GENERALIZATION OF A TOTAL LEAST SQUARES PROBLEM IN FREQUENCY DOMAIN SYSTEM
Advisor: István Kollár
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.
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
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 =
where the constraint P = is included (, ).
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.
K λ = ( JP) t JP − λ ( P t AP −1)
Differential this, we get that
= J t JP − λ AP = 0 and J t JP = AP (4)
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 (, ).
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:
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
K = ( JTP ) t JTP = ( JP ) t JP (6)
It can be seen that solving problem (6) with constraint P = , we solve a problem which differs
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
 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
 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.