modeling & control of magnetic levitation system

By Marwan K. Abbadi

Advisor: Dr. W. AnakwaDate: March 11th 2003

Modeling & Control Modeling & Control of of

Magnetic Levitation SystemMagnetic Levitation System

Modeling & Control Modeling & Control of of

Magnetic Levitation SystemMagnetic Levitation System

OutlineOutline

• Overview • Project Description

- Objective

- Functional Description - System Block diagram- Mathematical Model- Software Flowchart

• Project Status • Updated timeline• References

OverviewOverviewOverviewOverview

Magnetic Levitation Systems• DefinitionDefinition

• NatureNature: non-linear and open-loop unstable

Objective of the project:• ModelingModeling: Derive mathematical equations that describe

the system dynamics and validate the model experimentally

• ControllerController: Design stabilizing controller and implement the controller equations on an 8051 micro-controller to suspend the ball at a desired position

Functional Description

Inputs: Desired vertical position of the ball Disturbances could be

• Externally: Varying the disturbance input to the plant• Internally: From the internal system such as power

supply fluctuation

Output: • Actual ball position

Feedback: • Analog signal corresponding to the ball position

System LayoutSystem Layout

HW + SWController

Plant 8051UCPlant 8051UC

Ball position: 22.5 Ball position: 22.5 mmmm‘‘C’= RestartC’= Restart

Mathematical ModelMathematical Model

Coil ResistanceR

Coil inductance= L

+e-

Input Voltage

Current= i

Photo-detectorCells

Photo-emitterCells

There are two sets of equations that govern the dynamics of the system:

• Electrical equation- relates the ball’s suspension X with the input voltage e.

Obtaining KVL around circuit

e = R. i + L.di/dt - (Lo. x0.i/x^2)dx/dt (1)

Mathematical ModelMathematical Model

• Mechanical equation- relates the resultant force applied to the ball with the distance from the electromagnet

From the force diagram

F= GF- EF = m.g - C (i/x)2 (2)

Gravitation forceGravitation forceGF = mGF = m**gg

ElectromagneElectromagnetic forcetic force

EF= C (EF= C (ii//xx))22

Mathematical Model

• Linearization of the mathematical model

To develop a linear model of the plant about a specified operating point Xo.

• Method of linearization

Use of Taylor series expansion to linearize the model about an operating point

Fmag= C(I/x)^2Delta F = 2C{ io/Xo2 delta I - io^2/Xo^3 delta X)Where Xo, io are the parameters at the linearized operating point

Mathematical Model(Experimental determination of Variables)

• Magnetic force constant- Plant is highly sensitive to temperature.- Several attempts were taken to calculate the plant’s constant.

Method:- An analog controller was connected to stabilize the plant.- The gain of the controller was adjusted till the ball slightly suspended upwards. The ball was assumed to be at mechanical equilibrium(F=0)- The position of the ball and current flowing through coil were recorded and substituted in the previous equation.- C was found to be 1.477x10^-4 N.m2.A-2


• Coil inductance vs. Ball distance from electromagnet

• Exhibits a non-linear relationship• The coil inductance was measured at different ball

positions.• Found to be fairly constant about the operating range

of 18 to 27mm. (Full-range inductance variation=210uH)

• After linearization, the inductance was set to be constant( = 296.74mH)


• Coil inductance vs. Ball distance from the electromagnetCoil Inductance vs Ball Distance y = 0.001x2 - 0.0761x + 298.12

R2 = 0.9958

295.5

296

296.5

297

297.5

298

298.5

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

Ball distance from the coil (mm)

Co

il In

du

cta

nce (

mH

)

Series2

Poly. (Series2)


• Calibrating the sensor• Highly sensitive to opaque objects

Method of calibration:• A transparent tube-calibrator was used to place the ball

at different positions.• Assumption taken that the calibrator’s interference is

small. This assumption had to be verified experimentally by removing the ball and placing the tube-calibrator between the sensor plates.

• The relationship between the position of the ball and the sensor’s voltage was found to be linear as expected.


Sensor voltage vs. Ball distance about the operating point


• Combining all the equations and substituting the measured constants, the overall plant’s transfer function was obtained analytically.

X(s) 0.4838-------=

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

E(S) 0.006232 s^3 + 0.4386 s^2 - 5.29 s - 372.3


• Root locus sketch of the plants’ poles (Notice the plant’s zeroes are located @ infinity)


• Step response of the open-loop transfer function.


• Next step in the mathematical modeling is verifying the model experimentally, which is the current task being performed.

Closed-loop frequency response data were measured and recorded for graphical analysis. More data needs to be measured to finalize the modeling, and obtain the actual transfer function of the plant.

Software FlowchartSoftware Flowchart

Multiplexed 8-bit A/DConverter

Error SignalE= Xi-Xo

ActualPosition, XoDesired

positionXi

ControlAlgorithm

u(k)=f [e(k)]Samplederror Signal

e(k)

8-bit D/AConverter

u(k)

u(t)

LCDdisplaying desired and

actual ball position

Software

8051 micro-controller• Developed assembly language modules for sampling an input

signal via the A/D converter and producing it at the D/A converter. User-interface module is fully completed.

• Managed to code a software skeleton for implementing a hypothetical digital filter (code modification will be done later for the actual filter).

• Verified the digital filter code by running it on the simulator, but was unsuccessful with real-time signals.

• Task not fully accomplished.

Software

Signal sampled through the A/D and produced via the D/A converter.(Freq=7Khz)

Updated Project timeline

March• Finalize the modeling phase and obtain the validated

transfer function of the plant.

• Design a stabilizing controller and verify its response using Matlab

• Adjust the controller to account for stability, noise rejection, phase margin and overshoot specifications.


April• Design hardware components required by the

controller.

• Finish coding and debugging the digital filter.

• Test the overall software and verify its operation.

• Interface the hardware and software and verify the overall system performance.



May• Continue debugging the overall system.

ReferencesReferences

Barie and Chiason, International Journal of System Sciences, 1996, vol 27

D’Azzo and Houpis, Linear control system analysis and design: Conventional and Modern.

Dempsey, Bradley Univ. EE431-EE432 Lecture notes, 2003.

Grabbe, Ramo and Wooldridge, Handbook of automation, computation and control, vol.1

Wong, IEEE Trans. on Education, 1986, vol. E29#4.

The End

http://cegt201.bradley.edu/projects/proj2003/maglev

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modeling & control of magnetic levitation system

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