Optimal Control Function Research Articles

Advanced Model with a Trajectory Tracking Stabilisation System and Feasible Solution of the Optimal Control Problem

In this study, we consider the extended optimal control problem and search for a control function in the class of feasible functions for a real control object. Unlike the classical optimal control problem, the control function should depend on the state, not time. Therefore, the control synthesis problem for the initial-state domain should be solved, instead of the optimal control problem with one initial state. Alternatively, an optimal trajectory motion stabilisation system may be constructed. Both approaches—control and trajectory motion stabilisation system syntheses—cannot be applied to real-time control, as the task is too complex. The minimum threshold of quality criteria is searched for in the space of mathematical expression codes. Among other problems, the search space is difficult to define and the gradient is hard to determine. Therefore, the advanced control object model is used to obtain a feasible control function. The advanced model is firstly obtained before solving the optimal control problem and it already includes a trajectory motion stabilisation system; in particular, this stabilisation system is synthesised in advance at the control system design stage. When the optimal control problem appears, it is solved in real time in the classical statement, and a control function is searched for as a function of time. The advanced control object model also uses the reference model to generate the optimal trajectory. The search for the optimal control function is performed in real time and considers the synthesised stabilisation system of motion along a determined trajectory. Machine learning control via symbolic regression, namely, the network operator method, is used to directly solve the control synthesis problem. An example solution of the optimal control problem, with an advanced model moving in the environment with obstacles for a group of two mobile robots, is presented. The obtained solution is a control function for a reference model that generates a trajectory from a class of trajectories stabilised with the object’s control system.

CONTROL SYSTEMS SYNTHESIS FOR ROBOTS ON THE BASE OF MACHINE LEARNING BY SYMBOLIC REGRESSION

This paper presents a novel numerical method for solving the control system synthesis problem through the application of machine learning techniques, with a particular focus on symbolic regression. Symbolic regression is used to automate the development of control systems by constructing mathematical expressions that describe control functions based on system data. Unlike traditional methods, which often require manual programming and tuning, this approach leverages machine learning to discover optimal control solutions. The paper introduces a general framework for machine learning in control system design, with an emphasis on the use of evolutionary algorithms to optimize the generated control functions. The key contribution of this research lies in the development of an algorithm based on the principle of small variations in the baseline solution. This approach significantly enhances the efficiency of discovering optimal control functions by systematically exploring the solution space with minimal adjustments. The method allows for the automatic generation of control laws, reducing the need for manual coding, which is especially beneficial in the context of complex control systems, such as robotics. To demonstrate the applicability of the method, the research applies symbolic regression to the control synthesis of a mobile robot. The results of this case study show that symbolic regression can effectively automate the process of generating control functions, significantly reducing development time while improving accuracy. However, the paper also acknowledges certain limitations, including the computational demands required for symbolic regression and the challenges associated with real-time implementation in highly dynamic environments. These issues represent important areas for future research, where further optimization and hybrid approaches may enhance the method's practicality and scalability in real-world applications.

On the optimal control function diagrams in the problem of the movement of a platform with oscillators

We consider the problem of the time-optimal movement of a rigid body moving translationally along a horizontal straight line and carrying n-linear oscillators. The only control force is applied to the platform and is limited in magnitude, there is no friction. The system is transferred from a state of rest to a specified distance with vibration damping. The evolution of optimal control functions depending on the distance of movement is investigated. A general approach to constructing a visual diagram reflecting such evolution is proposed. To do this, a geometric interpretation of the necessary optimality conditions is used as properties of an auxiliary “control” curve in n-dimensional space. Numerical examples of constructing diagrams of optimal control functions for a platform with three oscillators are given.

Optimal control model for the spread of Streptococcus suis in human population

Streptococcus suis infection is a zoonotic disease that can spread from pigs to humans. The infection can cause permanent deafness in people. Although it is considered as a neglected zoonotic pathogen, the prevalence of this infection has not measurably decreased in recent years. In this research, we proposed mathematical models of S. suis infection in both human and pig populations. Disinfectant sterilization in swine farms and educational campaigns were considered for cooperative control strategies. The aim of this study was to derive an optimal control strategy by using optimal control techniques associated with an existing epidemic model. Equilibrium points and the basic reproduction numbers were analyzed to determine the effect of control parameters in the constant case. For the optimal control problem, we showed numerically that the optimal control functions effectively reduced the number of infectious individuals within a finite time. However, the suggested control functions had more impact on the human group than the swine group. The sensitivity analysis implied that the control of swine population should be focused on the surveillance of weaning piglets.

