一类带随机跳跃的完全耦合的线性二次随机控制问题

研究一类带随机跳跃的完全耦合的线性二次随机控制问题. 得到了最优控制的显式解, 并可以证明最优控制是唯一的. 引入了一类推广的黎卡提方程并讨论了其可解性. 利用这一类推广的黎卡提方程的解, 得到了上述带随机跳跃的最优控制问题的线性状态反馈调节器.     

时变时滞不确定性系统的状态反馈控制器设计

研究一类不确定性动态系统.这类系统具有多重时变状态时滞和多重时变控制输 入时滞,其不确定性满足范数有界条件.采用黎卡提方程方法,得到了这类不确定性时滞系 统可状态反馈镇定的充分条件.通过解一个特定的黎卡提不等式,即可得到镇定已知系统的 控制器.     

基于单领航者相对位置测量的多AUV协同导航系统定位性能分析

自主水下航行器 (Autonomous underwater vehicle, AUV) 的协同导航是解决水下导航定位问题的重要方法, 其中导航系统的定位误差增长特性是衡量其定位性能的关键指标. 本文针对单领航者相对位置测量的多 AUV 协同导航系统, 利用扩展卡尔曼滤波方法建立了导航系统的整体定位误差关于相对位置量测误差的传递方程. 在此基础上, 通过求解系统定位误差随时间演化的代数黎卡提方程, 得到了其在稳态情形下的方差上界估计. 理论分析表明, 单领航 AUV 协同导航系统的整体定位误差有界收敛且与初始化滤波方差无关, 具有良好的综合性能. 最后, 仿真实例验证了文中理论分析结果的正确性.     

李雅普诺夫函数优化:正交矩阵构造方案EI北大核心CSCD

本文研究了李雅普诺夫函数的优化问题.提出了一种正交矩阵构造方案,用于求解黎卡提不等式中的最优李雅普诺夫函数.通过分析系统H_(∞)范数的几何特征,本文将黎卡提不等式转换为近似等式,进而给出了最优李雅普诺夫函数的存在条件.基于所给最优李雅普诺夫函数存在条件,所提正交矩阵构造方案利用旋转变换,将非线性方程组的求解问题转换为幅值和角度的线性优化问题,进而实现李雅普诺夫函数参数的优化.研究结果弥补了目前的研究无法求解最优李雅普诺夫函数的不足,对系统性能分析和非保守控制的设计具有建设性.算例验证了所提正交矩阵构造方案的有效性.     

船舶航向非线性H∞逆优化输出反馈控制

为抑制船舶航向非线性优化控制中模型参数摄动和由状态观测器引入的不确定观测误差,提出了一种非线性H∞逆优化控制算法.首先,基于无源理论设计观测器以实现海浪滤波,该观测器无需海浪扰动的方差信息从而减少了观测器参数数量.然后,考虑模型参数摄动对观测误差的影响,给出了描述系统局部(全局)性态的局部(全局)H∞优化性能指标.在以广义黎卡提方程(GARE)对局部优化问题的求解的基础上,应用逆优化方法将全局H∞优化问题转化为构造闭环系统的Lyapunov函数问题,得到同时满足两种指标的优化控制器,并证明了稳定性.仿真结果证明了该算法的有效性.     

多电机驱动系统的保性能鲁棒Funnel控制

本文针对含参数不确定性的多电机驱动系统,提出一种基于最优保性能鲁棒的Funnel控制方法实现系统的规定跟踪性能.该控制方法通过构造Funnel函数对误差系统进行变换,并设计自适应反步控制器保证变换后系统的稳定性即可使跟踪误差的瞬态和稳态响应均被限制在给定的Funnel边界内.然而由于系统中存在的参数不确定性会影响系统的规定控制性能,本文在Funnel控制基础上又设计了最优保性能鲁棒控制器.它是通过将参数不确定性系统的保性能鲁棒控制问题转化为标称系统的最优控制问题,并求解新的黎卡提方程而得到的.因此所设计的控制器不但消除了参数不确定性对系统的影响并且能够使系统的性能指标达到一确定的上界.最后,对四电机驱动系统进行了仿真和实验验证,说明所提出控制方法的有效性.     

