Introduction to Modeling and Simulation

5Stability

5.1 Introduction

In this chapter we consider the problem of stability of equilibrium states of nonlinear systems. In this context, stability refers to the response of a system if it is perturbed away from an equilibrium state. Perturbations, or disturbances, are always present in physical systems. Gusts of wind, ocean currents, tripped circuit breakers, temperature fluctuations, and so on cause a system state to deviate from an equilibrium and it is important to understand the effect of such deviations on the behavior of the system.

The intuitive notions of a stable and an unstable equilibrium are represented in Figure 5.1 showing a ball rolling in a bowl under the influence of gravity. If the ball on the left is perturbed slightly, it will stay close to the equilibrium position, whereas any perturbation of the ball on the right will cause it to roll away. We say that the equilibrium on the left is stable and the equilibrium on the right is unstable. In addition, if there is friction present, the ball on the left will settle back to the equilibrium position, which we call asymptotic stability.

In this chapter we present a rigorous treatment of stability of equilibria for systems that matches this intuitive notion of stability. We introduce the notion of stability in the sense of Lyapunov or Lyapunov stability. We also discuss an important extension of Lyapunov stability, known as LaSalle's invariance principle.

Previously, we defined exponential stability for a linear ...