A perturbation approach to the stabilization of nonlinear cascade systems with time-delay

2003 ◽  
Author(s):  
W. Michiels ◽  
R. Sepulchre ◽  
D. Roose ◽  
L. Moreau
Keyword(s):  
Time Delay ◽  
Perturbation Approach ◽  
Cascade Systems

RobustH∞Control for Non-Minimum Phase Switched Cascade Systems with Time Delay

2014 ◽  
Vol 17 (5) ◽  
pp. 1590-1599 ◽  
Author(s):  
Shengzhi Zhao ◽  
Georgi M. Dimirovski ◽  
Ruicheng Ma
Keyword(s):  
Time Delay ◽  
Minimum Phase ◽  
Cascade Systems

Correcting for Ray Refraction in Velocity and Attenuation Tomography: A Perturbation Approach

1982 ◽  
Vol 4 (3) ◽  
pp. 201-233 ◽  
Author(s):  
Stephen J. Norton ◽  
Melvin Linzer
Keyword(s):  
Time Delay ◽  
Ray Tracing ◽  
Perturbation Expansion ◽  
Index Of Refraction ◽  
Perturbation Approach ◽  
First Order ◽  
Phase Cancellation ◽  
Straight Line ◽  
Beam Displacement ◽  
Delay Correction

In velocity and attenuation tomography, ray refraction leads to errors in time-of-arrival, as well as to errors in attenuation due to phase cancellation and lateral beam displacement. Some authors have proposed iterative techniques based on numerical ray tracing to correct for these effects. In this paper, we consider an alternative approach using a perturbation analysis of refraction. This approach requires neither iteration nor numerical integration of the ray equation. Assuming that the index of refraction deviates from its mean on the order of the small quantity ε, we derive expressions for the refracted ray path whose departure from a straight line is first order in ε. Using this result, we obtain a perturbation expansion of the path integral of the refractive index along the refracted ray and derive a time-delay correction of order ε2 arising from the deviation of the refracted ray from a straight line. The expression for the first-order path is also used to obtain explicit corrections for phase-cancellation and beam-displacement errors that affect attenuation measurements when transducers of finite extent are employed. In addition, because of refraction, large aperture transducers are susceptible to an arrival-time uncertainty in a time-of-flight measurement; a first-order expression for the maximum value of this uncertainty is derived. In both two and three dimensions, the perturbation approach is much simpler computationally than numerical ray tracing methods. Computer simulated reconstructions are presented which clearly show the improvement that can be achieved with the second-order time delay correction.

DELAY-INDUCED BIFURCATIONS IN A NONAUTONOMOUS SYSTEM WITH DELAYED VELOCITY FEEDBACKS

2004 ◽  
Vol 14 (08) ◽  
pp. 2777-2798 ◽  
Author(s):  
JIAN XU ◽  
PEI YU
Keyword(s):  
Control System ◽  
Time Delay ◽  
Feedback Control ◽  
Periodic Solutions ◽  
Stability Boundary ◽  
Primary Resonance ◽  
Perturbation Approach ◽  
Feedback Control System ◽  
Trivial Equilibrium ◽  
Functional Differential

This paper investigates the bifurcations due to time delay in the feedback control system with excitation. Based on an self-sustained oscillator, the delayed velocity feedback control system is proposed. For the case without excitation, the stability of the trivial equilibrium is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or periodic solutions may occur. For the case with excitation, the main attention is focused on the effect of time delay on the obtained periodic solution when primary resonance occurs in the system under consideration. To this end, the control system is changed to be a functional differential equation. Functional analysis is carried out to obtain the center manifold and then a perturbation approach is used to find periodic solutions in a closed form. Moreover, the unstable regions for the limit cycles are also obtained, predicting the occurrence of some complex behaviors. Numerical simulations are employed to find the routes leading to quasi-periodic motions as the time delay is varied. It has been found that: (i) Time delay can be used to control bifurcations; and (ii) time delay can be applied to generate bifurcations. This indicates that time delay may be used as a "switch" to control or create complexity for different applications.

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