Energy-efficient train control: From local convexity to global optimization and uniqueness

The optimal driving strategy for a train is essentially a power–speedhold–coast–brake strategy unless the track contains steep grades in which case the speedhold mode must be interrupted by phases of power for steep uphill sections and coast for steep downhill sections. The Energymiser^®  device is used on freight and passenger trains in Australia and the United Kingdom to provide on-board advice for drivers about energy-efficient driving strategies. Energymiser^®  uses a specialized numerical algorithm to find optimal switching points for each steep section of track. Although the algorithm finds a feasible strategy that satisfies the necessary optimality conditions there has been no direct proof that the corresponding switching points are uniquely defined. We use a comprehensive perturbation analysis to show that a key local energy functional is convex with a unique minimum and in so doing prove that the optimal switching points are uniquely defined for each steep section of track. Hence we also deduce that the global optimal strategy is unique. We present two examples using realistic parameter values.     

Simulated annealing for optimal ship routing

In this article we present a simulated annealing based algorithm for the determination of optimal ship routes through the minimization of a cost function defined as a weighted sum of the time of voyage and the voyage comfort (safety is taken into account too). This cost function is dependent on the wind speed and its direction as well as on the wave height and its direction. The constructed algorithm at the beginning discretizes an initial route and then optimizes it by considering small deviations, which are accepted or rejected by utilizing the simulated annealing technique. Using calculus of variations, we prove a key theorem which tremendously accelerates the convergence of the proposed algorithm. For an illustration of the advantages of the constructed method, both computational and real experiments have been carried out which are presented and discussed.     

Some observations on optimal frequency selection in DVFS-based energy consumption minimization

In recent years, the issue of energy consumption in parallel and distributed computing systems has attracted a great deal of attention. In response to this, many energy-aware scheduling algorithms have been developed primarily using the dynamic voltage-frequency scaling (DVFS) capability which has been incorporated into recent commodity processors. Majority of these algorithms involve two passes: schedule generation and slack reclamation. The former pass involves the redistribution of tasks among DVFS-enabled processors based on a given cost function that includes makespan and energy consumption, while the latter pass is typically achieved by executing individual tasks with slacks at a lower processor frequency. In this paper, a new slack reclamation algorithm is proposed by approaching the energy reduction problem from a different angle. Firstly, the problem of task slack reclamation by using combinations of processors’ frequencies is formulated. Secondly, several proofs are provided to show that (1) if the working frequency set of processor is assumed to be continues, the optimal energy will be always achieved by using only one frequency, (2) for real processors with a discrete set of working frequencies, the optimal energy is always achieved by using at most two frequencies, and (3) these two frequencies are adjacent/neighbouring when processor energy consumption is a convex function of frequency. Thirdly, a novel algorithm to find the best combination of frequencies to result the optimal energy is presented. The presented algorithm has been evaluated based on results obtained from experiments with three different sets of task graphs: 3000 randomly generated task graphs, and 600 task graphs for two popular applications (Gauss-Jordan and LU decomposition). The results show the superiority of the proposed algorithm in comparison with other techniques.     

Characteristic model-based all-coefficient adaptive control for automatic train control systems

Safe and reliable automatic train control is a primary consideration for any advanced rail transit system. This paper introduces the characteristic model-based modeling method into ATC system and develops an all-coefficient adaptive control. Two characteristic models,namely speed characteristic model and position characteristic model,are established for analyzing both train traction and cruising dynamical characteristics.Control strategies are proposed using single speed feedback and speed/position bi-feedback. Numerical simulations are carried out to verify the effectiveness of the proposed strategies,showing that control objective and required dynamic performance are well satisfied. Moreover,the system shows robustness against time-varying model uncertainties and unknown operational environment dynamics.     

基于CBTC的列车自动驾驶控制算法

基于通信的列车控制系统是列车控制系统技术的发展方向。讨论了基于CBTC的列车自动控制系统的基本结构和功能，指出传统控制方法不能适应列车运行参数的非线性和时变性，而采用智能控制方法较为有效，对基于专家系统的ATO系统、基于模糊控制的ATO系统和基于模糊神经网络的ATO系统等几种速度控制算法进行了分析和建模。     

Minimizing control variation in nonlinear optimal control

In any real system, changing the control signal from one value to another will usually cause wear and tear on the system’s actuators. Thus, when designing a control law, it is important to consider not just predicted system performance, but also the cost associated with changing the control action. This latter cost is almost always ignored in the optimal control literature. In this paper, we consider a class of optimal control problems in which the variation of the control signal is explicitly penalized in the cost function. We develop an effective computational method, based on the control parameterization approach and a novel transformation procedure, for solving this class of optimal control problems. We then apply our method to three example problems in fisheries, train control, and chemical engineering.     

