How to multiply transfer function in matrix form?

I want to multiply transfer function in matrix form in order to calculate gain for compensator

which requires to do following calculation:

Dc = - K * (sI - A + BK + LC)^(-1) *L

where K and L are matices designed with pole placement,

A is a matrix for my state space

I is an Identity matrix of a size defined by A

s is frequency domain variable.

My challenge/problem here is that:

Whenever I multiply K matrix with the following matrix

the pole and zero increases as well.

I somehow can't find a way to make 'Dc' into the form of Dc1

Here is my code:

% define A B C for state space

A = [0 1; 0 0];
B = [0 ; 1];
C = [1 0];

% design pole for control
pc = [-0.7071067812+0.7071067812j -0.7071067812-0.7071067812j];
K = place(A,B,pc);  %calculate control gain
K = real(K)

% design pole for estimator 
% 4 times the frequecy of controller 
% but not to large to decrease bandwith for noise
pe = [-2.5+4.330127019j -2.5-4.330127019j];
Lt = place(A',C',pe);   %calculate estimator gain
L = Lt'

% define s and I before calculate overall gain
s = tf('s');

[row, column] = size(A)
I = eye(row)

sI = s*I

INV = sI-A+B*K+L*C

X = INV\L

Dc = -mtimes(K,X)

Dc1=-40.4*(s+0.619)/(s+3.21+4.77j)/(s+3.21-4.77j)
rlocus(Dc);
bode(Dc);
margin(Dc);

Answer 1

[Andrew Lee]

just found a way to solve this, but " .*" seem to caue error message

Full code below:

% define A B C for state space

A = [0 1; 0 0];
B = [0 ; 1];
C = [1 0];

% design pole for control
pc = [-0.7071067812+0.7071067812j -0.7071067812-0.7071067812j];
K = place(A,B,pc);  %calculate control gain
K = real(K)

% design pole for estimator 
% 4 times the frequecy of controller 
% but not to large to decrease bandwith for noise
pe = [-2.5+4.330127019j -2.5-4.330127019j];
Lt = place(A',C',pe);   %calculate estimator gain
L = Lt'

% define s and I before calculate overall gain
s = tf('s');

[row, column] = size(A)
I = eye(row)

sI = s*I

INV = sI-A+B*K+L*C

X = INV\L

Dc = tf('s')
Dc = 0*Dc

for i = 1:row
    Dc = Dc + K(1:i).*X(i:1)
    
end

Dc

Dc1=-40.4*(s+0.619)/(s+3.21+4.77j)/(s+3.21-4.77j)
rlocus(Dc);
bode(Dc);
margin(Dc);

Answer 2

[Paul]

The result you seek can be found by (after running the code in the original question):

>> minreal(Dc)

ans =
 
      -40.36 s - 25
  ---------------------
  s^2 + 6.414 s + 33.07
 
Continuous-time transfer function.

The transfer function algebra in the code is introducing artificial poles and zeros. There is probably a better way solve the whole problem from the start. Such as:

> tf(ss(A-B*K-L*C,L,-K,0))

ans =
 
      -40.36 s - 25
  ---------------------
  s^2 + 6.414 s + 33.07
 
Continuous-time transfer function.