Autocontrol Eng

8/13/2019 Autocontrol Eng

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Copyright Hanyang University

1-90

. Control

Ch. 1 Introduction

Ch. 2 Basic Control Actions

Ch. 3 Root-locus

Ch. 4 Frequency Response

Ch. 5 Compensator Design

Ch. 6 Modern Control


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Ch. 1 IntroductionControl : to decide input to get the desired outputControl System : collection of components to have desired response

(Plant, Controller, Sensor, Actuator)Process(Plant) : the object of control

Open-loop ControlClosed-loop Control

- Closed-loop Control : Measure and feedback the output andcompare with the desired output (feedback control)

-

-


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- Open-loop ControlNo feedback of the output.Feedforward using the prior knowledge of the system


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Example of Control Systems- Liquid level control

- Driving a car

- Toaster

OpenlooporClosedloop ?

.


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Motor speed controlOpen-loop

Closed-loop


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Traffic light controlOpen-loop : fixed intervalClosed-loop : change interval depending on the traffic

Open-loop Closed-loop (feedback) Simple less expensive Inaccurate Sensitive to

parameter changeexternal disturbance

Stable

Complicated Expensive Accurate less Sensitive to

parameter changeexternal disturbance

May be unstable

Special considerations required


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Control System Design Procedure

Use a detailed model to predict the real responses and performance


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mathematical model- Represent the characteristics and behavior of a system in a

mathematical form- Have to be detail to represent the real characteristics- If too complicated, not good for controller design simple model- Compromise between the two

Modeling procedure- Derive the governing equations- convert to Laplace Transform, Transfer function

State-space eqn.Block diagramSignal processing model


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Linearization

pointoperating:),( 00 yx

Taylor Series Expansion of f(x) around ),( 00 yx

2

02

2

00 )(

!2

1)()()(

0

xx

x

fxx

x

fxfxf

xx

y 0y

)(

)(ofionapproximatLinear:)(

00

00

0

xxKyy

xfyxxx

fyy

xx

Neglect higher-order terms


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Transfer functionLinear, Time-invarient, differential eqn Transfer function, G(s)

)(

)()(

input

outputsG

:

describes characteristics of the system, independent of input

Example :

kbsmssF

sYsG

2

1

)(

)()(

)()()( sFsGsY

fkyybym

)()()( 2 sFsYkbsms

Let initial conditions zero


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n-th order differential equations => transfer functionxbxbxbxbyayayaya mmmmnnnn

1)1(1)(01)1(1)(0

nn

nn

mm

mm

asasasa

bsbsbsb

sX

sYsG

1

1

10

1

1

10

)(

)()(

: zero (): pole ()

)(

)(

)())((

)())((

)(

21

21

0

0

sD

sN

pspsps

zszszsA

asa

bsbsG

n

m

n

n

m

m

0)( sN izs

0)( sD ips

dif s

integs

1


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Operations of Block Diagram Cascade Connection

Feedback connection

Error ,

)()()( 2 sXsGsY )()()( 1 sFsGsX

)()()()( 21 sFsGsGsY

)()()()(

)()()()(

sEsGsHsF

sYsHsFsE

)()()(1

)(

)()()(

sFsHsG

sG

sEsGsY


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: Closed-loop Transfer function

Basic Blocksconstant :

integrator :

differentiator :

Summer(summing point) :

)()(1

)(

)(

)(

sHsG

sG

sF

sY

)()( sKFsY )()( tKfty

)(

1

)( sFssY

t

dttfty0

)()(

)()( ssFsY )()( tfty

21)( FFsY )()( 21 tftfy

)(sG

)()( sHsG

: feed forward T. F: open-loop T.F.

Where,


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State-Space ApproachState variable : min number of variables to represent the states of asystem

Ex :

In Matrix form,

Fkzzbzm

zyzx

Fuzx

Let

2

1

equationstate1

1)(1

212

21

2

1

um

xm

kx

m

kx

xx

um

kzzbm

zx

zx

1Output xy

2

1

2

1

2

1

01

10

10

x

xy

u

mx

x

m

b

m

kx

x


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State-space representation

VectorStatex

x

x

2

1

,

State equation Laplace transform transfer functionL

zero i. c.

