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

January 17, 2007
  • Linear higher-order systems
  • 35G35
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).