Wavelet basis expansion-based Volterra kernel function identification through multilevel excitations

Volterra series is a powerful mathematical tool for nonlinear system analysis, which extends the convolution integral for linear system to nonlinear system. There is a wide range of nonlinear engineering systems and structures which can be modeled as Volterra series. One question involved in modeling a functional relationship between the input and output of a system using Volterra series is to identify the Volterra kernel functions. In this article, a wavelet balance method-based approach is proposed to identify the Volterra kernel functions from observations of the in- and outgoing signals. The basic routine of the approach is that, from the system outputs under multilevel excitations, the Volterra series outputs of different orders are first estimated with the wavelet balance method, and then the Volterra kernel functions of different orders are separately estimated through their corresponding Volterra series outputs by expanding them with four-order B-spline wavelet on the interval. The simulation studies verify the effectiveness of the proposed Volterra kernel identification method.

[1]  Richard J. Prazenica,et al.  Volterra Kernel Identification Using Triangular Wavelets , 2004 .

[2]  T. Vinh,et al.  Nonlinear Behaviour of Structures Using the Volterra Series—Signal Processing and Testing Methods , 2004 .

[3]  Wagner Caradori do Amaral,et al.  CHOICE OF FREE PARAMETERS IN EXPANSIONS OF DISCRETE-TIME VOLTERRA MODELS USING KAUTZ FUNCTIONS , 2005 .

[4]  Taiho Koh,et al.  Second-order Volterra filtering and its application to nonlinear system identification , 1985, IEEE Trans. Acoust. Speech Signal Process..

[5]  Jianping Cai,et al.  Comparison of multiple scales and KBM methods for strongly nonlinear oscillators with slowly varying parameters , 2004 .

[6]  Richard J. Prazenica,et al.  Multiwavelet Constructions and Volterra Kernel Identification , 2006 .

[7]  W. Reiss.  Nonlinear distortion analysis of p-i-n diode attenuators using Volterra series representations , 1984 .

[8]  Yibing Shi,et al.  An Approach to Locate Parametric Faults in Nonlinear Analog Circuits , 2012, IEEE Transactions on Instrumentation and Measurement.

[9]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[10]  Vimal Singh,et al.  Perturbation methods , 1991 .

[11]  Bing Li,et al.  Identification of a crack in a beam based on the finite element method of a B-spline wavelet on the interval , 2006 .

[12]  V. J. Mathews,et al.  Volterra and general polynomial related filtering , 1993, IEEE Winter Workshop on Nonlinear Digital Signal Processing.

[13]  C. Cowan,et al.  Non-linear system modelling: Concept and application , 1984, ICASSP.

[14]  Philip M. Morse,et al.  Methods of Mathematical Physics , 1947, The Mathematical Gazette.

[15]  Hoda Moodi,et al.  On identification of nonlinear systems using Volterra kernels expansion on Laguerre and wavelet function , 2010, 2010 Chinese Control and Decision Conference.

[16]  W. Cai,et al.  A fast wavelet collocation method for high-speed circuit simulation , 1999 .

[17]  A. Cohen.  Numerical Analysis of Wavelet Methods , 2003 .

[18]  N. Soveiko,et al.  Steady-state analysis of multitone nonlinear circuits in wavelet domain , 2004, IEEE Transactions on Microwave Theory and Techniques.

[19]  Shijun Liao,et al.  Comparison between the homotopy analysis method and homotopy perturbation method , 2005, Appl. Math. Comput..

[20]  Matjaž Perc,et al.  Chaos in temporarily destabilized regular systems with the slow passage effect , 2006 .

[21]  M. Nakhla,et al.  Wavelet harmonic balance , 2003, IEEE Microwave and Wireless Components Letters.

[22]  Kenneth C. Chou,et al.  Representation of Green's Function Integral Operators Using Wavelet Transforms , 2000 .

[23]  M. Perc.  Visualizing the attraction of strange attractors , 2005 .

[24]  W. Silva,et al.  Identification of Nonlinear Aeroelastic Systems Based on the Volterra Theory: Progress and Opportunities , 2005 .

[25]  W. Rugh.  Nonlinear System Theory: The Volterra / Wiener Approach , 1981 .

[26]  M. Perc.  The dynamics of human gait , 2005 .

[27]  Bernhard Schölkopf,et al.  A Unifying View of Wiener and Volterra Theory and Polynomial Kernel Regression , 2006, Neural Computation.

[28]  Z. Lang,et al.  Frequency domain analysis of a dimensionless cubic nonlinear damping system subject to harmonic input , 2009 .

[29]  Richard J. Prazenica,et al.  Multiresolution Methods for Reduced-Order Models for Dynamical Systems , 2001 .

[30]  Nalinaksh S. Vyas,et al.  Application of Volterra and Wiener Theories for Nonlinear Parameter Estimation in a Rotor-Bearing System , 2001 .

[31]  Ricardo J. G. B. Campello,et al.  Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions , 2007, Autom..

[32]  Chen Xuefeng,et al.  A new wavelet-based thin plate element using B-spline wavelet on the interval , 2007 .

[33]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[34]  Xuan Zeng,et al.  A wavelet-balance approach for steady-state analysis of nonlinear circuits , 2002 .

[35]  M. Schetzen.  The Volterra and Wiener Theories of Nonlinear Systems , 1980 .

[36]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

[37]  V. Marmarelis.  Identification of nonlinear biological systems using laguerre expansions of kernels , 1993, Annals of Biomedical Engineering.

[38]  Daniel Coca,et al.  Non-linear system identification using wavelet multiresolution models , 2001 .

[39]  Matjaz Perc,et al.  Synchronization of Regular and Chaotic oscillations: the Role of Local Divergence and the Slow Passage Effect - a Case Study on calcium oscillations , 2004, Int. J. Bifurc. Chaos.

[40]  Keith Worden,et al.  A harmonic probing algorithm for the multi-input Volterra series , 1997 .

[41]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[42]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[43]  J.S. Kenney,et al.  An Approximation of Volterra Series Using Delay Envelopes, Applied to Digital Predistortion of RF Power Amplifiers With Memory Effects , 2008, IEEE Microwave and Wireless Components Letters.

[44]  M. Perc,et al.  Detecting chaos from a time series , 2005 .

[45]  Stephen A. Billings,et al.  Wavelet based non-parametric NARX models for nonlinear input–output system identification , 2006, Int. J. Syst. Sci..

[46]  Matjaž Perc,et al.  Introducing nonlinear time series analysis in undergraduate courses , 2006 .

[47]  I. Hunter,et al.  The identification of nonlinear biological systems: Volterra kernel approaches , 1996, Annals of Biomedical Engineering.

[48]  Daniella E. Raveh,et al.  Computational-fluid-dynamics-based aeroelastic analysis and structural design optimization—a researcher’s perspective , 2005 .

[49]  M. Perc.  Nonlinear time series analysis of the human electrocardiogram , 2005 .

[50]  Per Christian Hansen,et al.  Rank-Deficient and Discrete Ill-Posed Problems , 1996 .

[51]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[52]  G. Vossoughi,et al.  Dynamic analysis of microrobots with Coulomb friction using harmonic balance method , 2012 .

[53]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[54]  Zhike Peng,et al.  Feasibility study of structural damage detection using NARMAX modelling and Nonlinear Output Frequency Response Function based analysis , 2011 .

[55]  Daniel Coca,et al.  Continuous-Time System Identification for Linear and Nonlinear Systems Using Wavelet Decompositions , 1997 .

[56]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[57]  E. Bedrosian,et al.  The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs , 1971 .

[58]  W. Cai,et al.  An adaptive wavelet method for nonlinear circuit simulation , 1999 .