线性系统稳定性与最优性的关系(Ⅱ)

我们在[1]中提出并讨论了与R.E.Kalman的最优控制反问题不同的另一类线性最优控制的反问题:任给一渐近稳定的定常线性系统和一个非负二次型性能指标,问是否可以从该稳定系统中分解出一个状态反馈,使得这个状态反馈就是给定指标下的最优控制。本文将上述问题加以推广,对线性离散系统和线性时变系统得出了相应的结论,进而得出[1]中提出的渐近稳定系统和最优系统之间的对应关系是一切线性系统的内在特征。     

非线性广义系统最优控制的最大值原理：无限维情形

§1.引言 对于无限维非线性最优控制问题,[1]—[3]在一定条件下证明了最大值原理。在有限维情形,[4]讨论了线性广义系统的二次型指标最优问题。关于有限维非线性广义系统的讨论见[5],[6]。而对于无限维非线性广义系统的最优控制问题,目前尚无讨论。本文利用Ekeland变分原理[7]—[10]和Fattorini引理,对具有一般目标泛函的无限维广义系统的最优控制问题给出了最大值原理。     

广义分散控制系统的结构脉冲固定模

一、引言广义分散控制系统日益受到人们的重视.从80年代后期开始,Chang、王朝珠等人各自提出并讨论了这类系统的脉冲固定模;文[3—7]和[10]从不同的角度讨论了脉冲固定模的特征.本文将从系统结构上讨论脉冲固定模及其结构特征.     

线性脉冲控制系统的能控性与最优控制

首先讨论了一类线性随机脉冲控制系统的精确能控性质,给出了该类控制系统的脉冲精确能控的等价的代数判据.然后提出了一个确定性的二维线性脉冲控制系统的时间-脉冲强度最优控制问题;利用动态规划原理,给出了脉冲最优控制的反馈形式和值函数的显式表达式;说明了值函数在整个平面上是连续的,在左右两个半平面的内部还是连续可微的.     

广义LQ最优控制问题的极小化序列

将经典LQ问题的评价泛函中关于控制变量的二次型推广为一类偶次多项式,证明了这类广义LQ无约束最优控制问题的一个等价扩张逼近可由一列半径递增的球约束最优控制问题加以实现.进而利用P0ntryagin极值原理建立相应的球约束最优控制问题的二次规划,并通过Canonical倒向微分流及不动点定理,求解常微分方程边值问题,得到球约束最优控制问题的最优值.随着约束球半径趋于无穷大,形成原广义LQ最优控制问题的一个极小化序列,从而得到原问题的最优值.     

分数阶时变广义系统稳定性分析

基于Caputo分数阶导数,研究了分数阶时变广义线性系统和分数阶时变广义非线性系统的稳定性问题.首先利用相关不等式,给出了一个时变广义线性系统无脉冲且稳定的充分条件.然后,通过慢子系统来判断快子系统的变化,并利用Riccati方程,建立了分数阶时变广义非线性系统是渐近稳定的判定准则.最后,给出了算例和Simulink仿真结果,以说明结论的正确性.     

向量测度在分布参数系统最优控制理论中的应用

Lagaue, J. P.首先利用向量测度的值域定理[2,8]证明时间最优控制的开关原理;我们在[3]中进一步讨论了线性系统时间最优控制的一般问题,给出了最优时间的计算公式。在无限维空间取值的向量测度值域定理是Uhl, J. J.给出的。我们在[5,6,7]中利用它讨论了分布参数系统的时间最优控制问题。它们表明,向量测度是研究时间最优控制的有力工具。     

线性系统稳定性与最优性的关系——线性最优控制反问题的一种提法

本文提出并讨论了与R.E.Kalman提出的线性系统最优控制反问题不同的最优控制的另一类反问题:任给一渐近稳定系统=x及指标,问能否将分解成为=A+BKT,使u=KTx在指标J下为系统=Ax+Bu的最优控制?本文给出了上述问题的解答并得出了渐近稳定系统与最优系统之间的某种对应关系.     

广义控制系统状态观测器的结构

在广义控制系统的设计中,经常会遇到状态反馈问题.如极点配置和二次性能指标的最优控制都是状态反馈.可是在实际系统中,直接量测的是输出,而状态通常不能直接量测.从而知道状态反馈是物理上不能实现的.已知在正常系统中,解决这一问题的方法是重构状态——即通过一个被称为状态观测器的动态系统来实现它.因此对广义系统亦可期望通过状态观测器来获得状态反馈中的状态.文献[1]曾为广义系统具体设计了     

二维粘弹性问题的广义差分法及其数值模拟

本文利用广义差分法讨论了二维粘弹性问题.建立了二维粘弹性问题的广义差分格式,给出了一种新的初始值近似,证明了广义差分解的最优L~p估计和W~1,p(2≤p≤∞)模估计,同时得到了广义的Ritz—Volterra投影和广义差分解之间的超收敛的W~1,p(2≤p≤∞)模的误差估计.最后给出了一个数值算例以验证该方法的可行性.     

