基于动态规划的约束优化问题多参数规划求解方法及应用

结合动态规划和单步多参数二次规划, 提出一种新的约束优化控制问题多参数规划求解方法. 一方面能得到约束线性二次优化控制问题最优控制序列与状态之间的显式函数关系, 减少多参数规划问题求解的工作量; 另一方面能够同时求解得到状态反馈最优控制律. 应用本文提出的多参数二次规划求解方法, 建立无限时间约束优化问题状态反馈显式最优控制律. 针对电梯机械系统振动控制模型做了数值仿真计算.     

基于拟线性化和Haar 函数的最优控制问题的直接求解方法

针对有约束条件的非线性最优控制问题,提出一种基于拟线性化和Haar函数的数值求解方法.首先将最优控制问题转化为一系列的二次规划问题,并使用系数未知的Haar函数对问题中的状态变量进行近似;然后应用拟线性化法将原非线性最优控制问题转化为相应的一系列受限的二次最优控制问题进行求解;最后基于所提出的方法对2个受限非线性最优控制问题进行求解,并通过仿真结果表明了采用所提出的算法求解最优控制问题的有效性.     

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.     

基于Radau伪谱法的非线性最优控制问题的收敛性

在过去的10年里,伪谱方法(如Legendre伪谱法、Gauss伪谱法、Radau伪谱法)逐步成为求解不同领域中非线性最优控制问题的一种高效、灵活的数值解法.本文从最优控制问题解的存在性、收敛性以及解的可行性3个方面对采用Radau伪谱法求解一般非线性最优控制问题解的收敛性进行研究.证明了原最优控制问题的离散解存在、存在收敛到原最优控制问题解上的离散解和离散形式的收敛解是原最优控制问题的最优解.在此基础上,证明了Radau伪谱法的收敛性.本文结论与现有文献相比,去掉了一些必要条件,更适合一般的非线性时不变系统.     

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.     

挡位离散型车辆经济性加速策略的伪谱法优化

加速过程中, 车辆的油耗与驾驶员的操作策略密切相关. 本文通过最优控制方法定量化地研究了挡位离散型车辆的经济性加速策略. 将加速策略的辨识构建为一个Bolza型最优控制问题(Optimal control problem, OCP), 设计了考虑加速距离影响的经济性定量评价指标. 该问题含有离散型控制变量, 隶属于混合整型最优控制问题, 且性能函数和状态方程呈现强非线性. 为高效地求解该问题, 结合变速器挡位切换规律, 将该整型问题转化为多段光滑问题的协同优化, 采用Legendre伪谱拼接法实现变速器挡位、换挡时机、发动机力矩的数值求解. 解析分析了经济性加速策略的形成机理, 总结了实用化的经济性加速度选择策略和挡位切换规律. 仿真验证了所求策略的节油潜力.     

含扩散项不可靠生产系统最优生产控制的数值求解

针对含扩散项不可靠随机生产系统最优生产控制的优化命题, 采用数值解方法来求解该优化命题最优控制所满足的模态耦合的非线性偏微分HJB方程. 首先构造Markov链来近似生产系统状态演化, 并基于局部一致性原理, 把求解连续时间随机控制问题转化为求解离散时间的Markov决策过程问题, 然后采用数值迭代和策略迭代算法来实现最优控制数值求解过程. 文末仿真结果验证了该方法的正确性和有效性.     

CG伪谱法在欠驱动航天器姿态机动最优控制中的应用

使用Chebyshev-Gauss(CG)伪谱法研究带动量轮和推力器的欠驱动航天器姿态最优控制问题.基于欧拉姿态角和动量矩定理导出两类航天器姿态运动模型,采用Clenshaw-Curtis积分近似得到性能指标函数中的积分项,应用重心拉格朗日插值逼近状态变量和控制变量,将连续最优控制问题离散为具有代数约束的非线性规划(NLP)问题,通过序列二次规划(SQP)算法求解.数值仿真结果表明,对两类欠驱动航天器的姿态机动最优控制均能达到设计控制要求,得到的姿态最优曲线与验证得到的曲线几乎完全重叠.     

线性混杂系统优化控制的Monte Carlo统计预测方法

针对含扩散项的线性混杂切换系统优化控制问题, 为降低优化求解的计算复杂性, 提出了Monte Carlo统计预测方法. 首先通过数值求解技术把连续时间优化控制问题转化为离散时间的Markov决策过程问题; 然后在若干有限状态子空间内, 利用反射边界技术来求解相应子空间的最优控制策略; 最后根据最优控制策略的结构特性, 采用统计预测方法来预测出整个状态空间的最优控制策略. 该方法能有效降低求解涉及大状态空间及多维变量的线性混杂切换系统优化控制的计算复杂性, 文末的仿真结果验证了方法的有效性.     

求解最优控制问题的混合变量变分方法及其航天控制应用

针对非线性最优控制导出的Hamiltonian系统两点边值问题,提出一种以离散区段右端状态和左端协态为混合独立变量的数值求解方法, 将非线性Hamiltonian系统两点边值问题的求解通过混合独立变量变分原理转化为非线性方程组求解.所提出的算法综合了求解最优控制 的"直接法"和"间接法"的特征,既满足最优控制理论的一阶必要条件,又不需要对协态初值的准确猜测,避免了求解大规模非线性规划问题. 通过两个航天控制算例讨论了本文算法的精度和效率等问题.与近年来在航空航天控制中备受关注的高斯伪谱方法相比较,本文算法无论是在 精度还是效率上都具有明显的优势.     

Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution

In this article, the Legendre wavelet operational matrix of integration is used to solve boundary ordinary differential equations with non-analytic solution. Although the standard Galerkin method using Legendre polynomials does not work well for solving ordinary differential equations in which at least one of the coefficient functions or solution function is not analytic, it is shown that the Legendre wavelet Galerkin method is very efficient and suitable for solving this kind of problems. Several numerical examples are given to illustrate the efficiency and performance of the presented method.     

A numerical approach of optimal control for generalized delay systems by general Legendre wavelets

A general Legendre wavelets approach is presented. The delay function is expanded by general Legendre wavelets. The general operational matrix of integration is introduced. By the general Legendre wavelets, the linear-quadratic problem of generalized delay systems are transformed into the optimization problem of multivariate functions. The approximate solutions of the optimal control and state as well as the optimal value of the objective functional are derived. The numerical examples demonstrate that the algorithms are valid.     

小推力轨道转移快速优化设计

在研究电推进系统中,为满足小推力转移轨道高精度在线生成的要求,伪光谱方法在电推进小推力轨道转移优化设计中的应用。首先对小推力航天器轨道转移最优控制问题模型进行无量纲化处理,以提高优化算法求解精度。然后采用基于勒让德-高斯-兰伯特配置点的勒让德伪光谱方法,将最优控制问题离散成约束参数优化问题,再利用适于求解大尺度非线性规划问题的TOMLAB/SNOPT优化软件包进行求解。通过数值仿真计算,求解生成了满足各类约束条件的小推力转移轨道,并利用余向量映射定理及极小值原理验证了所得轨道转移控制量的最优性。结果表明,勒让德伪光谱优化算法具有对初始猜测值不敏感、收敛速度快、精度高等优点。     

关于一般耦合矩阵方程的迭代对称解

本文构造了一个有效的迭代方法(CGL)去求解一般耦合矩阵方程的对称解.若一般耦合矩阵方程关于对称解相容,则对于任意给定的初始对称矩阵组,利用所构造的迭代算法,都能在有限步迭代出所求问题的一组对称解,若选用一些特殊的初值,则可获得矩阵方程的极小范数对称解.最后的数值例子表明了所给算法的有效性.     

