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.

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.

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 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.

Implementing model predictive control (MPC) in practice faces many subtle but prevalent problems, including modeling errors, solver errors, and actuator faults. In essence, the real control input applied to the system always deviates from the ideal one based on a perfect controller, resulting in an imperfect controller. In this letter, we provide a general analysis to quantify the suboptimality of MPC for Lipschitz-continuous nonlinear systems due to imperfect control inputs in terms of dynamic regret. Based on a general assumption about how the imperfect controller may improve over time, sublinear regret upper bounds are established for cases where the closed-loop system under the ideal controller is Lipschitz-contractive (i.e., its Lipschitz constant is smaller than one). In addition, we also discuss how the regret scales when the closed-loop system under the oracle controller is not Lipschitz-contractive. The results provide insights into designing suitable MPC strategies, especially for learning-based MPC.

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.

In this paper, we present a control-oriented benchmark of a network of dynamical systems representing an abstraction of the European Economic Area (EEA) electricity network. In the network each node represents a country of the EEA as an equivalent electrical area with specific generation and load features. The benchmark has been developed to provide the research community with a tool to assess non-centralized control strategies over a standardized case study. The Load Frequency Control (LFC) problem in the presence of renewable energy sources is considered, where the objective is to maintain a nominal operating frequency of the electricity network despite the presence of variations in the load request, and renewable energy production. A hybrid implementation of Energy Storage Systems (ESSs) with different operating modes is considered in the network to support energy generation. We test the features of the system through control simulations with centralized Model Predictive Control (MPC), and a Distributed MPC (DMPC) based on the Alternating Direction Method of Multipliers (ADMM). The benchmark is provided together with a long-term access repository containing both the data, and the scripts to access and process the data.

In this chapter, we explore model predictive control of fuel cell electric vehicles (FCEVs), a type of vehicle that utilizes the chemical energy of hydrogen to generate electricity for their power train. Since vehicles are typically utilized for mobility purposes only for a fraction of the time, the energy stored in the onboard hydrogen tanks of these vehicles also can be used to supply power when they are parked. Our focus is on examining how to control the operation of FCEVs, ensuring the fulfilment of constraints imposed by both electrical and transportation networks while minimizing operational costs.

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.

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.

The partitioning problem is a key problem for distributed control techniques. The problem consists in the definition of the subnetworks of a dynamical system that can be considered as individual control agents in the distributed control approach. Despite its relevance and the different approaches proposed in the literature, no generalized technique to perform the partitioning of a network of dynamical systems is present yet. In this article, we introduce a general approach to partitioning for distributed control. This approach is composed by an algorithmic part selecting elementary subnetworks, and by an integer program, which aggregates the elementary components according to a global index. We empirically evaluated our approach on a distributed predictive control problem in the context of power systems, obtaining promising performances in terms of reduction of computation speed and resource cost, while retaining a good level of performance.

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.

In this paper, the disjunctive and conjunctive lattice piecewise affine (PWA) approximations of explicit linear model predictive control (MPC) are proposed. Training data consisting of states and corresponding affine control laws are generated in a control invariant set, and redundant sample points are removed to simplify the construction of lattice PWA approximations. Resampling is proposed to guarantee the equivalence of lattice PWA approximations and optimal MPC control law at the sample points. Under certain conditions, the disjunctive lattice PWA approximation constitutes a lower bound, while the conjunctive version formulates an upper bound of the original optimal control law. The equivalence of the two lattice PWA approximations then guarantees error-free approximations in the domain of interest, which is tested through a statistical guarantee. The performance of the proposed approximation strategy is tested through two simulation examples, and the results show that error-free lattice PWA approximations can be obtained with low offline complexity and small storage requirements. Besides, the online complexity is less compared with the state-of-the-art method.

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.

Infinite-horizon optimal control of constrained piecewise affine (PWA) systems has been approximately addressed by hybrid model predictive control (MPC), which, however, has computational limitations, both in offline design and online implementation. In this article, we consider an alternative approach based on approximate dynamic programming (ADP), an important class of methods in reinforcement learning. We accommodate nonconvex union-of-polyhedra state constraints and linear input constraints into ADP by designing PWA penalty functions. PWA function approximation is used, which allows for a mixed-integer encoding to implement ADP. The main advantage of the proposed ADP method is its online computational efficiency. Particularly, we propose two control policies, which lead to solving a smaller-scale mixed-integer linear program than conventional hybrid MPC, or a single convex quadratic program, depending on whether the policy is implicitly determined online or explicitly computed offline. We characterize the stability and safety properties of the closed-loop systems, as well as the suboptimality of the proposed policies, by quantifying the approximation errors of value functions and policies. We also develop an offline mixed-integer-linear-programming-based method to certify the reliability of the proposed method. Simulation results on an inverted pendulum with elastic walls and on an adaptive cruise control problem validate the control performance in terms of constraint satisfaction and CPU time.

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.

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.

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.

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.

This paper proposes tackling safety-critical stochastic Reinforcement Learning (RL) tasks with a sample-based, model-based approach. At the core of the method lies a Model Predictive Control (MPC) scheme that acts as function approximation, providing a model-based predictive control policy. To ensure safety, a probabilistic Control Barrier Function (CBF) is integrated into the MPC controller. To approximate the effects of stochasticies in the optimal control formulation and to fulfil the probabilistic CBF condition, a sample-based approach with guarantees is employed. Furthermore, to counterbalance the additional computational burden due to sampling, a learnable terminal cost formulation is included in the MPC objective. An RL algorithm is deployed to learn both the terminal cost and the CBF constraint. Results from a numerical experiment on a constrained LTI problem corroborate the effectiveness of the proposed methodology in reducing computation time while preserving control performance and safety.