PBH Test matrix
By John Peck •
Can the matrix occurring from PBH test have rank greater than the dimension n of the state space? What does it mean for the controllability of the system?
$\endgroup$ 31 Answer
$\begingroup$No. As mentioned in the comments, the PBH matrix is defined to be
$$ K = [(A-\lambda I) B].$$
Since $A$ is $n\times n$ and $B$ is $n \times m$, it follows that $K$ is $n \times (n+m) $. Since the rank of a matrix cannot exceed the lesser of the column or row numbers, it follows that the rank of $K$ is at most $n$.
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