  The paper addresses the model reduction problem for linear and nonlinear
systems using the notion of least squares moment matching. For linear systems,
the main idea is to approximate a transfer function by ensuring that the
interpolation conditions imposed by moment matching are satisfied in a least
squares sense. The paper revisits this idea using tools from output regulation
theory to provide a new time-domain characterization of least squares moment
matching. It is shown that least squares moment matching can be characterized
in terms of an optimization problem involving an invariance equation and in
terms of the steady-state behavior of an error system. This characterization,
in turn, is then used to define a nonlinear enhancement of the notion of least
squares moment matching and to develop a model reduction theory for nonlinear
systems based on the notion of least squares moment matching. Parameterized
families of models achieving least squares moment matching are determined both
for linear and nonlinear systems. The new parameterizations are shown to admit
natural geometric and system-theoretic interpretations. The theory is
illustrated by worked-out numerical examples.