不满足匹配条件的非确定性系统的鲁棒滑模控制

针对一类非确定性时变系统实施滑模控制,该系统在状态矩阵和控制输入矩阵中都 包含不满足匹配条件的不确定项.给出了能使系统鲁棒镇定的切换面设计方法和相应的控制 律的形式.同时,也给出了系统渐近稳定的充分条件.设计过程中的代数黎卡提方程和线性矩 阵不等式形式简单,易于求解.仿真结果证明了理论的正确性.     

有界随机测量时滞的网络控制系统的最优估计（英文）

在假设测量没有丢包的情况下,研究了带有随机测量时滞的网络控制系统的最优估计问题.利用已知的时滞分布概率,建立新的模型来描述随机时滞测量.进一步将带有时滞的测量等价成每个通道是单时滞的多通道测量,从而利用新息重组方法,通过求解黎卡提方程求解最优估计器.最后给出仿真实例验证了该算法的有效性.     

有界随机测量时滞的网络控制系统的最优估计

在假设测量没有丢包的情况下, 研究了带有随机测量时滞的网络控制系统的最优估计问题. 利用已知的时 滞分布概率, 建立新的模型来描述随机时滞测量. 进一步将带有时滞的测量等价成每个通道是单时滞的多通道测 量, 从而利用新息重组方法, 通过求解黎卡提方程求解最优估计器. 最后给出仿真实例验证了该算法的有效性.     

非线性H∞导弹纵向自动驾驶仪设计

状态相关黎卡提方程（SDRE）需要在每个采样点求解矩阵Riccati方程,计算成本非常高,不利于实时控制器的实现。将SDRE方法与鲁棒H∞方法相结合,针对导弹纵向自动驾驶仪设计SDRE H∞控制器,利用一种多层回归神经网络结构来进行在线求解代数Riccati方程,从而获得控制器增益。神经网络学习方法采用常规梯度法。仿真结果表明导弹纵向跟踪控制性能好;神经网络在线求解代数Riccati方程有效降低计算成本,且利用硬件实现神经网络求解器。     

H∞滤波问题数值求解的精细积分算法

有限时间H∞滤波的Riccati方程和滤波方程分别为非线性矩阵微分方程和线性变系 数微分方程,而且Riccati微分方程解的存在性还依赖于参数 γ-2,因此求这些方程的数值解一 般比较困难.按照结构力学与最优控制的模拟关系,Riccati方程解存在的临界参数 γ-2cr对应于 广义Rayleigh商的一阶本征值.因此可以用精细积分法结合扩展的Wittrick-Williams(W-W) 算法计算 γ-2cr .并求解Ricclati方程,滤波微分方程的解也可以由精细积分法计算.     

On the solution of linear boundary value problems by extended invariant imbedding

Unstable linear boundary value problems can be solved by the method of Invariant Imbedding in a stable manner. Instead of integration of the system equations this method requires the integration of a matrix Riccati equation, which depends on the boundary values of the problem. The dimension of the Riccati equation is determined by a suitable decoupling of the system equations. Invariant Imbedding now fails, if this decoupling does not correspond with the boundary condition. In addition, the Riccati equation has to be solved once more for each new boundary condition. An extension algorithm is defined, which maps the boundary value problem into a problem of double dimension. This “extended” boundary value is solved by a modified Invariant Imbedding. The resulting “Extended (Dual) Invariant Imbedding” is always applicable and does not depend on the boundary conditions. The corresponding “extended” Riccati equation has to be integrated only once “offline”. If the boundary condition is changed, only systems of linear equations have to be solved “online”.     

Computational structural mechanics. Optimal control and semi-analytical method for PDE

Based on the generalized variational principle the analysis of a substructural chain is considered, and the 1:1 relationship between the structural analysis problem and the linear quadratic optimal control problem is then introduced. Hence, the algebraic Riccati equation can be solved in two ways; the upper-bound and lower-bound iterative methods. The theory and methods of structural analysis problems can then be transferred to the linear quadratic optimal control problems. As to the continuous coordinate, and/or continuous-time problems, it can be shown that the linear quadratic control problem also corresponds to the semi-analytical method of the elliptic partial differential equation. It is hoped that the unified method of these disciplines will lead to further progress.     

