NUST Institutions Library Catalogue

Mathematical modeling aims to analyze dynamical systems, which is an essential part of
control systems engineering. The need for more rigorous mathematical models is growing
as models get more complicated. Simulation becomes computationally tedious for largescale
systems containing lumped parameter systems, such as ordinary differential equations,
distributed parameter systems, such as partial differential equations, etc. Dealing with these
conditions is made easier with model order reduction. Studying these complex and largescale
systems is a difficult task; as a result, reduced-order models are needed to make the
analysis easier.
Multidimensional models, which contain many independent variables (e.g., time), are critical
in control system theory. Researchers have focused their attention on significant issues
such as stability, decomposition, factorization, modeling, analysis, and model reduction.
These issues are not as straightforward as they are for one-dimensional models due to the
fact that fundamental algebraic concepts do not apply directly to multidimensional models.
Similarly, two-dimensional models, which is a sub-class of multidimensional models,
encompass many real-time physical systems such as hydraulics, images, filters, nuclear reactors,
etc. Due to their complex structure, two-dimensional models are difficult to work with;
additionally, simulation, analysis, design, and control become more difficult as the model’s
order increases. Additionally, fundamental algebraic theorems such as stability, decomposition,
factorization, and model order reduction are only possible when two-dimensional
models can be decomposed into two one-dimensional sub-models. In comparison, while
there is a large amount of literature on one-dimensional modeling and analysis systems (including
model reduction), the majority of findings do not directly apply to two-dimensional
models. If separable denominator two-dimensional models are demonstrated, the separable
denominator form must satisfy the minimal rank-decomposition criteria to decompose
the two-dimensional models into two one-dimensional sub-models. On the other hand, the
majority of real-world applications do not exist in a separable denominator form.
The model order reduction challenge seeks to provide a simple-to-measure alternative to
the original large-scale stable model that exhibits the same responses. In reduced-order models,
the model order reduction aims to retain essential properties of the original large-scale
model, such as stability and input-output response. The model order reduction contributed
significantly to the control system, primarily through simulation of complicated systems
such as large-scale complex integrated circuits, robotics systems, communication systems, controller reduction, etc. Similarly, extensive study has been conducted over the last few
decades on the model order reduction of large-scale systems, with several model order reduction
approaches developed. However, these strategies are limited by the absence of essential
attributes of the original system in reduced-order systems, such as stability, considerable
approximation error, and the lack of a priori error bounds.
This dissertation aims to explore model reduction techniques for one- and twodimensional
systems that utilize balanced structure. Additionally, two distinct transformations
for two-dimensional Roesser models are proposed.
Firstly, the model reduction problem is formulated for the one-dimensional continuous
and discrete-time systems (includes weighted and limited intervals scenarios). New techniques
are proposed for the one-dimensional continuous and discrete-time systems based
on weighted and limited intervals Gramians. Proposed techniques provide the stability of
reduced-order models for certain double-sided weights and intervals of interest. Furthermore,
proposed techniques offer a priori error bounds expressions for the weighted and
limited intervals scenarios. Proposed techniques yield mostly low truncation error when
compared to the state-of-the-art existing model reduction techniques.
Secondly, the model reduction problem is formulated for the discrete-time twodimensional
causal recursive separable denominator and generalized systems (includes
weighted and limited intervals scenarios). Model reduction algorithms for two-dimensional
causal recursive separable denominator systems are developed in the first step. A priori
error bounds formulations for two-dimensional CRSD based weighted and limited intervals
scenarios are constructed in the second step. Furthermore, proposed techniques yield
mostly low truncation error when compared to the state-of-the-art existing model reduction
techniques.
Finally, two different transformations are proposed, which transforms generalized twodimensional
systems into causal recursive separable denominator model form and twodimensional
diagonalized model form, respectively. Proposed MOR are employed on transformed
two-dimensional transformed realizations, which also provide a priori error bounds
formulations for weighted and limited intervals scenarios. Compared to existing state-ofthe-
art two-dimensional model reduction strategies, the proposed techniques produce mostly
modest truncation errors.