{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/26845"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/26845","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Multidimensional Linear Systems and Robust Control","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 Schützenberger and Fliess originally developed for \"recognizable series\" to our systems.","abstract_html":"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 <span class=\"etd-inline-math\">H<sup>\\infty</sup></span> 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 <span class=\"etd-inline-math\"><strong>z</strong> = (z<sub>1</sub>, \\dots, z<sub>d</sub>)</span>. We then consider the output feedback stabilization problem for a plant <span class=\"etd-inline-math\">P(<strong>z</strong>)</span>. By assuming that <span class=\"etd-inline-math\">P(<strong>z</strong>)</span> admits a double coprime factorization, then a set of stabilizing controllers <span class=\"etd-inline-math\">K(<strong>z</strong>)</span> can be parametrized by the Youla parameter <span class=\"etd-inline-math\">Q(<strong>z</strong>)</span>. By doing so, one can convert such a problem to the model matching problem with performance index <span class=\"etd-inline-math\">F(<strong>z</strong>)</span>, affine in <span class=\"etd-inline-math\">Q(<strong>z</strong>)</span>. Then, with <span class=\"etd-inline-math\">F(<strong>z</strong>)</span> as the design parameter rather than <span class=\"etd-inline-math\">Q(<strong>z</strong>)</span>, one has an interpolation problem for <span class=\"etd-inline-math\">F(<strong>z</strong>)</span>. Incorporation of a tolerance level on <span class=\"etd-inline-math\">F(<strong>z</strong>)</span> 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 <span class=\"etd-inline-math\">\\mathcal{F}<sub>d</sub></span>. 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 <span class=\"etd-inline-math\">T(z) = \\sum<sub>v \\in \\mathcal{F}<sub>d</sub></sub>T<sub>v</sub>z<sup>v</sup></span> where <span class=\"etd-inline-math\">z<sup>v</sup> = z<sub>i<sub>n</sub></sub> \\dotsm z<sub>i<sub>1</sub></sub></span> if <span class=\"etd-inline-math\">v = g<sub>i<sub>n</sub></sub> \\dotsm g<sub>i<sub>1</sub></sub> \\in \\mathcal{F}<sub>d</sub></span> and <span class=\"etd-inline-math\">z<sub>i</sub>z<sub>j</sub> \\neq z<sub>j</sub>z<sub>i</sub></span> 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 &quot;recognizable series&quot; to our systems.","abstract_has_math":true,"creators":["Malakorn, Tanit"],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Electrical and Computer Engineering","degree_department":"Electrical and Computer Engineering","school":null,"contributors":[],"advisors":[],"committee_chairs":["Ball, Joseph A.","Baumann, William T."],"committee_members":["VanLandingham, Hugh F.","Jacobs, Ira","Day, Martin V."],"year":2003,"date_issued":"2003-04-10","date_published":"2003-04-10","updated_at":"2026-07-22T22:18:46Z","subjects":["noncommutative d-D linear systems","model matching form","Linear Operator Inequality (LOI)","H^{infty} control problem","interpolation theory","minimal realization"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-04142003-144447"],"render_values":[{"text":"etd-04142003-144447","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/26845","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Ball, Joseph A.","Baumann, William T."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["VanLandingham, Hugh F.","Jacobs, Ira","Day, Martin V."]},{"key":"dc:contributor.department","label":"Department","values":["Electrical and Computer Engineering"]},{"key":"dc:creator","label":"Author","values":["Malakorn, Tanit"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T20:09:35Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T20:09:35Z","2004-04-16"]},{"key":"dc:date.issued","label":"Date","values":["2003-04-10"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Electrical and Computer Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["noncommutative d-D linear systems","model matching form","Linear Operator Inequality (LOI)","H^{infty} control problem","interpolation theory","minimal realization"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-04142003-144447"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/26845"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 Schützenberger and Fliess originally developed for \"recognizable series\" to our systems."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Multidimensional Linear Systems and Robust Control"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Ball, Joseph A.","Baumann, William T."],"dc:contributor.committeemember":["VanLandingham, Hugh F.","Jacobs, Ira","Day, Martin V."],"dc:contributor.department":["Electrical and Computer Engineering"],"dc:creator":["Malakorn, Tanit"],"dc:date.accessioned":["2014-03-14T20:09:35Z"],"dc:date.available":["2014-03-14T20:09:35Z","2004-04-16"],"dc:date.issued":["2003-04-10"],"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 Schützenberger and Fliess originally developed for \"recognizable series\" to our systems."],"dc:description.degree":["Ph. D."],"dc:identifier.other":["etd-04142003-144447"],"dc:identifier.uri":["http://hdl.handle.net/10919/26845"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["noncommutative d-D linear systems","model matching form","Linear Operator Inequality (LOI)","H^{infty} control problem","interpolation theory","minimal realization"],"dc:title":["Multidimensional Linear Systems and Robust Control"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Electrical and Computer Engineering"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:18:46Z"}