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.     

关于Sontag-Type控制的注记

基于控制李雅普诺夫函数的Sontag-Type控制是仿射系统鲁棒镇定中的重要控制律.首先揭示该控制律本质上是一种变结构控制且闭环的切换面总可达,受此启发并为了相对容易地构造控制李雅普诺夫函数,运用零状态可检测概念定义弱控制李雅普诺夫函数,并证明了基于弱控制李雅普诺夫函数的Sontag-Type控制的优化镇定性.文中还证明,在温和条件下,基于弱控制李雅普诺夫函数的Sontag＿Type控制为仿射系统的输入到状态镇定控制.     

李雅普诺夫稳定性理论中V函数的构造研究

李雅普诺夫稳定性理论,在自动控制等方面有着很重要的作用,李雅普诺夫第二判别方法,可以直接判定微分方程组的稳定性,应用非常广泛,但是如何构造满足特定条件下的李雅诺夫函数V,则是微分方程组稳定性理论要解决的课题。本文通过实例分析总结,研究出了几种实用性强的李雅普诺夫函数V的构造形式和方法。     

新型Lyapunov函数方法的模糊控制器设计

研究了在新型Lyapunov方法下的带有不确定性的T—S模糊模型稳定性及控制器的设计,给出了输入为零时系统稳定的充分条件,并验证了该条件比常用李雅普诺夫函数具有更大的松弛性。基于新型李雅普诺夫函数,对带有不确定性模糊模型设计了线性矩阵不等式形式的模糊控制器,该方法不必知道某一时刻被激活的规则数,同时把一个公共矩阵的寻找分解为p个矩阵的寻找。仿真实验证明,通过应用改进的李雅普诺夫函数设计的模糊控制器,具有良好的鲁棒性,控制效果良好。     

基于分段多项式李雅普诺夫函数的模糊系统的稳定性分析

本文采用最大型分段多项式李雅普诺夫函数研究了多项式模糊系统的闭环稳定性问题.首先,本文设计了与分段李雅普诺夫函数对应的切换模糊控制器,提出了多项式模糊模型稳定的平方和条件,同时证明了最大型分段多项式李雅普诺夫函数在函数切换点的稳定性.然后,设计了相应的路径跟踪优化算法,对本文非凸的稳定条件进行迭代求解.最后,通过两个算例进行仿真与比较,说明并验证了本文所提出结论的可行有效性.     

系统H∞范数计算:Lyapunov函数的直接优化方法

研究了李雅普诺夫函数的选择对求解系统H_∞范数的影响,提出了一种李雅普诺夫函数的直接优化方法,该方法通过优化黎卡提不等式中的李雅普诺夫函数,给出了H_∞范数的通用解析表达式,实现了二阶系统H_∞范数的精确求解.不同于需要繁琐优化过程的线性矩阵不等式（Linear matrix inequality,LMI）方法,本文提供了一种有效的途径以直接求解系统H_∞范数.     

李雅普诺夫指数谱的研究与仿真

李雅普诺夫指数在研究动力系统的分岔、混沌运动特征中起着重要的作用。该文介绍了李雅普诺夫指数的定义、计算原理及数值计算的实现方法，应用Matlab软件，编制了计算李雅普诺夫指数的程序，分析了一维、二维和三维非线性动力系统中分岔和混沌与李雅普诺夫指数之间的关系。绘制了逻辑斯蒂映射、埃农映射和洛仑兹方程的分岔图，并计算了相应的李雅普诺夫指数。实例的计算机仿真表明李雅普诺夫指数是研究分岔、混沌运动，解决工程实际问题的有效方法之一。     

基于李雅普诺夫量子系统控制方法的状态调控

从多方面对基于李雅普诺夫的量子系统控制方法进行了系统深入地研究, 包括该方法与最优控制的关系; 李雅普诺夫函数与性能指标之间的关系; 几种常用李雅普诺夫函数下的控制所能解决的问题, 适用范围和所存在的问题等. 在此基础上, 结合量子系统本身所具有的特点, 分别针对本征态, 叠加态和混合态的制备与调控目标, 总结出多种不同控制问题的改进方案. 对不同改进方案的设计思想, 所能解决的问题, 物理意义及其适用范围等进行剖析, 系统化了一套基于李雅普诺夫稳定性理论对量子系统进行状态调控的设计方法.     

带有输出约束的柔性关节机械臂预设性能自适应控制

针对带有输出约束和模型不确定的柔性关节机械臂系统,提出一种基于时变障碍李雅普诺夫函数的预设性能自适应控制方法.通过构造指数衰减的时变约束边界,提出时变正切型障碍李雅普诺夫函数,能够同时适用于约束与非约束情况,进而拓宽传统对数型障碍李雅普诺夫函数的适用范围.此外,通过预先设置时变边界函数的相关参数,使得系统输出在初始阶段具有较小的超调量和较快的跟踪速度,并能够满足系统的稳态性能要求.在此基础上,结合反演法设计反馈控制律,保证系统的输出约束性能和轨迹跟踪精度.最后,基于李雅普诺夫稳定性定理证明所有闭环信号能够达到一致最终有界,并给出数值仿真对比验证所提出方法的有效性.     

