On the stability of two-dimensional state-space systems: A special case

The stability of two-dimensional state-space systems can be determined by knowing the zero manifolds of the characteristic equation
\det(I- z_{1}A_{1} - z_{2}A_{2}) = 0
or, equivalently, the eigenvalues of the matrix
(A_{1} + e^{J\omega}A_{2})
for all real ω. We propose a new simple test for the special case when the matrices A
1
and A
2
are simultaneously reducible to upper (or lower) triangular form. This test can also be used to simplify the general problem if these matrices are partially reducible to the triangular form.