)()()(

)()()(

)()()(

1

1

sUDBAsICsY

sBuAsIsX

sBusXAsI

Transfer function :

DuCxyBuAxx

0,01,10

,10

DC

m

B

m

b

m

kA

)()()(

Bu(s)AX(s)sX(s)

sDusCXsY

DuCxy

BuAxx

DBAsICsu

sYsG 1)(

)(

)()(

Space spanned by state vectors State space


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Ch. 2 Basic Control Actions Basic elements of feedback control system

actuator : motor, hydraulic(pneumatic) motor, heatersensor : position encoder, potentiometer, LVDT

velocity tachometerpressure transducer, accelerometer, themometer

controller : computes control input uusing the error signal and sends out tothe actuator

Basic industrial controllers1.Two-position (on-off)

On-off with differential gap


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2.Proportional Control(P-control)

3. Integral Control

4.P-I Control (Proportional plus Integral)

5.P-D Control (Proportional plus Derivative)

6.P-I-D Control

)( yyKu dp

dtyyKu di )(

)1

1()(sT

Ks

KKsG

i

pi

pc

)1()( sTKsKKsG ddpc


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)1

1(

)(

sTsT

K

s

KsKKsG

d

i

p

idpc


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Derivative Control increases damping. improve stability

P-control under external disturbances

pp KbsJs

bsJs

KbsJssDsE

2

2

2

1

1

1

)()(

Assume reference input, r(t)=0 ( 0)( sR )

step disturbances

TsD d)(

Steady state error

p

dd

ps

sss

K

T

s

Ts

KbsJs

ssEe

20

0

1lim

)(lim


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PI control under external disturbances

i

p

p

i

ip

T

KsKbsJs

s

bsJssT

sTKbsJs

sD

sE

23

2

2

1)1(1

1

)(

)(

s

TsD

sR

d

)(

0)(

0lim23

0

s

Ts

T

KsKbsJs

se d

i

p

p

sss

* Integral Control reduces the steady-state error,But may cause instability.


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Stability criterion

Closed-loop T. F. )()(

)(

)(

0

0

sA

sB

asa

bsb

sR

sCnn

m

m

A(s)=0 closed-loop polesStability Real[Closed-loop poles] < 0 : asymptotically stable

0 : stableRouths Stability Criterion

Determine stability without computing the Poles

0110

n

nn asasa Condition 1 : 0

i

a If any 0ia , there exists at least one root with positive or zero

real partCondition 2 : decide the number of unstable poles


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Example

)10)(1(

107

ss

s

F

Y , sF1 tt eety 10

3

2

3

11)(

Dominant pole: located close to the imaginary axis :

Can predict the response of a higher-order system by looking at thedominant pole

Fast, well-damped system

21 n

n

4.0

time)(settling4

st


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Example

i) P-I control )1()(s

KKsG ic

innn

in

KKsKss

KsK

R

Y2223

2

)1(2

)(

For stability, innn KKK 22 )1(2 0,

12

ini KK

K

KK

ii) Integral controls

KsG ic )(

022223 ninn Ksss

niK 20

iii) P-control KsGc )( 0)1(2

22 nn Kss

1K

22

2

2 nn

n

ss


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Steady-state Error Open-loop T.F.

)1()1(

)1()1()(

1

sTsTs

sTsTKsG

p

N

ma

: Type N system

Type : no. of free integrators

)(lim

)(1

1

)(

)(

0ssEe

sGsR

sE

sss

Step input

ssR

1)( 1)( tr

ps

ssKGsGe 1

1

)0(1

1

)(1

1

lim0 consterrorpositionstatic:)0()(lim

0GsGK

sp

Type 0 system

)1(

)1()()0(

s

i

pT

sTKsGKGK

"2Type

"1Type

system0Type


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Type 1 or higher system )0(GKp

higheror1type:0

0type:1

1

ss

ss

e

Ke


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Performance Index

Quantitative measure of the performance of a control system

0

)( dtefJ 1. I.S.E (Integral square Error)

0

2dteJ

2. I.T.S.E (Integral of Time-multiplied square Error)

0

2dtteJ

3. I.A.E (Integral Absolute Error)

0

dteJ 4. I.T.A.E (Integral of Time-Multiplied Absolute Error)

0

dtetJ


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5.others

dteJor

dteeJ

]e[

][

0

0

22

minimize J by choosing an appropriate controller and controlparameter.


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Ch. 3. Root-Locus

Rouths stability Criterion : determine only stabilityRoot Locus method : plots trace of C.L. poles on s-plane as K varies.

stability + help to select controller and gain K+ predict the system dynamics

Root LocusLocus of C. L. poles in the s-plane as gain K is varied

Example

For stability 0K i) K1 , s=-1 1 Kj

Kss

K

sR

sC

2)(

)(2

Kss

Kss

11,

02

21

2


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Root LocusFeedback Control system(negative feedback)

)()(1

)(

)(

)(

sHsG

sG

sR

sC

closed-loop poles C.E : 0)()(1 sHsG 1)()(

T.Floop-Open

sHsG traces of s satisfying the C.E.

