# Subdiagonal pivot structures and associated canonical forms under state isometries

We consider a linear state space system described by a quadruple of matrices (A,B,C,D), with 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. We define a pivot vector with pivot on the i-th position as a column vector with positive i-th entry and zeros in each k-th entry with k>i. Note that the first (i-1) entries of such a vector are arbitrary. We will say that the n x (m+n) matrix [B|A] has a pivot structure if for each i=1,2,..,n the matrix contains a pivot vector with pivot on the i-th position. A principal result of our presentation will be that the pair (A,B) is controllable if [B|A] has a pivot structure such that all the pivots in the matrix A lie below the main diagonal of A, and that if this is not the case, a counterexample can be found: then a non-controllable pair exists which has the given pivot structure. Pivot structures for [B|A] such that all pivots in A lie below the main diagonal of A, will be called “subdiagonal pivot structures”. The result then says that a pivot structure for [B|A] guarantees controllability if and only if the pivot structure is subdiagonal. In the presentation we will consider how this can be used to obtain canonical forms for state space systems under state isometries (i.e. orthogonal or unitary state transformations) and how a simple recursive, and numerically stable algorithm can be constructed that determines whether a pair (A,B) is controllable or not and which puts a controllable pair in a local canonical form under state isometry, with subdiagonal pivot structure. We will make remarks about the effect of model reduction by truncation on linear systems in a canonical form associated with a subdiagonal pivot structure. An interesting relation with R.J.Ober’s original balanced canonical form ,, (constructed in CUED in the 1980’s) will be mentioned. The subdiagonal pivot structures presented here were found using the approach of constructing local canonical forms for lossless state space systems as presented in  (related to the so-called “tangential Schur algorithm”). If time permits we will discuss the relation of subdiagonal pivot structures with the so-called “staircase forms” as presented in .

The presentation is based on joint work with M. Olivi (INRIA, France) and R.L.M. Peeters (Univ Maastricht).

References:

 R.J. Ober, “Balanced realizations for Finite and Infinite Dimensional Linear Systems”, PhD thesis CUED , supervisor J.M. Maciejowski, Cambridge, 1987

 J.M. Maciejowski and R.J. Ober, “Balanced Parametrizations and Canonical Forms for System Identification”, Proc.IFAC Identification and System Parameter Estimation, Beijing, 1988, pp. 701-708.

 R.J. Ober, “Balanced realizations:canonical form, parametrization, model reduction”, Int. J. Control, vol. 46, pp. 263—280, 1987.

 B. Hanzon, M. Olivi, R.L.M.Peeters, “Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm”,Linear Algebra and Its Applications, vol. 418, pp. 793-820, 2006.

 R.L.M. Peeters, B. Hanzon, M.Olivi, “Canonical lossless state-space systems: Staircase forms and the Schur algorithm” Lin.Alg and Its Appl., vol. 425, pp. 404-433, 2007.

 B.Hanzon, M.Olivi, R.L.M. Peeters, “Subdiagonal pivot structures and associated canonical forms under state isometries”, under preparation.

This talk is part of the CUED Control Group Seminars series.