The problem of finding a reduced order model, optimal in the H2 sense, to a given
system model is a fundamental one in control system analysis and design. The addition of
a H∞ constraint to the H2 optimal model reduction problem results in a more practical
yet computationally more difficult problem. Without the global convergence of homotopy
methods, both the H2 optimal and the combined H2 / H∞ model reduction problems are
very difficult.
For both problems homotopy algorithms based on several formulations input normal
form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization
are developed and compared here. For the H2 optimal model order reduction problem,
these numerical algorithms are also compared with that based on Hyland and Bernstein's
optimal projection equations.
Both the input normal form and Ly form are very efficient compared to the over
parametrization formulation and the optimal projection equations approach, since they
utilize the minimal number of possible degrees of freedom. However, they can fail to exist
or be very ill conditioned. The conditions under which the input normal form and the Ly
form become ill conditioned are examined.
The over-parametrization formulation solves the ill conditioning issue, and usually is
more efficient than the approach based on solving the optimal projection equations for
the H2 optimal model reduction problem. However, the over-parametrization formulation
introduces a very high order singularity at the solution, and it is doubtful whether this
singularity can be overcome by using interpolation or other existing methods.
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