Sparsity-promoting polynomial response surface: A new surrogate model for response prediction

Computation-intensive analyses/simulations are becoming increasingly common in engineering design problems. To improve the computation efficiency, surrogate models are used to replace expensive simulations of engineering problems. This paper proposes a new high-fidelity surrogate modeling approach which is called the Sparsity-promoting Polynomial Response Surface (SPPRS). In the SPPRS model, a series of Legendre polynomials is selected as basis functions, and its number is compatible with the sample size so as to enhance the expression ability for complex functional relationships. The coefficients associated with basis functions are estimated using a “sparsity-promoting” regression approach which is an ensemble of two techniques: least squares and ℓ₁-norm regularization. As a result, only these basis functions relevant to explain the function relationship are picked out, and that dedicates to ease the problem of overfitting for training points. With the sparsity-promoting regression approach, such a surrogate model intends to capture both the global trend of the functional variation and a reasonable local accuracy in the neighborhood of training points. Additionally, Latin hypercube design (LHD) is proved conducive to improving the predictive capability of our model. The SPPRS is applied to seven benchmark test functions and a complex engineering problem. The results illustrate the promising benefits of this novel surrogate modeling technique.     

一种新型神经网络模型在数据挖掘中的应用

针对传统神经网络存在的问题,提出一种新型神经网络模型,即Legendre前向神经网络模型,进而在该模型的基础上提出一种基于伪逆的权值直接确定法(即一步确定法),并将其应用到数据挖掘中.仿真结果显示该方法不仅能更快地收敛,而且可以达到更高的工作精度.     

Modelling fingerprint ridge orientation using Legendre polynomials

The estimation of fingerprint ridge orientation is an essential step in every automatic fingerprint verification system. The importance of ridge orientation can be deflected from the fact that it is inevitably used for detecting, describing and matching fingerprint features such as minutiae and singular points. In this paper we propose a novel method for fingerprint ridge orientation modelling using Legendre polynomials. One of the main problems it addresses is smoothing orientation data while preserving details in high curvature areas, especially singular points. We show that singular points, which result in a discontinuous orientation field, can be modelled by the zero-poles of Legendre polynomials. The models parameters are obtained in a two staged optimization procedure. Another advantage of the proposed method is a very compact representation of the orientation field, using only 56 coefficients. We have carried out extensive experiments using a state-of-the-art fingerprint matcher and a singular point detector. Moreover, we compared the proposed method with other state-of-the-art fingerprint orientation estimation algorithms. We can report significant improvements in both singular point detection and matching rates.     

Discontinuous Legendre wavelet Galerkin method for reaction–diffusion equation

This paper proposes a novel numerical method, that is, discontinuous Legendre wavelet Galerkin technique for solving reaction–diffusion equation (RDE). Specifically, variational formulation and corresponding numerical fluxes of this type equation are devised by utilizing the advantages of both Legendre wavelet bases and discontinuous Galerkin approach. Furthermore, adaptive algorithm, stability and error analysis of this method have been discussed. Especially, the distinctive features of the presented approach are easy to cope with a variety of boundary conditions and able to effectively approximate solution of the RDE with less execution and storage space. Finally, numerical tests affirm better accuracy for a range of benchmark problems and demonstrate the validity and utility of this approach.     

Piecewise fractional Legendre functions for nonlinear fractional optimal control problems with ABC fractional derivative and non-smooth solutions

In this study, to solve fractional problems with non-smooth solutions (which include some terms in the form of piecewise or fractional powers), a new category of basis functions called the orthonormal piecewise fractional Legendre functions is introduced. The upper bound of the error of the series expansion of these functions is obtained. Two explicit formulas for computing the Riemann–Liouville and Atangana–Baleanu fractional integrals of these functions are derived. A direct method based on these functions and their fractional integral is proposed to solve a family of optimal control problems involving the ABC fractional differentiation whose solutions are non-smooth in the above expressed forms. By the proposed technique, solving the original fractional problem turns into solving an equivalent system of algebraic equations. The established method accuracy is studied by solving some examples.