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The research at the System Modelling Group at the Hamilton Institute is focused on System Theory and Nonlinear Dynamic System Identification.
Systems Theory
Engineering systems are becoming ever more complex, varied and disparate in nature requiring extensions to the standard analysis framework for linear systems. Other than the standard cases, the consistent mathematical formulation of feedback systems is still unclear. For instance, it has long been known that there are difficulties in extending the singly infinite time-axis case to the doubly infinite time-axis case and recently there has been considerable interest in resolving these difficulties.
The construction of consistent frameworks for feedback systems is being investigated. In addition to the above doubly infinite time-axis case, a framework, which is sufficiently general to include continuous time and/or discrete time systems, is sought. Both single-rate and multi-rate systems are considered. Key issues are the appropriate choice for the classes of signals and systems. By choosing richer classes, more general frameworks for the analysis of feedback systems can be constructed. Frameworks with the class of signals chosen to be the generalised function spaces of the distributions and ultra-distributions are being investigated. For the standard cases, these frameworks are required to preserve the existing formalism in both the time domain and the frequency domain.
System Identification Using Gaussian Process Prior Models
There has been strong recent interest in the application of the Gaussian process prior methodology to model nonlinear relationships from noisy data. It has been shown to be a flexible approach that, in many circumstances, outperforms competing methods such as neural networks. The basic idea is to place a prior over the space of nonlinear functions using a Gaussian process. (The Gaussian process is defined by mean and covariance functions that depend on a small set of hyperparameters, the values of which are typically determined by maximising the likelihood of some training data.) The prior is then conditioned on data to obtain a refined Gaussian process, the posterior. It provides a full probabilistic description for the nonlinear relationship; in particular, the joint probability distribution for the values of the nonlinear relationship is available at any set of values of the explanatory variable.
The application of Gaussian process prior models to the identification of nonlinear dynamic systems is being investigated. There are several key issues that must be addressed. The choice of covariance function used to define the prior must be tailored to suit dynamic rather than static relationships. Fast and memory efficient algorithms are required to handle large amounts of data. The data is typically measured as a time series: both the time series and state space aspects of the data should be exploited. Elements of the explanatory variable consist of measured data and noise is thus also present on the explanatory variable. Both local information in the form of linear models and non-local information in the form of transient trajectories must be integrated. A methodology for identifying nonlinear dynamic systems based on Gaussian process prior models that deals with all the above issues is being developed.
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