Optimal Control Problem Research Articles

Fractional logarithmic Chebyshev cardinal functions: Application for Caputo–Hadamard fractional optimal control problems

This study investigates a specific type of fractional optimization problems, subject to a dynamical system including the Caputo–Hadamard fractional differentiation. In this way, a novel class of basis functions known as the fractional logarithmic Chebyshev cardinal functions is introduced. An operational matrix regarding the Hadamard fractional integral of these fractional functions is derived and employed to develop an efficient numerical method for the provided problem. The established algorithm converts the solution of the primary fractional optimization problem into the solution of an algebraic system of equations by representing the state and control variables via the introduced fractional functions. Two test problems are considered to confirm the effectiveness of the suggested method. The derived outcomes of solving these examples highlight the satisfactory accuracy and efficiency of the designed method.

Pressure-Driven Batch Distillation Optimal Control for Acetone-Methanol Separation

Over the last fifteen years, immense progress has been made in the research of pressure-swing batch distillation. The challenge lies in the fact that certain pressure-sensitive azeotropic mixtures cannot be separated in a regular open batch mode with an acceptable outcome. Throughout most of this text, findings contradict previously established facts. The separation of acetone-methanol by pressure-swing batch distillation in a mixed double system, consisting of a regular and inverted column, is the process under investigation. In this work, a comprehensive global solution to the optimal control problem is derived in the form of a sequential synthesis of controlled trajectories. A thorough analysis of the control variables, with and without parametric sensitivity of manipulated variables, on the optimal control pattern with variable reflux is presented. Most authors did not achieve significant results by simultaneously optimizing reflux and liquid division ratios or by using a "variable pressure gap." This study challenges previous findings, emphasizing the importance of simultaneous optimization of all three factors mentioned. During this study, the optimal reflux strategy through cyclic operation was extended to optimize energy expenditure over a fixed time horizon. Moreover, the proposed scheme offers a high level of operational flexibility, allowing units to be operated independently or as a system. Additionally, the facility of connecting additional devices is highlighted by the significant difference in output temperatures and the potential for using evaporated fluids consecutively. Lastly, the research provides a solid basis for future investigations into internal and external heat integration, potentially leading to conclusions about expanding the network using well-known heat cycles.

How optimal is the minimum-time manoeuvre of an artificial race driver?

Minimum-lap-time optimal control problems (MLT-OCPs) are a popular tool to assess the best lap time of a vehicle on a racetrack. However, MLT-OCPs with high-fidelity dynamic vehicle models are computationally expensive, which limits them to offline use. When using autonomous agents in place of an MLT-OCP for online trajectory planning and control, the question arises of how far the resulting manoeuvre is from the maximum performance. In this paper, we improve a recently proposed artificial race driver (ARD) for online trajectory planning and control, and we compare it with a benchmark MLT-OCP. The novel challenge of our study is that ARD controls the same high-fidelity vehicle model used by the benchmark MLT-OCP, which enables a direct comparison of ARD and MLT-OCP. Leveraging its physics-driven structure and a new formulation of the g-g-v performance constraint, ARD achieves lap times comparable to the offline MLT-OCP (few milliseconds difference). We analyse the different trajectories resulting from the ARD and MLT-OCP solutions, to understand how ARD minimises the effect of local execution errors in search of the minimum-lap-time. Finally, we show that ARD consistently maintains its performance when tested on unseen circuits, even with unmodelled changes in the vehicle's mass.

Schwarz decomposition for parallel minimum lap-time problems: evaluating against ADMM

The Minimum Lap Time Problem (MLTP) remains a significant area of research, particularly in the motorsport context. This form of Optimal Control Problem (OCP) aims to minimise lap times on a specific track with a given vehicle. Various complexities in both vehicle and track models are employed across the literature to address optimal trajectory planning. While previous works have tackled MLTP as a singular task using a serial approach, the increasing model complexity and horizon length demands the utilisation of parallelisation techniques. This paper introduces a novel application of the Overlapping Schwarz Decomposition algorithm to address the MLTP. The algorithm divides the problem into smaller sub-problems based on different sectors of a track, distributing them among multiple processors. We validate and compare the Schwarz approach against a serial approach and the Alternating Direction Method of Multipliers (ADMM) in solving MLTP with over 2.5 million variables. Despite the general efficiency improvement of parallelisation compared to the serial approach, the Schwarz algorithm demonstrates superior speed, accuracy and robustness compared to ADMM. As a result of our findings, it emerges as the preferred choice when large-scale MLTPs need to be solved.

