Conditions For Optimal Control Research Articles

Studying the Optimal Frequency Control Condition for Electric Vehicle Fast Charging Stations as a Dynamic Load Using Reinforcement Learning Algorithms in Different Photovoltaic Penetration Levels

This study investigates the impact of renewable energy penetration on system stability and validates the performance of the (Proportional-Integral-Derivative) PID-(reinforcement learning) RL control technique. Three scenarios were examined: no photovoltaic (PV), 25% PV, and 50% PV, to evaluate the impact of PV penetration on system stability. The results demonstrate that while the absence of renewable energy yields a more stable frequency response, a higher PV penetration (50%) enhances stability in tie-line active power flow between interconnected systems. This shows that an increased PV penetration improves frequency balance and active power flow stability. Additionally, the study evaluates three control scenarios: no control input, PID-(Particle Swarm Optimization) PSO, and PID-RL, to validate the performance of the PID-RL control technique. The findings show that the EV system with PID-RL outperforms the other scenarios in terms of frequency response, tie-line active power response, and frequency difference response. The PID-RL controller significantly enhances the damping of the dominant oscillation mode and restores the stability within the first 4 s—after the disturbance in first second. This leads to an improved stability compared to the EV system with PID-PSO (within 21 s) and without any control input (oscillating more than 30 s). Overall, this research provides the improvement in terms of frequency response, tie-line active power response, and frequency difference response with high renewable energy penetration levels and the research validates the effectiveness of the PID-RL control technique in stabilizing the EV system. These findings can contribute to the development of strategies for integrating renewable energy sources and optimizing control systems, ensuring a more stable and sustainable power grid.

Evaluating optimal quality assurance and quality control conditions of activation measurements at the accelerator-based boron neutron capture therapy system employing a lithium target

Evaluating neutron output is important to ensure proper dose delivery for patients in boron neutron capture therapy (BNCT). It requires efficient quality assurance (QA) and quality control (QC) while maintaining measurement accuracy. This study investigated the optimal measurement conditions for QA/QC of activation measurements using a high-purity germanium (HP-Ge) detector in an accelerator-based boron neutron capture therapy (AB-BNCT) system employing a lithium target. The QA/QC uncertainty of the activation measurement was evaluated based on counts, reproducibility, and standard radiation source uncertainties. Measurements in a polymethyl methacrylate (PMMA) cylindrical phantom using aluminum-manganese (Al–Mn) foils and aluminum-gold (Al–Au) foils and measurements in a water phantom using gold wire with and without cadmium cover were performed to determine the optimal measurement conditions. The QA/QC uncertainties of the activation measurements were 4.5% for Au and 4.6% for Mn. The optimum irradiation proton charge and measurement time were determined to be 36 C and 900 s for measurements in a PMMA cylindrical phantom, 7.0 C and 900 s for gold wire measurements in a water phantom, and 54 C and 900 s at 0–2.2 cm depth and 3,600 s at deeper depths for gold wire measurements with cadmium cover. Our results serve as a reference for determining measurement conditions when performing QA/QC of activation measurements using HP-Ge detectors at an AB-BNCT employing a lithium target.

Optimality Conditions for Optimal Control of the Monodomain Model with Pointwise Control and State Constraints

Optimality Conditions for Optimal Control of the Monodomain Model with Pointwise Control and State Constraints

Second-order sufficient conditions in optimal control of evolution systems

Second-order sufficient conditions in optimal control of evolution systems

Optimal boundary control for a competitive population system with size structure and time delay in a polluted environment

In this paper, we consider the optimal boundary control problem of a two-species competitive system with time delay and size structure in a polluted environment. First, the well posedness of the system is studied by using the characteristic line method and the fixed point principle. Second, the necessary conditions for optimal boundary control are obtained by conjugate system and normal cone property. Finally, the existence and uniqueness of the optimal strategy are proved by Ekeland variational principle.

