Pontryagin's Minimum Principle Research Articles

Assistive Power Buffer Control via Adaptive Dynamic Programming

Power buffers are power electronic converters, with large capacitors, that decouple volatile loads and a low-inertia distribution network in a DC microgrid. In this work, a set of distributed optimal control policies enable power buffers to reciprocally assist each other during abrupt load changes. While the majority of existing control paradigms are localized, enabling communication among buffers extends their effective range of assistance and helps them minimize a shared objective in a cooperative fashion. The control law's weights surfaces are learned for a mesh of reference loads of each power buffer. Hamilton-Jacobi-Bellman equation is solved by a continuous-time adaptive dynamic programming (ADP) approach with off-policy learning to directly provide a feedback controller, instead of existing approaches that obtain open-loop policies via Pontryagin's minimum principle. This paper presents the first attempt in using ADP techniques for the control of power buffers that respects their original nonlinear dynamics, overcoming the limitations of previous approaches based on small-signal analysis. Compared to the current literature, the proposed approach provides trained controllers that are known a priori, avoiding player-by-player solutions or real-time optimization procedures that could degrade performances or become computationally intensive. Hardware-in-the-loop emulations of a low-voltage DC microgrid validates the proposed approach.

A robust pseudospectral method for numerical solution of nonlinear optimal control problems

In the present paper, a robust pseudospectral method for efficient numerical solution of nonlinear optimal control problems is presented. In the proposed method, at first, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality which are led to the Hamiltonian boundary value problem are derived. Then, utilizing a pseudospectral method for discretization, the nonlinear optimal control problem is converted to a system of nonlinear algebraic equations. However, the need to have a good initial guess may lead to a challenging problem for solving the obtained system of nonlinear equations. So, an optimization approach is introduced to simplify the need of a good initial guess. Numerical findings of some benchmark examples are presented at the end and computational features of the proposed method are reported.

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Abstract In this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

Time optimal control of two-level quantum systems

The objective of time-optimal control that helps to minimize relaxation losses, is the evolution of a quantum state from a given initial mixed state to a final target mixed state in minimum time. In this paper, we study a time-optimal control problem of the dynamic of a pure two-level system with unbounded control using Pontryagin's minimum principle and obtain the minimal time for some initial and final states. The results will apply to basically all qubit systems that one can consider such as NMR spectroscopy, trapped ions, superconducting qubits, etc. We also show that these results hold for pure states, and only the direction nˆ is important in the evolution of a quantum state. In this work, the problem of computing minimum time to produce any unitary transformation Uf∈SU(2) is reduced to finding the minimum time to steer the system from an initial to a final state.

Solving optimal control problems using the Picard’s iteration method

In this paper, the Picard’s iteration method is proposed to obtain an approximate analytical solution for linear and nonlinear optimal control problems with quadratic objective functional. It consists in deriving the necessary optimality conditions using the minimum principle of Pontryagin, which result in a two-point-boundary-value-problem (TPBVP). By applying the Picard’s iteration method to the resulting TPBVP, the optimal control law and the optimal trajectory are obtained in the form of a truncated series. The efficiency of the proposed technique for handling optimal control problems is illustrated by four numerical examples, and comparison with other methods is made.

A Minimum Principle-Based Algorithm for Energy-Efficient Eco-Driving of Electric Vehicles in Various Traffic and Road Conditions

This paper presents an optimization algorithm for energy-efficient driving of electric vehicles. The algorithm is based on Pontryagin's Minimum Principle by considering the driving mission as an optimal control problem. On a planned route, diverse application cases may occur due to various physical traits such as powertrain (e.g., switching between motor and generator operation), environmental variations (e.g., terrain grade change), traffic laws (e.g., speed limits), and safety concerns (e.g., headway to the leading vehicle). The proposed algorithm handles these real-world challenges by considering the perturbation to the system dynamics and the constraints to the control and state variables.

