Stability Theory

The Oxford English Dictionary's definition of stable is not easily moved, changed or destroyed”. Most of us have an intuitive notion of stability which corresponds more or less with this definition. However, in order to build a theory of stability it is necessary to be more precise about terms like “not easily moved or changed”. We need to define the basic class of objects to which the notion of stability is applied and also specify the type of perturbations which are considered. In this chapter we study the stability of

state trajectories

under the influence of perturbations in the

initial state

and in the next two chapters we consider perturbations in the

system parameters

. The stability of output trajectories under the influence of perturbations in the input signal will be discussed in Volume II. The development of modern stability theories was initiated by

Maxwell

(1868) [364] and

Vyshnegradskiy

(1876) [511] in their work on governors, but the importance of the concept of stability in many other scientific fields was soon recognized and now it is a cornerstone of applied mathematics. For example, the prediction of instabilities from a mathematical model has in many instances led to a confirmation that the model adequately represents the corresponding physical process. In 1923 G.I. Taylor [492], using the Navier-Stokes equations, showed that the flow of a viscous fluid between rotating cylinders would become unstable at a particular value of a parameter, now known as the Taylor number. He confirmed this experimentally and so increased confidence in modelling viscous fluid flows by the Navier-Stokes equations. Perhaps more relevant to this text is the fact that almost all control system designs are founded on a stability requirement and our treatment of the subject will be slanted in this direction.