The partitioning problem is of central relevance for designing and implementing non-centralized Model Predictive Control (MPC) strategies for large-scale systems. These control approaches include decentralized MPC, distributed MPC, hierarchical MPC, and coalitional MPC. Partitioning a system for the application of non-centralized MPC consists of finding the best definition of the subsystems, and their allocation into groups for the definition of local controllers, to maximize the relevant performance indicators. The present survey proposes a novel systematization of the partitioning approaches in the literature in five main classes: optimization-based, algorithmic, community-detection-based, game-theoretic-oriented, and heuristic approaches. A unified graph-theoretical formalism, a mathematical re-formulation of the problem in terms of mixed-integer programming, the novel concepts of predictive partitioning and multi-topological representations, and a methodological formulation of quality metrics are developed to support the classification and further developments of the field. We analyze the different classes of partitioning techniques, and we present an overview of their strengths and limitations, which include a technical discussion about the different approaches. Representative case studies are discussed to illustrate the application of partitioning techniques for non-centralized MPC in various sectors, including power systems, water networks, wind farms, chemical processes, transportation systems, communication networks, industrial automation, smart buildings, and cyber–physical systems. An outlook of future challenges completes the survey.

This paper presents a deep learning framework for analyzing on-board vibration response signals in infrastructure health monitoring. The proposed WaveletInception–BiGRU network uses a Learnable Wavelet Packet Transform (LWPT) for early spectral feature extraction, followed by one-dimensional Inception-Residual Network (1D Inception-ResNet) modules for multi-scale, high-level feature learning. Bidirectional Gated Recurrent Unit (BiGRU) modules then integrate temporal dependencies and incorporate operational conditions, such as the measurement speed. This approach enables effective analysis of vibration signals recorded at varying speeds, eliminating the need for explicit signal preprocessing. The sequential estimation head further leverages bidirectional temporal information to produce an accurate, localized assessment of infrastructure health. Ultimately, the framework generates high-resolution health profiles spatially mapped to the physical layout of the infrastructure. Case studies involving track stiffness regression and transition zone classification using real-world measurements demonstrate that the proposed framework significantly outperforms state-of-the-art methods, underscoring its potential for accurate, localized, and automated on-board infrastructure health monitoring.

PieceWise Affine (PWA) approximations for nonlinear functions have been extensively used for tractable, computationally efficient control of nonlinear systems. However, reaching a desired approximation accuracy without prior information about the behavior of the nonlinear systems remains a challenge in the function approximation and control literature. As the name suggests, PWA approximation aims at approximating a nonlinear function or system by dividing the domain into multiple subregions where the nonlinear function or dynamics is approximated locally by an affine function also called local mode. Without prior knowledge of the form of the nonlinearity, the required number of modes, the locations of the subregions, and the local approximations need to be optimized simultaneously, which becomes highly complex for large-scale systems with multi-dimensional nonlinear functions. This paper introduces a novel approach for PWA approximation of multi-dimensional nonlinear systems, utilizing a hinging hyperplane formalism for cut-based partitioning of the domain. The complexity of the PWA approximation is iteratively increased until reaching the desired accuracy level. Further, the tractable cut definitions allow for different forms of subregions, as well as the ability to impose continuity constraints on the PWA approximation. The methodology is explained via multiple examples and its performance is compared to two existing approaches through case studies, showcasing its efficacy.

This work analyzes how the trade-off between the modeling error, the terminal value function error, and the prediction horizon affects the performance of a nominal receding-horizon linear quadratic (LQ) controller. By developing a novel perturbation result of the Riccati difference equation, a novel performance upper bound is obtained and suggests that for many cases, the prediction horizon can be either 1 or +∞ to improve the control performance, depending on the relative difference between the modeling error and the terminal value function error. The result also shows that when an infinite horizon is desired, a finite prediction horizon that is larger than the controllability index can be sufficient for achieving a near-optimal performance, revealing a close relation between the prediction horizon and controllability. The obtained suboptimality performance upper bound is applied to provide novel sample complexity and regret guarantees for nominal receding-horizon LQ controllers in a learning-based setting. We show that an adaptive prediction horizon that increases as a logarithmic function of time is beneficial for regret minimization.

Learning-based control with safety guarantees usually requires real-time safety certification and modifications of possibly unsafe learning-based policies. The control barrier function (CBF) method uses a safety filter (SF) containing a constrained optimization problem to produce safe policies. However, finding a valid CBF for a general nonlinear system requires a complex function parameterization, which in general makes the policy optimization problem difficult to solve in real time. For nonlinear systems with nonlinear state constraints, this paper proposes the novel concept of state-action CBFs (SACBFs), which do not only characterize the safety at each state but also evaluate the control inputs taken at each state. SACBFs, in contrast to CBFs, enable a flexible parameterization, resulting in a SF that involves a convex quadratic optimization problem, which significantly alleviates the online computational burden. We propose a learning-based approach to synthesize SACBFs. The effect of learning errors on the effectiveness of SACBFs is addressed by constraint tightening and introducing a new concept called contractive-set CBFs. This ensures formal safety guarantees for the learned CBFs and control policies. Simulation results on an inverted pendulum with elastic walls validate the proposed CBFs in terms of constraint satisfaction and CPU time.

Dispersion modeling is crucial for marine environmental modeling and management. However, operational applications require a practical balance between model accuracy and computational efficiency. To address this challenge, we develop and validate a generalized cell-based model (CBM) framework for contaminant dispersion. The framework enhances physical realism through a novel three-dimensional (3D) transport model and a formulation for chemical reactions. Additionally, a new discretization-based approach is proposed to robustly relate the CBM’s diffusion coefficient to its partial differential equation counterpart, improving performance in scenarios with sharp gradients of the concentration level. The proposed framework’s favorable trade-off between accuracy and efficiency is demonstrated in a comparative simulation study, where the 3D CBM reduces computation time from 14.72 s to 0.06 s compared to finite-element methods (FEM), with a relative Root Mean Square Error (RMSE) of 7.67%. To demonstrate its practical applicability, the proposed framework is validated using ocean current and nitrate concentration data from the Copernicus Marine Environment Monitoring Service. After identifying a key model parameter from the data, the model’s forward predictions accurately reproduce the observed nitrate concentration patterns, confirming its suitability for operational scenarios.

Temperature plays a critical role in performance and stability of anaerobic digestion processes, subject to frequent meteorological fluctuations. However, state-of-the-art modeling and process control approaches for anaerobic digestion often do not consider the temporal dynamics of the temperature, which can influence microbial communities, kinetics, and chemical equilibrium, and consequently, biogas production efficiency. Therefore, to account for anaerobic digesters operating under fluctuating meteorological conditions, the Anaerobic Digestion Model no. 1 (ADM1) is mechanistically extended in this paper to incorporate temporal changes into temperature-dependent parameters by defining inhibition functions for microbial activities using the cardinal temperature model, and accounting for the lag in microbial adaptation to temperature fluctuations using a time-lag adaptation function. Thereafter, given that temperature fluctuations are a significant disturbance, a control framework based on Model Predictive Control (MPC) is developed to regulate the feeding flow rate and to ensure stable production rates despite temperature disturbances without relying on direct temperature control. An adaptive MPC approach is formulated based on a linear input–output model, where the parameters of the linear model are updated online to capture the nonlinear dynamics of the process and frequent changes in the dynamics accurately. In addition, a fuzzy logic system is employed to assign a reference trajectory for the production rate based on the temperature and its rate of change. Integrating this fuzzy logic system with the MPC controller enhances the production rate on warm days and avoids the operational failure in production on cold days. Additionally, to enhance biogas production rates, the feasibility of utilizing a portion of the produced biogas for external heating purposes is also investigated. It is demonstrated that by utilizing the proposed MPC approach, the additional amount of feed for the digester to produce methane required for a self-consumption biogas-fueled heating system can be calculated according to the meteorological variations. This enhances the process performance and stability. Finally, a thermally optimized dome digester semi-buried in the ground, operating under climate conditions of The Netherlands is considered as a case study to validate the extended model in agreement with biological and physicochemical behaviors of real-world applications, and to demonstrate the effectiveness of the proposed control system in handling temperature changes and enhancing performance.

