magnetic levitation system - mechatronics2020.net · magnetic levitation system k. craig 5 •...

Magnetic Levitation System K. Craig 1

Magnetic Levitation System

Electromagnet

Infrared LED

Phototransistor

Levitated Ball


Electromagnet

Infrared LED

Phototransistor

Vsensor ≈ 2.5 V

At Equilibrium

Levitated Ball

m = 0.008 kg

r = 0.0062 m

Equilibrium Conditions

gap0 = 0.0053 m

i0 = 0.31 A

gap

i


Emitter

Detector


Emitter Circuit

Detector

Circuit

Power Supply

Capacitors

to Ground

Buffer Op-Amp

Buffer Op-Amp

Power MOSFET

with Diode

To Electromagnet

Analog Sensor PWM Gnd


Microcontroller Board

Analog Sensor

Gnd PWM


• Electromagnet Actuator

– Current flowing through the coil windings of the

electromagnet generates a magnetic field.

– The ferromagnetic core of the electromagnet provides

a low-reluctance path in the which the magnetic field

is concentrated.

– The magnetic field induces an attractive force on the

ferromagnetic ball.

Electromagnetic ForceProportional to the square

of the currentand

inversely proportional to the square of the gap

distance


– The electromagnet uses a ¼ - inch steel bolt as the

core with approximately 3000 turns of 26-gauge

magnet wire wound around it.

– The electromagnet at room temperature has a

resistance R = 34 Ω and an inductance L = 154 mH.


• Ball-Position Sensor

– The sensor consists of an infrared diode (emitter) and

a phototransistor (detector) which are placed facing

each other across the gap where the ball is levitated.

– Infrared light is emitted from the diode and sensed at

the base of the phototransistor which then allows a

proportional amount of current to flow from the

transistor collector to the transistor emitter.

– When the path between the emitter and detector is

completely blocked, no current flows.

– When no object is placed between the emitter and

detector, a maximum amount of current flows.

– The current flowing through the transistor is converted

to a voltage potential across a resistor.


– The voltage across the resistor, Vsensor, is sent through

a unity-gain, follower op-amp to buffer the signal and

avoid any circuit loading effects.

– Vsensor is proportional to the vertical position of the ball

with respect to its operating point; this is compared to

the voltage corresponding to the desired ball position.

– The emitter potentiometer allows for changes in the

current flowing through the infrared LED which affects

the light intensity, beam width, and sensor gain.

– The transistor potentiometer adjusts the phototransistor

current-to-voltage conversion sensitivity and allows

adjustment of the sensor’s voltage range; a 0 - 5 volt

range is required as an analog input to the

microcontroller.


Ball-Position SensorLED Blocked: esensor = 0 V

LED Unblocked: esensor = 5 V

Equilibrium Position: esensor ≈ 2.5 V

Ksensor ≈ 1.6 V/mm Range ± 1mm

Emitter Current = 10 mADetector Voltage = 0-5 V



Block DiagramFeedback Control System

to Levitate Steel Ball

about an Equilibrium Position

Corresponding to Equilibrium Gap

gap0 and Equilibrium Current i0

From Equilibrium:

As i ↑, gap ↓, & Vsensor ↓

As i ↓, gap ↑, & Vsensor ↑


Magnetic Levitation System Derivation

2

2

if gap,i C

gap

gap

m m

m

2

m m m core gap object return path

m

core object return path

22 2gap 0 gap

mgap0 gap m 0 gap gap

0 gap

0 gap2

field

NiNeglect

N iN N L i

Define: constant

x A NN NL

xA A x

A

A1 1W L x i

2 2

2

2

0 gap gap

2 2

2 2 2

e 0 gap 1

0 gap gap 2 gap

Ni

A x

1 dL(x) 1 1 if i A N i K

2 dx 2 A x K x


At Static Equilibrium:

Equation of Motion:

2

2

img C

x

Linearization:

Magnetic Levitation SystemControl System Design

Measure the gap from theelectromagnet with

x positive ↓

2

3 2

2 i 2 i ˆˆ ˆmx C x C ix x

2 2

2 3 2

i 2 i 2 i ˆˆ ˆmx mg C C x C ix x x

2 2 2

2 2 3 2

i i 2 i 2 i ˆˆC C C x C ix x x x

2

2

imx mg C

x


Use of Experimental Testing in Multivariable Linearization

0 00 0

m

m 0 0 0 0

i ,yi ,y

f f (i, y)

f ff f i , y y y i i

y i


2

2

img C

x

m 0.008

g 9.81

x 0.0053

i 0.31

C 2.29E 5

ˆx 3695x 63iˆ ˆ 2

x 63ˆ

ˆ s 3695i

2

3 2

2 i 2 i ˆˆ ˆmx C x C ix x

SI Units


Basic Component

Equations

(Constitutive Equations)

