New Liapunov Function for Nonlinear Time-Varying Systems

[+] Author and Article Information
A. K. Newman

The Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pa.

J. Basic Eng 90(2), 208-212 (Jun 01, 1968) (5 pages) doi:10.1115/1.3605081 History: Received August 01, 1967; Online November 03, 2011


A new method of generating Liapunov functions is described that is useful for time-varying nonlinear differential equations. Constant coefficient quadratic forms are often used as Liapunov functions for linear constant coefficient differential equations. However, if the differential equation coefficients are time-varying and nonlinear, better results are usually obtained by using variable quadratic forms as Liapunov functions. This variation in V often introduces undesirable terms in V̇, which are cancelled by modifying V by subtracting integrals of certain partial derivatives of V with respect to the dependent variables. With few restrictions, V is proved to remain positive definite after the modification. The method often directly extends a Liapunov function useful for constant coefficient differential equations to cover the case when the coefficients are time-varying and nonlinear. Two examples are presented, including the incremental circuit for a time-varying nonlinear transmission line with hysteresis and the equations for an N-body collision avoidance problem.

Copyright © 1968 by ASME
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