SolutionsManualfor:

LinearState-SpaceControlSystems

RobertL.WilliamsIIandDouglasA.Lawrence

OhioUniversity

submittedtoWileyDecember,2006

TableofContents

NUMERICALEXERCISESSOLUTIONS.....................................................................................4

CHAPTER14

CHAPTER25

CHAPTER38

CHAPTER412

CHAPTER516

CHAPTER618

CHAPTER722

CHAPTER824

CHAPTER926

ANALYTICALEXERCISESSOLUTIONS....................................................................................30

CHAPTER13()

CHAPTER238

CHAPTER346

CHAPTER459

CHAPTER566

CHAPTER670

CHAPTER775

CHAPTER879

CHAPTER987

CONTINUINGMATLABEXERCISESSOLUTIONS....................................................................91

CONTINUINGMATLABEXERCISE1..................................................................................................91

CONTINUINGMATLABEXERCISE2..................................................................................................95

CONTINUINGMATLABEXERCISE3..................................................................................................99

CONTINUINGMATLABEXERCISE4.................................................................................................103

CONTINUINGEXERCISESSOLUTIONS....................................................................................108

CONTINUINGEXERCISE1-THREE-MASSTRANSLATIONALMECHANICALSYSTEM............................109

Open-LoopSystem.......................................................................................................................109

Open-LoopResponse...................................................................................................................109

Controllability&Observability....................................................................................................110

CanonicalRealizations................................................................................................................110

StabilityAnalysis..........................................................................................................................112

DynamicShaping.........................................................................................................................112

ControllerDesign........................................................................................................................113

ObserverDesign..........................................................................................................................115

CONTINUINGEXERCISE2-INVERTEDPENDULUM...........................................................................117

Open-LoopSystem.......................................................................................................................117

Open-LoopResponse...................................................................................................................117

Controllability&Observability....................................................................................................118

CanonicalRealizations................................................................................................................118

StabilityAnalysis..........................................................................................................................119

DynamicShaping.........................................................................................................................120

ControllerDesign........................................................................................................................120

LQRDesign..................................................................................................................................121

CONTINUINGEXERCISE3-ROBOTJOINT/LINKCONTROL..................................................................123

Open-LoopSystem.......................................................................................................................123

Open-LoopResponse...................................................................................................................124

Controllability&Observability....................................................................................................124

CanonicalRealizations................................................................................................................124

StabilityAnalysis..........................................................................................................................125

DynamicShaping.........................................................................................................................126

ControllerDesign........................................................................................................................126

ObserverDesign..........................................................................................................................727

CONTINUINGEXERCISE4-BALL/BEAMSYSTEM..............................................................................129

Open-LoopSystem.......................................................................................................................729

Open-LoopResponse...................................................................................................................130

Controllability&Observability....................................................................................................130

CanonicalRealizations................................................................................................................130

StabilityAnalysis..........................................................................................................................131

DynamicShaping.........................................................................................................................132

ControllerDesign........................................................................................................................133

ObserverDesign..........................................................................................................................133

LQRDesign..................................................................................................................................134

CONTINUINGEXERCISE5-PROOF-MASSACTUATORSYSTEM...........................................................136

Open-LoopSystem.......................................................................................................................136

Open-LoopResponse...................................................................................................................137

Controllability&Observability....................................................................................................137

CanonicalRealizations................................................................................................................137

Minimality....................................................................................................................................138

StabilityAnalysis..........................................................................................................................138

DynamicShaping.........................................................................................................................139

ControllerDesign........................................................................................................................140

ObserverDesign..........................................................................................................................141

NumericalExercisesSolutions

Chapter1

NEl.la

01'

A=C=[10]0=0

-6-2B=:

NEl.lb

r

010

A=B=C=[31]D=0

-6-21

NE1.lc

01°一一°]

