2011Feedback Control Systems

Modeling and control of Magnetic levitation system

Table of contents :

INTRODUCTION ………………………………………………………………………………………………………… 3

Abstract……………………………………………………………………………………………………………………….. 4

System Analysis and Design…………………….……………………………………………………………………. 5

Dynamic Model Analysis………………………………………………………..…………………………………….. 5

Parameters of experimental system……………………………………………..………………………………… 9

MATLAB and simulink modeling………………………………………………………………….…………….… 10

Magnetic levitation system before stabilizing it………………………………………………………..…..... 10

the phase-lead compensated …………………………..………………………………………………….……….. 13

Magnetic levitation system after stabilizing it……..……………………………………………….…..….... 13- Kc <

8.3584………………………………………………………………………………………………………… 16

- Kc > 8.358…………………………………………………….…………..……………………………..……... 17

- Kc > 8.358 ………………………………………………………………….………………………………..….…… 18

Things to know about Magnetic Levitation systems………………………………………………….………… 19- Transportation

advantages………………………………………………………………………………………19- Magnetic

Bearing……………………………………………………………………………………………………20

- Facts………………………………………………………………………………………………………………………21

- Conclusion……………………………………………………………………………………………………………..22

- References……………………………………………………………………………………………………………..23

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Introduction

Magnetic levitation is a technology that has been first introduced back in the 1900’s.

Levitation is the process by which an object is suspended against gravity, in a stable position, without physical contact. For levitation on Earth, a force is needed to be directed vertically upwards and equal to the gravitational force,and for any small displacement of the levitating object, a returning force should appear to stabilize it.

Magnetic levitation has been very useful due to its various advantages .The most important advantage is its ability to eliminate frictional losses since there are no mechanical contacts. Magnetic levitation has been used in many applications which include high-speed maglev trains.

Abstract

This project deals with modelling and controlling a Magnetic levitation system.

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The main objective of the project is to balance the gravitational force of the steel ball due to its weight and the electromagnetic force using current and a sensor as controlling units to monitor the position of the ball . Also since this system represents a non-linear system , we derived a linearized equation.

Matlab and simulink were used to model the system and also a phase lead compensator was added to stabilize it.

To model the idea of magnetic levitation we will use a steel ball that must stay suspended in the air;

stabilized.

System Analysis and Design

A MLS is a quite complex system since it presents nonlinearities and it is naturally unstable. The easiest way to handle this system is to consider only small variations around a given operation point where it should remain stable . This allows one to apply a well-established linear control systems.

Dynamic Model Analysis

The following is how we derived the equations for this model.

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The steel ball stays suspended in the air by counteracting the ball’s gravitational force by the 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 and where m is the weight of the ball, and g is the gravitationalConstant.

The electromagnetic force produced by current i(t) and where L(x) is the inductance of the electromagnet coil with the ball. is as follows:

If L is the inductance without the ball and L0 is the incremental inductance with the ball, then the inductance of the electromagnet coil is

Combining (3) with (2) we get the electromagnetic force as

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The ball’s position will affect the inductance of the electromagnet coil and these changes are nonlinear. Moreover the balance point between the electromagnetic force and gravity is of course naturally unstable.

Therefore we use Taylor’s series to linearalize the equation.

I0 equals the current of the electromagnet coil when the ball is at X0. When the magnetic force balances the gravitational force on the ball, the acceleration of the ball is zero, and Equation (1) becomes

And from Taylor’s series we neglect the higher order terms and take the first 2 terms only therefore the control force f becomes the following equation in oreder to keep the ball balanced

In the relation between Equations (1), (5), (6), and (7), we get

The formula of the relation between the voltage and current of the electromagnet coil is

Where R is the coil resistance. If L0 is far less than L, Equation (9) can be simplified as

The position of the ball can be detected using a sensor which is given the following equation

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Where β is the sensor gain.

Combining the equation (7), (8), (10), and (11), and finding their La Place with the voltage of the electromagnet as the input and the ball position displacement as the output, the open loop transfer function, G(s).

Laplace of Eq. (7) and (8): ms²X(s)=-k1 I(s) +k2 X(s)

Laplace Eq. (9) :V(s)=RI(s) + LsI(s)

Laplace Eq. (10) :Vx(s) = BX(s)

We simplify this equation further considering 1/(s+ R/L) as 1 which is achieved by a technique is called the virtual pole cancellation method.

Let K1=2C (I 0/X0 2) and K2=2C (I0 2/X0 3),

Therefore the transfer function is simplified to

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The following is a parameters table that has been achieved experimentally

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For instance the XO is measured by the height gauge as the ball intersects the light beam of the photodetector.The weight of the ball, m, is measured on the scale.The sensor gain β, is found by displacing the ball 1 mm and measuring the output of thePhotodetector. To find IO, a pedestal (a support) was used to set the ball at XO, and the coil current was increased until the ball just lifted off the pedestal. Substituting the measured parameters, XO, IO, m, and the gravitational constant, into Equation (6) yields the parameter C. The calculated parameter C and the measured parametersXO and IO are used to determine the parameters K1 and K2 in Equation (13)

MATLAB and simulink modelling

The following is the block diagram for an Magnetic levitation system before stabilizing it.

