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On characterizations of the input-to-state stability property. (English) Zbl 0877.93121
Summary: We show that the well-known Lyapunov sufficient condition for “input-to-state stability” (ISS) is also necessary, settling positively an open question raised by several authors during the past few years. Additional characterizations of the ISS property, including one in terms of nonlinear stability margins, are also provided.

MSC:
93D25Input-output approaches to stability of control systems
93C10Nonlinear control systems
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References:
[1] Z.-P. Jiang, A. Teel and L. Praly, Small gain theorem for ISS systems and applications, to appear in: Math. Control Signals Systems. · Zbl 0836.93054
[2] Lin, Y.; Sontag, E. D.; Wang, Y.: A smooth converse Lyapunov theorem for robust stability. IMA preprint # 1192 (1993) · Zbl 0856.93070
[3] Praly, L.; Jiang, Z. -P.: Stabilization by output feedback for systems with ISS inverse dynamics. Systems control lett. 21, 19-34 (1993) · Zbl 0784.93088
[4] Sontag, E. D.: Smooth stabilization implies coprime factorization. IEEE trans. Automat. control 34, 435-443 (1989) · Zbl 0682.93045
[5] Sontag, E. D.: Some connections between stabilization and factorization. Proc. IEEE conf. Decision and control, 990-995 (1989)
[6] Sontag, E. D.: Further facts about input to state stabilization. IEEE trans. Automat. control 35, 473-476 (1990) · Zbl 0704.93056
[7] Tsinias, J.: Sontag’s ”input to state stability condition” and global stabilization using state detection. Systems control lett. 20, 219-226 (1993) · Zbl 0768.93063
[8] Tsinias, J.: Versions of sontag’s input to state stability condition and the global stabilizability problem. SIAM J. Control optim. 31, 928-941 (1993) · Zbl 0788.93076