How do I detect if a zero pole cancelation has happened when looking for a zero in a MIMO transfer function?

Question

Consider a transfer function for a MIMO control system:

$G(s)=\begin{pmatrix} \frac{2}{s+1} & \frac{3}{s+2}\\ \frac{1}{s+1}& \frac{1}{s+1} \end{pmatrix}$

I want to find its poles and zeros. For the poles, I know that I have to look at the roots of the pole polynomial. So, I have to look at the common divisor for the minors of order one and two, and I find that the poles are:

$s=-1 $ (multiplicity 2)

and

$s=-2 $ (multiplicity 1)

then I look for the zeros, by looking for the value that makes the transfer function matrix lose rank, so for the value such that:

$det[G(s)]=0$

and I find that there is a zero at $s=+1$.

By studying this topic, I have found that when looking for zeros, we need to be careful when looking for the value of $s$ that make the determinant equal to zero, since we could not see zero-pole cancelations, or we may miss some zeros of the system.

But, how do I know if I have missed some zero or that a zero-pole cancelation has happened?

The only way I know to find a zero is to look at the lost of rank, and I can do so by looking at the deretminant. But if a zero-pole cancelation has happened, how do I detect it?

In theory I should have a lost of controllability and/or observability if this has happened. But I am confused on how to work with zeros even if i have been trying to go deeper on this topic for days.

Can somebofy please help me?

Answer 1

It is true that a matrix loses rank is equivalent to its determinant being zero. And you are correct to suspect that by calculating determinant, pole-zero cancellation might happen. For this reason, this method is generally not preferred, and you need to resort to other methods like MacFarlane-Karcanias method, or reducing the MIMO matrix to Smith-McMillan form. If you have a state-space realization, you can use the Rosenbrock system matrix.

Other reasons to avoid calculating determinants are:

1. It is not defined for non-square systems.
2. Even if the system is square and there is no zero, determinant can still be zero because of linear dependency between rows or zeros.

The method that you used to calculate poles is related to the MacFarlane-Karcanias method, where you need to find the least common denominator of the minors of all orders. For more information, see MacFarlane & Karcanias 1976. Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex-variable theory.