基于开关控制的线性奇异系统的二次状态反馈镇定问题

主要讨论基于开关控制的线性奇异系统的二次状态反馈镇定问题.利用二次反馈镇定的概念,给出了线性奇异系统基于异步开关控制的二次状态反馈镇定问题可解的两个充分条件.进一步,对于带有范数有界的不确定项的奇异线性系统,给出了其可以基于异步开关控制的二次状态反馈鲁棒镇定的可解性条件.     

Stability analysis and optimal control of stochastic singular systems

In this paper, problems of stability and optimal control for a class of stochastic singular systems are studied. Firstly, under some appropriate assumptions, some new results about mean-square admissibility are developed and the corresponding LMI sufficient condition is given. Secondly, finite-time horizon and infinite-time horizon linear quadratic (LQ) control problems for the stochastic singular system are investigated, in which the coefficients are allowed to be random in control input and quadratic criterion. Some results involving new stochastic generalized Riccati equation are discussed as well. Finally, the proposed LQ control model for stochastic singular systems provides an appropriate and effective framework to study the portfolio selection problem in light of the recent development on general stochastic LQ problems.     

Optimal control of singular systems via piecewise linear polynomial functions

A method for finding the optimal control of linear singular systems with a quadratic cost functional using piecewise linear polynomial functions is discussed. The state variable, state rate, and the control vector are expanded in piecewise linear polynomial functions with unknown coefficients. The relation between the coefficients of the state rate with state variable is provided and the necessary condition of optimality is derived as a linear system of algebraic equations in terms of the unknown coefficients of the state and control vectors. A numerical example is included to demonstrate the validity and the applicability of the technique. Copyright © 2002 John Wiley & Sons, Ltd.     

A separation theorem for stochastic singular linear quadratic control problem with partial information

In this paper, we provide a separation theorem for the singular linear quadratic (LQ) control problem of Itô-type linear systems in the case of the state being partially observable. Above all, the Kalman-Bucy filtering of the dynamics is given by means of Girsanov transformation, by which the suboptimal feedback control of the LQ problem is determined. Furthermore, it is shown that the well-posedness of the LQ problem is equivalent to the solvability of a generalized differential Riccati equation (GDRE).     

奇异系统的最优调节器

奇异系统(singular systems)在工程、经济、生态等领域方面,有着广泛的背景与深远的意义.近几年来,作为控制系统的一个非常活跃的分支出现于自动控制界.国内外许多学者在这方面做了不少工作.文献[1]讨论了奇异系统使二次性能指标取极小的最优调节问题.利用奇异系统解     

奇异脉冲系统的奇异H_∞控制

主要研究奇异脉冲系统的奇异H∞控制问题.当系统不满足正则条件时,给出奇异脉冲系统的奇异H∞控制问题可解的充分条件,控制律使得闭环系统在保证内稳定的条件下达到干扰衰减.     

Instability Degree and Singular Subspaces of Integral Isotropic Cones of Linear Systems of Differential Equations

The instability degree of linear systems of differential equations is estimated in terms of the dimensions of completely singular subspaces of integral cones of these systems. Special attention is given to the case where the linear system under study has first integrals of the type of nonsingular quadratic forms. General results are applied to a well-known problem concerning the gyroscopic stabilization of unstable equilibria of a mechanical system.     

Linear Hamiltonian Systems: Quadratic Integrals,Singular Subspaces and Stability

A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.     

Accumulation of switchings in distributed parameter problems

In recent times, optimal control theory for distributed parameter systems has been actively studied; among them, an important place is occupied by the class of systems describing oscillation processes. This work studies linear control distributed parameter systems of hyperbolic type. The minimization problem of a quadratic functional on the trajectories of the system is considered. By using the Fourier method, the problem is reduced to studying optimal solutions for a countable control system of ordinary differential equations. For Galerkin’s approximations of this system, it is proved that the optimal control is a chattering control, i.e., it has infinitely many switchings on a finite interval of time. The construction of the optimal synthesis uses the results of the theory of singular regimes and regimes with with more and more frequent switchings. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

位势问题边界元法中几乎奇异积分的正则化

将一种通用算法应用于平面位势问题边界元法中近边界点几乎奇异积分的正则化。对线性单元,位势问题近边界点的几乎强和超奇异积分可归纳为两种形式。通过分部积分,将引起奇异的积分元素变换到积分号之外,从而对这两种积分分别给出了无奇异的正则化计算公式。除了线性元,二次元也应用于该算法。与近边界点临近的二次单元划分为两段线性单元,该算法仍然适用。算例证明了方法的有效性和精确性。对曲线边界问题,联合二次元和线性元可提高计算结果精确度。