A Note About the Stability of a String of LTI Systems

2002 ◽  
Vol 124 (3) ◽  
pp. 472-475 ◽  
Author(s):  
Swaroop Darbha
Keyword(s):  
Transfer Function ◽  
Weak Interaction ◽  
Error Propagation ◽  
Transfer Functions ◽  
Lp Norms ◽  
The Stability

This note concerns the stability of a string of LTI systems; it is shown here that, if the “weak interaction” condition among systems in the string is satisfied (i.e., the sum of the ∞-norms of the interaction/error propagation transfer functions is less than unity), then the string is Lp stable for every p⩾1. Since the ∞-norm of a transfer function is the smallest of all its induced Lp norms, the result presented here enables one to obtain a tighter estimate of the geometric rate of attenuation of the states of systems in the string.

Transfer Function Modeling of Distributed Piezoelectric Vibration Energy Harvesters

2009 ◽  
Author(s):  
Chin An Tan ◽  
Heather L. Lai
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Vibration Energy ◽  
System Response ◽  
Distributed Parameter ◽  
Domain Modeling ◽  
Vibration Energy Harvesting ◽  
Energy Harvesters ◽  
Adjoint Systems ◽  
The Stability

Extensive research has been conducted on vibration energy harvesting utilizing a distributed piezoelectric beam structure. A fundamental issue in the design of these harvesters is the understanding of the response of the beam to arbitrary external excitations (boundary excitations in most models). The modal analysis method has been the primary tool for evaluating the system response. However, a change in the model boundary conditions requires a reevaluation of the eigenfunctions in the series and information of higher-order dynamics may be lost in the truncation. In this paper, a frequency domain modeling approach based in the system transfer functions is proposed. The transfer function of a distributed parameter system contains all of the information required to predict the system spectrum, the system response under any initial and external disturbances, and the stability of the system response. The methodology proposed in this paper is valid for both self-adjoint and non-self-adjoint systems, and is useful for numerical computer coding and energy harvester design investigations. Examples will be discussed to demonstrate the effectiveness of this approach for designs of vibration energy harvesters.

scholarly journals On the Essential Instabilities Caused by Fractional-Order Transfer Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
Farshad Merrikh-Bayat ◽  
Masoud Karimi-Ghartemani
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Characteristic Equation ◽  
Stability Condition ◽  
Transfer Functions ◽  
Branch Points ◽  
The Stability ◽  
Unstable Branch

The exact stability condition for certain class of fractional-order (multivalued) transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the transfer function, in the systems under consideration, the branch points must also come into account as another kind of singularities. It is shown that a multivalued transfer function can behave unstably because of the numerator term while it has no unstable poles. So, in this case, not only the characteristic equation but the numerator term is of significant importance. In this manner, a family of unstable fractional-order transfer functions is introduced which exhibit essential instabilities, that is, those which cannot be removed by feedback. Two illustrative examples are presented; the transfer function of which has no unstable poles but the instability occurred because of the unstable branch points of the numerator term. The effect of unstable branch points is studied and simulations are presented.

Thermoacoustic Stability Analysis of an Annular Combustion Chamber With Acoustic Low Order Modeling and Validation Against Experiment

2005 ◽  
Author(s):  
Jan Kopitz ◽  
Andreas Huber ◽  
Thomas Sattelmayer ◽  
Wolfgang Polifke
Keyword(s):  
Transfer Function ◽  
Network Model ◽  
Transfer Functions ◽  
Model Simulation ◽  
Scaling Methods ◽  
Annular Combustor ◽  
The Stability ◽  
Operating Points ◽  
Low Order

A low order acoustic network model is used to examine the stability of an annular combustor for different operating points. The results obtained by this approach are compared against experimental data from a full annular combustor. This annular combustor, in contrast to commonly used single burners or sector rigs, was used to include also 2-dimensional effects like the influence of circumferential modes, which can occur in practical gas turbine applications. The influence of the flame enters the network model simulation through an experimentally measured flame transfer function in terms of the response of heat release to acoustic velocity fluctuations. This flame transfer function, which has been measured at a stable operating point, is then used as a basis for the determination of flame transfer functions at other operating points by means of scaling methods. The transition to instability is thereby simulated by determination of the complex eigen modes, applying methods from control theory. The analytically determined stability behavior is compared to the experimentally measured one, with the aim to enhance and validate the network model approach as a means of predicting combustion instabilities in early design stages.

