scholarly journals New Robust Projection to Latent Structure

2022 ◽  
pp. 211-232
Author(s):  
Jing Wang ◽  
Jinglin Zhou ◽  
Xiaolu Chen
Keyword(s):  
Nonlinear Systems ◽  
Equilibrium Point ◽  
Local Structure ◽  
Latent Structure ◽  
Series Method ◽  
Structure Feature ◽  
Small Uncertainty ◽  
System Nonlinearity ◽  
Secondary Feature

AbstractIn many actual nonlinear systems, especially near the equilibrium point, linearity is the primary feature and nonlinearity is the secondary feature. For the system that deviates from the equilibrium point, the secondary nonlinearity or local structure feature can also be regarded as the small uncertainty part, just as the nonlinearity can be used to represent the uncertainty of a system (Wang et al. 2019). So this chapter also focuses on how to deal with the nonlinearity in PLS series method, but starts from an different view, i.e., robust PLS. Here the system nonlinearity is considered as uncertainty and a new robust $$\mathrm{L}_1$$ L 1 -PLS is proposed.

The Response Statistics, Jump and Bifurcation of Nonlinear Dynamical Systems Subjected to White Noise and Combined Sinusoidal and White Noise Excitation

2007 ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Nonlinear Dynamical Systems ◽  
Gaussian White Noise ◽  
Nonlinear Dynamical ◽  
White Noise Excitation ◽  
Bladed Disk Assembly ◽  
Noise Frequency ◽  
System Nonlinearity ◽  
Non Gaussian

The prediction of the response of nonlinear systems subjected to stochastic parametric, narrowband and wideband or coloured external excitation is of importance in the field of structural and rotor dynamics. The transitional probability density function (pdf) for the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker-Planck (FP) and backward Kolmogorov equations. This paper presents efficient numerical solution of the FP equation for the pdf of response for general nonlinear systems subjected to external white noise and combined sinusoidal and white noise excitation. The effect of intensity of white noise, frequency and amplitude of sinusoidal excitation and level of system nonlinearity on the non-Gaussian nature of response caused by the system nonlinearity are investigated. Stochastic behaviours like stability, jump, bifurcation are examined as the system parameters change. The finite element (FE) scheme is used to solve the FP equation and obtain the statistics of a two degree-of-freedom linear system representative of the vibration of gas turbine tip-shrouded bladed disk assembly subjected to Gaussian white noise excitation as an illustrative example.

scholarly journals Towards Design of Quasilinear Gurvits Control Systems

2019 ◽  
Vol 18 (3) ◽  
pp. 678-705
Author(s):  
Anatoly Gaiduk
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Control ◽  
Equilibrium Point ◽  
Control Systems ◽  
Design Problem ◽  
Systems Design ◽  
Nonlinear Control Systems ◽  
Proved Theorem ◽  
Control Systems Design ◽  
Nonlinear Plants

The design problem of control systems for nonlinear plants with differentiated nonlinearity is considered. The urgency of this problem is caused by the big difficulties of practical design of nonlinear control systems with the help of the majority of known methods. In many cases, even provision by these methods of just stability of equilibrium point of a designing system represents a big challenge. Distinctive feature of the method of nonlinear control systems design considered below is the use of the nonlinear plants models represented in a quasilinear form. This form of the nonlinear differential equations exists, if nonlinearities in their right parts are differentiated across all arguments. The quasilinear model of the controlled plant allows reducing the design problem to the solution of an algebraic equations system, which has the unique solution if the plant is controlled according to the controllability condition provided in the article. This condition is similar to the controllability condition of the Kalman’s criterion. Procedure of the nonlinear control systems design on a basis of the plant’s quasilinear models is very simple. Practically, it is close to the known polynomial method of the linear control systems design. The equations of the nonlinear systems designed with application of the plant’s quasilinear models also can be represented in the quasilinear form. The basic result of this article is the proof of the theorem and the corollary from it about conditions of the asymptotical stability at whole of the equilibrium point of the nonlinear control systems designed on a basis of the plant’s quasilinear models. For the proof of the theorem and consequence, the properties of simple matrixes and known theorems of stability of the indignant systems of the differential equations are used. A way of the stability research of the equilibrium point of the quasilinear control systems based on the proved theorem is illustrated by numerical examples. Computer simulation of these systems verifies correctness of the hypoyhesis of the proved theorem. Obtained results allow applying the method of nonlinear systems design on a basis of the quasilinear models for creation of various control systems for plants in power, aviation, space, robotechnical and other industries.

scholarly journals On the Existence and Stability of Periodic Solutions of Airy’s Equation with Elastic Coefficients

2021 ◽  
pp. 3864-3873
Author(s):  
U.E. Obasi ◽  
B.O. Osu ◽  
C.P. Ogbogbo
Keyword(s):  
Equilibrium Point ◽  
Direct Method ◽  
Series Method ◽  
Lyapunov Direct Method ◽  
Elastic Coefficient ◽  
Eigenvalue Approach ◽  
Power Series Method ◽  
Existence And Stability ◽  
Stability Method ◽  
Instability Regions

In this paper, the existence and stability of periodic solutions of a certain second order differential equation with elastic coefficient were investigated using power series method, eigenvalue approach and lyapunov direct method. Existence of analytical solution which is independent of time was achieved using the power series method. Eigenvalue approach and Lyapunov direct method were used to investigate the stability of the resulting solution. Periodic solution was obtained using the eigenvalues of the resulting matrix. The first stability method further examined stability of the equilibrium point by considering the intervals around the origin and it’s discriminate. The equilibrium points for the intervals and the discriminate were unstable because the real part of the characteristics root is zero. Unstable equilibrium point was also obtained for the second stability method using the energy function and time derivative around the equilibrium point. The two unstable results indicated that there were highly instability regions with a strictly positive elastic coefficient. The highly instability regions were confirmed by the presence of elastic coefficient which reduces oscillation with an increase in amplitude. Furthermore, numerical simulations for existence and stability of Airy’s equation at different values of the elastic coefficient were illustrated in order to demonstrate the behaviour of the solutions which extends some results in literature.

Regularized Semi-Supervised Metric Learning with Latent Structure Preserved

2021 ◽  
pp. 2150013
Author(s):  
Qianying Wang ◽  
Ming Lu ◽  
Meng Li ◽  
Fei Guan
Keyword(s):  
Local Structure ◽  
Metric Learning ◽  
Latent Structure ◽  
Data Sets ◽  
Knn Classifier ◽  
Public Data ◽  
Low Dimension ◽  
The Hierarchical Structure ◽  
Split Function ◽  
The Given

Metric learning is a critical problem in classification. Most classifiers are based on a metric, the simplest one is the KNN classifier, whose outcome is directly decided by the given metric. This paper will discuss semi-supervised metric learning. Most traditional semi-supervised metric learning algorithms preserve the local structure of all the samples (including labeled and unlabeled) in the input space, when making the same labeled samples together and separating different labeled samples. In most existing methods, the local structure is calculated by the Euclidean distance which uses all the features. As we all know, high dimensional data lies on a low dimension manifold, and not all the features are discriminative. Thus, in this paper, we try to explore the latent structure of the samples and use the more discriminative features to calculate the local structure. The latent structure is learned by clustering random forest and cast into similarity between samples. Based on the hierarchical structure of the trees and the split function, the similarity is obtained from discriminant features. Experimental results on public data sets show our algorithm outperforms the traditional similar related algorithms.

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