National Instruments Xmath

Chapter 3 Multiplicative Error Reduction

Xmath Model Reduction Module 3-6 ni.com

2. With G(s)=D+C(sI – A)–1B and stable, with DD´ nonsingular and

G(jω) G'(–jω) nonsingular for all ω, part of a state variable realization

of a minimum phase stable W(s) is determined such that

W´(–s)W(s)= G(s)G´(–s) with

The state variable matrices in W(s) are obtained as follows. The

controllability grammian P associated with G(s) is first found from

AP+ PA´+BB´=0 then AW=A, B

W=PC´+BD´.

When G(s) is square, the algorithm checks to see if there is a zero or

singularity of G(s) close to the jω-axis (the zeros are given by the

eigenvalues of A–BD–1C and are computed reliably with the aid of

schur( )). If one is found, you are warned that results may be

unreliable. Next, a stabilizing solution Q is found for the following

Riccati equation:

The singriccati( ) function is used; failure of the nonsingularity

condition on G(jω)G´(–jω) will normally result in an error message

that no stabilizing solution exists. To obtain the best numerical results,

singriccati( ) is invoked with the keyword {method="schur"}.

Although DW, CW are not needed for the remainder of the algorithm,

they are simply determined in the square case by

with minor modification in the nonsquare case. The real point of the

algorithm is to compute P and Q; the matrix Q satisfies (square or

nonsquare case).

P, Q are the controllability and observability grammians of the transfer

function CW(sI –A)–1B. This transfer function matrix, it turns out, is

the strictly proper, stable part of θ(s)=W–T(–s)G(s), which obeys the

matrix all-pass property θ(s)θ´(–s)=I, and is the phase matrix

associated with G(s).

3. Compute ordered Schur decompositions of PQ, with the eigenvalues

of PQ is ascending and descending order. Obtain the phase matrix

Hankel singular values, that is, the Hankel singular values of the

Ws() DWCWsI Aw

–()

1–BW

QA A′QCB′WQ

–()′DD′()

1–CB

′Q–()++ 0=

DWD′CWD1–CB

′Q–()==

QA A′QC

′CW

++ 0=