The preceding chapters have developed a general theory pertaining to rational orthonormal bases. They have also presented a general framework for addressing system identification problems and have then provided a link between the two by illustrating the advantages of parameterizing models for system identification purposes using rational orthonormal bases. This chapter continues on these themes of exposing the links between rational orthonormal bases and the system identification problem. The main feature of this chapter is that it will progress to consider more general model structures, especially those not specifically formulated with respect to an orthonormal basis. At the same time that we generalize this aspect, we will focus another by considering only the noise-induced error. We will not discuss bias error, as it will be analysed in detail in the following chapters, and was also addressed in the previous one. Many of the underlying ideas necessary for the developments here have already been introduced, but in the interests of the reader who is seeking to ?dip into? chapters individually, a brief introduction to essential background ideas and notation will be made at the expense of some slight repetition. In precis then, the sole focus of this chapter is to examine and quantify noiseinduced estimation errors in the frequency domain, and there are four essential points to be made here in relation to this. 1. The quantification of noise-induced error (variance error) is equivalent to the quantification of the reproducing kernel for a particular function space Xn; 2. This function space Xn depends on the model structure, as well as on input and noise spectral densities. Hence, it is not independent of model structure; 3. The quantification of the reproducing kernel, and hence the quantification of variance error, depends crucially on the construction of a rational orthonormal basis for the afore-mentioned function space Xn; 4. The variance error quantifications that result are quite different to certain pre-existing ones, while at the same time are often much more accurate. This latter point will be established here empirically by simulation example.
Springer-Verlag New York, 2005. , 161 p.103-161 p.