Stability Analysis of Parametrically Excited Systems With Time-Delay

2003 ◽  
Author(s):  
Eric A. Butcher ◽  
Haitao Ma ◽  
Ed Bueler ◽  
Victoria Averina ◽  
Zsolt Szabo
Keyword(s):  
Time Delay ◽  
Integral Form ◽  
Original System ◽  
Time Interval ◽  
Linear Difference Equations ◽  
Direct Integral ◽  
Parametrically Excited ◽  
Stability Matrix ◽  
The Stability ◽  
Time Periodic

This paper presents a new technique for studying the stability properties of parametrically excited dynamic systems with time delay modeled by delay-differential equations (DDEs) with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the “infinite-dimensional Floquet transition matrix U”. Two different formulas for the computation of the approximate U, whose size is determined by the number of polynomials employed, are given. The first one, which results in a numerical stability matrix, uses the direct integral form of the original system in state space form while the second, which can give a symbolic stability matrix in terms of parameters, uses a convolution integral (variation of parameters) formulation. An extension of the method to the case where the delay and parametric periods are commensurate is also available. Numerical and symbolic stability charts are produced for several examples of time-periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance-modulated turning. The results indicate that this method is a effective way to study the stability of periodic DDEs.

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.

Model Reduction of a Parametrically Excited Drivetrain

2012 ◽  
Author(s):  
Thomas Pumhoessel ◽  
Peter Hehenberger ◽  
Klaus Zeman
Keyword(s):  
Stability Margin ◽  
Original System ◽  
Reduced Model ◽  
Parametrically Excited ◽  
System Parameters ◽  
Additional Stability ◽  
System Equations ◽  
The Difference ◽  
Set Up ◽  
The Stability

The complexity of engineering systems is continuously increasing, resulting in mathematical models that become more and more computationally expensive. Furthermore, in model based design, for example, system parameters are subject of change, and therefore, the system equations have to be evaluated repeatedly. Hence, there is a need for providing reduced models which are as compact as possible, but still reflect the properties of the original model in a satisfactory manner. In this contribution, the reduction of differential equations with time-periodic coefficients, termed as parametrically excited systems, is investigated using the method of Proper Orthogonal Decomposition (POD). A reduced model is set up based on the solution of the original system for a certain parametric combination resonance of the difference type, resulting in an additional stability margin of the trivial solution. It is shown that the POD reduced model approximates the stability behavior of the original system much better than a modally reduced model even if system parameters are subject of change.

Chaos Suppression via Integrative Time Delay Control

2020 ◽  
Vol 30 (14) ◽  
pp. 2050208
Author(s):  
Ayman A. Arafa ◽  
Yong Xu ◽  
Gamal M. Mahmoud
Keyword(s):  
Time Delay ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Time Interval ◽  
General Strategy ◽  
Normal Form Theory ◽  
Chaos Suppression ◽  
Form Theory ◽  
The Stability ◽  
Time Delayed Feedback

A general strategy for suppressing chaos in chaotic Burke–Shaw system using integrative time delay (ITD) control is proposed, as an example. The idea of ITD is that the feedback is integrated over a time interval. Physically, the chaotic system responds to the average information it receives from the feedback. The main feature of integrative is that the stability of the chaotic system occurs over a wider range of the space parameters. Controlling chaotic systems with ITD has not been discussed before as far as we know. Stability and the existence of Hopf bifurcation are studied which demonstrate that the switch stability occurs at critical values of the time delay. Employing the normal form theory and center manifold argument, an explicit formula is derived to determine the stability and the direction of the bifurcating periodic solutions. Numerically, the bifurcation diagram and the eigenvalues of the corresponding characteristic equations are computed to supply a clear interpretation for suppressing chaos via ITD. Furthermore, ITD method is compared with the time delayed feedback (TDF) control numerically. This comparison shows that the stability area with ITD is larger than TDF which demonstrates the feasibility and effectiveness of the ITD. Other examples of chaotic systems can be similarly investigated.

