GENERALIZATION OF A TOTAL LEAST SQUARES PROBLEM IN FREQUENCY DOMAIN SYSTEM IDENTIFICATION László Balogh Advisor: István Kollár I. Introduction In the last years the research area of the system identification in the frequency domain has been extended. As yet in this topic there are lots of known algorithms which have different properties. In this paper we would like to present a generalization of the socalled total least squares (TLS) algorithm. In the frequency domain system identification we would like to estimate a linear time invariant dynamic system with single input and single output. This means determination of the real coefficients of the transfer function. II. Preliminaries In this paper we assume that there are additive noises both on the input and the output. Ωk denotes the generalized frequency and k means the k th measure point in frequency domain. If we write the measured input and output to the place of the exact input and output in the model equation, then N (Ωk , P)U m (Ωk ) − D(Ωk , P )Ym (Ωk ) = 0 k =1, , F (1) Where N (Ωk , P) , D (Ωk , P) are the numerator and denominator of the transfer function. U m (Ωk ) and Ym (Ωk ) denotes the measured input and output, respectively. P notes the collected parameters of the transfer function and t is the transposing operator. The equation (1) is only true if the measured input and output are noiseless. Otherwise cannot be fulfill with any P , so it is an overdetermined set of linear equation. More details see in  and . We can formulate the parameter estimation as minimization of a socalled cost function: K = ( JP) t ( JP) where ( JP) k = N (Ωk , P)U m (Ωk ) − D (Ωk , P)Ym (Ωk ) (2) To avoid the trivial solution P = 0 , we fix the parameter P = . Considering this, the linear 1 equations can be solved by the TLS algorithm with by using singular value decomposition. The cost function can be written in form: ( JP) t JP K TLS = Pt P where the constraint P = is included (, ). 1 III. A generalization of the total least squares problem This method allows us to give us additional constraint on the parameter vector in the form of a bilinear expression. It is a generalization of fixing the norm (, ). Applications will be discussed in the next section. We use the cost function (2). A bilinear constraint can be written in the form P t AP = 1 (3) We use the Lagrange multiplier technique to consider (3) in the minimization of the cost function. Hence K λ = ( JP) t JP − λ ( P t AP −1) Differential this, we get that ∂K λ = J t JP − λ AP = 0 and J t JP = AP (4) ∂P Equation (4) is a generalized eigenvalue problem which can be solved effectively using generalized singular value decomposition (see ). IV. Using orthogonal polynomials The orthogonal polynomials are used to enhance the numerical condition of the problem (, ). ˆ Without details we note that using orthogonal polynomials is equivalent to a transformation. If P denotes a parameter vector in the new basis, we can write: ˆ P = TP (5) where T is the transformation matrix mentioned above. It is important top note that T has full rank in (5), so we can apply generalized singular value decomposition. Thus the corresponding cost function is: ˆ ˆ K = ( JTP ) t JTP = ( JP ) t JP (6) ˆ It can be seen that solving problem (6) with constraint P = , we solve a problem which differs 1 from the original one, unless we use rather the bilinear constraint mentioned above. In this case we choose matrix A so that we will solve the equivalent of the original problem in the new basis. A = T −t T −1 References  R. Pintelon et al.,”Parametric Identification of Transfer Function in the Frequency Domain--A Survey”, IEEE Transactions on Automatic Control, Vol. 39., No. 11, pp. 2245-2260, November 1994.  G. H. Golub and C. F. Van Loan, “Matrix Computations”, The John Hopkins University Press, Baltimore, USA, 1989.  S. Van Huffel and J. Vandewalle, “The Total Least Squares Problem - Computational Aspects and Analysis”, Society for Industrial and Applied Mathematics, Philadelphia, USA, 1991.  Yves Rolain, R. Pintelon, K. Q. Xu, and H. Vold, “On the Use of Orthognal Polynomials in High Order Frequency Domain System Identification and its Application to Modal Parameter Estimation”, Proceedings of the 33rd IEEE Conference on Decision and Control, Lake Buena Vista, Florida, USA, December 14-16, 1994, pp. 3365-3373.  R. Pintelon and J. Schoukens, “System identification. A frequency domain approach”, IEEE Press, to appear.
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