Collision-free and crossing-free trajectory design for second-order agents persistent monitoring

In this paper, we utilize second-order agents to address a one-dimensional persistent monitoring problem with potential agent collision and boundary-crossing problems. The objective is to minimize a sum of accumulated performance metric associated with targets over a finite time horizon by controlling the movements of agents. Different from the existing work where a collision avoidance algorithm is designed for avoiding collisions and boundary crossings, we consider these problems at the beginning of the problem formulation and further utilize Exterior Point Method (EPM) to propose a novel combinatorial objective function which guarantees a collision-free and crossing-free solution by penalizing all possible agents collisions and boundary crossings. According to Pontryagin Minimum Principle (PMP), we show that the optimal agent trajectories can be fully described by some parameters. Based on Infinitesimal Perturbation Analysis (IPA), a standard gradient descent method is proposed to obtain the optimal parameters. Our collision-free and crossing-free approach is computationally efficient as compared to the existing algorithm. Numerical examples are included to demonstrate the effectiveness of the proposed methods.

Time-optimal bang-bang control for the driven spin-boson system

In this paper we investigate the application of optimal control of a single qubit coupled to an ohmic heat bath. For the weak bath coupling regime, we derive a Bloch-Redfield master equation describing the evolution of the qubit state parametrized by vectors in the Bloch sphere. By use of the optimal control methodology we determine the field that generates an arbitrary single qubit rotation in minimum time. In particular we develop an efficient method for solving the time-optimal control problem, namely the implementation of $X$ gate, which consists of a rotation about the $\stackrel{P\vec}{ox}$ axis through angle $\ensuremath{\pi}$. The time-optimal control problem requires a bounded control. If upper and lower bounds are imposed for the external control, the optimal solution is of bang-bang type and switches from the upper to the lower values of the control bounds. We compare our results using the techniques of automatic differentiation to compute the gradient for the cost functional with some known results using Pontryagin's minimum principle. We use an alternative numerical approach for the optimization strategy.

A fast model predictive control allocation of distributed drive electric vehicles for tire slip energy saving with stability constraints

This paper proposes a fast model predictive control allocation (MPCA) approach to minimize the tire slip power loss on contact patches for distributed drive electric vehicles (DDEV). In this strategy, two assumptions are set up from a practical focus: (1) the vehicle acceleration and yaw rate are measurable by global position system (GPS)/ inertial navigation system (INS) and inertial measurement unit (IMU), respectively; (2) the longitudinal velocity, road adhesion factor, and vehicle yaw rate are arranged to be “already known” by advanced estimators. For the strategy design, a CarSim-embedded driver model and a linear quadratic regulator (LQR) based direct yaw moment controller, are respectively applied to calculate the desired longitudinal traction and yaw moment as a virtual input first. Then, a MPCA method is proposed to reasonably distribute the virtual input among four in-wheel motors in order to optimize the tire slip power loss and vehicle stability performance. To accurately characterize tire slip power loss in MPCA, a tire slip estimator is established for tire slip information acquirement. Moreover, addressing on the heavily computational challenge in MPCA, a modified continuation/generalized minimal residual (C/GMRES) algorithm is employed. Since the traditional C/GMRES algorithm cannot directly solve the inequality constraint problem, the barrier functions are applied for transforming the inequality constraints to equivalent cost. According to Pontryagin’s minimum principle (PMP) conditions, the existence and uniqueness for solution of the modified C/GMRES algorithm are strictly proved. Subsequently, a Karush–Kuhn–Tucker​ (KKT) condition based approach is developed to fast gain the optimally initial solution in C/GMRES algorithm for extending application. Finally, numerical simulation validations are implemented and demonstrate that the proposed MPCA can ensure the compatibility between the tire slip power loss reduction and vehicle stability in a computationally efficient way.

Vehicle Trajectory Specification in Presence of Traffic Lights with Known or Uncertain Switching Times

