Lie Bracket Research Articles

Privileged Coordinates and Nilpotent Approximation of Carnot Manifolds, I. General Results

In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies down the background for its sequel by clarifying a few points on privileged coordinates and the nilpotent approximation of Carnot manifolds. In particular, we give a description of all the systems of privileged coordinates at a given point. We also give an algebraic characterization of all nilpotent groups that appear as the nilpotent approximation at a given point. In fact, given a nilpotent group $G$ satisfying this algebraic characterization, we exhibit all the changes of variables that transform a given system of privileged coordinates into another system of privileged coordinates in which the nilpotent approximation is given by $G$.

Continuous Stabilizing Control for a Class Of Non-holonomic systems: Brockett System Example

A continuous adaptive controller is designed for a famous example of Brockett system which is an example of a class of non-holonomic systems. The controllability Lie Algebra of the Brockett’s system contains Lie brackets of depth one. It is shown that the closed loop system is globally uniformly asymptotically stable. The advantage of this method is that it does not require the conversion of the system model into a “chained form,” and thus does not rely on any special transformation techniques. The practical effectiveness of the controller is illustrated by numerical simulations.

On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties

In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields.

Global stability for linear system and controllability for nonlinear system in the dynamics model of diabetics population

This paper presents global stability for linear system and controllability for nonlinear system in dynamic model. We discuss dynamic model of diabetics population which is autonomous linear system. The model considers the development of individual from a healthy stage to a pre-diabetic stage, then stage of diabetes without complication, to the stage of diabetes with complication and diabetics become disabled. In the linear model is investigated global stability using quadratic forms of Lyapunov function. After investigating the global stability of the linear system, a nonlinear optimal control model is established. Control variable in the model is the prevention effort to reduce the number of pre-diabetic into diabetic without and with complication. Controllability of nonlinear control system is investigated using Lie Bracket method. The results show that the linear system is globally asymptotically stable and the nonlinear system is locally accessible.

A simplicial foundation for differential and sector forms in tangent categories

Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and results from differential geometry in an arbitrary tangent category, including generalizations of vector fields and their Lie bracket, vector bundles, and connections. In this paper we investigate differential and sector forms in tangent categories. We show that sector forms in any tangent category have a rich structure: they form a symmetric cosimplicial object. This appears to be a new result in differential geometry, even for smooth manifolds. In the category of smooth manifolds, the resulting complex of sector forms has a subcomplex isomorphic to the de Rham complex of differential forms, which may be identified with alternating sector forms. Further, the symmetric cosimplicial structure on sector forms arises naturally through a new equational presentation of symmetric cosimplicial objects, which we develop herein.

Weak Milstein scheme without commutativity condition and its error bound

This paper shows a discretization method of solution to stochastic differential equations as an extension of the Milstein scheme. With a simple method, we reconstruct weak Milstein scheme through second order polynomials of Brownian motions without assuming the Lie bracket commutativity condition on vector fields imposed in the classical Milstein scheme and show a sharp error bound for it. Numerical example illustrates the validity of the scheme.

Perturbed normalizers and Melnikov functions

Given m≥1 and a smooth family of planar vector fields (Xε)ε that is a perturbation of a period annulus, we provide a characterization, in terms of Lie brackets, of the property that the first (m−1) Melnikov functions of (Xε)ε vanish identically. The equivalent condition is the existence of a smooth family of planar vector fields (Uε)ε, called here perturbed normalizers of order m. We also provide an effective procedure for computing Uε when the first (m−1) Melnikov functions of (Xε)ε vanish identically. A formula for the derivative of the m-th order Melnikov function is given.

Quantum Super Nambu Bracket of Cubic Supermatrices and 3-Lie Superalgebra

We construct the graded triple Lie commutator of cubic supermatrices, which we call the quantum super Nambu bracket of cubic supermatrices, and prove that it satisfies the graded Filippov–Jacobi identity of 3-Lie superalgebra. For this purpose we use the basic notions of the calculus of 3-dimensional matrices, define the $$\mathbb Z_2$$ -graded (or super) structure of a cubic matrix relative to one of the directions of a cubic matrix and the super trace of a cubic supermatrix. Making use of the super trace of a cubic supermatrix we introduce the triple product of cubic supermatrices and find the identities for this triple product, where one of them can be regarded as the analog of ternary associativity. We also show that given a Lie algebra one can construct the n-ary Lie bracket by means of an $$(n-2)$$ -cochain of given Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov–Jacobi identity, thereby inducing the structure of n-Lie algebra. We extend this approach to n-Lie superalgebras.

