Optimal Hankel-Norm Approximation and H ∞-Minimization

Abstract

The notion of the Hankel norm was briefly introduced in Sects. 1.5.3 and 2.3.3. By a fundamental result of Nehari (to be discussed later), it will be shown in this chapter that the Hankel norm provides a meaningful measurement for functions which are essentially bounded on the unit circle |z| = 1. However, since the Hankel norm of any H∞function is zero, in contrast to the LP norms on |z| = 1, it is only a “semi-norm” in the sense that any two functions in L∞ (|z| = 1) with the same singular part have identical Hankel norms. In other words, in applying the Hankel norm, one has to take into account that an additive H∞ function must be determined by using a different method. This is not a serious draw-back in general. For instance, with the exception of an additive constant h0, the transfer function of a causal SISO linear system has zero analytic part. Since h0, can be easily determined and is really not very important, the Hankel norm is a very useful measurement for the study of causal linear systems. Indeed, as we have seen in Chap. 2, if the system is realizable in the sense that its transfer function is rational, then it is stable if and only if the Hankel norm of the transfer function has finite value. Moreover, the importance of this norm in the study of systems theory is apparent due to the fact that the best Hankelnorm (strictly proper) rational approximant with prescribed degree to the transfer function of a stable linear system along with the exact error of approximation can be described analytically. Important applications of best Hankel-norm approximation include system reduction, digital filter design, etc. This elegant analytical description is a fundamental result of the previously cited work (see Sect. 2.3.3) of Adamjan, Arov, and Krein [1971, 1978], usually known as the AAK approach. We will give a detailed constructive proof of the AAK theorem for finite-rank and real Hankel matrices in this chapter. A proof of the AAK’s general theorem is much more complicated and its discussion will be delayed to the next chapter. Two important applications, namely, system reduction and H ∞ minimization for SISO systems, will be discussed in this chapter. The study of the matrix-valued setting of the AAK theorem for MIMO systems will be delayed to Chap. 6.