Control The mathematical model of the magnetic levitation system

Model of the magnetic levitation system
The magnetic levitation system considered in this paper consists of a ferromagnetic ball suspended in a voltage-controlled magnetic field. Only the vertical motion is considered.
The objective is to keep the ball at a prescribed reference level. The schematic diagram of the system is shown in Figure 2.1. The dynamic model of the system can be written as


Schematic diagram of the magnetic levitation system.

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The mathematical model of the electromagnetic levitation system

One can build the mathematical model of the levitation system by writing
appropriate differential equations in accordance to the typical mechanical- and electrical principles. The way the components are appreciated in the approaching mode can lead to simpler or more complex alternatives.


The formula for the energetic balance within the system is:
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Basic mathematical model of magnetic levitation
In terms of the design, chosen part of the magnetic levitation model was simulated. We used the software for multidisciplinary system simulation – Dynast [9]. In the Fig. 6, a basic simulation model is shown. One of the significant parts of the model is the feedback circuit. It Decreasing / increasing of the duty cycle of the PWM
Approaching / taking away of the levitating object Decreasing / increasing of the output voltage of the Hall Effect sensor The place stabilization of the levitating object
The variation of the attraction force of the electromagnet means we simulated the output signal of the amplifier with the adjustable gain. According to the Fig. 6, there is a derivation block which is realized by passive components – resistors and capacitors. These components influence stability of the model. On this account, we have optimized the derivation block according to the results of the simulation. After the realization of the electromagnetic levitation device, we have compared real levels of the chosen parameters of the constructed device with the simulation model. The results of the simulation approximatelly correspondence with the real device.

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Model of the magnetic levitation system
The system dynamics describing the behaviour of the moving
ball is derived from the Newton’s laws:


where z denotes the position of the ball (as indicated figure 2),
m its mass, g the acceleration of gravity, i the coil current and
Fmag(i,z) the electromagnetic force applied to the ball. d
denotes a bounded perturbation.
Drawing up the energy balance of the whole system and under
the assumption that the magnetic core is non saturated (which
occurs because of the air gap), the electromagnetic force can
be expressed as following:
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Control Magnetic Levitation System - SYSTEM IDENTIFICATION

Magnetic Levitation SYSTEM IDENTIFICATION
A fundamental concept in science and technology is that of mathematical
modeling. System identification is conducted to obtain the plant transfer function
needed for the control design. Once a good model is obtained and verified, a
suitable control law can be implemented to compensate the plant instability and
improve performance.


Analytical Model
Analytical and experimental plant models were obtained for comparison
and verification. According to T. H. Wong, laboratory-scale maglev systems are
represented with electrical and mechanical equations [1]. Figure 3 shows the RLC
coil circuit that displaces the steel ball using electromagnetism.

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Determination of the Levitation System Model
The force/current/displacement relationship in the considered equipment given in Fig. 5
is extremely difficult to determine using an analytic method. Moreover, the obtained approximate
analytical expression f(x, i) is very complex for the further experimental purpose
[3]. However, the magnetic force characteristics may be experimentally calibrated as
a function of the applied current I and the ball position X. Namely, the experiment could
be consisted of resting the levitation metallic sphere on a non-magnetic stand directly
under the electromagnet. This special kind of xyz-stage (some solutions are shown in
Fig.8(a)-(c)) should be capable, for example, of 1mm incremental positioning and determining
the minimum current required to pick up the ball at various heights. Then the
model of the force/distance relationship can be determined by means of least squares fitting.
Note, that the validity of such obtained curve is limited to some range
Xmin ≤ X ≤ Xmax. At the moment, in the Laboratory of Automatic Control at the Faculty of
Electronic Engineering in Niš, the problem of the remote placement of the steel sphere
among the vertical axis of the electromagnet is still not realized adequately. Hence, this is
one of the basic problems in remote control of MagLev system in the underdeveloped
web-based laboratory at the Faculty, which was established in order to support learning in
automatic control. For now, as shown in Fig. 8, it is expected that the ball be placed along
the electromagnet vertical axis by the laboratory technician.
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Magnetic Levitation System
General Description
The process consists of a disk whose position can be controlled by a top and a bottom coil. Depending on which coil is used, this system can be either open loop stable (using the bottom coil) or unstable (using the top coil). Disk position is measured by laser sensors. The coil voltage is limited to [0, 3] V .

