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About the book/toolbox "Linear Fractional Representations"
Linear Fractional Representations
with a Toolbox for Use with Matlab
Author : Jean-François Magni
Affiliation : ONERA-CERT (DCSD), Toulouse, France
Free electronic publication December 2001.
Reference : IDDN.FR.001.370032.00.S.P.2001.000.31235
You may download the manual here
(1 Mb) and the toolbox here
(300 Kb).
Presentation of the toolbox manual
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Chapter 1 gives some hints for getting started with the toolbox.
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Chapter 2 gives some basic definitions such as
"LFT realization" and presents basic manipulations such as the "star
product" and so on. The objective of this chapter is to give general
guidelines for low order LFT generation. For this purpose, a "step by step"
or "object-oriented" approach to LFT generation is proposed.
From the discussions given in this chapter, it turns out that two main
steps are to be considered. First, realization must be done carefully,
especially taking advantage of parameter commutativity
(details in Chapter 3).
Then, some techniques reminiscent of "minimal realization"
for standard dynamic systems can be applied for further order reduction
(details in Chapter 4).
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Chapter 3. This chapter is devoted to LFT
realization. First, are considered the conversions from coprime
factorizations and from state-space realizations to input/poutput LFTs.
After this discussion, only the state-space form is considered.
The realization techniques that are treated
are:
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Morton's technique
([B. Morton, CDC-1985, pp233-238]),
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Horner factorization
([A. Varga and G. Looye, CACSD-1999, pp176-181])
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the tree decomposition
([J.C. Cockburn and B.G. Morton, Automatica 33(7), pp1263-1271]).
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a matrix-based approach
([C.M. Belcastro, Nasa/ TM-1998-206939]).
It is recommended to use the tree decomposition that is the most efficient
technique. The object-oriented realization technique described in
Chapter 2 is also an alternative way for low
order realization, but in this case the factorizations that reduce
the order must be done manually.
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Chapter 4. This chapter treats a generalization
to LFTs of "minimal realization" of standard dynamic systems.
Considering LFTs, the size of what is improperly called a "minimal
realization" depends on the initial realization considered before
order reduction. The reason for that is that parameter commutativity is not
taken into account, and in fact, minimality is truly attained only if
parameters do not commute.
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Before presenting specific
LFT order reduction techniques,
it is shown how standard system "minimal realization" techniques can
be applied to LFTs (1-D technique
[P. Lambrechts, J. Terlouw, S. Bennani and M. Steinbuch, ACC-1993, pp267-272])
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Then the generalized
Kalman decomposition
([R. D'Andrea and Khatri, S. ACC-1997, pp3557-3561])
and
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the generalized Gramians approaches
([C. Beck and J. Doyle,
IEEE-TAC, Vol 44(10),
October 1999, pp 1802-1813])
are briefly presented (n-D techniques).
These two last techniques lead to the so-called
"minimal realization", the last one offers the possibility of
further reduction by approximation. We conclude this chapter by
describing a technique that permits us to evaluate precisely LFT
approximation errors, and, by the way to model approximation errors.
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Chapter 5. This short chapter (that can be viewed as an
appendix) treats the problem of normalizing uncertainties. Normalization
must be considered after realization in order to avoid artificial
repetition of parameters. Normalization is usually required for
mu-analysis.
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Chapter 6. This chapter considers complex
uncertainties. Such uncertainties are usually used for
modelling neglected dynamics. In order to use mu-analysis for
performance analysis or for performance robustness analysis, it is also
convenient to consider artificial complex uncertainties.
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Chapter 7. This chapter describes the
use of LFTs for modelling the continuum of linearized models of a nonlinear
system. The dependency of parameters on the equilibrium
surface is also taken into account. However, we have to keep in mind
that such modelling leads to very high order LFTs. Finally is
considered the problem of transforming tables of data to LFTs
(e.g. derivatives in aeronautics).
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