A=001B=。C=[1000]0=0

-6—8-4J

NEl.ld

-01oo-「°】

00100

A=B=C=[641o]0=0

0001°

-66-44-11-10

NE1.2c

A=02]B=DIC=[1]0=0

NE1.2t

-01

A=B=「°〕C=[10]0=0

-10-3JL'J

NE1.2c

-01°：

A=001B=。C=[100]0=0

-2]

-5-3

NE1.2d

010000

-101()010100o-■()0

4=B=c=D=

000100001000

50-5-0.500.5

Chapter2

NE2.1

x(t)=1/5-1/5exp(-5/2t)timeconstantT=2/5

NE2.2

a.Transferfunction:

2s+7

sA2+7s+12

b.Transferfunction:

1

sA2+2s+3

c.Transferfunction:

1

sA2+12s+2

d.Transferfunction:

9s^2+38s-2

sA2-5s-2

NE2.3

a.characteristicpolynomial:[132]

eigenvalues:-2,-1

b.characteristicpolynomial:[12010]

eigenvalues:-0.5132,-19.4868

c.characteristicpolynomial:[1010]

eigenvalues:0+3.1623i,03.1623i

d.characteristicpolynomial:[1200]

eigenvalues:0,-20

NE2.4

phi(4)=

0.12940.0113

-0.0677-0.0059

x(4)=

0.2702

-0.1412

NE2.5

xl(t)=1/8+1/8exp(-4t)1/4exp(-2t)

NE2.8NE2.9

NE2.10

a.Ad=

-0.87690

0-9.1231

b.Ad=

-1.35890

07.3589

c.Ad=

-0.5000+3.1225i0

0-0.50003.1225i

d.Ad=

3.16230

0-3.1623

Chapter3

NE3.1a

P=

1-4

1-5

determinantofP-1

Systemisfullystate-controllable

NE3.1b

P=

1-4

00

determinantofP0

SystemisNOTfullystate-controllable

NE3.1C

P=

1-20

2-3

determinantofP37

Systemisfullystate-controllable

NE3.1d

P=

01

1-2

determinantofP-1

Systemisfullystate-controllable

NE3.1e

P=

12

-1-2

determinantofP0

SystemisNOTfullystate-controllable

NE3,2a

Ac=

01

-20一9

Be=

0

1

Cc=

92

NE3.2b

WecannotfindCCFfromtheformulasincethetransformationmatrixissingular.

NE3.2c

Ac=

0.00001.0000

-10.0000-2.0000

Be=

0.0000

1.0000

Cc=

12

NE3.2d

Ac=

0.00001.0000

-10.0000-2.0000

Be=

0

1

Cc=

1.00002.0000

NE3.2e

WecannotfindCCFfromtheformulasincethetransformationmatrixissingular.

