MECH466 L12 RLProof&PID

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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 12

Lecture 12

PID control (Preparation for Lab 3)

Root locus: Sketch of proofs

Course roadmap

Laplace transform

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

Root locus

Frequency domain

PID & Lead

PID & Lead--laglag

Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

PID controller

G(s) Kp

Ki/s

Kds

-r

r ee uu yy

Proportional

Proportional IntegralIntegral DerivativeDerivative

t

t--domain:domain:

s

s--domain:domain:

Notes on PID controller

Most popular in process and robotics industriesMost popular in process and robotics industries

Good performanceGood performance

Functional simplicity (Operators can easily tune.)Functional simplicity (Operators can easily tune.)

To avoid high frequency noise amplification, To avoid high frequency noise amplification, derivative term is implemented as

derivative term is implemented as

with

with ττddmuch smaller than plant time constant.much smaller than plant time constant.

PI controllerPI controller

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A simple example

We plot We plot y(ty(t) for step reference ) for step reference r(tr(t) with) with

P controllerP controller

PI controllerPI controller

PID controllerPID controller

G(s) Kp

Ki/s

Kds

-r

r ee uu yy

P controller

SimpleSimple

Steady state errorSteady state error

Higher gain gives Higher gain gives smaller error

smaller error

StabilityStability

Higher gain gives Higher gain gives faster and more

faster and more

oscillatory response

0 5 10 15 20

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 5 10 15 20

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

PI controller

Zero steady state Zero steady state error

error(provided that (provided that CL is stable.) CL is stable.)

StabilityStability

Higher gain gives Higher gain gives

faster and more

oscillatory response

0 5 10 15 20

0 0.5 1 1.5 2

PID controller

Zero steady state Zero steady state error (due to integral error (due to integral control)

control)

StabilityStability

Higher gain gives Higher gain gives more

more dampeddamped response

response

Too high gain Too high gain

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How to tune PID parameters

Empirical (ModelEmpirical (Model--free)free)

Trial and errorTrial and error

ZieglerZiegler--Nichols tuning rule (1942) (Appendix)Nichols tuning rule (1942) (Appendix)

Useful even if a system is too complex to modelUseful even if a system is too complex to model

Useful only when trialUseful only when trial--andand--error tuning is allowederror tuning is allowed

ModelModel--basedbased

Root locusRoot locus

Frequency response approachFrequency response approach

Useful only when a model is availableUseful only when a model is available

Necessary if a system has to work at the first trialNecessary if a system has to work at the first trial

PID controller realization

One example: Using OP ampOne example: Using OP amp

R

R22 R

R11

C

C22

-+

+

C

C11

-+

+

R

R33 R

R44

v

vii(t(t))

v

voo(t(t))

Exercise: Derive this! Exercise: Derive this!

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

Complex numbers (review)

RepresentationRepresentation

Cartesian formCartesian form

Polar formPolar form

Multiplication & division in the polar formMultiplication & division in the polar form Re Re Im

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What is Root Locus? (Review)

Consider a feedback system that has one Consider a feedback system that has one parameter (gain) K>0 to be designed. parameter (gain) K>0 to be designed.

Root locusRoot locusgraphically shows how poles of the graphically shows how poles of the closed

closed--loop system varies as K varies from 0 to loop system varies as K varies from 0 to infinity.

infinity.

L(s

L(s)) K

K

L(s

L(s): open): open--loop TFloop TF

RL sketching algorithm (review)

Step 0: Mark open-Step 0: Mark open-loop poles and zerosloop poles and zeros

Step 1: On the real axisStep 1: On the real axis

Step 2: AsymptotesStep 2: Asymptotes

Step 3: Breakaway pointsStep 3: Breakaway points

Step 4: Angles of departures and arrivalsStep 4: Angles of departures and arrivals

Examples of root locus

Re

Re ReRe ReRe ReRe

Re

Re ReRe ReRe ReRe

Don

Don’’t forget to put arrows!t forget to put arrows!

Characteristic equation & root locus

Characteristic equationCharacteristic equation

Root locus is obtained byRoot locus is obtained by

for a fixed K>0, finding roots of the characteristic for a fixed K>0, finding roots of the characteristic

equation, and

sweeping K over real positive numbers.sweeping K over real positive numbers.

A point “A point “ss””is on the root locus, if and only if L(sis on the root locus, if and only if L(s) ) evaluated for that

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Angle and magnitude conditions

Characteristic Characteristic eqeq. can be split into two conditions.. can be split into two conditions.

Angle conditionAngle condition

Magnitude conditionMagnitude condition

Odd number Odd number

For any point s,

this condition holds

for some positive K.

A simple example

Select a pointSelect a points=s=--2+j2+j Re

Re Im Im

s is on root locus.

Select a pointSelect a points=s=--1+j1+j

s is NOT on root locus.

Root locus: Step 0

Root locus is symmetric Root locus is symmetric w.r.tw.r.t. the real axis.. the real axis.

