Complex scaling circle criteria for Luré systems

ABSTRACT

This study reports alternative frequency-domain interpretation and implementation of the circle criteria for absolute stability in Luré systems by means of complex scaling and the argument principle. By Luré system, a feedback configuration with a nominal LTI model subject to sector nonlinearities is meant as usual. First, the complex scaling stability criterion is proved for asymptotic stability in LTI feedback connections, which dispenses open-loop poles and contour/locus pre-orientation and possesses bounded loci without prior frequency sweeping. Second, a novel frequency-domain interpretation for positive realness of transfer functions is developed and employed for claiming the complex scaling circle criteria, which accommodate various sector nonlinearities with unified conditions. The new circle criteria are conformable in both multivariable and scalar cases, implementable graphically and tractable numerically, besides being a frequency/complex-domain analytical technique. Third, the results also reveal several frequency-domain facts about Luré systems that remain unknown so far. Finally, numerical examples are included to illustrate the main results.

KEYWORDS:

• Stability
• positive realness
• return difference
• Luré problem
• sector nonlinearity

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Since (−∞, ∞) is not infinitely countable, ∪_{∀ω ∈ (−∞, ∞)}(·) is not well-defined by the set theory. This symbol is used for representing the region on the complex plane covered by all disks with ω ∈ (−∞, ∞). In the following discussion, ∪_{∀ω ∈ (−∞, ∞)}(·) is also termed the disks band.

Additional information

Funding

The work was supported by the Nature Science Foundation of China [grant number 61573001].