Pontryagin's Minimum Principle Research Articles

Optimal control for fast acquisition of equilibrium voltage for Li-ion batteries

Motivated by the recent electrode health diagnosis method that requires selective open-circuit voltage data, this paper presents a novel approach for the fast acquisition of battery equilibrium voltage using bidirectional current pulses. The proposed method was found by solving a minimum-time optimal control problem. Unlike the conventional pulse-relaxation method which requires a long relaxation period, the use of charge and discharge pulses reduces the testing time by 69%. The given problem is studied numerically and analytically. Analysis based on Pontryagin's minimum principle (PMP) demonstrates that the solution is either bang–bang or bang–off–bang control, depending on the constraints. Furthermore, the optimal control problem is reformulated into a finite-dimensional switching-time optimization problem, making it computationally tractable for on-board applications. Finally, experimental validation is presented for a 4.5 Ah NMC/graphite cell showing a 73% time reduction to reach a new equilibrium voltage.

Flight management system for hydrogen-powered aircraft in cruise

The minimization of the Direct Operating Cost (DOC) for hydrogen-powered aircraft is formulated in this paper as an optimal control problem and is solved based on Pontryagin's minimum principle. As a consequence, the optimum cruise flight speed is determined assuming cruising at a constant altitude. The optimization criterion corresponds to the minimization of a functional representing the trade-off between the cost of hydrogen fuel and time-dependent costs, which are related by a parameter denoted by cost index. The value of this parameter is introduced by a pilot into the Flight Management System (FMS) of the aircraft. The HY4 aircraft model is used to obtain numerical results for the proposed methodology.

Comparison of Two Energy Management Strategies Considering Power System Durability for PEMFC-LIB Hybrid Logistics Vehicle

For commercial applications, the durability and economy of the fuel cell hybrid system have become obstacles to be overcome, which are not only affected by the performance of core materials and components, but also closely related to the energy management strategy (EMS). This paper takes the 7.9 t fuel cell logistics vehicle as the research object, and designed the EMS from two levels of qualitative and quantitative analysis, which are the composite fuzzy control strategy optimized by genetic algorithm and Pontryagin’s minimum principle (PMP) optimized by objective function, respectively. The cost function was constructed and used as the optimization objective to prolong the life of the power system as much as possible on the premise of ensuring the fuel economy. The results indicate that the optimized PMP showed a comprehensive optimal performance, the hydrogen consumption was 3.481 kg/100 km, and the cost was 13.042 $/h. The major contribution lies in that this paper presents a method to evaluate the effect of different strategies on vehicle performance including fuel economy and durability of the fuel cell and battery. The comparison between the two totally different strategies helps to find a better and effective solution to reduce the lifetime cost.

Fuel Minimization of a Hybrid Electric Racing Car by Quasi-Pontryagin's Minimum Principle

This paper improves the fuel efficiency of a student-made parallel hybrid electric racing car whose internal combustion engine (ICE) either operates with peak efficiency or is turned off. The control to the ICE thus becomes a binary problem. Owing to the very limited computation resource onboard, the energy management strategy (EMS) for this car must have small time and space complexities. A computationally efficient controller that combines the advantages of dynamic programming (DP) and Pontryagin's minimum principle (PMP) is developed to run on a low-cost microprocessor. DP is employed offline to calculate the optimal speed trajectory, which is used as the reference for the online PMP to determine the real-time ICE on/off status and the electric motor (EM) torques. The normal PMP derives the optimal costate trajectory through solving partial differential equations. The proposed quasi-PMP (Q-PMP) method finds the costate from the value function obtained by DP. The fuel efficiency and computational complexity of the proposed controller are compared against several state of the art methods through both model-in-the-loop (MIL) and processor-in-the-loop (PIL) simulations. The new method reaches similar fuel efficiency as the explicit DP, but requires less than 1% onboard flash memory. The performance of the Q-PMP controller is compared between binary-controlled and continuously controlled ICEs. It achieves roughly 12% higher fuel efficiency for the binary ICE with only approximately 1/3 CPU utilization.

