Computation of Lyapunov functions and stability of interconnected systems

In this thesis, we investigate the problems of computation of Lyapunov functions and stability analysis of interconnected systems. In Chapter 1, preliminary results about stability, definitions of Lyapunov functions and triangulations are presented. In order to analyse stability of interconnected systems in Chapters 3 and 4, we introduce three small gain theorems. We propose a new approach of computing Lyapunov functions for dynamic systems without perturbations with an asymptotically stable equilibrium at the origin in Chapter 2. The proposed method constructs a continuous and piecewise affine (CPA) function on a compact subset of state space with the origin in its interior based on functions from classical converse Lyapunov theorems originally due to Yoshizawa, and then verifies if the vertex values satisfy linear inequalities for vertices in the subset excluding a small neighborhood of the origin. If the linear inequalities are satisfied, then the CPA function is a CPA Lyapunov function on the subset excluding a small neighborhood of the origin. Several examples are presented to show the feasibility of the approach. Since a maximal robust Lyapunov function for uniformly asymptotically stable systems can be obtained using Zubov's method, we present a new way of computing integral input-to- state stable (iISS) Lyapunov functions by Zubov's method and auxiliary systems in Chapter 3. For an iISS dynamic system with perturbation, we introduce an auxiliary system which is uniformly asymptotically stable. Then a robust Lyapunov function for the auxiliary system is computed by Zubov's method. We then prove that such a robust Lyapunov function is an iISS Lyapunov function for the original dynamic system with perturbation. We further state that the iISS Lyapunov function is a local input to state stable (ISS) Lyapunov function for the considered dynamic system with perturbations on a subset of the domain of attraction for the auxiliary system. Furthermore, stability of two interconnected iISS systems is investigated. For each subsystem, using our proposed method, iISS and ISS Lyapunov functions are constructed. Stability of the interconnected systems is then analyzed by the small gain theorem in comparison form and the small gain theorem in dissipative form, respectively. An academic example is shown to illustrate how this method is applied. In Chapter 4, we design a numerical algorithm for computing ISS Lyapunov functions for dynamic systems with perturbations. This algorithm relies on a linear optimization problem. If the linear optimization problem has a feasible solution, then the solution is proved to be a CPA ISS Lyapunov function on a spatial grid covering the given compact set excluding a small neighborhood of the origin. Since the interpolation errors are incorporated in the linear constraints, as in Chapter 2 the computed ISS Lyapunov function is a true ISS Lyapunov function rather than a numerical approximation. We prove that the linear optimization problem has a feasible solution if the system is ISS. Furthermore, we study stability of interconnected ISS systems. For each subsystem, an ISS Lyapunov function is computed by our proposed method. Since the obtained ISS Lyapunov functions satisfy linear inequalities, the stability of interconnected systems can be analyzed by the small gain theorem in linear form.