FracSym: Automated symbolic computation of Lie symmetries of fractional differential equations

In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by Buckwar & Luchko (1998) and Gazizov, Kasatkin & Lukashchuk (2007, 2009, 2011). The method has been generalised here to allow for the determination of symmetries for FDEs with n$n$ independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym (Jefferson and Carminati 2013) which uses routines from the MAPLE symmetry packages DESOLVII (Vu, Jefferson and Carminati, 2012) and ASP (Jefferson and Carminati, 2013). We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented.     

基于李代数的变形目标跟踪

动态几何变形是图像跟踪技术面临的突出难题之一. 本文提出基于李代数的变形目标跟踪方法, 用Gabor特征表征目标, 以仿射李群建立目标几何变形, 利用李代数和李群之间的指数映射将参数的最优化求解从欧氏空间转至光滑流形, 实现了对变形目标的稳定跟踪.从物理层面分析了目标跟踪过程中的参数几何变换的实质, 从理论上对在光滑流形上进行迭代求解的优点进行了详细分析, 并对其收敛性做出了证明.图像序列跟踪测试的对比实验表明, 本文方法较现有基于欧氏空间的算法在收敛速度、跟踪稳定性和精确性方面有显著提高.     

Algorithms of Dynkin diagrams in Lie group machine learning

This paper uses the geometric method to describe Lie group Machine Learning （LML） based on the theoretical framework of LML, which gives the geometric algorithms of Dynkin diagrams in LML. It includes the basic conceptions of Dynkin diagrams in LML, the classification theorems of Dynkin diagrams in LML, the classification algorithm of Dynkin diagrams in LML and the verification of the classification algorithm with experimental results.     

On a new method for finding generalized equivalence transformations for differential equations involving arbitrary functions

A new efficient method for finding generalized equivalence transformations for a class of differential equation systems via its related extended classical symmetries is presented. This technique can be further adapted to find the equivalence transformations for the mathematical model. It applies to classes of differential systems whose arbitrary functions involve all equations’ independent variables. As a consequence, any symbolic manipulation program designed to find classical Lie symmetries can also be used to determine generalized equivalence transformations and equivalence transformations, respectively, without any modification of the program. The method has been implemented as the maple routine gendefget and is based on the maple package desolv(by Carminati and Vu). The nonlinear stationary heat conduction parameter identification problem is considered as an example.     

一种新的李群分类器在手写体数字中的应用

李群是变换空间的一种基本表示理论。目前针对李群数据所设计的分类器较少,对多分类的效果也不是很好。以手写体数字的应用为背景,引入了支持向量机分类算法来处理李群数据。由于李群数据具有矩阵表现的形式,设计了一种矩阵高斯核函数,使得支持向量机能够处理矩阵数据。仿真结果表明,支持向量机方法在李群数据上具有很好的性能。     

Singular extremals on Lie groups

We investigate the space of singular curves associated to a distribution ofk-planes, or, what is the same thing, a nonlinear deterministic control system linear in controls. A singular curve is one for which the associated linearized system is not controllable. If a quadratic positive-definite cost function is introduced, then the corresponding optimal control problem is known as the sub-Riemannian geodesic problem. The original motivation for our work was the question “Are all sub-Riemannian minimizers smooth?” which is equivalent to the question “Are singular minimizers necessarily smooth?” Our main result concerns the singular curves for a class of homogeneous systems whose state spaces are compact Lie groups. We prove that for this class each singular curve lies in a lower-dimensional subgroup within which it is regular and we use this result to prove that all sub-Riemannian minimizers are smooth. A central ingredient of our proof is a symplectic-geometric characterization of singular curves formulated by Hsu. We extend this characterization to nonsmooth singular curves. We find that the symplectic point of view clarifies the situation and simplifies calculations.     

李群平均方法在单轮姿态和运动控制中的应用

针对一个单轮系统,采用欧拉群描述其位形空间,对于Wei-Norman方程中不能精确求解的参数,采用平均方法对其进行近似计算,从而使被控系统在满足李代数可控性秩条件下,在李群上的平均轨迹能以一定的精度逼近真实轨迹.通过对单轮系统的仿真实验与结果分析,证明了所用方法的有效性,并指出该方法更适合驱动一类量子力学系统的状态.     

