A behavioural approach to the pole structure of one-dimensional and multidimensional linear systems

A behavioural approach to the pole structure of one-dimensional and multidimensional linear systems

We use the tools of behavioural theory and commutative algebra to produce a new definition of a (finite) pole of a linear system. This definition agrees with the classical one, and allows a direct dynamical interpretation. It also generalizes immediately to the case of a multidimensional system. We make a natural division of poles into controllable and uncontrollable poles. When the behaviour in question has latent variables, we make a further division into observable and unobservable poles. In the case of a 1D state-space model, the uncontrollable and unobservable poles correspond respectively to the input and output decoupling zeros, whereas the observable controllable poles are the transmission poles. Most of these definitions can be interpreted dynamically in both the 1D and nD cases, and some can be connected to properties of kernel representations. We also examine the connections between poles, transfer matrices and their left and right MFDs. We find behavioural results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros, and an observable system is precisely one with no output decoupling zeros. We produce a decomposition of a behaviour as the sum of sub-behaviours associated with various poles. This is related to the integral representation theorem which describes every system trajectory as a sum of integrals of polynomial exponential trajectories.

627-61

Wood, J

9d32c83e-9d59-4c32-8ab2-a5f64ecc26e6

Oberst, U

def69490-a5e6-465c-aaeb-09d10acf13ea

Rogers, E

611b1de0-c505-472e-a03f-c5294c63bb72

Owens, D H

db24b8ef-282b-47c0-9cd2-75e91d312ad7

2000

Wood, J

9d32c83e-9d59-4c32-8ab2-a5f64ecc26e6

Oberst, U

def69490-a5e6-465c-aaeb-09d10acf13ea

Rogers, E

611b1de0-c505-472e-a03f-c5294c63bb72

Owens, D H

db24b8ef-282b-47c0-9cd2-75e91d312ad7

Wood, J, Oberst, U, Rogers, E and Owens, D H
(2000)
A behavioural approach to the pole structure of one-dimensional and multidimensional linear systems.
*SIAM Journal on Control and Optimization*, 38 (2), .

## Abstract

We use the tools of behavioural theory and commutative algebra to produce a new definition of a (finite) pole of a linear system. This definition agrees with the classical one, and allows a direct dynamical interpretation. It also generalizes immediately to the case of a multidimensional system. We make a natural division of poles into controllable and uncontrollable poles. When the behaviour in question has latent variables, we make a further division into observable and unobservable poles. In the case of a 1D state-space model, the uncontrollable and unobservable poles correspond respectively to the input and output decoupling zeros, whereas the observable controllable poles are the transmission poles. Most of these definitions can be interpreted dynamically in both the 1D and nD cases, and some can be connected to properties of kernel representations. We also examine the connections between poles, transfer matrices and their left and right MFDs. We find behavioural results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros, and an observable system is precisely one with no output decoupling zeros. We produce a decomposition of a behaviour as the sum of sub-behaviours associated with various poles. This is related to the integral representation theorem which describes every system trajectory as a sum of integrals of polynomial exponential trajectories.

## More information

Published date: 2000

Organisations:
Southampton Wireless Group

## Identifiers

Local EPrints ID: 250672

URI: http://eprints.soton.ac.uk/id/eprint/250672

PURE UUID: 9db95883-a5d8-4da0-9e0a-4558e1002725

## Catalogue record

Date deposited: 01 Mar 2004

Last modified: 27 Oct 2023 01:35

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## Contributors

Author:
J Wood

Author:
U Oberst

Author:
E Rogers
Author:
D H Owens

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