Model structure selection for switched NARX system identification: A randomized approach

The identification of switched systems is a challenging problem, which entails both combinatorial (sample-mode assignment) and continuous (parameter estimation) features. A general framework for this problem has been recently developed, which alternates between parameter estimation and sample-mode assignment, solving both tasks to global optimality under mild conditions. This article extends this framework to the nonlinear case, which further aggravates the combinatorial complexity of the identification problem, since a model structure selection task has to be addressed for each mode of the system. To solve this issue, we reformulate the learning problem in terms of the optimization of a probability distribution over the space of all possible model structures. Then, a randomized approach is employed to tune this distribution. The performance of the proposed approach on some benchmark examples is analyzed in detail.

[1]  René Vidal,et al.  Identification of Hybrid Systems: A Tutorial , 2007, Eur. J. Control.

[2]  Constantino M. Lagoa,et al.  A Sparsification Approach to Set Membership Identification of Switched Affine Systems , 2012, IEEE Transactions on Automatic Control.

[3]  Gérard Bloch,et al.  Piecewise smooth system identification in reproducing kernel Hilbert space , 2014, 53rd IEEE Conference on Decision and Control.

[4]  Gianluigi Pillonetto,et al.  A new kernel-based approach to hybrid system identification , 2016, Autom..

[5]  Fu-Lai Chung,et al.  Evolutionary segmentation of financial time series into subsequences , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[6]  Alberto Bemporad,et al.  A bounded-error approach to piecewise affine system identification , 2005, IEEE Transactions on Automatic Control.

[7]  Alex Waibel,et al.  Readings in speech recognition , 1990 .

[8]  A. Bemporad,et al.  Rao-Blackwellized sampling for batch and recursive Bayesian inference of Piecewise Affine models , 2020, Autom..

[9]  Luigi Piroddi,et al.  A randomised approach for NARX model identification based on a multivariate Bernoulli distribution , 2017, Int. J. Syst. Sci..

[10]  Gérard Bloch,et al.  Switched and PieceWise Nonlinear Hybrid System Identification , 2008, HSCC.

[11]  James M. Rehg,et al.  Learning and Inferring Motion Patterns using Parametric Segmental Switching Linear Dynamic Systems , 2008, International Journal of Computer Vision.

[12]  BakoLaurent.  Identification of switched linear systems via sparse optimization , 2011 .

[13]  Luigi Piroddi,et al.  A randomized two-stage iterative method for switched nonlinear systems identification , 2020 .

[14]  Alberto Bemporad,et al.  Identification of piecewise affine systems via mixed-integer programming , 2004, Autom..

[15]  Fen Wu,et al.  Switching LPV control designs using multiple parameter-dependent Lyapunov functions , 2004, Autom..

[16]  René Vidal,et al.  A continuous optimization framework for hybrid system identification , 2011, Autom..

[17]  Alberto Bemporad,et al.  Piecewise affine regression via recursive multiple least squares and multicategory discrimination , 2016, Autom..

[18]  N. Nguyen,et al.  Hidden Markov Model for Stock Trading , 2018 .

[19]  S. Billings.  Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains , 2013 .

[20]  Constantino M. Lagoa,et al.  Set membership identification of switched linear systems with known number of subsystems , 2015, Autom..

[21]  Peng Shi,et al.  $l_{2}-l_{\infty}$ Model Reduction for Switched LPV Systems With Average Dwell Time , 2008, IEEE Transactions on Automatic Control.

[22]  I. J. Leontaritis,et al.  Input-output parametric models for non-linear systems Part II: stochastic non-linear systems , 1985 .

[23]  W. P. M. H. Heemels,et al.  A Bayesian approach to identification of hybrid systems , 2004, IEEE Transactions on Automatic Control.

[24]  Manfred Morari,et al.  A clustering technique for the identification of piecewise affine systems , 2001, Autom..

[25]  A. Garulli,et al.  A survey on switched and piecewise affine system identification , 2012 .

[26]  Stephen P. Boyd,et al.  Fitting Jump Models , 2017, Autom..

[27]  Laurent Bako,et al.  Identification of switched linear systems via sparse optimization , 2011, Autom..

[28]  Gérard Bloch,et al.  Learning nonlinear hybrid systems: from sparse optimization to support vector regression , 2013, HSCC '13.

[29]  Biao Huang,et al.  An improved algebraic geometric solution to the identification of switched ARX models with noise , 2011, Proceedings of the 2011 American Control Conference.

[30]  Gérard Bloch,et al.  Hybrid System Identification: Theory and Algorithms for Learning Switching Models , 2019 .

[31]  Joao Xavier,et al.  Identification of switched ARX models via convex optimization and expectation maximization , 2015 .

[32]  Toshiharu Sugie,et al.  Identification of multiple mode models via Distributed Particle Swarm Optimization , 2011 .

[33]  René Vidal,et al.  Nonlinear hybrid system identification with kernel models , 2010, 49th IEEE Conference on Decision and Control (CDC).

[34]  D. Jacobson,et al.  Optimization of stochastic linear systems with additive measurement and process noise using exponential performance criteria , 1974 .

[35]  Michael I. Jordan,et al.  JOINT MODELING OF MULTIPLE TIME SERIES VIA THE BETA PROCESS WITH APPLICATION TO MOTION CAPTURE SEGMENTATION , 2013, 1308.4747.

[36]  René Vidal,et al.  Identification of Deterministic Switched ARX Systems via Identification of Algebraic Varieties , 2005, HSCC.

[37]  Henrik Ohlsson,et al.  Identification of switched linear regression models using sum-of-norms regularization , 2013, Autom..

[38]  Gérard Bloch,et al.  Reduced-Size Kernel Models for Nonlinear Hybrid System Identification , 2011, IEEE Transactions on Neural Networks.

[39]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[40]  Mari Ostendorf,et al.  From HMM's to segment models: a unified view of stochastic modeling for speech recognition , 1996, IEEE Trans. Speech Audio Process..