Conservative Structured Noncommutative Multidimensional Linear Systems: Realization Theory and Bounded Real Lemma

Wednesday, January 17, 2007 - 1:40pm - 2:30pm
EE/CS 3-180
Joseph Ball (Virginia Polytechnic Institute and State University)
By a noncommutative multidimensional linear system we mean a linear
discrete-time input/state/output system with evolution along a
finitely generated free semigroup. A formal Z-transform of the
input-output map results in a transfer function equal to a
formal power series in noncommuting indeterminates with operator (or
matrix) coefficients. If one imposes energy-balance inequalities and
additional structure to the system equations, the resulting transfer
function is a formal power series with the additional structure of
interest for analyzing the robust control problem for a plant with
linear-fractional-modeled time-varying structured uncertainty.
The Bounded Real Lemma for such systems is closely connected with work
of Paganini on the robust control of such systems. An abelianization
of the system equations leads to systems with evolution along a
multidimensional integer lattice with transfer function equal to a
linear-fractional expression in several commuting variables of
Givon-Roesser, Fornasini-Marchesini or other structured types. Connections
with the automata theory of Schuetzenberger, Fliess, Eilenberg and others
from the 1960s will also be discussed. This talk reports on joint
work of the speaker with Tanit Malakorn (Naresuan University,
Thailand) and Gilbert Groenewald (North West University, South Africa).
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