Canonical forms for stochastic nonlinear systems

In this paper, we obtain necessary and sufficient conditions for the existence of diffeomorphisms that transform stochastic nonlinear systems to various canonical forms. The main tool in our analysis is the so called invariance under transformation rule that directly relates the coordinate transformation for stochastic nonlinear systems to that for deterministic uncertain nonlinear systems. This invariance rule allows the utilization of the existing necessary and sufficient conditions for deterministic nonlinear systems in associated stochastic nonlinear systems.     

Local controllability for families of diffeomorphisms

We study groups and semigroups which are generated by analytic families of diffeomorphisms. The central notion is that of local controllability of a family of diffeomorphisms at a given point of the state manifold, which generalizes the familiar notion of local controllability of control systems with continuous, as well as discrete time. Lie theory methods are used. We systematically exploit the so called fast switching variations and properties of the jet spaces of curves on the state manifold.     

Registration of Anatomical Images Using Paths of Diffeomorphisms Parameterized with Stationary Vector Field Flows

Computational Anatomy aims for the study of variability in anatomical structures from images. Variability is encoded by the spatial transformations existing between anatomical images and a template selected as reference. In the absence of a more justified model for inter-subject variability, transformations are considered to belong to a convenient family of diffeomorphisms which provides a suitable mathematical setting for the analysis of anatomical variability. One of the proposed paradigms for diffeomorphic registration is the Large Deformation Diffeomorphic Metric Mapping (LDDMM). In this framework, transformations are characterized as end points of paths parameterized by time-varying flows of vector fields defined on the tangent space of a Riemannian manifold of diffeomorphisms and computed from the solution of the non-stationary transport equation associated to these flows. With this characterization, optimization in LDDMM is performed on the space of non-stationary vector field flows resulting into a time and memory consuming algorithm. Recently, an alternative characterization of paths of diffeomorphisms based on constant-time flows of vector fields has been proposed in the literature. With this parameterization, diffeomorphisms constitute solutions of stationary ODEs. In this article, the stationary parameterization is included for diffeomorphic registration in the LDDMM framework. We formulate the variational problem related to this registration scenario and derive the associated Euler-Lagrange equations. Moreover, the performance of the non-stationary vs the stationary parameterizations in real and simulated 3D-MRI brain datasets is evaluated. Compared to the non-stationary parameterization, our proposal provides similar results in terms of image matching and local differences between the diffeomorphic transformations while drastically reducing memory and time requirements.     

On Local Structural Stability of Differential 1-Forms and Nonlinear Hypersurface Systems on a Manifold with Boundary

In this paper we consider smooth differential 1-forms and smooth nonlinear control-affine systems with (n−1)-inputs evolving on an n-dimensional manifold with boundary. These systems are called hypersurface systems under the additional assumption that the drift vector field and control vector fields span the tangent space to the manifold. We locally classify all structurally stable differential 1-forms on a manifold with boundary. We give complete local classification of structurally stable hypersurface systems on a manifold with boundary under static state feedback defined by diffeomorphisms, which preserve the manifold together with its boundary. Date received: March 30, 2000. Date revised: October 30, 2000.     

Landmark Matching via Large Deformation Diffeomorphisms on the Sphere

This paper presents a methodology and algorithm for generating diffeomorphisms of the sphere onto itself, given the displacements of a finite set of template landmarks. Deformation maps are constructed by integration of velocity fields that minimize a quadratic smoothness energy under the specified landmark constraints. We present additional formulations of this problem which incorporate a given error variance in the positions of the landmarks. Finally, some experimental results are presented. This work has application in brain mapping, where surface data is typically mapped to the sphere as a common coordinate system.     

Geodesic Shooting for Computational Anatomy

Studying large deformations with a Riemannian approach has been an efficient point of view to generate metrics between deformable objects, and to provide accurate, non ambiguous and smooth matchings between images. In this paper, we study the geodesics of such large deformation diffeomorphisms, and more precisely, introduce a fundamental property that they satisfy, namely the conservation of momentum. This property allows us to generate and store complex deformations with the help of one initial “momentum” which serves as the initial state of a differential equation in the group of diffeomorphisms. Moreover, it is shown that this momentum can be also used for describing a deformation of given visual structures, like points, contours or images, and that, it has the same dimension as the described object, as a consequence of the normal momentum constraint we introduce.     

Optimization of Surface Registrations Using Beltrami Holomorphic Flow

In shape analysis, finding an optimal 1-1 correspondence between 3D surfaces within a large class of admissible bijective mappings is of great importance. Such a process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making it challenging to exhaustively search for the best mapping. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs)—complex-valued functions defined on surfaces with supremum norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map may be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated with it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.     

Identification and control of a nonlinear discrete-time system based on its linearization: a unified framework

This paper presents a unified theoretical framework for the identification and control of a nonlinear discrete-time dynamical system, in which the nonlinear system is represented explicitly as a sum of its linearized component and the residual nonlinear component referred to as a "higher order function." This representation substantially simplifies the procedure of applying the implicit function theorem to derive local properties of the nonlinear system, and reveals the role played by the linearized system in a more transparent form. Under the assumption that the linearized system is controllable and observable, it is shown that: 1) the nonlinear system is also controllable and observable in a local domain; 2) a feedback law exists to stabilize the nonlinear system locally; and 3) the nonlinear system can exactly track a constant or a periodic sequence locally, if its linearized system can do so. With some additional assumptions, the nonlinear system is shown to have a well-defined relative degree (delay) and zero-dynamics. If the zero-dynamics of the linearized system is asymptotically stable, so is that of the nonlinear one, and in such a case, a control law exists for the nonlinear system to asymptotically track an arbitrary reference signal exactly, in a neighborhood of the equilibrium state. The tracking can be achieved by using the state vector for feedback, or by using only the input and the output, in which case the nonlinear autoregressive moving-average (NARMA) model is established and utilized. These results are important for understanding the use of neural networks as identifiers and controllers for general nonlinear discrete-time dynamical systems.     

2D-Shape Analysis Using Conformal Mapping

The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this space every simple closed curve in the plane (a “shape”) is represented by a ‘fingerprint’ which is a diffeomorphism of the unit circle to itself (a differentiable and invertible, periodic function). More precisely, every shape defines to a unique equivalence class of such diffeomorphisms up to right multiplication by a Möbius map. The fingerprint does not change if the shape is varied by translations and scaling and any such equivalence class comes from some shape. This coset space, equipped with the infinitesimal Weil-Petersson (WP) Riemannian norm is a metric space. In this space, the shortest path between each two shapes is unique, and is given by a geodesic connecting them. Their distance from each other is given by integrating the WP-norm along that geodesic. In this paper we concentrate on solving the “welding” problem of “sewing” together conformally the interior and exterior of the unit circle, glued on the unit circle by a given diffeomorphism, to obtain the unique 2D shape associated with this diffeomorphism. This will allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this “space of shapes”. We then present an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes. The group of diffeomorphisms of S¹ acts as a group of isometries on the space of shapes and we show how this can be used to define shape transformations, like for instance ‘adding a protruding limb’ to any shape.     

基于虚拟模型的自适应解耦抗扰控制

针对一类多变量非线性耦合系统,提出了一种基于虚拟模型的非线性自适应控制器.首先将非线性系统线性化处理并将其作为虚拟模型,对该模型设计线性自适应控制律.然后将线性控制律分别应用在虚拟系统和受控的实际非线性系统上,根据两者的输出误差设计补偿控制律,以达到对实际被控对象进行自适应解耦抗扰的目的.利用李雅普诺夫稳定理论给出了控制系统稳定性条件.实验仿真验证了控制算法的有效性.     

Input linearization of nonlinear systems via pulse-width control

In this note, it is shown that a general nonlinear system can be transformed to an affine nonlinear system by virtue of the use of pulsewidth control. Therefore, the combined system from the pulse width control input to the nonlinear system output behaves as an affine (linear-in-control) system. By this way, a cumbersome nonlinear system model can be transformed to a simpler linear-in-control form without increasing the system dimension; this in turn enables simpler control design. Furthermore, using this methodology, some control design methods developed only for affine systems can be adapted to general nonlinear systems.     

