Pontryagin's Minimum Principle Research Articles

ANALYSIS AND OPTIMAL CONTROL OF TUBERCULOSIS DISEASE SPREAD MODEL WITH VACCINATION AND CASE FINDING CONTROL (CASE STUDY: SURABAYA CITY)

Tuberculosis is a contagious disease that infects humans. It is caused by the bacterium Mycobacterium tuberculosis (M.tb). It is the largest infectious disease in the world and has become a major global health problem. Therefore, efforts are being made to control the spread of tuberculosis disease through vaccination and case finding, with the aim of reducing the population of latently infected and actively infected individuals. This study discusses the mathematical model of tuberculosis disease spread, disease-free and endemic equilibrium points, and stability analysis around the equilibrium points. Then, using Pontryagin's minimum principle, the optimal control problem is solved numerically by the 4th-order Runge-Kutta method. Based on the analysis and simulation results, the system is asymptotically stable around the disease-free and endemic equilibrium points. Furthermore, optimal control in the form of vaccination of susceptible individuals is required to further suppress the rate of change of susceptible individuals into latent individuals, while control in the form of case finding on latently infected individuals is required until the 9th year to minimize the population size, while on actively infected individuals, it is required until the 8th year to minimize the population size. Providing optimal control resulted in a 100% increase in the susceptible population and a 100% reduction in the latent and infected populations.

Optimal Control of Transmission Dynamics of Meningitis Disease with Vaccination, Campaign, and Treatment Factors

Meningitis disease is an inflammation of the membranes that protect the brain and spinal cord. This disease can be fatal to life-threatening because symptoms can appear suddenly. Meningitis is also common in infants and young children and can cause complications if not treated promptly. To prevent infection of susceptible compartment and to speed recovery of infected compartments, vaccination, campaign, and treatment are needed to prevent the spread of the disease or at least reduce the number of carrier and infected individuals. Therefore, we propose a model of meningitis disease spread consisting of five compartments, namely susceptible, carrier, infected with and without symptoms, and recovered. Vaccination, campaign and treatment are included in the model mechanism as controls. This study aims to observe the effectiveness of vaccination, campaign, and treatment in inhibiting the spread of the disease and accelerating the recovery of infected individuals. The Pontryagin minimum principle is followed to derive the optimal control problem. Numerical simulation using Runge-Kutta of order four scheme with the forward-backward sweep approach is applied to visualize the curves of state and control variables. From the numerical simulations, it is found that vaccination, campaign, and treatment are reasonable to apply in order to bridle meningitis disease from the population.

Learning Fuel-Optimal Trajectories for Space Applications via Pontryagin Neural Networks

This paper demonstrates the utilization of Pontryagin Neural Networks (PoNNs) to acquire control strategies for achieving fuel-optimal trajectories. PoNNs, a subtype of Physics-Informed Neural Networks (PINNs), are tailored for solving optimal control problems through indirect methods. Specifically, PoNNs learn to solve the Two-Point Boundary Value Problem derived from the application of the Pontryagin Minimum Principle to the problem’s Hamiltonian. Within PoNNs, the Extreme Theory of Functional Connections (X-TFC) is leveraged to approximate states and costates using constrained expressions (CEs). These CEs comprise a free function, modeled by a shallow neural network trained via Extreme Learning Machine, and a functional component that consistently satisfies boundary conditions analytically. Addressing discontinuous control, a smoothing technique is employed, substituting the sign function with a hyperbolic tangent function and implementing a continuation procedure on the smoothing parameter. The proposed methodology is applied to scenarios involving fuel-optimal Earth−Mars interplanetary transfers and Mars landing trajectories. Remarkably, PoNNs exhibit convergence to solutions even with randomly initialized parameters, determining the number and timing of control switches without prior information. Additionally, an analytical approximation of the solution allows for optimal control computation at unencountered points during training. Comparative analysis reveals the efficacy of the proposed approach, which rivals state-of-the-art methods such as the shooting technique and the adaptive Gaussian quadrature collocation method.

