# 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).

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).

MSC Code:

35G35

Keywords: