H∞-norm-based optimization for the identification of gray-box LTI state-space model parameters

In this paper, the challenging problem of determining the unknown parameters of an identifiable LTI state-space representation of a stable system is addressed by resorting to a specific H∞-norm-based optimization algorithm. More specifically, by assuming the availability of a reliable fully-parameterized representation of the system to identify, the algorithm developed herein consists in restructuring this initial black-box representation of the system dynamics via the optimization of a dedicated maximum eigenvalue-based criterion. This study shows that this H∞-norm-based approach can be seen as a good solution for the identification of gray-box LTI state-space representations and, by extension, as an interesting alternative or a reliable initialization step for the standard output-error techniques.

[1]  Pierre Apkarian,et al.  Nonsmooth H∞ synthesis , 2005, IEEE Trans. Autom. Control..

[2]  Brett Ninness,et al.  On Gradient-Based Search for Multivariable System Estimates , 2008, IEEE Transactions on Automatic Control.

[3]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[4]  Georges Bastin,et al.  An estimator of the inverse covariance matrix and its application to ML parameter estimation in dynamical systems , 2001, Autom..

[5]  Pierre Apkarian,et al.  A Trust Region Spectral Bundle Method for Nonconvex Eigenvalue Optimization , 2008, SIAM J. Optim..

[6]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[7]  Pablo A. Parrilo,et al.  Initialization of Physical Parameter Estimates , 2003 .

[8]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[9]  Hugues Garnier,et al.  Direct continuous-time approaches to system identification. Overview and benefits for practical applications , 2015, Eur. J. Control.

[10]  D. Noll.  A PROXIMITY CONTROL ALGORITHM TO MINIMIZE NONSMOOTH AND NONCONVEX SEMI-INFINITE MAXIMUM EIGENVALUE FUNCTIONS , 2007 .

[11]  Naresh K. Sinha,et al.  Modern Control Systems , 1981, IEEE Transactions on Systems, Man, and Cybernetics.

[12]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[13]  Rik Pintelon,et al.  System Identification: A Frequency Domain Approach , 2012 .

[14]  V. Verdult,et al.  Filtering and System Identification: A Least Squares Approach , 2007 .

[15]  Edouard Laroche,et al.  Linear fractional LPV model identification from local experiments: An H∞-based optimization technique , 2013, 52nd IEEE Conference on Decision and Control.

[16]  Lennart Ljung,et al.  Estimate Physical Parameters by Black Box Modeling , 2003 .

[17]  Lennart Ljung,et al.  Kernel methods in system identification, machine learning and function estimation: A survey , 2014, Autom..

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

[19]  Liuping Wang,et al.  Identification of Continuous-time Models from Sampled Data , 2008 .

[20]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[21]  Marco Lovera,et al.  State space model identification: from unstructured to structured models with an H∞ approach , 2013 .

[22]  Aude Rondepierre,et al.  A Proximity Control Algorithm to Minimize Nonsmooth and Nonconvex Functions , 2008 .

[23]  P. Apkarian,et al.  Nonsmooth H ∞ synthesis , 2005 .

[24]  Guillaume Mercère,et al.  H∞-based LPV model identification from local experiments with a gap metric-based operating point selection , 2014, 2014 European Control Conference (ECC).

[25]  Lennart Ljung,et al.  Handling Certain Structure Information in Subspace Identification , 2009 .

[26]  José A. Ramos,et al.  Identification of Parameterized Gray-Box State-Space Systems: From a Black-Box Linear Time-Invariant Representation to a Structured One , 2014, IEEE Transactions on Automatic Control.