Verification of Aizerman´s conjecture for a class of third-order systems

Author

Bergen, A.R. ; Williams, I.J.

Author_Institution

University of California, Berkeley, CA, USA

Volume

7

Issue

3

fYear

1962

fDate

4/1/1962 12:00:00 AM

Firstpage

42

Lastpage

46

Abstract

The second method of Lyapunov is used to validate Aizerman\´s conjecture for the class of third-order nonlinear control systems described by the following differential equation: $tdot{e} + a_{2}\\ddot{e} + a_{1}dot{e} + a_{0}e + f(e)=0$ In this case, the stability of the nonlinear system may be inferred by considering an associated linear system in which the nonlinear function $f(e)$ is replaced by $ke$ . If the linear system is asymptotically stable for $k_{1} < k < k_{2}$ , then the nonlinear system will be asymptotically stable in-the-large for any $f(e)$ for which $k_{1} < frac{f(e)}{e} < k_{2}.$ The Lyapunov function used to prove this result is determined in a straightforward manner by considering the physical behavior of the system at the extreme points of the allowable range of $k$ .

Keywords

Differential equations; Gain; Linear systems; Lyapunov method; Nonlinear control systems; Nonlinear systems; Poles and zeros; Regulators; Research and development; Stability analysis;

fLanguage

English

Journal_Title

Automatic Control, IRE Transactions on

Publisher

ieee

ISSN

0096-199X

Type

jour

DOI

10.1109/TAC.1962.1105447

Filename

1105447