Finite-time optimal formation control of multi-agent systems on the Lie group SE(3)

This paper investigates the finite-time optimal formation problem of multi-agent systems on the Lie group SE(3), for the situation when the formation time and/or the cost function need to be considered. Under the condition that the formation time is given according to the task requirement, a finite-time optimal formation controller is proposed for the two-agent case to guarantee that the desired formation is achieved at the given time and the corresponding cost function is optimal. For the systems with multiple agents, the obtained finite-time optimal formation control law has second-order approximation accuracy. Finally, some numerical simulations are provided to illustrate the effectiveness of the theoretical results.     

Robust formation control of multiagent systems on the Lie group SE(3)

This paper addresses the robust formation control problem of multiple rigid bodies whose kinematics and dynamics evolve on the Lie group SE(3). First, it is assumed that all followers have access to the state information of a virtual leader. Then, a novel adaptive super‐twisting sliding mode control with an intrinsic proportional‐integral‐derivative sliding surface is proposed for the formation control problem of multiagent system using a virtual structure (VS) approach. The advantages of this control scheme are twofold: elimination of the chattering phenomenon without affecting the control performance and no requirement of prior knowledge about the upper bound of uncertainty/disturbance due to adaptive‐tuning law. Since the VS method is suffering from the disadvantages of centralized control, in the second step, considering a network as an undirected connected graph, we assume that only a few agents have access to the state information of the leader. Afterward, using the gradient of modified error function, a distributed adaptive velocity‐free consensus‐based formation control law is proposed where reduced‐order observers are introduced to remove the requirements of velocity measurements. Furthermore, to relax the requirement that all agents have access to the states of the leader, a distributed finite‐time super‐twisting sliding mode estimator is proposed to obtain an accurate estimation of the leader's states in a finite time for each agent. In both steps, the proposed control schemes are directly developed on the Lie group SE(3) to avoid singularity and ambiguities associated with the attitude representations. Numerical simulation results illustrated the effectiveness of the proposed control schemes.     

基于Lie理论的倒立摆系统的控制算法研究

该文通过能量反馈和最优控制相结合的方法实现倒立摆系统的自摆起和稳定控制。在摆起阶段采用能量反馈方法实现快速摆起，而在平衡稳定控制阶段，采用一种非线性系统微分几何方法一李理论，对倒立摆系统进行近似线性化，此种线性化方法使模型更多包含原系统主要的非线性部分，更能逼近实际系统，针对采用李理论得到的近似线性化模型，对倒立摆系统进行最优稳定控制设计。仿真和实时控制试验结果表明，文中提出的李理论近似模型线性化方法对于控制器设计结果是有效的，而且采用的能量反馈和最优控制相结合的联合控制策略能够成功实现倒立摆系统的自摆起和稳定控制过程。     

Motion on submanifolds of noninvariant holonomic constraints for a kinematic control system evolving on a matrix Lie group

For a control system on a matrix Lie group with one or more configuration constraints that are not left/right invariant, finding the combinations of (kinematic) control inputs satisfying the motion constraints is not a trivial problem. Two methods, one coordinate-dependent and the other coordinate-free are suggested. The first is based on the Wei–Norman formula; the second on the calculation of the annihilator of the coadjoint action of the constraint one-form at each point of the group manifold. The results are applied to a control system on SE(3) with a holonomic inertial constraint involving the noncommutative part in a nontrivial way. The difference in terms of compactness of the result between the two methods is considerable.     

Automatic estimation of asymmetry for gradient-based alignment of noisy images on Lie group

Many parametric image alignment approaches assume equality of the images to register up to motion compensation. In presence of noise this assumption does not hold. In particular, for gradient-based approaches, which rely on the optimization of an error functional with gradient descent methods, the performances depend on the amount of noise in each image. We propose in this paper to use the Asymmetric Composition on Lie groups (ACL) formulation of the alignment problem to improve the robustness in presence of asymmetric levels of noise. The ACL formulation, generalizing state-of-the-art gradient-based image alignment, introduces a parameter to weight the influence of the images during the optimization. Three new methods are presented to estimate this asymmetry parameter: one supervised (MVACL) and two fully automatic (AACL and GACL). Theoretical results and experimental validation show how the new algorithms improve robustness in presence of noise. Finally, we illustrate the interest of the new approaches for object tracking under low-light conditions.     