NE3.3

ApplythePopov-Belevitch-HautusRankTestforControllability

NE3.3a

TheeigenvaluesofAare:-5and-4

Fortheeigenvalue-5

-101

001

hasrank2

Fortheeigenvalue-4

001

011

hasrank2

Systemisfullystate-controllable

NE3.3b

TheeigenvaluesofAare:-5and-4

Fortheeigenvalue-5

-101

000

hasrank1

Fortheeigenvalue-4

001

010

hasrank2

SystemisNOTfullystate-controllable

NE3.3C

TheeigenvaluesofAare:-l+3iand-l-3i

Fortheeigenvalue-l+3i

-1.0000+3.0000i10.00001.0000

-1.00001.0000+3.0000i2.0000

hasrank2

Fortheeigenvalue-l-3i

-1.0000-3.0000i10.00001.0000

-1.00001.0000-3.0000i2.0000

hasrank2

Systemisfullystate-controllable

NE3.3d

TheeigenvaluesofAare:-l+3iand-l-3i

Fortheeigenvalue-l+3i

-1.0000+3.0000i-1.00000

10.00001.0000+3.0000i1.0000

hasrank2

Fortheeigenvalue-l-3i

-1.0000-3.00001-1.00000

10.00001.0000-3.0000i1.0000

hasrank2

Systemisfullystate-controllable

NE3.3e

TheeigenvaluesofAare:1and2

Fortheeigenvalue1

-101

10-1

hasrank1

Fortheeigenvalue2

001

11-1

hasrank2

SystemisNOTfullystate-controllable

Chapter4

NE4.1a

Q=

11

-4-5

determinantofQ-1

Systemisfullystate-observable

NE4.1b

Q=

10

-40

determinantofQ0

SystemisNOTfullystate-observable

NE4.1C

Q=

01

1-2

determinantofQ-1

Systemisfullystate-observable

NE4.1d

Q=

12

-20-3

determinantofQ37

Systemisfullystate-observable

NE4.Ie

Q=

11

11

determinantofQ0

SystemisNOTfullystate-observable

NE4.2a

Ao=

0-20

1-9

Bo=

9

2

Co=

01

NE4,2b

WecannotfindOCFfromtheformulasincethetransformationmatrixissingular.

NE4.2c

Ao=

0-10

1-2

Bo=

1

2

Co=

01

NE4.2d

Ao=

0.0000-10.0000

1.0000-2.0000

Bo=

1.0000

2.0000

Co=

0.00001.0000

NE4.2e

WecannotfindOCFfromtheformulasincethetransformationmatrixissingular.

NE4.3

ApplythePopov-Belevitch-HautusRankTestforObservability

NE4.3a

TheeigenvaluesofAare:-5and-4

Fortheeigenvalue-5

11

-10

00

hasrank2

Fortheeigenvalue-4

11

00

01

hasrank2

Systemisfullyobservable

NE4.3b

TheeigenvaluesofAare:-5and-4

Fortheeigenvalue-5

10

-10

00

hasrank1

Fortheeigenvalue-4

10

00

01

hasrank2

SystemisNOTfullyobservable

NE4.3c

TheeigenvaluesofAare:-l+3iand-l-3i

Fortheeigenvalue-l+3i

01.0000

-1.0000+3.0000i10.0000

-1.00001.0000+3.0000i

hasrank.2

Fortheeigenvalue-l-3i

01.0000

-1.0000-3.0000i10.0000

-1.00001.0000-3.0000i

hasrank2

Systemisfullyobservable

NE4.3d

TheeigenvaluesofAare:-l+3iand-l-3i

Fortheeigenvalue-l+3i

1.00002.0000

-1.0000+3.00001-1.0000

10.00001.0000+3.0000i

hasrank2

Fortheeigenvalue-l-3i

1.00002.0000

-1.0000-3.0000i-1.0000

10.00001.0000-3.0000i

hasrank2

Systemisfullyobservable

NE4.3e

TheeigenvaluesofAare:1and2

Fortheeigenvalue1

11

-10

10

hasrank2

Fortheeigenvalue2

11

00

11

hasrank1

SystemisNOTfullyobservable

Chapter5

NE5.1a

Transferfunction:

s+1

s^2+3s+2

1stateremoved,

a=

xl

xl-2

b=

ul

xl0.7071

c=

xl

yl1.414

d=

ul

yl0

NE5.1b

Transferfunction:

sA2+4s+5

A

s入3+7s2+17s+15

2statesremoved.

a=

xl

xl-3

b=

ul

xl0.1543

c=

xl

yl6.481

d=

ul

yl0

NE5.1C

Transferfunction:

s+3

sA3+7s"2+17s+15

1stateremoved.

a=

xlx2

xl1.398-5.077

x22.471-5.398

b=

ul

xl0.1943

x20.2686

c=

xlx2

yi-2.5621.854

d=

ul

yl0

NE5.1d

Transferfunction:

sA2+8s+25

s入4+15s入3+91s入2+255s+250

2statesremoved.

a=

xlx2

xl0.68131.215

x2-12.53-7.681

b=

ul

xl-0.001309

x20.03991

c=

xlx2

yi26.250.8612

d=

ul

yi0

NE5.1e

Transferfunction:

2sA2+9s+24

s入4+15s入3+91s入2+255s+250

a=

xlx2x3x4

xl000-250

x2100-255

x3010-91

x4001-15

b=

ul

xl24

x29

x32

x40

c=

xlx2x3x4

yi0001

d二

ul

yi0

NE5.IelookslikeOCFofNE5.Id;however,twonumbershavechangedandtherearenolonger

anycommonfactorsofthetransferfunctionnumeratoranddenominator.Thissystemis

alreadyminimal.

