Ivanov, Tzvetan
[UCL]
(eng)
Ever since mathematical models have been used to describe dynamical systems,
ideas related to approximation and estimation received a lot of attention. In
engineering applications two fundamentally different paradigms emerged: system
identification and model reduction. Both paradigms aim at providing their users
with models which are as simple as possible, and yet preserve those features of
the system, which are important for the application at hand.
In this thesis we follow a functional approach and consider both system
identification and model reduction in a common framework where linear algebra,
differential geometry, probability, statistics, and functional analysis find their
intersection.
In the context of system identification, with main focus on transfer function
estimation, we introduce the Crámer-Rao kernel function. The latter provides a
lower bound on the variance of any unbiased transfer function estimator. In many
cases the model structure forms a differentiable manifold which allows us to
circumvent parametrizations, for direct coordinate-free results. We demonstrate
how the Crámer-Rao kernel function can be used for optimal input design, which
aims at performance specification for a given frequency range. Moreover, we
establish necessary and sufficient conditions for the feasibility to estimate a
reduced order model, given samples from an experiment, with a singular
information matrix.
In the context of model reduction, with main focus on balanced truncation and
optimal Hankel norm approximation, we develop a novel state-realization where
the state-transition is given by a truncated Toeplitz operator which acts on minimal
norm past inputs. We show that every balanced matrix realization is a matrix
representation of our coordinate-free realization, in the basis given by the scaled
right singular vectors of the corresponding Hankel operator. We show that this
Hankel operator can be replaced by a unitary equivalent Toeplitz operator. This
replacement allows us to develop new algorithms for balancing and Hankel norm
approximation which are more efficient than the classical ones.
Bibliographic reference |
Ivanov, Tzvetan. A functional approach to system identification and model reduction. Prom. : Gevers, Michel ; Absil, Pierre-Antoine |
Permanent URL |
http://hdl.handle.net/2078.1/90773 |