An optimal regulation strategy with disturbance rejection for energy management of hybrid electric vehicles

The energy management problem of finding the optimal split between the different sources of energy in a charge-sustaining parallel HEV, ensuring stability and optimality with respect to a performance objective (fuel consumption minimization over a driving cycle), is addressed in this paper. The paper develops a generic stability and optimality framework within which the energy management problem is cast in the form of a nonlinear optimal regulation (with disturbance rejection) problem and a control Lyapunov function is used to design the control law. Two theorems ensuring optimality and asymptotic stability of the energy management strategy are proposed and proved. The sufficient conditions for optimality and stability are used to derive an analytical expression for the control law as a function of the battery state of charge/state of energy and system parameters. The control law is implemented in a simplified backward vehicle simulator and its performance is evaluated against the global optimal solution obtained from dynamic programming. The strategy performs within 4% of the benchmark solution while guaranteeing optimality and stability for any driving cycle.     

磁浮列车的非线性控制问题研究

建立了磁浮列车悬浮系统模型, 针对模型的非线性问题, 利用反馈线性化技术, 将系统的输入-状态线性化, 设计出一个P控制器和一个非线性补偿器, 得到了良好的仿真结果. 在此基础上, 进行了数字控制实验, 比较了仿真和实验结果. 实验表明反馈线性化技术可以解决磁浮列车非线性模型的控制问题.     

Optimal tracking and two-channel disturbance rejection under control energy constraint

In this paper, the optimal tracking problem under control energy constraint is studied. The disturbance in the upstream channel and down channel is adequately considered. Some new results are derived and it is shown that the performance limitation is tightly dependent on the non-minimum phase zeros and the unstable poles. In addition, the performance limitation, which is greater than that of the disturbance only in the down channel, depends on other new non-minimum phase zeros. Finally, the example explains and validates the conclusion.     

二次型最优控制问题中的权矩阵与最优控制律

二次型最优的权矩阵选择是一个包含大量经验与技巧的问题．对此，研究权矩阵与最优控制律的关系：一个最优控制律，其对应的性能指标（权矩阵）是否唯一的问题．不论是单输入还是多输入情形，在系统能控性指数大于2，对应权矩阵的对角线元素有2个不为零的元素，且满足一定条件时．该性能指标对应的最优控制律，必有另一性能指标与之对应．     

磁浮列车搭接结构的非线性解耦控制

磁浮列车的搭接结构是一个多输入多输出的非线性系统, 对该系统进行稳定控制是确保磁浮列车可靠性的关键. 本文采用微分几何方法对系统解耦; 采用反馈线性化方法对解耦后的系统进行线性化, 确保了系统的全局稳定性; 对线性化之后的系统采用极点配置的方法进行控制律设计, 使系统的动态性能达到一定的指标; 最后给出了应用此方法的一个实际例子.     

A control benchmark on the energy management of a plug-in hybrid electric vehicle

A benchmark control problem was developed for a special session of the IFAC Workshop on Engine and Powertrain Control, Simulation and Modeling (E-COSM 12), held in Rueil-Malmaison, France, in October 2012. The online energy management of a plug-in hybrid-electric vehicle was to be developed by the benchmark participants. The simulator, provided by the benchmark organizers, implements a model of the GM Voltec powertrain. Each solution was evaluated according to several metrics, comprising of energy and fuel economy on two driving profiles unknown to the participants, acceleration and braking performance, computational performance. The nine solutions received are analyzed in terms of the control technique adopted (heuristic rule-based energy management vs. equivalent consumption minimization strategies, ECMS), battery discharge strategy (charge depleting–charge sustaining vs. blended mode), ECMS implementation (vector-based vs. map-based), ways to improve the implementation and improve the computational performance. The solution having achieved the best combined score is compared with a global optimal solution calculated offline using the Pontryagin's minimum principle-derived optimization tool HOT.     

On the optimal control of a hydroelectric power plant

A class of optimal control problems with phase restrictions is investigated whose performance index and state equation are linear in the control variable. First some necessary conditions for optimality are proved and then they are used to get the optimal solution.     

On smooth optimal control determination

When using the Pontryagin Maximum Principle in optimal control problems the most difficult part of the numerical solution is associated with the non-linear operation of the maximization of the Hamiltonian over the control variables. For a class of problems, the optimal control vector is a vector function with continuous time derivatives. A method is presented to find this smooth control without the maximization of Hamiltonian. Three illustrative examples are considered.     