Chapter6

NE6.1a&NE6.3a

Eigenvalues:

-2+3.16i

-2-3.16i

LyapunovmatrixP:

2.020.04

0.040.13

Sylvester'sCriterion:

2.02

0.27

Systemisasymptoticallystable

NE6.1b&NE6.3b

Eigenvalues:

2+3.16i

2-3.16i

LyapunovmatrixP:

-2.020.04

0.04-0.13

Sylvester^sCriterion:

-2.02

0.27

Systemisunstable

NE6.1C&NE6.3c

Eigenvalues:

0

-4

???Errorusing==>lyap

Solutiondoesnotexistorisnotunique.

MATLABfunctionlyapfails.

Systemismarginallystableduetozeropole(andnopositivepoles).

NE6.1d&NE6.3d

Eigenvalues:

0+3.74i

0-3.74i

???Errorusing==>lyap

Solutiondoesnotexistorisnotunique.

MATLABfunctionlyapfails.

Systemismarginallystableduetozerorealpoles(andnopositiverealpoles).

NE6.2aSTABLENE6.2bUNSTABLE

NE6.2CMARGINALLYSTABLENE6.2dMARGINALLYSTABLE

Note:thebookdidnotgiveenoughinformationforNE6.2a-d;soforallcasesweassumed:

B=[0;l]

C=[10]

D[0]

NE6.4Given

f-fIC1=____I__-_s___2___=_(_s_—__2_)_(5__+_1_)=_(_s_+__1_)

$3+2s—-4s—8(5—2)($+2)~(s+2)~

a.Thesystemisbounded-input,bounded-outputstablebecausetheimpulseresponseh(t)=

satisfies

88Q

J|/z(r)|Jr<j(1+T)e~2Tdr=一<8

oo」

b.Thethree-dimensionalcontrollercanonicalformrealizationisspecifiedby

Theobservabilitymatrix

has|Q|=0,SOthisrealizationisnotobservable.Theeigenvaluesof&Fare-2,-2,2andsothis

realizationisnotasymptoticallystable.

c.Asecond-orderminimalrealization(incontrollercanonicalform)isspecifiedby

[11]

NE6.5Given

s~+s—2(s-l)(s+2)(s-l)

〃(s)

s'+2s--4s—8(s-2)(s+2>(5—2)(S+2)

a.Thesystemisnotbounded-input,bounded-outputstablebecausethezero-stateresponsetoaunitstep

inputis

y(t)

whichisunbounded.

b.Thethree-dimensionalobservercanonicalformrealizationisspecifiedby

()08-2

AZCF=BCCF=

1041CCCF=[0。1]

01-21

Thecontrollabilitymatrix

~-28-8

P=[BQCFA)CF^OCFAX:F3OCF]二124

1-14

Fare

has|P|=0,sothisrealizationisnotcontrollable.Theeigenvaluesof240c-2,-2,2andsothis

realizationisnotasymptoticallystable.

c.Asecond-orderminimalrealization(inobservercanonicalform)isspecifiedby

[01]

Chapter7

NE7.la(onepossiblesolution)

