TABLE OF CONTENTS The stability of a system implies that the small changes in the system input (either in system parameters or in initial conditions of the system) do not result in large changes in the system output. For a given control system, stability is one of the most important characteristics to be determined. In order to analyze the stability of linear time-invariant systems, many stability criteria, such as Nyquist stability criterion, Routh's stability criterion, etc., are available. However, for nonlinear systems and/or time-varying systems, such stability criteria do not apply.
In 1892, A.M. Lyapunov, the Russian mathematician, presented two methods (known as first and second methods of Lyapunov) in order to determine the stability of dynamic systems described by ordinary differential equations. By using the second method of Lyapunov, the stability of the dynamic system can be determined without actually solving the differential equations, that's why it is also referred to as the direct method of Lyapunov. This is quite advantageous because solving nonlinear and/or time-varying state equations is usually very difficult or sometimes may be impossible, too. Thus, the direct-method of Lyapunov is the most general method that can be employed for the determination of the stability of nonlinear and/or time-varying systems as it avoids the necessity of solving state equations.
Consider the general state equation for a time-invariant system
If the input vector u(t) is constant, we may write Equation (3.1) as
The system described by Equation (3.2)...