Initial tests perform system identification to quantify
the plant parameters and measure the strong
nonlinearities. Early experiments demonstrate application
of simple linear closed loop control to stabilize
and regulate the closed loop system about some
setpoint. It is shown that for the open loop stable
plant, (see front page) the system is stabilizable for all
positive gains but that for the unstable plant a minimum
gain (bandwidth) is necessary for closed loop
stability. Further tests vividly show the effect of the
nonlinear dynamics on closed loop tracking control
(see upper plots). By inverting these dynamics, rapid
and precise tracking control is demonstrated.


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Design and Implementation of a Controller for a
Magnetic Levitation System


System Analysis and Design
1. Dynamic Model Analysis
The magnetic levitation system in this paper is
illustrated in Figure 1a; it keeps a steel ball suspended
in the air by counteracting the ball’s weight with
electromagnetic force. x(t) is the distance between the
steel ball and the electromagnet. X0, the reference
position, is the proper levitation distance. The electromagnetic
force, f(x,t), acts on the ball, which can
be expressed as the following dynamic formula in an
upward direction according to Newton’s law.


Control system block diagram of the magnetic levitation
system.

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Design of Magnetic Levitation Controllers Using Jacobi Linearization, Feedback Linearization and Sliding Mode Control
THE goal of this project was to design three (one linear;two nonlinear) magnetic levitation controllers for the system shown in Fig. 1. Despite the fact that magnetic levitation systems are described by nonlinear differential equations [2], a simple approach would be to design a controller based on the linearized model (Jacobi linearization about a nominal operating point [6]). But this means the tracking performance deteriorates rapidly with increasing deviations from the nominal operating point. Nevertheless, a controller based on Jacobi linearization is a good ”litmus test” for our system identification, hence this controller is designed first.


II. SYSTEM IDENTIFICATION (BART AND KEVIN)
A. Physical Description of the System Components (KEVIN)
In Fig. 1, the different components are:


_ HV oltage to Current Inverter: This subsystem converts the output voltage from our controller into input current for the electromagnet .The reason for using this system is to separate the power amplifier (for a high current sink like the electromagnet) from the controller.


_ HSensor Electronics: For our Jacobi Linearization based controller, we use a simple red LED and a photoresistor. The reason for this is we emperically found that we need large gain values for the Hall Effect sensor solution. But for our nonlinear controllers, in order to sense a wider range of motion for the ball, we obtained data from two Melexis [4] Hall Effect sensors s1 and s2. We need two sensors instead of one because we emperically determined that the reading from a single sensor is
saturated by the magnetic field of the electromagnet. We are able to subtract the readings of the two sensors to get a nonlinear voltage function for the position of the ball.


_ Electromagnet: This is our plant, the model is derived below.


_ Controller: Designing this subsystem was the goal of this project. Again, as stated earlier, we were only able to design and implement the Jacobi linearization version.
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Control Modeling of Magnetic Levitation System

Magnetic Levitation System Modelling
Figure 2 below shows the simplified diagram of Maglev
system (adapted from Feedback Instrument Ltd.
manual).

Figure 2. Simplified Maglev System Diagram
The photo-sensors measure the ball’s position. Corresponding
to the ball’s position from the electromagnet,
the sensor generates a voltage output (Vsensor) obeying
the following relation:
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Magnetic Levitation System description and modelling
This work was concerned with the dynamics of the Feedback Magnetic Levitation System c° , which is depicted in Figure 1. Magnetic Levitation Circuit The infrared photo-sensor is assumed to be linear in the required range of operation, yielding a voltage y that is related to distance h as y = °h + y0, where the gain ° > 0 and the offset y0 are such that y 2 (−2V, +2V ). Current i is regulated by an inner control loop, and is linearly related to the input voltage u as i = ½u + i0 with ½ > 0 and i0 > 0.


