MECH466 L9 TimeResponse

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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 9

Lecture 9

Time responses of

1

1stst_-_-_{order and 2}_{order and 2}ndnd_-_-_{order systems}_{order systems}

Course roadmap

Laplace transform Laplace transform Transfer function Transfer function

Models for systems

•

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization

Linearization Modeling

Modeling AnalysisAnalysis DesignDesign

Time response

•

•TransientTransient •

•Steady stateSteady state

Frequency response

•

•Bode plotBode plot

Stability

•

•RouthRouth--HurwitzHurwitz

•

•NyquistNyquist

Design specs Design specs Root locus Root locus Frequency domain Frequency domain

PID & Lead

PID & Lead--laglag

Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Typical step response (review)

Performance measures (review)

Transient responseTransient response

Peak valuePeak value

Peak timePeak time

Percent overshootPercent overshoot

Delay timeDelay time

Rise timeRise time

Settling timeSettling time

Steady state responseSteady state response

Steady state errorSteady state error

Next, we will connect Next, we will connect these measures these measures with s

with s--domain.domain.

(Done) (Done) (Today

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First

-

order system

A A standard formstandard formof the firstof the first--order system:order system:

DC motor exampleDC motor example

DC motor example (cont

’

d)

If La<<Ra, we can obtain a 1stIf La<<Ra, we can obtain a 1st--order systemorder system

Remember that TF from motor voltage toRemember that TF from motor voltage to

motor motor speedspeedis 1stis 1st--oder (Lab #3)oder (Lab #3)

motor motor positionpositionis 2ndis 2nd--order (Lab #4)order (Lab #4)

Step response for 1

stst

-

order system

Input a Input a unit step functionunit step functionto a firstto a first--order system. order system. Then, what is the output?

Then, what is the output?

0

1

u(t

u(t)) y(ty(t))

0

(Partial fraction expansion) (Partial fraction expansion)

Meaning of K and T

K : K : GainGain

Final (steadyFinal (steady--state) valuestate) value

T : T : Time constantTime constant

Time when response Time when response

rises 63% of final value

Indication of Indication of speedspeedof of response (convergence)

response (convergence)

Response is faster as T Response is faster as T

becomes smaller.

0 1 2 3 4 5 6

0 0.2 0.4 0.6 0.8 1

K=1,T=1

Time

A

m

pl

it

ud

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DC gain for a general system

DC gain : DC gain : Final valueFinal valueof a unit step responseof a unit step response

For firstFor first--order systems, DC gain is K.order systems, DC gain is K.

For a For a general stable system Ggeneral stable system G, DC gain is G(0)., DC gain is G(0).

ExamplesExamples

Final value theorem Final value theorem

Settling time of 1

stst

-

order systems

Relation between time and exponential decayRelation between time and exponential decay

5% settling time is about 3T! 5% settling time is about 3T!

2% settling time is about 4T! 2% settling time is about 4T!

Step response for some K & T

0 5 10

0 1 2

K=1,T=1

Time

Am

p

li

tu

d

e

0 5 10

0 1 2

K=1,T=2

Time

Am

p

li

tu

d

e

0 5 10

0 1 2

K=2,T=1

Time

A

m

p

litu

d

e

0 5 10

0 1 2

K=2,T=2

Time

A

m

p

litu

d

e

System identification

Suppose that we have a Suppose that we have a ““blackblack--boxbox””systemsystem

Obtain step responseObtain step response

Can you obtain a transfer function? How?Can you obtain a transfer function? How? Unknown

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Summary for 1

stst

order systems

For 1For 1stst_{order systems, step responses have}_{order systems, step responses have}

SteadySteady--state gain: Kstate gain: K

Peak value: undefinedPeak value: undefined

Peak time: undefinedPeak time: undefined

Percent overshoot: undefinedPercent overshoot: undefined

Delay time: 0.7T (not required in this course)Delay time: 0.7T (not required in this course)

Rise time: 2.2T (not required in this course)Rise time: 2.2T (not required in this course)

Settling timeSettling time •

• 5%: 3T5%: 3T

•

• 2%: 4T2%: 4T

Second

-

order systems

A A standard formstandard formof the secondof the second--order systemorder system

DC motor position control exampleDC motor position control example

Closed

Closed--loop TFloop TF Motor

Motor

Amplifier

Step response for 2

ndnd

-

order system

Input a Input a unit step functionunit step functionto a 2ndto a 2nd--order system. order system. What is the output?

What is the output?

0

1

u(t

u(t)) y(ty(t))

0

DC gain DC gain

0 5 10 15

0 0.5 1 1.5 2

Step response for 2nd

-

order system

for various damping ratio

UndampedUndamped

UnderdampedUnderdamped

Critically dampedCritically damped

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Step response for 2nd

-

order system

Underdamped

case

Math expression of Math expression of y(ty(t) for ) for underdampedunderdampedcasecase

Damped natural frequency

0 5 1 0 1 5

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6

Peak value/time:

Underdamped

case

Properties of 2nd

-

order system

(5%)

(5%) (2%)(2%)

Some remarks

Percent overshoot depends on ζPercent overshoot depends on ζ, but NOT , but NOT ωωnn. .

(See the next slide.)

For the 2ndFor the 2nd--order transfer function, analytic order transfer function, analytic expressions of delay & rise time are hard to

expressions of delay & rise time are hard to

obtain (and not covered in this course).

Time constant is 1/(ζωTime constant is 1/(ζωnn), indicating ), indicating convergence speed.

convergence speed.

For ζFor ζ>1, we cannot define peak time, peak >1, we cannot define peak time, peak value, percent overshoot.

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P.O. vs. damping ratio

Pole locations of G

Poles (0<ζPoles (0<ζ<1)<1)

Damping ratioDamping ratio

Next, we clarify the influence of

pole locations on the step response.

Influence of real part of poles

Settling time Ts decreases.Settling time Ts decreases.

Ts Ts

Influence of

imag

. part of poles

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Influence of angle of poles

Over/underOver/under--shoot decreases.shoot decreases.

An example

Require 5% settling time Ts < Require 5% settling time Ts < TsmTsm(given):(given):

Re

Im

An example (cont

’

d)

Require PO < Require PO < POmPOm(given):(given):

Re

Im

An example (cont

’

d)

Combination of two requirementsCombination of two requirements

Re

Im

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A DC motor example (revisited)

DC motor position control exampleDC motor position control example

Design K Design K s.ts.t..

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

Closed

Closed--loop TFloop TF Motor

Motor

Amplifier

Summary and Exercises

Time response of Time response of

1st1st--order system is characterized byorder system is characterized by •

• Time constant and DC gainTime constant and DC gain

2nd2nd--order system is characterized byorder system is characterized by

•

• Damping ratio Damping ratio ζζ& & undampedundampednatural frequency natural frequency ωωnn

•

• Pole locationsPole locations

For transient responses of high order systems, For transient responses of high order systems, we need computer simulations.

we need computer simulations.

ExercisesExercises

Read Sections 4.1Read Sections 4.1--4.6.4.6.

Problems 4.2, 4.4, 4.6, 4.23, 4.24, 4.25, 4.26, 4.29.Problems 4.2, 4.4, 4.6, 4.23, 4.24, 4.25, 4.26, 4.29.

Announcements

Midterm examMidterm 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 yearLast year’’s midterm and exercise solutions in the s midterm and exercise solutions in the textbook are posted on Vista.

textbook are posted on Vista.

Lab #2 report should be submitted at the Lab #2 report should be submitted at the

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

from March.)