Lin out

out R

die e L

dt

e i R

KVLL

in out

L R out R

die L e 0

dt

i i i i 0

outout in

out out in

out in

out

in in

deLe e

R dt

LDe e e

R

LD 1 e e

R

1e 1 i R

L Le eD 1 D 1

R R

Reineout

iL

Iout = 0L

iR

KCL

outin out

ede L e 0

dt R

Electromagnet Model L = 154 mH R = 34 Ω


Magnetic Levitation System Control Design

Design a Feedback Controllerto Stabilize the Magnetic Levitation Plant

with Adequate Stability Margins

2

0.029 63

0.0045s 1 s 3695

voltage position

Note: Controller gain will need to be negative


101

102

103

104

-270

-225

-180

P.M.: Inf

Freq: NaN

Frequency (rad/s)

Phase (

deg)

-200

-180

-160

-140

-120

-100

-80

-60

G.M.: 66.1 dB

Freq: 0 rad/s

Unstable loop

Open-Loop Bode Editor for Open Loop 1 (OL1)

Magnitu

de (

dB

)

-800 -600 -400 -200 0 200 400-600

-400

-200

0

200

400

600

Root Locus Editor for Open Loop 1 (OL1)

Real Axis

Imag A

xis

Uncompensated Electromagnet + Ball System

2

in

x 0.029 63ˆ

e 0.0045s 1 s 3695

Note: Negative Controller

Gain Is Required


z = -50

p = -800

K = 52664

c

s 50G (s) 52664

s 800

Sample Control Design


• Nyquist Stability Criterion

– Key Fact: The Bode magnitude response corresponding to neutral

stability passes through 1 (0 dB) at the same frequency at which the

phase passes through180°.

– The Nyquist Stability Criterion uses the open-loop transfer function,

i.e., (B/E)(s), to determine the number, not the numerical values, of

the unstable roots of the closed-loop system characteristic equation.

– If some components are modeled experimentally using frequency

response measurements, these measurements can be used directly

in the Nyquist criterion.

– The Nyquist Stability Criterion handles dead times without

approximation.

– In addition to answering the question of absolute stability, Nyquist

also gives useful results on relative stability, i.e., gain margin and

phase margin.

– The Nyquist Stability Criterion handles stability analysis of complex

systems with one or more resonances, with multiple magnitude-

curve crossings of 1.0, and with multiple phase-curve crossings of

180°.


• Procedure for Plotting the Nyquist Plot

1. Make a polar plot of (B/E)(i) for - < . The magnitude

will be small at high frequencies for any physical system.

The Nyquist plot will always be symmetrical with respect to

the real axis.

2. If (B/E)(i) has no terms (i)k, i.e., integrators, as multiplying

factors in its denominator, the plot of (B/E)(i) for - < <

results in a closed curve. If (B/E)(i) has (i)k as a

multiplying factor in its denominator, the plots for + and -

will go off the paper as 0 and we will not get a single

closed curve. The rule for closing such plots says to connect

the "tail" of the curve at 0- to the tail at 0+ by

drawing k clockwise semicircles of "infinite" radius.

Application of this rule will always result in a single closed

curve so that one can start at the = - point and trace

completely around the curve toward = 0- and = 0+ and

finally to = +, which will always be the same point (the

origin) at which we started with = -.


3. We must next find the number Np of poles of B/E(s) that are

in the right half of the complex plane. This will almost

always be zero since these poles are the roots of the

characteristic equation of the open-loop system and open-

loop systems are rarely unstable.

4. We now return to our plot (B/E)(i), which has already been

reflected and closed in earlier steps. Draw a vector whose

tail is bound to the -1 point and whose head lies at the origin,

where = -. Now let the head of this vector trace

completely around the closed curve in the direction from =

- to 0- to 0+ to +, returning to the starting point. Keep

careful track of the total number of net rotations of this test

vector about the -1 point, calling this Np-z and making it

positive for counter-clockwise rotations and negative for

clockwise rotations.

