EE611

Solution to HW#20

1a)

» A

A =

-2.0000	0	3.0000	3.0000	0
0	-0.5000	0	-1.5000	0
0	1.5000	1.0000	1.5000	0
0	-1.5000	0	-0.5000	0
0	0	-3.0000	-3.0000	-2.0000

» B

B =

0	1.0000
-0.5000	-0.5000
-0.5000	0.5000
0.5000	0.5000
0	-1.0000

» C

C =

4	8	0	0	0
0	0	-4	-4	0

» R=[0 1 0 0 0;0 0 0 1 0;0 0 0 0 1]

R =

0	1	0	0	0
0	0	0	1	0
0	0	0	0	1

» Tro = inv([C;R])

Tro =

0.2500	0	-2.0000	0	0
0	0	1.0000	0	0
0	-0.2500	0	-1.0000	0
0	0	0	1.0000	0
0	0	0	0	1.0000

» Anew = inv(Tro)*A*Tro

Anew =

-2.0000	-3.0000	12.0000	-12.0000	0
0	1.0000	0	0	0
0	0	-0.5000	-1.5000	0
0	0	-1.5000	-0.5000	0
0	0.7500	0	0	-2.0000

» Bnew = inv(Tro)*B

Bnew =

-4.0000	0
0	-4.0000
-0.5000	-0.5000
0.5000	0.5000
0	-1.0000

» A11 = Anew(1:2,1:2)

A11 =

-2	-3
0	1

» A12 = Anew(1:2,3:5)

A12 =

12	-12	0
0	0	0

» A21 = Anew(3:5,1:2)

A21 =

0	0
0	0
0	0.7500

» A22 = Anew(3:5,3:5)

A22 =

-0.5000	-1.5000	0
-1.5000	-0.5000	0
0	0	-2.0000

» B1 = Bnew(1:2,:)

B1 =

-4	0
0	-4

» B2 = Bnew(3:5,:)

B2 =

-0.5000	-0.5000
0.5000	0.5000
0	-1.0000

» %Now set the eigenvalues of A22-Kro*A12

» Ao = A22'

Ao =

-0.5000	-1.5000	0
-1.5000	-0.5000	0
0	0	-2.0000

» Bo = A12'

Bo =

12	0
-12	0
0	0

» O = [Bo Ao*Bo Ao*Ao*Bo]

O =

12	0	12	0	12	0
-12	0	-12	0	-12	0
0	0	0	0	0	0

» rank(O)

ans =

1

» Tocf = [12 0 0;-12 1 0;0 0 1]

Tocf =

12	0	0
-12	1	0
0	0	1

» rank(Tocf)

ans =

3

» Aocf = inv(Tocf)*Ao*Tocf

Aocf =

1.0000	-0.1250	0
0	-2.0000	0
0	0	-2.0000

» Bocf = inv(Tocf)*Bo

Bocf =

1	0
0	0
0	0

» a1=Aocf(1,1)

a1 =

1

» b1=Bocf(1,:)

b1 =

1

0

» k1 = [11;0]

k1 =

11

0

» Kro = ([k1 [0 0;0 0]] * inv(Tocf))'

Kro =

0.9167	0
0	0
0	0

» eig(A22-Kro*A12)

ans =

-10

-2

-2

» Aro = (A22-Kro*A12)

Aro =

-11.5000	9.5000	0
-1.5000	-0.5000	0
0	0	-2.0000

» Bro = B2 - Kro*B1

Bro =

3.1667	-0.5000
0.5000	0.5000
0	-1.0000

» Cro = (A22-Kro*A12)*Kro - Kro*A11 + A21

Cro =

-8.7083	2.7500
-1.3750	0
0	0.7500

» Q1 = Tro(:,1:2)

Q1 =

0.2500	0
0	0
0	-0.2500
0	0
0	0

» Q2 = Tro(:,3:5)

Q2 =

-2	0	0
1	0	0
0	-1	0
0	1	0
0	0	1

b)