The main purpose of this work is to generate optimal trajectories for vehicles crossing a signalized junction, with traffic signals operated in either fixed-time or real-time (adaptive) mode. In the latter case, the next switching time is decided in real time based on the prevailing traffic conditions and is therefore uncertain in advance. The GLOSA (Green Light Optimal Speed Advisory) problem is addressed by using traffic lights information and calculating a trajectory and velocity profile for the vehicle based on the vehicle’s initial state (position and speed) and a fixed final destination state. At first, an appropriate optimal control problem is formulated and solved analytically via Pontryagin’s minimum principle (PMP) for the case of known switching times. Subsequently, for the case of real-time signals, availability of a time-window of possible signal switching times, along with the corresponding probability distribution, is assumed, and the problem is cast in the format of a stochastic optimal control problem and is solved numerically using stochastic dynamic programming (SDP) techniques. Application results, for various driving scenarios, of the deterministic approach, which considers the case of known switching times, and a comprehensive comparison of the stochastic GLOSA approach with a sub-optimal approach are presented. In particular, it is demonstrated that the proposed SDP approach achieves better average performance compared with the sub-optimal approach because of the better (probabilistic) information on the traffic light switching time.

Optimal Control of Lossy Energy Storage Systems With Nonlinear Efficiency Based on Dynamic Programming and Pontryagin's Minimum Principle

We consider energy storage systems having nonlinear efficiency functions, which are becoming increasingly important as shown in several recent works, and propose an optimal solution based on Pontryagin's minimum principle. A central challenge in such problems is the hard limits on the state variable, which restrict the use of the minimum principle. To address this challenge, we propose to include the capacity constraints in the objective function with a proper weighting constant. We show that this approach allows formulation of the problem based on the classical minimum principle, and eventually leads to an efficient optimal control strategy. The proposed solution is compared to a dynamic programming algorithm. Numeric experiments reveal that for lossless storage devices dynamic programming is beneficial, since it enables fast and accurate solutions when a low number of samples is used. However, for lossy storage devices the situation is the opposite, and the minimum principle provides faster and more accurate solutions, since its computational complexity is almost unaffected by changes in the system parameters.

Analysis of Optimal Dynamic Manipulation for Robotic Manipulator Based on Pontryagin’s Minimum Principle

A branch of dynamic manipulation is robotic throwing which recently plays a major role in the performance improvement of robotic manipulations. This paper aims at the optimal throwing using a robotic arm. Launching the captured object to a specified target by minimum control effort with the specified final time is the mission of the throwing robot in this study. Taking into account the indirect method based on the fundamental theorem of the calculus of variations, the optimal throwing problem is defined as the optimal control problem (OCP). By employing Pontryagin’s Minimum Principle (PMP), the optimality conditions are extracted and then considering the throwing equation as a moving-end boundary condition (BC), the corresponding OCP converts to a two-point boundary value problem (BVP). By applying the moving-end BCs, the best releasing condition is automatically obtained. Next, a simple algorithm is presented to solve the resulting BVP which leads us to the multiple optimal solutions that have been sorted according to the defined cost function values. This multiplicity of solutions allows for the selection of the most practical optimal solution. The feasibility of the derived framework is validated by some simulations and experiments done for a fabricated throwing robot.

Pontryagin’s minimum principle-based real-time energy management strategy for fuel cell hybrid electric vehicle considering both fuel economy and power source durability

Abstract This paper presents a real-time and approximately optimal energy management strategy (EMS) based on Pontryagin’s minimum principle (PMP), considering both fuel economy and power source durability. To develop the target strategy, performance degradation models are built for two power sources: a proton exchange membrane fuel cell and a lithium ion battery. This study provides an online co-state updating method for uncertain driving cycles. The method allows the battery’s state of charge to be controlled within a certain range and determines the nearly optimal hydrogen consumption. Furthermore, by incorporating a fuel cell power variation limiting factor with a weight coefficient into the PMP to suppress power changes, the durability of the fuel cell can be improved. The average daily operating cost (calculated based on the fuel consumption and power source degradation) is used to evaluate the trade-off between fuel economy and power source durability. Simulation results show that the fuel cell durability is improved with a slight increase in fuel consumption and battery degradation, and that the average daily operating cost is effectively reduced. A comparison with the results obtained by adopting a rule-based EMS and a dynamic programming-based EMS indicate the superiority of the proposed EMS.