Root Locuss : complexG(s)H(s) : complex function

Angle Condition ),21,0,(K360180)()( KSHsG Magnitude Condition 1)()( sHsG Traces of s satisfying the two conditions


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Note K = 0 K =

Departs from pole arrives at zeroSource Sink

Repel R.L. Attract R.L. pole pole real axis break away pt.

zero zero real axis break in pt.

break-away & break in locus real axis 90


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Double Pole Example

2)4)(2(

)1(11

sss

sKGH

Example2

)(1

s

asK

314

)1()4420(

180,60asymptotes3

14

A

mn


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Example

)4(

)1)(21(

122

2

ss

ssK

Example

-5 -4 -3 -2 -1 0 1 2 3 4-2

- 1 . 5

-1

- 0 . 5

0

0 . 5

1

1 . 5

2

R e a l A x i s

ImagAxis


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Ch. 4 Frequency Response

Steady-state response of a system to a sinusoidal input

- Frequency response is easily obtainedno need for modelingno need for characteristic eqn. .

no need for poles/zeros

At steady-state linear system : freq. of y is the same as freq. of xnonlinear system : y contains many freqs. depending

on amp of x


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T.F.)(

)()(

sX

sYsG

Input tAtx sin)( Output )sin()( tBty

shiftphase)(

)()(

ratio)ituderatio(magnampitude)(

)()(

jX

jYjG

jX

jYjG

A

B

T.Fsinusoidal:)()(

js

sGjG

Frequency Response Graphical expression : )( jG plot1.Bode diagram : Already covered2.Polar plot3.Log-magnitude vs. phase plot


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Polar plot (Nyquist plot))(Im)(Re)( jGjjGjG

i) 9011j

1)(

1)(

jjG

ssG

ii) jjGssG )()( iii)

2

2

2

2222

2

1

2

1

11

1

1

1)(

1

1)(

YY

T

Tj

TTjjG

sTsG

YX


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Nyquist stability Criterion

Char. Eq. : 0)()(1 sHsG

RHP)in0GH1of(zerospolesCLunstableofno.:ZplaneGHin1-ofntencirclemewise-clockofno.:N

GHofpolesunstableofno.:P

PNZ Example

)1)(1(

)()(21

sTsT

KsHsG

0-(iii)

0

090R(ii)

1800

0)(i

js

GH

eRs

GHKGH

js

j

stable0

1)-ofntencircleme(no0

0

Z

N

P


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Minimum phase systemNo O.L. poles or zeros in RHP

)1)(1(

)1)(1(

21 sTsTs

sTsTKGH ba

0P

For stability, N=0 Z=0polar plot 1

Relative stability1. Polar plot GM / PM

180

1

GMKm


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2. Gain Margin & Phase Margin in R-locus

K


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Gain margin : gain

180

)(

1

jwG

Kn

Phase margin : Phase lag

1)()(180

jwGn jwG Stable

0

0

n

nK


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Magnitude-phase plot


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Bandwidth

Cutoff frequency( 0w )db

jR

jC

jwR

jwC3

)(

)(

)(

)(

0

0 for 0ww cutoffw Frequency for which

707.02

1,3log20 input

outputdb

input

output Bandwidth = Range of frequency where system tracks

input sinusoids. Measure of speed of response ~rise time High bandwidth : better tracking properties for input of wide

frequency range But, too high bandwidth Noise problem

Higher cost

Closed-loop T.FG

G

1


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Ch. 5 Compensator DesignEffect of additional poles : root locus .Effect of additional zero : root locus .

Compensator : A controller to modify dynamics of a system to meetspecs.


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Lead compensator

)0,10(1

1

)( pzps

zs

Ts

Ts

KsG cc

Bode diagram of lead compensator ( )1.0,1 cK m : Maximum phase lead angle at nww

1

1sin m

Lead compensator : phase margin , Lead compensator :


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Example

4.18

41.47.41)(

s

ssGc

)4.18)(2(

)41.4()(

sss

sKsGGc

Frequency (rad/sec)

Phase(deg);Magnitude(d

B)

Bode Diagrams

-150

-100

-50

0

50

100

From: U(1)

10-2 10-1 100 101 102 103-200

-150

-100

-50

0

50

To:Y

(1)

Gc

G

GcG

Gc

G

GcG


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Lag compensator

),1(1

1

)( zpps

zsK

Ts

Ts

KsG ccc

Lag compensator : static error coeff. Steady state error

Frequency (rad/sec)

Phase(deg);Magnitude(dB)

Bode Diagrams

-20

-15

-10

-5

0From: U(1)

10-4 10-3 10-2 10-1 100-60

-40

-20

0

To:Y(1)

Example

]