5. In this final step we subtract Np-z from Np. This number will

always be zero or a positive integer and will be equal to the

number of unstable roots for the closed-loop.


• A system must have adequate

stability margins.

• Both a good gain margin and a

good phase margin are needed.

• Useful lower bounds: GM > 2.5,

PM > 30

Vector Margin is the distance to the -1

point from the closest approach of the

Nyquist plot. This is a single-margin

parameter and it removes all

ambiguities in assessing stability that

come from using GM and PM in

combination.


ω = ±∞

Np =1Np-z = 1

Np – Np-z = 0


ω = 0 rad/sGM = -4.23 dB

= 0.615

ω = 356 rad/sGM = 15.9 dB

= 6.237

ω = 86 rad/sPM = 32.5°


closed-loopBode plot


z = -50

p = -800

K = 3.2792E5


Neutral Stability


z = -50

p = -800

K = 1.0443E6


Np =1Np-z = -1

Np – Np-z = 2

ω = ±∞


z = -50

p = -800

K = 32323


Neutral Stability


z = -50

p = -800

K = 20095


Np =1Np-z = 0

Np – Np-z = 1

ω = ±∞


101

102

103

104

-270

-225

-180

P.M.: Inf

Freq: NaN

Frequency (rad/s)

Phase (

deg)

-200

-180

-160

-140

-120

-100

-80

-60

G.M.: 66.1 dB

Freq: 0 rad/s

Unstable loop


Magnitu

de (

dB

)

-800 -600 -400 -200 0 200 400-600

-400

-200

0

200

400

600


Real Axis

Imag A

xis

Uncompensated Electromagnet + Ball System

2

in

x 0.029 63ˆ

e 0.0045s 1 s 3695

Note: Negative Controller

Gain Is Required


100

101

102

103

104

105

-270

-225

-180

-135

P.M.: 25.3 deg

Freq: 201 rad/s

Frequency (rad/s)

Phase (

deg)

-140

-120

-100

-80

-60

-40

-20

0

20

G.M.: -7.78 dB

Freq: 0 rad/s

Stable loop


Magnitu

de (

dB

)

-300 -250 -200 -150 -100 -50 0 50 100

-500

-400

-300

-200

-100

0

100

200

300

400

500


Real Axis

Imag A

xis

c P D

s 30 NG (s) 132020 K K s

s 800 s N

Closed-Loop Poles: -888, -20.4, -56.9 ± 222i

KP = 4951 KD = 159 N = 800

Control

Design

PD


100

101

102

103

104

105

-270

-225

-180

-135P.M.: 30.1 deg

Freq: 163 rad/s

Frequency (rad/s)

Phase (

deg)

-150

-100

-50

0

50

G.M.: -6.55 dB

Freq: 21.7 rad/s

Stable loop


Magnitu

de (

dB

)

-250 -200 -150 -100 -50 0 50

-200

-150

-100

-50

0

50

100

150

200


Real Axis

Imag A

xis

Closed-Loop Poles: -959, -67 ± 185i, -12.8 ± 17.2i

2

Ic P D

s 38.28s 370.42 K NG (s) 113200 K K s

s s 896 s s N

KP = 4784 KI = 46798 KD = 121 N = 896

Control

Design

PID


C = 2.29E-5m = 0.008g = 9.81R = 34.1

L = 154.2E-3x0 = 0.0053

i0 = 0.31e0 = 10.57

Nonlinear System

e0

V Bias

StepSaturation

0 to 15 volts

Product

u2

MathFunction2

1

u

MathFunction1

u2

MathFunction

1/s

Integrator2

1/s

Integrator1

1/s

Integrator1/R

Gain2

R/L

Gain1 C/m

Gain

i

Current

-113200

ControllerGain

M

Control Effort

s +38.28s+370.422

s +896s2

Control

g

Constantx

Ball Position

Linear System

StepSaturation

-10.57 to 4.43 volts

x_hat

PerturbationPosition

-63

s +-36952

Magnet + Ball

0.0045s+1

0.029

LR Circuit

-113200

ControllerGain

s +38.28s+370.422

s +896s2

Control

i_hat

PerturbationCurrent

M_hat

PerturbationControl Effort

Comparison: Linear Plant vs. Nonlinear Plant


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2x 10

-3

time (sec)

Positio

n x

(m

)

Nonlinear & Linear Plant Response Comparison: 1 mm Step Command

Nonlinear Pant

Linear Plant

PD Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time (sec)

Curr

ent

i (A

)