Research on Equivalent Factor Boundary of Equivalent Consumption Minimization Strategy for PHEVs

The equivalent fuel consumption minimum strategy (ECMS) based on the Pontryagin's minimum principle (PMP) enables real-time energy management optimization of plug-in hybrid electric vehicles (PHEVs). However, it remains challenging to accurately determine the equivalent factor (EF). In this study, an analytical expression of the optimal EF boundary is addressed to facilitate more efficient search of the optimal EF. The deterministic expression of the optimal EF boundary is derived from the Hamilton equation of PMP. By combining the optimal EF boundary and differential evolution algorithm, a novel fusion adaptive ECMS (A-ECMS) is proposed to demonstrate the application of the proposed boundary in the framework of model predictive control (MPC). Two different simulations with and without prior knowledge of driving cycle were respectively conducted, and the simulation results verify the feasibility of optimal EF boundary and manifest that the energy savings by the proposed A-ECMS can reach more than 97% of dynamic programing, highlighting the superior performance of the proposed strategy.

Dynamic System Optimum Analysis of Multi-Region Macroscopic Fundamental Diagram Systems With State-Dependent Time-Varying Delays

This paper investigates the dynamic system optimum (DSO) problem with simultaneous route and departure time assignments for a general traffic network partitioned into multiple regions. Regional traffic congestion is modeled with a well-defined macroscopic fundamental diagram (MFD) mapping the trip completion rate to the vehicular accumulation. To overcome the limitation of inconsistent flow propagation between region boundaries and the corresponding travel time, the state-dependent regional travel time function is explicitly incorporated in the flow propagation of the conventional MFD dynamics. From a systems perspective, the traffic dynamics within a region can be regarded as a dynamic system with an endogenous time-varying delay depending on the system state. Equilibrium condition for the DSO problem is analytically derived through the lens of Pontryagin minimum principle and is compared against the static SO counterpart. The structure of path specific marginal cost is analyzed regarding the path travel cost and early-late penalty function. In contrast to existing analytical methods, the proposed method is applicable for general MFD systems without linearization of the MFD dynamics. Neither approximation of the equilibrium solution nor constant regional delay assumption is required. Numerical examples are conducted to illustrate the characteristics of DSO traffic equilibrium and the corresponding marginal cost together with other dynamic external costs.

Two-dimensional vehicular movement modelling at intersections based on optimal control

Modeling traffic flow at intersections is essential for the design, control, and management of intersections. A challenging feature of microscopic modeling vehicular movement at intersections is that drivers can choose among an infinite number of alternative traveling paths and speeds. This makes it fundamentally different from structured straight road sections with lanes. This study proposes a novel method to model the trajectories of vehicles in two-dimensional space and speed. Based on optimal control theory, it assumes drivers schedule their driving behavior, including the steering and acceleration, to minimize the predicted costs. The costs contain the running costs, which consist of the travel time and driving smoothness (longitudinally and laterally), and the terminal cost, which penalizes the deviations from the desired final state. Different than conventional methods, the vehicle motion dynamics are formulated in distance rather than in time. The model is solved by an iterative numerical solution algorithm based on the Minimum Principle of Pontryagin. The descriptive power, plausibility, and accuracy of the proposed model are investigated by comparing the calculated results under several cases, which can be solved from symmetry or analytically. The proposed model is further calibrated and validated using empirical trajectory data, and the quality of the predicted trajectory is confirmed. Qualitatively, the optimal trajectory changes in the range of the shortest path and smoothest path under different weights of the running cost. The proposed model can be used as a starting point and extended with more considerations of intersection operation in the real world for future applications.

Robust energy management of plug-in hybrid electric bus considering the uncertainties of driving cycles and vehicle mass

This paper proposes a robust energy management for plug-in hybrid electric buses (PHEBs) considering the uncertainties of driving cycles and vehicle mass. The goal is to improve the fuel economy of PHEB while weakening the sensitivity of Pontryagin’s minimum principle (PMP)-based energy management to driving conditions. Considering that the PMP can be controlled by constant, a robust co-state sequence of PMP is designed based on the Taguchi robust design, where the quality characteristic of fuel consumption is taken as smaller-the-better characteristic. Taking the sampled noise factors composed of driving cycles and stochastic vehicle mass into account, a reliable verification structure is constructed to verify the robustness of designed co-state sequence based on the Monte Carlo simulation. Combining the robust co-state sequence and charge sustaining control mode, a robust PMP control strategy is proposed for the optimal control of PHEB and the over-discharge protection of battery. The simulation results demonstrate that the proposed energy management strategy is effective, and is robust against the uncertainties. Compared with the rule-based energy management, the fuel economy can be averagely improved by 29.32%.

