Matrix RationalH2 Approximation:

a State-Space Approach using Schur Parameters

J.P. Marmorat, M. Olivi, B. Hanzon, R. Peeters

Sophia–Antipolis, France

1

Introduction

We present a method to compute astablerationalL2-approximation of

specified ordern to a givenmultivariabletransfer function.

• it works formultivariablesystems,

• it uses a niceparametrization of stable allpass systems, which

– takes into account thestability constraint

– ensures identifiability

– is well-conditionned

• it uses arecursive search on the degreewhich improves the chances

to reach the global minimum.

2

TheL2-criterion in state-space form

• F (z) = C(zIN −A)−1B +D, m× p given transfer function

• H(z) = γ(zIn −A)−1B +D, approximant at ordern

(A,B) input-normal pair:AA∗ + BB∗ = I.

L2-norm of the errorF −H:

J(A,B) = ‖F‖2 − Tr (γγ∗) ,

where

γ = CW,

andW solution to the Lyapunov equation:

AWA∗ + BB∗ = W.

3

Optimization set: stable-allpass systems

The two following sets (equivalence classes) are diffeomorphic:

• input-normal pairs(A,B)

• stable allpass functionsG(z) = D + C(zIn −A)−1B

up to a left constant unitary factor D C

B A

unitary matrix.

Alpay, Baratchart, Gombani[1];

4

Parametrization issue

Desirable properties:

• ensuresidentifiability

• a small perturbation of the parameterspreserves the stability and the

orderof the system?

• allows for the use ofdifferential tools.

→ Differentiable manifold.

Atlas of charts or overlapping canonical forms :

a collection oflocal parametrizationswith compatibility conditions

(changes of charts are smooth).

5

Atlases of charts

Two families can be found in the litterature

1. from a tangential Schur algorithm:

Gn(1/w̄)u = v, ‖v‖ < 1, GnLFT=⇒ Gn−1

Gn, . . . , Gk(wk,uk,vk)

=⇒ Gk−1, . . . , G0.

Alpay, Baratchart, Gombani[1]; Fulcheri, Olivi [2]

2. from state-space representations

Hanzon, Ober[3]

A parametrization that combines the two approaches:

Peeters, Hanzon, Olivi[4]

6

Encoding stable-allpass systems

A p× p stable allpass systems of degreen is encoded by

• w1, w2, . . . , wn points of the unit circle,

• u1, u2, . . . , un unit complexp-vectors,

• v1, v2, . . . , vn complexp-vectors,‖vi‖ < 1.

Thewi’s and theui’s definethe chartwhile thevi’s arethe Schur

parametersof the system in the chart.In a given chart, a system is

perfectly determined by its Schur parameters (identifiability).

7

Encoding stable-allpass systems (2)

The stable allpass system encoded in that way hasunitary realization

matrix Dn Cn

Bn An

computed by induction

Dk Ck

Bk Ak

=

Vk 0

0 Ik−1

1 0 0

0 Dk−1 Ck−1

0 Bk−1 Ak−1

U∗

k 0

0 Ik−1

,

whereAk is k × k, Dk is p× p, andD0 = Ip.

→ very nice numerical behavior

8

Encoding stable-allpass systems (3)

Uk andVk are the(p + 1)× (p + 1) unitary matrices:

Uk =

ξkuk Ip − (1 + ηkwk)uku∗k

ηkwk ξku∗k

Vk =

ξkvk Ip − (1− ηk) vkv∗k‖vk‖2

ηk − ξkv∗k

ξk =

√1− |wk|2√

1− |wk|2‖vk‖2, ηk =

√1− ‖vk‖2√

1− |wk|2‖vk‖2

9

Main steps of the algorithm

• finding an adapted chart:

realization in Schur form (A lower triangular)

A =

wn 0 · · · 0...

......

...

∗ · · · w1 0

, B =

√1− |wn|2 u∗n

...

→ v1 = v2 = . . . vn = 0 (Potapov factorization)

• optimization over the manifold

• recursive search on the degree (optional):

minimum of degreek → initial point of degreek + 1 (error

preserved)

10

The RARL2 software

This software computes astablerationalL2-approximation of specified

ordern to amultivariabletransfer function given in one of the following

forms:

• a realization

• a finite number ofFourier coefficients

• somepointwise valueson the unit circle.

It has been implemented using standard MATLAB subroutines. The

optimizer of the toolkit OPTIM is used to find a local minimum, given by

a realization, of the nonlinearL2-criterion.

http://www-sop.inria.fr/miaou/Martine.Olivi/me.html

11

Automobile gas turbine

Hung, MacFarlane[6]; Glover[5]; Yan, Lam[7]

12

Nyquist diagrams

2× 2; order12.

13

Approximants: order 1

14

Approximants: order 2

15

Approximants: order 3

16

Approximants: order 4

17

Approximants: order 5

18

Approximants: order 6

19

Approximants: order 7

20

Approximants: order 8

21

Hyperfrequency Filter

The problem:find a8th ordermodel of aMIMO (2× 2) hyperfrequency

filter, from experimental pointwise values in some range of frequencies

provided by the CNES (French space agency).

First stage (interpolation/completion):compute astable matrix transfer

function of high orderwhich approximates these data, given by a great

number(800)of Fourier coefficients.

PRESTO-HF: software byF. Seyfert;

HYPERION: software byJ. Grimm.

22

Data and approximant at order 8

Nyquist diagrams

23

Data and approximant at order 8

Bode diagrams

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References

[1] D. ALPAY, L. BARATCHART, AND A. GOMBANI , On the differentialstructure of matrix-valued rational inner functions, Operator Theory :Advances and Applications, 73 (1994), pp. 30–66.

[2] P. FULCHERI AND M. OLIVI , Matrix RationalH2-Approximation: aGradient Algorithm Based on Schur Analysis, SIAM J. Contr. and Opt.,36 (1998), pp. 2103–2127.

[3] B. HANZON AND R.J. OBER, Overlapping block-balanced canonicalforms for various classes of linear systems, Linear Algebra and its Ap-plications, 281 (1998), pp. 171-225.

[4] R.L.M. PEETERS, B. HANZON AND M. OLIVI , Balanced realizationsof discrete-time stable all-pass systems and the tangential Schur algo-rithm, in Proceedings of the ECC99, Karlsruhe, Germany, August 31-September 3.

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[5] K. GLOVER, All optimal Hankel-norm approximations of linear multi-variable systems and theirL∞-error bounds, Int. J. Control 39 (1984),pp. 1115-1193.

[6] Y.S. HUNG AND A.G.J. MACFARLANE, Multivariable feedback: aquasi-classical approach, Lect. Notes in Control Sci. 40, Springer- Verlag,1982.

[7] W.-Y. YAN AND J. LAM , An approximate approach toH2 optimal modelreduction, IEEE Trans. Autom. Control 44, No.7 (1999), pp 1341-1358.

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