Asymptotic Controllability and Lyapunov-like Functions Determined by Lie Brackets

For a given closed target we embed the dissipative relation that defines a control Lyapunov function in a more general differential inequality involving Hamiltonians built from iterated Lie brackets. The solutions of the resulting extended relation, here called degree-k control Lyapunov functions (k>=1), turn out to be still sufficient for the system to be globally asymptotically controllable to the target. Furthermore, we work out some examples where no standard (i.e., degree-1) smooth control Lyapunov functions exist while a smooth degree-k control Lyapunov function does exist, for some k>1. The extension is performed under very weak regularity assumptions on the system, to the point that, for instance, (set valued) Lie brackets of locally Lipschitz vector fields are considered as well.

Tensor-Centric Warfare II: Entropic Uncertainty Modeling

In the first paper of the tensor-centric warfare (TCW) series [1], we proposed a tensor model of combat generalizing earlier Lanchester-type systems with a particular emphasis on contemporary military thinking, including the distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance). In the present paper, we extend this initial tensor combat model with entropic Lie-derivative machinery in order to capture some aspects of this deep uncertainty, while, in the process, formalizing into our model military notion of symmetry and asymmetry in warfare as a commutator, also known as a Lie bracket. In doing so, we have sought to shift the question from the prediction of outcomes of combat, upon which previous combat models such as the Lanchester-type equations have been typically constructed, towards determining the uncertainty outcomes, using a rigorous analytical basis.

Formation Shape Control Based on Distance Measurements Using Lie Bracket Approximations

We study the problem of distance-based formation control in autonomous multi-agent systems in which only distance measurements are available. This means that the target formations as well as the sensed variables are both determined by distances. We propose a fully distributed distance-only control law, which requires neither a time synchronization of the agents nor storage of measured data. The approach is applicable to point agents in the Euclidean space of arbitrary dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic stability. Our approach involves sinusoidal perturbations in order to extract information about the negative gradient direction of each agent's local potential function. An averaging analysis reveals that the gradient information originates from an approximation of Lie brackets of certain vector fields. The method is based on a recently introduced approach to the problem of extremum seeking control. We discuss the relation in the paper.

Switching in Time-Optimal Problem with Control in a Ball

In this paper we analyze local regularity of time-optimal controls and trajectories for an $n$-dimensional affine control system with a control parameter, taking values in a $k$-dimensional closed ball. In the case of $k=n-1$, we give sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only isolated jump discontinuities.

Measure-theoretic Lie brackets for nonsmooth vector fields

In this paper we prove a generalization of the classical notion of commutators of vector fields in the framework of measure theory, providing an extension of the set-valued Lie bracket introduced by Rampazzo-Sussmann for Lipschitz continuous vector fields. The study is motivated by some applications to control problems in the space of probability measures, modeling situations where the knowledge of the state is probabilistic, or in the framework of multi-agent systems, for which only a statistical description is available. Tools of optimal transportation theory are used.

Rough Linearization By One-Dimensional Rough Paths

In this paper, we propose a linearization method of nonlinear control systems by using rough path analysis. The linearization is achieved by the addition of rough signals, which are designed by functions with unbounded variations such as a sinusoidal wave having infinitely-large frequency Therefore, the proposed method is considered to be an extension of the analysis of Lie bracket motions such as the approximation algorithms by Liu and Sussmann in 1990’s, or, the transverse function approach by Morin et al. in 2000’s. Note, however, that, the characteristic feature of the method is to derive controllable linear approximate models for nonholonomic systems even if no Lie bracket motion is generated. Instead, another motions appear based on the similar effects of Wong-Zakai correction terms in stochastic analysis, provided that any stochastic signals are never dealt with in this paper. The validity of the method is confirmed by a few simple examples of linearization problems for nonlinear control systems.

Adaptive Stabilization of Uncertain Non-Affine Systems with Nonlinear Parameterization

This paper investigates the problem of adaptive control for non-affine systems with nonlinear parameterization. Under the controllability-like conditions characterized by the Lie brackets of affine vector fields, we show that there is LgV-type adaptive controllers that globally asymptotically regulate the state of the nonlinearly parameterized system with stable free dynamics while maintaining global stability of the closed-loop system. LgV-type adaptive controllers are designed and their effectiveness is demonstrated by two interesting examples.

Higher-Order Analysis of Kinematic Singularities of Lower Pair Linkages and Serial Manipulators

Kinematic singularities of linkages are configurations where the differential mobility changes. Constraint singularities are critical points of the constraint mapping defining the loop closure constraints. Configuration space (c-space) singularities are points where the c-space ceases to be a smooth manifold. These singularity types are not identical and can neither be distinguished nor identified by simply investigating the rank deficiency of the constraint Jacobian (linear dependence of joint screws). C-space singularities are reflected by the c-space geometry. In a previous work, a kinematic tangent cone was introduced as an approximation of the c-space, defined as the set of tangents to smooth curves in c-space. Identification of kinematic singularities amounts to analyze the local geometry of the set of critical points. As a computational means, a kinematic tangent cone to the set of critical points is introduced in terms of Jacobian minors. Closed form expressions for the derivatives of the minors in terms of Lie brackets of joint screws are presented. A computational method is introduced to determine a polynomial system defining the kinematic tangent cone. The paper complements the recently proposed mobility analysis using the tangent cone to the c-space. This allows for identifying c-space and kinematic singularities as long as the solution set of the constraints is a real variety. The introduced approach is directly applicable to the higher-order analysis of forward kinematic singularities of serial manipulators. This is briefly addressed in the paper.

Quadratic obstructions to small-time local controllability for scalar-input systems

We consider nonlinear finite-dimensional scalar-input control systems in the vicinity of an equilibrium.When the linearized system is controllable, the nonlinear system is smoothly small-time locally controllable: whatever m>0 and T>0, the state can reach a whole neighborhood of the equilibrium at time T with controls arbitrary small in Cm-norm.When the linearized system is not controllable, we prove that: either the state is constrained to live within a smooth strict manifold, up to a cubic residual, or the quadratic order adds a signed drift with respect to it. This drift holds along a Lie bracket of length (2k+1), is quantified in terms of an H−k-norm of the control, holds for controls small in W2k,∞-norm and these spaces are optimal. Our proof requires only C3 regularity of the vector field.This work underlines the importance of the norm used in the smallness assumption on the control, even in finite dimension.

Motion planning problem with obstacle avoidance for step‐2 bracket‐generating systems

AbstractThis paper considers the obstacle avoidance problem for control‐linear systems satisfying the Hörmander condition with the first‐order Lie brackets. To solve the problem under consideration, we use a family of oscillating periodic control functions with state‐dependent coefficients. The proposed approach is shown to be applicable for different shapes of obstacles and various potential functions generating collision‐free paths. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Involutive flows and discretization errors for nonlinear driftless control systems

In a continuous-time nonlinear driftless control system, an involutive flow is a composition of input profiles that does not excite any Lie bracket. Such flow composition is trivial, as it corresponds to a “forth and back” cyclic motion obtained rewinding the system along the same path. The aim of this paper is to show that, on the contrary, when a (nonexact) discretization of the nonlinear driftless control system is steered along the same trivial input path, it produces a net motion, which is related to the gap between the discretization used and the exact discretization given by a Taylor expansion. These violations of involutivity can be used to provide an estimate of the local truncation error of numerical integration schemes. In the special case in which the state of the driftless control system admits a splitting into shape and phase variables, our result corresponds to saying that the geometric phases of the discretization need not obey an area rule, i.e., even zero-area cycles in shape space can lead to nontrivial geometric phases.

Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection

For an oriented 2-dimensional manifold Σ of genus g with n boundary components, the space Cπ1(Σ)/[Cπ1(Σ),Cπ1(Σ)] carries the Goldman–Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded Lie bialgebra (under the natural filtration) is described by cyclic words in H1(Σ) and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [13] using Kontsevich integrals and in [2] using solutions of the Kashiwara–Vergne problem.In this note, we give an elementary proof of this isomorphism over C. It uses the Knizhnik–Zamolodchikov connection on C\\{z1,…zn}. We show that the isomorphism naturally depends on the complex structure on the surface. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin [9]. Surprisingly, it turns out that a similar proof applies to cobrackets.Furthermore, we show that the formality isomorphism constructed in this note coincides with the one defined in [2] if one uses the solution of the Kashiwara–Vergne problem corresponding to the Knizhnik–Zamolodchikov associator.