有限时间二次型数值算法研究及其应用

为了实际需要和学术发展的要求,研究了以倒立摆为控制对象,通过闭环网络形成的反馈控制系统的随机传输时延的最优控制问题。在求解有限时间最优控制律过程中,通过矩阵Raccati方程的离散变换,利用Matlab中计算无限时间二次型最优控制器的LQR函数,从而求出有限时间LQR问题的数值解。通过仿真结果证明,研究的方法能够使倒立摆系统最终稳定,从而说明提出的算法对于求解有限时间LQR问题是有效的。     

Optimal control system design of a nuclear reactor by generalized spectral factorization

Optimal feedback control system design of a nuclear power plant has usually been conducted in the time domain by solving the Riccati equation. In this paper, the problem will be treated in the s-domain by means of the MacFarlane extension criterion (MEC). The algorithm is based on generalized spectral factorization (GSF), and the MEC is resolved by the Newton-Raphson method featuring rapid convergence and high precision. Whether the plant is stable or unstable, the initial matrix can be chosen effectively and a general computer code for finding the optimal feedback gain matrix of the multi-input system is presented. The control system design of a boiling water reactor (BWR) shows that the results obtained from the proposed method are consistent with those from solving the conventional Riccati equation. A satisfactory closed-loop multivariable system with a prescribed degree of stability can also be designed in the s-domain. The methodology of this paper can also be generalized to treat similar control system design problems in other engineering fields.     

Fault diagnosis and optimal fault-tolerant control for systems with delayed measurements and states

This note deals with the problems of fault diagnosis and fault-tolerant control for systems with delayed measurements and states. The main contribution consists in two aspects. First, by solving the Riccati equation and Sylvester equation, an optimal fault-tolerant control law is designed for systems with delayed measurements and states. The existence and uniqueness of the optimal fault-tolerant control law are proved. Second, the physically unrealizable problem of the optimal fault-tolerant control law is solved by proposing a novel fault diagnoser for systems with delayed measurements and states. Finally, a numerical example is given to demonstrate the feasibility and validity of the proposed schemes.     

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Principle of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.     

On the use of reduced models obtained through identification for feedback optimal control problems in a heat convection–diffusion problem

This paper deals with the use of reduced models for solving some optimal control problems. More precisely, the reduced model is obtained through the modal identification method. The test case which the algorithms is tested on is based on the flow over a backward-facing step. Though the reduction for the velocity fields for different Reynolds numbers is treated elsewhere [1], only the convection–diffusion equation for the energy problem is treated here. The model reduction is obtained through the solution of a gradient-type optimization problem where the objective function gradient is computed through the adjoint-state method. The obtained reduced models are validated before being coupled to optimal control algorithms. In this paper the feedback optimal control problem is considered. A Riccati equation is solved along with the Kalman gain equation. Additionally, a Kalman filter is performed to reconstruct the reduced state through previous and actual measurements. The numerical test case shows the ability of the proposed approach to control systems through the use of reduced models obtained by the modal identification method.     

An efficient sequential linear quadratic algorithm for solving nonlinear optimal control problems

We develop a numerically efficient algorithm for computing controls for nonlinear systems that minimize a quadratic performance measure. We formulate the optimal control problem in discrete-time, but many continuous-time problems can be also solved after discretization. Our approach is similar to sequential quadratic programming for finite-dimensional optimization problems in that we solve the nonlinear optimal control problem using sequence of linear quadratic subproblems. Each subproblem is solved efficiently using the Riccati difference equation. We show that each iteration produces a descent direction for the performance measure, and that the sequence of controls converges to a solution that satisfies the well-known necessary conditions for the optimal control.     

Duffing方程的辛精细积分方法研究

辛精细积分方法汲取了辛几何算法保持动力学系统辛结构的优点和精细积分方法高精度的数值优点,其实现过程中涉及到大量矩阵求逆运算.为减小辛精细积分方法的运算量,本文在辛精细积分算法之前先将非齐次方程近似齐次化,使得矩阵求逆部分不显含时间,降低矩阵求逆计算量,并将这一方法应用于无阻尼Duffing方程的数值分析.通过与经典四阶Runge-Kutta格式及精细积分方法对比,发现辛精细积分方法在数值精度、计算耗时、保持系统能量等方面明显优于Runge-Kutta格式.此外,与精细积分方法相比,辛精细积分方法在保持系统能量方面存在明显优势.