Civil Engineering Reference
In-Depth Information
This transformation effectively maps a point
in a
space to
a point
X
in the original
space.
Substituting this in (3.53)‚
premultiplying the first equation by
and using Equation (3.59)‚ we
obtain
Note that this is a reduced system since it is represented by
state variables
instead of the original
variables. This may be rewritten as
where we set a matrix that corresponds to the reduced
order system. In the Laplace domain‚ this implies that the transfer function
for this reduced-order system is given by
Using the eigendecomposition
we obtain
The matrix is diagonal and is hence easily invertible for small values
of the approximation order‚ As a result‚ this method is computationally
efficient. Since it does not directly work with circuit moments‚ it succeeds in
avoiding the numerical problems faced by AWE.
3.6.3
Passivity and PRIMA
Numerical problems are not the only issues that haunt AWE-like methods. A
major problem associated with solutions provided by AWE and simple Arnoldi-
or Lanczos-based methods is that they cannot guarantee that the resulting
network is
passive.
A passive network is one that always dissipates more energy
than it generates; pure RLC networks always possess this property. Stability
(which is equivalent to the statement that all poles of the transfer function must
lie in the left half plane) is a necessary
but not sufficient
condition for passivity‚
so that of the two‚ passivity is the stronger and more restrictive condition.
In particular‚ it is possible to show existence cases of a nonpassive but stable
system that‚ when connected to a passive and stable system‚ results in an
unstable system [KY98]. On the other hand‚ if a system is guaranteed to
be passive‚ any interconnection with any other passive system will also be
passive (and hence stable). This idea was described in the context of model
order reduction in [KY98]‚ and the idea of a split congruence transformation
was proposed. This was later incorporated into other model order reduction
algorithms‚ among which PRIMA is widely used.
