集装箱装卸摆动最优控制快速数值求解算法

为了实现起重机集装箱摆动最优控制,提出一种基于控制向量参数化(CVP)方法的最优控制问题快速求解算法.首先,建立了以摆动能量最小为目标的集装箱装卸最优控制数学模型.其次,采用光滑化惩罚函数路径约束处理方法降低了模型求解难度.进一步,针对控制向量参数化方法微分方程组求解耗时长难题,结合网格划分提出改进四阶Runge--Kutta方法的快速CVP算法加快了最优控制问题求解速度.仿真测试针对不同位置的集装箱装卸任务进行.数值测试结果显示,相较于其他变步长求解方法,改进方法在得到相近求解精度解的同时,求解耗时明显减少,表明本文方法在集装箱装卸最优控制方面的应用价值.     

离散时间最优控制—–评论动态规划

阐述离散时间最优控制的特点.对比3种求解离散时间最优控制的解法,即:1)用非线性规划求解离散时间最优控制;2)用无约束优化求解离散时间最优控制;3)动态规划及其数值解.1)和2)都适用于多维静态优化,计算效率较高,是高级方法.在名义上,3)为动态优化.实际上,3)为一维分段无约束静态优化,计算效率较低,是初级方法.本文并用数字实例进一步阐明动态规划及其数值解在求解方面较差,故动态规划及其数值解已失去实用价值.在求解离散时间最优控制问题方面,无法与非线性规划求解相匹敌.     

非线性动力学系统最优控制问题的保辛求解方法

基于对偶变量变分原理提出了求解非线性动力学系统最优控制问题的一种保辛数值方法.以时间区段一端状态和另一端协态作为混合独立变量,在时间区段内采用拉格朗日插值近似状态变量与协态变量,然后利用对偶变量变分原理并将非线性最优控制问题转化为非线性方程组的求解,最终得到求解非线性动力学系统最优控制问题的保辛数值方法.数值实验验证了本文算法在求解精度与求解效率上的有效性.     

计算机数控系统光滑时间最优轨迹规划

基于控制向量参数化(CVP)方法, 研究了计算机数控(CNC)系统光滑时间最优轨迹规划方法. 通过在规划问题中引入加加速度约束, 实现轨迹的光滑给进. 引入时间归一化因子, 将加加速度约束的时间最优轨迹规划问题转化为固定时间的一般性最优控制问题. 以路径参数对时间的三阶导数(伪加加速度)和终端时刻为优化变量, 并采用分段常数近似伪加加速度, 将最优控制问题转化为一般的非线性规划(NLP)问题进行求解. 针对加加速度、加速度等过程不等式约束, 引入约束凝聚函数, 将过程约束转化为终端时刻约束, 从而显著减少约束计算. 构造目标和约束函数的Hamiltonian函数, 利用伴随方法获得求解NLP问题所需的梯度.     

带不等式路径约束最优控制问题的惩罚函数法

控制变量参数化（Control variable parameterization,CVP）方法是目前求解流程工业中最优操作问题的主流数值方法,但如果问题中包含路径约束,特别是不等式路径约束时,CVP方法则需要考虑专门的处理手段.为了克服该缺点,本文提出一种基于L1精确惩罚函数的方法,能够有效处理关于控制变量、状态变量、甚至控制变量/状态变量复杂耦合形式下的不等式路径约束.此外,为了能使用基于梯度的成熟优化算法,本文还引进了最新出现的光滑化技巧对非光滑的惩罚项进行磨光.最终得到了能高效处理不等式路径约束的改进型CVP架构,并给出相应数值算法.经典的带不等式路径约束最优控制问题上的测试结果及与国外文献报道的比较研究表明:本文所提出的改进型CVP 架构及相应算法在精度和效率上兼有良好表现.     

求解量子系统时间最优控制问题的同伦算法研究

针对数值求解量子系统时间最优控制问题中反复调用梯度算法导致计算量大的问题,本文提出一类同伦算法用以快速求解量子系统的时间最优控制问题.与已有算法不同,这一算法通过引入同伦变量在减小终端时间的方向上搜索最优解.在这一算法中,可通过自由函数构造保真度函数对控制变量的梯度方向,也可通过方向函数引导算法的搜索方向,以加快算法的搜索速度.本文将这一算法用于求解量子系统态转移和门变换的时间最优控制问题.仿真结果表明这一算法的有效性.     

一类控制受约束非线性系统的基于单网络贪婪迭代DHP算法的近似最优镇定

提出一种贪婪迭代DHP (Dual heuristic programming)算法, 解决了一类控制受约束非线性系统的近似最优镇定问题. 针对系统的控制约束, 首先引入一个非二次泛函把约束问题转换为无约束问题, 然后基于协状态函数提出一种贪婪迭代DHP算法以求解系统的HJB (Hamilton-Jacobi-Bellman)方程. 在算法的每个迭代步, 利用一个神经网络来近似系统的协状态函数, 而后根据协状态函数直接计算系统的最优控制策略, 从而消除了常规近似动态规划方法中的控制网络. 最后通过两个仿真例子证明了本文提出的最优控制方案的有效性和可行性.     

