Articles

From Component Reliability to Dynamic Predictive Reliability: A Multilayer Framework for Cyber-Physical Systems

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The reliability of cyber-physical systems (CPS) cannot be reduced to the failure probability of individual hardware components. Modern industrial, urban, educational and energy CPS combine physical processes, sensors, communication networks, software services, control algorithms, cyber-security mechanisms, human operators and environmental conditions. This paper proposes an integral-predictive view of CPS reliability based on two complementary models: the multilayer integral reliability index of a cyber-physical system (MINKFS) and the dynamic cognitive-predictive reliability model (DKPN). The framework takes different types of signals, converts them into standardized layer indices, combines them using a weighted structure, and updates the current reliability estimate by adding the predicted chance of future failure.  The empirical part uses a synthetic industrial CPS dataset of 336 hourly observations and demonstrates how layer statistics, scenario analysis, operational regimes and criticality rankings can support early warning and preventive decision-making. The model transforms diverse inputs to normalized layer indexes, integrates these using a weighted integrative model, and adds the forecast probability of future failure to the present measure of reliability. The empirical part studies a fabricated data set related to industrial CPS which consists of 336 hourly measurements and provides a demonstration of how layer, scenario, regime and criticality information is used for detection and prevention. It was observed that the cyber-resilience layer had the lowest average reliability as well as the highest critical contribution, whereas communications and environment-energy layers were characterized by high volatility. Cyber event scenario resulted in the lowest predictive reliability and highest likelihood of failure. These results confirm the thesis that reliability of a CPS is better analyzed as a time-dependent, interpretable, and scenario-dependent feature instead of being just another component coefficient.

Automation Efficiency in Cyber-Physical Systems: Interaction Between Technical Performance and Algorithmic Intelligence

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The performance of a cyber-physical system cannot be judged reliably from a single engineering variable. In the investigated micro-CPS, response time, communication latency and throughput describe the technical layer, whereas prediction accuracy captures the contribution of the learning component. The present paper examines how these two dimensions interact and how their combined effect is reflected in an overall efficiency index. A quasi-experimental comparison was carried out for a logistics and warehouse control configuration equipped with connected sensing, real-time data exchange and machine-learning support, and for a functionally similar conventional automated system. The empirical analysis was based on six variables, which were first standardized using z-scores and then assigned to four domains: technical efficiency, algorithmic intelligence, resource and energy efficiency, and economic efficiency. A clear difference emerged between the two system architectures. The CPS produced a mean composite index of 0.954, whereas the conventional automated system had a mean value of −0.954. The regression results further showed that technical efficiency and algorithmic intelligence contributed most strongly to the overall index. Of particular interest was their interaction. The interaction between technical efficiency and algorithmic intelligence was statistically significant (B = 0.154, β = 0.267, p = 0.004).  This interaction included to the model increased R² from 0.93 to 0.95. The finding indicates that the contribution of algorithmic intelligence to overall system performance depends, at least partly, on the level of technical efficiency achieved by the system. Rather, their contribution becomes more pronounced when the control and communication infrastructure already performs reliably. The sensitivity analysis led to the same overall ordering of the two architectures under alternative weighting schemes. Within the scope of the available dataset, this provides additional evidence that the observed difference is not simply an artefact of the selected weights.

Predictive Pontryagin Optimal Control of Nonlinear Fully Distributed DCS and CPS under Reliability and Uncertainty Constraints

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The article provides an overview of the optimal control prediction framework for non-linear CPS and DCS in automation. The core issue is the fact that the present architecture of modern automation systems does not behave like a closed loop control system anymore. The design of these systems combines sensors, actuators, edge controllers, communication channels, digital twin, software-as-a-service, human involvement, and states of cyber-security. Thus, control design must not only address accuracy but should be a multicriteria design problem that accounts for reliability, uncertainty, probability mass, delay in communication, and energy costs. Four levels are considered in the proposed framework. The first level refers to forecasting based on hybrid ARIMA-ML model for a short horizon. The second level is concerned with estimating of risk states using Markov model/HMM.  The contribution is a simulation-ready mathematical architecture in which each node solves a local Hamiltonian problem using predicted states, neighbour information and reliability constraints, while the global CPS behavior emerges through networked local decisions. The paper formulates the nonlinear dynamics, cost functional, Hamiltonian conditions, reliability constraints and evaluation protocol for smart factories, smart grids, intelligent buildings and smart campuses. The framework is positioned as a bridge between predictive maintenance and optimal distributed automation control.

Residual-Regime Markov Modelling for Predictive Control in Cyber-Physical Systems

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Cyber-physical systems operate through a continuous interaction between physical processes, computational intelligence, communication networks and control mechanisms. Their behaviour is rarely fully deterministic: sensor noise, delayed communication, nonlinear dynamics, regime changes and early-stage failures create uncertainty that cannot be adequately represented by a single forecasting or control method. This paper proposes an original Residual-Regime Markov Forecast–Control Framework for cyber-physical systems. The framework brings together three modelling layers that play different roles. The ARIMA component handles the basic linear time‑series structure and keeps the model interpretable. On top of that, a machine‑learning layer works on the residuals to capture the nonlinear behaviour that ARIMA cannot. The final layer uses a Markov‑style state representation, turning the forecast errors, system signals and operating conditions into probabilistic regimes that describe how the system is likely to evolve. Unlike classical hybrid forecasting models that stop at prediction, the proposed approach links prediction to decision-making by using Markov transition probabilities, hidden-state belief updates and risk-aware policy selection. The main idea is that forecast errors are not treated only as modelling imperfections; instead, they are interpreted as early indicators of changing system regimes. A simulation-oriented evaluation design is presented for an industrial cyber-physical process with normal operation, peak load and degradation conditions. The proposed framework is expected to improve predictive maintenance, anomaly anticipation and control-policy selection by connecting statistical forecasting, data-driven correction and probabilistic decision logic in a single pipeline. The contribution of the paper lies in transforming hybrid forecasting into a regime-aware forecast–control architecture suitable for intelligent CPS monitoring and adaptive technical management.

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.

A Hybrid “ARIMA–ML Regression” Model for Enhanced Predictive Analysis in Cyber-Physical Systems: Conceptual framework and Simulation Evaluation

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This paper presents a hybrid ARIMA–machine learning (ARIMA–ML) regression framework designed to improve predictive accuracy in cyber‑physical systems (CPS). The approach brings together the strengths of classical statistical time‑series modelling and modern data‑driven techniques, allowing the model to capture both linear structures and nonlinear dynamics that commonly arise in CPS environments. A simulation‑based evaluation was conducted using a multivariate dataset generated from a MATLAB/Simulink CPS model, complemented by Python‑based machine learning components. The results show that the hybrid model consistently outperforms standalone ARIMA and ML approaches across multiple operational scenarios, including normal operation, peak load, and early‑stage failure conditions. Improvements were observed not only in RMSE and MAE but also in residual stability, prediction interval reliability, and statistical significance as confirmed by the Diebold–Mariano test. These findings suggest that hybrid modelling offers a practical and effective pathway for enhancing predictive maintenance, anomaly detection, and decision‑support capabilities in complex CPS environments. Future work will explore real‑time deployment, integration with edge computing platforms, and the use of more advanced learning architectures to further strengthen model adaptability and performance.