Reinforcement learning intermittent optimal formation control for multi-agent systems with disturbances

Abstract This paper investigates disturbance-resistant intermittent event-triggered optimal formation control problems of second-order multi-agent systems by using the reinforcement learning method, which takes into account the influence of network damage including denial-of-service (DoS) and deception attacks, stochastic noises, and unknown external disturbances. Firstly, we propose a novel disturbance observer based on adaptive control to estimate unknown external disturbances under an event-triggered mechanism. Secondly, by use of estimation of disturbances, an innovative intermittent event-triggered optimal formation algorithm is given. By applying theories such as Lyapunov stability and stochastic stability, sufficient conditions are derived to guarantee that all agents achieve the desired formation in mean square sense. Additionally, in the model-free case, the optimal controller is solved using the least squares method, which is computationally less complex than some existing approaches. Finally, the theoretical results are effectively validated through simulation examples.

Homotopy trust-region method for phase-field approximations in perimeter-regularized binary optimal control

We consider optimal control problems that have binary-valued control input functions and a perimeter regularization. We develop and analyze a trust-region algorithm that solves a sequence of subproblems in which the regularization term and the binarity constraint are relaxed by a non-convex energy functional. We show how the parameter that controls the distinctiveness of the resulting phase field can be coupled to the trust-region radius updates and be driven to zero over the course of the iterations in order to obtain convergence to points that satisfy a first-order optimality condition of the limit problem under suitable regularity assumptions. Finally, we highlight and discuss the assumptions and restrictions of our approach and provide the first computational results for a motivating application in the field of control of acoustic waves in dissipative media.

Features of the information and control system of the car's hybrid power plants

Abstract. Problem. Until now, a number of features of IBS power plants of electric and hybrid cars remain unexplored. The applied IIUS analysis and synthesis methods do not pay enough attention to the multi-criteria nature of the arising optimization tasks. Management adaptation methods to changingexternal operating conditions are insufficiently effective. These circumstances do not allow to fully reveal the potential possibilities of IISD power plants of electric and hybrid cars. This determines the relevance of improvement and development of new methods of modeling and optimization of IIUS control of power plants of electric and hybrid cars based on modern theory of automatic control, vector optimization, neural network and neuro-fuzzy adaptive methodology. A methodology for the construction of IIUS invariant to various power plants of electric and hybrid cars has been developed for operational management of modes according to quality, energy and other criteria. Results. The mathematicalsupport of the "Recovery" mode differs from classical problems of optimal control in that it allows changing the parameters of the control power plant, its model type, limit values for control influences, and phase coordinates. Originality. The core of the IICS GSU software is a knowledge base, in which procedural knowledge is presented in the form of frames that implement the algorithmic support of IICS GSU. The knowledge base has a stratified, hierarchical structure that reflects an integrated graph of the synthesis technology of the ZOU solution. The content of the frames of the knowledge base ensures the search for a solution to the ZOU problem in the state space. Frames contain slots that hold not only specific data but also the names of procedures that process them according to a given algorithm. Some frames contain slots filled with product rules. During the creation of the knowledge base of the research prototype, work was carried out on the unification of frame names, which makes it possible to quickly build the knowledge bases of working prototypes. Knowledge frames are a set of classes developed in a visual programming environment. Practical value. The resulting optimal program is worked out using simulation modeling modules, which include direct modeling of systems in multiple states of operation, and if it satisfies the specified requirements, the resulting data is converted into a file containing optimal control influences, which is stored in the database.

Optimal Control Allocation for Distributed Electric Propulsion in a Series/Parallel Partial Hybrid Powertrain

Abstract The SUbsonic Single Aft eNgine (SUSAN) Electrofan is a NASA concept transport aircraft representative of technology anticipated for a 2040 entry-into-service date. The powertrain consists of a single thrust-producing geared turbofan engine with generators driving a series/parallel partial hybrid power/propulsion system. The architecture includes 16 underwing contrarotating fans, eight on each side. The distributed fans can be used by the flight control system to augment or replace the rudder function. This paper sets up the optimal control problem of setpoint determination for individual wingfans in the distributed propulsion system, accounting for electrical string efficiencies, saturations, and failures. The solution minimizes power consumption while maintaining thrust and torque on the airframe for maneuvering. Additionally, thrust that would have been lost due to temporary fan speed or power saturation is optimally redistributed to maintain overall desired thrust and torque on the aircraft. A simulation of a coordinated turn utilizing the distributed electric propulsion for yaw rate control in a multiple wingfan failure scenario demonstrates the robustness of the powertrain design to failures and helps define its limitations.

Multi-constrained predictive optimal control for spacecraft attitude stabilization and tracking with performance guarantees

This paper investigates a novel predictive control approach for spacecraft attitude stabilization and tracking problems with consideration of performance constraints, angular velocity limitation and control saturation. First, the prescribed performance control (PPC) structure and barrier Lyapunov function (BLF) are utilized to design the cost function of optimal control problem. Subsequently, a multi-constrained predictive optimal controller for spacecraft attitude stabilization and tracking with performance guarantees is devised via exploiting an explicit receding horizon optimization control strategy under actuator saturation. Compared with the existing methods, the major merit of the proposed one lies in the semi-analytic solving scheme of optimal control problems with high computing efficiency and simultaneous handling of multiple constraints without increasing computational burden. Finally, two illustrative examples are employed to validate the effectiveness of the proposed control method.

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

Bifurcation Analysis and Optimal Control of an SEIR Model with Limited Medical Resources and Media Impact on Heterogeneous Networks

The spread of epidemic diseases is often considered to be bound up with human behavior. Both treatment and awareness-raising are seen as important preventive measures during an outbreak. In this paper, an SEIR model on heterogeneous network containing saturated treatment function with media information variable is constructed. Through mathematical analysis, the basic reproduction number and equilibriums of the system are derived. The global stability of disease-free equilibrium is further proved based on LaSalle’s invariant principle. It is also found that with a gradual increase of the delayed treatment parameter, the system exhibits a transition from forward bifurcation to backward bifurcation with the variation of the basic reproduction number. Finally, an optimal control problem is formulated with both treatment rate and media information dissemination rate as control variables and solved using the Pontryagin’s maximum principle. All the above results are verified by numerical experiments.

Modeling the Effects of Temperature in Enzymatic Biodiesel Synthesis of Jatropha Oil: An Optimal Control Approach.

Biodiesel, an alternative to diesel, is produced through the enzymatic transesterification of vegetable oil or animal fats. Enzymatic transesterification of oil is gaining more importance, as this method possesses no such disadvantages that are associated with the chemical process. Temperature is the most important factor in enzymatic transesterification for biodiesel synthesis. In this study, a mathematical model is developed to understand the effects of temperature on the enzymatic transesterification of Jatropha curcas oil. Reaction rates are expressed as a function of temperature using an appropriate function, and the effects of temperature on the overall enzymatic system are investigated using the mathematical model. A suitable temperature is determined using the mathematical model, keeping the other reaction conditions unchanged that gives maximum yield. Furthermore, an optimal control problem (OCP) is formulated to identify the temperature control strategy that maximizes the biodiesel yield while minimizing production costs. Simulated outcomes obtained from the mathematical model are analogized with the experimental results to make them acceptable. The optimal profile of temperature is determined from the developed OCP, which maximizes the biodiesel yield.

