Stability Analysis of Dynamical Systems

A focus of this project is in understanding fundamental stability properties of dynamical systems. We consider a variety of system models including nonlinear differential and difference equations, differential and difference inclusions, and large-scale systems comprised of nonlinear subsystems and a graph describing their interconnection structure.

We are actively looking at various stability concepts including KL-stability with respect to two measures, input-to-state stability (and its variants), and nonlinear L2-gain.

Applications

Belief Propagation

The field of forward error correction coding was revolutionised by the introduction of turbo and LDPC codes and their associated iterative decoders. Such iterative algorithms can be analysed in a dynamical systems framework, leveraging tools developed in our stability analysis research. In particular, the belief propagation algorithm can be viewed as a large-scale discrete-time system.

In a sense, the work described above looks to apply control theoretic tools to coding theory problems (or to the analysis of numerical algorithms used in coding theory). We are also looking to move results in the other direction; i.e., to use ideas developed in the coding and information theory community for the analysis of dynamical systems. The Extrinsic Information Transfer (ExIT) chart was developed in order to try to predict the convergence of the belief propagation algorithm for a particular code. This involves calculating (or estimating) the mutual information transfer between processing nodes on a bipartite graph. Similar concepts may prove useful in analysing the convergence and stability of more general nonlinear dynamical systems.

Power Systems

Modern electricity networks provide a rich source of interesting problems in the area of large scale system stability and performance. While many fundamental properties were described and analysed in the 1970s and early 1980s, new nonlinear control theoretic approaches have been subsequently developed which can provide better performance, wider stability margins, and improved efficiences. In addition, the advent of the "smart grid" and advances in power electronics mean more variables can be measured and directly controlled.