Multidimensional Linear Systems and Robust Control

This dissertation contains two parts: Commutative and Noncommutative Multidimensional ($d$-D) Linear Systems Theory. The first part focuses on the development of the interpolation theory to solve the $H\infty$ control problem for $d$-D linear systems. We first review the classical discrete-time 1D linear system in the operator theoretical viewpoint followed by the formulations of the so-called Givone-Roesser and Fornasini-Marchesini models. Application of the $d$-variable $Z$-transform to the system of equations yields the transfer function which is a rational function of several complex variables, say $z=(z1,\dots ,zd)$.

We then consider the output feedback stabilization problem for a plant $P(z)$. By assuming that $P(z)$ admits a double coprime factorization, then a set of stabilizing controllers $K(z)$ can be parametrized by the Youla parameter $Q(z)$. By doing so, one can convert such a problem to the model matching problem with performance index $F(z)$, affine in $Q(z)$. Then, with $F(z)$ as the design parameter rather than $Q(z)$, one has an interpolation problem for $F(z)$. Incorporation of a tolerance level on $F(z)$ then leads to an interpolation problem of multivariable Nevanlinna-Pick type. We also give an operator-theoretic formulation of the model matching problem which lends itself to a solution via the commutant lifting theorem on the polydisk.

The second part details a system whose time-axis is described by a free semigroup $Fd$. Such a system can be represented by the so-called noncommutative Givone-Roesser, or noncommutative Fornasini-Marchesini models which are analogous to those in the first part. Application of a noncommutative $d$-variable $Z$-transform to the system of equations yields the transfer function expressed by a formal power series in several noncommuting indeterminants, say $T(z)=\sum v\in FdTvzv$ where $zv=zin\cdots zi1$ if $v=gin\cdots gi1\in Fd$ and $zizj\ne zjzi$ unless $i=j$. The concepts of reachability, controllability, observability, similarity, and stability are introduced by means of the state-space interpretation. Minimal realization problems for noncommutative Givone-Roesser or Fornasini-Marchesini systems are solved directly by a shift-realization procedure constructed from appropriate noncommutative Hankel matrices. This procedure adapts the ideas of Schützenberger and Fliess originally developed for "recognizable series" to our systems.