Optimal quantized feedback control for linear quandratic Gaussian systems with input delay

AbstractThis paper is concerned with the optimal quantized feedback linear quadratic Gaussian (LQG) control problem for a discrete‐time stochastic system with input delay as well as the measurements to be quantized before transmitted to the controller. In this scenario, the system is presented with several choices of quantizers, along with the cost of using each quantizer. The objective is to jointly select the quantizers and synthesize the controller to maintain an optimal balance between control performance and quantization cost. It is shown that this problem can be decoupled into two optimization problems when the innovation signal is quantized instead of state: one for optimal controller synthesis and the other for optimal quantizer selection. More specifically, a necessary and sufficient condition is derived for the optimal control problem based on Pontryagin's maximum principle. On the other hand, the optimal quantizer selection policy is established by dealing with a certain Markov decision process (MDP).

Optimal Learning and Management of Threatened Species

Amid an unprecedented loss of biodiversity, a pressing issue is how to improve the efficiency of conservation with limited resources and information. Collecting data on species with a small population is costly and time consuming, and many high-stakes decisions need to be made based on limited data. We develop a partially observable Markov decision processes model with unknown parameters to jointly optimize the information collection and protection efforts for threatened species. The model takes into account uncertainties about the state, detectability, and dynamics of the species, and it adaptively adjusts the efforts of surveying and protection in real time. Although the standard formulation is intractable, we exploit the structure of ecological problems to identify a hybrid belief state in low dimensions, and we reformulate the stochastic dynamic program as a piecewise deterministic optimal control problem. This enables us to obtain structural insights into the optimal policy (some are in closed form) and find a near-optimal approximate policy with performance guarantee. In certain situations, areas where the species has never been found may be more likely to contain the species than areas where it has been previously found. We also conduct a case study on the conservation of the Hainan gibbon, the rarest primate species, in which we extend the model to optimize the spatiotemporal allocation of limited resources. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2019-05671]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01753 .

Emergency voltage control strategy for power system transient stability enhancement based on edge graph convolutional network reinforcement learning

Emergency control is essential for maintaining the stability of power systems, serving as a key defense mechanism against the destabilization and cascading failures triggered by faults. Under-voltage load shedding is a popular and effective approach for emergency control. However, with the increasing complexity and scale of power systems and the rise in uncertainty factors, traditional approaches struggle with computation speed, accuracy, and scalability issues. Deep reinforcement learning holds significant potential for the power system decision-making problems. However, existing deep reinforcement learning algorithms have limitations in effectively leveraging diverse operational features, which affects the reliability and efficiency of emergency control strategies. This paper presents an innovative approach for real-time emergency voltage control strategies for transient stability enhancement through the integration of edge-graph convolutional networks with reinforcement learning. This method transforms the traditional emergency control optimization problem into a sequential decision-making process. By utilizing the edge-graph convolutional neural network, it efficiently extracts critical information on the correlation between the power system operation status and node branch information, as well as the uncertainty factors involved. Moreover, the clipped double Q-learning, delayed policy update, and target policy smoothing are introduced to effectively solve the issues of overestimation and abnormal sensitivity to hyperparameters in the deep deterministic policy gradient algorithm. The effectiveness of the proposed method in emergency control decision-making is verified by the IEEE 39-bus system and the IEEE 118-bus system.

A numerical study of transformed mixed-integer optimal control problems

Time transformation is a ubiquitous tool in theoretical sciences, especially in physics. It can also be used to transform switched optimal control problems into control problems with a fixed switching order and purely continuous decisions. This approach is known either as enhanced time transformation, time-scaling, or switching time optimization (STO) for mixed-integer optimal control. The approach is well understood and used widely due to its many favorable properties. Recently, several extensions and algorithmic improvements have been proposed. We use an alternative formulation, the partial outer convexification (POC), to study convergence properties of (STO). We introduce the open-source software package ampl_mintoc (Sager et al., czeile/ampl_mintoc: Math programming c release, 2024, https://doi.org/10.5281/zenodo.12520490). It is based on AMPL, designed for the formulation of mixed-integer optimal control problems, and allows to use almost identical implementations for (STO) and (POC). We discuss and explain our main numerical result: (STO) is likely to result in more local minima for each discretization grid than (POC), but the number of local minima is asymptotically identical for both approaches.