Efficient mini-batch stochastic gradient descent with Centroidal Voronoi Tessellation for PDE-constrained optimization under uncertainty

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is frequently utilized in engineering, biology and finance. Although stochastic gradient descent is a well-known stochastic optimization technique for solving the approximate optimal control problem, it often exhibits a slow convergence rate due to the inherent variance in gradient approximation. To address this issue, we propose a more efficient stochastic optimization algorithm based on Centroidal Voronoi Tessellation sampling. This strategy reduces the error in gradient estimation compared to the conventional mini-batch stochastic gradient descent method. Our approach involves dividing the whole snapshot set into several Voronoi cells with low variance and extracting samples with good uniformity from each region to construct an unbiased estimation of the full gradient. Our method can economically and stably approximate numerical optimal control functions even with a fixed step size. Numerical results demonstrate that the proposed method is a reliable algorithm with great potential for applications in the field of optimization.

#1692 Prevalence and risk factors for persistent hyperparathyroidism in long-term kidney transplant recipients

Abstract Background and Aims Secondary hyperparathyroidism, associated with fractures and cardiovascular mortality in end-stage kidney disease, often resolves after kidney transplantation (KT). However, limited knowledge exists regarding persistent hyperparathyroidism (persist-PTH) after long-term KT. We aimed to define the prevalence and risk factors of persist-PTH in KT recipients at ≥3 years post-KT. Method A historical cohort study was conducted in adult KT recipients at Srinagarind Hospital, Khon Kaen University, Thailand, from 2015 to 2020. Persist-PTH is defined as serum parathyroid hormone (PTH) exceeding twice the upper limit of normal (>130 pg/mL) after ≥3 years post-KT. Results A total of 281 patients were included, with a median age of 41.4 years (interquartile range (IQR), 32.7-49.1), and 117 patients (41.6%) were female. The median post-KT follow-up was 71 months (IQR, 54-89) with a glomerular filtration rate (GFR) of 61.8 mL/min/1.73 m² (IQR, 48.2-76.9). The respective median pre-KT and post-KT PTH were 403 pg/mL (IQR, 196.3-862) and 96.7 pg/mL (IQR, 67.3-166). Post-KT PTH levels correlated with pre-KT PTH levels (r²=0.18; P<0.001). Persist-PTH occurred in 100 patients, with a prevalence of 35.6% (95% confidence interval (CI), 30.2-41.4%). Multivariate analysis revealed significant associations between persist-PTH and a longer dialysis vintage (odds ratio (OR) 1.13 per 6-month increase; 95% CI, 1.06-1.20; P<0.001), lower GFR (OR, 0.76 per 10 mL/min/1.73 m² increase; 95% CI, 0.63-0.92; P=0.004), and higher pre-KT PTH (HR, 1.06 per 50 pg/mL increase; 95% CI, 1.02-1.10; P=0.001). Furthermore, persist-PTH with hypercalcemia (>10.2 mg/dL) occurred in 44 patients (15.7%; 95% CI, 11.8-20.4%), and with hypophosphatemia (<2.5 mg/dL) in 23 patients (8.2%; 95% CI, 5.5-12.0%). Conclusion Persist-PTH occurred in over one-third of long-term post-KT recipients, particularly in those with extended dialysis vintage, impaired graft function, and elevated pre-KT PTH levels. This underscores the need for optimal pre-KT PTH control and vigilant graft function monitoring to prevent persist-PTH.

On the optimal control for the wave equation in heterogeneous materials

This article deals with the wave equation in heterogeneous materials. Under this premise, we obtain a result for the following optimal interior control problem: reach optimally (in finite time T) any point on a given trajectory (called the reference trajectory) determined by the solution of the wave equation in a homogeneous medium. In this article, we explicitly build an optimal control function using the decomposition into eigenmodes (or eigenfunctions) of the associated Laplacian operator, emphasizing that our approach is more advantageous than the classical numerical techniques when we need to control only some eigenmodes of the wave equation while ignoring others. As a main result, we obtain a useful expression for the optimal control solution based on the calculation of effective wave coefficients. This expression is given as a linear combination of control terms, each of them controlling an eigenmode associated to the heterogeneous system.

