., “Some connections between stabilization and factorization,” Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1989, IEEE Publica-tions, 1989, pp. 990-995.

    Google Scholar 

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

    Google Scholar 

  100. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  103. Sontag, E.D., “Stability and stabilization: Discontinuities and the effect of dis- turbances,” 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 

  104. Sontag, E.D., Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer, New York, 1990. Second Edition, 1998.

    Google Scholar 

  105. Sontag, E.D., “The ISS philosophy as a unifying framework for stability-like behavior,” in Nonlinear Control in the Year 2000 (Volume 2) (Lecture Notes in Control and Information Sciences, A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, eds.), Springer-Verlag, Berlin, 2000, pp. 443-468.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  108. Sontag, E.D., 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 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  113. Sontag, E.D., Y. Wang, “Lyapunov characterizations of input to output sta- bility,” SIAM J. Control and Opt. 39 (2001): 226-249.

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  115. Tanner, H.G., G.J. Pappas, V. Kumar, “Input-to-state stability on formation graphs,” in Proc. IEEE Conf. on Decision and Control, Las Vegas, December 2002.

    Google Scholar 

  116. Tanner, H.G., G.J. Pappas, V. Kumar, “Leader-to-formation stability,” IEEE Trans. on Robotics and Automation 20 (2004): 443-455.

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  118. Teel, A.R., L. Moreau, D. Nesic, “A note on the robustness of input-to-state stability,” in Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2001, IEEE Publications, 2001, pp. 875-880.

    Google Scholar 

  119. Teel, A.R., D. Nešić, 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 

  120. Teel, A.R., L. Praly, “On assigning the derivative of a disturbance attenuation control,” Mathematics of Control, Signals, and Systems 3 (2000): 95-124.

    Article  MathSciNet  Google Scholar 

  121. Su, W., L. Xie, Z. Gong, “Robust input to state stabilization for minimum- phase nonlinear systems,” Int J. Control 66 (1997): 825-842.

    Article  MATH  MathSciNet  Google Scholar 

  122. Tsinias, J., “Input to state stability properties of nonlinear systems and appli- cations to bounded feedback stabilization using saturation,” Control, Optimisation and Calculus of Variations 2 (1997): 57-85.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  125. Vorotnikov, V.I., “Problems of stability with respect to part of the variables,” Journal of Applied Mathematics and Mechanics 63 (1999): 695-703.

    Article  MathSciNet  Google Scholar 

  126. Willems, J.C., “Mechanisms for the stability and instability in feedback sys- tems,” Proc. IEEE 64 (1976): 24-35.

    Article  MathSciNet  Google Scholar 

  127. Xie, W., C. Wen, Z. Li, “Controller design for the integral-input-to-state stabi-lization of switched nonlinear systems: a Cycle analysis method,” Proc. IEEE Conf. Decision and Control, Las Vegas, Dec. 2002, IEEE Publications, 2002, pp. 3628-3633.

    Google Scholar 

  128. Xie, W., C. Wen, Z. Li, “Input-to-state stabilization of switched nonlinear systems,” IEEE Trans. Automatic Control 46 (2001): 1111-1116.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, 08854-8019, Piscataway, NJ, USA

    Eduardo D. Sontag

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  1. Dipartimento di Ingegneria dell'Informazione, Università di Siena, via Roma 56, 53100, Siena, Italy

    Paolo Nistri

  2. Dipartimento di Matematica Applicata “G. Sansone”, Università di Firenze, via di S. Marta 3, 50139, Firenze, Italy

    Gianna Stefani

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Sontag, E.D. (2008). Input to State Stability: Basic Concepts and Results. In: Nistri, P., Stefani, G. (eds) Nonlinear and Optimal Control Theory. Lecture Notes in Mathematics, vol 1932. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77653-6_3

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