Stability assessment of two-dimensional state-space systems

The two-dimensional Fornasini-Marchesini model [1], [2] is one of the most general state-spice models available. Although conditions for stability for this model are theoretically simple, actual numerical verification is not a trivial exercise. One way to overcome this problem is to compute the matrix norm
\max\parallel A_{1} + e^{jw}A_{2} \parallel_{1}
for real
w
, and the system is stable if this value is less than unity. We compute this value by transforming the two-dimensional system into a canonical form based on the generalized eigenstructure of the state matrices
A_{1}
and
A_{2}
.