The effect of governance structures on optimal control of two-patch epidemic models

Infectious diseases continue to pose a significant threat to the health of humans globally. While the spread of pathogens transcends geographical boundaries, the management of infectious diseases typically occurs within distinct spatial units, determined by geopolitical boundaries. The allocation of management resources within and across regions (the “governance structure”) can affect epidemiological outcomes considerably, and policy-makers are often confronted with a choice between applying control measures uniformly or differentially across regions. Here, we investigate the extent to which uniform and non-uniform governance structures affect the costs of an infectious disease outbreak in two-patch systems using an optimal control framework. A uniform policy implements control measures with the same time varying rate functions across both patches, while these measures are allowed to differ between the patches in a non-uniform policy. We compare results from two systems of differential equations representing transmission of cholera and Ebola, respectively, to understand the interplay between transmission mode, governance structure and the optimal control of outbreaks. In our case studies, the governance structure has a meaningful impact on the allocation of resources and burden of cases, although the difference in total costs is minimal. Understanding how governance structure affects both the optimal control functions and epidemiological outcomes is crucial for the effective management of infectious diseases going forward.

1498-P: Preemptive Insulin Supplementation for Islet Graft Protection, Optimal Blood Glucose Control, and Fetal Development during the Pregnancy in Insulin-Independent Patient after Islet Transplantation

Pregnancy increases metabolic demand for insulin, which may lead to the exhaustion of intraportally transplanted islets and postgestational hyperglycemia. Here we report a successful long-term preservation of islet graft function after two subsequent pregnancies while applying preemptive insulin supplementation. 29-year-old female with T1DM achieved long-term insulin independence with HbA1c <5.8% after a single intraportal islet transplant. Before conception, mycophenolate was replaced with azathioprine and tacrolimus target trough level was decreased to 4-6ng/ml. The patient became pregnant twice, 5 years and 7.5 years after her islet transplantation. During the first and second pregnancy patient preemptively supplemented with insulin up to 35-70u/day and up to 35 units daily, respectively to target fasting blood glucose below 90mg/ml. Postpartum the patient was successfully weaned off insulin maintaining optimal blood glucose control with HbA1c below 5.7%. Islet graft function before and after the pregnancy remained optimal as reflected by BETA-2 above 17 and comparable c-peptide secretion and peak blood glucose levels below 130mg/ml during the 8.5-year follow-up in Mixed Meal Tolerance Test. Both newborns were premature and delivered at 34 weeks with emergent c-sections due to preeclampsia. Unfortunately, the first newborn died due to necrotizing enterocolitis. The second child has been developing appropriately up-to-date three months after her birth. Our report confirms that insulin supplementation during pregnancy in insulin-independent patients after islet transplantation was safe and effective in maintaining optimal blood glucose control and islet graft function in the long term. It also underscores the high-risk nature of pregnancy in islet transplant recipients and the need for close clinical monitoring. Disclosure S.Gondek: None. L.Wang: None. M.Tibudan: None. R.Barth: None. J.Fung: None. P.Witkowski: Advisory Panel; Vertex Pharmaceuticals Incorporated, Novartis. M.Ogledzinski: None. W.Lin: None. K.Milejczyk: None. B.Juengel: None. L.Potter: None. P.J.Bachul: None. L.Basto: None. L.Perea: None.

Application of Deep Reinforcement Learning in Quantum Control

Machine learning technology based on artificial neural network has been successfully applied to solve many scientific problems. One of the most interesting areas of machine learning is reinforcement learning, which has natural applicability to optimization problems in physics. In the quantum control task, it is necessary to find a set of optimal control functions to transfer a quantum system from the initial state to the target state with the highest fidelity possible, which is essentially an optimization task. In this paper, we use Deep Deterministic Policy Gradient algorithm (DDPG) to study the quantum control tasks. We use the algorithm to control the transfer of several quantum systems from one state to another. The results show that DDPG algorithm can find a control strategy to make the fidelity of the final state and the target state of the quantum system be maximum value 1. The results show the potential of DDPG in quantum control.