A scalable, causal, adaptive energy management strategy based on optimal control theory for a fuel cell hybrid railway vehicle

A scalable, causal, adaptive optimal control-based energy management strategy for the fuel cell hybrid train is designed. As learned from the results of offline Pontryagin’s minimum principle (PMP)-based strategies, the convexity of the specific consumption curve is emphasized to improve the fuel economy. More important is that the dependency of the co-state on the state of charge (SoC) of batteries and the average fuel cell power is identified the first time. With the help of using the optimal control theory in a reverse way, a quantitative analytical formula is derived to determine the co-state based on the SoC and the average fuel cell power. The accuracy of the estimates, and the effectiveness of this strategy, under different weather, driving, and aging conditions, is validated by comparison to the results of offline PMP-based strategies. Thereby, a maximal deviation of the co-state average value compared to the offline results is 1.8%. An excellent fuel economy under a typical driving cycle of regional railway transports in Berlin, with only 0.03% more consumption for both summer and winter conditions, compared to the results of offline PMP, is resulted. Due to the model-based characteristics, the strategy can be scaled or transferred to other configuration systems or driving conditions without the loss of effectiveness.

A Real-Time Nonlinear Model Predictive Controller for Yaw Motion Optimization of Distributed Drive Electric Vehicles

This paper proposes a real-time nonlinear model predictive control (NMPC) strategy for direct yaw moment control (DYC) of distributed drive electric vehicles (DDEVs). The NMPC strategy is based on a control-oriented model built by integrating a single track vehicle model with the Magic Formula (MF) tire model. To mitigate the NMPC computational cost, the continuation/generalized minimal residual (C/GMRES) algorithm is employed and modified for real-time optimization. Since the traditional C/GMRES algorithm cannot directly solve the inequality constraint problem, the external penalty method is introduced to transform inequality constraints into an equivalently unconstrained optimization problem. Based on the Pontryagin's minimum principle (PMP), the existence and uniqueness for solution of the proposed C/GMRES algorithm are proven. Additionally, to achieve fast initialization in C/GMRES algorithm, the varying predictive duration is adopted so that the analytic expressions of optimally initial solutions in C/GMRES algorithm can be derived and gained. A Karush-Kuhn-Tucker (KKT) condition based control allocation method distributes the desired traction and yaw moment among four independent motors. Numerical simulations are carried out by combining CarSim and Matlab/Simulink to evaluate the effectiveness of the proposed strategy. Results demonstrate that the real-time NMPC strategy can achieve superior vehicle stability performance, guarantee the given safety constraints, and significantly reduce the computational efforts.

Accelerated Sizing of a Power Split Electrified Powertrain

<div class="section abstract"><div class="htmlview paragraph">Component sizing generally represents a demanding and time-consuming task in the development process of electrified powertrains. A couple of processes are available in literature for sizing the hybrid electric vehicle (HEV) components. These processes employ either time-consuming global optimization techniques like dynamic programming (DP) or near-optimal techniques that require iterative and uncertain tuning of evaluation parameters like the Pontryagin’s minimum principle (PMP). Recently, a novel near-optimal technique has been devised for rapidly predicting the optimal fuel economy benchmark of design options for electrified powertrains. This method, named slope-weighted energy-based rapid control analysis (SERCA), has been demonstrated producing results comparable to DP, while limiting the associated computational time by near two orders of magnitude. In this paper, sizing parameters for a power split electrified powertrain are considered that include the internal combustion engine size, the two electric motor/generator sizes, the transmission ratios, and the final drive ratio. The SERCA approach is adopted to rapidly evaluate the fuel economy capabilities of each sizing option in various driving missions considering both type-approval drive cycles and real-world driving profiles. While screening out for optimal sizing options, the implemented methodology includes drivability criteria along with fuel economy potential. Obtained results will demonstrate the agility of the developed sizing tool in identifying optimal sizing options compared to state-of-the-art sizing tools for electrified powertrains.</div></div>