Control of piecewise affine (PWA) systems under complex constraints faces challenges in guaranteeing both safety and online computational efficiency. Learning-based methods can rapidly generate control signals with good performance, but rarely provide safety guarantees. A safety filter is a modular method to improve safety for any controller. When applied to PWA systems, a traditional safety filter usually need to solve a mixed-integer convex program, which reduces the computational benefit of learning-based controllers. We propose a novel optimization-free safety filter designed to handle state constraints that involve a combination of polyhedra and ellipsoids. The proposed safety filter only utilizes algebraic and min-max operations to determine safe control inputs. This offers a notable advantage compared with traditional safety filters by allowing for significantly more efficient computation of control signals. The proposed safety filter can be integrated into various function approximators, such as neural networks, enabling safe learning throughout the learning process. Simulation results on a bicycle model with PWA approximation validate the proposed method regarding constraint satisfaction, CPU time, and the preservation of sub-optimality.

Co-optimization of both vehicle speed and gear position via model predictive control (MPC) has been shown to offer benefits for fuel-efficient autonomous driving. However, optimizing both the vehicle’s continuous dynamics and discrete gear positions may be too computationally intensive for a real-time implementation. This work proposes a learning-based MPC scheme to address this issue. A policy is trained to select and fix the gear positions across the prediction horizon of the MPC controller, leaving a significantly simpler continuous optimization problem to be solved online. In simulation, the proposed approach is shown to have a significantly lower computation burden and a comparable performance, with respect to pure MPC-based co-optimization.

In the backdrop of an increasingly pressing need for effective urban and highway transportation systems, this work explores the synergy between model-based and learning-based strategies to enhance traffic flow management by use of an innovative approach to the problem of ramp metering control that embeds Reinforcement Learning (RL) techniques within the Model Predictive Control (MPC) framework. The control problem is formulated as an RL task by crafting a suitable stage cost function that is representative of the traffic conditions, variability in the control action, and violations of the constraint on the maximum number of vehicles in queue. An MPC-based RL approach, which leverages the MPC optimal problem as a function approximation for the RL algorithm, is proposed to learn to efficiently control an on-ramp and satisfy its constraints despite uncertainties in the system model and variable demands. Simulations are performed on a benchmark small-scale highway network to compare the proposed methodology against other state-of-the-art control approaches. Results show that, starting from an MPC controller that has an imprecise model and is poorly tuned, the proposed methodology is able to effectively learn to improve the control policy such that congestion in the network is reduced and constraints are satisfied, yielding an improved performance that is superior to the other controllers.

Uncertainty in the behavior of other traffic participants is a crucial factor in collision avoidance for automated driving; here, stochastic metrics could avoid overly conservative decisions. This article introduces a stochastic model predictive control (SMPC) planner for emergency collision avoidance in highway scenarios to proactively minimize collision risk while ensuring safety through chance constraints. To guarantee that the emergency trajectory can be attained, we incorporate nonlinear tire dynamics in the prediction model of the ego vehicle. Further, we exploit max-min-plus-scaling (MMPS) approximations of the nonlinearities to avoid conservatism, enforce proactive collision avoidance, and improve computational efficiency in terms of performance and speed. Consequently, our contributions include integrating a dynamic ego vehicle model into the SMPC planner, introducing the MMPS approximation for real-time implementation in emergency scenarios, and integrating SMPC with hybridized chance constraints and risk minimization. We evaluate our SMPC formulation in terms of proactivity and efficiency in various hazardous scenarios. Moreover, we demonstrate the effectiveness of our proposed approach by comparing it with a state-of-the-art SMPC planner and we validate that the generated trajectories can be attained using a high-fidelity vehicle model in IPG CarMaker.