带ε误差限的近似最优控制

近似动态规划方法求解非线性系统最优控制, 需要迭代无限步才能得到最优控制律. 本文提出了一种ε–近似最优控制算法, 选择ε误差限, 通过自适应迭代不断逼近哈密顿– 雅可比– 贝尔曼(HJB)方程的解, 应用神经网络实现在有限步迭代后得到带ε误差限的近似最优控制律. 计算机仿真结果表明了该算法的有效性.     

自由时间最优控制问题的一种控制向量参数化方法

针对自由时间最优控制问题,提出一种控制向量参数化（CVP）方法.通过引入时间尺度因子,将自由时间最优控制问题转化为固定时间问题,并将终端时刻作为优化参数.基于CVP方法,最优控制问题被转化为一个非线性规划（NLP）问题.建立目标和约束函数的Hamiltonian函数,通过求解伴随方程获得目标和约束函数的梯度,采用序列二次规划（SQP）方法获得问题的数值解.对于控制有切换结构的优化问题,给出了一种网格精细化策略,以提高控制质量.补料分批反应器最优控制问题的仿真实验验证了所提出方法的有效性.     

线性–非二次最优控制问题的一种解法

讨论了求解状态终端无约束线性–非二次最优控制问题的拟Riccati方程方法, 并据此提出了计算无约束线性–非二次问题之数值解的方法; 然后将这个方法与一种能近似地化有约束问题为无约束问题的惩罚方法结合起来, 给出了一种算法, 可以计算状态终端有约束的线性–非二次最优控制问题之近似解.     

On solving hybrid optimal control problems with higher index DAEs

The paper deals with hybrid optimal control problems described by higher index differential–algebraic equations (DAEs). We introduce a numerical procedure for solving these problems. The procedure has the following features: it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of systems equations by an implicit Runge–Kutta method; the consistent initialization procedure is applied whenever control functions jumps, or state variables transition occurs. The procedure can cope with hybrid optimal control problems which are defined by DAEs with the index not exceeding three. Our approach does not require differentiation of some system equations in order to transform higher index DAEs to the underlying ordinary differential equations (ODEs). The presented numerical examples show that the proposed approach can be used to solve efficiently hybrid optimal control problems with higher index DAEs.     

LQ终端控制器设计的生成函数方法及其应用

利用哈密顿系统生成函数的性质求解LQ终端控制问题,并给出了相应的数值方法.针对现有文献中此类问题的最优控制律在终端时刻存在无穷大增益的情况,利用第二类生成函数的性质求解哈密顿系统两端边值问题并构造了无终端奇异性的时变最优控制律.然后根据哈密顿系统状态的正则变换性质导出了求解生成函数系数矩阵微分方程和计算时变控制律的矩阵递推格式.最后用所提出的方法研究了以能量均衡消耗为约束条件的卫星编队重构问题,设计了符合要求的闭环控制系统并给出了数值仿真结果.     

Weighted residual methods in optimal control

Weighted residual methods (WRM) afford a viable approach to the numerical solution of differential equations. Application of WRM results in the transformation of differential equations into systems of algebraic equations in the modal coefficients. This suggests that WRM can be used as a tool for reducing optimal control problems to mathematical programming problems. Thereby, the optimal control problem is replaced by the minimization of a cost function of static coefficients subject to algebraic constraints. The motivation for this approach lies in the profusion of sophisticated computational algorithms and digital computer codes for the solution of mathematical programming problems. In this note the solution of optimal control problems as mathematical programming problems via WRM is illustrated. The example presented indicates that reasonable accuracy is obtained for modest computational effort. While the simplest types of modes-polynomials and piecewise constants-are employed in this note, the ideas delineated can be applied in conjunction with cubic splines for the generation of computational algorithms of enhanced efficiency.     

High-resolution compact numerical method for the system of 2D quasi-linear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications

In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach.

     

Fast multilevel augmentation methods for nonlinear boundary value problems

We develop a multilevel augmentation method for solving nonlinear boundary value problems. We first describe the multilevel augmentation method for solving nonlinear operator equations of the second kind which has been proposed in our recent paper Chen et al. (2011) [16], and then apply it to solving the nonlinear two-point boundary value problems of second-order differential equations. The theoretical analysis of convergence order and computational complexity are proposed. Finally numerical experiments are presented to confirm the theoretical estimates and illustrate the efficiency of the method.     

A stochastically averaged optimal control strategy for quasi-Hamiltonian systems with actuator saturation

A stochastic optimal control strategy for quasi-Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method and stochastic dynamical programming principle. First, the partially completed averaged Itô stochastic differential equations for the energy processes of individual degree of freedom are derived by using the stochastic averaging method for quasi-Hamiltonian systems. Then, the dynamical programming equation is established by applying the stochastic dynamical programming principle to the partially completed averaged Itô equations with a performance index. The saturated optimal control consisting of unbounded optimal control and bounded bang-bang control is determined by solving the dynamical programming equation. Numerical results show that the proposed control strategy significantly improves the control efficiency and chattering attenuation of the corresponding bang-bang control.