Stability and Control of a Parametrically Excited Rotating Beam

1998 ◽  
Vol 120 (4) ◽  
pp. 462-470 ◽  
Author(s):  
S. C. Sinha ◽  
Dan B. Marghitu ◽  
Dan Boghiu
Keyword(s):  
Equations Of Motion ◽  
System Stability ◽  
Flexible Beam ◽  
Parametrically Excited ◽  
Stability And Control ◽  
Rotation Parameters ◽  
The Stability ◽  
And Control ◽  
Time Invariant ◽  
Time Periodic

In this paper the stability and control of a parametrically excited, rotating flexible beam is considered. The equations of motion for such a system contain time periodic coefficients. Floquet theory and a numerical integration are used to evaluate the stability of the linearized system. Stability charts for various sets of damping, parametric excitation, and rotation parameters are obtained. Several resonance conditions are found and it is shown that the system stability can be significantly changed due to the rotation. Such systems can be used as preliminary models for studying the flap dynamics and control of helicopter rotor blades and flexible mechanisms among other systems. To control the motion of the system, an observer based controller is designed via Lyapunov-Floquet transformation. In this approach the time periodic equations are transformed into a time invariant form, which are suitable for the application of standard time invariant controller design techniques. Simulations for several combinations of excitation and rotation parameters are shown.

scholarly journals Time-Delay Vibration Reduction Control of 3-DOF Vehicle Model with Vehicle Seat

2021 ◽  
Vol 11 (20) ◽  
pp. 9426
Author(s):  
Kaiwei Wu ◽  
Chuanbo Ren ◽  
Yuanchang Chen
Keyword(s):  
Time Delay ◽  
Performance Optimization ◽  
Low Frequency ◽  
Active Suspension ◽  
Random Excitation ◽  
Feedback Gain ◽  
Time Interval ◽  
On The Road ◽  
The Stability ◽  
Vehicle Seat

Vehicles driving on the road continuously suffer low-frequency and high-intensity road excitation, which can cause the occupant feelings of tension and dizziness. To solve this problem, a three-degree-of-freedom vehicle suspension system model including vehicle seat is established and a linear function equivalent excitation method is proposed. The optimization of the random excitation is transformed into the optimization of constant force in a discrete time interval, which introduces the adaptive weighted particle swarm optimization algorithm to optimize the delay and feedback gain parameters in the feedback control of time delay. In this paper, the stability switching theory is used for the first time to analyze the stability interval of 3-DOF time-delay controlled active suspension, which ensures the stability of the control system. The numerical simulation results show that the algorithm can reduce vertical passenger acceleration and vehicle acceleration, respectively, by 13.63% and 28.38% on average, and 29.99% and 47.23% on random excitation, compared with active suspension and passive suspension based on inverse control. The effectiveness of the method to suppress road random interference is verified, which provides a theoretical reference for further study of suspension performance optimization with time-delay control.

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.

On the stability of a class of nonlinear time delay systems

1999 ◽  
Author(s):  
Dan Ivancscu ◽  
Silviu-Iulian Niculcscu ◽  
Jcan-Michcl Dion ◽  
Luc Dugard
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Time Delay Systems ◽  
The Stability

Statistical Management of Food Production Processes

2020 ◽  
pp. 30-35
Author(s):  
N.A. Jurk ◽  
Keyword(s):  
Production Process ◽  
Control Chart ◽  
Food Production ◽  
Time Interval ◽  
Bakery Products ◽  
Shewhart Control ◽  
Manufactured Products ◽  
Mnemonic Technique ◽  
The Stability

The article presents scientific research in the field of statistical controllability of the food production process using the example of bakery products for a certain time interval using statistical methods of quality management. During quality control of finished products, defects in bakery products were identified, while the initial data were recorded in the developed form of a checklist for registering defects. It has been established that the most common defect is packaging leakage. For the subsequent statistical assessment of the stability of the production process and further analysis of the causes of the identified defect, a Shewhart control chart (p-card by an alternative feature) was used, which allows you to control the quality of manufactured products by the number of defects detected. Analyzing the control chart, it was concluded that studied process is conditionally stable, and the emerging defects are random. At the last stage of the research, the Ishikawa causal diagram was used, developed using the 6M mnemonic technique, in order to identify the most significant causes that affect the occurrence of the considered defect in bakery products. A more detailed study will allow the enterprise to produce food products that meet the established requirements.

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