88324 control system - solution assignment 5

8/9/2019 88324 Control System - Solution Assignment 5

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88-324Assignment5P5.2(a)Initially,assumingthatthecloseloopresponseisasecondorderfunction,whichhasthe

followingform,

2 1 Itfollowsthatthepolesarelocatedat:

tan If istheangleofthepolesinpolarform:Ifsettlingtimeistobewithin2%ofthefinalvalue,

0.02

0.020.6 0.6 6.52Butwealsorequire 10%


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(b)Thethirdrootshouldbeatleast10timesfartherinthelefthalfplaneinorderthatthe2ndorder

polesabovewerestilldominant.

Assumingthattrackingerroriszeroandsinceallthepolesarelocatedinthelefthalfplane,i.e.astable

system,wecanapplyfinalvaluetheorem,where

lim and

isthestep

responseof .So,wehave ,whichrequirethefollowingformofclosedloopresponse. lim 0 1 2 10 65.2(c)Sincetherequirementis20%overshoot,wehavethefollowing

tan tan 62.9

1 62.9 11 62.9 0.46 6.52 14.2So,thecloseloopfunctionbecomes,

65.2201.6 65.2 13.1 201.6G(s)canbederivedknowingthat,

1 , 1 65.2201.6 65.2 13.1 201.6 65.2201.6


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P5.4(a)Wefirstderivethecloseloopresponse

1 , 2 2 Again,wehave

2 2 2 , , 1

Butwerequirethat overshoot=5%

0.05 3wealsorequire 1.1, bothspecifications cannotbemetsimultaneously.nse,

So, 100 5 , Since obviously

(b) IfR(s)isthesteprespo

, 0 sin , 1

0 ,

So,wemayrewritethepeakvoltageexpressionas 100 100 1 1

Ifwerelaxbothspecificationsbyanamountof, wehave 1.1 0.05Solvingtheequation,wehave .07

wing.

0.05 10%SubstitutetheanswerintotheTpexpressi nwehavejustderivedandweget

=2

So,therequiredpercentageovershootandTpareasthefollo

1.1 2.3 o 1 2.9


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P5.16(a)BythesamemethodasP5.4,wederivedtheclosedloopTFasthefollowing.

12 12 12Fromthisequationand

2 Wehave 6 , 12 , Then,wemayfindthepeakvoltagetime:

1

12 3

Sinceovershoot


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AP5.2

5440 1 28 432 5440 1 5440 1 28 432 5440 5440

YoumayobtainsimilarplotusingMATLABbyapplyingastepinput toT(s)

As Zincreasesfrom0to0.1,theamountofovershootdecreasesand responseisfasterandmorestable.However,as Zisincreasedbeyond0.1,theovershootandTS(settlingtime)increase,althoughTr

the

continuestodecrease.


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AP5.3

1

2 1 1 1

1 2 2 1

BysimilarmethodfromAP5.3,thefollowingplotanddataforvarious pwasobtained.

As pincreases,theovershoot,TrandTsalsoincrease;addingthepolereducethesystemsstabilitywith

higheramountofovershoot.


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P6.1(a)

5 2,

1 25 0X=(010)/5=2 ,Sinceallelementsinthefirstcolumnispositive,thusthesystemisstable.

(b)

1 84 4 X=(48x4)/4=7, y=(1x04x0)/4=0, z=(4xy4xX)/X=4

Allelements thefirstcolumnispositive,thusthesystemisstable.

1 42 20 X=(1x20+2x4)/2=14, y=(1x02x0)/2=0, z=(2xy20X)/X=20

Therearetwosignchangesinthefirstcolumn,sotherearetworighthalfplaneroots,thusthesystemis

unstable.

4 8 4

in

(c)

2 4 20

(d)

2 10 8

1 2 81 10 0 0 0W=(102)/1=8, X=(08)/1=8, y=(x+10x8)/(8)=10, z=(0Xy)/y=8

Therearetwosignchangesinthefirstcolumn,sotherearetworighthalfplaneroots,thusthesystemis

unstable.


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(e)

3 2

1 3 1 2 0 0 0

W=(23)/1=1, X=(0K)/1=K, Y=(X2W)/W=2K, z=(0XY)/Y=K

ForthesystemtobestableY>0andz>0,thereforeK>0and2K>0,thus 00,u>0,weneedK>0andK>0,whicharecontradictoryconditions,

thusthesystemisunstableforallK.


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P6.3(a)

1 20.5 1

10.005 1And, 1

wemayderivetheclosedloopTFT(s)asthefollowing:

0.005 1

0.0025

0.5125

2.52

4.01 2

Thus,

cequation).

S4 0.0025 2.52 2+K

Then,weformtheRoutharrayforthedenominator(characteristi

S3 0.5125 4.01 0

S2 (0.011.3)/0.5125=2.5 2+K 0

S (1+0.5125K10)/2.5=3.60.205K1

S0 2+K

Forstabilit eneed3.60.205K>0, weneed2+K>

So,therange ofKforstabi

)LetK=9, wemayfindtherootsfromMATLABasbelow:

than10timesawayfroms2,3.So,thelatterpolesare

arerepresentedbythefollowing.

1

y,w soK


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P6.8(a)

Thecloseloopcharacteristicequationis

40.5 1 1 1 1 0

Expandthetermsandweget:

7 14 81 0FormingtheRoutharray:

S3 1 14

S2 7 8(1+K)

S1 X 0

S0 Y

4 81 77 1 12.86 1.143 , 81 Forastablesystem, x>0 and

Hence, 1


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Comparingcoefficientsandwegetthefollowingequations:

2 7

2 14

81 5Therefore, rootsareat:

S1=b=5, s2,3= 1 Solvingtheequationsandweget: b=5, n=2 and K=1.

= 11.73j

88324 control system - solution assignment 5

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