TABLE OF CONTENTS In the previous discussion the system was assumed to be noise-free. In practice this of course rarely happens. To discuss the treatment of more realistic circumstances, we now take the output-error estimation problem stated at the beginning of Chapter 7 in which the output is perturbed by an additive white-noise sequence. In the subsequent sections of this chapter it will be assumed that the noise sequences are ergodic (see Section 4.3.4).
Let the additive noise be denoted by ?(k), then the signal-generating system that will be considered can be written as
The data equation for this system is similar to (9.7) on page 296 and reads
where V i,s,N is a block Hankel matrix constructed from the sequence ?(k).
The next lemma shows that, in the limit of N ? ?, the result of Lemma 9.1 can be extended to the case in which the additive noise at the output is white.
Lemma 9.3 Given the minimal system (9.26) (9.27) with u(k), x(k), and ?(k) ergodic stochastic processes, with the input u(k) satisfying
and with ?(k) a white-noise sequence that is uncorrelated with u(k) and satisfies
the SVD of the matrix
is given by
where the n n diagonal matrix 
contains the nonzero singular values of the matrix 
and
The matrix U 1 in this SVD satisfies
Proof From the data equation it follows that
Using the fact that 
, we can...
