scholarly journals Modeling and Compensation for Hysteresis Nonlinearity of a Piezoelectrically Actuated Fast Tool Servo Based on a Novel Linear Model

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Zhiwei Zhu ◽  
Xiaoqin Zhou ◽  
Jieqiong Lin ◽  
Qiang Liu
Keyword(s):  
Fractional Calculus ◽  
Linear Model ◽  
Fractional Order ◽  
Analytical Description ◽  
Open Loop ◽  
Fast Tool Servo ◽  
Compensation Strategy ◽  
Hysteresis Nonlinearity ◽  
Series Of Experiments ◽  
Compensation Approach

In order to describe and compensate for complex hysteresis nonlinearities of piezoelectrically actuated fast tool servo (FTS), a novel Linear Fractional order Differentiation Hysteresis (LFDH) model is proposed in this paper. By means of the proposed LFDH model which is established on the fractional calculus theory, an analytical description of hysteresis behaviors of the FTS is derived. Furthermore, the LFDH model-based inverse compensation strategy is proposed to suppress the hysteresis effects of the FTS. Finally, a series of experiments are conducted to verify the effectiveness of the LFDH model and the corresponding compensation approach. The results demonstrate that the proposed LFDH model is efficient for describing hysteresis behaviors and the inverse compensation strategy can significantly suppress the inherent hysteresis of the FTS in open-loop operations.

A Fractional Calculus Perspective of PID Tuning

2003 ◽  
Author(s):  
Ramiro S. Barbosa ◽  
J. A. Tenreiro Machado ◽  
Isabel M. Ferreira
Keyword(s):  
Transfer Function ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Pid Controller ◽  
Open Loop ◽  
Controller Tuning ◽  
Pid Tuning ◽  
Loop Transfer Function ◽  
Pid Controller Tuning ◽  
Fractional Order Integrator

This paper gives an interpretation of the classical PID controller tuning based on the fractional calculus theory. The PID parameters are calculated according with the specifications of an elementary system whose open-loop transfer function is a fractional order integrator (FOI). The performances of the two systems are compared and illustrated through the frequency and time responses.

scholarly journals Tilt Active Vibration Isolation Using Vertical Pendulum and Piezoelectric Transducer with Parallel Controller

2021 ◽  
Vol 11 (10) ◽  
pp. 4526
Author(s):  
Lihua Wu ◽  
Yu Huang ◽  
Dequan Li
Keyword(s):  
Piezoelectric Transducer ◽  
Vibration Isolation ◽  
Vertical Vibration ◽  
Open Loop ◽  
Negative Effects ◽  
Voltage Controller ◽  
Hysteresis Nonlinearity ◽  
Passive Vibration Isolation ◽  
Active Vibration ◽  
Active Vibration Isolation

Tilt vibrations inevitably have negative effects on some precise engineering even after applying horizontal and vertical vibration isolations. It is difficult to adopt a traditional passive vibration isolation (PVI) scheme to realize tilt vibration isolation. In this paper, we present and develop a tilt active vibration isolation (AVI) device using a vertical pendulum (VP) tiltmeter and a piezoelectric transducer (PZT). The potential resolution of the VP is dependent on the mechanical thermal noise in the frequency bandwidth of about 0.0265 nrad, which need not be considered because it is far below the ground tilt of the laboratory. The tilt sensitivity of the device in an open-loop mode, investigated experimentally using a voltage controller, is found to be (1.63±0.11)×105 V/rad. To compensate for the hysteresis nonlinearity of the PZT, we experimentally established the multi-loop mathematical model of hysteresis, and designed a parallel controller consisting of both a hysteresis inverse model predictor and a digital proportional–integral–differential (PID) adjuster. Finally, the response of the device working in close-loop mode to the tilt vibration was tested experimentally, and the tilt AVI device showed a good vibration isolation performance, which can remarkably reduce the tilt vibration, for example, from 6.0131 μrad to below 0.0103 μrad.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

scholarly journals Nonlinear Viscoelastic-Plastic Creep Model of Rock Based on Fractional Calculus

2022 ◽  
Vol 2022 ◽  
pp. 1-7
Author(s):  
Erjian Wei ◽  
Bin Hu ◽  
Jing Li ◽  
Kai Cui ◽  
Zhen Zhang ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Engineering Application ◽  
Creep Model ◽  
Nonlinear Viscoelastic ◽  
Rock Creep ◽  
Order Change ◽  
Creep Constitutive Model ◽  
Plastic Creep

A rock creep constitutive model is the core content of rock rheological mechanics theory and is of great significance for studying the long-term stability of engineering. Most of the creep models constructed in previous studies have complex types and many parameters. Based on fractional calculus theory, this paper explores the creep curve characteristics of the creep elements with the fractional order change, constructs a nonlinear viscoelastic-plastic creep model of rock based on fractional calculus, and deduces the creep constitutive equation. By using a user-defined function fitting tool of the Origin software and the Levenberg–Marquardt optimization algorithm, the creep test data are fitted and compared. The fitting curve is in good agreement with the experimental data, which shows the rationality and applicability of the proposed nonlinear viscoelastic-plastic creep model. Through sensitivity analysis of the fractional order β2 and viscoelastic coefficient ξ2, the influence of these creep parameters on rock creep is clarified. The research results show that the nonlinear viscoelastic-plastic creep model of rock based on fractional calculus constructed in this paper can well describe the creep characteristics of rock, and this model has certain theoretical significance and engineering application value for long-term engineering stability research.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals An Improved Adaptive Feedforward Cancellation for Trajectory Tracking of Fast Tool Servo Based on Fractional Calculus

2011 ◽  
Vol 15 ◽  
pp. 315-320 ◽  
Author(s):  
Xiaoqin Zhou ◽  
Zhiwei Zhu ◽  
Shaoxin Zhao ◽  
Jieqiong Lin ◽  
Jianli Dou
Keyword(s):  
Fractional Calculus ◽  
Trajectory Tracking ◽  
Fast Tool Servo ◽  
Adaptive Feedforward

scholarly journals The Asymptotic Behaviours of a Class of Neutral Delay Fractional-Order Pantograph Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Baojun Miao ◽  
Xuechen Li
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Asymptotic Estimates ◽  
Stability Of Solutions ◽  
Neutral Delay ◽  
Estimates Of Solutions ◽  
Fractional Delay ◽  
Asymptotic Behaviours

By using fractional calculus and the summation by parts formula in this paper, the asymptotic behaviours of solutions of nonlinear neutral fractional delay pantograph equations with continuous arguments are investigated. The asymptotic estimates of solutions for the equation are obtained, which may imply asymptotic stability of solutions. In the end, a particular case is provided to illustrate the main result and the speed of the convergence of the obtained solutions.

Fractional Order Dynamics in the Trajectory Planning of Redundant and Hyper-Redundant Manipulators

2003 ◽  
Author(s):  
Fernando B. M. Duarte ◽  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Trajectory Planning ◽  
Free Space ◽  
Trajectory Optimization ◽  
Generalized Inverse ◽  
Resolution Of Singularities ◽  
Control Algorithms ◽  
Redundant Manipulators ◽  
Kinematic Control

Redundant manipulators have some advantages when compared with classical arms because they allow the trajectory optimization, both on the free space and on the presence of obstacles, and the resolution of singularities. For this type of arms the proposed kinematic control algorithms adopt generalized inverse matrices but, in general, the corresponding trajectory planning schemes show important limitations. Motivated by these problems this paper studies the pseudoinverse-based trajectory planning algorithms, using the theory of fractional calculus.

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