The working excursion of u is limited between −3V (corresponding to a null coil current) and +5V (saturation value). Rates of change larger than 50V/s for u cannot be implemented by the current driver along its entire working range.

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MODELLING AND SIMULATION OF A MAGNETIC LEVITATION SYSTEM
Abstract. The electromagnetic levitation system (MLS) is a mechatronic system already
acknowledged and accepted by the field experts. Due to a synergic integration of the sensorial elements, the control subsystem and the actuating subsystem, the mentioned levitation system becomes an especially recommended subject in the academic curricula for mechatronic study programmes. This paper intends to initiate the investigation of different modelling, simulation and control possibilities for a magnetic levitation system starting from a real, physical reference model.
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Control Model of Magnetic Levitation System

MODEL OF THE MAGNETIC LEVITATION SYSTEM
Figure shows the experiment model for the
Maglev system that has been carried out in a
project at National key lab for Digital Control &
System Engineering, Vietnam.

Diagram of magnetic levitation


The Maglev system in the model contains two
feedback sensors. One is a small current sense
resistor in series with the coil. The other is a
phototransitor embedded in the chamber pedestal
and providing the ball position signal. After
amplifying, both current sensor and phototransitor
are wired to analog inputs of card PCI-1711. The
control signal from the computer is sent to the
controllable voltage source through the analog
output of card PCI-1711.
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EXPERIMENTAL APPARATUS AND CONTROL MODEL
Figure 1 shows the single-axis magnetic levitation
system used in the experiment. The levitation object is a
ping-pong ball with a permanent magnet attached inside it
to provide an attractive force. The attraction force is
controlled by means of a computer-controlled electromagnet
mounted directly above the ball. A light source and a
linear image sensor (LIS, Hamamatsu S5462-512Q) are
used to determine the displacement of the ball. There are
512 photo diode cells in LIS, and the length of each cell is
0.05mm. The light source and the sensor are tuned such
that the outputs of the photocells are saturated when the

Schematic diagram of the magnetic levitation system.


ball does not cover the cells. A comparator is used to
compare the outputs of each cell with a preset voltage to
judge whether the cell is saturated or not. We then obtain
the levitation displacement of the ball by utilizing a counter
to count the numbers of saturated cells. The sampling rate
used is 200 Hz. This low sampling rate is used due to the
bandwidth limitation of LIS. The control computer is an
industrial personal computer with a Pentium processor
and an Advantech PCL818H analog I/O and counter
board.
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MAGNETIC LEVITATION SYSTEM
In this section, a physical maglev system and its components are described. Presenting system equations nonlinear and linear models are developed for the plant. The schematic of the MAGLEV plant is presented in Fig. 3.1 below:

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MODEL OF THE MAGNETIC LEVITATION SYSTEM
Diagram of the magnetic levitation system.

Consider the magnetic levitation system shown in Fig. 1. this
is a popular gravity-based one degree-of-freedom magnetic
levitation system, in which an electromagnet exerts attractive
force to levitate a steel ball (in some references a steel plate is
levitated). The system dynamics can be described in the
following equations
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Control Magnetic Levitation System Control - NEURAL NETWORK

AN INTRODUCTION TO THE USEOF NEURAL NETWORKS IN CONTROL SYSTEMS
SUMMARY
The purpose of this paper is to provide a quick overview of neural networks and to explain how they can be used in control systems. We introduce the multilayer perceptron neural network and describe how it can be used for function approximation. The backpropagation algorithm (including its variations) is the principal procedure for training multilayer perceptrons; it is briefly described here. Care must be taken, when training perceptron networks, to ensure that they do not overfit the training data and then fail to generalize well in new situations. Several techniques for improving generalization are discussed. The paper also presents three control architectures: model reference adaptive control, model predictive control, and feedback linearization control. These controllers demonstrate the variety of ways in which multilayer perceptron neural networks can be used as basic building blocks. We demonstrate the practical implementation of these controllers on three applications: a continuous stirred tank reactor, a robot arm, and a magnetic levitation system.


