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 dc.contributor.author Malakorn, Tanit en_US dc.date.accessioned 2014-03-14T20:09:35Z dc.date.available 2014-03-14T20:09:35Z dc.date.issued 2003-04-10 en_US dc.identifier.other etd-04142003-144447 en_US dc.identifier.uri http://hdl.handle.net/10919/26845 dc.description.abstract 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 $\mathbf{z} = (z_{1}, \dots, z_{d})$. 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. en_US dc.publisher Virginia Tech en_US dc.relation.haspart ETD.pdf en_US dc.rights In Copyright en dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ en dc.subject noncommutative $d$-D linear systems en_US dc.subject model matching form en_US dc.subject Linear Operator Inequality (LOI) en_US dc.subject H^{infty} control problem en_US dc.subject interpolation theory en_US dc.subject minimal realization en_US dc.title Multidimensional Linear Systems and Robust Control en_US dc.type Dissertation en_US dc.contributor.department Electrical and Computer Engineering en_US dc.description.degree Ph. D. en_US thesis.degree.name Ph. D. en_US thesis.degree.level doctoral en_US thesis.degree.grantor Virginia Polytechnic Institute and State University en_US thesis.degree.discipline Electrical and Computer Engineering en_US dc.contributor.committeemember VanLandingham, Hugh F. en_US dc.contributor.committeemember Jacobs, Ira en_US dc.contributor.committeemember Day, Martin V. en_US dc.identifier.sourceurl http://scholar.lib.vt.edu/theses/available/etd-04142003-144447/ en_US dc.contributor.committeecochair Ball, Joseph A. en_US dc.contributor.committeecochair Baumann, William T. en_US dc.date.sdate 2003-04-14 en_US dc.date.rdate 2004-04-16 dc.date.adate 2003-04-16 en_US
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