Characteristic equation isCharacteristic equation isananequation with real equation with real coefficients. Hence, if a complex number is a root, its coefficients. Hence, if a complex number is a root, its complex conjugate is also a root.

complex conjugate is also a root.

The number of branches = order of The number of branches = order of L(sL(s))

If If L(sL(s)=)=n(s)/d(sn(s)/d(s), then Ch. ), then Ch. eqeq. is . is d(s)+Kn(sd(s)+Kn(s)=0, which )=0, which has roots as many as the order of

has roots as many as the order of d(sd(s).).

Mark poles of L with Mark poles of L with ““xx””and zeros of L with and zeros of L with ““oo””..

Re Re Im Im

Root locus: Step 1

-

1

RL includes all points on real axis to the left of an RL includes all points on real axis to the left of an odd number of real poles/zeros.

odd number of real poles/zeros.

Re Re Im

Im Test point

Test point

0

0 00 00

Re Re Im

Im

180

180 00 00

Not satisfy angle condition!

Satisfy angle condition!

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Root locus: Step 1

-

1 (cont

’

d)

RL includes all points on real axis to the left of an RL includes all points on real axis to the left of an odd number of real poles/zeros.

odd number of real poles/zeros.

Re Re Im

Im

180

180 180180 00

Re Re Im

Im

180

180 180180 180180 Not satisfy angle condition!

Not satisfy angle condition!

Satisfy angle condition!

Root locus: Step 1

-

2

RL originates from the poles of L, and terminates RL originates from the poles of L, and terminates at the zeros of L, including infinity zeros.

at the zeros of L, including infinity zeros.

s: Poles of

s: Poles of L(sL(s)) s: Zeros of s: Zeros of L(sL(s))

Root locus: Step 2

-

1

Number of asymptotes = relative degree (r) of L:Number of asymptotes = relative degree (r) of L:

Angles of asymptotes areAngles of asymptotes are

Root locus: Step 2

-

1 (cont

’

d)

For a very large s,For a very large s,

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Root locus: Step 2

-

2

Intersections of asymptotesIntersections of asymptotes

Proof for this is omittedProof for this is omittedand not required in this and not required in this course.

course.

Interested students should read the proof in Interested students should read the proof in Appendix L.1 at

Appendix L.1 at www.wiley.com/colege/nisewww.wiley.com/colege/nise..

Root locus: Step 3

Breakaway points are among roots ofBreakaway points are among roots of

Suppose that s=b is a breakaway point.

Root locus: Step 4

RL departs from a pole RL departs from a pole ppjjwith angle of departurewith angle of departure

RLRLarrives at a zero arrives at a zero zzjjwith with angle of arrivalangle of arrival

(No need to memorize these formula.)

Root locus: Step 4 (cont

’

d)

Sketch of proof for angle of departureSketch of proof for angle of departure

Im Im

Re Re

For s to be on root locus,

due to

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Root locus: Step 4 (cont

’

d)

Sketch of proof for Sketch of proof for angle of arrivalangle of arrival

Im Im

Re Re

For s to be on root locus,

due to

due to angle conditionangle condition

Summary and exercises

PID controlPID control

Most popular controller in industryMost popular controller in industry

Simple controller structure Simple controller structure

Simple controller tuningSimple controller tuning

ModelModel--free methods for design are available.free methods for design are available.

Sketch of proofs for root locus algorithmSketch of proofs for root locus algorithm

Angle condition is important, and will be used in Angle condition is important, and will be used in

controller design.

Exercises: Problems 8.1, 8.2, 8.3, 8.6, 8.7.Exercises: Problems 8.1, 8.2, 8.3, 8.6, 8.7.

Lab #3 starts this week.Lab #3 starts this week.

Ziegler

-

Nichols PID tuning rules

Step response method (for only stable systems)Step response method (for only stable systems)

t

y(t

y(t))

Open

Open--loop step responseloop step response

Steepest tangent

PID parameters

Ziegler

-

Nichols PID tuning rules

Ultimate sensitivity methodUltimate sensitivity method

t

y(t

y(t)) Closed

Closed--loop step responseloop step response with a gain controller

with a gain controller

Increase gain and find

Increase gain and find KcKc

generating oscillation

(marginally stable case).

PID parameters

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0 2 4 6 8 10 12 14 16 18 20 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Example revisited

Step response methodStep response method Ultimate sensitivityUltimate sensitivity

0 2 4 6 8 10 12 14 16 18 20 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

P

PI

PID

P

PI

PID

Open

Open-

-

loop step response for

“

“step response method

step response method”

”

0 2 4 6 8 10

0 0.2 0.4 0.6 0.8 1

0.79

--0.280.28

Closed

-loop step responses for

-

loop step responses for

“

Ultimate sensitivity method”

Ultimate sensitivity method

”

0 5 1 0

0 0 .5 1

0 5 1 0

0 0 .5 1

0 5 1 0

0 1 2

0 5 1 0

0 0 .5 1 1 .5

Kp Kp=1=1

Kp Kp=2=2

Kp Kp=4=4

Kp

Kp=8=8 KcKc=8=8

Tc