Delay-independent small gain theorem and frequency domain interpretation of the Lyapunov-Krasovskii method for stability of nonlinear delay systems

2001 ◽  
Author(s):  
Pierre-Alexandre Bliman
Keyword(s):  
Frequency Domain ◽  
Delay Systems ◽  
Small Gain Theorem ◽  
Small Gain ◽  
Nonlinear Delay

Coordinate Transformations and Logical Operations for Minimizing Conservativeness in Coupled Stability Criteria

1994 ◽  
Vol 116 (4) ◽  
pp. 643-649 ◽  
Author(s):  
J. E. Colgate
Keyword(s):  
Frequency Domain ◽  
Unit Circle ◽  
Stability Criterion ◽  
Coordinate Transformation ◽  
Coordinate Transformations ◽  
Stability Criteria ◽  
Small Gain Theorem ◽  
Logical Operations ◽  
Small Gain ◽  
Canonical Coordinate

Often it is desirable to guarantee that a manipulator will remain stable when contacting any member of some set of environments. Coupled stability criteria based on passivity may be used to provide such a guarantee, but may be arbitrarily conservative depending on the environment set. In this paper, two techniques for reducing conservativeness are introduced. The first is based on a canonical coordinate transformation which enables an environment set viewed in the frequency domain to be conformally mapped to the interior of the unit circle. A stability criterion is then derived via the small gain theorem. The second technique uses logical combinations of such criteria to reduce conservativeness further. Both techniques are illustrated with nontrivial examples.

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