help regarding controllability of input matrix

I have a state space equation defined as:

X(dot) =A x(t)+ B u(t)

as given:

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.

1. Now how do I calculate the values of C in order to make this system controllable.
2. 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.

My solution:

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.

Answer 1

[M]

Is the matrix V completely free ?

If you can actually add any input to your system, why don't you simply define

  C = eye(3)

But of course you have to see if it's realistic in regard with your actual system.

[aayam goch]

Well the matrix V is free.

Will the controllability of the system do have controllability of full rank if I do define

C=eye (3) ?

How do I check the controllability for these two inputs?

[M]

You can define a new input vector

[u v]

and the corresponding input matrix

[B C]