Automatica 41 (2005) 1557–1565 www.elsevier.com/locate/automatica Brief paper KernelmethodsforsubspaceidentificationofmultivariableLPVand bilinearsystems  VincentVerdult ∗ , MichelVerhaegen Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, NL-2628 CD Delft, The Netherlands Received 18 September 2003; received in revised form 22 December 2004; accepted 22 March 2005 Available online13 June 2005 Abstract 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.  2005 Elsevier Ltd. All rights reserved. Keywords: System identification; Subspace methods; State-space methods; Linear parameter-varying systems; Bilinear systems 1. Introduction Subspace identification methods are widely used for the identification of linear time-invariant systems. Unlike the classical identification methods, subspace methods do not require a particular parameterization; this makes them nu- merically attractive and especially suitable for multivari- able systems. In recent years subspace identification meth- ods have also been developed for linear parameter-varying (LPV) and bilinear systems. LPV systems are widely used in control, especially in gain-scheduling (Shamma & Athans, 1991)androbustcontroltechniques(Zhou,Doyle,&Glover, 1996). Bilinear systems form an important class of nonlinear  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Tomas McKelvey under the direction of Editor T. Soderstrom. ∗ Corresponding author. Tel.: +31152785768; fax: +31152786679. E-mail address: v.verdult@dcsc.tudelft.nl (V. Verdult). 0005-1098/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.03.027 systems for which a considerable body of theoretical results has been obtained over the years (Bruni, Dipillo, & Koch, 1974; Mohler & Kolodziej, 1980). The first subspace methods that were developed for the identification of bilinear systems were based on the assumptio that the input to the system is a white-noise sequence (Favoreel, De Moor, & Van Overschee, 1999; Verdult, 2002). Although white-noise inputs provide a good excitation of the dynamical system, in several appli- cations the input cannot be taken equal to a white-noise sequence; it might be that only sum-of-sine inputs or step signals with a minimum duration are allowable or pos- sible. Therefore, bilinear subspace identification methods that can deal with more general input signals are of in- terest. Favoreel (1999) and Chen and Maciejowski (2000) described such bilinear subspace identification methods. We recently described how these subspace methods can be extended to identify LPV systems with affine param- eter dependence ( Verdult & Verhaegen, 2002; Verdult, 2002).