National Instruments Xmath

Chapter 1 Introduction

• Suppose the transfer-function matrix corresponds to a discrete-time

system, with state variable dimension n. Then the infinite Hankel

matrix,

has for its singular values the n nonzero Hankel singular values,

together with an infinite number of zero singular values.

The Hankel singular values of a (stable) all pass system (or all pass matrix)

are all 1.

Slightly different procedures are used for calculating the Hankel singular

values (and so-called weighted Hankel singular values) in the various

functions. These procedures are summarized in Table1-2.

Table 1-2. Calculating Hankel Singular Values

(balance( )) For a discussion of the balancing algorithm, refer to

the Internally Balanced Realizations section; the

Hankel singular values are given by

diag(R1/2)=HSV

balmoore( ) For a discussion of the balancing algorithm, refer to

the Internally Balanced Realizations section; the

matrix SH yields the Hankel singular values through

diag(SH)

hankelsv( ) real(sqrt(eig(p*q)))

ophank( ) Calls hankelsv( )

redschur( ) Computes a Schur decomposition of P*Q and then

takes the square roots of the diagonal entries

bst( )

mulhank( )

wtbalance( )

fracred( )

Same as redschur( ) except either P or Q can be

a weighted grammian

CB CAB CA2B

CAB CA2B

CA2B