A tensor-based volterra series black-box nonlinear system identification and simulation framework

Tensors are a multi-linear generalization of matrices to their d-way counterparts, and are receiving intense interest recently due to their natural representation of high-dimensional data and the availability of fast tensor decomposition algorithms. Given the input-output data of a nonlinear system/circuit, this paper presents a non-linear model identification and simulation framework built on top of Volterra series and its seamless integration with tensor arithmetic. By exploiting partially-symmetric polyadic decompositions of sparse Toeplitz tensors, the proposed framework permits a pleasantly scalable way to incorporate high-order Volterra kernels. Such an approach largely eludes the curse of dimensionality and allows computationally fast modeling and simulation beyond weakly non-linear systems. The black-box nature of the model also hides structural information of the system/circuit and encapsulates it in terms of compact tensors. Numerical examples are given to verify the efficacy, efficiency and generality of this tensor-based modeling and simulation framework.

[1]  Gérard Favier,et al.  Parametric complexity reduction of Volterra models using tensor decompositions , 2009, 2009 17th European Signal Processing Conference.

[2]  Joel R. Phillips,et al.  Projection-based approaches for model reduction of weakly nonlinear, time-varying systems , 2003, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

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

[4]  Lawrence T. Pileggi,et al.  Compact reduced-order modeling of weakly nonlinear analog and RF circuits , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[5]  O. Nelles.  Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models , 2000 .

[6]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[7]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[8]  A. Kibangou,et al.  Nonlinear system modeling and identification using Volterra‐PARAFAC models , 2012 .

[9]  Lijun Jiang,et al.  STAVES: Speedy tensor-aided Volterra-based electronic simulator , 2015, 2015 IEEE/ACM International Conference on Computer-Aided Design (ICCAD).

[10]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[11]  Biao Huang,et al.  System Identification , 2000, Control Theory for Physicists.

[12]  Lennart Ljung,et al.  System identification (2nd ed.): theory for the user , 1999 .

[13]  Roland Badeau,et al.  Fast orthogonal decomposition of Volterra cubic kernels using oblique unfolding , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[14]  Yang Zhang,et al.  Compact model order reduction of weakly nonlinear systems by associated transform , 2016, Int. J. Circuit Theory Appl..

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

[16]  Ngai Wong,et al.  Symmetric tensor decomposition by an iterative eigendecomposition algorithm , 2014, J. Comput. Appl. Math..

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

[18]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

[19]  Ngai Wong,et al.  Autonomous Volterra Algorithm for Steady-State Analysis of Nonlinear Circuits , 2013, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[20]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[21]  Ngai Wong,et al.  A Constructive Algorithm for Decomposing a Tensor into a Finite Sum of Orthonormal Rank-1 Terms , 2014, SIAM J. Matrix Anal. Appl..

[22]  Sanjit K. Mitra,et al.  Kronecker Products, Unitary Matrices and Signal Processing Applications , 1989, SIAM Rev..

[23]  C. Loan.  The ubiquitous Kronecker product , 2000 .