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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

基于能量最小化的图像融合

图像融合是指联合两个或两个以上的图像通过某种算法得到一幅更高质量的新图像.提出了一种融合全色图像和光谱图像的方法即能量最小化方法,能量主要由两个部分组成.第1部分保证了相关细节信息的注入.第2部分保持了多光谱图像的低频信息.另外,能量还可以包含高分辨率光谱图像的先验知识和其它一些约束条件.     

Constrained optimal estimation and control

A requirement of many multi-input multi-output control and estimation problems is a special structure of the gain matrix. This paper presents a constrained optimal method by which it is possible to constrain some elements of the gain matrix and to optimize the behaviour using the remaining parameters. Only linear constraints are considered and an algorithm based on a modified Riccati equation is obtained. The asymptotic solution for the steady state can be reduced to a sequence of classical Riccati equations by using an iterative approach. Numerical techniques have been found and computer programmes developed. Numerical examples are given.     

Some second-order necessary conditions in optimal control

In optimal control problems any extremal arc which trivially satisfies the Maximum Principle, that is a first-order control variation produces no change in cost, is called singular. Higher-order conditions are then needed to check the optimality of such arcs. Using the Volterra series associated with the variation of the cost functional gives a new context for analyzing singular optimal control problems. A basic optimality criterion for a fixed terminal time Mayer problem is obtained which allows one to derive the necessary conditions for optimality in terms of Lie brackets of vector fields associated with the dynamics of the problem.     

A stochastic method for the energy management in hybrid electric vehicles

There are many approaches addressing the problem of optimal energy management in hybrid electric vehicles; however, most of them optimise the control strategy for particular driving cycles. This paper takes into account that the driving cycle is not a priori known to obtain a near-optimal solution. The proposed method is based on analysing the power demands in a given receding horizon to estimate future driving conditions and minimise the fuel consumption while cancelling the expected battery energy consumption after a defined time horizon. Simulations show that the proposed method allows charge sustainability providing near-optimal results.     

Design, implementation, and experimental validation of optimal power split control for hybrid electric trucks

Hybrid electric vehicles require an algorithm that controls the power split between the internal combustion engine and electric machine(s), and the opening and closing of the clutch. Optimal control theory is applied to derive a methodology for a real-time optimal-control-based power split algorithm. The presented strategy is adaptive for vehicle mass and road elevation, and is implemented on a standard Electronic Control Unit of a parallel hybrid electric truck. The implemented strategy is experimentally validated on a chassis dynamo meter. The fuel consumption is measured on 12 different trajectories and compared with a heuristic and a non-hybrid strategy. The optimal control strategy has a fuel consumption lower (up to 3%) than the heuristic strategy on all trajectories that are evaluated, except one. Compared to the non-hybrid strategy the fuel consumption reduction ranged from 7% to 16%.     

The time-invariant linear-quadratic optimal control problem

This paper provides a review of one of the basic problems of systems theory—the general time-invariant optimal control problem involving linear systems and quadratic costs. The problem includes on one hand the regulator problem of optimal control and on the other, the theory of linear dissipative systems, itself central to network theory and to the stability theory of feedback systems. The theory is developed using simple properties of dynamical systems and involves a minimum of ‘hard’ analysis or algebra. It includes a full existence theory of the matrix quadratic equation, of interest in its own right.     

Lossless convexification of a class of optimal control problems with non-convex control constraints

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem, which is referred to as the lossless convexification of the optimal control problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. The solution approach is demonstrated on a number of planetary soft landing optimal control problems.