Linear and nonlinear system identification using separable least-squares
The idea of the separable least squares (SLS for short), introduced by \textit{G. H. Golub} and \textit{V. Pereyra} [SIAM J. Numer. Anal. 10, 413-432 (1973; Zbl 0258.65045)], is as follows. Suppose one has to determine the unknown vectors \(a\) and \(b\) that minimize the function \( J(a,b)=\|y-F(a)b\|\) where \(y\) is a known vector and \(F\) is a matrix function. Then, instead of minimizing \(J(a,b)\), minimize the functional: \( \widehat{J}(a)=\|y-F(a)F(a)^{\dagger }y\|\) (where \(()^{\dagger }\) denotes the pseudo-inverse) and then set \(b=F( \overline{a})^{\dagger }y\), where \(\overline{a}\) is the optimum value. The intuitive motivation stems obviously from the properties of the pseudo inverse. As recalled by the authors, the SLS method works for matrices with well behaved rank. But is there also any advantage (besides, obviously, the reduced number of parameters at the expense of a more complex function)? The authors prove that, if the error is negligible, then the method is better conditioned than the straight least squares method. The SLS technique is applicable for problems that can be parameterized separating the unknowns in two vectors \(a\) and \(b\) in such a way that the functional to be minimized has the form of \(J(a,b)\) above, provided the rank condition is satisfied and a good initial estimate is known (to achieve better conditioning). Under suitable conditions the authors work out parameterizations complying with these conditions for the identification of linear dynamic systems as well as for nonlinear dynamic systems of Wiener type. The latter consist of a series connection of a linear system and a static nonlinearity. They are known to provide good approximations for general classes of nonlinear systems. Numerical results are provided and two of the examples refer to industrial plants. Finally the authors give the reference for obtaining the software they used from the World Wide Web.
scientific article; zbMATH DE number 1343408
scientific article