## Equivalence transformations in linear systems theory

thesis

posted on 01.06.2018 by Paul Fretwell#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

There is growing interest in infinite frequency structure of
linear systems, and transformations preserving this type of
structure. Most work has been centred around Generalised
State Space (GSS) systems. Two constant equivalence
transformations for such systems are Rosenbrock's Restricted
System Equivalence (RSE) and Verghese's Strong
Equivalence (str.eq.). Both preserve finite and infinite
frequency system structure. RSE is over restrictive in
that it is constrained to act between systems of the same
dimension. While overcoming this basic difficulty str.eq.
on the other hand has no closed form description. In this
work all these difficulties have been overcome. A constant
pencil transformation termed Complete Equivalence (CE) is
proposed, this preserves finite elementary divisors and
non-unity infinite elementary divisors. Applied to GSS
systems CE yields Complete System Equivalence (CSE)
which is shown to be a closed form description of str.eq.
and is more general than RSE as it relates systems of
different dimensions.
Equivalence can be described in terms of mappings of the
solution sets of the describing differential equations
together with mappings of the constrained initial
conditions. This provides a conceptually pleasing
definition of equivalence. The new equivalence is termed
Fundamental Equivalence (FE) and CSE is shown to be a
matrix characterisation of it.
A polynomial system matrix transformation termed Full
Equivalence (fll.e.) is proposed. This relates general
matrix polynomials of different dimensions while preserving
finite and infinite frequency structure. A definition of
infinite zeros is also proposed along with a generalisation
of the concept of infinite elementary divisors (IED) from
matrix pencils to general polynomial matrices. The IED provide an additional method of dealing with infinite zeros.

### Funding

### Science and Engineering Research Council.

### History

#### School

- Science

#### Department

- Mathematical Sciences