Mathematical Modeling of Linear Dynamic Systems and Stochastic Signals: A Comprehensive Approach

Abstract

Mathematical models serve as the backbone of our understanding and control of dynamic systems and stochastic signals. They provide a structured framework to describe, analyze, and predict the behavior of these systems and signals. Despite their widespread application in fields such as control theory, signal processing, and econometrics, a comprehensive understanding of these models and their interrelationships remains a challenge. This research presents a comprehensive study of mathematical models used to describe linear dynamic systems and stochastic signals. The paper first explores the various models for linear dynamic systems, including Ordinary Differential Equations (ODEs), State-Space Models, Transfer Function Models, and Discrete-Time Models. Each model's applicability, strengths, and limitations in describing continuous-time and discrete-time systems are discussed in detail. The second part of the paper delves into stochastic signals, focusing on Random Process Models, Markov Models, Gaussian Process Models, Autoregressive Models (AR), Moving Average Models (MA), and combinations of AR and MA models such as ARMA and ARIMA. The paper elucidates how these models capture the inherent randomness in signals and their utility in predicting future states.The research aims to provide a holistic understanding of these mathematical models, highlighting their significance in various fields.