A Component-Sizing Methodology for a Hybrid Electric Vehicle Using an Optimization Algorithm

Many leading companies in the automotive industry have been putting tremendous effort into developing new powertrains and technologies to make their products more energy efficient. Evaluating the fuel economy benefit of a new technology in specific powertrain systems is straightforward; and, in an early concept phase, obtaining a projection of energy efficiency benefits from new technologies is extremely useful. However, when carmakers consider new technology or powertrain configurations, they must deal with a trade-off problem involving factors such as energy efficiency and performance, because of the complexities of sizing a vehicle’s powertrain components, which directly affect its energy efficiency and dynamic performance. As powertrains of modern vehicles become more complicated, even more effort is required to design the size of each component. This study presents a component-sizing process based on the forward-looking vehicle simulator “Autonomie” and the optimization algorithm “POUNDERS”; the supervisory control strategy based on Pontryagin’s Minimum Principle (PMP) assures sufficient computational system efficiency. We tested the process by applying it to a single power-split hybrid electric vehicle to determine optimal values of gear ratios and each component size, where we defined the optimization problem as minimizing energy consumption when the vehicle’s dynamic performance is given as a performance constraint. The suggested sizing process will be helpful in determining optimal component sizes for vehicle powertrain to maximize fuel efficiency while dynamic performance is satisfied. Indeed, this process does not require the engineer’s intuition or rules based on heuristics required in the rule-based process.

Electric thruster mode-pruning strategies for trajectory-propulsion co-optimization

Application of idealized constant-specific-impulse, constant-thrust electric thruster performance models or curve-fitted polynomials is quite common for spacecraft trajectory design. However, incorporation of realistic performance models of multi-mode electric thrusters leads to notable challenges, and at the same time, offers unprecedented system-level optimization opportunities. In this paper, a framework is developed for co-optimization of 1) spacecraft trajectory, 2) operation modes of multi-mode propulsion systems, and 3) solar array size. The selection of the most optimal operation modes is in accordance to Pontryagin's minimum principle for solving a payload-mass-maximization problem. The novelty of the work lies in solving a mixed-integer trajectory optimization problem featuring user-defined constraints on the maximum number of operating modes used along the trajectory. Utility of the framework is demonstrated through optimization of a multi-year trajectory from Earth to comet 67P/Churyumov–Gerasimenko using an SPT-140 Hall thruster with 21 operating modes; the results are interesting and of significant practical utility.

Predictive Energy Management of Hybrid Electric Vehicles via Multi-Layer Control

This paper presents predictive energy management of hybrid electric vehicles (HEVs) via computationally efficient multi-layer control. First, we formulate an optimization problem by considering driveability and a penalty for using service brakes in the objective function to optimize gear, engine on/off, engine clutch state, and power-split decisions subject to constraints on the battery state of charge (SOC) and charge sustenance. Then, we split it into two control layers, including a supervisory control in a higher layer and a local power-split control in a lower layer. In the supervisory layer, a gear and powertrain mode manager (PM) is designed, and optimal gear, engine on/off and clutch states are obtained by using a combination of dynamic programming (DP) and Pontryagin's minimum principle (PMP). Moreover, a real-time iteration Secant method is proposed to calculate optimal battery costate such that the constraint on charge sustenance is satisfied. In the local controller layer, a linear quadratic tracking method (LQT) is used to optimally split power between the engine and the electric machine and keep battery SOC within its bounds.

Optimal Control of Mathematical Models on The Dynamics Spread of Drug Abuse

This article examines the optimal control of a mathematical model of the spread of drug abuse. This model consists of five population classes, namely susceptible to using drugs (S), light-grade drugs (A), heavy-grade drugs (H), medicated drugs (T), and Recovery from drugs (R). The system is solved using the Pontryagin minimum principle and numerically by the forward-backward sweep method. Numerical simulations of the optimal problem show that with the implementation of anti-drug campaigns and strengthening of self-psychology through counseling, the spread of drug abuse can be eradicated more quickly. The implementation of campaigns and strengthening of self-psychology through large amounts of counseling needs to be done from the beginning then the proportion can be reduced until a certain time does not need to be given anymore. The use of control in the form of strengthening efforts to self-psychology through counseling means that it needs to be done in a longer time to prevent the spread of drug abuse.

