嫦娥三号软着陆轨道设计与控制策略问题评析

针对2014年全国大学生数学建模竞赛A题"嫦娥三号软着陆轨道设计与控制策略"问题,根据全国评阅的具体情况,首先介绍了问题的要求和评阅基本要点;然后给出几种有代表性的解法和模型;最后对参赛论文中存在的较普遍的问题作了分析。     

基于高斯伪谱法的月球软着陆轨道的数值分析

月球软着陆是月球探测技术发展的关键所在,针对月球软着陆过程中的着陆轨道设计、着陆过程最优控制的实际问题进行定量分析,运用高斯伪谱法将问题转化为求解满足一定性能指标和约束条件的最优控制问题以确定月球软着陆的最优控制策略;分析着陆区域数字高程图,得到月球最优着陆区域.     

基于配点GA-SQP算法的月球软着陆轨道优化设计

月球软着陆是月球探测中的一项关键技术.针对这项技术,提出基于直接配点法和GA-SQP法的月球软着陆轨道优化算法.算法通过建立月球软着陆动力学模型并进行归一化处理,利用直接配点法将月球软着陆轨道优化问题离散为非线性规划问题,应用SQP方法可求解该优化问题.同时考虑到SQP方法对初值敏感、收敛半径小及容易陷入局部极值等不足,提出了应用GA为SQP方法提供迭代初值的算法,提高了算法的稳定性.最后给出了相关的仿真结果.     

基于在险价值的物流网络规划模糊两阶段模型与精确求解方法

针对物流网络规划问题中顾客需求和运输成本的不确定性,使用在险价值量化投资风险,建立了以投资损失的在险价值最小化为目标的模糊两阶段物流网络规划模型。对于模型中不确定参数均为规则模糊数的这一类模糊两阶段规划模型,本文通过理论分析和证明将其转化为等价的确定一阶段规划模型进行求解,从而将无穷维的优化问题转化为有限维的经典优化问题,降低了计算难度且得到了模型的精确解。不同规模的数值实验证实了所提出模型及其求解方法的有效性。     

一个物体运动的优化模型及应用

建立了一类具广泛应用价值的物体运动非线性泛函优化模型,包括目标泛函,决策函数,约束条件,可行函数空间.决策函数是能量消耗分配函数,可行函数空间中的能量消耗分配函数确定目标泛函值,该模型的最优解是使目标泛函值最大的能量分配函数.这个非线性泛函优化模型,表述了一类物体运动能量转化为机械功的实际问题.例如机动车行驶中如何控制燃料消耗方式,使燃油消耗最少.运动员在赛跑中如何分配体能消耗使成绩最好等.该文从非线性泛函变分及优化理论角度对该模型进行了定量探讨.所得结果可应用于物体运动功能转化相关实际问题中.该文也提出了若干公开问题.     

混合动力公交车能量控制策略的优化模型

针对混合动力公交车在循环工况内功率需求的特点,建立了未来功率需求贝叶斯预测模型;利用2-阶段随机动态规划模型将大规模的随机动态规划问题简化为多个小规模的随机动态规划问题和一个确定型动态规划问题;对于随机动态规划模型的求解,给出了稀疏表示的降维方法,将复杂的泛函极值问题转化为常规的随机动态优化问题,并采用分布估计算法和计算资源最优配置算法的计算机仿真优化算法对随机动态优化问题进行求解;给出了基于查表的在线控制策略,为模型的实际应用进行了有益的探索。     

基于时间满意度的应急物资储备库选址问题

基于应急物资配送过程中时间因素的重要性,将时间满意度引人应急物资储备库选址问题中.针对时间满意度为线性分段函数,建立了以时间满意度最小的需求点的时间满意度尽量大以及系统总费用最小为目标的双目标混合整数规划模型,对目标函数的最小最大值问题进行转化,在此基础上构造新的优化模型,并设计了相应的启发式算法求解.最后通过算例说明算法的可行性和有效性.     

面向节能的单/多列车优化决策问题

主要研究了单/多列车运行优化控制问题.首先建立了面向节能的单列车能耗优化模型,采用节能控制策略对模型进行求解,得出节能运行的速度距离曲线;其次对多列车多区段的节能运行进行优化控制设计,以再生能量利用最大化为目标,分别建立多列车能耗优化通用模型和高峰/非高峰情形下优化模型,利用模拟退火算法求解模型,得出使总能耗最低的列车运行方案;最后针对晚点情况下追踪运行的多列车运行优化控制问题,分别建立随机和非随机晚点情况下实时控制模型,求解得到耗能最少的列车运行曲线.     

