EE611

Solution to HW#18

1a)

» a=[-2 0 3 3 0; 0 -.5 0 -1.5 0;0 1.5 1 1.5 0;0 -1.5 0 -.5 0;0 0 -3 -3 -2]

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

» A=[-2 0 3 3 0; 0 -.5 0 -1.5 0;0 1.5 1 1.5 0;0 -1.5 0 -.5 0;0 0 -3 -3 -2]

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=[0 1;-.5 -.5;-.5 .5;.5 .5;0 -1]

B =

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

» C=[4 8 0 0 0;0 0 -4 -4 0]

C =

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

» M=[B A*B A*A*B A*A*A*B A*A*A*A*B]

M =

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

» rank(M)

ans =

2

» Tccf = [M(:,1:2) [1;0;0;0;0] [0;1;0;0;0] [0;0;1;0;0]]

Tccf =

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

» rank(Tccf)

ans =

5

» Accf = inv(Tccf)*A*Tccf

Accf =

1	0	0	-3	-3
0	1	0	0	3
0	0	-2	0	0
0	0	0	-2	0
0	0	0	0	-2

» Ac = [1 0;0 1]

Ac =

1	0
0	1

» Ac_bar = [-2 0 0;0 -2 0;0 0 -2]

Ac_bar =

-2	0	0
0	-2	0
0	0	-2

» eig(Ac_bar)

ans =

-2

-2

-2

» %ALL uncontrollable eigenvalues are stable!!

» %Therefore, system is stabilizable!!

b)

» Bccf = inv(Tccf)*B

Bccf =

1	0
0	1
0	0
0	0
0	0

» Bc = [1 0;0 1]

Bc =

1	0
0	1

» %want eigenvalues : {-5, -5}

» %This is a trivial problem since Ac and Bc are identity matrices!!

» Kc = [6 0; 0 6]

Kc =

6	0
0	6

» %check:

» eig(Ac-Bc*Kc)

ans =

-5
-5

» K = [Kc [0 0 0; 0 0 0]] * inv(Tccf)

K =

0	0	0	12	6
0	0	0	0	-6

» %check eigenvalues of entire system:

» eig(A-B*K)

ans =

-2.0000

-5.0000

-2.0000

-5.0000

-2.0000

c)

» yref = [10;-4]

yref =

10

-4

» inv([C [0 0;0 0];A B])*[eye(2);zeros(5,2)]

ans =

0	-0.2500
0.1250	0.1250
0.1250	-0.1250
-0.1250	-0.1250
0	0.2500
0.2500	0
0	0.2500

» Nx = ans(1:5,:)

Nx =

0	-0.2500
0.1250	0.1250
0.1250	-0.1250
-0.1250	-0.1250
0	0.2500

» Nu = ans(6:7,:)

Nu =

0.2500	0
0	0.2500

d)

» %settling time:

» ts = -4/(-2)

ts =

2

e)

» Ao = A'

Ao =

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

» Bo = C'

Bo =

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

» O = [Bo Ao*Bo Ao*Ao*Bo Ao*Ao*Ao*Bo Ao*Ao*Ao*Ao*Bo]

O =

4	0	-8	0	16	0	-32	0	64	0
8	0	-4	0	20	0	-28	0	68	0
0	-4	12	-4	-12	-4	36	-4	-60	-4
0	-4	0	-4	0	-4	0	-4	0	-4
0	0	0	0	0	0	0	0	0	0

» rank(O)

ans =

3

» Tccf = [O(:,1:3) [1;0;0;0;0] [0;0;0;0;1]]

Tccf =

4	0	-8	1	0
8	0	-4	0	0
0	-4	12	0	0
0	-4	0	0	0
0	0	0	0	1

» rank(Tccf)

ans =

5

» Accf = inv(Tccf)*Ao*Tccf

Accf =

0.0000	0	2.0000	0	0
0	1.0000	0	-0.7500	0.7500
1.0000	0	-1.0000	0	0
0	0	0	-2.0000	0
0	0	0	0	-2.0000

» Ac = [0 0 2; 0 1 0;1 0 -1]

Ac =

0	0	2
0	1	0
1	0	-1

» Ac_bar = [-2 0;0 -2]

Ac_bar =

-2	0
0	-2

» %All unobservable eigenvalues are in LHP

» %Therefore, system is detectable!!

» Bccf = inv(Tccf)*Bo

Bccf =

1	0
0	1
0	0
0	0
0	0

» Bc = [1 0;0 1;0 0]

Bc =

1	0
0	1
0	0

» eig(Ac)

ans =

1

-2

1

» %Ac is not cyclic!! Pick Kf:

» Kf = [1 1 1;1 1 1];

» eig(Ac-Bc*Kf)

ans =

-2.0000

1.0000

-1.0000

» Anew = Ac-Bc*Kf;

» %pick Ks:

» ks = [0;1];

» bs = Bc*ks

bs =

0

1

0

» Ms = [bs Anew*bs Anew*Anew*bs];

» rank(Ms)

ans =

3

» poly(Anew)

ans =

1.0000

2.0000

-1.0000

-2.0000

» Apv = [0 1 0;0 0 1;2 1 -2]

Apv =

0	1	0
0	0	1
2	1	-2

» bpv = [0;0;1]

bpv =

0

0

1

» poly([-10 -10 -10])

ans =

1

30

300

1000

» Kpv = [1000+Apv(3,1) , 300+Apv(3,2) , 30+Apv(3,3)]

Kpv =

1002

301

28

» eig(Apv-bpv*Kpv)

ans =

-9.9999

-10.0000 + 0.0001i

-10.0000 - 0.0001i

» Tpv_inverse = [bpv Apv*bpv Apv*Apv*bpv]*inv(Ms)

Tpv_inverse =

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

» Kc = Kf + ks * Kpv * Tpv_inverse

Kc =

1

1

1

-244 29 -756

» eig(Ac-Bc*Kc)

ans =

-10.0000 + 0.0001i

-10.0000 - 0.0001i

-9.9999

» K = [Kc [0 0; 0 0]] * inv(Tccf)

K =

0	0.1250	0.1250	-0.3750	0
0	-30.5000	-73.1667	65.9167	0

» Ko = K'

Ko =

0	0
0.1250	-30.5000
0.1250	-73.1667
-0.3750	65.9167
0	0

» %check:

» eig(Ao - Bo*Ko')

ans =

-10.0001 + 0.0002i

-10.0001 - 0.0002i

-9.9998

-2.0000

-2.0000