A real-time energy management strategy for fuel cell vehicle based on Pontryagin’s minimum principle

This paper proposes a Pontryagin's minimum principle (PMP) energy management strategy (EMS) based on driving cycle recognition for fuel cell vehicle powertrains, aiming to minimize hydrogen consumption and fuel cell degradation. Firstly, the neural network-based driving cycle recognizer is optimized using the tuna swarm optimization (TSO) algorithm and trained under four typical driving cycles. Then, the optimal co-state variables for the four driving cycles are obtained by iteration. Finally, the co-state variables are dynamically updated based on real-time driving cycle recognition results. Comparative analysis demonstrates that the PMP-DCR effectively improves fuel cell lifetime and vehicle economy under short-distance driving cycles. Based on the combined driving cycle, the proposed PMP-DCR EMS exhibits similar economy performance to optimal dynamic programming (DP) EMS, reducing equivalent hydrogen consumption by 13.8% and 9.2%, and decreasing fuel cell degradation rates by 93% and 8.7% in comparison to the conventional power-following and PMP EMS, respectively.

Research on energy-saving control of agricultural hybrid tractors integrating working condition prediction.

To address the issues of tractors using too much fuel and not being energy efficient, a predictive control strategy based on Pontryagin's minimum principle integrating working condition prediction is proposed for agricultural hybrid tractors. The Dongfanghong 1804 tractor is being used for research. Firstly, the main parameters of the hybrid drive system are determined and modeled. Secondly, based on the adaptive cubic exponential forecasting method, the working condition information for a period of time in the future is predicted through historical working condition information. Furthermore, combining the predicted working conditions information, the goal is to minimize the total energy consumption cost of the entire machine. Motor power and diesel engine power are control variables. The battery state of charge is a state variable. Subsequently, a predictive control strategy based on Pontryagin's minimum principle integrating working condition prediction is proposed. Finally, the simulation test is carried out based on the MATLAB simulation platform. Research indicates: under plowing conditions, compared with the power following control strategy, the proposed predictive control strategy can effectively manage the performance of the diesel engine and motor, ensuring they operate at their most efficient level. The total energy consumption costs of the power following control and predictive control strategies are 37.17 China Yuan (CNY) and 33.67 CNY, respectively. The cost of energy used is decreased by 9. 42%, which helps make tractor field plowing more efficient and economical.

Research on Energy Distribution Strategy of Tandem Hybrid Tractor Based on the Pontryagin Minimum Principle

In order to study the energy distribution of the tandem hybrid tractor and achieve optimal fuel economy under the whole operating condition, an energy distribution strategy based on PMP (Pontryagin minimum principle) is proposed. The performance parameters of the relevant power components are obtained by constructing the test bench of the tandem hybrid tractor. At the same time, the mathematical model of energy distribution is established from the aspects of energy distribution objective function, state variable, and variable constraint. Based on the external characteristic test of the hub motor and the resistance analysis, the transport operation condition of the tractor is selected as the simulation target condition. The ADVISOR2002 and Simulink software are used to jointly simulate the three energy distribution strategies: the thermostat type, the power-following type, and the PMP type. The simulation results show that, compared with the other two, the fuel economy and battery power loss of the PMP-based energy distribution strategy are significantly improved. The fuel consumption per 100 km is decreased by 32.91% and 26.10%, respectively, which verifies the feasibility of the strategy. This study is of great significance for improving the production efficiency and reducing fuel consumption of hybrid tractors.

Model order reduction based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems

In this paper, two model order reduction methods based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems are studied. The spatial discrete scheme of the cost function subject to a parabolic equation is obtained by Galerkin approximation, and then the coupled ordinary differential equations of the optimal original state and adjoint state with initial value and final value conditions are obtained by Pontryagin's minimum principle. For this original system, we propose two kinds of model order reduction methods based on the differential recurrence formula and integral recurrence formula of Laguerre orthogonal polynomials, respectively, and they are totally different from these existing researches on model order reduction only for initial value problems. Furthermore, we prove the coefficient-matching properties of the outputs between the reduced system and the original system. Finally, two numerical examples are given to verify the feasibility of the proposed methods.

