State-Dependent Scaling Problems and Stability of Interconnected iISS and ISS Systems

Abstract

This paper addresses the problem of establishing stability of nonlinear interconnected systems. This paper introduces a mathematical formulation of the state-dependent scaling problems whose solutions directly provide Lyapunov functions proving stability properties of interconnected dissipative systems in a unified manner. Stability criteria are interpreted as sufficient conditions for the existence of solutions to the state-dependent scaling problems. Computing solutions to the problems is straightforward for systems covered by classical stability criteria. It, however, could be too difficult for systems with strong nonlinearity. The main purpose of this paper is to demonstrate the effectiveness beyond formal applicability by focusing on interconnected integral input-to-state stable (iISS) systems and input-to-state stable (ISS) systems. This paper derives small-gain-type theorems for interconnected systems involving iISS systems from the state-dependent scaling formulation. This paper provides solutions and Lyapunov functions explicitly. The new framework seamlessly generalizes the ISS smallgain theorem and classical stability criteria such as the smallgain theorem, the passivity theorems, the circle, and Popov criteria. State-dependence of the scaling is crucial for effective treatment of essential nonlinearities, while constants are sufficient for classical nonlinearities.Index Terms-Dissipation, input-to-state stability, integral input-to-state stability, Lyapunov function, nonlinear interconnected system, small-gain condition.

Revisiting the IISS Small‐Gain Theorem through Transient Plus ISS Small‐Gain Regulation

“…In fact, when there exists i ∈ {1,2} such that lims→•ai(s) < • holds, the previously existing results show only the iISS of the interconnected system [6,10]. Hence, in contrast to the GAS case, the condition lims→•a2(s) > lims→• 2(s) is not sufficient for resorting to the ISS small-gain argument in the presence of external inputs.…”

“…For w ∈ K\K•, its inverse w -1 is defined on the finite interval [0,limt→•w (t)) since the continuous function w is strictly increasing and w (0) = 0. Following the convention employed in [6,10], in this paper, s < limt→•w (t) or…”

“…The property (6) implied by the small-gain condition (4) again plays a key role in implementing the idea of a transient plus the ISS smallgain argument. The proof consists of two parts.…”

Dynamic vs static scaling: an existence result

“…It has been used, for example, to establish a Lyapunov proof of the reduction principle arising in center manifold theory, see Carr (1981); Khalil (2002); in the study of stability properties of interconnected systems, see Jiang et al (1996Jiang et al ( , 1994; Sontag and Teel (1995); Angeli and Astolfi (2007); Ito (2006); Ito and Jiang (2009); in the design of stabilizing control laws for cascaded or feedback interconnected systems, see Mazenc and Praly (1996); Jankovic et al (1996), and in adaptive control systems, Krstic et al (1995); Jiang (1999); Astolfi et al (2008). Informally, the idea of Lyapunov function scaling can be described as follows.…”

ISS of interconnected impulsive systems with and without time-delays

“…The first results on the ISS property for the delay-free case were given for two coupled continuous systems in Jiang et al [1994] and for an arbitrarily large number (n ∈ N) of coupled systems in Dashkovskiy et al [2007], using a smallgain condition. Lyapunov versions of the ISS small-gain theorems were proved in Jiang et al [1996] (two systems) and (n systems), for the ISDS property in Dashkovskiy and Naujok [2010], for LISS in Dashkovskiy and Rüffer [2010] and for iISS in Ito [2006] (two systems) and Ito et al [2009] (n systems), where Lyapunov functions for the overall system are constructed. ISS for interconnected hybrid systems was investigated in Nesic and Teel [2008] (two systems) and Dashkovskiy and Kosmykov [2009] (n systems).…”