A Contribution to Liapunov’s Second Method: Nonlinear Autonomous Systems

[+] Author and Article Information
G. P. Szegö

RIAS, Baltimore, Md.; Control and Information Systems Laboratory, Purdue University, Lafayette, Ind.

J. Basic Eng 84(4), 571-578 (Dec 01, 1962) (8 pages) doi:10.1115/1.3658713 History: Received July 31, 1961; Online November 04, 2011


The stability of nonlinear autonomous systems with nonlinearity representable in polynomial form is investigated. For the case of locally stable systems the following theorem is presented: A sufficient condition for local stability of the system ẋ = X (x ) is the existence of a definite function v = φ(x ) such that dv/dt = θ(x )g[ξ(x )], where θ(x ) is a semidefinite function not identically equal to zero on a solution of ẋ = X (x ), g(x ) is such that g(0) = 0 and sign g(u) ≠ sign g(−u), and ξ(x ) = 0 is a closed surface. A procedure for constructing Liapunov functions based upon the use of a generating v-function is developed. Such a generating v-function may have the form:

v(x) = x A (x)x
where A(x ) = {aij (xi , xj )}, and aij = aji . The coefficients aij (xi , xj ) can be computed in order to obtain dv/dt of the wanted form. Particular emphasis is given to the case of systems with limit cycles and, as an example, the limit cycle of the van der Pol equation is identified with good approximation. It is also analytically proved that outside a closed algebraic curve, circumscribing the limit cycle, the system is asymptotically stable.

Copyright © 1962 by ASME
Your Session has timed out. Please sign back in to continue.





Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In