Control strategy of HIV/AIDS model with different stages of infection of subpopulation

We formulated mathematical model of HIV/AIDS with two different stages of infection subpopulation with Antiretroviral (ARV) treatment as control strategy. We applied optimal control theory to minimize HIV-infected subpopulation using the Pontryagins-Minimum Principle. Numerical solution was conducted by solving the optimally system using the sweep backward and forward method. The results showed that by giving ARV in the model could decrease the infected subpopulation significantly.

Kendali Optimal Penyebaran Penyakit Influenza H1N1 Tiga Strain dengan Vaksinasi dan Pengobatan

H1N1 influenza is a respiratory tract infection with common symptoms, namely fever, headache, coughing, and sore throat. H1N1 influenza has several strains known to infect humans, birds, and pigs. Some efforts to overcome the infection of the H1N1 influenza virus are vaccination and treatment. This Research discusses the optimal control in the form of vaccination and treatment applied to the model H1N1 Influenza disease spread. The method used to solve this optimal control problem is the Pontryagin Minimum Principle followed by searching numerical solutions using the Runge-Kutta Forward-Backward Sweep method. Numerical simulations were conducted to compare the spread of H1N1 Influenza before and after optimal control efforts were given. Based on the simulation result, it was shown that giving optimal control in the form of vaccination to susceptible individuals and treatment in individuals infected with three strains could reduce the number of individuals infected with H1N1 Influenza disease .

Pontryagin Neural Networks with Functional Interpolation for Optimal Intercept Problems

In this work, we introduce Pontryagin Neural Networks (PoNNs) and employ them to learn the optimal control actions for unconstrained and constrained optimal intercept problems. PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed for tackling optimal control problems via the Pontryagin Minimum Principle (PMP) application (e.g., indirect method). The PMP provides first-order necessary optimality conditions, which result in a Two-Point Boundary Value Problem (TPBVP). More precisely, PoNNs learn the optimal control actions from the unknown solutions of the arising TPBVP, modeling them with Neural Networks (NNs). The characteristic feature of PoNNs is the use of PINNs combined with a functional interpolation technique, named the Theory of Functional Connections (TFC), which forms the so-called PINN-TFC based frameworks. According to these frameworks, the unknown solutions are modeled via the TFC’s constrained expressions using NNs as free functions. The results show that PoNNs can be successfully applied to learn optimal controls for the class of optimal intercept problems considered in this paper.

Suboptimal Control and Targeted Constant Control for Semi-Random Epidemic Networks

Compared with traditional models, semi-random epidemic network models may be more reasonable to describe the real dynamics of many epidemics. In this paper, we first investigate the optimal control problem (OCP) of semi-random epidemic networks. By using the Pontryagin's minimum principle, we obtain the optimal control strategy aimed to minimize the total epidemic incidence and control cost. We then define a centrality index which can measure average control strength of the optimal control. Based on this index, the OCP is converted into a static OCP (SOCP), whose solution is utilized to design a nonidentical constant control (NCC). NCC is suboptimal as it is optimal on a subset of the whole control set, and is determined by only the network's clustering coefficient and initial condition. We finally propose an effective targeted constant quarantine control by using this centrality index. The results uncover the relationship between the optimal control and the network's topological structure, provide a convenient method to determine suboptimal control, and present a strategy for targeted constant control. This paper can help to design effective control strategies for more general epidemic networks in the real world.

The stability of marine ecological environment under the optimal control of switching forward system

In order to analyze the stability of marine ecological environment, the study on the stability of marine ecological environment under the optimal control of switching forward system was designed and proposed. The minimum principle method of the control unconstrained variational method and the control constrained Pontryagin minimum principle are extended to the optimal switching problem. A two-stage algorithm is proposed to solve the nonlinear optimization problem of fixed switched sequences. In the first stage, the switching time is first determined, and then the cut-off time is obtained under this condition. In the second stage, the optimal solution is obtained by changing the switching time. It should be pointed out that in the first stage, the variational method and the control constraint minimum principle method are extended to the switching system, and then in the second stage, when the switching time changes, the genetic algorithm is used to obtain the optimal solution, so as to achieve the ecological environment of the ship stability analysis. The simulation results show that the marine ecological environment stability performance under the optimal control of the switching forward system is practical and effective.

