Nonparametric Identification of Hammerstein Systems Using Orthogonal Basis Functions as Ersatz Nonlinearities

2013 ◽  
Author(s):  
Khaled F. Aljanaideh ◽  
Dennis S. Bernstein
Keyword(s):  
Fourier Series ◽  
Theoretical Analysis ◽  
Orthogonal Basis ◽  
Basis Functions ◽  
Inner Product ◽  
Fourier Series Expansion ◽  
Unknown Input ◽  
Hammerstein System ◽  
Basis Expansion ◽  
Input Nonlinearity

In this paper, we present a technique for estimating the input nonlinearity of a Hammerstein system by using multiple orthogonal ersatz nonlinearities. Theoretical analysis shows that by replacing the unknown input nonlinearity by an ersatz nonlinearity, the estimates of the Markov parameters of the plant are correct up to a scalar factor, which is related to the inner product of the true input nonlinearity and the ersatz nonlinearity. These coefficients are used to construct and estimate the true nonlinearity represented as an orthogonal basis expansion. We demonstrate this technique by using a Fourier series expansion as well as orthogonal polynomials. We show that the kernel of the inner product associated with the orthogonal basis functions must be chosen to be the density function of the input signal.

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

The Surface Variational Principle: A Different Perspective for Structural Acoustic Modeling

1997 ◽  
Author(s):  
J. H. Ginsberg
Keyword(s):  
Fourier Series ◽  
Variational Principle ◽  
Vibration Analysis ◽  
Basis Functions ◽  
Acoustic Modeling ◽  
Normal Velocity ◽  
Convergence Properties ◽  
Variational Functional ◽  
Acoustic Interaction ◽  
System Equations

Abstract This paper surveys the development and application of the surface variational principle (SVP) governing the acoustic interaction between surface pressure and normal velocity. SVP is analogous to the method of assumed modes for vibration analysis, in that it represents the response in terms of a sequence of basis functions that are globally defined. The system equations governing the series coefficients are obtained by requiring that the value of the variational functional be stationary. In the wavenumber-based version of SVP, the pressure and velocity are represented by dual range Fourier series. A brief description of the steps entailed in formulating the SVP equations and coupling them to the equations for an elastic structure is provided. Then the computational requirements of an SVP analysis relative to conventional boundary element and finite element techniques are discussed. This is followed by an example illustrating the convergence properties of SVP. Another example is used to highlight the physical interpretation of the SVP representation of surface response. The evolution of the present version of SVP is surveyed, along with some of its applications. The paper closes with a brief discussion of possible future applications of the method.

scholarly journals Optimal multistage relaxation with automatic parameter selection

2021 ◽  
Author(s):  
Alvin Wong
Keyword(s):  
Fourier Series ◽  
Relaxation Method ◽  
Accurate Solution ◽  
Convergence Time ◽  
Fourier Series Expansion ◽  
End User ◽  
Local Flow ◽  
Flow Problems ◽  
User Expertise ◽  
Complex Fourier Series

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

scholarly journals Optimal multistage relaxation with automatic parameter selection

2021 ◽  
Author(s):  
Alvin Wong
Keyword(s):  
Fourier Series ◽  
Relaxation Method ◽  
Accurate Solution ◽  
Convergence Time ◽  
Fourier Series Expansion ◽  
End User ◽  
Local Flow ◽  
Flow Problems ◽  
User Expertise ◽  
Complex Fourier Series

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

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