Stability of a Time-Delayed System With Parametric Excitation

2006 ◽  
Vol 129 (2) ◽  
pp. 125-135 ◽  
Author(s):  
Nitin K. Garg ◽  
Brian P. Mann ◽  
Nam H. Kim ◽  
Mohammad H. Kurdi
Keyword(s):  
Time Delay ◽  
Parametric Excitation ◽  
Mathieu Equation ◽  
Periodic Coefficient ◽  
Single Element ◽  
New Approach ◽  
Delayed System ◽  
Representative Problem ◽  
The Stability ◽  
Time Periodic

This paper investigates two different temporal finite element techniques, a multiple element (h-version) and single element (p-version) method, to analyze the stability of a system with a time-periodic coefficient and a time delay. The representative problem, known as the delayed damped Mathieu equation, is chosen to illustrate the combined effect of a time delay and parametric excitation on stability. A discrete linear map is obtained by approximating the exact solution with a series expansion of orthogonal polynomials constrained at intermittent nodes. Characteristic multipliers of the map are used to determine the unstable parameter domains. Additionally, the described analysis provides a new approach to extract the Floquet transition matrix of time periodic systems without a delay.

scholarly journals Full Characterization of Act-and-wait Control for First-order Unstable Lag Processes

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1209-1233 ◽  
Author(s):  
T. Insperger ◽  
P. Wahi ◽  
A. Colombo ◽  
G. Stépán ◽  
M. Di Bernardo ◽  
...  
Keyword(s):  
Time Delay ◽  
Feedback Systems ◽  
Periodic Control ◽  
First Order ◽  
Full Characterization ◽  
The Stability ◽  
And Control ◽  
Time Periodic ◽  
Special Case

Act-and-wait control is a special case of time-periodic control for systems with feedback delay, where the control gains are periodically switched on and off in order to stabilize otherwise unstable systems. The stability of feedback systems in the presence of time delay is a challenging problem. In this paper, we show that the act-and-wait type time-periodic control can always provide deadbeat control for first-order unstable lag processes with any (large but) fixed value of the time delay in the feedback loop. A full characterization of this act-and-wait controller with respect to the system and control parameters is given based on performance and robustness against disturbances.

The Stability Analysis and Planning for Multivariable Linear Time-Delay Systems and its Application in Supervision Oilfield

2013 ◽  
Vol 709 ◽  
pp. 727-730
Author(s):  
Ren Wang ◽  
Xue Kun Qi ◽  
Long Xing
Keyword(s):  
Time Delay ◽  
Exponential Stability ◽  
Linear Time ◽  
Control Process ◽  
Delay Systems ◽  
Sufficient Condition ◽  
Water Separation ◽  
Delayed Systems ◽  
Delayed System ◽  
The Stability

Based on Lyapunovs theory of stability, analyze on a control process of oil-water separation with time-delay, coupling and uncertainty in the unite station. a equation of sufficient condition for multi-variable Linear Delayed Systems with delayed independent stabilization is derived and another forms are also given. Based on this, several simple criterions for judging independent stabilization from linear delayed system are presented. We also discuss the exponential stability for delayed system and also provide the sufficient condition of exponential stability with any appointed convergent rate and its corresponding deductions. Using these conditions, we can choose a set of suitable parameters to reduce the conservation. By calculating and being compared with methods of the literature, the results show that our methods have less conservation.

Semi-Discretization of Delayed Dynamical Systems

2001 ◽  
Author(s):  
Tamás Insperger ◽  
Gábor Stépán
Keyword(s):  
Dynamical Systems ◽  
Mathieu Equation ◽  
Special Kind ◽  
Delayed Systems ◽  
The Past ◽  
Linear Discrete System ◽  
Approximate System ◽  
Discretization Technique ◽  
The Stability ◽  
Time Periodic

Abstract An efficient numerical method is presented for the stability analysis of linear retarded dynamical systems. The method is based on a special kind of discretization technique with respect to the past effect only. The resulting approximate system is delayed and time-periodic in the same time, but still, it can be transformed analytically into a high dimensional linear discrete system. The method is especially efficient for time varying delayed systems, including the case when the time delay itself varies in time. The method is applied to determine the stability charts of the delayed Mathieu equation with damping.

Faraday waves on a cylindrical fluid filament – generalised equation and simulations

2018 ◽  
Vol 857 ◽  
pp. 80-110 ◽  
Author(s):  
Sagar Patankar ◽  
Palas Kumar Farsoiya ◽  
Ratul Dasgupta
Keyword(s):  
Linear Theory ◽  
Body Force ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Mathieu Equation ◽  
Faraday Waves ◽  
Stability Chart ◽  
Fluid Filament ◽  
The Stability ◽  
Time Periodic

We perform linear stability analysis of an interface separating two immiscible, inviscid, quiescent fluids subject to a time-periodic body force. In a generalised, orthogonal coordinate system, the time-dependent amplitude of interfacial perturbations, in the form of standing waves, is shown to be governed by a generalised Mathieu equation. For zero forcing, the Mathieu equation reduces to a (generalised) simple harmonic oscillator equation. The generalised Mathieu equation is shown to govern Faraday waves on four time-periodic base states. We use this equation to demonstrate that Faraday waves and instabilities can arise on an axially unbounded, cylindrical capillary fluid filament subject to radial, time-periodic body force. The stability chart for solutions to the Mathieu equation is obtained through numerical Floquet analysis. For small values of perturbation and forcing amplitude, results obtained from direct numerical simulations (DNS) of the incompressible Euler equation (with surface tension) show very good agreement with theoretical predictions. Linear theory predicts that unstable Rayleigh–Plateau modes can be stabilised through forcing. This prediction is borne out by DNS results at early times. Nonlinearity produces higher wavenumbers, some of which can be linearly unstable due to forcing and thus eventually destabilise the filament. We study axisymmetric as well as three-dimensional perturbations through DNS. For large forcing amplitude, localised sheet-like structures emanate from the filament, suffering subsequent fragmentation and breakup. Systematic parametric studies are conducted in a non-dimensional space of five parameters and comparison with linear theory is provided in each case. Our generalised analysis provides a framework for understanding free and (parametrically) forced capillary oscillations on quiescent base states of varying geometrical configurations.

