LS-SVM approximate solution for affine nonlinear systems with partially unknown functions
Guoshan Zhang Shiwei Wang Yiming Wang Wanquan Liu
By using the Least Squares Support Vector Machines (LS-SVMs), we develop a numerical approach to find an approximate solution for affine nonlinear systems with partially unknown functions. This approach can obtain continuous and differential approximate solutions of the nonlinear differential equations, and can also identify the unknown nonlinear part through a set of measured data points. Technically, we first map the known part of the affine nonlinear systems into high dimensional feature spaces and derive the form of approximate solution. Then the original problem is formulated as an approximation problem via kernel trick with LS-SVMs. Furthermore, the approximation of the known part can be expressed via some linear equations with coefficient matrices as coupling square matrices, and the unknown part can be identified by its relationship to the known part and the approximate solution of affine nonlinear systems. Finally, several examples for different systems are presented to illustrate the validity of the proposed approach.
keywords: Least Squares Support Vector Machines (LS-SVM) coupling square matrices. approximate solutions measured data points affine nonlinear systems
Lyapunov method for stability of descriptor second-order and high-order systems
Guoshan Zhang Peizhao Yu

In this study, the stability problem of descriptor second-order systems is considered. Lyapunov equations for stability of second-order systemsare established by using Lyapunov method. The existence of solutions for Lyapunov equations are discussed and linear matrixinequality condition for stability of second-order systems aregiven. Then, based on the feasible solutions of the linear matrixinequality, all parametric solutions of Lyapunov equations are derived.Furthermore, the results of Lyapunov equations and linear matrixinequality condition for stability of second-ordersystems are extended to high-order systems. Finally, illustratingexamples are provided to show the effectiveness of the proposed method.

keywords: Stability Lyapunov equations second-order systems linear matrix inequality

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