## Multidimensional Linear Systems and Robust Control

##### Abstract

We then consider the output feedback stabilization problem for a plant $P(\mathbf{z})$. By assuming that $P(\mathbf{z})$ admits a double coprime factorization, then a set of stabilizing controllers $K(\mathbf{z})$ can be parametrized by the Youla parameter $Q(\mathbf{z})$. By doing so, one can convert such a problem to the model matching problem with performance index $F(\mathbf{z})$, affine in $Q(\mathbf{z})$. Then, with $F(\mathbf{z})$ as the design parameter rather than $Q(\mathbf{z})$, one has an interpolation problem for $F(\mathbf{z})$. Incorporation of a tolerance level on $F(\mathbf{z})$ 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 $\mathcal{F}_{d}$. 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 $T(z) = \sum_{v \in \mathcal{F}_{d}}T_{v}z^{v}$ where $z^{v} = z_{i_{n}} \dotsm z_{i_{1}}$ if $v = g_{i_{n}} \dotsm g_{i_{1}} \in \mathcal{F}_{d}$ and $z_{i}z_{j} \neq z_{j}z_{i}$ 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 SchÃ¼tzenberger and Fliess originally developed for "recognizable series" to our systems.

##### Collections

- Doctoral Dissertations [11240]