Modified ADRC Design of Permanent Magnet Synchronous Motor Based on Improved Memetic Algorithm

In this paper, a novel modified auto disturbance rejection control (ADRC) design of a permanent magnet synchronous motor based on the improved memetic algorithm (IMA) is proposed. Firstly, there is an obvious system ripple caused by the defect that the optimal control function used in traditional ADRC cannot be differentiable and smooth at the segment point; aiming at weakening the system ripple effectively, the proposed method constructs a novel differentiable and smooth optimal control function to modify the ADRC design. Furthermore, aiming at improving the integration parameters optimization effect effectively, a novel improved memetic algorithm is proposed for obtaining the optimal parameters of ADRC. Specifically, an IMA with high-quality balance based on an adaptive nonlinear decreasing strategy for the convergence factor, Gaussian mutation mechanism, improved learning mechanism with the high-quality balance between competitive and opposition-based learning (OBL) and an elite set maintenance mechanism based on fusion distance is proposed so that these strategies can improve the optimization precision by a large margin. Finally, the experiment results of the PMSM speed control practical cases show that the ADRC based on IMA has an apparent better optimization effect than that of fuzzy PI, traditional ADRC based on the genetic algorithm and an improved ADRC based on improved moth-flame optimization.

Optimal control by deep learning techniques and its applications on epidemic models

We represent the optimal control functions by neural networks and solve optimal control problems by deep learning techniques. Adjoint sensitivity analysis is applied to train the neural networks embedded in differential equations. This method can not only be applied in classic epidemic control problems, but also in epidemic forecasting, discovering unknown mechanisms, and the ideas behind can give new insights to traditional mathematical epidemiological problems.

Forward-Backward Sweep Method for the System of HJB-FP Equations in Memory-Limited Partially Observable Stochastic Control

Memory-limited partially observable stochastic control (ML-POSC) is the stochastic optimal control problem under incomplete information and memory limitation. To obtain the optimal control function of ML-POSC, a system of the forward Fokker-Planck (FP) equation and the backward Hamilton-Jacobi-Bellman (HJB) equation needs to be solved. In this work, we first show that the system of HJB-FP equations can be interpreted via Pontryagin's minimum principle on the probability density function space. Based on this interpretation, we then propose the forward-backward sweep method (FBSM) for ML-POSC. FBSM is one of the most basic algorithms for Pontryagin's minimum principle, which alternately computes the forward FP equation and the backward HJB equation in ML-POSC. Although the convergence of FBSM is generally not guaranteed in deterministic control and mean-field stochastic control, it is guaranteed in ML-POSC because the coupling of the HJB-FP equations is limited to the optimal control function in ML-POSC.

Optimal Boundary Control of a Distributed Heterogeneous Vibrating System with Given States at Intermediate Times

The problem of optimal boundary control of a distributed heterogeneous vibrating system governed by the one-dimensional wave equation with piecewise constant characteristics is considered. It is assumed that each homogeneous segment is traveled by a wave over the same time. The control is performed via displacements of both ends. The cost functional is specified on the whole time interval. A constructive approach is proposed for finding an optimal control function that transfers the vibrations from an initial state through multipoint intermediate states to a terminal state over a given time interval. The results are illustrated by an example.

Research on Servo Control Algorithm of Photoelectric Stabilized Platform Based on Sliding Mode Active Disturbance Rejection

Aiming at the problems of parameters to be modified and the indistinct physical meaning of the photoelectric stabilization platform control based on the principle of active disturbance rejection, a sliding mode active disturbance rejection control algorithm is proposed. The mathematical model of the control system of the two-axis optoelectronic stabilization platform is established. The external disturbance and various order states of the airborne optoelectronic stabilization platform are estimated in actual time by using the extended state observer, and the optimal control function is constructed by the sliding mode reaching law to complete the nonlinear reactive reactance. The optimization of the disturbance controller structure is carried out, and the algorithm is verified by Matlab simulation. The simulation results show that after the photoelectric stabilization platform adopts the sliding mode ADRC controller, it not only retains the advantages of strong nonlinear ADRC resistance to load disturbance, but also can effectively suppress the external disturbance and realize the high-precision position of the photoelectric stabilization platform control.

