Stability theory for hybrid systems: Specific contributions include solving the CQLF problem for finite sets of second order systems; relating intersections of convex cones to spectral conditions on matrix products; developing spectral versions of the circle criterion; developing tests for the existence of diagonal Lyapunov functions tests for the stability of positive dynamic systems. Our work on using full-rank decompositions for descriptor systems has led to a number of new results for this class of systems. Some of our most recent work established a unifying framework in which many classical stability results can be easily explained. Some of this work is summarised in SIAM Review.
Stochastic matrices and congestion control of networks: We pioneered the use of stochastic matrices in the study and design of congestion control algorithms. This work has been applied in the design of new networking congestion control algorithms, in automotive control, and in the design of algorithms to manage the smart grid. From a theoretical perspective, this work has led to a number of fundamental results in linear algebra and in stochastic processes, and led to the proof of the ergodicity of networks of a network of long-lived AIMD flows. This book is a good starting point for this work. Our congestion control algorithms (HTCP) have been implemented in Linux and FreeBSD and we have also won best paper awards at IWQoS and at ACM SIGMETRICS (student) for joint work with Rade Stanojevic.
Spectral methods and passivity: Our work on CQLFs established a connection between intersections of convex cones and the spectrum of products of matrices. Other work on stability theory established a link between the spectrum of products of matrices and rational polynomial functions. Taken together, and by developing new realizations of inverse transfer functions, these results gave rise to a number of important results to determine when linear normal and descriptor systems are passive. A major result that was derived involves reducing a certain classes of generalized eigenvalue problems to conventional eigenvalue problems. This gave rise to a series of tests to establish passivity. A related result, based on full rank decompositions, gives rise to a technique to reduce a descriptor system to a lower dimensional ODE. Many of these ideas are exploited in our recent paper on SPRification.
Smart Cities/Connected Mobility: Since 2012, I have been working in the area of collaborative mobility for smarter cities. Our focus has been in applying system theoretic and queuing theory ideas to the design of new mobility concepts and to sharing-economy systems. In this context, our main contributions have been to show that many classical problems can be solved as distributed resource-allocation/feedback-control problems involving bespoke queue management systems. This work has won several awards and has also attracted media attention. Current theoretical work involves signalling for behavioural change, prediction under feedback, and large scale city-wide identification techniques; see here for some recent papers. Some of the more applied work can be found here, and our recent book here. We have also won best paper awards at ICCVE and the ERTICO European Intelligent Transport Systems (ITS) Congress.