Robust optimal control of regular languages

This paper presents an algorithm for robust optimal control of regular languages under specified uncertainty bounds on the event cost parameters of the language measure that has been recently reported in literature. The performance index for the proposed robust optimal policy is obtained by combining the measure of the supervised plant language with uncertainty. The performance of a controller is represented by the language measure of the supervised plant and is minimized over the given range of event cost uncertainties. Synthesis of the robust optimal supervisory control policy requires at most n iterations, where n is the number of states of the deterministic finite-state automaton (DFSA) model, generated from the regular language of the unsupervised plant behavior. The computational complexity of the control synthesis method is polynomial in n.     

An optimal control based approach to designing plantwide control system architectures

An approach to designing decentralized plantwide control system architectures is presented. The approach is based on splitting the optimal controller gain matrix that results from solving an output optimal control problem into feedback and feedforward parts. These two parts are then used to design and evaluate decentralized control systems. Results for the application of the methodology to a realistic, 4 by 4 reactor with recycle process are given. For this system, the optimal control based approach suggests feedback pairings that are significantly different than those suggested by the steady state RGA. The approach presented can give an indication if MPC is preferred over a decentralized approach to plantwide control. Comparison of the results produced by the best decentralized plantwide system and a model predictive control system are presented.     

Stochastic optimal control of networked control systems with control packet dropouts

This paper considers the stochastic optimal control problem for networked control systems(NCSs)with control packet dropouts.The proportional plus up to the third-order derivative(PD3)compensation strategy is adopted to compensate for control packet dropouts at the actuator by using the past control packets stored in the buffer.Based on the strategy,a new NCS structure model with packet dropouts is provided,where the packet dropout is assumed to obey the Bernoulli random binary distribution.In terms of the given model,the stochastic optimal control law is proposed. Numerical examples illustrate the effectiveness of the results.     

离散线性信息融合最优跟踪控制

提出一种有限时间离散线性最优跟踪控制问题的新解法--信息融合估计解法.基于信息融合估计理论,推导出协状态融合滤波方程和控制量融合估计值,由此获得最优融合控制律及二次性能指标最小值.从理论上证明了信息融合估计解法与传统解法的等同性,从信息融合的角度建立了有限时间离散线性最优跟踪控制系统,从而统一了最优控制问题和最优估计问题,电机系统的控制仿真结果验证了该解法的有效性以及与传统解法的等同性.     

利用距离与内能极小平滑链接Bézier曲线

目的 对于满足低阶连续的链接Bézier曲线,提高曲线之间的连续性以达到平滑的目的,需要对曲线的控制顶点进行相应调整。因此,可根据具体的目标对需要调整的控制顶点进行优化选取,使得平滑后的链接曲线满足相应的要求。针对这一问题,给出了3种目标下优化调整控制顶点的方法。方法 首先对讨论的问题进行描述,分别指出链接Bézier曲线从C⁰连续平滑为C¹连续和从C¹连续平滑为C²连续两种情形需调整的控制顶点;然后分别给出两种情形下,以新旧控制顶点距离极小为目标、曲线内能极小为目标、新旧控制顶点距离与曲线内能同时极小为目标,对链接Bézier曲线进行平滑的方法,最后对3种极小化方法进行对比,并指出了不同方法的适用场合。结果 数值算例表明,距离极小化方法调整后的控制顶点偏离原控制顶点的距离相对较小,适合于控制顶点取自于实物时的应用场合;内能极小化方法获得的链接曲线内能相对较小,适合于要求曲线能量尽可能小的应用场合;距离与内能同时极小化方法兼顾了新旧控制顶点的距离和链接曲线的内能,适合于对两个目标都有要求的应用场合。结论 提出的方法为链接Bézier曲线的平滑提供了3种有效手段,且易于实现,对其他类型链接曲线的平滑具有参考价值。     

Hierarchical optimal force-position control of complex manufacturing processes

A hierarchical optimal controller is developed in this paper to regulate the machining force and axis positions, simultaneously, in a micro end milling process. The process is divided into two levels of decision making. The bottom level includes the measurable states, which in this work comprises the axis positions. The top level includes the higher order objectives, which can be derived from the bottom level objectives by an aggregation relationship. In this work, the top level's objective is to regulate the machining force. A series of simulations were conducted in which the weighting between the top and the bottom level objectives is adjusted within the feasible range. The results demonstrated that excellent tracking of both axis positions and machining force are achieved during the steady state regardless of the weighting. However, the transient performance of the system could be systematically shaped to achieve better performance of either objective. For the purpose of comparison a decentralized optimal controller was constructed and simulated for the feasible range of controller weights. When the axis position errors were weighted heavily, both controllers were able to regulate the axis errors well, while the hierarchical controller had smaller machining force errors. When the machining force errors were weighted heavily, although the machining force error decreased for the decentralized controller the axis position errors increased significantly. However, with heavy machining force weighting, the hierarchical controller was able to manipulate the axial errors in a way that while the machining force error was reduced, the contour error (i.e., smallest deviation from the tool tip to the desired contour) remained small.