05.2 Controllability 2
Once we get the basic intuition from 05.1 Controllability 1, how do we know that if a system is controllable? There are couple of ways actually to do this. They are as follows:
First: Using MATLAB and LTI Object
We need to create an LTI object using state-space realization 03.1 Creating a LTI Object in MATLAB - Part 1 and 03.2 Creating a LTI Object in MATLAB - Part 2 (Conversion)
Then use the following code-snippet
sys = ss(A,B,C,D)
Q = ctrb(sys)
Second: Create the Q matrix and determine the rank of the matrix
You can form the controllability matrix by yourself with the following formula \(Q = \mathbf{ \begin{bmatrix}B & AB & A^2B & ... & A^{N-1}B \end{bmatrix}}\) $Q$ is of $N\times M$ $B$ is of $N \times M$ and rest of the elements of $Q$ matrix is of $N \times M$
rank_of_Q = rank(Q)