Energy Economization Using Connectivity-Based Eco-Routing and Driving for Fleet of Battery Electric Vehicles

Energy economizing using Connectivity-based eco-Routing and Driving (CeRD) is a key to conserve the resources of a fleet. This work takes maximum advantage by advising optimal energy consumption based route and velocity. Product delivery to several universities by group of Battery Electric Vehicles (BEVs), is considered. Problem contains certain constraints i.e., BEV load capacity, time window, resources restriction, speed bounds, battery charging/discharging limits and traffic. Simulation of Urban Mobility (SUMO) is used for the estimation of traffic and routes. Selection of route is based on energy consumption of traffic’s velocity. Once the traffic based energy consumptions are available, Dijkstra’s algorithm is then applied to select the route with least energy consumption. Clarke and Wright (C&W) algorithm is then applied to find the optimal sequence of visiting universities. Pontryagin’s Minimum Principle (PMP) is then applied to find the optimal velocity profiles for BEVs on the optimized routes. Results have shown 60-70% reduction in energy consumption along with improvement in charging rates as compared to the vehicles without CeRD assistance in different scenarios. Furthermore, results are validated for distance and time using online route planner, for energy efficiency using electric vehicles database and for energy consumptions using driver model based forward simulator.

A Predictive PMP Strategy for Plug-In Hybrid Electric Buses Considering Motor Thermal Characteristics and Loss Distribution

Energy management strategy (EMS) is the key to improving the fuel economy of the plug-in hybrid electric bus (PHEB). Some of the current researches mainly concentrate upon reducing fuel consumption, but do not fully consider the influence of thermal dynamics of the vehicle motor. In order to solve the shortage of this research, a predictive EMS considering electromagnetic thermal control is proposed in this paper. First, integrated with the actual working conditions and environment, the distribution of the loss of the main part of the motor under two extreme working conditions, namely, low-speed overload and high speed, is studied. Then the motor thermal model is built to study the maximum temperature variation rule in the actual operating condition. Finally, the motor thermal model is combined with the model prediction control (MPC) strategy based on Pontryagin’s minimum principle (PMP) strategy to ensure fuel economy and avoid motor overheating. The research results have guiding significance for the improvement and design of subsequent EMSs considering the influencing factors of motor temperature rise.

Bilevel Predictive Control for HEVs Integrating Energy and Cabin Thermal Comfort

This paper proposes an integrated energy and cabin thermal management strategy for a connected hybrid electric vehicle that utilizes a vehicle speed preview informed by traffic connectivity. The proposed strategy employs a bilevel model predictive control framework over long and short predictive horizons to improve fuel economy while ensuring the vehicle’s traction energy and cabin thermal demands are satisfied. The cabin thermal load planning is developed with sensitivity to the vehicle speed of an air conditioning (AC) system, and the proportion of traction energy supplied between the engine and battery is optimized using the Pontryagin minimum principle. Under standard and on-road driving cycles, the simulation results indicate that over 18% and 22% reduction in fuel consumption can be saved compared to a baseline strategy and an equivalent consumption minimization strategy, respectively. Specifically, the electricity consumption of the AC system is reduced by 13% via thermal load planning of the cabin.

DYNAMIC SYSTEM OF TUBERCULOSIS MODEL USING OPTIMAL CONTROL IN SEMARANG CITY INDONESIA

Tuberculosis is a disease which is very contagious among human. To prevent this from happening, Semarang city government has enacted vaccination for exposed individuals and treatment for the infected individuals. Vaccination and treatment are forms of control that will be applied to dynamic model systems of Tuberculosis. The present paper will describe epidemic model of Tuberculosis with control using Pontryagin Minimum Principle to find optimal solution of the control with fixed time and free end point. The optimal control will aim to reduce or minimize the number of infected populations. Numerical calculation is carried out with MATLAB software programming to illustrate and compare the graph of the dynamic model with and without optimal control. The results of dynamic modeling of Tuberculosis with control state that vaccination and treatment have succeeded in reducing the population of infected individuals.

