Model Order Reduction of Discrete-Time Interval Systems

Abstract

Operational systems available in the real-time world studied through their well-established mathematical representations are recognized to be a challenging task. This assignment is then processed via system identification contributing to the derivation of a vast number of differential and difference equations resulting in state variable or transfer function of higher order, making the overall study and analysis of the system cumbersome. Here, rises the trouble for researchers working on a particular system via numerical investigations. This analysis of composite nature is stated to be time-consuming and is inconvenient especially if a repeated assessment is desired. Thus, demanding a technique or an algorithm to cut down the order of the system to a user-friendly approximate model which is readily available for their investigation. The method emerged to the practice for deriving the simplified version of the higher order system known as Model Order Reduction (MOR). It aims to preserve fewer of the essential characteristics such as stability, transient and steady state response, etc. of the original system. This requirement is concerned with the limited computational, accuracy, and storage capabilities. Nevertheless, the reduced models are believed to be a replica of primary systems. The derived reduced models are then substituted in place of the original complex systems making them simple for either study or analysis or simulation or control. Throughout the thesis, the label of System, and Model are used for the higher and lower order mathematical representations respectively. The acceptance of formulating the MOR algorithms grabbed a consistent interest from every arena from theoretical to practical from electrical to mechanical systems accommodating every smaller to larger subjects. The practice resulted in the emergence of various algorithms and techniques observed through the bulk of literature both in time and frequency domains ranging from continuous-time to discrete-time domains. In-depth study of the system dynamics showcased the presence of uncertainty within the system. These uncertainties are the consequence of un-modeled dynamics, sensor noises, disturbances, manual errors, parameter variations, actuator constraints and many others. Thus, systems having coefficients of uncertain nature are known as uncertain systems. And when these systems are bounded by a finite range or boundary is designated as Interval Systems. In literature, researcher assigns them either Interval or Uncertain systems. In this Model Order Reduction of Discrete-Time Interval Systems thesis, they are Interval Systems for convenience. The presence of uncertainties in the system results in foster inconvenience for the systems simulation, design and implementation. Additionally, these errors significantly affect the stability and performance of the structure. Since the uncertainty in the system cannot be ignored or neglected, the study proceeded towards the order reduction of such systems which is the prime objective of this thesis. As the study of larger systems matured from non-interval systems to the present day interval systems, the arena for MOR also enhanced from non-interval to interval systems. Again, literature retrieves various algorithms for the approximation of such systems. From the available series of reduction methodologies, the researcher can observe that there are multiple algorithms for interval systems in the continuous-time domain, whereas discrete-time domain bear only fewer algorithms, leaving a possible arena for work. The observation affirms the prime motivation for the author to work through Model Order Reduction of Discrete-Time Interval Systems. The present work elaborates the development of MOR techniques for discretetime interval systems classified into two categories. 1) Routh Approximation Approach. It constitutes the methods based on Routh approximation. Since the Routh approximation is not applicable to discrete-time systems directly; an appropriate discrete-transformation performs its execution. The proposed algorithms under this approach are Gamma-Delta Approximation, Arithmetic Operator or Multiplicative approach; Novel Arrangement of Routh array; Simplified Interval Structure Approximation; Advanced Routh Approximation Method (A-RAM), Extended Direct Routh Approximation Method (E-DRAM), Routh Approximant, Routh and Pade Approximation, Direct Truncation and Pade Approximation amalgamated with Routh Approximation, 2) Another category of the proposed algorithms in the thesis considers Assorted Approach, developed from the necessary mathematical adaptations. The methods formulated in this category are Non-Computational Technique, Classical Differentiation Technique, Direct Truncation Method, Routh approximation and Direct Truncation combined with Mikhailov Stability Criterion. Validation of the proposed algorithms is through examples available in the literature. Their comparison with the existing techniques via widely accepted performance tools as the weighted square error and step response depict the effectiveness. Frequency domain response performs the stability check of the derived reduced model. MATLAB environment is used to verify the algorithms. Model Order Reduction of Discrete-Time Interval Systems The thesis is spread over six chapters commencing with Chapter 1 letting in the introduction, motivation, literature survey, thesis contribution and the road map to the thesis write-up. Chapter 2 reports the preliminaries required for understanding the argument for MOR of interval system. The necessities integrated include the interval arithmetic, stability check algorithm, performance analysis tools, problem statement and the desired transformation. Chapter 3 and Chapter 4 describes the order reduction techniques based on Routh approximation approach and Assorted approach respectively. Chapter 5 discusses two separate assessments from the results. One is an evaluation of the two frequently used discrete transformation techniques namely linear and bilinear transformation for order reduction of interval systems. Another is the overall analysis of the proposed algorithms observed throughout the thesis including the limitations if discovered any. Lastly, the conclusion in Chapter 6 conveys the inevitable development of successful algorithms of varied forms applicable to linear discrete-time interval systems. These algorithms are stated to be computationally effortless and uncomplicated to access with a promise to retain dynamic characteristics. This chapter also addresses three possible future works.