Please solve all parts Compute and concatenate the controllability and observabi

Question

Please solve all parts

Compute and concatenate the controllability and observability matrices yourself to practice how this is done. Use MATLAB to compute sub matrices and assemble them to check the ranks. Later, check your answers by ctrb and obsv commands to make sure you have been right: System below has 1 input and 2 outputs (how do you know?), answer the following questions: a. Is the system controllable? b. Is it observable? c. Is it observable from the 1^st element of the output vector alone? d. Is it observable from the 2^nd element of the output vector alone? {dx/dt = Ax + Bu y = Cx + Du; A = [-3 1 0 2 -3 2 0 1 -3]; B = [1 0 0]; C = [0 0 1 1 0 0]; D = 0

Explanation / Answer

>> A=[-3 1 0;2 -3 2;0 1 -3]

A =

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

>> B=[1;0;0]

B =

     1
     0
     0

>> C=[0 0 1;1 0 0]

C =

     0     0     1
     1     0     0

>> D=0

D =

     0

>> X=ctrb(A,B)

X =

     1    -3    11
     0     2   -12
     0     0     2

>> C0=[B A*B A^2*B]

C0 =

     1    -3    11
     0     2   -12
     0     0     2

>> rank(X)

ans =

     3

Hence the system is controllable:

>> Y=obsv(A,C)

Y =

     0     0     1
     1     0     0
     0     1    -3
    -3     1     0
     2    -6    11
    11    -6     2

>> OB=[C; C*A; C*A^2]

OB =

     0     0     1
     1     0     0
     0     1    -3
    -3     1     0
     2    -6    11
    11    -6     2

>> rank(Y)

ans =

     3

Hence the system is observable.

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