Problem in observibility and controlibility function ctrbf,obsvf
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After doing similarity transform, transfer function of a system taking controllable modes should be same as original. But it is not showing same results.
>> clear >> A = [-.5 1 0;-1 -.5 0; 0 1 0]
A =
-0.5000 1.0000 0
-1.0000 -0.5000 0
0 1.0000 0
>> b=[1;2;0]
b =
1
2
0
>> c =[0 0 1]
c =
0 0 1
>> [Abar,Bbar,Cbar,T,k]=ctrbf(A,b,c)
Abar =
0 0 0
0.4472 -0.5000 1.3416
0.6667 -0.7454 -0.5000
Bbar =
0
0.0000
2.2361
Cbar =
0.7454 0.6667 0
T =
-0.5963 0.2981 0.7454
0.6667 -0.3333 0.6667
0.4472 0.8944 0
k =
1 1 0
>> c*inv([s 0 0;0 s 0;0 0 s]-A)*b ??? Undefined function or variable 's'.
>> s=sym('s'); >> c*inv([s 0 0;0 s 0;0 0 s]-A)*b
ans =
(2*(4*s + 2))/(s*(4*s^2 + 4*s + 5)) - 4/(s*(4*s^2 + 4*s + 5))
>> Cbar*inv([s 0 0;0 s 0;0 0 s]-Abar)*Bbar
ans =
(4*s + 2)/(13510798882111488*(4*s^2 + 4*s + 5)) + 40/(20*s^2 + 20*s + 25)
>> T*A*T'
ans =
0 0 0
0.4472 -0.5000 1.3416
0.6667 -0.7454 -0.5000
>> Ac=[Abar(2,2) Abar(2,3);Abar(3,2) Abar(3,3)]
Ac =
-0.5000 1.3416
-0.7454 -0.5000
>> Bc=[Bbar(2,1);Bbar(2,3)] ??? Attempted to access Bbar(2,3); index out of bounds because size(Bbar)=[3,1].
>> Bc=[Bbar(2,1);Bbar(3,1)]
Bc =
0.0000
2.2361
>> Cc=[Cbar(1,2);Cbar(1,3)]
Cc =
0.6667
0
>> Cc=[Cbar(1,2) Cbar(1,3)]
Cc =
0.6667 0
>> Cc*inv([s 0 ;0 s]-Ac)*Bc
ans =
(4*s + 2)/(13510798882111488*(4*s^2 + 4*s + 5)) + 8/(4*s^2 + 4*s + 5)
>>
1 commentaire
Walter Roberson
le 27 Mar 2011
It would be easier on readers if you were to cut out the places you know you made mistakes.
Réponse acceptée
Teja Muppirala
le 28 Mar 2011
This difference is just from round-off errors. Notice the second element in Bbar is 1.1102e-016. That is why you get that very large denominator (13510798882111488) in the symbolic expression
say Bbar(2) = 0, then everything works fine.
A = [-.5 1 0;-1 -.5 0; 0 1 0];
b=[1;2;0];
c =[0 0 1];
[Abar,Bbar,Cbar,T,k]=ctrbf(A,b,c)
syms s
Bbar(2) = 0;
c*( (s*eye(3)-A) \b)
Cbar*( (s*eye(3)-Abar) \Bbar)
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Teja Muppirala
le 28 Mar 2011
Just a suggestion, but when doing symbolic calculations using 's' as a transfer function variable, you might want to explicitly make s a transfer function instead of just an ordinary symbolic variable.
s = tf('s')
instead of
s = sym('s')
This will allow you to actually treat those expressions like transfer functions.
G1 = c*( (s*eye(3)-A) \b) G2 = Cbar*( (s*eye(3)-Abar) \Bbar)
bode(G1) etc...
There is also a function MINREAL that you can use to cancel out poles and zeros.
minreal(G1,1e-6)
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