control systems engineering lecture

7/27/2019 Control Systems Engineering Lecture

1/14

Lecture4

ControlSystemsIV

CSY401T

A.A.Yusuff

Wednesday, August 21, 13


2/14

DesignviaFrequencyResponse

Gainadjustmentforcesustouseatransient

responseandessforthepointsontheRL

CascadecompensatorchangetheRL

FrequencyresponsedesignusesBodeplots

insteadofRL

2



3/14

ComparisonB/WRL&Freq.Resp.

3

Frequencyresponsemethod RootLocusmethod

*Neednocomputertodesign

*Usesasympto9cBodeplot

*Computerisonlyusedforverifica9on

*Notasintui9veasRL-methods

(somewhatofanart)*Reshapeopenloopresponsetomeet

bothPM(%OS)andBW(Ts,Tp)requirements

* Can design deriva9ve compensa9on to

speed up the design (Lead design)& atthe

same 9me get Ess to be met by lead

compensatoralone.*Essisbuiltintothecompensatordesign

*Requiresacomputerduringdesign

*Repeatedtrialsneededtofinda

goodsolu9on

*Caniden9fyspecificloca9onof

rootsgivingustherequired

specifica9on

*Aninfinitenumberofsolu9onis

possibleforleadcompensatordesign

*Requiresnumeroustriestoget

requiredEss

PMisrelatedto%OS BWisrelatedto,dampingra9o,Ts,Tp



4/14

UnderlyingConcepts

4

Stability,Transientresponse,Steadystateerror

%OSIncreasingPM

IncreasesspeedofresponseBW

Ess Whenlowfreq.magresponseisincreased,

evenifhighfreq.responseisaenuated

Lagcompensa9onLeadcompensa9on

Lag-Leadcompensa9on



5/14

5

Transientresponseviagainadjustment

%OSischangedbyvaryingthePM.Fora

desired%OS.Weonlyhavetomakea

gainadjustment

Procedurefordetermininggainto

meet%OSrequirement

Welluseopenloopfreq.response.

Weassumedominant2ndorderpoles.

DesignProcedure

(1)DrawBodemagnitudeandphaseplot

forconvenientvalueofgain(2)DeterminerequiredPMfrom%OS

(3)FindonBodephasediagramthat

yieldsrequiredPM

(4)ChangegainbyABtoforce

magnitudecurvetogothrough0dBat

GainABisrequiredgaintoenforce

requiredPM

= ln(%OS/100)q2 + ln2(%OS/100)

m = atan1

0

@2

q2+p1 + 44

1

ADesign via Frequency Response

M(dB)

\ ARequiredincrease in gain

Phase (degrees)i.

* M

**

1 D\

logo

log (o

-180

Bod e plots showing gain adjustment for a desired phase margin

628 Chapter 11 Design via Frequency ResponseM(dB)

\ ARequired

increase in gainPhase (degrees)

i.

* M

**

1 D\

logo

log (o

-180FIGURE 11.1 Bo de plots show ing gain adju stm ent for a desire d phas e marg inWednesday, August 21, 13


6/14


7/14

7

LagCompensa9on

WhatdoesalagCompensatordo?

1.Improvesthesta9cerrorconstant

byincreasingonlythelowfrequencygainwithoutcrea9nginstability

2.IncreasesPMtoyielddesired

transientresponse

*UncompensatedsystemisunstablesinceM>0

@-180deg.

*Lagcompensatorreduceshighfrequencygain

andleaveslowfrequencygainunchanged*Lowfrequencygaincanbemadelargew/o

instability

*Stabiliza9oneffectisduetogain


8/14

8

. . . , . = . -poles. Equation (10.73) yields a 59.2 phase margin for a damping ratio of 0.6.

Desiredpositionff(.v)

Preamplifier

- * * K

Poweramplifier100(s + 100)

Motorandload1(s + 36)

Shallvelocity1s

Shaltpositioncm

FIGURE 11. 2 System for Ex am ple 11.1%OS=.5 tenfold improvement in steady-state error

K=583

0dB

-24.2dB,@.61

.61rad/s

0.61rad/s

-20dB/dec

w1

-24.2dB,0.61rad/s

-20dB/dec=(0dB-(-24.2dB))/(log(w1)-log(0.61))



9/14

Gc(s) = s +1

T

s + 1T

, > 1

|Gc(s)| = 1

9

DesignProcedureforlagcompensator

1.Choosegaintosa9sfyEssspecifica9onandplotBodeplotforthevalueofgain

2.Findthefreq.wherePMis15-12deggreaterthanPMthatyieldsdesiredtransient

response.(The5-12degcompensatesforthelagcompensatorphasecontribu9onatthePM

freq.)