Interval riccati equation-based and non-probabilistic dynamic reliability-constrained multi-objective optimal vibration control with multi-source uncertainties

Structural vibration control is used to ensure the safety of structures in service and has become an active research area in the past few years. In this study, uncertainties in structural dynamics are considered to propose an interval linear quadratic regulator (LQR) method based on an interval Riccati equation. Multi-objective optimization with non-probabilistic dynamic reliability as a constraint is performed to design an optimal uncertain vibration control system. To alleviate the difficulty of quantifying the probabilistic uncertainty and accurately characterize uncertainty information for small samples, multi-source uncertainties are regarded as interval parameters. The bounds on these uncertainties are determined, and the deterministic and uncertain parts of an interval state-space equation for vibration control are then formulated using an expanded-order matrix and the interval analysis method. Next, the interval perturbation method with a matrix series is used to develop a novel solution method for the linear inhomogeneous time-invariant equation in the interval state-space equation for the vibration control system and is used to accurately predict the interval bounds of the state responses. The conventional LQR method from modern control theory is used to propose a novel optimal control method based on a novel interval Riccati equation using the Lyapunov equation and interval uncertainty propagation. The proposed interval control method can be used to estimate the uncertain bounds of the controlled gain and cost function. A non-probabilistic dynamic reliability model is established to evaluate the reliability index of the system, which can overcome the limitations of the large computational cost of determining the probabilistic reliability. The deterministic controlled cost and uncertain states are considered as two optimization objectives with the proposed reliability constraint. The resulting constrained multi-objective optimal vibration control problem is designed and solved using c-DPEA. A flowchart is presented as a clear overview of the uncertain vibration control method, and the effectiveness of the proposed method is validated using two numerical examples. Different optimal control designs are obtained using different reliability constraints for quantitatively analyzing the safety of the control system.

Compactness results for a Dirichlet energy of nonlocal gradient with applications

AbstractWe prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem.

Malware containment with immediate response in IoT networks: An optimal control approach

The exponential growth of Internet of Things (IoT) devices has triggered a substantial increase in cyber-attacks targeting these systems. Recent statistics show a surge of over 100 percent in such attacks, underscoring the urgent need for robust cybersecurity measures. When a cyber-attack breaches an IoT network, it initiates the dissemination of malware across the network. However, to counteract this threat, an immediate installation of a new patch becomes imperative. The time frame for developing and deploying the patch can vary significantly, contingent upon the specifics of the cyber-attack. This paper aims to address the challenge of pre-emptively mitigating cyber-attacks prior to the installation of a new patch. The main novelties of our work include: (1) A well-designed node-level model known as Susceptible, Infected High, Infected Low, Recover First, and Recover Complete (SIHILRFRC). It categorizes the infected node states into infected high and infected low, according to the categorization of infection states for IoT devices, to accelerate containment strategies for malware propagation and improve mitigation of cyber-attacks targeting IoT networks by incorporating immediate response within a restricted environment. (2) Development of an optimal immediate response strategy (IRS) by modeling and analyzing the associated optimal control problem. This model aims to enhance the containment of malware propagation across IoT networks by swiftly responding to cyber threats. Finally, several numerical analyses were performed to fully illustrate the main findings. In addition, a dataset has been constructed for experimental purposes to simulate real-world scenarios within IoT networks, particularly in smart home environments.

Optimal control of Volterra integral equations of third kind using Krall–Laguerre Polynomials

A novel method has been introduced to dissolve optimal control problems in systems governed by third-kind Volterra integral equations. This method utilizes Krall–Laguerre Polynomials as a basis function for expanding functions. By transforming the optimal control problem governed by third-kind Volterra integral equations (OCVIE3k) into a nonlinear programming problem (NLP), the solution process is significantly simplified. This approach involves converting the original problem into a more manageable form, which can be solved using established optimization techniques. The effectiveness and reliability of this proposed method are evaluated by comparing its outcomes to exact solutions when available.