Application - Magnetic Levitation System
Now we will demonstrate the predictive controller by applying it to a simple test problem. In this test problem, the objective is to control the position of a magnet suspended above an electromagnet, where the magnet is constrained so that it can only move in the vertical direction, as shown in Figure

Magnetic Levitation System
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NEURAL NETWORK-BASED ROBUST TRACKING CONTROL FOR MAGNETIC LEVITATION SYSTEM
ABSTRACT
This paper proposes a robust tracking controller with bound estimation based on neural network for the magnetic levitation system. The neural network is to approximate an unknown uncertain nonlinear dynamic function in the model of the magnetic levitation system. And the robust control is proposed to compensate for approximation error from the neural network. The weights of the neural network are tuned on-line and the bound of the approximation error is estimated by the adaptive law. The stability of the proposed controller is proven by Lyapunop theory. The robustness effect of the proposed controller is verified by the simulation and experimental results for the magnetic levitation system.
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Control Magnetic Levitation System Control - SLIDING MODE CONTROL

SLIDING MODE CONTROL OF A MAGNETIC
LEVITATION SYSTEM
Introduction
Magnetic levitation systems have practical importance in many engineering systems such
as in high-speed maglev passenger trains, frictionless bearings, levitation of wind tunnel
models, vibration isolation of sensitive machinery, levitation of molten metal in induction
furnaces, and levitation of metal slabs during manufacturing. The maglev systems
can be classified as attractive systems or repulsive systems based on the source of levitation
forces. These kind of systems are usually open-loop unstable and are described by
highly nonlinear differential equations which present additional difficulties in controlling
these systems. Therefore, it is an important task to construct high-performance feedback
controllers for regulating the position of the levitated object.
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H∞CONTROL AND SLIDING MODE CONTROL OF MAGNETIC LEVITATION SYSTEM
ABSTRACT
In this paper, H∞disturbance attenuation control and sliding mode
disturbance estimation and compensation control of a magnetic levitation
system are studied. A magnetic levitation apparatus is established, and its
model is measured. Then the system model is feedback linearized. A H∞
controller is then designed. For comparison, a sliding mode controller and a
PID controller also were designed. Some experiments were performed to
compare the performance of the H∞controller, the sliding mode controller and
the PID controller.
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High performance variable structure control of
a magnetic levitation system
Abstract- In this paper the position-tracking problem of a
voltage-controlled magnetic levitation system is considered. It is
well known that the control problem is quite complicated and
challenging duo to inherent nonlinearities associated with the
electromechanical dynamics. A sliding mode control is employed
for controlling the system. The proposed controller exhibits
satisfactory robustness in response to parameter uncertainties.
Simulation results reveal the effectiveness of the proposed robust
controller.
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ADVANCED SLIDING MODE STABILIZATION OF A LEVITATION SYSTEM
Abstract
Levitation bearings are intrinsically unstable, nonlinear and
highly uncertain systems. In this paper, we focus our attention
on sliding mode controllers which allow robust design and
more particularly on second order sliding mode control which
appears very relevant with respect to the process structure.
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Control Magnetic Levitation System Control