3.ChooselagcompensatorforwhichcompositeBodediagramgoesthrough0dBatthefreqin

(2)asfollows:

(a)Drawcompensatorhigh.freq.asymptotestoyield0dBatfreq.foundin(2)

(b)Selectupperbreakfreq.onedecadelowerthanfreq.foundin(2)

(c)Selectlowfreq.asymptotestobe0dB(d)Connectthecompensatorhighandlowfreq.asymptoteswith-20dB/declinetolocate

lowerbreakfreq.

4.ResetsystemgainKtocompensateforaenua9oninlagnetworksoastokeepsta9cerror

constantthatsameasthatfoundinstep(1)



10/14

0dB MagdB

log(!1) log(!s)= 20dB/decade

Lag

10



11/14

11

LeadCompensa9on

WhendesigningleadcompensatorwithBodeplotswewanttochangethephasediagram

sothatforanincreaseinPMwereduce%OSandincreasingthesystemgaincrossover

freq.whichresultsinafastertransientresponse.

Visualizingleadcompensa9on

*LeadcompensatorincreasesBWbyincreasingthe

gaincrossoverfreq.

*Atthesame9methephasediagramisraisedat

higherfreq.

*ThisresultsinincreasingPMandahigherPMfreq.

*Inthe9medomain%OSdecreases(PMincreases)

withsmallerTp(PMfreq.increases)results.

-UncompensatedsystemhassmallPMand

lowPMfreq.

-Usingphaseleadcompensator,weraisedphaseplotathigherfreq.

-Atthesame9megaincrossoverfreq.inthe

magnitudeplotincreases

-ThisyieldalargerPM,ahigherPMfreq.and

BWincreases

636 Chapter 11 Design via Frequency ResponseMfdB)

Phase (degrees)0

CompensatorCompensatedC system *- logoUncompensatedsystemCompensator

CompensatedsystemUncompensatedsystem

-270

FIGURE 11.7 Visualizing lead compensationLead Compensator Frequency Response



12/14

!max =1

Tp

|Gc(j!max)| =1p

max = sin1

1

1 +

12

Advantagesoff-responsedesignoverRLdesign

*WecanimplementaEssandthendesignforthetransientresponse

*Specifica9onoftransientresponsewithEssconstraintiseasiertoimplementwith

f-responsetechniquesthanwithRL.

*No9cethatini9alsloperemainsunaffectedbytransientdesign(Ini9alslope

determineEss)

*Peakofphasecurvevaryinmaximumangle,andinfreq.wheremaximumangleoccurs.

*DCgainissettounitybyfactor

11.4 Lead Com pensat ion 6 3 720181614

5 12j ? 106420

/ > .,^y* *H-^*ffr

i i i ! ^^ -p - C i .**

^/3=0.2

" /3=0.3"/3=0.4 " " = 0.5

0.1 10 100

60504030 ., ^

/3=0/ "

\ ^ -

A)

Jj=t'**iz?A

3=).3

).2 ,\ N

S

11.4 Lead Com pensa tion 6 3 7201816145 12j ? 106420

/ > .,^y* *H

-^*ffr

i i i ! ^^ -p - C i .**^

/3=0.2"/3=0.3"

/3=0.4 " " = 0.5

0.1 10 100

6050403020100 ^

d*j^= 1^L


13/14

|Gc(j!max)| =1p

!max =1

Tp

c(s) = 1

s +1

T

s + 1T

13

DesignProcedureofLeadcompensator

max = sin1

1

1 +

Wednesday, August 21, 13 in Eq. (11.2) and th e quanti ty ft for the lead network in Eq. (11.6). For our design, a. . . . . ,


14/14

Gc(s) = Glead(s)Glag(s) =

s + 1

T1

s + T1

!s + 1 1

T2

s + 1T2

!

14

LagLeadCompensa9on

*Firsttermproducesleadcompensa9on.Secondtermproduceslagcompensa9on

*Constraintfollowedhereisthatsinglevaluereplacesthequan9tyforthelag

networkandthequan9tyfortheleadnetwork

*Forourdesignandmustbereciprocalofeachother

and f3 must be reciprocals of each other. An example of the frequency response ofthe passive lag-lead is shown in Figure 11.11.We are now ready to enumerate a design procedure.

o- 5

- 1 0* - M

- 2 5- 3 0- 3 5

X : \: \ / = 1> :;^SNN3Q\ 4 0 X5 U V0vX*= 2: : ^ *

$'/

/ ^

,. /0.001 0.01 0.1 1

Frequency (rad/s)10 100

0.001 0.01 0.1 1Frequency (rad/s)IGURE 11.11 Sam ple frequency response curves for a lag-lead compen sator, Gc{s) = [(s + l)(s + 0.1)]/ Gr+y)(*+^i


control systems engineering lecture

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