The growing volume of available infrastructural monitoring data enables the development of powerful data-driven approaches to estimate infrastructure health conditions using direct measurements. This paper proposes a deep learning methodology to estimate infrastructure physical parameters, such as railway track stiffness, using drive-by vibration response signals. The proposed method employs a long short-term memory (LSTM) feature extractor accounting for temporal dependencies in the feature extraction phase, and bidirectional long short-term memory (BiLSTM) networks to leverage bidirectional temporal dependencies in both the forward and backward paths of the drive-by vibration response in the condition estimation phase. In addition, a framing approach is employed to enhance the resolution of the monitoring task to the beam level by segmenting the vibration signal into frames equal to the distance between individual beams, centering the frames over the beam nodes. The proposed LSTM-BiLSTM model offers a versatile tool for various bridge and railway infrastructure conditions monitoring using direct drive-by vibration response measurements. The results demonstrate the potential of incorporating temporal analysis in the feature extraction phase and emphasize the pivotal role of bidirectional temporal information in infrastructure health condition estimation. The proposed methodology can accurately and automatically estimate railway track stiffness and identify local stiffness reductions in the presence of noise using drive-by measurements. An illustrative case study of vehicle–track interaction simulation is used to demonstrate the performance of the proposed model, achieving maximum mean absolute percentage errors of 1.7% and 0.7% in estimating railpad and ballast stiffness, respectively.

In practice, achieving a balance between accuracy, stability, and computational efficiency in modeling contaminant dispersion in marine environments remains challenging due to complex physical dynamics and numerical constraints. To address these challenges, an enhanced cell-based model (CBM) is developed and applied to simulate pollutant transport in the ocean. The CBM discretizes the spatial domain into uniform cells, resulting in a naturally parallelizable structure, and characterizes the transport process by incorporating both water flow-driven convection and diffusion effects. Moreover, two approaches are proposed for estimating the diffusion coefficient, and their performance is compared to a first-order upwind scheme finite-difference method (FDM) solution. Finally, the CBM is comprehensively compared with both the FDM and the finite-element method (FEM) solvers under varying spatial and temporal resolutions. Simulation results show that the CBM is less affected by the Courant-Friedrichs-Lewy (CFL) conditions and demonstrates stable convergence where the FDM fails or requires stricter settings. In addition, the CBM offers a favorable trade-off between accuracy and computational efficiency under coarse configurations. These results indicate that the CBM provides a reliable foundation for dynamic modeling and integration with learning-based frameworks in marine environment simulations.

The European Economic Area Electricity Network Benchmark (EEA-ENB) is a multi-area power system representing the European network of transmission systems for electricity to facilitate the application of distributed control techniques. In the EEA-ENB, we consider the Load Frequency Control (LFC) problem in the presence of Renewable Energy Sources (RESs), and Energy Storage Systems (ESSs). RESs are known to cause instability in power networks due to their inertialess and intermittent characteristics, while ESSs are introduced as a resource to mitigate the problem. In the EEA-ENB, particular attention is dedicated to Distributed Model Predictive Control (DMPC), whose application is often limited to small and homogeneous test cases due to the lack of standardized large-scale scenarios for testing, and due to the large computation time required to obtain a centralized MPC action for performance comparison with DMPC strategies under consideration. The second problem is exacerbated when the scale of the system grows. To address these challenges and to provide a real-world-based and control-independent benchmark, the EEA-ENB has been developed. The benchmark includes a centralized MPC strategy providing performance and computation time metrics to compare distributed control within a repeatable and realistic simulation environment.

Learning-based control, in particularReinforcement Learning (RL) reinforcementReinforcement learning, and optimization-based control, in particular model predictive control, each have their advantages and disadvantages for online, real-timeOptimal control optimal controlOptimal control of systems with complex dynamicsDynamic. However, both approaches are highly complementary and therefore there is an increased interest in combining their advantages in an integrated approach. In this chapter, we provide an overview of recent results, challenges, and opportunities on an integrated learning-based and optimization-based control approach. We focus in particular on piecewise affine systems as they are an extension of linear systemsLinear systems that can model or approximate hybridHybrid or nonlinearNonlinearbehaviorBehavior and as they still allow for effective numerical solutionSolution approaches.

Model mismatch often presents significant challenges in model-based controller design. This paper investigates model predictive control (MPC) for uncertain linear systems with input constraints, where the uncertainty is characterized by a parametric mismatch between the true system and its estimated model. The main contributions of this work are twofold. First, a theoretical performance bound is derived using relaxed dynamic programming. This bound provides a novel insight into how the prediction horizon and modeling errors affect the suboptimality of the MPC controller to the oracle infinite-horizon optimal controller, which has complete knowledge of the true system. Second, sufficient conditions are established under which the nominal MPC controller, which relies solely on the estimated system model, can stabilize the true system despite model mismatch. Numerical simulations are presented to validate these theoretical results, demonstrating the practical applicability of the derived conditions and bounds. These findings offer practical guidelines for achieving desired modeling accuracy and selecting an appropriate prediction horizon in designing certainty-equivalence MPC controllers for uncertain linear systems.