Type 1 system : step input 0sse

-0.24+j0.86


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Ramp input 11 v

ssK

e (too big) 1vK

ccs

cs

v K

sssT

s

Ts

sKsGssGK

)15.0)1(

1

1

1

lim)()(lim00

vK cK .To reduce sse , increase cK to 5 for example

If)15.0)(1(

5,5)( 1

sss

GGGsG cc

Lag compensator control

)15.0)(1)(1100(

)110(5)()(

100

1

)10

1(5.0

1100

)110(5)(

ssss

ssGsG

s

s

s

ssG cc


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Frequency (rad/sec)

Phase(deg);Magnitude(dB)

Bode Diagrams

-200

-100

0

100 From: U(1)

10-2 10-1 100 101 102 103-300

-200

-100

0

To:Y(1)

Gc

G

GcG

Gc

G

GcG

Summary

ps

zskGc

)( Lead compensator : adds phase lead at crossoverw

increase without decreasing nw Lag compensator : adds low frequency gain

rcompensatolagpz

rcompendatoleadpz

:

:

Improve sse Disturbance reject


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PID controlPID Turning

)1

1()( dsi

pc TsT

KSG

Parameter : dip TTK ,, trial & errorZieglerNichols rule

Plant step response parameter

1) (if no integrator) plant response parameter

LLL

T

L

L

T

LT

TTK dp

0.521.2PID

03.0

0.9PI

0P

i


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2)P() Kp 0 output crp KK

crP :

crcrcr

crcr

cr

lp

PPK

PK

K

TTK

0.1250.50.6PID

02.1

10.45PI

00.5P

i

1.Ziegler-Nichols turning 25 overshoot

2.plant model tuning3. .

gain initial tuning trial & error


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Example

0

21

0

1

3

02

2

1

2

1

cP

ux

x

x

x

2

0

1,

3

02

AB

BA

ableuncontroll0

lecontrollab0)det(

cP 2x

Observability ()Output y state variable x

observable0)det(

nn:

1

Q

CA

CA

C

Q

n

22 3xx


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Optimal Control

CXY

BuAXX

min ft

dttuXgJ0

),,( Control Law : state feedback ( state )

Kxu

x

xKKxKxKxKu

n

nnn

1

12211

Closed-loop systemXAXBKABKXAXX *)(

Problem J feedback gain Ki

e.g. desired 0dX 0X

ft T

dtXXJ0

)( XXPXX

dt

d TT )( P symmetric):n(n


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Performance Index

2

2

2

2

221211

2

12

2

)0()0(

K

KK

PPP

PXXJ T

2,1 min2 JKWhen

Optimal Control with Energy term

Control gain Pbr

KT 1

whereP

is obtain fromequationRiccati:0

1 QPPbb

rPAPA

TT

factorgweightin:r)(:..

:

energy

2

0dtruXQXJIP

KXuControl

buAXX

T

1

1)0(xAssume


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Ch. 6 Modern Control1.Linear Quadratic Control System model

0),()()( tttuBtxAtx

Performance index

T

t

TT

T

dttuRtutxQtx

TxTSTxtJ

0

))()()()((2

1

)()()(2

1)( 0

0,0,0)( RQTS

Riccati EducationQSBRBSASSAS

TT

1

Algebraic Riccati Equation(ARE) (When T )QSBRBSASSA

TT 1

0

Feedback control gainSBRK

T1


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2. Adaptive Control () .

1) Direct Adaptive control1. -Output error controller gain adjust.

2) Indirect Adaptive control- Plant model parameter adjust controller gain .

1) Gain scheduling2) Model Reference Adaptive Systems3) Self-Tuning Regulator4) Dual Control

Controller

DesiredReference model

Process

Adjustmentrule

ym

yr u

e

y

+-


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3. Intelligent ControlControl strategy and decision are made at a symbolic level throughperception and reasoning and learning.

Signal Symbol Control

Pattern recognition Neural network

Rule based control Fuzzy logic control

A. Neural NetworkBiological neurons are believed to be the structural constituents of the

brain. A neural network can:

Learn by adapting its synaptic weights to changes in the

surrounding environments

Handle imprecise, fuzzy, noisy, and probabilistic information

Generalize from known tasks or examples to unknown ones


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Membership Function

Neural NetworkControl

B. Fuzzy logic control Control Decision system Boolean Logic Fuzzy logic

Fuzzy Set Membership Function Fuzzy Logic


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Fuzzy controller Fuzzifier Fuzzy Inference Engine Fuzzy Rule Base Defuzzifier

Merits of Fuzzy Controllera. Linguistic control rulesb. Highly nonlinearc. Fewer rules

OutputDefuzzifier

FuzzyInference Engine

Fuzzy

Rule Base

Input 1

FuzzifierInput 2

< Block Diagram of fuzzy logic controller >

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Autocontrol Eng