Minimality and Local State Decompositions of a
Nonlinear State Space Realization using Energy Functions
J.M.A. Scherpen and
W.S. Gray,
IEEE Transactions on Automatic Control, AC-45, 11 (2000) 2079-2086.
Abstract:
In this paper we develop a set of sufficient conditions in terms of
controllability and observability functions under which a given state
space realization of a formal power series is minimal. Specifically, it
is shown that positivity of these functions,
in addition to a stability requirement and a
few technical conditions,
implies minimality. In doing so, connections are established between
Hamilton-Jacobi type optimal control theory and the well
known necessary and
sufficient conditions for minimality in terms of Kalman type rank
conditions on the accessibility and observability distributions.
Then, as an application of this new minimality result,
an existing theory of similarity invariants for nonlinear systems
is extended.
Using the nonlinear analogue of the Kalman decomposition,
connections are established between minimality, singular value functions,
balanced realizations,
and various notions of reachability and observability for nonlinear systems.
Keywords:
minimal realizations, similarity invariants, energy functions, formal power
series, nonlinear systems
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