Stability Analysis of Nonlinear Systems with Linear Programming

The Lyapunov theory of dynamical systems is the most useful general theory for studying the stability of nonlinear systems. It includes two methods, Lyapunov's indirect method and Lyapunov's direct method. Lyapunov's indirect method states that the dynamical system ˙ x = f (x), (1) where f (0) = 0, has a locally exponentially stable equilibrium point at the origin, if and only if the real parts of the eigenvalues of the Jacobian matrix of f at zero are all strictly negative. Lyapunov's direct method is a mathematical extension of the fundamental physical observation , that an energy dissipative system must eventually settle down to an equilibrium point. It states that if there is an energy-like function V for (1) that is strictly decreasing along its trajectories, then the equilibrium at the origin is asymptotically stable. The function V is then said to be a Lyapunov function for the system. A Lyapunov function provides via its preimages a lower bound of the region of attraction of the equilibrium. This bound is non-conservative in the sense, that it extends to the boundary of the domain of the Lyapunov function. Although these methods are very powerful they have major drawbacks. The indirect method delivers a proposition of purely local nature. In general one does not have any idea how large the region of attraction might be. It follows from the direct method, that one can extract important information regarding the stability of the equilibrium at the origin if one has a Lyapunov function for the system, but it does not provide any method to gain it. In this thesis we will tackle these drawbacks via linear programming. The advantage of using linear programming is that algorithms to solve linear programs, like the simplex algorithm used here, are fast in practice. A further advantage is that open source and commercial software to solve linear programs is readily available. Part I contains mathematical preliminaries. In Chapter 1 a brief review of the theory of continuous autonomous dynamical systems and some stability concepts of their equilibrium points is given. We will explain why such systems are frequently encountered in science and engineering and why the concept of stability for their equilibrium points is so important. We will introduce Dini derivatives, a generalization of the classical derivative, and we will prove Lyapunov's direct method with less restrictive assumptions of the Lyapunov function than usually done in textbooks …