Compensation Algorithm for Voltage Dip at Transmission End of Distribution Network Considering Uncertainty of Wind Power

Due to the random interference of voltage fluctuation, the power consumption value is large when actually compensating voltage quality. In view of this problem, a compensation algorithm for sudden voltage drop at the transmission end of distribution network considering the uncertainty of wind power is proposed. Build the Internet of things monitoring architecture for the multi-parameter operation of the transmission end of the distribution network, design a multi-channel acquisition circuit based on this, initially obtain the multi-parameter operation of the transmission end of the distribution network, and build a parameter identification equation to identify the operating parameters of the transmission end of the distribution network. Based on the obtained multi parameters of distribution network equipment operation, a compensation strategy is established. Based on the bus of the regional power station, the power flow calculation method of the regional power grid is constructed, and the topological structure of the regional voltage control points is obtained. The points in the structure are used as the voltage quality evaluation points, and the optimal control function relationship is constructed. Combined with different voltage control results, the sudden drop compensation of the voltage operation state at the transmission end of the distribution network is finally completed. The experimental results show that the controllable voltage value of the compensation algorithm is higher, and the network loss rate is lower, which has a good compensation effect.

Rotating mirror servo system control based on modified sliding mode-active disturbance rejection controller

In order to improve the tracking accuracy and robustness of the rotating mirror servo system, a modified sliding mode-active disturbance rejection control (MSM-ADRC) strategy is proposed. Firstly, the structure and working principle of the rotating mirror servo system are analysed, and its mathematical model is established to prepare for the design of the controller. Then, a MSM-ADRC is proposed to reduce the influence of unknown disturbance and improve the tracking accuracy. Among them, the modified sliding mode extended state observation (MSM-ESO) is designed by replacing the traditional nonlinear function with the designed optimal control function, which enhances the observation accuracy of the system state quantity and total disturbance. Meanwhile, an improved approach law is proposed, and an improved sliding mode nonlinear error feedback control law (MSM-NLSEF) is designed based on this approach law, which improves the convergence speed and accuracy of the control law. In addition, the stability of the designed MSM-ESO and MSM-NLSEF is proved. Finally, the proposed control method is validated by simulation and experimental comparison with other state-of-the-art controllers. Results reveal that the proposed control method has satisfying tracking performance and strong disturbance rejection ability.

Near-optimal control of a stochastic partial differential equation SEIR epidemic model under economic constraints

The outbreak of a disease has negative effects on the economy, and economic activities affect the spread of a disease. In this paper, based on the relationship between economic activities and infectious diseases, we establish a SEIRG (susceptible-exposure-infective-recovered-economic) model. Applying the near-maximum condition of a Hamiltonian function, we provide sufficient and necessary conditions for the near optimal control problem of the SEIRG system. In addition, we obtain the optimal control function of the research system. Finally, we develop an algorithm for a near optimal control problem and use numerical simulations to illustrate the effects of vaccination and treatment.

Near-optimal control of dynamical systems with neural ordinary differential equations

Optimal control problems naturally arise in many scientific applications where one wishes to steer a dynamical system from an initial state x 0 to a desired target state in finite time T. Recent advances in deep learning and neural network–based optimization have contributed to the development of numerical methods that can help solve control problems involving high-dimensional dynamical systems. In particular, the framework of neural ordinary differential equations (neural ODEs) provides an efficient means to iteratively approximate continuous-time control functions associated with analytically intractable and computationally demanding control tasks. Although neural ODE controllers have shown great potential in solving complex control problems, the understanding of the effects of hyperparameters such as network structure and optimizers on learning performance is still very limited. Our work aims at addressing some of these knowledge gaps to conduct efficient hyperparameter optimization. To this end, we first analyze how truncated and non-truncated backpropagation through time affect both runtime performance and the ability of neural networks to learn optimal control functions. Using analytical and numerical methods, we then study the role of parameter initializations, optimizers, and neural-network architecture. Finally, we connect our results to the ability of neural ODE controllers to implicitly regularize control energy.

Minimum Superstability of Stochastic Ternary Antiderivations in Symmetric Matrix-Valued FB-Algebras and Symmetric Matrix-Valued FC-⋄-Algebras

Our main goal in this paper is to investigate stochastic ternary antiderivatives (STAD). First, we will introduce the random ternary antiderivative operator. Then, by introducing the aggregation function using special functions such as the Mittag-Leffler function (MLF), the Wright function (WF), the H-Fox function (HFF), the Gauss hypergeometric function (GHF), and the exponential function (EXP-F), we will select the optimal control function by performing the necessary calculations. Next, by considering the symmetric matrix-valued FB-algebra (SMV-FB-A) and the symmetric matrix-valued FC-⋄-algebra (SMV-FC-⋄-A), we check the superstability of the desired operator. After stating each result, the superstability of the minimum is obtained by applying the optimal control function.