Study on energy management strategy of fuel cell patrol vehicle hybrid power system

The slow dynamic response of fuel cells has limited the development of fuel cell vehicles to a certain extent, and reasonable energy management strategies are one of the critical issues that need to be resolved to promote the industrialization of fuel cell vehicle. Based on the complex operating conditions of fuel cell patrol vehicles, this paper establishes a model of the vehicle and the component, exploring the impact of energy management strategies on vehicle performance. The simulation results indicate that Fuzzy logic control (FLC) causes significant fluctuations in fuel cell output power under both light and heavy load conditions. Pontryagin minimum principle (PMP) and Adaptive Pontryagin minimum principle (APMP) maintain a constant lower output power near a fixed value and change slightly according to demand. Comprehensively analyzing the two operating conditions, PMP and APMP can stably maintain State of charge (SOC) around the reference value and converge toward the reference value, and the SOC of the FLC strategy is generally lower than the reference value. Under light and heavy load conditions, the average efficiency of the APMP fuel cell system reaches 50.44% and 49.58%, respectively, which is similar to the PMP strategy and is respectively improved by 0.67% and 0.32% compared to the FLC strategy.

Pontryagin's Minimum Principle-Guided RL for Minimum-Time Exploration of Spatiotemporal Fields.

This article studies the trajectory planning problem of an autonomous vehicle for exploring a spatiotemporal field subject to a constraint on cumulative information. Since the resulting problem depends on the signal strength distribution of the field, which is unknown in practice, we advocate the use of a model-free reinforcement learning (RL) method to find the solution. Given the vehicle's dynamical model, a critical (and open) question is how to judiciously merge the model-based optimality conditions into the model-free RL framework for improved efficiency and generalization, for which this work provides some positive results. Specifically, we discretize the continuous action space by leveraging analytic optimality conditions for the minimum-time optimization problem via Pontryagin's minimum principle (PMP). This allows us to develop a novel discrete PMP-based RL trajectory planning algorithm, which learns a planning policy faster than those based on a continuous action space. Simulation results: 1) validate the effectiveness of the PMP-based RL algorithm and 2) demonstrate its advantages, in terms of both learning efficiency and the vehicle's exploration time, over two baseline methods for continuous control inputs.

Design for six sigma-based energy management for a plug-in fuel cell electric bus considering the noise disturbance

This paper proposes an energy management strategy for plug-in fuel cell electric buses (PFCEBs) using a discontinuous control method based on Pontryagin's minimum principle (D-PMP) algorithm.

Optimal Control of Brushless Doubly Fed Wind Power Generator under Zero-Voltage Ride-Through

In the grid-connected operation dynamics of brushless doubly fed generators (BDFGs), a dip in the grid voltage is equivalent to suddenly adding a reverse voltage source at the parallel node. By deriving the expressions of the transient current of power winding (PW), control winding (CW), and rotor winding (RW) of a BDFG in the complex frequency domain under a natural state, it was concluded that the overshoot and oscillation time are affected by the CW voltage, the drop degree and phase of the grid voltage, and the rotor speed. Therefore, an optimal control strategy is proposed. A state model with the CW current as the state variable was constructed using the Pontryagin minimum principle. The finite-time integral value of the square of the electromagnetic torque was set as the objective function to achieve the minimum value that could suppress the overshoot and oscillation of the electromagnetic torque, and the optimal CW voltage command value was directly solved to accelerate the convergence of the BDFG’s physical quantities, thereby reducing the amplitude. Finally, the feasibility of the optimal control algorithm was verified using tests on an experimental platform.

Optimal tracking control of an Autonomous Underwater Vehicle: A PMP approach

This paper addresses the optimal tracking control problem of an Autonomous Underwater Vehicle (AUV) using Pontryagin’s Minimum Principle (PMP). The formulation uses a choice of Hamiltonian function using the desired control objective with AUV dynamics acting as dynamic constraints. We first develop PMP based on AUV kinematics and then extend it to the dynamics to arrive at optimal thrusts and moments. Necessary conditions for optimality are derived for both the models using PMP, which results in optimal trajectories that simultaneously minimize the tracking error as well as the control cost, thereby arriving at energy optimality. It was observed that the adjoint variables (costates) are indeed the momenta in the inertial and body-fixed frames. At the kinematic level, this forms a stable solution. The developed methodology is applied to both 2D and 3D AUV model with disturbances due to inputs and ocean currents. Numerical simulations are carried out with the derived control laws for a given trajectory tracking target. Quantitative evaluation of the performance and comparison of the controller is done using Mean Square Error(MSE) and Total Variation(TV) measures. The proposed control laws are found to achieve the desired control objectives.

