Whereas a linear system has only one type of behaviour everywhere in the phase plane, a nonlinear system may exhibit qualitatively different types of behaviour in different regions. Let Σ be a nonlinear system having k qualitatively different types of behaviour in the phase plane. Suppose the system Σ to be linearised in each of the k regions, to yield linear approximations Σι, ..., Σk. It is a reasonable proposition that a knowledge of the behaviour of the system Σ in each of the k regions, can be obtained from a knowledge of the behaviour of each of the k linear approximations. The extent to which the proposition is valid was determined in a doctoral thesis in 1892 by A. M. Lyapunov. (Available in translation as Lyapunov (1966)). The approach outlined above and described below is sometimes called Lyapunov's first method.

Inspec keywords: nonlinear control systems; approximation theory; Lyapunov methods; linearisation techniques

Other keywords: nonlinear second order system; qualitative behaviour determination; linear approximations; Lyapunov first method; linearisation

Subjects: Nonlinear control systems; Stability in control theory