# Structural and behavioural analyses to linear multivariable control systems

thesis

posted on 2012-09-28, 10:43 authored by Liansheng TanThis thesis is devoted to a number of structural and behavioural problems in linear multivariable
control system theory.
The first problem addresses the subject of determination of the finite and infinite frequency
structure of a rational matrix. A novel method is proposed that determines the finite and infinite
frequency structure of any rational matrix. Some neat and numerically stable algorithms
are developed to implement this method.
The second problem concerns the resol vent decompositions of a regular polynomial matrix
and solutions of regular polynomial matrix descriptions (PMDs). Regarding these fundamental
is'sues, three contributions are made therein. Firstly, based on a general resolvent
decomposition a complete solution of regular PMDs is presented that takes into account both
the non-zero initial conditions of the pseudo state and the non-zero initial conditions of the
input. Secondly, two special resolvent decompositions are proposed, both of which are applied
to formulate the solution of the regular PMDs. The first one is formulated in terms of
the finite, infinite, and the generalised infinite Jordan pairs, which is a refinement of the results
given by Gohberg et al. [74] and Vardulakis [25]. The second resolvent decomposition
is proposed on the Weierstrass canonical form of the generalised companion matrix of the
polynomial matrix. Thirdly, a new characterization of the impulsive free initial conditions of
regular PMDs is given and the relationship between the finite and infinite frequency structure
of a regular polynomial matrix and its generalised companion matrix is determined.
In the third problem a generalization of the chain-scattering representation for general
plants is presented. Through the notion of input-output consistency, the conditions under
which the generalised chain-scattering representation and the dual generalised chain-scattering
representation exist are proposed. Some algebraic system properties of the GCSRs and DGCSRs
are studied.
The fourth problem is devoted to a new notion of realization of behaviour. We introduce
a notion realization of behavior which is shown to be a generalization of the classical concept
of a realization of transfer function. By using this approach, the input-output structures
of the generalized chain-scattering representations and the dual generalized chain-scattering
representations are investigated in a behavioral theory context. The last problem is devoted to the subjects of system wellposedness and internal stability.
We present certain generalisations to the classical concepts of wellposedness and internal stability.
The input consistency and output uniqueness of the closed-loop system in the standard
control feedback configurations are discussed. Based on this, a number of notions are introduced
such as fully internal wellposedness, externally internal wellposedness, and externally
internal stability, which characterize the rich input-output and stability features of the general
control systems in a general setting. On the basis of these notions the extended JL control
problem is defined in a general setting.

## History

## School

- Science

## Department

- Mathematical Sciences

## Publisher

© Liansheng Tan## Publication date

1999## Notes

A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.## Language

- en