Generation of a Class of Equivalent Networks and its Sensitivities

This paper presents a method of generating a class of equivalent networks from an initial network
\eta^{(0)}
such that a transfer function
H(p)
kept invariant throughout the transformation. The class of networks to be considered is chosen so that the sensitivities of
H(p)
to changes in the network elements of the initial network given by
S_{e_{k}}H(p)= \frac{\partial H (p)}{\partial e_{k}}\frac{e_{k}}{H(p)}
and the sensitivities of all of the generated equivalent networks can be obtained by simple matrix multiplication. Partial differentiation is avoided. The equivalent networks are generated from
\eta^{(0)}
by transforming the vector of network variables and the input vector in the equilibrium equations defined below. The transformation is performed in such a manner that the equivalent networks
\eta^{(1)}
are generated by congruence, transforming the matrices
M
and
N
that appear in the system of equations
M \dot{x} = -Nx + bu_{in}
w = d^{t}x
. The single-output transfer function
H(p)
is given by
d^{t} (pM + N)^{-1} b
. Provided that the equations possess specified properties, the sensitivities are easily obtained and can be applied to the problem of sensitivity minimization. Furthermore, if
\phi
is the sum
\sum |S_{e_{k}}|^{2}
, then the partial derivatives of
\phi
with respect to the transformation parameters are readily obtained. Again only matrix multiplication is required.