现代控制理论研究报告(英文)

moderncontroltheory

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Experiment4SolvingSimilarTransformationandCanonicalFormofState-SpaceModelandItsRealizationUsingMatlab

Objectives:

1.GraspingtheexpressionofdiagonalandJordancanonicalformandthemethodsofsolutionstotransfermatricescorrespondingtotheabovematricesofLTIsystembyexperiment.

2.Graspingthecontrollability,observabilitycriterionsandcontrollable,observabledecompositionsofthesystembyprogramminganddebugging.

3.Reinforcingtheunderstandingaboutthemethodsoftransferfunctiontocontrollableandobservablecanonicalform.

Requirements:

Answeringthefollowingquestionthroughanexample:

Thestate-spacemodelofsametransferfunctionisuniqueornot.

Solution:

Wechoosethetransferfunctionis:

1.diagonalcanonicalform:

executethem-file:

num=[1]

den=[18.52012.5]

[A,B,C,D]=tf2ss(num,den)

G=ss(A,B,C,D);

[V,D]=eig(A)

G1=ss2ss(G,D)

Obtain:

D=

-5.000000

0-2.50000

00-1.0000

a=

x1x2x3

x1-8.5-40-62.5

x20.500

x300.40

b=

u1

x1-5

x20

x30

c=

x1x2x3

y1-00-1

d=

u1

y10

2.Jordancanonicalform

num=[1]

den=[18.52012.5]

[A,B,C,D]=tf2ss(num,den)

G=ss(A,B,C,D);

[V,J]=jordan(A)

G1=ss2ss(G,V)

Obtain:

V=

2.50000.1667-1.6667

-0.5000-0.16670.6667

0.10000.1667-0.2667

a=

x1x2x3

x1-104-648.8-854.8

x221131.8174.8

x3-4.2-26.89-36.29

b=

u1

x12.5

x2-0.5

x30.1

c=

x1x2x3

y1165

d=

u1

y10

Now,weshoulddetermineitscontrollability,observabilityandconductthecontrollable,observabledecomposition

3.Determinethecontrollabilityandobservability

num=[1]

den=[18.52012.5]

[A,B,C,D]=tf2ss(num,den)

G=ss(A,B,C,D);

Tc=ctrb(A,B)

n=size(A);

ifrank(Tc)==n(1)

disp('Thesystemiscontrolled')

else

disp('Thesystemisnotcontrolled')

end

Obtain:

Tc=

1.0000-8.500052.2500

01.0000-8.5000

001.0000

Thesystemiscontrolled

num=[1]

den=[18.52012.5]

[A,B,C,D]=tf2ss(num,den)

G=ss(A,B,C,D)

To=obsv(A,C)

n=size(A);

ifrank(To)==n(1)

disp('Thesystemisobservability')

else

disp('Thesystemisnotobservability')

end

Obtain:

To=

001

010

100

Thesystemisobservability

4.Controllabledecomposition

num=[1];

den=[18.52012.5];

[A,B,C,D]=tf2ss(num,den);

G=ss(A,B,C,D);

[Ac,Bc,Cc,Tc,Kc]=ctrbf(A,B,C)

G1=ss2ss(G,Tc)

Obtain:

Tc=

00-1

0-10

100

a=

x1x2x3

x1010

x200-1

x312.520-8.5

b=

u1

x10

x20

x31

c=

x1x2x3

y1-1-00

d=

u1

y10

5.Observabledecomposition

num=[1];

den=[18.52012.5];

[A,B,C,D]=tf2ss(num,den);

G=ss(A,B,C,D);

[Ao,Bo,Co,To]=obsvf(A,B,C)

G1=ss2ss(G,To)

Obtain:

To=

-100

010

00-1

a=

x1x2x3

x1-8.520-12.5

x2-100

x30-10

b=

u1

x1-1

x20

x30