Development of Model Order Reduction Techniques for 1-D and 2-D Systems

Abstract

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 large scale 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 large scale 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 two dimensional 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 two dimensional 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 two dimensional systems into causal recursive separable denominator model form and two dimensional 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.