Observer gain matrix calculations give warning when replicated with Matlab

Hi,

I am trying to replicate on Matlab an exercise on the book Modern Control Engineering of Ogata Chapter 10.7.

When using the state space values of the book I can calculate the observer gain matrix Ke. When calculating the state space values on Matlab using the tf2ss, I am getting a warning for the observer gain matrix Ke.

The transfer function, desired closed loop poles and desired observer poles are below.

Gp=tf(1, [1 0 1 0])

Gp =
 
     1
  -------
  s^3 + s
 
Continuous-time transfer function.

s1=-1+i; %Desired closed loop poles
s2=-1-i; %Desired closed loop poles
s3=-8; %Desired closed loop poles
s4=-4; %Desired observer poles
s5=-4; %Desired observer poles
J=[s1 s2 s3];
L=[s4 s5];

On the book, the state space matrix A, B, C, D is given as below. The same values I find when I manually calculate the state space matrix.

A=[0 1 0; 0 0 1; 0 -1 0]

A = 3×3

0 1 0 0 0 1 0 -1 0

B=[0; 0; 1]

B = 3×1

0 0 1

C=[1 0 0]

C = 1×3

1 0 0

 
Aaa=[0]

Aaa = 0

Aab=[1 0]

Aab = 1×2

1 0

Aba=[0; 0]

Aba = 2×1

0 0

Abb=[0 1; -1 0]

Abb = 2×2

0 1 -1 0

Ba=[0]

Ba = 0

Bb=[0; 1];
 
K=acker(A,B,J)

K = 1×3

16 17 10

 
Ke=acker(Abb',Aab',L')'

Ke = 2×1

8 15

When calculating the state space values using the tf2ss, I get for observer gain matrix Ke a wanring “Warning: Matrix is singular to working precision” and the ouput is Ke=[Nan; Inf].

[num den] = tfdata(Gp, 'v');
[A B C D]=tf2ss(num,den)

A = 3×3

0 -1 0 1 0 0 0 1 0

B = 3×1

1 0 0

C = 1×3

0 0 1

D = 0

Aaa=[0]

Aaa = 0

Aab=[-1 0]

Aab = 1×2

-1 0

Aba=[1; 0]

Aba = 2×1

1 0

Abb=[0 0; 1 0]

Abb = 2×2

0 0 1 0

Ba=[1]

Ba = 1

Bb=[0; 0];
 
K=acker(A,B,J)

K = 1×3

10 17 16

 
Ke=acker(Abb',Aab',L')'

Warning: Matrix is singular to working precision.

Ke = 2×1

NaN Inf

The difference on two methods are the state space values A, B, C , D which also make different values Aaa, Aab, Aba Abb.

Any idea why I get warning when trying to calculate the observer gain matrix Ke.

Answer 1

[Paul]

Hi NGiannis,

Here's the plant model and desired pole locations

Gp=tf(1, [1 0 1 0]);
s1=-1+i; %Desired closed loop poles
s2=-1-i; %Desired closed loop poles
s3=-8; %Desired closed loop poles
s4=-4; %Desired observer poles
s5=-4; %Desired observer poles
J=[s1 s2 s3];
L=[s4 s5];

State space realization via tf2ss

[num den] = tfdata(Gp, 'v');
[A B C D]=tf2ss(num,den);

Parttion for reduced order observer:

Aaa = A(1,1);
Aab = A(1,2:3);
Aba = A(2:3,1);
Abb = A(2:3,2:3);
Ke=acker(Abb',Aab',L')'

Warning: Matrix is singular to working precision.

Ke = 2×1

NaN -Inf

The problem arises because Abb and Aab are not an observable pair; the observability matrix is clearly not full rank because the second column is all zeros.

obsv(Abb,Aab)

ans = 2×2

-1 0 0 0

I believe that this approach assumes that the first state is available for feedback via the output, but in this realization it's the third state that is the output

C

C = 1×3

0 0 1

Compare to the C matrix in your original formulation where C(1) = 1.

[NGiannis]

Thank you Paul