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Mathematical Preliminaries
Published in Edgar N. Sanchez, Fernando Ornelas-Tellez, Discrete-Time Inverse Optimal Control for Nonlinear Systems, 2017
Edgar N. Sanchez, Fernando Ornelas-Tellez
REMARK 2.1 The connection between the existence of a Lyapunov function and the input-to-state stability is that an estimate of the gain function γ in (2.17) is γ=α1−1∘α2∘ρ, where o means composition4 of functions with α1 and α2 as defined in (2.23) [58].
On robustness of finite-time stability of homogeneous affine nonlinear systems and cascade interconnections
Published in International Journal of Control, 2022
Youness Braidiz, Denis Efimov, Andrey Polyakov, Wilfrid Perruquetti
One of the most popular robust stability properties, which was introduced in Sontag (1989), is the concept of input-to-state stability (ISS). This framework has become indispensable for various branches of nonlinear control theory, such as design of nonlinear observers (Arcak & Kokotović, 2001), robust stabilisation of nonlinear systems (Freeman & Kokotovic, 2008), etc. However, sometimes it is impossible to ensure the ISS behaviour of a closed loop system globally, and its local variant (LISS) is frequently used. Moreover, another relaxation of the ISS concept, known as integral input-to-state stability (iISS), has been proposed in Sontag (1998). The following interpretation of these notions is possible: while the state of an ISS system is small if inputs are small, the state of an iISS system is small if inputs have a finite energy. Moreover, every ISS system is necessarily iISS, but the converse is not true. It has been shown that ISS property (resp., iISS, LISS) are equivalent to the existence of a smooth ISS (resp., iISS, LISS) Lyapunov function (Sontag & Wang, 1995), which allows this framework to be widely used.
On robustness against disturbances of passive systems with multiple invariant sets
Published in International Journal of Control, 2021
N. F. Barroso, R. Ushirobira, D. Efimov, A. L. Fradkov
For the analysis of robustness, the input-to-state stability (ISS) framework (Sontag & Wang, 1995, 1996) is one of the most popular, and its development for multistable systems in terms of usual Lyapunov dissipation inequalities has been obtained in Angeli and Efimov (2015). Next, other useful stability concepts got their extension for this class of systems: the notion of detectability or output-to-state stability (OSS) was generalised in Forni and Angeli (2016b) and the integral input-to-state stability (iISS) (Angeli et al., 2000; Liberzon et al., 1999; Sontag, 1998) was extended in Forni and Angeli (2017). Specifically, a notion of iISS as the conjunction of global attractiveness with zero disturbances (0-GATT) and the uniform bounded-energy bounded-state properties (UBEBS) providing again an equivalent characterisation in terms of Lyapunov/LaSalle-like dissipation inequalities was introduced. Further research along the lines of multistable systems addressed the analysis and synthesis on specific problems such as conditions of synchronisation (Ahmed et al., 2016), stability of nonlinear cascades and feedback interconnections (Forni & Angeli, 2016a) or periodic systems (Efimov et al., 2017).
Stochastic input-to-state stability for impulsive switched stochastic nonlinear systems with multiple jumps
Published in International Journal of Systems Science, 2019
Lijun Gao, Meng Zhang, Xiuming Yao
The concepts of input-to-state stability (ISS), originally introduced in Sontag (1989, 1998), play a central role in characterising the effects of external inputs to modern nonlinear control theory. The ISS means that no matter what the initial state is, the state of the system will eventually become small if the external inputs are uniformly small. In recent years, ISS properties have received much attention, and thus, various related results have been obtained for different types of dynamical systems, such as discrete systems, hybrid systems and impulsive systems (Chen & Zheng, 2011; Hespanha, Liberzon, & Teel, 2008; Peng & Deng, 2017; Wu, Tang, & Zhang, 2016).