L2-optimallinear system identification Structured Total Least Squares for SISO systems

We treat the extension of Total Linear Least Squares to single-input single-output dynamic systems. It is shown that the L2-optimal model for a given dataset has some remarkable properties: The given data sequence can be decomposed into two sequences, the L2-approximations and the residuals, which are orthogonal to each other (in diagonal inner products derived from given weights). It is shown how certain block Rankel matrices Wi constructed from the weighted residuals are always rank deficient and hence, the residuals themselves can be considered as being generated by a linear system. One can find completions Wicomof the block RankeImatrices Wi with the L2-approximations, such that W,comWT = 0 (which is a more general type of orthogonality than the first one mentioned). , The results of this paper are just a special application of a general theory to approximate in a least squares sense, a given structured matrix, by one which has the same structure and is rank deficient. The main result of this theory says that, when the matrix structure is affine, the solution is generated in terms of a Riemannian SVD, which is a 'nonlinear' generalized singular value decomposition (see De Moor B., Linear Algebra and its Applications, Vo1.188-189,pp.163-207, 1993). °The following text presents research results obtained within the framework of the Belgian program me on Interuniversity Attraction Poles (IUAP-17 and IUAP-SO) initiated by the Belgian State -Prime Minister's Office Seience Policy Programming, the NFWO under contract nr. G.0292.9S (Differential geometry and matrix algorithms), the European SCIENCE project ERNSI (European Resea.rch Network for System Identification) and the European Human Capital and Mobility network SIMONET (Systems and Modelling Network). The seientific responsibility is assumed by its authors, one of which (BDM) is a. senior research assoeiate with the NFWO. 0-7803-1968-0/94$4.0001994 IEEE Berend Roorda Tinbergen Instituut Oostmaaslaan 950-952 NL 3063 DM Rotterdam The Netherlands tel: 31/10/4088932 fax: 31/10/4527347 [email protected] 1 IntroductÎon and notatÎon Let Wk E JR2, k = 0,..., N be a given vector sequence of data where the scalar sequences Uk and Yk are defined as Wk = (Uk Ykl (a superscript capital T denotes 'transpose'). One could consider the first component of Wk to be an input sequence Uk and the second component to be an output sequence Yk or the ot her way around. Our task now is to find least squares approximations Vk of the Uk, and Zk of the Yk, such that Vk and Zk are related by a linear model of given order n with real coefficients: