Control Problems Research Articles

A new method for solving fractional optimal control problems

In this article, we present a novel approach that utilizes the operational matrix method to solve fractional optimal control problems. We address the challenge of missing initial conditions for the co-state function by incorporating the linear shooting method. Our objective is to provide a method that not only enhances computational efficiency and reduces cost but also improves accuracy and usability. With our new approach, we can calculate the coefficients of the expansion solution without the need to solve an algebraic system. These coefficients can be generated explicitly or through an iterative process. We provide a proof of the uniform convergence of the approximate series of functions to the unique solution of the optimal control problem. To demonstrate the effectiveness of our proposed method, we solve several numerical examples and compare the results with those obtained by other researchers. Additionally, we extend the application of this new method to solve problems with multiple states, thereby expanding its scope to a wider range of problem domains. This allows us to address diverse scenarios using the same efficient and accurate approach. Through our research, we contribute to the development of a powerful and versatile method for solving fractional optimal control problems, offering significant advantages over existing techniques.

A Trajectory Planning Method for Capture Operation of Space Robotic Arm Based on Deep Reinforcement Learning

Abstract In order to deal with the complex dynamics and control problems involved in space debris removal, a trajectory planning technique for a spatial robotic arm based on twin delayed DDPG (TD3) in deep reinforcement learning is proposed, and it can accomplish an end-to-end control effect comparable to that of human hand gripping objects. The trajectory planning method for capturing space debris by a floating-base space robotic arm is realized using a space robotic arm task simulation platform built on MuJoCo and using trajectory planners, trajectory trackers, and joint and end-effector control strategies formulated with seven different weighted reward functions. This makes it easier to complete spacecraft in-orbit servicing and maintenance missions. The experiment results demonstrate that the capture strategy can maintain a capture success rate of more than 99%, and debris capture can be mostly finished in three stages when taking the stability of the floating base into consideration by continuously modifying the trajectory.

The implicit inversion method for calculating the forward dynamics input Jacobian

AbstractThis paper presents the implicit inversion method (IIM), a recursive method to evaluate the Jacobian of the forward dynamics w.r.t. the system inputs, using intermediate results obtained from an O(n) forward dynamics algorithm. The resulting coefficient matrix, called the inertia-weighted input matrix (IWIM), can be used to significantly improve the performance of solving optimal control problems that take into account system dynamics for only the current time step. As the relationship between inputs and accelerations appears fixed within a time step, this matrix can be evaluated in the initialization step of the optimization. This means that the forward dynamics only needs to be solved once at the initialization of the optimization, rather than having to solve the equations in every iteration of the optimization. The method presented in this paper especially targets a case where the forward dynamics are calculated using an O(n) method and takes advantage of variables that are already known through the evaluation of that method. These quantities allow us to obtain the inertia-weighted input matrix without having to convert the system to its generalized coordinate form. Exploiting the shape of the resulting equation, it is even possible to avoid an explicit inversion of any matrices in the process. Finally, runtime comparisons between three different methods to calculate the IWIM are made for several examples.

An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems

An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems

A necessary conditions for optimal singular control of McKean-Vlasov stochastic differential equations driven by spatial parameters local martingale

In this paper, we focus on stochastic singular control problems involving McKean-Vlasov stochastic differential equations driven by a spatially parameterized continuous local martingale. The drift coefficient in these equations depends on the state of the solution process and its law. The control variable consists of two components: an absolutely continuous control and a singular one. Firstly, under Lipschitz conditions, we establish the existence and uniqueness of its strong solution. Next, we derive the necessary conditions for optimal singular control under the assumption that the control domain is convex. These optimality conditions differ from the classical ones in the sense that here the adjoint equation is a McKean-Vlasov backward stochastic differential equation driven by a continuous local martingale with spatial parameters. The proof of our result is based on the derivatives of the local martingale with respect to spatial parameters and the derivative of the drift coefficient with respect to the probability measure.