A two-strain avian–human influenza model with environmental transmission: Stability analysis and optimal control strategies

On the basis of the WHO legitimated fear that there will be an avian influenza virus strain capable of mutating once it reaches the human population and sustains human-to-human transmissions, we formulate an hypothetical mathematical model which accounts for the environmental transmission and mutation of an avian influenza virus having the ability to spill over into the human population and become a highly pathogenic strain. We compute the basic reproduction number of the model and use it to study the existence and stability of equilibrium points. We derive conditions for the global asymptotic stability of any of the three equilibrium. The model is extended to incorporate six relevant time-dependent controls, and use the Pontryagin’s maximum principle to derive the necessary conditions for optimal disease control. Finally, the optimal control problem is solved numerically to show the effect of each control parameter and their combination. The incremental cost-effectiveness ratios are calculated to investigate the cost-effectiveness of all possible combinations of the control strategies. This study suggests that quarantine infected humans might be the most cost-effective strategy to control avian influenza transmissions with the virus mutation.

Model‐free distributed optimal control for general discrete‐time linear systems using reinforcement learning

AbstractThis article proposes a novel data‐driven framework of distributed optimal consensus for discrete‐time linear multi‐agent systems under general digraphs. A fully distributed control protocol is proposed by using linear quadratic regulator approach, which is proved to be a sufficient and necessary condition for optimal control of multi‐agent systems through dynamic programming and minimum principle. Moreover, the control protocol can be constructed by using local information with the aid of the solution of the algebraic Riccati equation (ARE). Based on the Q‐learning method, a reinforcement learning framework is presented to find the solution of the ARE in a data‐driven way, in which we only need to collect information from an arbitrary follower to learn the feedback gain matrix. Thus, the multi‐agent system can achieve distributed optimal consensus when system dynamics and global information are completely unavailable. For output feedback cases, accurate state information estimation is established such that optimal consensus control is realized. Moreover, the data‐driven optimal consensus method designed in this article is applicable to general digraph that contains a directed spanning tree. Finally, numerical simulations verify the validity of the proposed optimal control protocols and data‐driven framework.

Time‐optimal feedback control of nonlocal Hilfer fractional state‐dependent delay inclusion with Clarke's subdifferential

In this article, we discuss the optimal feedback control of nonlocal Hilfer fractional state‐dependent delay inclusion of order with Clarke's subdifferential. Firstly, we establish the existence of mild solutions for this class of equations by employing the Krasnoselskii's fixed‐point theorem, without using Lipschitz conditions. Subsequently, we develop set of sufficient assumptions, utilizing Cesari property and Filippov theorem, to ensure the existence of feasible pairs for the feedback control systems. Eventually, using the Mazur theorem and Fatou's lemma, we derive the optimal state–control pair. The study also introduces and explores the problem of time‐optimal feedback control. In conclusion, an application is provided to enhance the theoretical findings.

Parametrized guaranteed cost method with prescribed performance via critic‐only ADP for uncertain nonlinear systems and disturbances

AbstractIn this paper, a novel parametrized guaranteed cost control (PGCC) method with prescribed performance control (PPC) is proposed for nonlinear systems with unknown bounded uncertainties and external disturbances described by Euler‐Lagrange (EL) equations. The proposed method is capable of ensuring a robust optimal performance in the presence of dynamics constraints without the specific information of nonlinear perturbation. As a consequence, the conditions in the form of disturbances and uncertainties are also relaxed. The augmented model is converted by transforming the uncertainties and disturbances in the original system to a mismatched term using asymmetric PPC. By designing and modifying the novel guaranteed cost function to account for the maximum value of the nonlinear perturbation, an infinite‐horizon PGCC problem is proposed by the unconstrained stationary optimal control problem. The relevant proposed linear parametrized Hamilton‐Jacobian‐Bellman (PHJB) equation is approximated to solve by a critic‐only neural network (NN) with efficient computationally. Uniformly ultimately bounded stability is guaranteed via a Lyapunov‐based stability analysis. Finally, numerical simulation results demonstrate the effectiveness of the proposed control scheme.

Second-order optimality conditions for the bilinear optimal control of a degenerate equation

The main purpose of this paper is the study of second-order optimality conditions for the bilinear control of a strongly degenerate parabolic equation. The equation is degenerate at the boundary of the spatial domain. The well-posedness of the state equation, as well as weak maximum principles are established. We prove some differentiability properties of the control-to-state operator and the existence of optimal solutions. Finally, we derive first- and second-order optimality conditions for the system.

Second-Order Sufficient Conditions in the Sparse Optimal Control of a Phase Field Tumor Growth Model with Logarithmic Potential

This paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn-Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term like the L1-norm in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before.

Optimality conditions for optimal control of non-well-posed p-Laplacian elliptic equations

Optimality conditions for optimal control of non-well-posed p-Laplacian elliptic equations

Existence of the optimal controls for a controlled elliptic system with an $ L^0 $ term in the cost functional

Optimal control problems with an $ L^0 $ term in the cost functional are considered. The $ L^0 $ term measures the size of the support of the control, and brings difficulty for studying the optimal control problems. By studying the existence and necessary conditions of optimal controls for a series of approximation problems and corresponding relaxed problems, the existence of optimal controls for a controlled elliptic system with an $ L^0 $-cost is obtained. Necessary conditions and the $ L^\infty $-regularity of optimal controls are showed in the paper as well.

Necessary and sufficient optimal control conditions for networked control systems with input delay and Markovian packet dropouts

AbstractThis article studies the optimal control for networked control systems (NCSs) subject to Markovian packet dropouts and input delay. In this scenario, Markovian packet dropouts occur in two channels, that is, from sensor to controller and from controller to actuator. Meanwhile, the input delay arises in the channel from controller to actuator. The contributions of this paper are as the following two aspects: Firstly, based on Pontryagin's maximum principle, the solution to the forward and backward stochastic difference equations (FBSDEs) is obtained. Secondly, in virtue of the obtained solution to the FBSDEs, sufficient and necessary optimal control conditions are derived and the explicit expression of the optimal controller is presented. The validity of the results is illustrated by numerical examples.

Model based optimal magnetic feedback control of multiple resistive wall modes with discrete coil and sensor arrays

A model-based optimal control method for multiple resistive wall mode (RWM) feedback stabilization has been developed and tested in plasma experiments at the EXTRAP T2R reversed field pinch (RFP) device. The controller is designed to target issues that arise in connection with RWM magnetic feedback stabilization systems based on discrete control coil and sensor arrays in tokamak and reversed field pinch devices. Multi-mode control capabilities in these systems is limited due to coupling of modes induced by the control system. The coupling originates from the generation of side-band control field harmonics and from aliasing of multiple harmonics in the sensor measurements. These couplings naturally leads to a multiple-input, multiple-output (MIMO) control problem. A model based state space controller has been designed based on a relatively simple physics model of the RWM plasma response. The physics model, which is applicable for the high aspect ratio RFP, is the basis for Fourier decoupling of the MIMO control problem into a set of single-input, multiple-output (SIMO) systems using the discrete Fourier transform (DFT). The linear, time-invariant physics model allows for the design of a state space model with states representing physical quantities; in this case the Fourier harmonics of the radial field at the resistive wall. Since the states cannot be directly measured, a Kalman filter is used for estimation of the states from the aliased sensor array measurements. A linear–quadratic (LQ) optimal state controller has been implemented. Design parameters, such as the LQ control cost function state weights and the Kalman filter input error covariances have been used to optimize the control operation in various ways. The controller has enhanced multi-mode control capabilities compared to earlier designs. For example it allows the prioritizing of suppression of one of the multiple magnetic field Fourier harmonics produced by a given control current DFT component. The controller has been tested in plasma experiments at EXTRAP T2R device, utilizing a newly installed extended sensor array, and the enhanced capabilities for multiple RWM feedback stabilization has been demonstrated.

Modelling and analysis of an HIV model with control strategies and cost-effectiveness

The Human Immunodeficiency Virus (HIV) remains a significant public health concern, despite global efforts to reduce new infections and increase access to antiretroviral treatment. While pre-exposure prophylaxis (PrEP) has shown promise in reducing new infections, its overall effectiveness is hindered by challenges in adherence. In this paper, a novel model was proposed. The basic properties of the model were established and the reproduction number for the model without control was computed. We conducted a sensitivity analysis to highlight the need for optimal control analysis and hence proposed an optimal control model that incorporates PrEP, condom use, and antiretroviral treatment to capture the dynamics of HIV transmission. The existence of the optimal control was mathematically proven and Pontryagin’s maximum principle was employed to establish the necessary optimal control conditions. Utilizing demographic and epidemiological parameters specific to South Africa, we conducted numerical simulations. The results demonstrate the community-level benefits of implementing a combination of control measures for HIV reduction. Additionally, a cost-effectiveness analysis was performed to determine the most effective and economically viable control strategies. We anticipate that the findings from this analysis will provide valuable insights for policymakers in designing effective HIV control programmes.

