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Multidimensional Linear Systems and Robust Control
dc.contributor.author  Malakorn, Tanit  en_US 
dc.date.accessioned  20140314T20:09:35Z  
dc.date.available  20140314T20:09:35Z  
dc.date.issued  20030410  en_US 
dc.identifier.other  etd04142003144447  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 discretetime 1D linear system in the operator theoretical viewpoint followed by the formulations of the socalled GivoneRoesser and FornasiniMarchesini 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 NevanlinnaPick type. We also give an operatortheoretic 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 timeaxis is described by a free semigroup $\mathcal{F}_{d}$. Such a system can be represented by the socalled noncommutative GivoneRoesser, or noncommutative FornasiniMarchesini 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 statespace interpretation. Minimal realization problems for noncommutative GivoneRoesser or FornasiniMarchesini systems are solved directly by a shiftrealization 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/etd04142003144447/  en_US 
dc.contributor.committeecochair  Ball, Joseph A.  en_US 
dc.contributor.committeecochair  Baumann, William T.  en_US 
dc.date.sdate  20030414  en_US 
dc.date.rdate  20040416  
dc.date.adate  20030416  en_US 
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Doctoral Dissertations [14723]