Tackling inventory losses of small accessories in engineering manufacturing: a case study of Euro Manufacturing

Learning outcomes After completion of the case study, the students will be able to understand ABC analysis and develop a systematic approach using PDCA, analyze processes, technology, employee training and supplier relationships when analyzing shrink and developing solutions, evaluate how technology improves production inventory control and visibility and recognize the importance of fostering a culture of employee accountability and ownership to minimize inventory loss and improve overall operational efficiency. Case overview/synopsis On June 2, 2023, sitting in his office in Karachi, Pakistan, Khan Aamir, the manager of store and inventory at Euro Manufacturing, found himself immersed in a cloud of confusion. The incessant loss of inventory items, particularly the nut bolts and small accessories, had become a perplexing challenge. To address these losses and provide a cycle count report to the director of supply chain, Aamir, manager of store and inventory, was given the responsibility to take action. He was looking for a comprehensive approach to address the current problems and prevent further losses in the future. This case study examines the various reasons for the losses, including theft, inadequate inventory control methods, human error and problems with suppliers. It highlights the importance of established procedures, the use of technology (such as barcode scanning, radio-frequency identification tagging and inventory management software) and the cultivation of a culture of accountability among employees. Complexity academic level This case study is developed for class discussion in the course of operations management or supply chain management. This case study is suitable for use with undergrad students. This case study can be taught in a module on operations management or supply chain management, as part of a broader course in business management or industrial engineering. Supplementary materials Teaching notes are available for educators only. Subject code CSS: 9: Operations and logistics.

Well-posedness of the Optimal Control Problem Related to Degenerate Chemo-attraction Models

This paper delves into the mathematical analysis of optimal control for a nonlinear degenerate chemotaxis model with volume-filling effects. The control is applied in a bilinear form specifically within the chemical equation. We establish the well-posedness (existence and uniqueness) of the weak solution for the direct problem using the Faedo Galerkin method (for existence), and the duality method (for uniqueness). Additionally, we demonstrate the existence of minimizers and establish first-order necessary conditions for the adjoint problem. The main novelty of this work concerns the degeneracy of the diffusive term and the presence of control over the concentration in our nonlinear degenerate chemotaxis model. Furthermore, the state, consisting of cell density and chemical concentration, remains in a weak setting, which is uncommon in the literature for solving optimal control problems involving chemotaxis models.

Optimality Conditions in Control Problems with Random State Constraints in Probabilistic or Almost Sure Form

In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. Although the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them with a model based on robust constraints with respect to the (compact) support of the given distribution. Funding: The authors thank the Deutsche Forschungsgemeinschaft [Projects B02 and B04 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”] for support. C. Geiersbach acknowledges support from the Deutsche Forschungsgemeinschaft [Germany’s Excellence Strategy–the Berlin Mathematics Research Center MATH+ Grant EXC-2046/1, Project 390685689]. R. Henrion acknowledges support from the Fondation Mathématique Jacques Hadamard [Program Gaspard Monge in Optimization and Operations Research, including support to this program by Electricité de France].

Optimal Power Model Predictive Control for Electrochemical Energy Storage Power Station

Aiming at the current power control problems of grid-side electrochemical energy storage power station in multiple scenarios, this paper proposes an optimal power model prediction control (MPC) strategy for electrochemical energy storage power station. This method is based on the power conversion system (PCS) grid-connected voltage and current to establish a power prediction model for energy storage power stations, achieving a one-step prediction of the power of the power station. The power prediction error is used as a power regulation feedback quantity to correct the reference power input. Considering the state of charge (SOC) constraint of the battery, partition the SOC into different states. Using SOC as the power regulation feedback, the power of the battery compartment can be adjusted according to the range of the battery SOC to prevent SOC from exceeding the limit value, simultaneously calculating the power loss of the energy storage power station to improve the energy efficiency. The objective function is to minimize the power deviation and power loss of the power station. By solving the objective function, the optimal switching voltage vector of the converter output is achieved to achieve optimal power control of the energy storage power station. The simulation results in various application scenarios of the energy storage power station show that the proposed control strategy enables the power of the storage station to quickly and accurately track the demand of grid scheduling, achieving the optimal power control of the electrochemical energy storage power station.

Existence of Optimal Pairs for Optimal Control Problems with States Constrained to Riemannian Manifolds

Existence of Optimal Pairs for Optimal Control Problems with States Constrained to Riemannian Manifolds

Remarks on control and inverse problems for PDEs

Remarks on control and inverse problems for PDEs

Optimal Control Problems of a Class of Nonlinear Degenerate Parabolic Equations

Optimal Control Problems of a Class of Nonlinear Degenerate Parabolic Equations

Gradient Flows for Regularized Stochastic Control Problems

Gradient Flows for Regularized Stochastic Control Problems

New Existence Results for Evolutionary Quasi-Hemi-Variational Inequalities with Semi-monotone Maps and Applications to Optimal Control Problems