基于参数化水平集法的材料非线性子结构拓扑优化

针对利用传统水平集法进行非线性结构拓扑优化计算过程复杂及计算效率低等问题,将参数化水平集方法引入材料非线性结构拓扑优化中。通过全局径向基函数插值初始水平集函数,建立了以插值系数为设计变量、结构的应变能最小为目标函数、材料用量为约束条件的材料非线性结构拓扑优化模型,利用有限元分析对材料非线性结构建立平衡方程,并用迭代法求解。同时,采用子结构法划分设计区域为若干个子区域,将全自由度平衡方程的求解分解为缩减的平衡方程和多个子结构内部位移的求解,减小了计算成本。算例表明,这种处理非线性关系的方法可以在保证数值稳定的同时提高计算效率,得到边界清晰、结构合理的拓扑优化构形。     

广义锥函数类的变度量法

本文针对广义锥函数类模型建立的算法,把n维空间的极小化问题,转化为m维空间的极小化问题。同时,建立了基于广义锥函数模型求解一般无约束问题的算法。     

Optimal feedback control of bilinear systems

Feedback synthesis of optimal constrained controls for single-input bilinear systems is considered. Quadratic cost functionals (with and without quadratic control penalization) are modified by the inclusion of additional nonnegative state penalizing functions in the respective cost integrands. The latter functions are chosen so as to regularize the problems, in the sense that feedback solutions of particularly simple form are obtained. Finite and infinite time horizon problem formulations are treated, and associated aspects of feedback stabilization of bilinear systems are discussed.     

Bang-bang control of a plant with one positive real pole and one negative real pole

In this paper, a relay-controlled plant, possessing one positive real pole and one negative real pole, is studied. First, a regulating system is considered; we investigate the design of the controller for simultaneous reduction of error and error derivative to zero. It is seen that this can be achieved provided the initial values of error and error derivative fall inside an admissible (or controllable) region. Attention is then focused on the tracking system. It is shown that, provided the input to the system is of a form similar to the impulse response of the plant, and provided the initial values of error and error derivative of the system fall inside the same admissible region, then the controller used for regulating the system can also be used for tracking the system so as to force the output of the plant to follow the input perfectly; in both cases with at mostone switching reversal of the relay.     

Computational Method for Time-Optimal Switching Control

An efficient algorithm, called the time-optimal switching (TOS) algorithm, is proposed for the time-optimal switching control of nonlinear systems with a single control input. The problem is formulated in the arc times space, arc times being the durations of the arcs. A feasible switching control, or as a special case bang-bang control, is found using the STC method previously developed by the authors to get from an initial point to a target point with a given number of switchings. Then, by means of constrained optimization techniques, the cost being considered as the summation of the arc times, a minimum-time switching control solution is obtained. Example applications of the TOS algorithm involving second-order and third-order systems are presented. Comparisons are made with a well-known general optimal control software package to demonstrate the efficiency of the algorithm.     

Efficient robust optimization for robust control with constraints

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.     

Finding candidate singular optimal controls: A state of the art survey

Pontryagin's maximum principle gives no information about a singular optimal control if the problem is linear. This survey shows how candidate singular optimal controls may be found for linear and nonlinear problems. A theorem is given on the maximum order of a linear singular problem.This paper is based in part on the research undertaken by the author at the Hatfield Polytechnic, Hatfield, Hertfordshire, England, for the Ph.D. Degree.     

The Kumar-Becker-Lin scheme revisited

The Kumar-Becker-Lin scheme introduces a slowly vanishing cost bias in the parameter estimation part of self-tuning control in order to improve its performance. This paper establishes the a.s. optimality of a variant of this scheme for Markov chains on a countable state space when the action space is compact metric and the parameter space is a compact subset ofR ^m .     

Model-based and model-free control of flexible-link robots: A comparison between representative methods

The paper presents a comparative study on representative methods for model-based and model-free control of flexible-link robots. Model-based techniques for the control of flexible-link robots can come up against limitations when an accurate model is unavailable, due to parameters uncertainty or truncation of high order vibration modes. On the other hand, several research papers argue that suitable model-free control methods result in satisfactory performance of flexible-link robots. In this paper two model-free approaches of flexible-link robot control are examined: (i) energy-based control, and (ii) neural adaptive control. The performance of the aforementioned methods is compared to the inverse dynamics model-based control, in a simulation case study for planar 2-DOF manipulators.     

Time-optimal control of multivariable systems near the origin

This article deals with bang-bang solutions of linear time-optimal control problems. Linear multivariable systems are considered which have one or more control components. It is shown in which way the control components act together to make the system achieve the ultimate aim (namely, the origin in the state space) as quickly as possible. The theory only applies to initial positions sufficiently near the origin. Criteria are given which give the number of switches per control component.Asymptotic dependences of the switching times and the final time on the distance of the initial position from the origin are established. The theory provides a numerical procedure to calculate the time-optimal control. These calculations are very simple. Basic to the proof of these results is a generalized implicit function theorem due to Artin (Ref. 1).     

A data-driven switching control approach for braking systems with constraints

Braking control is of paramount importance in guaranteeing driving safety and comfort, but it is a well-known challenging task, due to the highly nonlinear and road condition-dependent behavior of the vehicle. Existing braking controllers typically rely on accurate models of the vehicle dynamics and the vehicle–road interaction, which are quite difficult to be retrieved in practice. In the wake of the data-driven control paradigm, we propose a model-free and fully data-based braking control method. The architecture of our scheme is two-layered, featuring: an inner switching controller, directly designed from data to match a given closed-loop behavior, and an outer predictive reference governor, exploited to enforce constraints and possibly improve the overall braking performance. The effectiveness of the approach is shown in a simulation environment, by providing a sensitivity analysis to the main tuning knobs of the method.     

Optimal projection equations for reduced-order modelling,estimation, and control of linear systems with multiplicative white noise

The optimal projection equations for quadratically optimal reduced-order modelling, estimation, and control are generalized to include the effects of state, control, and measurement dependent noise.