How do I find the the zero state diretion and the zero input direction of a transfer matrix in Matlab?

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Flavio Clarizia 2020-2-11

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https://ww2.mathworks.cn/matlabcentral/answers/504811-how-do-i-find-the-the-zero-state-diretion-and-the-zero-input-direction-of-a-transfer-matrix-in-matla

评论： Star Strider 2020-2-19

I want to compute using matlab the Rosenbrock System Matrix, and in particular I am trying to do a an analysis similar to the one at the end of http://people.duke.edu/~hpgavin/ce263/zeros.pdf . So, starting from a transfer matrix I would like to show that if there is a zero at a certain position, there is a zero blocking property.

To do so, as written in the link, I have to find the zeros state direction and the zero input direction, and I am having troubles doing do.

I am considering a different system from the one in the link, so I am trying to do a similar analysis applied to another system, and I am doing this:

s = tf('s');
P = 1/(s+5);
C = 6/s;
S = 1/(1+P*C);
T = P*C/(1+P*C);
G_2 = [T S; S -S]; %transfer matrix

Now, I have that my transfer matrix is :

G_2

and for it I would like to do the analysis to find the zero state diretion and the zero input direction, but I am having troubles doing so.

Can somebody please help me? Thanks in advance.

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Answer 1

Star Strider 2020-2-11

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https://ww2.mathworks.cn/matlabcentral/answers/504811-how-do-i-find-the-the-zero-state-diretion-and-the-zero-input-direction-of-a-transfer-matrix-in-matla#answer_414988

在 MATLAB Online 中打开

I have not done anything with Rosenbrock System Matrices since my multivariable control course in graduate school. We used the Maciejowski textbook, that I still have.

To reproduce that analysis:

s = tf('s');
P = [2/(s^2+3*s+2) 2*s/(s^2+3*s+2); -2*s/(s^2+3*s+2) -2/(s^2+3*s+2)]
S = ss(P)
Smr = minreal(S)
A = Smr.A
B = Smr.B
C = Smr.C
D = Smr.D
trz = tzero(A,B,C,D)
RSM_1 = [eye(3)-A B; -C D]
rRSM_1 = rank(RSM_1)

producing:

P =
 
  From input 1 to output...
             2
   1:  -------------
       s^2 + 3 s + 2
 
           -2 s
   2:  -------------
       s^2 + 3 s + 2
 
  From input 2 to output...
            2 s
   1:  -------------
       s^2 + 3 s + 2
 
            -2
   2:  -------------
       s^2 + 3 s + 2
 
Continuous-time transfer function.
S =
 
  A = 
       x1  x2  x3  x4
   x1  -3  -2   0   0
   x2   1   0   0   0
   x3   0   0  -3  -2
   x4   0   0   1   0
 
  B = 
       u1  u2
   x1   2   0
   x2   0   0
   x3   0   2
   x4   0   0
 
  C = 
       x1  x2  x3  x4
   y1   0   1   1   0
   y2  -1   0   0  -1
 
  D = 
       u1  u2
   y1   0   0
   y2   0   0
 
Continuous-time state-space model.
1 state removed.
Smr =
 
  A = 
            x1       x2       x3
   x1   -1.222  -0.4444  -0.5556
   x2    1.556   -2.889   -1.111
   x3   -1.556   0.8889  -0.8889
 
  B = 
           u1      u2
   x1       1  0.3333
   x2      -1   1.667
   x3       1  0.3333
 
  C = 
               x1          x2          x3
   y1           1           1  -6.939e-16
   y2     -0.3333      0.3333      -1.333
 
  D = 
       u1  u2
   y1   0   0
   y2   0   0
 
Continuous-time state-space model.
A =
      -1.2222     -0.44444     -0.55556
       1.5556      -2.8889      -1.1111
      -1.5556      0.88889     -0.88889
B =
            1      0.33333
           -1       1.6667
            1      0.33333
C =
            1            1  -6.9389e-16
     -0.33333      0.33333      -1.3333
D =
     0     0
     0     0
     
trz =
     1
RSM_1 =
       2.2222      0.44444      0.55556            1      0.33333
      -1.5556       3.8889       1.1111           -1       1.6667
       1.5556     -0.88889       1.8889            1      0.33333
           -1           -1   6.9389e-16            0            0
      0.33333     -0.33333       1.3333            0            0
rRSM_1 =
     4