An equivalent linearization method for nonlinear Van der Pol oscillator subjected to random vibration using orthogonal functions

This paper deals with the idea of the orthogonal functions in the equivalent linearization of the nonlinear systems. Block Pulse (BP) function gives effective tools to approximate complex problems. The aim of this work is on using properties of the BP function as an orthogonal function in process of linearization. The BP functions have been used to propose an equivalent linearization method in the time domain to determine the unknown linearization coefficients. The accuracy of the proposed method compared with the other equivalent linearization approaches, including the regulation linearization and the dual criterion linearization methods. This study exploited the nonlinear Van der Pol oscillator system under stationary random excitation to demonstrate the feasibility of the proposed method. The validity of the analytical method is verified by applying different values of nonlinearity and intensity of excitation. Besides, by comparing the mean square responses and frequency response functions of the linearized systems for a wide range of nonlinearity depicted the present method is in agreement with other methods.     

System reliability analysis with saddlepoint approximation

System reliability is usually estimated through component reliability, which is commonly computed by the First Order Reliability Method (FORM). The FORM is computationally efficient, but may not be accurate for nonlinear limit-state functions. An alternative system reliability analysis method is proposed based on saddlepoint approximation. Unlike the FORM that linearizes limit-state functions in a transformed random space, the proposed method linearizes the limited-state functions without any transformation. After the linearization, the joint probability density of limit-state functions is estimated by the multivariate saddlepoint approximation. Without the nonnormal-to-normal transformation, the present method is more accurate than the FORM when the transformation increases the nonlinearity of limit-state functions. As demonstrated in the two examples, the new method is also as efficient as the FORM.     

一种核正交局部敏感辨别分析算法

为了发掘嵌入在人脸样本的非线性结构信息,把核方法和基向量正交化思想引入局部敏感分析算法中,提出一种新的人脸识别算法-核正交局部敏感辨别分析(Kernel based Orthogonal Locality Sensitive Discriminant Analysis).并给出了算法的推导过程及计算步骤.首先用核方法提取人脸样本的非线性信息,并将其投影至高维非线性空间,然后采用局部敏感辨别分析做线性映射,最后采用施密特正交化方法得到正交的基向量,从而使算法更好地描述人脸非线性流形结构特征.在ORL和YaleB人脸库的人脸识别实验证明了所提算法的有效性.     

Comparative study on linearized robot models

For representing the input—output behaviour of a robot manipulator by a linear time-invariant model, four direct linearization schemes are: (i) state linearization, (ii) linearization based on an identification method, (iii) linearization based on neglecting velocity-dependent and gravity terms and (iv) linearization based on neglecting the velocity-dependent term only (rate linearization). In order to make an appropriate choice of linear model for the development of real-time control, these schemes are extensively studied in this paper. It is shown that the rate linearization method leads to a satisfactory tradeoff between computation, accuracy, and stability. In the case of high velocity motions, a combination of state linearization and rate linearization is proposed.     

Predictive control based on local linear fuzzy models

This paper deals with predictive control based on fuzzy models. A novel algorithm (LOLIMOT) is proposed for the construction of Takagi-Sugeno fuzzy models. The rule consequents are optimized by a local orthogonal least-squares method that selects the significant regressors. The rule premises are optimized by a tree construction algorithm which partitions the input space in hyper-rectangles. A generalized predictive controller (GPC) and a dynamic matrix controller (DMC) are designed. Both controllers require the extraction of a linear model from the Takagi-Sugeno fuzzy model. For the GPC a new technique called local dynamic linearization is proposed that exploits the special structure of the local linear models. The DMC is based on the evaluation of a step response. The effectiveness of both the identification algorithm and the predictive controllers is shown by application to temperature control of an industrial-scale cross-flow heat exchanger.     

色噪声激励的时滞非线性系统瞬态响应研究

建立了色噪声与时滞联合作用的非线性系统模型,提出求解其瞬态概率密度的高效近似算法。利用等价变换将时滞系统简化为非时滞系统;通过线性化方法和随机平均原理得到原系统振幅过程的平均It?随机微分方程和相应的Fokker-Planck-Kolmogorov（FPK）方程。基于退化线性系统导出一组正交基,在该基空间内进行Galerkin变分得到近似瞬态概率密度。将该方法应用到受色噪声激励的双时滞Duffing-Van Der Pol振子得到理论解,采用蒙特卡罗模拟（MCS）验证理论解的正确性。分析了色噪声参数和时滞参数对瞬态响应的影响。研究结果表明：所提理论方法可有效求解受色噪声激励的时滞非线性系统的瞬态概率密度;算法求解效率高于MCS;色噪声和时滞均明显影响了系统瞬态响应。     

A computational procedure for response statistics-based optimization of stochastic non-linear FE-models

An approach for an efficient solution of response statistics-based optimization problems of non-linear FE systems under stochastic loading is presented. A sequential approximate optimization approach, where approximate stochastic analyses are used during portions of the optimization process, is implemented in the proposed formulation. In this approach, analytical approximations of the performance functions in terms of the design variables are considered during the optimization process. The analytical approximations are constructed by combining a mixed linearization approach with a stochastic response sensitivity analysis. The state of the system is defined in terms of the statistical second-moment characteristics of the structural response. The stochastic loading and the response of the system are represented by an orthogonal series expansion of the corresponding covariance matrices. In particular, a truncated Karhunen-Loève (K-L) expansion is applied. The system of non-linear equations is replaced by a statistical equivalent linear system. The evaluation of the K-L vectors is carried out by an efficient procedure that combines local linearization, modal analysis and static response of higher structural modes. An illustrative example is presented that shows the efficiency of the proposed methodology: it considers a building finite element model enforced with non-linear hysteretic devices and subject to a stochastic ground acceleration. Two types of problems are considered: a minimum structural weight design problem and an optimal non-linear device design problem.     

