A Normal Form for Two-Input Flat Nonlinear Discrete-Time Systems

Discrete-time Flatness-based Control of a Gantry Crane

This article addresses the design of a discrete-time flatness-based tracking control for a gantry crane and demonstrates the practical applicability of the approach by measurement results. The required sampled-data model is derived by an Euler-discretization with a prior state transformation in such a way that the flatness of the continuous-time system is preserved. Like in the continuous-time case, the flatness-based controller design is performed in two steps. First, the sampled-data system is exactly linearized by a discrete-time quasi-static state feedback. Subsequently, a further feedback…

On the normal bundle of Levi-flat real hypersurfaces

Let $X$ be a connected complex manifold of dimension $\geq 3$ and $M$ a smooth compact Levi-flat real hypersurface in $X$. We show that the normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature along the leaves. This generalizes a result obtained by Brunella.

On the flat remainder in normal forms of families of analytic planar saddles

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter.

一种用于前向平坦性的对偶几何检验方法

A Dual Geometric Test for Forward-Flatness

本文针对离散时间系统的前向平坦性，提出了一种基于唯一可积余分布序列的对偶几何检验方法，计算效率高且便于与现有动态反馈线性化研究比较，并用两个示例进行了验证。

关于具有两个输入的x-平坦控制仿射系统的三角形式

On Triangular Forms for x-Flat Control-Affine Systems With Two Inputs

本文提出了一种适用于具有两个输入的x-平坦控制仿射系统的三角形式，并改进了识别x-平坦输出的算法。

单立柱堆垛机采样数据模型的平坦性分析

Flatness Analysis for the Sampled-data Model of a Single Mast Stacker Crane

本文证明单立柱堆垛机非线性连续时间模型的Euler离散化具有平坦性，通过将子系统转化为线性时变离散时间控制器规范型构造平坦输出，并基于此实现静止位置间轨迹规划与前馈控制计算。

Exceptional Discrete Mapping Class Group Orbits in Moduli Spaces

Let $M$ be a four-holed sphere and $Γ$ the mapping class group of $M$ fixing $\partial M$. The group $Γ$ acts on the space ${\mathcal M}_{\mathcal B}(SU(2))$ of SU(2)-gauge equivalence classes of flat SU(2)-connections on $M$ with fixed holonomy on $\partial M$. We give examples of flat SU(2)-connections whose holonomy groups are dense in SU(2), but whose $Γ$-orbits are discrete in ${\mathcal M}_{\mathcal B}(SU(2))$. This phenomenon does not occur for surfaces with genus greater than zero.

沿某类 Levi-平坦超曲面的转移函数线性化

Linearization of transition functions along a certain class of Levi-flat hypersurfaces

本文研究通过悬挂构造的 Levi-平坦超曲面上转移函数的规范化问题，借助圆周动力学与线性化定理给出获得规范形的充分条件，形成 Arnol'd 定理的几何类比。

Normal form for travelling kinks in discrete Klein-Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $ϕ^4$ lattice and the discrete sine--Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutio…

Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms

We consider the problem of the continuation with respect to a small parameter $ε$ of spatially localised and time periodic solutions in 1-dimensional dNLS lattices, where $ε$ represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal form algorithm in order to investigate the continuation and the linear stability of degenerate localised periodic orbits on lower and full dimensional invariant resonant tori. We recover results already existing in the literature and provide new insightful ones, b…