- ▪E.D. Sontag, B.P. Ingalls, "A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections", J. Franklin Inst., vol. 339, no. 2, pp. 211–229, 2002. pdf
Abstract
A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.
- ▪E.D. Sontag, "Asymptotic amplitudes, Cauchy gains, an associated small-gain principle, and an application to inhibitory biological feedback", In Proc.\ IEEE Conf.\ Decision and Control, Las Vegas, Dec.\ 2002, IEEE Publications, pp. 4318–4323, 2002.
- ▪E.D. Sontag, "Asymptotic amplitudes and Cauchy gains: A small-gain principle and an application to inhibitory biological feedback", Systems Control Lett., vol. 47, no. 2, pp. 167–179, 2002. pdf
Abstract
The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops. An application in biology (MAPK signaling) is worked out in detail.