New Existence Results for Evolutionary Quasi-Hemi-Variational Inequalities with Semi-monotone Maps and Applications to Optimal Control Problems

Fractional Legendre wavelet approach resolving multi-scale optimal control problems involving Caputo-Fabrizio derivative

Fractional Legendre wavelet approach resolving multi-scale optimal control problems involving Caputo-Fabrizio derivative

Prescribed-time tracking control of Mars entry vehicle with output constraints

This paper studies the prescribed-time output-constrained tracking control problems for Mars entry vehicle systems. By using the properties of the time-varying high-gain function and output-constrained conditions, time-varying barrier Lyapunov functions are constructed. Based on such Lyapunov functions, time-varying adaptive controllers are designed to solve the considered problem. It is proved that the proposed controllers can drive the errors converge to zero in a prescribed time under unknown disturbances, and the control signals are bounded. Both the symmetric and asymmetric cases are considered. Finally, the effectiveness of the proposed methods is illustrated by several numerical simulations.

Asymptotic method for solving the problem of transition process optimization in a three-tempo singularly perturbed system

The problem of constructing a transition process with minimal energy costs for a linear singularly perturbed system containing three groups of variables with significantly different rates of change is considered. Asymptotic approximations to solving this problem are constructed in the form of an open-loop and feedback controls. The main advantage of the proposed computational procedures is that the original problem is split into three unperturbed optimal control problems of lower dimension.

State-feedback control for full-state constrained nonlinear systems in a prescribed time

This study focuses on the unresolved control problems pertaining to full-state constrained nonlinear systems within a prescribed time. Initially, two Lyapunov theorems are established using a time-varying scaling function. These theorems facilitate the achievement of prescribed-time stability while ensuring that constraints are not violated. Building upon these theorems and incorporating completely known nonlinear terms, a state-feedback controller is developed and analyzed by employing an asymmetric barrier Lyapunov function (BLF). The analysis demonstrates that the resulting controller enables the tracking error to converge to the origin within a prescribed time while subsequently maintaining it at the origin without violating any constraints. Additionally, as an extension of the proposed prescribed-time control approach, the adaptive control problem for parameterized constrained nonlinear systems is addressed. The designed adaptive controller ensures boundedness of all signals, maintains the states within the specified constraints, and achieves convergence of the tracking error to the origin within an arbitrarily prescribed time. Finally, simulation examples are presented to validate the effectiveness of the proposed control scheme.

Consensus control and vibration suppression for multiple flexible nonlinear Timoshenko manipulators under undirected communication topology

In this paper, consensus control and vibration suppression problems are considered for the flexible nonlinear Timoshenko manipulator multi-agent system. The multi-agent system comprises multiple identical flexible Timoshenko manipulators, which can realize cooperation utilizing local information exchange between various agents. The bending deformation and shear deformation generated by the practical robotic manipulators during motion both affect the control effect, so the flexible Timoshenko manipulator is used to describe the dynamic characteristics. The flexible Timoshenko manipulator is a typical distributed parameter system, the deformation states are not only related to time but also to position, and the partial differential equations (PDEs) models are studied in this paper. Based on the PDEs models, the consensus control scheme is developed under the undirected communication topology structure, which can realize joint angle consensus tracking and vibration suppression of all the flexible Timoshenko manipulator agents simultaneously. Finally, the control effect is verified by the numerical simulations.

Machine learning & conventional approaches to process control & optimization: Industrial applications & perspectives

Technologies based on Artificial Intelligence (AI) and Machine Learning (ML) concepts are advancing at a rapid pace. The new paradigms are challenging the status-quo of mature automation and control technologies in industry. Autonomous operation is a frequently stated goal of AI evangelists and technology providers. This white paper gives an overview of the current state of art of advanced process control and optimization technologies. It also provides a brief summary of the AI and ML-based approaches that address the closed-loop control and optimization space. Some results from industrial implementations are shared for both conventional and AI/ML-based approaches. Experience from four industrial applications is shared, covering rigorous model-based, machine learning and hybrid approaches to real-time optimization and control problems. The applications range from unit-based control & optimization to refinery & network wide optimization. A set of high-level requirements that need to be satisfied regardless of the underlying technology for closed-loop autonomous operations is reviewed. The article concludes with some future directions and perspectives highlighting areas where the emerging technologies may have significant impact in industry.