DomPole=-2

Poles2=-2-20

Poles3=-2-20-21

Poles4=-2-20-21-22

NE7.lb(onepossiblesolution)

zeta=-0.6671

wn=-1.4990

den2=1.00002.00002.2469

DomPoles=

-1+1.12i

-1-1.12i

Poles3=-1-1.121-1+1.12i-10

Poles4=-1-1.12i-1+1.12i-10-11

NE7.1C

ITAE2=1725

Poles2i=

-3.5000+3.5707i

-3.5000-3.5707i

ITAE3=18.7553.75125

Poles3i=

-2.6048+5.3405i

-2.6048-5.3405i

-3.5405

ITAE4=110.585337.5625

Poles4i=

-2.1199+6.3150i

-2.1199-6.3150i

-3.1301+2.0707i

-3.1301-2.0707i

NE7.2a&NE7.3a

K=3.33-4.33

NE7.2b&NE7.3b

K=141

NE7.2C&NE7.3C

K=149

NE7.2d&NE7.3d

K=12130

Chapter8

NE8.la(onepossiblesolution)

DomPoleO=-20

Poles20=-20-200

Poles30=-20-200-201

Poles40=-20-200-201-202

OM010J50^6^ol0^04045tt5

time(sec)

NE8.lb(onepossiblesolution)

DomPolesO=

-10.0000+11.1665i

-10.0000-11.1665i

Poles3=

1.0e+002*

-0.1000-0.1117i-0.1000+0.1117i-1.0000

Poles4=

1.0e+002*

-0.1000-0.1117i-0.1000+0.1117i-1.0000-1.1000

NE8.1C

Poles2iO=

-35.0000+35.707H

-35.0000-35.707H

Poles3iO=

-26.0475+53.405H

-26.0475-53.405H

-35.4050

Poles4iO=

-21.1991+63.1496i

-21.1991-63.1496i

-31.3009+20.7069i

-31.3009-20.7069i

NE8.2a&NE8.3a

L=

420.33

-385.33

NE8.2b&NE8.3b

L=

82

1338

NE8.2c&NE8.3C

L=

90

1994

NE8.2d&NE8.3d

L=

208

30

Chapter9

NE9.1a

KLQR=

0.4169-0.0093

-0.00930.1234

NE9.1b

KLQR=

1.10320.0828

0.08280.0725

NE9.1C

KLQR=

6.56690.0828

0.08281.0796

NE9.1d

KLQR=

21.9240.3

240.32694.9

NE9.2ai

KLQR=

0.3724-0.0153

-0.01530.1221

NE9.2aii

KLQR=

0.4505-0.0053

-0.00530.1241

NE9.2bi

KLQR=

1.10230.0822

0.08220.0721

NE9.2bii

KLQR=

1.10370.0830

0.08300.0727

NE9.2ci

KLQR=

4.70360.0822

0.08220.7630

NE9.2cii

KLQR=

9.22630.0830

0.08301.5271

NE9.2di

KLQR=

11.1123.5

123.51426.8

NE9.2dii

KLQR=

43.5472.3

472.35198.0

NE9.3TheparametervaluesarethesameasinExample9.5exceptthatnowR=p1forvariablep

andwefixcr=lyieldingtheperformanceindex

J=i￡[^2(0+p2?2(0]jx2(1)

TheHamiltonianmatrixisgivenby

i

02

Hp

0

fromwhichwecomputethematrixexponential

{{e-lp+e,lp)

eHl

}(e-'lp+e',p)

Thisyieldsthe

X。)i(e-c-^P+ec-n/p)「1

A⑴1

古［（2+現3）。+（夕-1）""。］

fromwhichweconstruct

(j+l)e-"T"0-(0_l)e"fS_(/?+l)-(p-l)e2<f-|)；p

-=第Jp+?+(p-De**。-^(p+D+Cp-lk20-1^

Theassociatedfeedbackgainis=P⑴,yieldingthetime-varyingclosed-loopstateequation

x(t)=-$PQ)x(f)x(0)=%

Theclosed-loopstateresponsefortheinitialstateJC(O)=1andp=0.1,1,and1()isshownbelow.

Regulationperformanceimproveswithdecreasingpcorr