MECH466 L8 TDomSpecSSError

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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 8

Lecture 8

Time

Time--domain specificationsdomain specifications Steady

Steady--state errorstate error

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 •

•Routh-Routh-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

What we did and what we do next

We have learned stability.We have learned stability.

Definition in time domainDefinition in time domain

Condition in sCondition in s--domaindomain

RouthRouth--Hurwitz criterion to check the conditionHurwitz criterion to check the condition

Stability is a necessary requirement, but not Stability is a necessary requirement, but not

sufficient in most control problems.

Specifications other than stabilitySpecifications other than stability

How to evaluate a system quantitatively in time domain?How to evaluate a system quantitatively in time domain?

How to give specifications in time domain?How to give specifications in time domain?

What are the corresponding conditions in sWhat are the corresponding conditions in s--domain?domain?

Time response

We would like to analyze a system property by We would like to analyze a system property by applying a

applying a test inputtest inputr(tr(t) and observing a time ) and observing a time response

response y(ty(t).).

Time response is divided asTime response is divided as

System

Transient (natural) response

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Example of transient &

steady

-

state responses

Transient responseTransient response

SteadySteady--state state respresp..

Step Response

Time (sec)

A

m

pl

it

ude

0 2 4 6 8 10 12

0 0.5 1 1.5 2 2.5 3

Step response

Step response

Time (sec)

Time (sec) 2008/09 MECH466 : Automatic Control 6

Usage of time responses

ModelingModeling

Some parameters in the system may be estimated by Some parameters in the system may be estimated by

time responses.

AnalysisAnalysis

Evaluate transient and steadyEvaluate transient and steady--state responses state responses (Satisfactory or not?)

(Satisfactory or not?)

DesignDesign

Given design specs in terms of transient and steadyGiven design specs in terms of transient and steady-

-state responses, design controllers satisfying all the

state responses, design controllers satisfying all the

design specs.

Typical test inputs

Step function Step function

(Most popular)

Ramp function Ramp function

Parabolic Parabolic function function Sinusoidal input

Sinusoidal input will be dealt with will be dealt with

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Steady state value for step input

Suppose that Suppose that G(sG(s) is stable) is stable..

By the final value theorem:By the final value theorem:

Step response converges to some finite value, Step response converges to some finite value,

called

calledsteady state value steady state value ..

G(s

G(s))

Steady

Steady-

-state error for input

state error for input u(t

u(t)

)

Peak value, peak time, percent

overshoot

Delay, rise, and settling times

Delay time

Delay time: time to reach 0.5 : time to reach 0.5 yyssss

Rise time

Rise time: time to rise from 0.1y: time to rise from 0.1yssssto 0.9yssto 0.9yss

Settling time

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An example revisited

For the example in a previous slide,For the example in a previous slide,

SteadySteady--state error : 2state error : 2

Delay time around 1.5 secDelay time around 1.5 sec

Rise time around 5 secRise time around 5 sec

Settling time around 6 secSettling time around 6 sec

Step Response Time (sec) A m pl it ude

0 2 4 6 8 10 12

0 0.5 1 1.5 2 2.5 3 Remark

Remark: There is no peak in : There is no peak in this case, so peak value, peak

this case, so peak value, peak

time and percent overshoot

cannot be defined.

Remarks on time

-

domain responses

Speed of responseSpeed of responseis measured byis measured by

Rise time, delay time, peak time and settling timeRise time, delay time, peak time and settling time

Relative stabilityRelative stabilityis measured byis measured by

Percent overshootPercent overshoot

Typically Typically ……..

Fast response Fast response ÆÆLarge percent overshootLarge percent overshoot

Large percent overshoot Large percent overshoot ÆÆsmall stability marginsmall stability margin

We need to take tradeWe need to take trade--off between response off between response speed and stability.

speed and stability.

Performance measures

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

We will connect We will connect these measures these measures with s

with s--domain.domain.

(Today

(Today’’s lecture)s lecture) (Next lecture) (Next lecture)

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 •

•Routh-Routh-HurwitzHurwitz •

•NyquistNyquist

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

PID & Lead PID & Lead--laglag

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Steady

Steady-

-state error

state error

Suppose that we want output Suppose that we want output y(ty(t) to track ) to track r(tr(t).).

Error Error

SteadySteady--state errorstate error

Final value theorem Final value theorem (Suppose CL system is stable!!!)

(Suppose CL system is stable!!!)

Unity feedback! Unity feedback! We assume that the We assume that the CL system is stable! CL system is stable!

Error constants

StepStep--error (positionerror (position--error) constanterror) constant

RampRamp--error (velocityerror (velocity--error) constanterror) constant

ParabolicParabolic--error (accelerationerror (acceleration--error) constanterror) constant

KpKp, , KvKv, Ka :, Ka :ability to reduce steadyability to reduce steady--state errorstate error

Steady

-state error for step

-

state error for step r(t

r(t)

)

Kp Kp

Steady

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Steady

Steady-

-state error for parabolic

state error for parabolic r(t

r(t)

)

Ka Ka

System type

System type of CL systemSystem type of CL systemis defined as the is defined as the

order (number) of poles of

order (number) of poles of G(sG(s) at s=0.) at s=0.

ExamplesExamples

type 1

type 2

type 3

Zero steady

-state error

-

state error

If error constant is infinite, we can achieve zero If error constant is infinite, we can achieve zero

steady

steady--state error. (Accurate tracking)state error. (Accurate tracking)

For step For step r(tr(t))

For ramp For ramp r(tr(t))

For parabolic For parabolic r(tr(t))

Example 1

CL system of type 2CL system of type 2

Characteristic equationCharacteristic equation

CL system is NOT stable for any K.CL system is NOT stable for any K.

e(te(t) will not converge. (Don) will not converge. (Don’’t use todayt use today’’s results s results if CL system is not stable!!!)

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Example 2

By By RouthRouth--Hurwitz criterion, CL is stable Hurwitz criterion, CL is stable iffiff

Step Step r(tr(t))

Ramp Ramp r(tr(t))

Parabolic Parabolic r(tr(t))

Example 3

By By RouthRouth--Hurwitz criterion, we can show that CL Hurwitz criterion, we can show that CL system is stable.

system is stable.

Step Step r(tr(t))

Ramp Ramp r(tr(t))

Parabolic Parabolic r(tr(t))

A control example

ClosedClosed--loop stable?loop stable?

Compute error constantsCompute error constants

Compute steady state errorsCompute steady state errors

Summary and Exercises

Time response and time domain specificationsTime response and time domain specifications

SteadySteady--state errorstate error

For For unity feedbackunity feedback(STABLE!) systems, CL system (STABLE!) systems, CL system type determines if the steady

type determines if the steady--state error is zero.state error is zero.

The key tool is the The key tool is the final value theoremfinal value theorem!!

ExercisesExercises

Read Sections 7.1Read Sections 7.1--7.4.7.4.

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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