Internal stability and loop-transformations: an overview on LFTs, Möbius transforms and chain scattering

Z. Szabó, Peter J Seiler Jr, J. Bokor

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

In robust control problems it is often convenient to consider maps between controller sets that are defined by Möbius transformations. Moreover, these loop-transformations are also intimately related to different factorizations, that simplify the structure of the problem. Starting from a fundamental observation that relates internal stability of the control loops to mere stability of a specific LFT, the paper provides an exhaustive answer to the question under which conditions the internal stability of a control loop is preserved by performing a loop-transformation. As a main result it is shown that Möbius transformations defined by unimodular matrices preserves the internal stability of the loop and an explicit formula is also given that relates the two loops. This result is formulated in a general context that includes LTV or LPV systems as well. The paper also provides an overview on the different transformation techniques, as Möbius transformations, LFTs, different generalized chain scattering (CSD) transforms, and their interrelations. This knowledge gives a solid basis for those approaches that use an input-output framework in combination with the IQC analysis and synthesis techniques in the solution of linear parameter varying (LPV) design problems.

Original languageEnglish (US)
Pages (from-to)7547-7553
Number of pages7
JournalIFAC-PapersOnLine
Volume50
Issue number1
DOIs
StatePublished - Jul 1 2017

Fingerprint

Scattering
Robust control
Factorization
Controllers

Cite this

Internal stability and loop-transformations : an overview on LFTs, Möbius transforms and chain scattering. / Szabó, Z.; Seiler Jr, Peter J; Bokor, J.

In: IFAC-PapersOnLine, Vol. 50, No. 1, 01.07.2017, p. 7547-7553.

Research output: Contribution to journalArticle

@article{67f3a5d536984eb28c429091fb1e2b7b,
title = "Internal stability and loop-transformations: an overview on LFTs, M{\"o}bius transforms and chain scattering",
abstract = "In robust control problems it is often convenient to consider maps between controller sets that are defined by M{\"o}bius transformations. Moreover, these loop-transformations are also intimately related to different factorizations, that simplify the structure of the problem. Starting from a fundamental observation that relates internal stability of the control loops to mere stability of a specific LFT, the paper provides an exhaustive answer to the question under which conditions the internal stability of a control loop is preserved by performing a loop-transformation. As a main result it is shown that M{\"o}bius transformations defined by unimodular matrices preserves the internal stability of the loop and an explicit formula is also given that relates the two loops. This result is formulated in a general context that includes LTV or LPV systems as well. The paper also provides an overview on the different transformation techniques, as M{\"o}bius transformations, LFTs, different generalized chain scattering (CSD) transforms, and their interrelations. This knowledge gives a solid basis for those approaches that use an input-output framework in combination with the IQC analysis and synthesis techniques in the solution of linear parameter varying (LPV) design problems.",
author = "Z. Szab{\'o} and {Seiler Jr}, {Peter J} and J. Bokor",
year = "2017",
month = "7",
day = "1",
doi = "10.1016/j.ifacol.2017.08.1190",
language = "English (US)",
volume = "50",
pages = "7547--7553",
journal = "IFAC-PapersOnLine",
issn = "2405-8963",
publisher = "IFAC Secretariat",
number = "1",

}

TY - JOUR

T1 - Internal stability and loop-transformations

T2 - an overview on LFTs, Möbius transforms and chain scattering

AU - Szabó, Z.

AU - Seiler Jr, Peter J

AU - Bokor, J.

PY - 2017/7/1

Y1 - 2017/7/1

N2 - In robust control problems it is often convenient to consider maps between controller sets that are defined by Möbius transformations. Moreover, these loop-transformations are also intimately related to different factorizations, that simplify the structure of the problem. Starting from a fundamental observation that relates internal stability of the control loops to mere stability of a specific LFT, the paper provides an exhaustive answer to the question under which conditions the internal stability of a control loop is preserved by performing a loop-transformation. As a main result it is shown that Möbius transformations defined by unimodular matrices preserves the internal stability of the loop and an explicit formula is also given that relates the two loops. This result is formulated in a general context that includes LTV or LPV systems as well. The paper also provides an overview on the different transformation techniques, as Möbius transformations, LFTs, different generalized chain scattering (CSD) transforms, and their interrelations. This knowledge gives a solid basis for those approaches that use an input-output framework in combination with the IQC analysis and synthesis techniques in the solution of linear parameter varying (LPV) design problems.

AB - In robust control problems it is often convenient to consider maps between controller sets that are defined by Möbius transformations. Moreover, these loop-transformations are also intimately related to different factorizations, that simplify the structure of the problem. Starting from a fundamental observation that relates internal stability of the control loops to mere stability of a specific LFT, the paper provides an exhaustive answer to the question under which conditions the internal stability of a control loop is preserved by performing a loop-transformation. As a main result it is shown that Möbius transformations defined by unimodular matrices preserves the internal stability of the loop and an explicit formula is also given that relates the two loops. This result is formulated in a general context that includes LTV or LPV systems as well. The paper also provides an overview on the different transformation techniques, as Möbius transformations, LFTs, different generalized chain scattering (CSD) transforms, and their interrelations. This knowledge gives a solid basis for those approaches that use an input-output framework in combination with the IQC analysis and synthesis techniques in the solution of linear parameter varying (LPV) design problems.

UR - http://www.scopus.com/inward/record.url?scp=85031824759&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85031824759&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2017.08.1190

DO - 10.1016/j.ifacol.2017.08.1190

M3 - Article

AN - SCOPUS:85031824759

VL - 50

SP - 7547

EP - 7553

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

SN - 2405-8963

IS - 1

ER -