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Comments on integral variants of ISS. (English) Zbl 0902.93062
Summary: This note discusses two integral variants of the input-to-state stability (ISS) property, which represent nonlinear generalizations of $L^{2}$ stability, in much the same way that ISS generalizes $L^{\infty}$ stability. Both variants are equivalent to ISS for linear systems. For general nonlinear systems, it is shown that one of the new properties is strictly weaker than ISS, while the other one is equivalent to it. For bilinear systems, a complete characterization is provided of the weaker property. An interesting fact about functions of type $K{\cal L}$ is proved as well.

93D25Input-output approaches to stability of control systems
93C10Nonlinear control systems
Full Text: DOI
[1] D. Angeli, E.D. Sontag, Y. Wang, A characterization of integral input to state stability, submitted. · Zbl 0979.93106
[2] W. Hahn, Stability of Motion, Springer, Berlin, 1967. · Zbl 0189.38503
[3] J. Hale, Ordinary Differential Equations, 2nd ed.,Krieger, Malabar, 1980. · Zbl 0433.34003
[4] Isidori, A.: Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems. Systems control lett. 28, 115-122 (1996) · Zbl 0877.93055
[5] M. Krstić, I. Kanellakopoulos, P.V. Kokotović, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. · Zbl 0763.93043
[6] Praly, L.; Wang, Y.: Stabilization in spite of matched unmodelled dynamics and an equivalent definition of input-to-state stability. Math. control signals systems 9, 1-33 (1996) · Zbl 0869.93040
[7] E.D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control AC-34 (1989) 435--443. · Zbl 0682.93045
[8] E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer, New York, 1990. · Zbl 0703.93001
[9] Sontag, E. D.; Wang, Y.: On characterizations of the input-to-state stability property. Systems control lett. 24, 351-359 (1995) · Zbl 0877.93121
[10] Sontag, E. D.; Wang, Y.: New characterizations of the input to state stability property. IEEE trans. Automat. control 41, 1283-1294 (1996) · Zbl 0862.93051
[11] J. Tsinias, Sontag’s ”input to state stability condition” and global stabilization using state detection, Systems Control Lett. 20 (1993) 219--226. · Zbl 0768.93063