# controllability

Dear All,
I have an academic probelm. I tried really hard, but i'm not looking through it.
I should proof that if and only if the system x(t)=Ax(t)+Bu(t) is
controllable, then the system z(t)=(A-beta*I)z(t)+Bu(t) is controllable. beta is a real constant.
Thanks tuna
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tuna wrote:

Give us more information -- what have you done, and do you know the definition of a controllable system?
I would assume that this would involve performing the test for controllability on the modified system, then applying some rules of matrix algebra to the result to show continued controllability. I suspect that if it were me I'd have more problems with the matrix math than the pure control theory (matrix linear algebra makes my head spin, but it's usually worth it to work through).
--

Tim Wescott
Wescott Design Services
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There is not much more information. That's all and yes I know the definition for the controllability. The contrallable gramain C=[B AB ... A^(n-1)B] must have full rank(C)=n. And I tried also through the time variant definition with the transition matrix... but my math is to bad.
Tim Wescott schrieb:

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tuna wrote:

Write the Gramian for both systems, expand the second system. Remember what a full rank matrix is and what conditions on the columns of the matrix this implies. You know something about the columns of the Gramian foe the first system, perhaps this means you can say something about the columns of the second system?
I hope this leads you in the right direction.
-Matt
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no no no no! that is not definition but *condition*
--
Mikolaj

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oh sorry, of course you are right...
the definition is that you can lead the state from an given initial state x_0 with an continuous input signal u(t) after a arbitrary time to x=0. Then the system is controllable in between t_0 and this time.
Maybe it is a bit confusing, but i don't have the book in front of me... but the definition goes in this direction.
Mikolaj schrieb: