Computation of Lyapunov functions and stability of interconnected systems

Lyapunov function by checking the validity of the linear inequalities from Theorems 2.1.4and 2.2.4.Theorems 2.1.7 and 2.2.8 demonstrate that this method will always succeed if the CPA function has enough structure, i.e., if the triangulation has a sufficient number of vertices, and if the Yoshizawa construction meets certain conditions.Stability of two interconnected systems and estimate of the domain of attraction For higher dimensional systems, the direct computation of Lyapunov functions by the above methods becomes very expensive.In order to avoid expensive computation, we consider the whole system as a set of interconnected subsystems.Each subsystem is considered as a dynamic system with perturbations by treating other states' influence as perturbation.We then study stability of the whole system in terms of the stability of the subsystems and their interconnection.In [78], it is summarized that one can construct a scalar or vector Lyapunov function for the whole system by imposing certain conditions on Lyapunov functions for each free subsystem (i.e., systems without inputs).In [107], small-gain-type theorems with linear gains are proposed to study general interconnected systems.If the spectral radius of the gain matrix is less than one, then the whole system is asymptotically stable.In this work, we will analyse stability of interconnected systems by iISS or ISS small gain theorems.The concept of input to state stability (ISS) was first introduced by Sontag [88] in the late 1980s and has soon turned out to be one of the most influential concepts for characterizing stability of nonlinear systems with perturbations.Various types of ISS small gain theorem were then proposed such as [17,18,19,57,59,57] where stability analysis of interconnected systems is presented.Another notion playing an important part in investigating stability of interconnected systems is integral input to state stability (iISS).The concept was first proposed in [92].The properties of iISS are described in [4].iISS small gain theorems used to analyse the stability of interconnected systems were established e.g., in [51,53].In [13], ISS Lyapunov functions in implication formulation for dynamic systems with perturbations were obtained by the introduction of a suitable auxiliary system and Zubov's method for perturbed systems proposed in [9].Stability of interconnected systems is then investigated by an ISS small gain theorem.Inspired by this idea, we propose a new technique for computing ISS Lyapunov functions in dissipative form as introduced in [73].Based on this result, we consider how to construct iISS and ISS Lyapunov functions by Zubov's method for perturbed systems in Chapter 3.In [3], the stability of two interconnected one dimensional systems is investigated.This result lays a foundation for the stability analysis of two iISS interconnected systems.Therefore, we restrict our attention to stability analysis of two interconnected systems in Chapter 3. We assume each subsystem is iISS.By introducing an auxiliary system for each subsystem which is uniformly asymptotically stable, we construct a robust Lyapunov function for the auxiliary system by Zubov's method for perturbed system.We then in Proposition 3.3.8prove that such a robust Lyapunov function for the auxiliary system is a local iISS Lyapunov function for a fixed subsystem.Based on iISS Lyapunov functions for subsystems obtained by our proposed approach, we study stability of the whole system by a small gain theorem in comparison form, cf.Theorem 3.4.3.Moreover, an estimate of the domain of attraction of interconnected systems can be obtained.who encouraged and helped me in one way or another during my Ph.D study.I express my thanks to my colleagues for their help from the Dynamic and Control group at the University of Wrzberg where I spent the first year of my Ph.D study.I also wish to thank all my colleagues from Chair of Applied Mathematics at University of Bayreuth for help, discussion and very nice work atmosphere.Special thanks go