Articles

A Stochastic Framework for Fully Distributed Control Systems and CPS: From Local State Transitions to Global Uncertainty Propagation

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The transition from hierarchical automation toward fully distributed Distributed Control Systems (DCS) and Cyber-Physical Systems (CPS) creates a new class of engineering problems in which local intelligence, networked coordination and physical dynamics must operate under uncertainty. In these systems, control is no longer concentrated in a single supervisory unit. Instead, sensors, controllers, actuators, edge devices and cyber agents cooperate through local decisions and partial information. This article develops a stochastic framework for examining fully distributed DCS/CPS by linking three levels of analysis: how local states shift through Markov transitions, how short‑term decisions accumulate over time through the Chapman–Kolmogorov relation, and how uncertainty spreads in continuous processes through the Fokker–Planck equation. All these aspects indicate that a distributed system is something much more than a mere configuration of communication; it is an adaptive, stochastic controller organism, where its global functioning arises from many small local decisions that modify probabilities and paths. The design should then consider how these local decisions affect one another over time, rather than simply interconnecting devices. The proposed framework is then analyzed from the perspectives of stability, resilience, communication delay, cyber-security, scalability, energy efficiency and digital-twin-based prediction. The result is a theoretical foundation suitable for smart factories, smart grids, intelligent buildings, autonomous transport systems and future smart urban infrastructures. The added interpretative value of the framework is that each equation is treated not only as a formal mathematical relation, but also as a design logic. Markov probabilities are interpreted as local operational tendencies, Chapman-Kolmogorov composition as the logic of accumulated distributed decisions, and Fokker-Planck dynamics as the evolution of confidence, risk and uncertainty in the whole cyber-physical network.

 

Conceptualization of Markov Processes in Cyber-Physical Systems: Modelling, Prediction, and Control

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Cyber-Physical Systems (CPS) bring together physical processes with computing, communication, and control. They often operate in environments full of uncertainty, noise, and constant change, which makes traditional deterministic models struggle to capture how these systems really behave. This work introduces a more flexible framework based on Markov processes that helps model, predict, and control CPS in a more realistic way. By viewing system behaviour as probabilistic transitions between states, it becomes easier to analyze uncertainty and understand how the system evolves over time. The study looks at discrete-time Markov chains and expands the discussion to Hidden Markov Models (HMMs) and Markov Decision Processes (MDPs), allowing both visible and hidden aspects of system dynamics to be represented. It outlines a well-defined process for defining states, calculating transition probabilities, and making forecasts. The paper explores, in addition to that, the use of control techniques based on the use of probability theory and shows that these methods have a greater level of robustness compared to traditional control techniques. An example is given to show how this model improves performance and flexibility. All in all, Markov modelling is a good starting point for dealing with the challenges in CPSs, paving the way for integration with other tools.