Minimal realizations of the matrix transition lie group for bilinear control systems: Explicit results

The paper gives the explicit construction of two minimal realizations for the matrix transition Lie group associated to bilinear control systems: Magnus' exponentiation and Wei-Norman's factorization. Explicit formulas are derived for both representations as well as the connecting relation between them.     

Type synthesis of a class of spatial lower-mobility parallel mechanisms with orthogonal arrangement based on Lie group enumeration

Type synthesis of lower-mobility parallel mechanisms (PMs) has drawn extensive interests, particularly two main approaches were established by using the reciprocal screw system theory and Lie group theory, respectively. Although every above approach provides a universal framework for structural design of general lower-mobility PMs, type synthesis is still a comparably difficult task for the PMs with particular geometry or required to fulfill some specified tasks. This paper aims at exploring a simple and ef...     

多体系统动力学Lie 群微分⁃代数方程约束稳定方法

针对多体系统动力学微分-代数方程求解问题,研究基于Lie群表达的约束稳定方法.首先引入新的Lagrange乘子,结合位移约束、速度级约束和加速度级约束方程,构造了新的Lie群微分-代数方程.然后使用向后差商隐式方法和CG(Crouch-Grossman)方法,对微分–代数方程进行离散求解,得到精确度较高的动力学仿真结果.该方法在精确保持各级约束方程的同时,保持旋转矩阵的正交性,并且使系统总能量误差较小.     

The Garden of Eden theorem for cellular automata on group sets

ABSTRACT

We prove the Garden of Eden theorem for big-cellular automata with finite set of states and finite neighbourhood on right amenable left homogeneous spaces with finite stabilisers. It states that the global transition function of such an automaton is surjective if and only if it is pre-injective. Pre-Injectivity means that two global configurations that differ at most on a finite subset and have the same image under the global transition function must be identical. The theorem is proven by showing that the global transition function of an automaton as above is surjective if and only if its image has maximal entropy and that its image has maximal entropy if and only if it is pre-injective. Entropy of a subset of global configurations measures the asymptotic growth rate of the number of finite patterns with growing domains that occur in the subset.     

Robotic problems over a configurational manifold of vector parameters and dual vector parameters

In this paper, a diversity of problems arising in manipulator modelling and control are treated on the group configurational manifoldQ _ct in place of the standard configurational spaceQ. This possibility is a result of the vector-parametrization of the SO(3) Lie group. The simple composition law of vector parameters as well others of their nice properties reduce the computational burden (in comparison with the methods used up to now) in solving direct and inverse kinematic problems, both in dynamic modelling and full simulation of the motion of a manipulator system. This fact, proved in relation with the standard configurational space approach, becomes stronger over the group manifold where all kinematical equations are pure algebraic and the differential equations of motion feel the group structure. The suggested approach gives the opportunities for the powerful methods of Lie group theory to be involved in controlability and observability of the treatment of manipulator systems. The introduction of a dual vector parameter provides an interesting interplay of special geometrical considerations and special algebraic structures in this important area of the application of linear algebra.Research partially supported by Bulgarian NSF grant No. MM 63/91 and MM 445/94.     

基于微分博弈的追逃问题最优策略设计

本文设计了基于线性二次型微分博弈的多个攻击者、多个防御者和单个目标的追逃问题最优策略. 首先, 针对攻防双方保持聚合状态的情形, 基于攻击方内部、防御方内部以及双方之间的通信拓扑, 分别给出了目标沿固定轨迹运动和目标采取逃跑时攻防双方的最优策略. 其次, 针对攻防双方保持分散状态的情形, 利用二分图最大匹配算法分配相应的防御者与攻击者, 将多攻击者、多防御者追逃问题转化为多组两人零和微分博弈, 并求解出了攻防双方的最优策略. 最后, 数值仿真验证了所提策略的有效性.