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SUMMARY:Subdiagonal pivot structures and associated canonical forms under 
 state isometries - Professor Bernard Hanzon (School of Mathematical Scienc
 es\, University College Cork)
DTSTART:20080617T130000Z
DTEND:20080617T140000Z
UID:TALK12401@talks.cam.ac.uk
CONTACT:Dr Guy-Bart Stan
DESCRIPTION:We consider a linear state space system described by a quadrup
 le of matrices (A\,B\,C\,D)\,\nwith A: n x n\, B: n x m\, C: p x n\, D: p 
 x m. The system can be in discrete time or in continuous time.\nWe define 
 a pivot vector with pivot on the i-th position as a column vector with pos
 itive i-th entry and \nzeros in each k-th entry with k>i. Note that the fi
 rst (i-1) entries of such a vector are arbitrary. \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 princi
 pal result of our presentation\nwill be that the pair (A\,B) is controllab
 le if [B|A] has a pivot structure such that all 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-controllable pair exists which 
 has the given pivot structure.\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 result then says that a pivot structure for [B|A]
  guarantees controllability if and \nonly if the pivot structure is subdia
 gonal. \nIn the presentation we will consider how this can be used to obta
 in canonical forms for state space systems \nunder state isometries (i.e. 
 orthogonal or unitary state transformations) and  how a simple recursive\,
 \nand numerically stable algorithm can be constructed that determines whet
 her a pair (A\,B) is controllable or not and \nwhich puts a controllable p
 air in a local canonical form under state isometry\, with subdiagonal pivo
 t structure. \nWe will make remarks about the effect of model reduction by
  truncation on linear systems in a canonical form \nassociated with a subd
 iagonal pivot structure. \nAn interesting relation with R.J.Ober's origina
 l balanced canonical form [1]\,[2]\,[3] (constructed in CUED in the 1980's
 ) \nwill be mentioned.\nThe subdiagonal pivot structures presented here we
 re found using the approach of constructing \nlocal canonical forms for lo
 ssless state space systems as presented in [4] (related to the so-called \
 n"tangential Schur algorithm"). If time permits we will discuss the relati
 on of subdiagonal pivot structures with the\nso-called "staircase forms" a
 s presented in [5].\n\nThe presentation is based on joint work with M. Oli
 vi (INRIA\, France) and R.L.M. Peeters (Univ Maastricht).\n\nReferences:\n
 \n[1] R.J. Ober\, "Balanced realizations for Finite and Infinite Dimension
 al Linear Systems"\, \nPhD thesis CUED\, supervisor J.M. Maciejowski\, Cam
 bridge\, 1987\n\n[2] J.M. Maciejowski and R.J. Ober\, "Balanced Parametriz
 ations and Canonical Forms for System\nIdentification"\, Proc.IFAC Identif
 ication and System Parameter 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 discr
 ete-time stable all-pass systems \nand the tangential Schur algorithm"\,Li
 near 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 sys
 tems: Staircase forms and the Schur algorithm" \nLin.Alg and Its Appl.\, v
 ol. 425\, pp. 404-433\, 2007.\n\n[6] B.Hanzon\, M.Olivi\, R.L.M. Peeters\,
  "Subdiagonal pivot structures and associated canonical forms under state 
 isometries"\,\nunder preparation.\n
LOCATION: Cambridge University Engineering Department\, Lecture Room 4
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