A state space embedding approach to approximate feedback linearization of single input nonlinear control systems

This contribution presents a numerical approach to approximate feedback linearization which transforms the Taylor expansion of a single input nonlinear system into an approximately linear system by considering the terms of the Taylor expansion step by step. In the linearization procedure, higher degree terms are taken into account by using a state space embedding such that the corresponding system representation has not to be computed in every linearization step. Linear matrix equations are explicitly derived for determining the nonlinear change of coordinates and the nonlinear feedback that approximately linearize the nonlinear system. If these linear matrix equations are not solvable, a least square solution by applying the Moore–Penrose inverse is proposed. The results of the paper are illustrated by the approximate feedback linearization of an inverted pendulum on a cart. Copyright © 2006 John Wiley & Sons, Ltd.     

New spatial basis functions for the model reduction of nonlinear distributed parameter systems

The selection of spatial basis functions is important for the model reduction of nonlinear distributed parameter systems (DPSs). Such a selection will significantly affect the accuracy and efficiency of modeling. The current study proposes new spatial orthogonal basis functions for the model reduction of nonlinear DPSs. Each new spatial basis function is a linear combination of the orthogonal eigenfunctions of such systems. The basis function transformation matrix is obtained using the balanced truncation method, which results in a straightforward derivation of the transformation matrix and low computation cost. This performance is proven theoretically. A numerical example is used to demonstrate the effectiveness of the proposed method.     

正交化近邻关系保持的降维及分类算法

针对近邻关系保持嵌入（NPE）算法易于受到降低后的维数影响,而且性能依赖于正确的维数估计的问题,提出了一种正交化的近邻关系保持的嵌入降维方法——ONPE。ONPE方法是使用数据点间的近邻关系来构造邻接图,假设每个数据点都能由其近邻点的线性组合表示,则可以通过提取数据点的局部几何信息,并在降维中保持提取的局部几何信息,迭代地计算正交基来得到数据的低维嵌入坐标。同时,在ONPE算法的基础上,利用局部几何信息,提出了一种在低维空间中使用标签传递（LNP）的分类算法——ONPC。其是假设高维空间中的局部近邻关系在降维后的空间中依然得到保持,并且数据点的类别可由近邻点的类别得到。在人工数据和人脸数据上的实验表明,该算法在减少维数依赖的同时,能有效提高NPE算法的分类性能。     

Structure analysis via orthogonal functions

The use of orthogonal functions to analyse the structure of a system is investigated. Applying the definitions of observability and controllability to a system that is approximated with the help of orthogonal functions, it is shown that the concepts of the state space and the space of orthogonal functions are equivalent, provided that two weak conditions are met. This result ensures that the observability and controllability properties remain invariant under the transformation introduced by the approximation. Furthermore, new criteria to test observability and controllability are given in terms of the coefficient matrix of the orthogonal expansion. Because this test does not require the knowledge of the system matrices A, B and C, the results derived may be used for the identification of systems. It is demonstrated that all the results obtained remain true, even for an approximation with low accuracy. These properties allow the application of orthogonal functions for the analysis of systems     

Application of input — State of the system transformation for linearization of some nonlinear generators

The paper presents an application of the transformation input — state of the system, built on the basis of elements of the theory of differential geometry allowing linearization of a certain class of nonlinear generators to a linear form. The necessary conditions fulfilled by nonlinear system undergoing linearization process are presented. Numerical solutions of the nonlinear equations of state together with linearized system obtained from direct transformation of the state space are included.     

基于广义正交多项式的近似系统解的存在唯一性

通过在投影空间中的基交换,导出了一种基于广义正多项多,展开的线性时变系统的新的近似模型。基于这一模,提出了近似系统 的存在唯一性判据及其实现方案。证明了对任意 线性时变系统,都可以构造出基于ＧＯＰｓ展开的近似系统,使得对任意阶的这种近似系统的解总是存在唯一的,最后,给出两个简单的例子说明了本文方法的可行性 。     

基于系数矩阵变换的最优MPRM求解方法

针对多输出布尔函数,给出一种求解混合极性Reed-Muller(MPRM)的系数矩阵变换算法。以MPRM中的乘积项数为化简标准,采用穷举策略进行极性空间搜索,求解最优MPRM。在MCNC和ISCAS基准电路上的测试结果表明,与采用列表技术相比,该系数矩阵变换算法能平均缩短55.8%的最优MPRM求解时间。     