An aspect of the turnpike property. Long time horizon behavior

The turnpike phenomenon concerns the structure of the optimal control and the optimal state of dynamic optimal control problems for long time horizons. The focus is regularly placed on the study of the interior of the time interval. Classical turnpike results state how the solution of the dynamic optimal control problems approaches the solution of the corresponding static optimal control problem in the interior of the time interval.
 In this paper we look at a new aspect of the turnpike phenomenon. We show that for problems without explicit terminal condition, for large time horizons in the last part of the time interval the optimal state approaches a certain limit trajectory that is independent of the terminal time exponentially fast. For large time horizons also the optimal state in the initial part of the time interval approaches exponentially fast a limit state.

Experimental study on preparation of α-hemihydrous gypsum by modification of desulfurized gypsum

In the process of preparing α-hemihydrous gypsum from desulfurized gypsum under atmospheric pressure, the optimum conditions of crystallization transformation and modification of desulfurized gypsum were studied. The effects of the type and concentration of composite salts (CaCl2, KCl, MgCl2 mixed solution) on the crystallization and modification of desulfurized gypsum were studied. The results were further verified by scanning electron microscopy (SEM) and differential scanning calorimetry (DSC/TGA), so as to determine the ideal directional crystallization control index. The experimental results show that temperature, type and concentration of salt solution are the most sensitive factors in the crystallization process of desulfurized gypsum. The optimal control conditions for obtaining α-hemihydrous gypsum with good crystalline form are temperature 95℃~98℃, concentration of composite salt solution (CaCl2: MgCl2: KCl = 25:5:1) is 25%~30%, and slurry concentration is greater than 20%.

Optimal boundary control of a model thin-film fiber coating model

This paper considers the control of fluid on a solid vertical fiber, where the fiber radius is larger than the film thickness. The fluid dynamics is governed by a fourth-order partial differential equation (PDE) that models this flow regime. Fiber coating is affected by the Rayleigh–Plateau instability that leads to breakup into moving droplets. In this work, we show that control of the film profile can be achieved by dynamically altering the input flux to the fluid system that appears as a boundary condition of the PDE. We use the optimal control methodology to compute the control function. This method entails solving a minimization of a given cost function over a time horizon. We formally derive the optimal control conditions, and numerically verify that subject to the domain length constraint, the thin film equation can be controlled to generate a desired film profile with a single point of actuation. Specifically, we show that the system can be driven to both constant film profiles and traveling waves of certain speeds.

Validation of a closed-loop aim point management system at the Jülich solar tower

Due to modeling errors and disturbances caused by clouds, Solar Power Tower (SPT) plants consider high safety margins during operation leading to a significant loss of revenues. This conservatism contradicts the expansion of SPT plants despite their potential. In this work, an aim point management system is deployed at the open-volumetric receiver of the Jülich solar tower. It employs All Sky Imager (ASI)-based nowcasts as well as flux density and temperature measurements. While the temperature measurements provide feedback for an outer receiver control loop, the flux density measurements offer feedback for an inner heliostat field control loop of a cascade control. The receiver controller estimates flux density setpoints for the inner control loop. By comparing these flux density setpoints with measured flux densities, a static optimal control technique compensates for modeling errors and increases the intercept. This control requires a fast aim point optimization, the enhanced ant-colony optimization meta-heuristic. Furthermore, a feed forward controller compensates for disturbances due to clouds by scaling the heliostats’ reflected power used in the system model by DNI nowcasts. At the Jülich solar tower, the static optimal control compensates for changes in the allowable flux density in less than three control steps. The cascade control controls the subreceivers in a narrow temperature range of 15K, which has never been achieved before at this tower.