Design of a Robust Controller for a Magnetic Levitation System
Abstract
A Magnetic Levitation System (Maglev) is considered as a good test-bed for the design and analysis of control systems since it is a nonlinear unstable plant with practical uses in high-speed transportation and magnetic bearings. The objective of this project is to design a robust controller and implement it on a test-bed to help students learn the robust control design. In this project a robust controller for a maglev system is designed, using H-infinity optimization [3]. Complete mathematical models of the electrical, mechanical and magnetic systems are also developed. The design and simulations are performed under a Matlab/Simulink platform. Wincon control software of Quanser Inc. [7] is used to establish the link between the Matlab/Simulink models and the actual magnetic levitation system.
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Design and Implementation of a Controller for a
Magnetic Levitation System
Abstract
This paper reports on the design of a controller for keeping a steel ball suspended in the air. In
the ideal situation, the magnetic force produced by current from an electromagnet will counteract the weight of the steel ball. Nevertheless, the fixed electromagnetic force is very sensitive, and there is noise that creates acceleration forces on the steel ball, causing the ball to move into the unbalanced region. The main function of this controller is to maintain the balance between the magnetic force and the ball’s weight. According to the analytical method, the mathematical models of this magnetic levitation system were established with the goal of designing the control system. System linearization and phaselead compensation were employed to design the controller of this unstable nonlinear system. The algorithm
proposed in this paper provides a robust closed-loop magnetic levitation system which can stabilize the system over a large range of variations of the suspended mass. The design methods of this system are presented in this paper. And lastly, the hardware is implemented for a scientific demonstration.
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PREDICTIVE CONTROL OF A MAGNETIC LEVITATION SYSTEM WITH EXPLICIT TREATMENT OF OPERATIONAL CONSTRAINTS
Abstract.
This paper concerns the application of a predictive control methodology to the stabilization and referencefollowing operation of a magnetic levitation process. From a control engineering point of view, the problem is challenging owing to the nonlinear and unstable nature of the plant, the required positioning accuracy and the operational restrictions on the manipulated and controlled variables during transients.


The formulation employed in this work is based on a linear prediction model obtained by linearizing the plant dynamics around the center of the working range of the position sensor.
Offset-free tracking is achieved by augmenting the cost function with a term associated to the integral of the tracking error. Operational constraints on the input (current in the electromagnet coil) and output (width of the air gap between the electromagnet core and the suspended object) of the process are enforced in the optimization process. The optimal control sequence is implemented in a receding-horizon strategy, in which the optimization is repeated at every sampling instant, by taking into account the new sensor readings. The design and validation of the predictive control loop are carried out
by using physical parameters from a real magnetic levitation process. The results obtained by simulation show that the explicit treatment of operational constraints, especially those related to the input variation rate, is fundamental to an appropriate control of the system.
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MAGNETIC LEVITATION SYSTEM
IN CONTROL ENGINEERING EDUCATION
Abstract.
This paper deals with the magnetic levitation control system of a metallic
sphere, which is an interesting and visually impressive equipment for demonstrating
many intricate problems. In order to stimulate future research, after short description
of the system operation in analogue and digital mode, several open problems in areas
of electrical and control engineering are offered. Also, the paper presents some initial
outcomes in creating a laboratory environment for remote monitoring of the magnetic
levitation equipment.
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Modeling and Control of a Magnetic Levitation System
ABSTRACT
Magnetic levitation technology has been receiving increasing attention
because it helps eliminate frictional losses due to mechanical contact. Some
engineering applications include high-speed maglev trains, magnetic bearings and
high-precision platforms. The objectives of this project are to model and control a
laboratory-scale magnetic levitation system. The control algorithm is
implemented using assembly language on Intel 8051 microprocessor to levitate
and stabilize a spherical steel ball at a desired vertical position.
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Inverse Model Based Adaptive Control of Magnetic Levitation System
ABSTRACT
This paper presents, an adaptive finite impulse response
(FIR) filter based controller used for the tracking
of a ferric ball under the influence of magnetic
force. The adaptive filer is designed online as approximate
inverse system. To stabilize the open-loop unstable
and highly nonlinear magnetic levitation system,
PID controller is designed using polynomial approach.
To improve the stability, an adaptive FIR filter
is added along side the PID controller while the
use of the proposed controller has improved tracking.
Since adaptive FIR filters are inherently stable so the
controller remains stable. Experimental results are included
to highlight the excellent position tracking performance.


AFIR addition to improve the stability

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