MECH466 L10 TResponseEx

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2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

Lecture 10

Time responses: 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

Typical step response (review)

Performance measures (review)

Transient response

Peak valuePeak value

Peak timePeak time

Percent overshootPercent overshoot

Delay timeDelay time

Rise timeRise time

Settling timeSettling time

Steady state response

Steady state errorSteady state error

Connect Connect

these measures these measures with s

with s--domain.domain.

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A DC motor example (revisited)

DC motor position control example

Design K

s.t

.

Percent overshoot is 5%.Percent overshoot is 5%.

Closed

Closed--loop TFloop TF

Motor Motor Amplifier

Amplifier

(Incorrect in last lecture slide) (Incorrect in last lecture slide)

Allowable pole locations (review)

Require PO <

POm

(given):

Re Re Im Im

Problem 4.16

Derive the relationship for damping ratio as a

function of percent overshoot, from

Problem 4.23

For each pair of second

-

-order specifications that

order specifications that

follow, find the location of

the second

-

order pair of

poles.

a) PO = 12%, Ts = 0.6 sec

b) PO = 10%,

Tp

= 5 sec

c) Ts = 7 sec,

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Problem 4.23 (cont

Problem 4.23 (cont’

’d)

d)

Ideas

From a given percent overshoot, we can compute a From a given percent overshoot, we can compute a corresponding damping ratio

corresponding damping ratio ζζ..

From a given settling time Ts, we can compute From a given settling time Ts, we can compute

From a given peak time From a given peak time TpTp, we can compute, we can compute

Problem 4.25

For the system below,

Find the transfer function from Find the transfer function from X(sX(s) to ) to F(sF(s))

Find DC gain, Find DC gain, ζζ, , ωωnn, T, Tss(2%), (2%), TTpp, PO, PO

Problem 4.25 (cont

’d)

d)

Find DC gain,

ζ,

ζ

,

ω

nn

, T

s s

(2%)

,

T

pp

, PO

DC gain G(0)=1/33

Problem 7.10

For the unity feedback system, check and

compute

StabilityStability

2% settling time for 2% settling time for u(tu(t))

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Problem 7.10 (cont

’d)

d)

Stability

Characteristic equationCharacteristic equation

Since all the coefficients of a 2

Since all the coefficients of a 2ndnd_{order polynomial}_{order polynomial}

have the same sign, the CL system is stable.

2% settling time

Time constant isTime constant is

2% settling time is 2% settling time is

Problem 7.10 (cont

’

d)

Steady

-

state errors

For 5u(t) (step)For 5u(t) (step)

For 5tu(t) (ramp)For 5tu(t) (ramp)

For 5t*For 5t*tu(ttu(t) (parabolic) ) (parabolic)

Problem 7.29

Design the unity feedback system.

SteadySteady--state error for unit ramp input is 1/10state error for unit ramp input is 1/10

ClosedClosed--loop poles are at loop poles are at

Problem 7.35

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Problem 7.35 (cont

’d)

d)

Closed

-

loop transfer function

Problem 7.35 (cont

’

d)

From the two requirements,

For CL stability, K

For CL stability, K11has to be positive.has to be positive.

Announcements

Midterm exam

February 12 (Thursday), 9:40February 12 (Thursday), 9:40--10:40am.10:40am.

Please come to CEME1202 at/before 9:30am.Please come to CEME1202 at/before 9:30am.

Policy: Closed book, No lecture slide, No calculatorPolicy: Closed book, No lecture slide, No calculator

Last year

’

s midterm and exercise solutions in the

textbook are posted on Vista.

Lab #2 report should be submitted at the

beginning of your Lab #3 session. (Lab 3 starts

from March.)