Predictive cruise control for hybrid electric vehicles based on hierarchical convex optimization

Predictive Cruise Control (PCC) achieves automated driving with improved vehicle economy by incorporating road gradient information from high-precision maps. However, existing research on PCC has primarily centered around conventional gasoline vehicles. Hybrid Electric Vehicle (HEV) offer superior fuel efficiency compared to their traditional counterparts. To further enhance vehicle economy by integrating the PCC system with HEVs, this paper proposes a hierarchical control architecture for HEV-PCC based on convex optimization. Firstly, considering spatial slope information from the road ahead, this paper introduces a Time-Space Integration Model Predictive Control (TSI-MPC) based on the principles of kinetic energy. Secondly, the paper combines a current regression model with the Pontryagin Minimum Principle (PMP) to present a Regression Analysis - Pontryagin Minimum Principle (RA-PMP) approach for the direct analytical solution of the energy management problem. Finally, the strategy is validated through simulation analysis and Hardware-in-Loop (HIL) testing on various real road scenarios. The result indicate that, in comparison to the Equivalent Consumption Minimization Strategy (ECMS) with constant speed cruising (CC), this method achieves an average fuel economy improvement of 9.45%.

Integrated Optimal Energy Management and Sizing of Hybrid Battery/Flywheel Energy Storage for Electric Vehicles

This paper presents an integrated optimal Energy Management Strategy (EMS) and sizing of a high-speed Flywheel Energy Storage System (FESS) in a battery electric vehicle (EV). The methodology aims at extending the battery cycle life and drive range by relegating fast dynamics of the power demand to the FESS. For the EMS, the battery power and FESS energy are considered as weighted objectives of an optimization problem that are established using Pontryagin's Minimum Principle (PMP). In order to derive the optimal FESS size, the sizing algorithm is targeted to minimize the battery degradation level and increase the FESS energy interaction with the system. The performance of the proposed methodology is assessed using some driving cycles including ECE-15, UDDC, HWFET, and US06. Comparing the proposed battery/FESS to the battery-only topology, the battery SoH and life cycle are improved considerably for different driving cycles.

Competitive information propagation considering local-global prevalence on multi-layer interconnected networks

The popularity of online social networks (OSNs) promotes the co-propagation of multiple types of information. And there exist inevitably competitive interactions between these information, which will significantly affect the spreading trend of each information. Besides, the coupled topology of multi-layer interconnects exhibited in OSNs will also increase the research complexity of information propagation dynamics. To effectively address these challenges, we propose a novel competitive information propagation model on multi-layer interconnected networks, where the tendency of an individual to become a positive or negative spreader depends on the weighted consideration of local and global prevalence. Then the basic reproduction number is calculated via next-generation matrix method. And under the critical conditions of the basic reproduction number, the asymptotic stability of information-free and information-endemic equilibria is theoretically proven through Lyapunov stability theory. Besides, an optimal control problem involving two heterogeneous controls is formulated, aiming at achieving the best suppression performance of negative information with the minimum control cost. According to Cesari theorem and Pontryagin minimum principle, the existence and analytical formulation of optimal solutions are derived. Extensive numerical experiments are conducted to prove the correctness of our theoretical results, and evaluate the effectiveness of our proposed control strategies. This study can provide useful insights into the modeling and control of multiple information propagation considering multi-layer network topology and individual adaptive behavior.

Adaptive energy management strategy of plug-in hybrid electric bus

In this study, the online application of a new energy management strategy (EMS) is proposed for a power-split plug-in hybrid electric bus (PHEB) to improve its system's fuel economy and reduce power battery degradation. The new EMS employs Pontryagin's minimum principle (PMP) and the power battery state of charge (SOC) feedback method to adjust the covariates online. A model-free adaptive control (MFAC) EMS was designed and constructed according to the compact form dynamic linearization model to further obtain accurate SOC feedback; the time–domain cumulative as well as covariate upper and lower limit constraints were used to obtain a real-time MFAC-Ⅲ strategy for online applications. The average reduction in battery capacity loss on all ten routes was 8.19 %, resulting from the correlation between the MFAC-III EMS and MFAC EMS; evaluations conducted on both strategies indicated that the MFAC-III outperforms the MFAC strategy in hardware-in-the-loop.