On minimum-fuel control of affine nonlinear systems

The minimum-fuel control problem is investigated for a class of multiinput affine nonlinear systems whose associated Lie algebra is nilpotent. After preliminary notions and definitions are presented, emphasis is on the solution to the optimal control problem of the Lie algebra generated by certain system vector fields is nilpotent. Consequences of the maximum principle are deduced for such systems     

A method to obtain the lie group associated with a nilpotent lie algebra

According to Ado and Cartan Theorems, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the Lie algebra associated with the general linear group of matrices. We show in this paper a method to obtain the simply connected Lie group associated with a nilpotent Lie algebra, by using unipotent matrices. Two cases are distinguished, according to the nilpotent Lie algebra is or not filiform.     

Generic nonlinear stabilization of systems with matching algebraic structure

The paper demonstrates that using algebraic methods for the construction of time varying stabilizing controls for general controllable systems which are affine in the control is not only computationally feasible, but delivers generic feedback laws. A single feedback control law can be stabilizing for all systems which have the same algebraic structure and also for systems that can be adequately approximated by this structure. The systems considered are not limited to those whose controllability Lie algebra is nilpotent or even finite dimensional. The stabilizing controls are constructed by the help of an open-loop control problem on an associated Lie group which is posed as a trajectory interception problem in the logarithmic coordinates of flows.     

Finite‐Time Optimal Tracking Control for Dynamic Systems on Lie Groups

This paper investigates the problem of finite‐time optimal tracking control for dynamic systems on Lie groups for the situation when the tracking time and/or the cost functions need to be considered. The specific results are illustrated on SE(3) (the specific Euclidean groups of rigid body motions). The tracking time is given according to task requirements in advance. By using Pontryagin's maximum principle (PMP) on Lie groups and the backstepping method, a finite‐time optimal tracking control law is designed to track a desired reference trajectory at the given time. Simultaneously, the corresponding cost functions are guaranteed to be optimal. Compared with existing results of optimal control on Lie groups, it is noteworthy that we consider the finite‐time tracking control for dynamic systems rather than kinematic systems. Furthermore, the obtained optimal control law is described by explicit formulations, which is significant for practical applications.     

Controllability for single-input mechanical control systems with dissipation

Within the affine connection framework of Lagrangian control systems , based on the results of Sussmann on small- time locally controllability of single- input affine nonlinear control systems , the controllability results for mechanical control systems with single- input are extended to the case of the systems with isotropic damping ,where the Lagrangian is the kinetic energy associated with a Riemannian metric. A sufficient condition of negative small- time locally controllability for the system is obtained. Then ,it is demonstrated that such systems are small- time locally configuration controllable if and only if the dimension of the configuration manifold is one. Finally , two examples are given to illustrate the results. Lie bracketting of vector fields and the symmetric product show the advantages in the discussion.     

Controllability for single-input mechanical control systems with dissipation

Within the affane connection framework of Lagrangian control systems, based on the results of Sussmann on small-time locally controllability of single-input affine nonlinear control systems, the controllability results for mechanical control systems with single-input are extended to the case of the systems with isotropic damping, where the Lagrangian is the kinetic energy associated with a Riemannian metric, A sufficient condition of negative small-time locally controllability for the system is obtained.Then,it is demonstrated that such systems are small-time locally configuration controllable if and only if the dimemion of the configuration manifold is one. Finally, two examples are given to illustrate the results. Lie bracketting of vector fields and the symmetric product show the advantages in the discussion.     

Lie algebraic canonical representations in nonlinear control systems

This paper is the applied counterpart to previous results [5] for linear-analytic control systems. It is mainly concerned with two canonical representations of the exponential type. They exhibit the Lie algebraic structure of the system in such a form that results on weak controllability are easily derived in an algebraic manner. The first representation is a single exponential of a canonical Lie series in Hall's basis of the Lie algebra of vector fields. The second one is a factorization in terms of simpler exponentials of Hall's basic vectors. Both of them exhibit, as canonical coefficients, an infinite set of characteristic parameters which are a minimal representation of the input paths, when no drift occurs in the system (or, equivalently, in the weak control case). The weak controllability theorem is easily derived from these results, in a purely algebraic way.     

航天器姿轨控制研究综述:微分几何控制方法

作为空间任务顺利执行的关键技术,航天器姿态和轨道控制具有典型的非线性特征,其本质是对在矩阵李群SO (3)和SE (3)上演化的动态系统进行控制.与传统的参数化模型相比,航天器的矩阵李群模型对姿轨表征具有全局性、非奇异性和唯一性等固有优势,为控制设计提供了数学简洁、精度高、适用性好的模型基础.因此,近年来基于微分几何控制方法直接对其进行系统分析和控制设计的研究逐渐兴起,取得了一系列突破性成果.鉴于此,首先,系统地论述微分几何控制理论在航天器姿轨控制应用中的研究现状和进展;然后,面向空间任务的实际需要,分别对基于矩阵李群模型的单体航天器姿态控制、姿轨耦合控制、网络化航天器集群姿轨协同控制3个技术方向加以讨论;最后,对上述研究领域中存在的难题和挑战进行总结,并对未来发展方向进行展望.     

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.     

On the optimal control of bilinear systems and its relation to Lie algebras

The optimal control of bilinear systems is considered and related to the Lie algebra generated by the system matrices. Interesting results obtain when this Lie algebra is nilpotent.     

Some second-order necessary conditions in optimal control

In optimal control problems any extremal arc which trivially satisfies the Maximum Principle, that is a first-order control variation produces no change in cost, is called singular. Higher-order conditions are then needed to check the optimality of such arcs. Using the Volterra series associated with the variation of the cost functional gives a new context for analyzing singular optimal control problems. A basic optimality criterion for a fixed terminal time Mayer problem is obtained which allows one to derive the necessary conditions for optimality in terms of Lie brackets of vector fields associated with the dynamics of the problem.     

自旋量子系统的可控李代数计算

动态系统可控性同是经典和量子控制中研究的一个基本问题,本文研究了单自旋和双自旋量子系统的可控李代数的计算.首先基于量子系统可控的等价性条件,通过单量子系统Hamiltonian算符的李括号运算,给出了与基系数相关的系统可控的充要条件.然后利用Cartan分解方法构造了李代数su(4)的矩阵基,同时根据可控性基本定理提出了Hamiltonian算符多重李括号的计算方法及系统的可控性判据.     

Controllability and motion algorithms for underactuated Lagrangiansystems on Lie groups

We provide controllability tests and motion control algorithms for underactuated mechanical control systems on Lie groups with Lagrangian equal to kinetic energy. Examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimension of the configuration space. Local controllability properties of these systems are characterized, and two algebraic tests are derived in terms of the symmetric product and the Lie bracket of the input vector fields. Perturbation theory is applied to compute approximate solutions for the system under small-amplitude forcing; in-phase signals play a crucial role in achieving motion along symmetric product directions. Motion control algorithms are then designed to solve problems of point-to-point reconfiguration, static interpolation and exponential stabilization. We illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies, satellites and underwater vehicles     

Approximation of almost solvable bilinear systems

The complexity of the Wei-Norman integration for a bilinear system can be reduced by approximating the associated Lie algebra by one of lower dimension. An algorithm is given for approximating a class of bilinear systems in this way in the case their Lie algebras are almost solvable.     

Asymptotic expansions and Lie algebras for some nonlinear filtering problems

State estimation problems for systems involving small parameters are treated by both analytical and Lie algebraic approximation techniques. An asymptotic expansion for the unnormalized conditional density corresponding to the case of observations of a Gauss-Markov process through a (weak) polynomial nonlinearity is computed and a convergence result is derived. The expansion is related to certain approximations of the associated estimation Lie algebra. The convergence result is based on arguments used recently to prove existence and uniqueness and to estimate the tail behavior of solutions to nonlinear filtering problems with unbounded coefficients. These arguments are, in turn, adapted from the analytical theory of parabolic PDE's. Simulation results are presented to further assess the performance of the resulting approximate filters.     

Learning the Lie groups of visual invariance

A fundamental problem in biological and machine vision is visual invariance: How are objects perceived to be the same despite transformations such as translations, rotations, and scaling? In this letter, we describe a new, unsupervised approach to learning invariances based on Lie group theory. Unlike traditional approaches that sacrifice information about transformations to achieve invariance, the Lie group approach explicitly models the effects of transformations in images. As a result, estimates of transformations are available for other purposes, such as pose estimation and visuomotor control. Previous approaches based on first-order Taylor series expansions of images can be regarded as special cases of the Lie group approach, which utilizes a matrix-exponential-based generative model of images and can handle arbitrarily large transformations. We present an unsupervised expectation-maximization algorithm for learning Lie transformation operators directly from image data containing examples of transformations. Our experimental results show that the Lie operators learned by the algorithm from an artificial data set containing six types of affine transformations closely match the analytically predicted affine operators. We then demonstrate that the algorithm can also recover novel transformation operators from natural image sequences. We conclude by showing that the learned operators can be used to both generate and estimate transformations in images, thereby providing a basis for achieving visual invariance.     

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.     

From Chronological Calculus to Exponential Representations of Continuous and Discrete-Time Dynamics: A Lie-Algebraic Approach

In this paper, formal exponential representations of the solutions to nonautonomous nonlinear differential equations are derived. It is shown that the chronological exponential admits an ordinary exponential representation, the exponent being given by an explicitly computable Lie series expansion. The results are then used to describe controlled dynamics, dynamics under sampling and forced discrete-time dynamics. The study emphasizes the role of Lie algebra techniques in nonlinear control theory and specifies structural similarities between nonautonomous differential equations, dynamics under sampling and forced discrete-time dynamics up to hybrid ones.     

Optimization on Lie manifolds and pattern recognition

Several pattern recognition problems can be reduced in a natural way to the problem of optimizing a nonlinear function over a Lie manifold. However, optimization on Lie manifolds involves, in general, a large number of nonlinear equality constraints and is hence one of the hardest optimization problems. We show that exploiting the special structure of Lie manifolds allows one to devise a method for optimizing on Lie manifolds in a computationally efficient manner. The new method relies on the differential geometry of Lie manifolds and the underlying connections between Lie groups and their associated Lie algebras. We describe an application of the new Lie group method to the problem of diagnosing malignancy in the cytological extracts of breast tumors. The diagnosis method that we present has a mean sensitivity of 98.086% and a predictive index of 0.0602, making it the most accurate and reliable diagnostic method reported thus far.     

Motion control of drift-free, left-invariant systems on Lie groups

In this paper we address the constructive controllability problem for drift-free, left-invariant systems on finite-dimensional Lie groups with fewer controls than state dimension. We consider small (ϵ) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems which require up to (p-1) iterations of Lie brackets to satisfy the Lie algebra controllability rank condition. In the cases p=2,3, we give algorithms for constructing these controls as a function of structure constants that define the control authority, i.e., the actuator capability, of the system. The algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with O(ϵ^P) accuracy in general (exactly if the Lie algebra is nilpotent). The methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs