LYAPUNOV'S DIRECT METHOD

PHYSICAL ORIGINES AND DIFFICULTIES

The basic philosophy of Lyapunov's direct method is the mathematical extension of a fundamental physical observation : if the total energy of a mechanical or electrical system is continuously dissipated, then the system, whether linear or nonlinear, must eventually settle down to an equilibrium point. Thus, we may conclude the stability of a system by examining the variation of a single scalar function.

Let us consider the nonlinear mass-damper-spring system in Figure 1

Figure 1: A nonlinear mass-damper-spring system

The dynamic equation of this system is :

(1)

with  representing non-linear dissipation or damping, and representing a non-linear spring term. If the mass is pulled away from the natural length of the spring by a large distance, and then released. Will the resulting motion be stable? It is very difficult to answer this question using the definition of stability, because the general solution of this non-linear equation is not available. The linearization method cannot be used either because the motion starts outside the linear range. However, examination of the system energy can tell us a lot about the motion pattern.

The total mechanical energy of the system is the sum of its kinetic energy and its potential energy

(2)

Comparing the definitions of stability and mechanical energy, one can easily see some relations between the mechanical energy and the stability concepts described earlier :

• Zero energy corresponds to the equilibrium point
• Asymptotic stability implies the convergence of mechanical energy to zero
• Instability is related to the growth of mechanical energy
These relations indicate that the value of a scalar quantity, the mechanical energy, indirectly reflects the magnitude of the state vector, and furthermore, that the stability properties of the system can be characterised by the variation of the mechanical energy of the system.

The rate of energy variation during the system's motion is obtained easily by differentiating the first equality in (2) and using (1):

(3)

This equation implies that the energy of the system, starting from some initial value, is continuously dissipated by the damper until the mass settles down. Physically, it is easy to see that the mass must finally settle down at the natural length of the spring, because it is subjected to a non-zero spring force at any position other than the natural length.

The direct method of Lyapunov is based on a generalization of the concepts in the above mass-spring-damper system to more complex systems. Faced with a set of non-linear differential equations, the basic procedure of Lyapunov's direct method is to generate a scalar "energy-like" function for the dynamic system, and examine the time variation of the scalar function. In this way, conclusion may be drawn on the stability of the set of differential equations without using the difficult stability definitions or requiring explicit knowledge of solutions.