Nonlinear Plant

Linear Plant

PD Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

time (sec)

Contr

ol E

ffort

M (

volts)


Nonlinear Plant

Linear Plant

PD Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2x 10

-3

time (sec)

Positio

n x

(m

)


Nonlinear Plant

Linear Plant

PID Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time (sec)

Curr

ent

i (A

)


Nonlinear Plant

Linear Plant

PID Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

time (sec)

Contr

ol E

ffort

M (

volts)


Nonlinear Plant

Linear Plant

PID Control


Complete System: Electromagnet + Ball + PWM Voltage ControlC = 2.29E-5m = 0.008g = 9.81R = 34.1

L = 154.2E-3x0 = 0.0053

i0 = 0.31e0 = 10.57

Identical Controller - PID Format

e0

V BiasPWM

>=

SwitchTransistorMOSFET

0

Supply VoltageSwitch Off

15

Supply VoltageSwitch ON

Step1 mmstep

command

5

Set amplitude

to 5V

Saturation0 to 15 volts

Saturation0 to 1 amp

>

RelationalOperatorReference

Signal 4000Hz

Product

PID(s)

PID Controller

u2

MathFunction2

1

u

MathFunction1

u2

MathFunction

1/s

Integrator2

1/s

Integrator1

1/s

Integrator1/R

Gain2

R/L

Gain1 C/m

Gain

i

Current

1

Convert Boolean

into Double

-113200

ControllerGain2

1/3

-1

ControllerGain

s +38.3s+370.42

s +896s2

Controller

M

ControlEffort

g

Constantx

Ball Position


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2x 10

-3

time (sec)

Positio

n x

(m

)

Nonlinear Plant & PWM Voltage Control: 1 mm Step Command

PD Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time (sec)

Curr

ent

i (A

)


PD Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

time (sec)

Contr

ol E

ffort

M (

volts)


PD Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2x 10

-3

time (sec)

Positio

n x

(m

)

Nonlinear Plant & PWM Voltage Control 1 mm Step Command

PID Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time (sec)

Curr

ent

i (A

)


PID Control


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

time (sec)

Contr

ol E

ffort

M (

volts)


PID Control


Emitter Circuit

Detector

Circuit

Power Supply

Capacitors

to Ground

Buffer Op-Amp

Buffer Op-Amp

Power MOSFET

with Diode

To Electromagnet

Analog Sensor PWM Gnd


Microcontroller Board

Analog Sensor

Gnd PWM


Arduino Microcontroller Implementation

With Simulink Autocode Generator

Arduino Discrete PiD ControlMagnetic Levitation System

PWMTs = sample period = 0.001

Operating point is 0.0053 m gap and corresponds to sensor reading of 2.5 VSensor gain is 1.6V/mm around operating point + or - 1 mm

volts = 1600*m - 5.98m = (volts + 5.98)/1600

0.0053 m gap

Saturation0 to 15 volts

PID(s)

PID Controller

1/1600

Gain1

1/1600

Gain

Pin 10

Digital Output

-1

ControllerGain2

1/3

5.98

Constant

2.5

CommandedPosition

Volts

10.57

Bias Voltage

Pin 0

Analog Input

255/5

8-Bit D/A

5/1023

10-Bit A/D


Closed-Loop System

Block Diagram

LM 258

Low-Power

Dual Op-Amp

Unity-Gain Buffer Op-Amp

ein = eout and in phase


Power MOSFET TO-220

N-Channel, 60 V, 0.07 Ω, 16 A


eoutein

R2

R1 +V

-V-

+

RS

RMLM

Electromagnet

Voltage-to-Current Converter

2M in

1 2 S

R 1i e

R R R

OPA544High-Voltage, High Current

Op Amp

Assume Ideal Op-Amp

Behavior

e e R1 = 49KΩ, R2 = 1KΩ, RS = 0.1Ω

Alternative: Analog Power Stage


Non-Ideal

Op-Amp Behavior

o

Ae e e

s 1

e1

out 1 M M

1 S

1out 1 M M

S

M M Sout 1

S

e e L s R i

e R i

ee e L s R

R

L s R Re e

R

eoutein

R2

R1 +V

-V-

+

RS

RMLM

Electromagnet

+

-

Saturation

Σ

2in

1 2

Re

R R

e1

i

S

1

R

A

s 1

S

M M S

R

L s R R

eout e1

magnetic levitation system - mechatronics2020.net · magnetic levitation system k. craig 5 •...

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