变转速风力机额定风速以上的非线性控制 ——恒功率输出控制问题

讨论了额定风速以上时变转速风力机的控制问题,系统采用基于微分几何理论下的浆距角和触发角双输入控制,变转 速风机是一个带有大范围风速的多输入产多输出 非线性系统,通过坐标变换,使非线性系统转化为一个标准的线性系统。     

1种多面体描述的非线性预测控制器综合

针对工业过程中的非线性计算量大,实时性低等问题,提出了1种计算非线性预测控制的新方法。该方法将神经网络与线性微分包含(LDI)相结合对非线性系统建模,从而将非线性系统转换成多面体描述的线性时变系统。对于多面体描述系统的各个顶点构成的多个线性模型,在线求得不同状态下的控制器。最后通过证明多面体描述的线性系统的稳定性来保证原非线性的稳定性。通过仿真看出此算法在处理复杂系统的控制问题具有良好的控制效果。     

Nonlinear neural controller with neural Smith predictor

This paper proposes a new nonlinear neural controller with a neural Smith predictor for time-delay compensation of nonlinear processes. Similar to the conventional linear PID controller, the nonlinear neural network based controller uses the system error, the integral of the system error, and the derivative of the system error as its inputs but the mapping from the inputs to the output of the controller is nonlinear. Finally simulation results are presented. On leave from Guilin Institute of Electronic Technology, PRC     

基于非线性电路频域核估计和神经网络的故障诊断

提出一种基于非线性电路频域核分析和神经网络的故障诊断方法．主要研究非线性系统频谱的获取,非线性系统频谱特征的提取及基于非线性系统频谱特征的故障诊断．利用Volterra频域核估计辨识非线性系统,通过系统广义频率响应函数的估算提取电路特征,将其预处理后作为递归神经网络的输入样本,利用神经网络的分类功能对电路的工作模式作出故障决策、最后。给出故障诊断实例验证了该方法的有效性．     

基于神经网络的非线性系统近似线性化

神经网络具有同时逼近某一函数及其高阶导数的功能，这一结果为神经网络在非线性系统中的应用提供了可行的工具，本文提出了一种利用网络的近似功能的非线性系统的近似线性方法，无论系统是否满足可积条件，神经网络都可实现其对各条件的近似职分，从而构造满足系统近似线性化的反馈控制，对球－杆系统的仿真结果显示了这种方法的有效性。     

An exploratory study of chaos in human-Machine system dynamics

The human-machine system behavior and performance are dynamic, nonlinear, and possibly chaotic. Various techniques have been used to describe such dynamic and nonlinear system characteristics. However, these techniques have rarely been able to accommodate the chaotic behavior of such a nonlinear system. Therefore, this study proposes the use of nonlinear dynamic system theory as one possible technique to account for the dynamic, nonlinear, and possibly chaotic human-machine system characteristics. It briefly describes some of the available nonlinear dynamic system techniques and illustrates how their application can explain various properties of the human-machine system. A pilot's heart interbeat interval (IBI) and altitude tracking error time series data are used in the illustration. Further, the possible applications of the theory in various domains of human factors for on-line assessment, short-term prediction, and control of human-machine system behavior and performance are discussed.     

一种改进的神经网络非线性预测控制

从建立神经网络非线性预测模型出发,针对BP网络存在收敛速度慢,容易陷入局部最小的缺点,该文在BFGS拟牛顿法的基础上,提出了一种基于并行拟牛顿优化算法的并行拟牛顿神经网络。该并行拟牛顿优化算法采用两个含有不同参数的拟牛顿校正公式,在每次迭代过程中,利用这两个不同的校正公式得到相应的搜索方向,并通过不精确搜索法求取最优步长,最后根据一性能指标取最优的一个搜索方向和相应的步长对网络各层之间的权值进行修正。Matlab仿真结果表明,同BP神经网络和BFGS拟牛顿神经网络相比,该神经网络具有收敛速度快、模型精度高的特点,更适合于实时非线性控制。     

J-inner-outer factorization, J-spectral factorization, and robustcontrol for nonlinear systems

The problem of expressing a given nonlinear state-space system as the cascade connection of a lossless system and a stable, minimum-phase system (inner-outer factorization) is solved for the case of a stable system having state-space equations affine in the inputs. The solution is given in terms of the stabilizing solution of a certain Hamilton-Jacobi equation. The stable, minimum-phase factor is obtained as the solution of an associated nonlinear spectral factorization problem. As an application, one can arrive at the solution of the nonlinear H_∞-control problem for the disturbance feedforward case     

神经网络在线投影算法及非线性建模应用

针对神经网络难以在线学习的缺点,把神经网络当作结构已知的非线性系统,权系数的学习看成非线性系统的参数估计,基于新估计准则的非线性系统在线参数估计投影算法,给出前馈神经网络的一种在线运行投影学习算法.理论上证明该算法的全局收敛性,讨论算法参数的物理意义和取值范围.通过2个非线性时变系统的神经网络建模应用的仿真,验证算法的全局收敛性和在线运行能力.