  We provide an analytical framework for balanced realization model order
reduction of linear control systems which depend on an unknown parameter.
Besides recovering known results for the first order corrections, we obtain
explicit novel expressions for the form of second order corrections for
singular values and singular vectors. The final result of our procedure is an
order reduced model which incorporates the uncertain parameter. We apply our
algorithm to the model order reduction of a linear system of masses and springs
with parameter dependent coefficients.
