help regarding controllability of input matrix
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I have a state space equation defined as:
X(dot) =A x(t)+ B u(t)
A=[-1 0 1;
0 0 1;
-1 0 1]
B=[1 0 1]'
If I do check the controllability of (A,B) the system is not controllable as it does not have full rank.
Now, I add another function as:
X=A x(t)+ B u(t) + C v(t)
where v(t) is another input in the same system.
- Now how do I calculate the values of C in order to make this system controllable.
- Another problem is, If I assume I can only measure the state x1(t), I would like to design a full-state estimator to estimate x2(t) and x3(t) and verify that the estimated states do in fact track the true states and the output does in fact follow the step input.
I tried calculating the determinant of whole the system including C but got stuck in part one.
In the second part, I was not sure how to solve for the problem to go on.