In the first part of the thesis, we shall give detailed proofs that a number of classes of linear dynamical systems are diffeomorphic to each other. We shall also prove the diffeomorphisms between the state bundles associated with these classes of linear systems and also between their associated principal fibre bundles. The main implication of these results is that these classes of linear systems have identical topological and differential geometric properties. We shall discuss how these results can be used to deduce the geometric properties of other classes of systems, to construct overlapping parametrizations and Riemannian gradient identification algorithms.
In the second part of the thesis, we shall propose a system identification algorithm for the identification of stable linear time-invariant state space systems from input/output data. The algorithm consists of two steps. In the first step, a state space model is obtained from the input/output data using either model reduction based system identification methods or subspace methods. This initial state space estimate is then supplied to a parameter optimization algorithm to find an optimal model. The novelty of this approach lies in the use of balanced parametrizations for stable systems in parameter estimation. We shall use three different algorithms to obtain an initial state space model. They are Kung's algorithm, a model reduction algorithm due to Al-Saggaf and N4SID (which stands for Numerical Algorithm for Subspace State Space System Identification). We shall also discuss a few issues related to using balanced parametrizations for system identification such as model structure selection and gradient computations. Two case studies on applying the proposed algorithm to two sets of industrial data will be presented. The first set of data comes from a very detailed simulation model of a fractional distillation column. The second set comes from an industrial drying process.