MECH466 L22 DesignExample

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2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

University of British Columbia Lecture 22

Lecture 22

Design examples

Course roadmap

Laplace transform Laplace transform

Transfer function Transfer function

Models for systems Models for systems •

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization Linearization

Modeling

Modeling AnalysisAnalysis DesignDesign

Time response Time response •

•TransientTransient •

•Steady stateSteady state

Frequency response Frequency response •

•Bode plotBode plot

Stability Stability •

•RouthRouth--HurwitzHurwitz •

•NyquistNyquist

Design specs Design specs

Root locus Root locus

Frequency domain Frequency domain

PID & Lead PID & Lead--laglag

Design examples Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Outline

System description and control objectiveSystem description and control objective

ModelingModeling

AnalysisAnalysis

Stability analysis via Stability analysis via RouthRouth--Hurwitz criterionHurwitz criterion

SteadySteady--state error analysisstate error analysis

Design of leadDesign of lead--lag compensatorslag compensators

Root locusRoot locus

Frequency responseFrequency response

Radio telescope antenna

Large & parabolicLarge & parabolic

Receive radio wave Receive radio wave

Radio astronomyRadio astronomy

TrackingTracking

Collecting dataCollecting data

SatelliteSatellite

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Antenna azimuth position control

(Taken from

(Taken from NiseNise’’ssbook)book)

Antenna Antenna

Potentiometer Potentiometer Potentiometer

Potentiometer

Motor Motor

Outline

ModelingModeling

AnalysisAnalysis

Antenna azimuth position control

Block diagram

Closed

-

loop transfer function

(3)

Outline

ModelingModeling

AnalysisAnalysis

Stability of feedback system

RouthRoutharrayarray

CL system is stable

Outline

ModelingModeling

AnalysisAnalysis

Steady

-

state error

(4)

Steady

-

state error (cont

’

d)

For unit step:For unit step:

For unit ramp:For unit ramp:

For unit parabolic:For unit parabolic:

Outline

ModelingModeling

AnalysisAnalysis

Gain design via root locus

Find K required for Find K required for 25% overshoot.

25% overshoot.

For 2nd order systems

By magnitude

condition

Step response

(5)

Lead

-

lag design via root locus

Design specificationsDesign specifications

25% overshoot25% overshoot

2% settling time 2 seconds2% settling time 2 seconds

Kv=20Kv=20

In the previous gain design,In the previous gain design,

2% settling time 4.8 seconds2% settling time 4.8 seconds

KvKv=2.49 =2.49 (Verify this by yourself!)(Verify this by yourself!) Not satisfactory!

Not satisfactory!

Lead compensator design

Locate desired polesLocate desired poles

Due to Due to ““2% settling time 2 seconds2% settling time 2 seconds””

Angle conditionAngle condition

Re Re Im

Im

Lead compensator design

Fix z=Fix z=--2 (rather arbitrarily), 2 (rather arbitrarily), and obtain p

and obtain p s.ts.t. .

Find KFind K

Im Im

Lag compensator design

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Step and ramp responses

0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

9.5 9.6 9.7 9.8 9.9 10 9

9.2 9.4 9.6 9.8 10

Gain compensated Gain compensated

Lead compensated Lead compensated

Lead

Lead--lag compensatedlag compensated

Ramp reference Ramp reference

Outline

ModelingModeling

AnalysisAnalysis

10-2 10-1 100 101 102

-100 -50 0

-250 -200 -150 -100

Open

-

loop frequency response

GM=68.41dB

PM=88.68deg

PM=88.68deg

Gain, PM, and step response

Change preamplifier gain K to 20, 40, 60.Change preamplifier gain K to 20, 40, 60.

0.2 0.4 0.6 0.8 1 1.2 1.4

K=20 (PM=67) K=20 (PM=67)

K=40 (PM=53) K=40 (PM=53)

K=60 (PM=45) K=60 (PM=45)

Step responses

(7)

Lead

-

lag design via freq. response

Design specifications (same as before)Design specifications (same as before)

2% settling time 2 seconds 2% settling time 2 seconds

KvKv=20=20

One possible procedureOne possible procedure 1.

1. Use the gain to satisfy Use the gain to satisfy KvKv..

2.

2. Use lead compensator to increase PM (and gain Use lead compensator to increase PM (and gain

crossover frequency).

3.

3. Use lag compensator to decrease gain crossover Use lag compensator to decrease gain crossover

frequency (to increase PM).

