Abstract
In this paper, the direct method of Lyapunov is used to study the stability of a Liénard-type nonlinear system. The system is given by n second-order differential equations. To establish the procedure for constructing Lyapunov function, the system is rewritten by a transformation. Then, a stability criterion for the system, which introduces a new type Lyapunov function, is presented. While the positive matrix P appeared in the Lyapunov function is obtained by solving matrix equations, the possibility of existence of this P is discussed from the point of view of the condition that the Luré type Lyapunov function for the well-known nonlinear feedback control systems exists. Thus, the construction procedure developed in this paper, for the new type Lyapunov function, is systematic and the result obtained for the Liénard-type nonlinear system corresponds to the Luré type Lyapunov function for the nonlinear feedback control systems.
This new technique is applied to the simple system given by so-called Liénard's equation. In this system, it is shown that the conventional energy function and the Lyapunov function given by generalized Zubov's method are special cases of the Lyapunov function proposed in this paper.
