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CATEGORIES:CUED Control Group Seminars
SUMMARY:Subdiagonal pivot structures and associated canoni
cal forms under state isometries - Professor Berna
rd Hanzon (School of Mathematical Sciences\, Unive
rsity College Cork)
DTSTART;TZID=Europe/London:20080617T140000
DTEND;TZID=Europe/London:20080617T150000
UID:TALK12401AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/12401
DESCRIPTION:We consider a linear state space system described
by a quadruple of matrices (A\,B\,C\,D)\,\nwith A:
n x n\, B: n x m\, C: p x n\, D: p x m. The syste
m can be in discrete time or in continuous time.\n
We define a pivot vector with pivot on the i-th po
sition as a column vector with positive i-th entry
and \nzeros in each k-th entry with k>i. Note tha
t the first (i-1) entries of such a vector are arb
itrary. \nWe will say that the n x (m+n) matrix [B
|A] has a pivot structure if for each i=1\,2\,..\,
n the \nmatrix contains a pivot vector with pivot
on the i-th position. A principal result of our pr
esentation\nwill be that the pair (A\,B) is contro
llable if [B|A] has a pivot structure such that al
l the pivots in the matrix A lie \nbelow the main
diagonal of A\, and that if this is not the case\,
a counterexample can be found: then a \nnon-contr
ollable pair exists which has the given pivot stru
cture.\nPivot structures for [B|A] such that all p
ivots in A lie below the main diagonal of A\, will
be called "subdiagonal \npivot structures". The r
esult then says that a pivot structure for [B|A] g
uarantees controllability if and \nonly if the piv
ot structure is subdiagonal. \nIn the presentation
we will consider how this can be used to obtain c
anonical forms for state space systems \nunder sta
te isometries (i.e. orthogonal or unitary state tr
ansformations) and how a simple recursive\,\nand
numerically stable algorithm can be constructed th
at determines whether a pair (A\,B) is controllabl
e or not and \nwhich puts a controllable pair in a
local canonical form under state isometry\, with
subdiagonal pivot structure. \nWe will make remark
s about the effect of model reduction by truncatio
n on linear systems in a canonical form \nassociat
ed with a subdiagonal pivot structure. \nAn intere
sting relation with R.J.Ober's original balanced c
anonical form [1]\,[2]\,[3] (constructed in CUED i
n the 1980's) \nwill be mentioned.\nThe subdiagona
l pivot structures presented here were found using
the approach of constructing \nlocal canonical fo
rms for lossless state space systems as presented
in [4] (related to the so-called \n"tangential Sch
ur algorithm"). If time permits we will discuss th
e relation of subdiagonal pivot structures with th
e\nso-called "staircase forms" as presented in [5]
.\n\nThe presentation is based on joint work with
M. Olivi (INRIA\, France) and R.L.M. Peeters (Univ
Maastricht).\n\nReferences:\n\n[1] R.J. Ober\, "B
alanced realizations for Finite and Infinite Dimen
sional Linear Systems"\, \nPhD thesis CUED\, super
visor J.M. Maciejowski\, Cambridge\, 1987\n\n[2] J
.M. Maciejowski and R.J. Ober\, "Balanced Parametr
izations and Canonical Forms for System\nIdentific
ation"\, Proc.IFAC Identification and System Param
eter Estimation\, Beijing\, 1988\, pp. 701-708.\n\
n[3] R.J. Ober\, "Balanced realizations:canonical
form\, parametrization\, model reduction"\, \nInt.
J. Control\, vol. 46\, pp. 263--280\, 1987.\n\n[4
] B. Hanzon\, M. Olivi\, R.L.M.Peeters\, "Balanced
realizations of discrete-time stable all-pass sys
tems \nand the tangential Schur algorithm"\,Linear
Algebra and Its Applications\, vol. 418\, pp. 793
-820\, 2006.\n\n[5] R.L.M. Peeters\, B. Hanzon\, M
.Olivi\, "Canonical lossless state-space systems:
Staircase forms and the Schur algorithm" \nLin.Alg
and Its Appl.\, vol. 425\, pp. 404-433\, 2007.\n\
n[6] B.Hanzon\, M.Olivi\, R.L.M. Peeters\, "Subdia
gonal pivot structures and associated canonical fo
rms under state isometries"\,\nunder preparation.\
n
LOCATION: Cambridge University Engineering Department\, Lec
ture Room 4
CONTACT:Dr Guy-Bart Stan
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