基于Hamilton-Jacobi方程的飞行器机动动作可达集分析

为了给驾驶员完成标准机动动作提供决策支持, 提出一种使用哈密尔顿-雅克比(Hamilton-Jacobi)方程求解机动动作可行状态空间的研究方法.使用关键点将机动动作划分为不同阶段, 将各关键点的标准状态约束作为目标集, 逆时间求解目标集对应的可达集得到各阶段的边界状态范围, 目标集和可达集均由零水平集表示.使用该方法得到斤斗动作三维度运动模型下各阶段的可达集及斤斗动作的可行状态空间, 为了使运动模型的控制量与驾驶员实际操纵更为接近, 构建了以迎角变化率为控制量的四维度运动模型, 在此基础上对斤斗动作各阶段的可达集进行了分析.     

具有自适应噪声边界的Tube可达集鲁棒预测控制

针对过程噪声设定边界与真实噪声边界失配的有界干扰离散线性不确定系统,提出一种具有自适应噪声边界的Tube可达集鲁棒模型预测控制方法.首先,该算法引入基于MIT规则的自适应集员滤波在线估计系统状态和噪声边界.其次,基于估计值,通过迭代自适应集员滤波的时间更新部分计算出预测时域内闭环不确定系统状态的可达集.最后,用可达集代替不变集并根据Tube鲁棒模型预测控制策略,给出了实际不确定系统的控制律,确保系统状态鲁棒渐近稳定,并收敛于终端干扰不变集.仿真结果验证了该控制方法的有效性.     

双曲型守恒律方程的小波解法

本文基于Hamilton-Jacobi方程的小波Galerkin近似和微分算子的小波表示,讨论一维双曲型守恒律方程初值问题的Daubechies小波解．由于小波在空间和时间上的局部性,本方法适用于处理具有奇异解的问题,可以有效的防止数值振荡．数值试验的结果表明,本方法是可行的．     

一种基于AOS格式的多相水平集快速分割方法

采用迎风格式的水平集算法实现需要在曲线演化过程中重新初始化水平集函数的要求,为保证算法的稳定,时间步长选取较小值,算法运行速度较慢。文中基于无须重新初始化的水平集方法,在算法数值实现中引入AOS半隐格式,对基于不同统计模型的水平集分割算法给出统一的数值实现。以二相水平集分割算法为基础提出一种新的多相水平集分割方法。该方法采用一个水平集函数进行多次演化实现多区域分割,其优点包括：1)采用AOS半隐格式,该格式无条件稳定,可采用较大的时间步长;2)对多个统计模型进行统一处理;3)采用单一的水平集函数进行演化,减少水平集演化方程的数量,算法更加灵活。实验结果表明,该方法具有较快的分割速度,对具有多个区域的图像能够进行较准确的分割。     

一类离散时间线性切换系统的二次鲁棒镇定

针对一类具有范数有界时变不确定性的离散时间线性切换系统,研究了其二次稳定化状态反馈控制律的设计问题.利用多李亚普诺夫函数法推导了在任意切换下二次稳定化控制律存在的充分条件,该条件被进一步等价地表示成线性矩阵不等式的可解性问题.同时它的解提供了二次稳定化控制律的一个参数化表示.仿真结果验证了所提方法的有效性.     

三维图像多相分割的变分水平集方法

变分水平集方法是图像分割等领域出现的新的建模方法,借助多个水平集函数可有效地实现图像多相分割.但在区域/相的通用表达、不同区域内图像模型的表达、通用的能量函的设计、高维图像分割中的拓展研究等方面仍是图像处理的变分方法、水平集方法、偏微分方程方法等研究的热点问题.文中以三维图像为研究对象,系统地建立了一种新的三维图像多相分割的变分水平集方法.该方法用n-1个水平集函数划分n个区域,并基于Heaviside函数设汁出区域划分的通用的特征函数;其能量泛函包括通用的区域模型、边缘检测模型和水平集函数为符号距离函数的约束项3部分;最后,针对所得到的曲面演化方程,采用半隐式差分格式进行离散,并对多种类型三维图像进行分割验证了所提出模型的通用性和有效性.     

输入状态稳定的鲁棒预测控制

以有界干扰非线性系统为研究对象,设计一种基于近似可达集的鲁棒预测控制方法。该方法以鲁棒控制不变集作为终端约束集,采用一种简单的三次多项式逼近预测控制的待优化控制律,通过在线优化求解三次多项式的各项系数,并从理论上证明了所设计的鲁棒预测控制律可以使系统输入状态稳定。最后通过仿真实例验证了所提出的鲁棒预测控制方法的可行性和有效性。     

具有饱和执行器线性系统的综合PLCöVSC 控制

基于LQ理论和变结构方法,导出一种带有饱和执行机构和不确定性线性系统的混合PLC／VSC控制律．引入一组嵌套的Lyapunov级集,并根据分段Lyapunov函数导出控制律,该控制律含有不连续控制项和趋近率项．利用Lyapunov函数方法分析了闭环系统的稳定性．结合简化单摆系统验证了设计方法的有效性,在抗干扰能力和减少超调量上反映出所提出的控制器具有比PLC和PLC／LHG更佳的性能．     

