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A convex optimisation approach to Youla's broadband matching theory

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HAL Id: hal-01909618

https://hal.inria.fr/hal-01909618

Submitted on 31 Oct 2018

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A convex optimisation approach to Youla’s broadband

matching theory

David Martínez Martínez, Gibin Bose, Fabien Seyfert, Martine Olivi, L

Baratchart, S Bila, F Ferrero

To cite this version:

David Martínez Martínez, Gibin Bose, Fabien Seyfert, Martine Olivi, L Baratchart, et al.. A convex optimisation approach to Youla’s broadband matching theory. 27th ERNSI Workshop in System Identiifcation, Sep 2018, Cambridge, United Kingdom. �hal-01909618�

(2)

D. Martínez Martínez

1,2

G. Bose

1,3

F. Seyfert

1

M. Olivi

1

L. Baratchart

1

S. Bila

2

F. Ferrero

3

1INRIA Sophia-Antipolis Méditerranée 2XLIM Université de Limoges 3Université Côte d'Azur

This work was supported by the Frech ANR (National Research Agency) with the collaboration of CNES (Centre National d'Études Spatiales) and DGA (Direction Générale de l'Armement)

INTRODUCTION

UNMATCHED LOAD

Reflection of the load

goes through the filter

UNWANTED REFLECTIONS

MATCHING FILTER

Filter input signal &

reject reflections

NO REFLECTION

RESULTS

THEORETICAL RESULTS & NUMERICAL IMPLEMENTATION

CONCEPT: OVERALL DESIGN + DE-EMBEDDING

YOULA'S DE-EMBEDDING CONDITIONS

Synthesize global system instead of matching filter

BRINGING OPTIMISATION INTO CONTEXT

FILTER

F

21

F

11

PUMA

project.inria.fr/puma

PRACTICAL USE

Choice of global response: Butterword / Tchebyshev

Interpolation conditions on :

Rigid approach: Not optimisation friendly

CONCEPT

Darlington equivalent

+

design: global system

+

load de-embbeding

matching filter

optimisation

non convex

techniques

no optimality

guaranteed

CLASSICAL APPROACH

?

S

22

F

22

L

22

-5

-10

-15

0

Magnitu

de (dB)

1.6 (GHz)

1.55

S

22

F

22

L

22

-5

-10

-15

0

Magnitu

de (dB)

900 (MHz)

870

S

22

F

22

L

22

-5

-10

-15

0

Magnitu

de (dB)

1.3

1.21

1.55

1.6 (GHz)

SINGLE-INTERVAL RESULTS

DUAL-INTERVAL RESULTS

EXISTENCE OF FACTOR B

CONVEX OPTIMISATION PROBLEM

Modulus of UP is obtained from the filtering function P/R

SDP with constrains on positive polynomials

POSITIVITY: GRAM MATRIX PARAMETRISATION

define

A NON-LINEAR SEMI-DEFINITE PROGRAM

AT THE OPTIMUM IS SINGULAR AND deg(B)<M

NEVANLINNA-PICK INTERPOLATION

Existence of inner function B satisfying interpolation conditions

Practical implementation of Youla's characterisation

CONCAVE OPERATOR CONVEX SET

Allow handling of Pick matrix by augmented lagrangian techniques

Interpolation conditions at load transmission zeros allow de-embbeding

Optimisation problem:

MINIMISE REFLECTION

Magnitu

de

A CONVEX OPTIMISATION APPROACH TO

YOULA'S BROADBAND MATCHING THEORY

relaxed

set of functions:

Références

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