一类稳定矩阵的共同二次李雅谱诺夫函数的构造

Nerendra 和 Balakrishnan对一组两两可交换的稳定矩阵提出了计算其共同二次李雅谱诺夫函数的方法。本文修改了此方法,并运用于一组两两不可交换的稳定矩阵的共同二次李雅谱诺夫函数的计算。     

FEEDBACK STABILIZATION OF NONHOLONOMIC CONTROL SYSTEMS WITH DRIFT

This paper presents a differential geometric approach for feedback stabilization of nonholonomic control systems with drift and its applicability is tested on two different systems possessing different algebraic structures: a system with six state variables and three controls, and a knife edge example. The approach is universal in the sense that it is independent of the vector fields determining the motion of the system, or of the choice of a Lyapunov function. The proposed feedback law is as a composition of a standard stabilizing feedback control for a Lie bracket extension of the original system and a periodic continuation of a specific solution to an open loop control problem stated for an abstract equation on a Lie group, an equation which describes the evolution of flows of both the original and extended systems. The open loop problem is solved as a trajectory interception problem in logarithmic coordinates of flows.     

A geometric approach to feedback stabilization of nonlinear systems with drift

The paper presents an approach to the construction of stabilizing feedback for strongly nonlinear systems. The class of systems of interest includes systems with drift which are affine in control and which cannot be stabilized by continuous state feedback. The approach is independent of the selection of a Lyapunov type function, but requires the solution of a nonlinear programming satisficing problem stated in terms of the logarithmic coordinates of flows. As opposed to other approaches, point-to-point steering is not required to achieve asymptotic stability. Instead, the flow of the controlled system is required to intersect periodically a certain reachable set in the space of the logarithmic coordinates.     

Stability and Stabilization of Block-cascading Switched Linear Systems

The main purpose of this paper is to investigate the problem of quadratic stability and stabilization in switched linear systems using reducible Lie algebra. First, we investigate the structure of all real invariant subspaces for a given linear system. The result is then used to provide a comparable cascading form for switching models. Using the common cascading form, a common quadratic Lyapunov function is (QLFs) is explored by finding common QLFs of diagonal blocks. In addition, a cascading Quaker Lemma is proved. Combining it with stability results, the problem of feedback stabilization for a class of switched linear systems is solved.     

线性中立型多时滞系统李代数稳定性判据

研究了线性中立型多时滞微分系统的稳定性。从矩阵李代数可解性角度,推导出新的简单的时滞独立稳定性判据。该新判据的重要意义和优越性在于首次突破了以往相关文献的稳定性判据在应用上受条件mΣj=1‖Cj‖<1或ρ(mΣj=1︱Cj︱)<1的限制,从而首次成功确定了在mΣj=1‖Cj‖≥1和ρ(mΣj=1︱Cj︱)≥1的情形下中立型多时滞微分系统的渐近稳定性。最后,通过两个例子显示了新判据的优越性。     

A Lie group formulation of Kane’s equations for multibody systems

We propose a Lie group approach to formulate the Kane’s equations of motion for multibody systems. This approach regards the set of rigid body transformations as the special Euclidean group SE(3). By expressing rigid body displacements as exponential maps generated from the Lie algebra se(3), it subsequently manipulates rigid body kinematics as convenient matrix operations. With this approach, all the individual quantities involved in Kane’s equations can be computed explicitly in an intrinsic manner, and the motion equations can be obtained systematically and efficiently. An example is presented to illustrate its use and effectiveness.     

Stability analysis of interconnected systems with possibly unstable subsystems

A direct method is presented for stability analysis of nonlinear interconnected dynamical systems. A new scalar Lyapunov function is considered as weighted sum of individual Lyapunov functions for each free subsystem and individual scalar functions related separately to each connection. Sufficient conditions are obtained for asymptotic stability of the equilibrium state by testing the definity of two constant square matrices whose dimension is equal to the number of subsystems. This method can assure stability of systems with possible unstable subsystems. A simple numerical example is included to illustrate this theory.     

Stabilization of Nonholonomic Robotic Systems Using Adaptation and Homogeneous Feedback

This paper considers the problem of stabilizing nonholonomic robotic systems in the presence of uncertainty regarding the system dynamic model. It is proposed that a simple and effective solution to this problem can be obtained by combining ideas from homogeneous system theory and adaptive control theory. Thus each of the proposed control systems consists of two subsystems: a (homogeneous) kinematic stabilization strategy which generates a desired velocity trajectory for the nonholonomic system, and an adaptive control scheme which ensures that this velocity trajectory is accurately tracked. This approach is shown to provide arbitrarily accurate stabilization to any desired configuration and can be implemented without knowledge of the system dynamic model. Moreover, it is demonstrated that exponential rates of convergence can be achieved with this methodology. The efficacy of the proposed stabilization strategies is illustrated through extensive computer simulations with nonholonomic robotic systems arising from explicit constraints on the system kinematics and from symmetries of the system dynamics.     

一类四维幂零向量场的超规范形及应用

研究一类具有对称性质的四维幂零向量场的超规范形问题,并将其应用于具有实际工程背景的高维非线性动力学模型的简化.发展与完善由Sanders、Baider和KOW提出的规范形进一步简化的理论,利用线性次数函数、多重李括号与分块矩阵的新记号表示相结合的方法,分别获得四维幂零向量场3次、5次截断的超规范形的一般形式,并将超规范形理论应用于研究环型桁架卫星天线模型的化简问题.本文通过引入并完善大尺寸分块矩阵的新记号表示方法,获得一种处理大尺寸分块矩阵运算的新方法,简化繁琐的大尺寸矩阵的运算,为后续的研究带来便利条件.     

非完整约束多智能体系统基于屏障控制函数的分布式协同控制

本文设计了一种基于屏障控制函数(CBF)的分布式协同控制算法,实现了领航–跟随者框架下非完整约束多智能体系统的连通性与编队控制.首先,通过将连通性保持问题建模为系统约束,定义了该约束的调零屏障函数(ZBF).其次,在此基础上,构建李雅普诺夫函数与角速度输入之间的关系,对跟随者智能体设计了基于调零屏障函数的协同控制算法,其中线速度控制器保证跟随者的速度的跟踪与队形的跟踪,而梯度型角速度控制器实现跟随者角度的矫正.然后,利用调零屏障函数不变集相关引理证明了连通性约束集为正不变集,若初始时刻连通,则跟随者智能体始终与领航者保持连通性.同时,本文提出的算法实现编队误差的渐近收敛.本文中的队形适用常见的固定队形编队需求,也适用于领航者是动态(有线速度和角速度)的情况.最后,通过数值仿真进一步验证了该算法在不同队形需求下的有效性.     

The Minkowski–Lyapunov equation for linear dynamics: Theoretical foundations

We consider the Lyapunov equation for the linear dynamics, which arises naturally when one seeks for a Lyapunov function with a uniform, exact decrease. In this setting, a solution to the Lyapunov equation has been characterized only for quadratic Lyapunov functions. We demonstrate that the Lyapunov equation is a well-posed equation for strictly stable dynamics and a much more general class of Lyapunov functions specified via Minkowski functions of proper C$C$-sets, which include Euclidean and weighted Euclidean vector norms, polytopic and weighted polytopic (1,∞$1,\infty$)-vector norms as well as vector semi-norms induced by the Minkowski functions of proper C$C$-sets. Furthermore, we establish that the Lyapunov equation admits a basic solution, i.e., the unique solution within the class of Minkowski functions associated with proper C$C$-sets. Finally, we provide a characterization of the lower and upper approximations of the basic solution that converge pointwise and compactly to it, while, in addition, the upper approximations satisfy the classical Lyapunov inequality.     

二维齐次高阶临界系统的稳定性判别算法

本文给出了一种判别二维高次齐次非线性临界系统的稳定性的算法,这一算法对中心流形是２维高次的系统通过构造李亚普诺夫函数来判稳有良好的效果,依据这种算法编制成的程序对随机产生的４０个临界非线性系统,均顺利地找到了相应的李亚普诺夫函数。     

二维非线性临界解析动态系统的局部渐近稳定性

考察具有一对共轭纯虚数特征值的二维非线性临界解析动态系统的局部渐近稳定性. 首先在非奇异线性坐标变换和时间尺度变换下, 将其化成标准形式. 之后, 运用形式级数法的思想, 通过构造多组线性方程组,给出了确定该系统的李雅普诺夫函数的方法, 并得到了判别系统局部渐近稳定和不稳定的充分条件. 最后通过示例说明该判别条件的有效性.     

Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form

This paper further studies stabilization of nonlinear systems represented by a Takagi-Sugeno (T-S) discrete fuzzy model. By extending a nonquadratic Lyapunov function and applying a nonparallel distributed compensation (non-PDC) law, a new stabilization conclusion is presented that is a generalization of some previous results in the literature. The new conclusion under a special situation is also suitable for a PDC law, which presents to be more relaxed than some existing results. An example is provided to demonstrate the effectiveness of the new conclusions.     

A REDUCE program for the normalization of polynomial Hamiltonians

The program LINA01 is proposed for the direct and the inverse normalization of Hamiltonian systems and for the calculation of formal integrals of motion of them. The calculations required in LINA01 are made on the basis of Lie canonical transformation method. The program package of LINA01 is written on REDUCE.

Program summary

Title of program:LINA01Catalogue identifier:ADUVProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUVProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputer: IBM PC PENTIUM 4/2.40 GHz 512 MbOperating systems under which the program has been tested: Windows XPProgramming language used: REDUCE vs. 3.7No. of lines in distributed program, including test data, etc.:485No. of bytes in distributed program, including test data, etc.:4320Distribution format:tar.gz Nature of physical problem. The transformation bringing a given Hamiltonian function into the normal form (namely, the normalization) is one of the conventional methods for non-linear Hamiltonian systems [A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Springer-Verlag, Berlin, 1983; G.D. Birkhoff, Dynamical Systems, A.M.S. Colloquium Publications, New York, 1927; F. Gustavson, Astron. J. 71 (1966) 670; G.I. Hori, Astron. Soc. Japan 18 (1966) 287; A. Deprit, Cel. Mech. 1 (1969) 12; A.A. Kamel, Cel. Mech. 3 (1970) 90]. Recently, beyond classical mechanics, the normal form method has been applied to quantization of chaotic Hamiltonian systems with the aim of finding quantum signature of chaos [L.E. Reichl, The Transition to Chaos. Conservative Classical Systems: Quantum Manifestations, Springer, New York, 1992]. Besides those utilities, the normalization requires quite cumbersome algebraic calculations of polynomials, so that the computer algebraic approach is worth studying to promote further investigations around the normalization together with the ones around the inverse normalization. Method of solution. The canonical transformation proceeding the normalization is expressed in terms of the Lie transformation power series, which is also referred to as the Hori-Deprit transformation. After (formal) power series expansion as above, the fundamental equation of the normalization is solved for the normal form together with the generating function of transformation recursively from degree-3 to the degree desired to be normalized. The generating function thus obtained is applied to the calculation of (formal) integrals of motion. Restrictions due to the complexity of the problem. The computation time rises in a combinatorial manner as the desired degree of normalization does. Especially, such a combinatorial growth of computation is more significant in the inverse normalization than in the direct one. The hardware (processor and memory, for example) available for the computation may restrict either the degree of normalization or the computation time.     

具有有色噪声的线性随机系统的矩稳定性(英文)

研究具有有色噪声的线性随机系统的矩稳定性,得到了无需计算矩阵方根的矩阵特征值代数判据,并用实例演示了文中的方法。