Gain+lead

compensation

10-2 10-1 100 101 102 -200

-100 0 100

10-2 10-1 100 101 102 -250

-200 -150 -100 -50

0 1 2 3 4 5

0 0.5 1 1.5 2

G(s G(s))

KG(s

KG(s) (PM=13)) (PM=13)

KC

KCLeadLead(s)G(s(s)G(s) (PM=50)) (PM=50)

Step responses

Open

Open--loop Bode plotloop Bode plot

0.5 1 1.5 2

Gain+lead

/lag compensation

KG(s

KG(s) (PM=13)) (PM=13)

KC

KCLeadLead(s)C(s)CLagLag(s)G(s(s)G(s) ) (PM=69)

(PM=69)

10-2 10-1 100 101 102 -200

-100 0 100

-150 -100 -50

G(s G(s))

Step responses

Open

Open--loop Bode plotloop Bode plot

Hard disk drives

Data storage mediaData storage media

ApplicationsApplications

ComputerComputer

Mobile audio playerMobile audio player

Mobile phoneMobile phone

Video cameraVideo camera

Car navigation, etc.Car navigation, etc.

(8)

Control mechanism

Disk spins ~5,400rpmDisk spins ~5,400rpm

Head slider flies over Head slider flies over

the disk <100nm height

Input: VCM actuatorInput: VCM actuator

Output: Head positionOutput: Head position

Track seeking:Track seeking:Head Head

moves from one track to

another.

Track following:Track following:Head Head

follows a specified track.

follows a specified track. VCM

VCM

Spindle

Spindle

Head

Tracks

Pivot

Pivot ArmArm

Samsung HG501LJ Samsung HG501LJ

Modeling & Block diagram

Flexure Flexure

Head Head

VCM VCM

Arm Arm

Sensor Sensor ( (H(sH(s)=1))=1) Controller

Controller C(s C(s))

Load Load

G

G2(s)2(s) Y(sY(s):): Actual head

Actual head

position

R(s

R(s):):

Desired head

position

Coil Coil G G11(s)(s)

D(s

D(s): Torque disturbance): Torque disturbance

Transfer functions

Open loop transfer functionOpen loop transfer function

From From R(sR(s) (reference) to ) (reference) to Y(sY(s))

From From D(sD(s) (disturbance) to ) (disturbance) to Y(sY(s))

Specifications in this example

Track seeking:Track seeking:For unit step reference For unit step reference r(tr(t):):

(1) Percent overshoot

(1) Percent overshoot< 5%< 5% (2) 1% settling time

(2) 1% settling time< 50ms< 50ms

Track following:Track following:For unit step disturbance For unit step disturbance d(td(t):):

(3) Maximum response

(3) Maximum response< 0.005< 0.005

40 40

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(1)%% (2)(2)msms (3)(3) PM degPM deg ωωggrad/srad/s

4.6

4.6 470470 0.00520.0052 ₆₅₆₅ _9.1_9.1 C(s

(9)

0 0.5 1 1.5 2

0 0.005 0.01 0.015 0.02

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Step responses for gain

C(s

)

C(s C(s)=10)=10

C(s C(s)=40)=40 From

From R(sR(s) to ) to Y(sY(s)) From From D(sD(s) to ) to Y(sY(s))

C(s C(s)=80)=80 P.O. should be <5% P.O. should be <5% Settling time should be <50ms Settling time should be <50ms

Max. response to unit Max. response to unit d(td(t) )

should be <0.005 should be <0.005

We cannot find

a satisfactory gain

a satisfactory gain C(sC(s).).

(sec)

(sec) (sec)(sec) _2008/09 _{MECH466 : Automatic Control} ₃₄

A satisfactory lead

-

lag

C(s

)

100 101 102 103 -100

-50 0 50

Gain plot

100 101 102 103 -250

-200 -150 -100 -50

Phase plot

0 0.05 0.1 0.15 0.2 0

0.5 1 1.5

y(t) for step r(t)

0 0.05 0.1 0.15 0.2 0

1 2 3 4 5x 10

-3 _{y(t) for step d(t)}

Lead

Lead C(sC(s))

Gain

Gain C(sC(s))

Step response for

r(t

)

0.4 0.6 0.8 1

y(t) for step r(t)

P.O should be <5% P.O should be <5% Settling time should be <50ms Settling time should be <50ms

Summary

Case studyCase study

Antenna azimuth position control Antenna azimuth position control

(

(““Control Systems EngineeringControl Systems Engineering””by by NiseNise.).)

Hard disk drive control Hard disk drive control

(

(““Modern Control SystemsModern Control Systems””by by DorfDorfand Bishop)and Bishop)

•

• ModelingModeling •

• AnalysisAnalysis •

• Design of leadDesign of lead--lag compensators (A lag compensators (A MatlabMatlabcode is code is posted on