Structural Dynamics in Machine-Tool Chatter: Contribution to Machine-Tool Chatter Research—2

1965 ◽  
Vol 87 (4) ◽  
pp. 455-463 ◽  
Author(s):  
G. W. Long ◽  
J. R. Lemon
Keyword(s):  
Transfer Function ◽  
Machine Tool ◽  
Stability Theory ◽  
Transfer Functions ◽  
Phase Relationship ◽  
Structure Response ◽  
Machine Tool Structure ◽  
Machine Tool Chatter ◽  
The Stability ◽  
Tool Chatter

This paper is one of four being presented simultaneously on the subject of self-excited machine-tool chatter. Transfer-function theory is applied to obtain a representation of the dynamics of a machine-tool structure. The stability theory developed to investigate self-excited machine-tool chatter requires such a representation. Transfer functions of simple symmetric systems are derived and compared with measurements. When measured frequency-response data of more complex structures are obtained, it provides a very convenient means of data interpretation and enables one to develop the significant equations of motion that define the structure response throughout a specified frequency range. The transfer function presents the phase relationship between structure response and exciting force at all frequencies in the specified range. This knowledge of phase is essential to the proper application of the stability theory and explains the “digging-in” type of instability that is often encountered in machine-tool operation. The instrumentation used throughout these tests is discussed and evaluated. The concept of developing dynamic expressions for machine-tool components and joining these together through properly defined boundary conditions, thereby building up the transfer function of the complete machine-tool structure, is suggested as an area for further study.

A Unified Transfer Function (UTF) Approach for the Modeling and Stability Analysis of Long Slender Bars in 3-D Turning Operations

1993 ◽  
Vol 115 (2) ◽  
pp. 193-204 ◽  
Author(s):  
I. N. Tansel
Keyword(s):  
Stability Analysis ◽  
Transfer Function ◽  
Transfer Functions ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Tool Geometry ◽  
Turning Operations ◽  
Chip Area ◽  
The Stability ◽  
Unique Relationship

A new approach is introduced to model 3-D turning operations that are used for the stability analysis of long slender bars. This approach utilizes the unique relationship between externally created feed direction tool displacements (input) and the resultant thrust direction workpiece vibrations (output) to estimate stability limits in three-dimensional turning operations from the data of a single dynamic cutting test. In this paper, this unique relationship is referred to as the “Unified Transfer Function ” (UTF) and its expressions are derived from conventional cutting and structural dynamics transfer functions. For the stability analysis, the uncut chip area variations of oblique cutting are represented by a linear model having different coefficients at different depths of cuts. These coefficients are found by using a tool geometry simulation program. An iterative procedure is developed for the stability analysis. The proposed approach considers in-process structural and cutting dynamics and can be automatically implemented without any input from the operator for the traverse turning of a long slender bar. Experimental studies have validated the proposed modeling and stability analysis techniques. The UTFs can also be used to monitor machine tool structure, tool wear, and the machinability of the material.

Model-based feedforward control for suppressing torque oscillation of electric power steering system

2021 ◽  
pp. 095440702110583
Author(s):  
Duo Fu ◽  
De-Zheng Li ◽  
Wen-Bin Shangguan
Keyword(s):  
Transfer Function ◽  
Electric Power ◽  
Transfer Functions ◽  
Controller Design ◽  
Feedforward Control ◽  
Stability Control ◽  
Electric Power Steering ◽  
Power Steering ◽  
Gain Margin ◽  
The Stability

Oscillation suppression is essential for the stability design of electric power steering (EPS) systems. The stability controller module in EPS controller is the key to solve the stability control problem of EPS system. This paper proposes a new method of stability analysis and stability controller module design for EPS systems. Furthermore, the dynamic characteristics of the EPS system are analyzed, and two critical factors on the resulting EPS stability, that is, large assist and variable assist gain are investigated experimentally. The transfer function from steering torque to sensor torque is redefined. A new transfer function is proposed for measuring the effect of variable assist gain on system performance. Based on the above factors and transfer functions, constraints on the stability controller design are proposed. Then the optimal parameters in the controller are obtained by maximizing an objective function including phase margin, gain margin, and crossover frequency. It is concluded from simulations and bench tests that the proposed stability controller can significantly reduce the torque oscillation of the EPS system.

scholarly journals Developed Transfer Function Allows Hydro Generators to Enter the Full Range of Ancillary Services Market

Energies ◽  
2022 ◽  
Vol 15 (2) ◽  
pp. 397
Author(s):  
Paulius Cicėnas ◽  
Virginijus Radziukynas
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Renewable Energy Sources ◽  
Storage System ◽  
Full Range ◽  
Parametric Identification ◽  
Energy Storage System ◽  
Ancillary Services ◽  
Ancillary Services Market ◽  
The Stability

As the number of available renewable energy sources has increased annually, there has been a corresponding rise in the levels of pollution created by traditional electricity generation, ultimately contributing to breaking down the stability of the electrical system at large. Therefore, there is an increasing need to integrate the use of nonpolluting electricity sources, such as pumped storage hydropower plants (PSHP), to ensure the stability of the power system and to maintain the frequency of the system from year-to-year. This paper addresses the issue of PSHP being unsuitable for providing Frequency Containment Reserve (FCR) services and proposes real measurements of the aggregation approach to obtain different data arrays. Based on this, the proposed methodology is orientated toward obtaining transfer functions that were developed using the parametric identification models, and the efficiency of these functions was thoroughly investigated. The proposed transfer function in this paper, in combination with battery energy storage system (BESS) technologies, would allow PSHP technologies to occupy a space in the ancillary services market by providing FCR, Frequency Restoration Reserve (FRR), and Replacement Reserve (RR) services. The performance of the function activated in the BESS is positively validated using the Simulink modeling environment.

Analytic integration of contrast transfer functions

1988 ◽  
Vol 46 ◽  
pp. 822-823
Author(s):  
Peter Rez
Keyword(s):  
Wave Function ◽  
Transfer Function ◽  
Transfer Functions ◽  
Spherical Aberration ◽  
Continuous Distribution ◽  
Objective Lens ◽  
Direction Parallel ◽  
Scattering Angle ◽  
Analytic Integration ◽  
Image Position

In high resolution microscopy the image amplitude is given by the convolution of the specimen exit surface wave function and the microscope objective lens transfer function. This is usually done by multiplying the wave function and the transfer function in reciprocal space and integrating over the effective aperture. For very thin specimens the scattering can be represented by a weak phase object and the amplitude observed in the image plane is1where fe (Θ) is the electron scattering factor, r is a postition variable, Θ a scattering angle and x(Θ) the lens transfer function. x(Θ) is given by2where Cs is the objective lens spherical aberration coefficient, the wavelength, and f the defocus.We shall consider one dimensional scattering that might arise from a cross sectional specimen containing disordered planes of a heavy element stacked in a regular sequence among planes of lighter elements. In a direction parallel to the disordered planes there will be a continuous distribution of scattering angle.

Trombone Transfer Functions: Comparison Between Frequency-Swept Sine Wave and Human Performer Input

2012 ◽  
Vol 37 (4) ◽  
pp. 447-454
Author(s):  
James W. Beauchamp
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Cutoff Frequency ◽  
Sine Wave ◽  
Nonlinear Propagation ◽  
Linear Filter ◽  
Wave System ◽  
General Effect ◽  
Filter Characteristic ◽  
High Pass

Abstract Source/filter models have frequently been used to model sound production of the vocal apparatus and musical instruments. Beginning in 1968, in an effort to measure the transfer function (i.e., transmission response or filter characteristic) of a trombone while being played by expert musicians, sound pressure signals from the mouthpiece and the trombone bell output were recorded in an anechoic room and then subjected to harmonic spectrum analysis. Output/input ratios of the signals’ harmonic amplitudes plotted vs. harmonic frequency then became points on the trombone’s transfer function. The first such recordings were made on analog 1/4 inch stereo magnetic tape. In 2000 digital recordings of trombone mouthpiece and anechoic output signals were made that provide a more accurate measurement of the trombone filter characteristic. Results show that the filter is a high-pass type with a cutoff frequency around 1000 Hz. Whereas the characteristic below cutoff is quite stable, above cutoff it is extremely variable, depending on level. In addition, measurements made using a swept-sine-wave system in 1972 verified the high-pass behavior, but they also showed a series of resonances whose minima correspond to the harmonic frequencies which occur under performance conditions. For frequencies below cutoff the two types of measurements corresponded well, but above cutoff there was a considerable difference. The general effect is that output harmonics above cutoff are greater than would be expected from linear filter theory, and this effect becomes stronger as input pressure increases. In the 1990s and early 2000s this nonlinear effect was verified by theory and measurements which showed that nonlinear propagation takes place in the trombone, causing a wave steepening effect at high amplitudes, thus increasing the relative strengths of the upper harmonics.

Structural-parametric model of an electroelastic actuator for nanomechanics

2020 ◽  
pp. 3-11
Author(s):  
S.M. Afonin
Keyword(s):  
Transfer Function ◽  
Piezoelectric Actuator ◽  
Transfer Functions ◽  
Piezoelectric Effect ◽  
Parametric Model ◽  
Elastic Compliance ◽  
Parametric Models ◽  
Piezo Actuator ◽  
Structural Scheme ◽  
Model Transfer

Structural-parametric models, structural schemes are constructed and the transfer functions of electro-elastic actuators for nanomechanics are determined. The transfer functions of the piezoelectric actuator with the generalized piezoelectric effect are obtained. The changes in the elastic compliance and rigidity of the piezoactuator are determined taking into account the type of control. Keywords electro-elastic actuator, piezo actuator, structural-parametric model, transfer function, parametric structural scheme

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