Period-m Motions to Chaos in a Periodically Forced, Duffing Oscillator with a Time-Delayed Displacement

2014 ◽  
Vol 24 (10) ◽  
pp. 1450126 ◽  
Author(s):  
Albert C. J. Luo ◽  
Hanxiang Jin
Keyword(s):  
Time Delay ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Initial Time ◽  
Periodic Motions ◽  
Delayed System ◽  
The Stability ◽  
Analytical Bifurcation Trees

In this paper, periodic motions in a periodically excited, Duffing oscillator with a time-delayed displacement are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed through eigenvalue analysis. The time-delayed displacement is from the feedback control of displacement. The analytical bifurcation trees of period-1 motions to chaos in the time-delayed Duffing oscillator are presented through asymmetric period-1 to period-4 motions. Stable and unstable periodic motions are illustrated through numerical and analytical solutions. From numerical illustrations, the analytical solutions of stable and unstable period-m motions are relatively accurate with AN/m < 10-6 compared to numerical solutions. From such analytical solutions, any complicated solutions of period-m motions can be obtained for any prescribed accuracy. Because time-delay may cause discontinuity, the appropriate time-delay inputs (or initial conditions) in the initial time-delay interval should satisfy the analytical solution of periodic motions in the time-delayed dynamical systems. Otherwise, periodic motions in such a time-delayed system cannot be obtained directly.

Parametric Oscillations and Stability of a Periodically Charged Liquid Droplet

1973 ◽  
Vol 51 (13) ◽  
pp. 1443-1445 ◽  
Author(s):  
G. N. Ionides
Keyword(s):  
Incompressible Fluid ◽  
Electric Charge ◽  
Liquid Droplet ◽  
Mathieu Equation ◽  
Parametric Oscillations ◽  
The Stability ◽  
Time Periodic ◽  
Constant Charge

The stability of an incompressible fluid droplet, carrying a time periodic electric charge, is considered. It is shown that this electrohydrodynamic problem is described by a Mathieu equation. As a result, the amplitude of the charge required to induce instability may be much smaller than in the corresponding case of a constant charge. The calculations are applicable to the dynamics of raindrops in electrified clouds.

Stabilization of Inverted Coupled Pendula Using Parametric Excitation

1997 ◽  
Author(s):  
Reiko Osada ◽  
Chikara Sato
Keyword(s):  
Parametric Excitation ◽  
Inverted Pendulum ◽  
Mathieu Equation ◽  
Stability Diagram ◽  
Affirmative Answer ◽  
Stable Region ◽  
View Point ◽  
Limited Region ◽  
Resonant Systems ◽  
The Stability

Abstract Parametric stabilization of a single inverted pendulum has been extensively studied using the Mathieu equation and its corresponding stability diagram. The inverted single pendulum may be stabilized using parametric excitation at a specified frequency and amplitude given by a narrow stable region in the Mathieu diagram. Coupled pendula with parametric excitation or corresponding resonant systems have been studied from mathematical view point (Cesari, 1959; Gambill, 1955; Richards, 1983), from electrical view point (Sato, 1962a; Sato, 1962b; Sato, 1971; Sato, 1975) and from mechanical view point (Sato, 1995). Coupled pendula with parametric excitation have been studied within a limited region by some researchers, including the authors. A study of inverted coupled pendula with parametric excitation has not been performed as far as the authors know. Usually it is assumed that inverted coupled pendula are unstable in the absence of any other stabilizing mechanism such as feedback. One question is whether the inverted coupled pendula could be stabilized only by parametric excitation? The present paper gives an affirmative answer to this question in a limited and finite region. The stability is also examined using the differential equations and other methods.

Efficient Stability Chart Computation for General Delayed Linear Time Periodic Systems

2013 ◽  
Author(s):  
Daniel Bachrathy ◽  
Gabor Stepan
Keyword(s):  
Linear Time ◽  
Order Approximation ◽  
Stability Boundary ◽  
Periodic Coefficient ◽  
Periodic Systems ◽  
Stability Chart ◽  
Matlab Package ◽  
The Stability ◽  
Cutting Processes ◽  
Time Periodic

The determination of the stability of systems with time delay is of high importance in many industrial and research applications, like cutting processes, wheel shimmy, traffic jams and even in neural systems, human balancing. A user friendly numerical method was implemented to analyse the general form of delayed linear time periodic systems with time periodic coefficients. The goal is to create a freeware Matlab package which is able to determine automatically the so-called stability chart, which illustrates the parameter range for which the given linear system is stable. The user has to define the governing equation by the time periodic coefficient matrices, the corresponding time delays, the orders of time derivatives of the general coordinate vector, as well as the range of the parameters and the resolution of the stability chart. The method is optimized for 2 parameters, which is a typical case in engineering applications, but 1 and 3 parameter stability charts are also supported and tested. The stability is analysed in frequency domain based on the Nrth order approximation of Hill’s infinite determinant. The parameter points where the number of unstable Floquet multipliers changes are computed by the Multi Dimensional Bisection Method. From these parameter points, another algorithm selects the stability boundary lines. The algorithm is tested by means of numerous examples.

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