Feedback loop G(s) =T(s)/ (1-T(s) H(s))

Proof

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MATLAB CODE

[A, B, C, D]=linmod('esraa'); SYS = ss (A, B, C, D)SYS= tf(SYS)sys=mineral(SYS) num=[784]den=[1 0 -653.3 ]rlocus(num,den)sys=tf(num,den)[k,poles]=rlocfind(sys)grid onResultsselected_point = 0.0711 + 0.0776i

k = 0.8333

poles =

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0 + 0.0312i 0 - 0.0312i

Using SISOTOOL

C(s)=0.833

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The open-loop bode and phase plot.

Conclusion

The system is unstable for any chosen gain.

Therefore we stabilize the system using phase lead compensator

The transfer function of the phase-lead compensated controller is shown as:

The values for a=20 and for b=200 selected in order to cancel the unstable pole of the transfer function. b should be greater than a by 10 times.

To get Kc, we need to use the Routh- Hurwitz stability test to verify the closed-loop transfer

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The outcome of the system is a stable system where Kc>0 so the system can control a ball of any weight.

-Magnetic levitation system after stabilizing it

MATLab and simulink for the phase-lead compensated controller

MATLAB CODE

]A, B, C, D=[linmod ('esraa') ;

SYS = ss(A,B,C,D)

SYS= tf(SYS)

sys=minreal(SYS)

num=[784 1.568e004]

den=[1 200 -653.3 -1.307e005]

rlocus(num,den)

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sys=tf(num,den)

]k,poles=[rlocfind(sys)

grid on

002- 051- 001- 05- 0 0506-

04-

02-

0

02

04

0653.06.067.078.0529.0

569.0

489.0

699.0

53.06.067.078.0529.0569.0

489.0

699.0

520557001521051571002

sucoL tooR

sixA laeR

sixA

yra

niga

mI

Result

Transfer function:

784 s + 15680

--------------------------------

s^3 + 200 s^2 - 653.3 s - 130700

Select a point in the graphics window

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selected_point=

-0.0592 - 0.5590i

k=

8.3584

poles=

-164.0506

-35.8882

-0.0612

=8.3584The Gain Margin of the stabilized system is GM

In here we used SISOTOOL to demonstrate the different ranges of Kc and stability of the system.

Kc < 8.3584

Unstable system

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Step response

Kc=8.358 =GM

Marginally stable system

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step response

Kc>8.3

=Kc20

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Step response

Things to know about Magnetic Levitation systems!

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Transportation advantages

-Because there is no mechanical contact between the vehicles and the guide way, the vehicles such as trains can have very high speed trips.

- Maglev has high reliability and is not affected by weather conditions.- Maglev gives petroleum independence - with respect to air and auto because of

Maglev being electrically powered. - The power needed for levitation is usually not a large percentage of the overall

consumption.- Maglev systems produce less pollution if compared to air and auto since it is powered

by electricity - Maglev has a higher capacity than air travel with at least 12,000 passengers per hour

in each direction. - Maglev has high safety - both perceived

and actual, based on foreign experience and has improved comfort due to the absence of air turbulence which ensures a consistently smooth ride.

Magnetic Bearing

A magnetic bearing is a bearing which supports a load using magnetic levitation. Magnetic bearings support moving machinery without physical contact, for example, they can levitate a

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rotating shaft and permit relative motion with very low friction and no mechanical wear. They are in service in such industrial applications as electric power generation, petroleum refining, machine tool operation and natural gas pipelines.

Magnetic bearings are used in turbo molecular pumps where oil-lubricated bearings are a source of contamination. Magnetic bearings support the highest speeds of any kind of bearing; they have no known maximum relative speed.

Facts

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The Shinkansen , also known as the bullet train has highest recorded speed of a Maglev train is 581 kilometres per hour (361 mph), achieved in Japan in 2003 .

Lineup of JR East Shinkansen trains, October 2009

The most well known implementation of high-speed maglev technology currently operating commercially is the Shanghai Maglev Train, Shanghai, China which can transport people

30 km (19 mi) to the airport in just 7 minutes 20 seconds, achieving a top speed of 431 km/h (268 mph), averaging 250 km/h (160 mph).

Shanghai Maglev Train,

Conclusion

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In this project we have modelled the Magnetic Levitation system. We used equations to relate the factors that define such system. This includes the position and displacement of an object , the current and force of the electromagnetic and the weight of the object .

The equations showed a non-linear system, therefore it was linearized.

When the system was modelled, we observed that the system is unstable, therefore in ordered to stabilize the system we put a phase lead compensator in order to stabilize it.

Magnetic levitation system have many advantages in the transportation world as it has very high speed because of its non-contacting wheels therefore no resistance due to friction .

References:

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http://faculty.kfupm.edu.sa/EE/aliawami/Ali%20Web%20Files%20EE432-032/MLS%20Root%20locus.pdf

http://inventors.about.com/library/inventors/blrailroad3.htm

http://cegt201.bradley.edu/projects/proj2003/maglev/docs/PDF/final.pdf

http://en.wikipedia.org/wiki/Shinkansen

Reference: http://www.seminarprojects.com/Thread-magnetic-levitation-full-report#ixzz1MEbu5r00

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