Efficient Identification of Error-in Variables Switched Systems Based on Riemannian Distance-Like Functions

This paper considers the problem of identifying error in variables switched affine models from experimental input/output data. Since this problem is generically NP hard, several relaxations have been proposed in the past. While these relaxations work well for low dimensional systems with few subsystems, they scale poorly with both the number of subsystems and their memory. As an alternative, in this paper we present a computationally efficient method based on embedding the data in the manifold of positive semidefinite matrices, and using a manifold metric to detect switches and identify subsystems. The main result of the paper shows that, under dwell-time assumptions, the method is guaranteed to identify the system, for suitably low noise scenarios. In the case of larger noise levels, consistent numerical experience shows that the proposed method outperforms existing ones. These results are illustrated with a non-trivial practical example: action segmentation.

[1]  J. Ben Rosen,et al.  Total Least Norm Formulation and Solution for Structured Problems , 1996, SIAM J. Matrix Anal. Appl..

[2]  Abdollah Homaifar,et al.  Switched linear system identification based on bounded-switching clustering , 2015, 2015 American Control Conference (ACC).

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

[4]  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.

[5]  Constantino M. Lagoa,et al.  Robust identification of switched affine systems via moments-based convex optimization , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[6]  Stéphane Lecoeuche,et al.  A recursive identification algorithm for switched linear/affine models , 2011 .

[7]  Yin Wang,et al.  Subspace Clustering with Priors via Sparse Quadratically Constrained Quadratic Programming , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

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

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

[10]  S. Sastry,et al.  An algebraic geometric approach to the identification of a class of linear hybrid systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[11]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[12]  Necmiye Ozay,et al.  An exact and efficient algorithm for segmentation of ARX models , 2016, 2016 American Control Conference (ACC).

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

[14]  Mehrtash Tafazzoli Harandi,et al.  From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices , 2014, ECCV.

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

[16]  Le Thi Hoai An,et al.  A Difference of Convex Functions Algorithm for Switched Linear Regression , 2014, IEEE Transactions on Automatic Control.

[17]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[18]  Yin Wang,et al.  Efficient Temporal Sequence Comparison and Classification Using Gram Matrix Embeddings on a Riemannian Manifold , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[19]  Biao Huang,et al.  An Iterative Algebraic Geometric Approach for Identification of Switched ARX Models with Noise , 2016 .

[20]  Biao Huang,et al.  Matrix-wise approach for identification of multi-mode Switched ARX models with noise , 2012, 2012 American Control Conference (ACC).

[21]  B. Mohar.  Some applications of Laplace eigenvalues of graphs , 1997 .

[22]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

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

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

[25]  Fabien Lauer.  Estimating the probability of success of a simple algorithm for switched linear regression , 2013 .

[26]  Yin Wang,et al.  A convex optimization approach to semi-supervised identification of switched ARX systems , 2014, 53rd IEEE Conference on Decision and Control.

[27]  H. Fang,et al.  Identification of Switched Linear Systems via Sparse Optimization , 2015 .

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

[29]  Chao Feng,et al.  Hybrid system identification via sparse polynomial optimization , 2010, Proceedings of the 2010 American Control Conference.

[30]  Fabien Lauer.  Global optimization for low-dimensional switching linear regression and bounded-error estimation , 2018, Autom..

[31]  Anoop Cherian,et al.  Jensen-Bregman LogDet Divergence with Application to Efficient Similarity Search for Covariance Matrices , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[33]  Olivier Gehan,et al.  A real-time identification algorithm for switched linear systems with bounded noise , 2016, 2016 European Control Conference (ECC).

[34]  R. Bhatia.  Positive Definite Matrices , 2007 .

[35]  Roland Tóth,et al.  An SDP approach for l0-minimization: Application to ARX model segmentation , 2013, Autom..

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

[37]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

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