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Abstract

This dissertation addresses the problem of parameter estimation of stochastic,
discrete-time, state-space models from input and output data. This class
of problems can be found in many modern engineering applications, where
the relationships in the real world processes have to be inferred from the
observed data and available prior knowledge. This problem is notoriously
difficult because to estimate the model parameters one should dispose of full
information on the internal system states. These are not directly observed and
therefore also have to be inferred from the data. However, their estimation
requires knowledge of the model parameters. This leads to the problem of
joint estimation of both model parameters and hidden states.
In this dissertation, the framework for solving this problem is the expectationmaximization
(EM) algorithm. It enables the computation of maximum likelihood
estimates of the model parameters through recursive interplay between
the parameter and the state estimation.
The central issue of the approach is related to the quality of estimation of
internal states. Namely, state transitions in nonlinear systems modify the underlying
state distributions so that a normally distributed initial state eventually
renders into a distribution that can be far from normal. Apart from some
special cases, no closed-form solutions are available. These distributions have
to be suitably approximated in the expectation step of the EM algorithm, as
they are essential for further maximization of the likelihood function. In the
maximization step, the expected value of the likelihood function is maximized
with respect to the model parameters. As the likelihood is a nonlinear function
of random variables, computation of its expected value is again related
to the problem of approximating the distributions of random variables.
The aim of this dissertation is to explore the performance of the EM algorithm
under different approximation schemes, namely the sequential Monte
Carlo (SMC) based approximations and the unscented transformation (UT).
The SMC tends to reconstruct the entire probability density function, but it
is limited by computational load, which increases dramatically with increasing
number of states. This dissertation proposes a novel implementation of
the expectation-maximization algorithm, which we refer to as the UTEM algorithm,
which is entirely based on the unscented transformation. The UT
guarantees that the computational complexity is limited and the algorithm
can thus be used for estimation of high dimensional models or applied to the
problems, where computational load has to be low.
In this dissertation we provide a detailed performance analysis of the UTEM
algorithm on simulated nonlinear dynamic systems. The focus of the estimation
are the unknown model parameters, including the parameters of the
system noise. We demonstrate that the proposed algorithm can solve the
problem of joint estimation but has some limitations, dictated by the approximation
errors. It seems that the convergence of the UTEM algorithm can be
endangered mostly by divergence in noise covariance parameters originating
in the insufficient quality of approximation. If the noise covariances are fixed,
the estimation of the rest of the parameters converges much more reliably.
In the second part of the dissertation, the UTEM algorithm is applied to a relatively
new problem domain dealing with estimation of the remaining useful life
of mechanical systems. More particularly, the case study dealing with a gear
transmission systems is addressed. The condition of the gear is assessed based
on the vibrational signals. The relation between the damage progression in the
gear and vibrational signature is described by a stochastic state-space model.
Two model candidates are investigated: a simple linear black-box model and
a more elaborated non-linear gray model. In both cases the unknown model
parameters are estimated with the expectation-maximization algorithm. The
resulting estimates are used to predict the future evolution of the damage and
the remaining useful life (RUL) of the gearbox by applying Monte Carlo simulations.
Surprisingly enough, no convincing advantage in the performance of
the nonlinear model compared to the linear one has been observed. Actually,
both of them provide reliable and useful estimates of the RUL. Hence timely
alerts to the operators to plan maintenance actions can be issued well ahead
of final failure. This feature represents a promising extension of functionality
to the existing condition monitoring systems.