基于N步不变集的时滞切换系统的饱和控制

构造离散时滞切换系统的不变集,提出基于N步不变集的切换控制器设计方法,估计执行器饱和非线性的吸引域范围。首先,考虑时滞的影响,选取依赖于时滞的Lyapunov函数,构造时滞切换系统的不变集,并将其表达为若干个椭球集的凸组合,椭球集的个数与时滞常数相关。其次,在系统的前N个采样时刻,分别施加不同的饱和约束,求解得到一组椭球集,椭球集的个数与常数N相关,而每一步计算得到的椭球集均为时滞切换系统的不变集。再将N个不变集用一组凸包系数拟合,即可获取较大的吸引域估计。最后,在满足平均驻留时间约束的条件下设计切换律,并设计状态反馈控制器,保证闭环系统渐近稳定。控制器的求解转化为线性矩阵不等式的可行性问题。仿真结果验证了所提方法的可行性和有效性。     

非线性预测控制终端约束集的优化

为保证预测控制的稳定性,经典的策略是在预测控制的优化问题中加入终端约束集和终端惩罚函数,并保证终端约束集是一个在终端控制律作用下的正不变集,终端惩罚函数是受控系统的局部控制Lyapunov函数.本文提供了一种求解非线性系统终端约束集、终端控制律和终端惩罚函数的新策略.通过在优化问题中引入新的变量来降低求解终端约束条件的...     

A proximal characterization of the reachable set

We show that the graph of the reachable set of a control system given by a differential inclusion is uniquely characterized by a Hamilton-Jacobi equation involving proximal normals.     

A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear     

Overapproximating Reachable Sets by Hamilton-Jacobi Projections

In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). We have developed a numerical tool—based on the level set methods of Osher and Sethian—for computing these sets, and we can accurately calculate them for a range of continuous and hybrid systems in which control inputs are pitted against disturbance inputs. The cost of our algorithm, like that of all convergent numerical schemes, increases exponentially with the dimension of the state space. In this paper, we devise and implement a method that projects the true reachable set of a high dimensional system into a collection of lower dimensional subspaces where computation is less expensive. We formulate a method to evolve the lower dimensional reachable sets such that they are each an overapproximation of the full reachable set, and thus their intersection will also be an overapproximation of the reachable set. The method uses a lower dimensional HJI PDE for each projection with a set of disturbance inputs augmented with the unmodeled dimensions of that projection's subspace. We illustrate our method on two examples in three dimensions using two dimensional projections, and we discuss issues related to the selection of appropriate projection subspaces.     

Properties of the reachable set of control systems

We investigate some properties of the reachable set of a control system. Representing the system as a differential inclusion and using proximal Hamilton–Jacobi equation we describe its graph. We work in infinitely dimensional Hilbert space and use one sided Lipschitz approach. The funnel equation is considered in the last section. That equation describes the reachable set in arbitrary Banach space. We consider also the autonomous case and prove the existence of a limit of the reachable set.     

基于可达集的鲁棒模型预测控制

设计了一种基于可达集的鲁棒模型预测控制算法.首先确定了一个鲁棒不变集,并将此不变集用作模型预测控制的终端约束集;接着采用终端约束集对可达集的包含度作为优化指标;最后,采用预测时域逐渐减小的控制策略以保证在线优化存在可行解.从理论上证明了吸引域内的任意点在有限时域内都会被引导至终端约束集并始终停留在此集之内,并由仿真算例验证了本文所设计鲁棒模型预测控制算法的可行性.     

Determination of optimal feedback terminal controllers for general boundary conditions using generating functions

Given a nonlinear system and a performance index to be minimized, we present a general approach to expressing the finite time optimal feedback control law applicable to different types of boundary conditions. Starting from the necessary conditions for optimality represented by a Hamiltonian system, we solve the Hamilton-Jacobi equation for a generating function for a specific canonical transformation. This enables us to obtain the optimal feedback control for fundamentally different sets of boundary conditions only using a series of algebraic manipulations and partial differentiations. Furthermore, the proposed approach reveals an insight that the optimal cost functions for a given dynamical system can be decomposed into a single generating function that is only a function of the dynamics plus a term representing the boundary conditions. This result is formalized as a theorem. The whole procedure provides an advantage over methods rooted in dynamic programming, which require one to solve the Hamilton-Jacobi-Bellman equation repetitively for each type of boundary condition. The cost of this favorable versatility is doubling the dimension of the partial differential equation to be solved.     

Topological shape optimization of geometrically nonlinear structures using level set method

Using the level set method, a topological shape optimization method is developed for geometrically nonlinear structures in total Lagrangian formulation. The structural boundaries are implicitly represented by the level set function, obtainable from “Hamilton-Jacobi type” equation with “up-wind scheme,” embedded into a fixed initial domain. The method minimizes the compliance through the variations of implicit boundary, satisfying an allowable volume requirement. The required velocity field to solve the Hamilton-Jacobi equation is determined by the descent direction of Lagrangian derived from an optimality condition. Since the homogeneous material property and implicit boundary are utilized, the convergence difficulty is significantly relieved.     

Simultaneous optimization of the material properties and the topology of functionally graded structures

A level set based method is proposed for the simultaneous optimization of the material properties and the topology of functionally graded structures. The objective of the present study is to determine the optimal material properties (via the material volume fractions) and the structural topology to maximize the performance of the structure in a given application. In the proposed method, the volume fraction and the structural boundary are considered as the design variables, with the former being discretized as a scalar field and the latter being implicitly represented by the level set method. To perform simultaneous optimization, the two design variables are integrated into a common objective functional. Sensitivity analysis is conducted to obtain the descent directions. The optimization process is then expressed as the solution to a coupled Hamilton-Jacobi equation and diffusion partial differential equation. Numerical results are provided for the problem of mean compliance optimization in two dimensions.