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In our rapidly
developing technological world, complex and large scale first and
second order
systems are becoming more and more encountered with an increasing
application
in different fields of engineering. To handle these systems, it is
advisable to
calculate a reduced order model that approximates the behaviour of the
original
system and simplifies the tasks of simulation, optimization, prediction
and
controller design.
The reduction
methods based on moment matching using the Krylov subspaces are
nowadays among
the best choices, where the moments are the coefficients of the Taylor series
expansion
of the transfer function around a certain point. This method finds the
reduced
order model in a relatively short time with a good numerical accuracy,
via a
projection whose columns form bases of particular Krylov subspaces.
Lately, the idea of
order reduction has been coupled to Laguerre orthonormal series as an
alternative to the Taylor
series. Instead of moment matching, this alternative approach matches
some of
the coefficients of the Laguerre series of the reduced and original
models’
transfer functions.
In this
presentation, the order reduction of first order systems using Laguerre
series
expansion is reviewed and then this approach is generalized to reduce
second
order systems while preserving the structure, where the so called
second order
Krylov subspaces are used.
Furthermore,
the equivalence between the Laguerre-based model reduction
and moment matching around a certain point is shown. The proof of the
equivalence is based on showing that the subspaces used for projection
in both
methods are exactly equal if the parameter of the Laguerre series and
the
expansion point in moment matching are chosen to be the same. This
fact is also generalized to include a general family of coefficients
known as generalized Markov parameters.
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