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, z_d). 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 \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_vz^v where z^v = z_in \dotsm z_i1 if v = g_in \dotsm g_i1 \in \mathcal{F}_d and z_iz_j \neq z_jz_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.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Electrical and Computer Engineering

Department dc:contributor.department

Electrical and Computer Engineering

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2003

Author dc:creator

• Malakorn, Tanit

Chairs dc:contributor.committeechair

• Ball, Joseph A.
• Baumann, William T.

Committee members dc:contributor.committeemember

• VanLandingham, Hugh F.
• Jacobs, Ira
• Day, Martin V.

• noncommutative d-D linear systems
• model matching form
• Linear Operator Inequality (LOI)
• H^{infty} control problem
• interpolation theory
• minimal realization

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

etd-04142003-144447

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/26845