Maintenance is a necessary to keep assets, in this case, a pavement system, in good condition. Spending too much on maintenance is not efficient, while not spending enough may cause the condition to drop below a desired level. Therefore, in this paper, a conceptual approach, based on systems and control theory, is developed to improve the efficiency of maintenance of a pavement system, compared to the currently used maintenance approach where often only fixed bounds of the condition determine whether or not a maintenance action is required.  A state-space description of the condition of the pavement system is chosen for predicting the future evolution of the health condition. This allows the use of a moving-horizon optimisation approach, to determine optimal anticipative maintenance actions. Furthermore, in this approach, the maintenance cost and the condition of the pavement system are optimised. This model-based approach can be applied in practice as well-documented data, from which degradation models can be constructed, are often available. In this paper, we also show how degradation models from the literature can be converted for use in the proposed condition-based maintenance approach. Note that because of the general character of the proposed maintenance optimisation approach, the degradation model and the chosen optimisation method that are used as illustration in this paper can easily be replaced by another one, depending on the needs of the user. A case study is performed, where a representative situation is considered using the developed approach and the maintenance approach currently used in practice. This case study shows how the approach works and what the cost reduction can be assuming that the models are accurate. The paper ends with a discussion and recommendations.

Online model predictive control (MPC) for piecewise affine (PWA) systems requires the online solution to an optimization problem that implicitly optimizes over the switching sequence of PWA regions, for which the computational burden can be prohibitive. Alternatively, the computation can be moved offline using explicit MPC; however, the online memory requirements and the offline computation can then become excessive. In this work we propose a solution in between online and explicit MPC, addressing the above issues by partially dividing the computation between online and offline. To solve the underlying MPC problem, a policy, learned offline, specifies the sequence of PWA regions that the dynamics must follow, thus reducing the complexity of the remaining optimization problem that solves over only the continuous states and control inputs. We provide a condition, verifiable during learning, that guarantees feasibility of the learned policy’s output, such that an optimal continuous control input can always be found online. Furthermore, a method for iteratively generating training data offline allows the feasible policy to be learned efficiently, reducing the offline computational burden. A numerical experiment demonstrates the effectiveness of the method compared to both online and explicit MPC.

Game-theoretic models are frequently used to analyse the effects of information availability and quality on supply chain decision-making. Although information asymmetry plays a vital role in shaping sustainable supply chains, a comprehensive review of these models within this context remains lacking. This study conducts a systematic literature review and performs an in-depth content-based analysis of 73 peer-reviewed journal articles, categorising them based on their assumptions regarding supply chain structure, information structure, and interaction among supply chain members. We find that researchers are extending traditional supply chain models to address emerging challenges and opportunities driven by sustainable practices under asymmetric information. However, the research remains in a preliminary phase, with most models relying on simplified settings – typically dyadic, single-product, and single-period frameworks – and focusing primarily on demand and cost information asymmetries. Multilateral information structures and non-contractual coordination mechanisms remain largely unexplored. Theoretical advancements have considerably outpaced empirical validation, revealing a critical gap in the integration of real-world practices. Our findings highlight the importance of information sharing and coordination mechanisms in achieving sustainability outcomes and improving supply chain performance. These insights enrich the theoretical discourse on information asymmetry in sustainable supply chains.

This paper presents a novel approach for distributed model predictive control (MPC) for piecewise affine (PWA) systems. Existing approaches rely on solving mixed-integer optimization problems, requiring significant computation power or time. We propose a distributed MPC scheme that requires solving only convex optimization problems. The key contribution is a novel method, based on the alternating direction method of multipliers, for solving the non-convex optimal control problem that arises due to the PWA dynamics. We present a distributed MPC scheme, leveraging this method, that explicitly accounts for the coupling between subsystems by reaching agreement on the values of coupled states. Stability and recursive feasibility are shown under additional assumptions on the underlying system. Two numerical examples are provided, in which the proposed controller is shown to significantly improve the CPU time and closed-loop performance over existing state-of-the-art approaches.