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

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.

Answer 1

[Star Strider]

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

Verifying the analysis in the zeros PDF.

[Flavio Clarizia]

Thanks a lot, it works.

[Star Strider]

As always, my pleasure!

The reason your code is throwing the error is because this assignment:

null = null(RSM_1)

creates ‘null’ as a variable, overshadowing the MATLAB null function. Then the next time you call the null function, you will get that error.

Do this instead:

RSM_1_null = null(RSM_1)

and the problem will not recur.

[Flavio Clarizia]

Hi, sorry if I bother again with this question, but I have tried to do the simulation of my problem using the rest of the code in the paper, so I have done the following:

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

S = ss(G_2)
Smr = minreal(S)
A = Smr.A
B = Smr.B
C = Smr.C
D = Smr.D
trz = tzero(A,B,C,D)
RSM_1 = [0*eye(2)-A B; -C D]
rRSM_1 = rank(RSM_1)

null = null(RSM_1)


%verifying zero blocking property
x0 = [-0.4388; -0.1482];
u0 = [-0.0000; 0.8863];
t = linspace(0,5);
u = exp(t);
[y,x] = lsim(A,B*u0,C,D*u0,u,t,x0);
plot(t,y)
axis([0 5 -1 1])
xlabel('time, seconds')
title('response to x_{0} and u(t)=u_{0}e^{t}')

In order to verify the zero blocking property as it is done in the paper. But when I do this, I obtain:

which does not make sense since the output should go to zero.

I know this answer should be closed, so sorry if I write again here. Can I ask you why does this happen to me even if the code is almost the same? Thanks in advance.

[Star Strider]

When I run your code in R2019b (although without the axis call), I get this result:

I do not understand the problem.

[Flavio Clarizia]

Thank you!

[Star Strider]

As always, my pleasure!