In proceedings IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Princeton, March 2000.

An energybased Lyapunov function for physical systems 
Marwan U. Bikdash and Richard A. Layton

A new expression of the first law of thermodynamics is obtained for nonholonomic lumpedparameter systems.
The input power due to the constraints is expressed in the framework of Lagrangian differentialalgebraic equations
using Lagrange multipliers. General procedures to prove the stability of autonomous systems and to design control laws
for them using the method of Lyapunov are then outlined and discussed. The approach is based on (a) postulating a Lyapunov
function candidate consisting of the total energy stored in the system (expressed in terms of momenta not flows)
augmented by a simple positive definite function of the displacements, (b) computing its timederivative using the first
law of thermodynamics, and (c) designing control laws as to make the timerate of the candidate function negative definite.
The procedure is applicable to systems where the control actuation appears through effort sources (such as voltage sources
and applied forces), or flow sources (such as current or velocity sources).

©2000 International Federation of Automatic Control

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