A stability inequality for a class of nonlinear feedback systems

Author

Dewey, A.G. ; Jury, E.I.

Author_Institution

University of California, Berkeley, CA, USA

Volume

11

Issue

1

fYear

1966

fDate

1/1/1966 12:00:00 AM

Firstpage

54

Lastpage

62

Abstract

For some systems, the Popov stability criterion fails to verify Aizerman\´s conjecture, that is, when the Popov sector is not equal to the linear (Hurwitz) sector. In these cases, the question of stability for a nonlinearity which exceeds the Popov sector, but which is included in the Hurwitz sector, is unanswered. This paper provides a partial answer to this question by taking into account the slope of the nonlinear function. By constraining this slope to the interval $[-k_{1}, k_{2}]$ and the nonlinearity to the sector $[0, k]$ , the following stability inequality is obtained $\\Re (1 + j\\omega q)G(j\\omega ) + 1/k +\\mu\\omega ^{2}{1 + (k_{2}-k_{1})\\Re G(j\\omega )-k_{1}k_{2}| G(j\\omega )|^{2}} > 0$ where μ is a non-negative parameter. For $\\mu=0$ this inequality reduces to the Popov criterion. Two examples are given, in the first of which the sector is extended up to the linear limit. The Popov theorem concerned only the zero-input response of the nonlinear feedback system~ whereas here a restricted class of inputs to the system is allowed.

Keywords

Nonlinear systems; Popov stability; Control system analysis; Delay systems; Feedback; Frequency domain analysis; Information analysis; Laboratories; Nonlinear control systems; Stability analysis; Stability criteria; Time varying systems;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.1966.1098236

Filename

1098236