An averaging technique for stability analysis

Abstract

A technique is developed whereby the stability of autonomous or nonautonomous second order nonlinear systems can be accurately determined in a straightforward manner. The differential equations describing the system under consideration are transformed into polar coordinates and the average distance of the phase point from the point in question (usually the singular point whose stability is to be investigated) is calculated a function of time. This relationship is expressed in the form of a new set of ordinary differential equations. These equations can then be integrated to yield a prediction of the behaviour of the system in future time, by providing an approximate analytical solution to the original set of differential equations. No assumptions about small or large parameters are necessary for the application of this averaging technique. In addition to stability determination, this technique proves the existence or nonexistence of limit cycles and shows how fast a phase point approaches a singular point or a limit cycle. This information is of great importance in the design of control systems and in improving the quality of control. This averaging technique is applied to the analysis and control of a continuous stirred tank reactor (CSTR), and is used to investigate the behaviour of a CSTR under the effect of some forcing functions. It is shown small perturbations introduced into an asymptotically stable CSTR can produce large oscillations. A system comprising of a series of CSTR's is successfully analyzed with this averaging technique.