基于双线性滤波器的自适应有源消声算法的研究

利用双线性滤波器对非线性信号处理方面的优势,同时结合李雅普诺夫稳定性理论,提出BF-LMS算法。该算法首先定义李雅普诺夫函数,然后对双线性滤波器的权系数进行自适应调整,并且根据李雅普诺夫稳定性理论,使误差渐近趋近于零,从而达到能够避免传统噪声控制方法中对噪声信号统计特性的依赖性,最终使系统达到渐近稳定。     

非线性时变离散大系统的稳定性

本文应用标量李雅普诺夫函数法研究了线性时变离散大系统及非线性时变离散大系统的稳定性分解。同时分别给出了分解系数与稳定域的参数估计公式。     

Robust stability of discrete systems

The objective of this paper is to show how to choose a Liapunov function to obtain the best and sometimes exact estimates of the degree of exponential stability for linear time-invariant discrete systems. The choice is interesting because it is also shown that it provides the largest robustness bounds on non-linear time-varying perturbations which can be established by either norm-like or quadratic Liapunov functions. By applying the results obtained to large-scale systems, where the role of perturbations is played by the interconnections among the subsystems, the least conservative stability conditions are derived for the overall system which are available in the context of vector Liapunov functions and M-matrices.     

Multilevel stabilization of large-scale systems: A spinning flexible spacecraft

Multilevel stabilization schemes for high order linear systems are proposed in the framework of the decomposition-aggregation method used so far for stability analysis of large-scale systems by vector Liapunov functions. In one scheme, the output feedback is applied and the control parameters are determined for low order subsystems to produce a stable aggregate model for the overall system. When system inputs are decentralized such that each subsystem has a distinct input, and states are available for control, another scheme is proposed which has a multilevel structure with local and global controllers. Both stabilization schemes are aimed at the maximization of structural parameters which couple the subsystems.An 11th-order linear model of a spinning flexible spacecraft is stabilized by the proposed decomposition-aggregation method using both the passive and the active stabilization devices. The method can take advantage of the special structural features of the model to provide the best estimate of the stability region for a salient parameter which couples the wobble and the spin motion of the spacecraft.     

非线性大系统在有限域中的鲁棒稳定性

本文研究了含有不确定参数的非线性大系统的稳定性,当参数向量受到限制时,这种稳定 性称为鲁棒(粗壮)稳定性.在状态空间的含原点的有限域中,利用李亚普诺夫函数进行集结 线性化,可将大系统化为线性系统,由于线性化的任意性,得到的线性系统不是唯一的.文 中阐明了利用图论方法建立的大系统的多层结构可以简化稳定性判据,帮助选择局部反馈,以 实现大系统的镇定.     

Multi-parameter sensitivity analysis of linear dynamic systems through the second method of Liapunov †

Using the second method of Liapunov, a new procedure for performing sensitivity calculations in the stability analysis of linear dynamic systems is developed. Defining stability measure in terms of a Liapunov function and its time derivative, the paper formulates a method for the calculation of the sensitivity of stability measure to variations in system parameters. The method provides a basis for optimal stability design of linear dynamic systems, the criterion of optimality being that the sensitivity of the stability measure to parameter variations be zero. The feasibility and usefulness of the method are illustrated by numerical examples involving one and two variable parameters. Comparison of the time responses of the output variable obtained by using zero-sensitivity parameter settings with those resulting from the use of the minimum integral-square error criterion demonstrates the superiority of the proposed method. The procedure is simple and lends itself readily to programming on a digital computer.     

关于非綫性控制系統的稳定性问題

本文利用李亚普诺夫方法研究非线性控制系统的全局稳定性、大范围稳定和其吸引区的寻求以及一个品质问题.对非线性元件采用了分段线性的模型,从而使得到的条件中也包含有非线性特性的参数.     

On the attractivity of imbedded systems

A new approach designed to aid in the construction of Liapunov functions is proposed. It is based on the notion of partial properties. A given dynamic system is imbedded into an augmented system whose attractivity with respect to a part of variables is deduced by means of an auxiliary function. This leads to an exact procedure for computation of attractivity region estimates for dynamic systems. An important feature of the method is the great reduction of conservativeness of the region estimate in the case of the overlapping decomposition of large-scale systems. When applied to a multimachine power system with non-uniform damping and transfer conductances, the method gives an enlarged estimate of the region of transient stability.     

On absolute stability and the aizerman conjecture

Necessary and sufficient Liapunov-like conditions for both absolute stability and the Aizerman conjecture are established in the framework of stationary, continuous-time Lurie-Postnikov systems with multiple nonlinearities, thus providing theoretical solutions for these problems in the Liapunov sense. The solutions are based on the use of functional families instead of a single Liapunov function.Directly applicable results of the paper are new algebraic criteria for absolute stability. One of them is proved by using a new form of a tentative system Liapunov function. Also an estimate of the asymptotic stability domain is determined for all admissible nonlinearities.The results do not impose any requirement on the linear part of the system (such as its stability or controllability or observability).Three illustrative examples are worked out.     

Stability of a class of interconnected systems†

Sufficient conditions are given for exponential stability of the equilibrium of systems of interconnected components taken from a class of nonlinear, time-varying, multi-terminal, differential, algebraic and mixed components. The analysis is based on the mathematical models of the unconstrained components and the linear constraints imposed by the interconnections between the components. A Liapunov function is constructed for the interconnected system from Liapunov functions for the individual components. A parameter vector appearing in the Liapunov function is selected in a prescribed optimal manner. The Liapunov function so constructed can be used to establish a lower bound on the rate of decay of the system. An example of a ninth-order, non-linear system is included to demonstrate the application to design.     

A multi-machine power system Liapunov function using the generalized Popov criterion

A certain mathematical model of a multi-machine power system, which is suitable for transient stability studies and allows for machine damping, is Bhown to satisfy the generalized Popov criterion, within a limited range. Hence it is possible to use a systematic method to construct a Lur'e-type Liapunov function for the system. The Liapunov function obtained in this paper contains damping terms whose weight can be adjusted by a variable parameter. The variation will have corresponding effects on the estimates of the system's stability region.