Kernel methods for subspace identification of multivariable LPV and bilinear systems

Subspace identification methods for multivariable linear parameter-varying (LPV) and bilinear state-space systems perform computations with data matrices of which the number of rows grows exponentially with the order of the system. Even for relatively low-order systems with only a few inputs and outputs, the amount of memory required to store these data matrices exceeds the limits of what is currently available on the average desktop computer. This severely limits the applicability of the methods. In this paper, we present kernel methods for subspace identification performing computations with kernel matrices that have much smaller dimensions than the data matrices used in the original LPV and bilinear subspace identification methods. We also describe the integration of regularization in these kernel methods and show the relation with least-squares support vector machines. Regularization is an important tool to balance the bias and variance errors. We compare different regularization strategies in a simulation study.

[1]  R. Mohler.  An Overview of Bilinear System Theory and Applications , 1980 .

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

[3]  G. Wahba.  Spline models for observational data , 1990 .

[4]  Michael Athans,et al.  Guaranteed properties of gain scheduled control for linear parameter-varying plants , 1991, Autom..

[5]  Per Christian Hansen,et al.  Truncated Singular Value Decomposition Solutions to Discrete Ill-Posed Problems with Ill-Determined Numerical Rank , 1990, SIAM J. Sci. Comput..

[6]  Dietmar Bauer,et al.  Choosing Integer Parameters in Subspace Methods: A Survey on Asymptotic Results , 2003 .

[7]  Gene H. Golub,et al.  Matrix computations , 1983 .

[8]  Bart De Moor,et al.  Subspace identification of bilinear systems subject to white inputs , 1999, IEEE Trans. Autom. Control..

[9]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[10]  L. Ljung,et al.  Overtraining, regularization and searching for a minimum, with application to neural networks , 1995 .

[11]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[12]  Per Christian Hansen,et al.  Regularization Tools Version 3.0 for Matlab 5.2 , 1999, Numerical Algorithms.

[13]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[14]  Thomas Kailath,et al.  Linear Systems , 1980 .

[15]  B. Moor,et al.  Subspace identification for linear systems , 1996 .

[16]  C. Bruni,et al.  Bilinear systems: An appealing class of "nearly linear" systems in theory and applications , 1974 .

[17]  Michel Verhaegen,et al.  Subspace identification of multivariable linear parameter-varying systems , 2002, Autom..

[18]  V. Verdult,et al.  Identification of multivariable bilinear state space systems based on subspace techniques and separable least squares optimization , 2001 .

[19]  J. Maciejowski,et al.  An improved subspace identification method for bilinear systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[20]  Bernhard Schölkopf,et al.  A tutorial on support vector regression , 2004, Stat. Comput..

[21]  V. Verdult,et al.  A Kernel method for subspace identification of multivariable bilinear systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[22]  R. Shah,et al.  Least Squares Support Vector Machines , 2022 .