basic control problem


I have a basic question from ogata. i hope you know the answer.
consider the system defined by matrices:
A=[0 1 0;
0 0 1;
-6 -11 -6]
B=[0;
0;
1]
C=[c1 c2 c3]
Except for the obvious choice of c1=c2=c3, find an example set of c1,
c2, c3 that will make the system unobservable??
To check the observability, i have formed observ.matrix
Obsv=[C;
CA;
CA^2]
the determinant of the "Obsv" must be nonzero.
and it must be full rank.
When i formed the "Obsv" matrix, such complicated matrix was become.
Do you have any suggestions for c1, c2 and c3?
Thank you
Reply to
temcon
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Homework?
Reply to
Tim Wescott
I'm studying for the exam.
Reply to
temcon
Can't you use software?
A = {{0, 1, 0}, {0, 0, 1}, {-6, -11, -6}}; cmat = {{c1, c2, c3}}; obsrv = {Flatten[cmat], Flatten[cmat . A], Flatten[cmat . MatrixPower[A, 2]]}; Det[obsrv]
c1^3 - 6*c1^2*c2 + 11*c1*c2^2 - 6*c2^3 + 14*c1^2*c3 - 48*c1*c2*c3 + 36*c2^2*c3 + 49*c1*c3^2 - 66*c2*c3^2 + 36*c3^3
Reduce[% == 0, {c1, c2, c3}]
c3 == -(c1/9) + c2/3 || c3 == -(c1/4) + c2/2 || c3 == -c1 + c2
--Nasser
Reply to
Nasser Abbasi
Ah. See my response to your original post, in a minute.
Reply to
Tim Wescott
If the system matrix were in diagonal form you'd have no problem, right? Because if the C matrix has a zero in it, then the corresponding mode (which only maps to one state) will be unobservable.
So one way you could do this is to find a similarity transformation to a diagonal form, zero out one or more columns in the C matrix, then transform the system back to the one you have.
(it worked for me, just like magic).
Reply to
Tim Wescott
Stretch my old ring out.
I Am Kirk Johnson. "Anal Stretching, Wrenching & Expanding Specialist"
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Reply to
Kirk Johnson

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