Control of multi-axis robots with elastic joints via cascade compensation using the method of exact linearization

This paper presents a new method for the position control of industrial robots with elastic joints and where the dynamic of each actuator is described by a simplified model. The inverse dynamic of the system is computed and used to compensate the nonlinear terms and decouple the system through a coordinate transformation and nonlinear feedback (exact linearization). To simplify the algorithm for the inverse dynamic and hence reduce its computation, each actuator-link pair of the robot is considered as a 2-input 2-output nonlinear system, a link subsystem and an actuator subsystem. A cascade compensation using the exact linearization is then applied to each subsystem, thereby avoiding the computation of the first and second partial derivatives of the inverse of the inertia matrix and the vector of the coriolis and centrifugal forces. This gives a formalism that is relatively simple and efficient for symbolical computation, which is very important for the maintenance of accuracy. Similarly, a cascade linear controller is constructed for each subsystem of the resulting linear decoupled 2-input 2-output system. The basis vector functions for the coordinate transformation are so chosen that only one state of the link subsystem can theoretically not be measured directly or indirectly. To estimate this state, an observer with linear error dynamic is constructed. The applicability of this observer to this general case is also proved. Simulation results using the first three links of Puma 560 are finally presented.     

非线性大系统在有限域中的鲁棒稳定性

本文研究了含有不确定参数的非线性大系统的稳定性,当参数向量受到限制时,这种稳定性称为鲁棒(粗壮)稳定性.在状态空间的含原点的有限域中,利用李亚普诺夫函数进行集结线性化,可将大系统化为线性系统,由于线性化的任意性,得到的线性系统不是唯一的.文中阐明了利用图论方法建立的大系统的多层结构可以简化稳定性判据,帮助选择局部反馈,以实现大系统的镇定.     

基于自适应切空间的MRI图像配准

微分同胚是一种光滑可逆的变换,在MRI图像配准中可以保证图像形变后的拓扑结构保持不变,同时避免出现不合理的物理现象。为了在空间变换中获得更合理的同胚映射,高维空间中数据的非线性结构被考虑,基于流形学习方法提出一种自适应切空间的MRI图像配准算法。首先,把MRI数据构造成对称正定（SPD）的协方差矩阵,然后形成李群;接着,利用样本点邻域的局部切空间来表示李群的几何结构的非线性;接下来,在流形上用自适应邻域选择的方法形成的线性子空间去逼近局部切空间,提高切空间的局部线性化程度,从而最大限度地保留流形的局部非线性结构,得到最优的同胚映射。仿真数据和临床数据的实验结果显示,与传统的非参数微分同胚配准算法相比,该算法在高维稠密形变场上获得更高的拓扑保持度,最终提高图像配准精度。     

Exponential trajectory tracking control in the workspace of a class of flexible robots

In this paper, we present a control strategy that ensures the exponential stability of the tracking error in the virtual joint space of a class of mechanical systems made up of rigid links that form a chain that ends with a flexible beam. Virtual joints are defined so as to be related kinematically to the workspace. Thus, when the inverse kinematics is nonsingular, trajectory tracking in the virtual joint space is equivalent to trajectory tracking in the workspace. The method proposed in this paper calls for the transformation of the trajectory from the virtual joint space to the joint and deformation space. The robot is a non-minimum-phase system in the virtual joint space. However, this transformation, which involves the dynamics of the flexible part, can be solved using a causal–anticausal iterative approach. The controller is then designed using an input–output feedback linearization scheme, with regard to the joints, and two linear control laws with regard to the joint and to the deformation variable tracking errors. Analysis based on the passivity theorem, hierarchical systems stability, and linear matrix inequalities then allows us to determine the controller gains that ensure that the tracking errors in the virtual joint space are well damped and exponentially stable. Finally, the strategy is validated by simulating a controller that incorporates the proposed laws and that drives a two-link manipulator that has one rigid and one flexible link. The simulation results demonstrate the good performance of the proposed control system. © 1998 John Wiley & Sons, Inc.     

Grapho-analytical method for the determination of self-sustained oscillations for a class of automatic nonlinear systems

A grapho-analytical method for the determination of self-sustained oscillations in a nonlinear system having two nonlinear memoryless energyless elements, separated by a linear device is proposed. The method relies on the graphical solution of the system equations, derived by harmonic linearization of the nonlinear functions that describe the two nonlinear elements.     

基于核主成分分析的异常轨迹检测方法

针对现有算法不能有效应用于多因素轨迹异常检测的问题,提出基于核主成分分析(KPCA)的异常轨迹检测方法。首先,为了改善轨迹特征提取的效果,采用KPCA对轨迹数据进行空间转换,将非线性空间转换到高维线性空间;其次,为了提高异常检测的准确率,采用一类支持向量机对轨迹特征数据进行无监督学习和预测;最终检测出具有异常行为的轨迹。采用大西洋飓风数据对算法进行测试,实验结果表明,该算法能够有效提取出轨迹特征,并且与同类算法相比,该算法在多因素轨迹异常检测方面具有更好的检测效果。