An Intrinsic Material Tailoring Approach for Functionally Graded Axisymmetric Hollow Bodies Under Plane Elasticity

One of the main requirements in the design of structures made of functionally graded materials is their best response when used in an actual environment. This optimum behaviour may be achieved by searching for the optimal variation of the mechanical and physical properties along which the material compositionally grades. In the works available in the literature, the solution of such an optimization problem usually is obtained by searching for the values of the so called heterogeneity factors (characterizing the expression of the property variations) such that an objective function is minimized. Results, however, do not necessarily guarantee realistic structures and may give rise to unfeasible volume fractions if mapped into a micromechanical model. This paper is motivated by the confidence that a more intrinsic optimization problem should a priori consist in the search for the constituents’ volume fractions rather than tuning parameters for prefixed classes of property variations. Obtaining a solution for such a class of problem requires tools borrowed from dynamic optimization theory. More precisely, herein the so-called Pontryagin Minimum Principle is used, which leads to unexpected results in terms of the derivative of constituents’ volume fractions, regardless of the involved micromechanical model. In particular, along this line of investigation, the optimization problem for axisymmetric bodies subject to internal pressure and for which plane elasticity holds is formulated and analytically solved. The material is assumed to be functionally graded in the radial direction and the goal is to find the gradation that minimizes the maximum equivalent stress. A numerical example on internally pressurized functionally graded cylinders is also performed. The corresponding solution is found to perform better than volume fraction profiles commonly employed in the literature.

The reduction of CO2 emission using the optimal control approach

Indonesia is the country that has the largest tropical forest with a wide variety of biodiversity. Forests are natural resources that have an important role in human life. However, the existence of forest is currently decreasing along with the increasing of industrialization. The depletion of forest area and the expansion of industrialization can cause the increasing of CO2 emission. Therefore, it is necessary to control forest resources and industrial development, thus the level of CO2 emission can be minimized. In this study, the dynamics of CO2 emission, forest area and industrialization are modelled mathematically. In order to reduce the level of CO2 emission, we design two control variables in the model which are reforestation and government policy. The optimal control theory using Pontryagin Minimum Principle is applied to find the optimal level of control variables. Furthermore, numerical simulation is given to illustrate the performance of the optimal control variables. Based on the simulation result, the level of CO2 emission with controls is less than the level of CO2 emission without controls. This result suggests that providing reforestation and government policy can reduce the concentration of CO2.

Analytical Solution to Equivalent Consumption Minimization Strategy for Series Hybrid Electric Vehicles

Analytical optimal solution to the energy management of hybrid electric vehicles is of interest from theoretical and practical perspectives. Particularly, effort has been made to derive analytical solution to the energy management problem for series hybrid electric vehicles using Pontryagin's minimum principle (PMP). However, admissibility of the system input was not fully explored in determining the optimal input candidates. In this paper, the analytical solution for the same problem is found by partitioning the positive power demand set into four subsets, where the solution is derived for each case separately according to the corresponding admissible input set. The analytical solution is verified through comparison with numerical solution for a series hybrid electric wheel loader, and two different drive cycles are considered for this purpose. From the proposed solution, effective equivalence factor bounds are found and used to construct an adaptive equivalent consumption minimization strategy. The proposed strategy and the analytical solution are implemented together for the same vehicle to demonstrate their effectiveness in dealing with real-world applications. Simulations are performed for 12 drive cycles, and the results are compared to the ones achieved by PMP-based optimal control where the optimization is done numerically. Simulation results suggest that the proposed methodology is relatively fast and has satisfactory performance in presence of drive cycle uncertainty. It is observed that the proposed method fulfills charge sustenance, and the achieved fuel consumption figures are very close to the optimal benchmarks found by the non-causal method.

Development of Economic Velocity Planning Algorithm for Plug-in Hybrid Electric Vehicle

The energy consumption of a plug-in hybrid electric vehicle (PHEV) is directly related to its velocity. Considering the influence of road condition information such as traffic velocity, road speed limit and slope on the energy economy of PHEV, the economic velocity planning algorithm is designed in this study. Firstly, based on the actual vehicle test data, the polynomial fitting energy consumption models with the input parameters of vehicle speed and acceleration for charging of depleting mode and charging of sustaining mode are developed. The economic velocity planning model based on traffic flow velocity and road speed limit on flat roads is established adopting the Pontryagin’s minimum principle (PMP). Afterwards, the road gradient information is introduced into the economic velocity planning algorithm based on the equivalent acceleration. Finally, the planned economic velocity and traffic flow speed are compared by the hardware-in-the-loop energy consumption tests. The results show that compared with the traffic flow speed, the economic velocity improves the vehicle energy economy by more than 4%, and the output torques of the engine and driving motor are more stable, which are more conducive to the energy economy of the vehicle. The designed economic velocity planning algorithm based on road information for PHEV improves the adaptability of PHEV control strategy to actual road conditions, and expands the optimization dimension of energy-saving driving. Thus, it has high practical application value.

Development of an Efficient Thermal Electric Skipping Strategy for the Management of a Series/Parallel Hybrid Powertrain

In recent years, the development of hybrid powertrain allowed to substantially reduce the CO2 and pollutant emissions of vehicles. The optimal management of such power units represents a challenging task since more degrees of freedom are available compared to a conventional pure-thermal engine powertrain. The a priori knowledge of the driving mission allows identifying the actual optimal control strategy at the expense of a quite relevant computational effort. This is realized by the off-line optimization strategies, such as Pontryagin minimum principle—PMP—or dynamic programming. On the other hand, for an on-vehicle application, the driving mission is unknown, and a certain performance degradation must be expected, depending on the degree of simplification and the computational burden of the adopted control strategy. This work is focused on the development of a simplified control strategy, labeled as efficient thermal electric skipping strategy—ETESS, which presents performance similar to off-line strategies, but with a much-reduced computational effort. This is based on the alternative vehicle driving by either thermal engine or electric unit (no power-split between the power units). The ETESS is tested in a “backward-facing” vehicle simulator referring to a segment C car, fitted with a hybrid series-parallel powertrain. The reliability of the method is verified along different driving cycles, sizing, and efficiency of the power unit components and assessed with conventional control strategies. The outcomes put into evidence that ETESS gives fuel consumption close to PMP strategy, with the advantage of a drastically reduced computational time. The ETESS is extended to an online implementation by introducing an adaptative factor, resulting in performance similar to the well-assessed equivalent consumption minimization strategy, preserving the computational effort.

Prospects on Solving an Optimal Control Problem with Bounded Uncertainties on Parameters using Interval Arithmetics

An interval method based on the Pontryagin Minimum Principle is proposed to enclose the solutions of an optimal control problem with embedded bounded uncertainties. This method is used to compute an enclosure of all optimal trajectories of the problem, as well as open loop and closed loop enclosures meant to enclose a concrete system using an optimal control regulator with inaccurate knowledge of the parameters. The differences in geometry of these enclosures are exposed, as well as some applications. For instance guaranteeing that the given optimal control problem will yield a satisfactory trajectory for any realization of the uncertainties or on the contrary that the problem is unsuitable and needs to be adjusted.

Dynamics of COVID-19 spreading model with social media public health awareness diffusion over multiplex networks: Analysis and control

Social media plays a crucial role in controlling the epidemic spreading by creating awareness. Epidemic spreading processes over the multiplex network where epidemic spreading in one layer and awareness spreading in another layer, are well suitable to study the spreading of infection in a small region. Using the Microscopic Markov chain approach (MMCA), the dynamical interplay between the spreading of epidemics over the physical contact layer and human awareness diffusion over the virtual contact layer over a multiplex network framework is analyzed. SEIR process for epidemic spreading and UAU process for awareness diffusion are assumed to study the spreading process of COVID-19. The basic-reproduction number [Formula: see text] is determined using the next generation matrix approach. The stability of disease-free equilibrium (DFE) and endemic equilibrium (EE) is analyzed. Using Pontryagins Minimum Principle, optimal parametric values are determined analytically for testing and cleaning the virus-contaminated surfaces. The impact of various parameters over the epidemic spread is analyzed numerically. Through numerical illustrations, the effect of early identification of exposed people and immediate cleaning of virus-contaminated surfaces over [Formula: see text] studied. The sensitivity analysis is performed to show the importance of wearing face-mask, following social distancing, washing hands frequently and, avoiding unnecessary travel. Through numerical simulations, the results obtained coincide with real data of COVID-19 spreading in the city Chennai, India. The results obtained show that in order to control or discard the spreading of COVID-19, each people should follow the self-preventive measures without relaxation till the epidemic comes nearer to the disease free state.