How do I test stability of a MIMO system?

Question

Let's say I have a system similar to two interconnected IIR filters described like this:

\begin{align} x_1(t)&=a_{11} x_1(t-1)+a_{12} x_1(t-2) +a_{13} x_2(t-1) + a_{14} x_2(t-2)+y_1(t)\\ x_2(t)&=a_{21} x_2(t-1)+a_{22} x_2(t-2) +a_{23} x_1(t-1) + a_{24} x_1(t-2)+y_2(t) \end{align}

where $x$ are the outputs and $y$ are the inputs.

How would I go about testing the stability of such a system?

I myself have pretty much no background in MIMO systems, so I'm not really sure where even to start.

For a SISO system, I'd start with poles and zeros and go on from there, but I'm not sure how such a graph might looks like in the MIMO case.

Answer 1

Let's try to make your system into a state space representation with state: $$ \mathbf{x}(t) = \left [ \begin{array}{c} x_1(t)\\ x_2(t)\\ x_1(t-1)\\ x_2(t-1)\\ \end{array} \right ] $$ so that $$ \mathbf{x}(t) = \mathbf{A} \mathbf{x}(t-1) + \mathbf{y}(t) $$ where $$ \mathbf{A} = \left [ \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \end{array} \right ] $$ and $$ \mathbf{y}(t) = \left [ \begin{array}{c} y_1(t)\\ y_2(t)\\ 0\\ 0 \end{array} \right ] $$ or you could make $\mathbf{y}$ $2 \times 1$ and introduce a $\mathbf{B}$ matrix.

The system may be unstable if the eigenvalues of $\mathbf{A}$ are outside the unit circle.