Dynamics and Control of Hybrid Mechanical Systems (DyCoHyMS)

The project is aimed at a better understanding of the dynamics and control of a large class of hybrid dynamical systems (HDS) - the systems that combine both continuous- and discrete-time dynamics. Possible applications can be found in traditional mechanical systems with dry friction (of particular interest are systems with 2-dimensional friction), CD/DVD players, pick-and-place machines and robots that should operate with/without contacts with the environment, flexible manufacturing lines, synchronization in complex networks where the interconnections between the subsystems can suddenly change, etc. HDS have recently attracted considerable attention, though a theory of HDS is far from being complete. The main direction of our research is based on the studies of HDS with inputs and particularly on the convergency properties of such systems. The convergency property of dynamical systems had been studied since 60s after pioneering works of the Russian mathematicians Pliss, Demidovich and Yakubovich. Recently this subject was rediscovered because of numerous potential applications in control technology. Briefly this property means that the system behavior is determined by input signals rather than by initial conditions that are ``forgotten' during the system evolution. Traditionally the analysis of convergency is based upon linearization of the system equations and this approach can not work for the HDS. To avoid this difficulty we plan to use Lyapunovs direct method that should be modified to study stability of the system solutions with respect to each other rather than to study stability of invariant sets. The applicants believe that the convergency property (recently also called contraction, or incremental stability) of HDS being analyzed in detail will allow to address the following important practical problems: -- performance based design/control of HDS. The switchings in control systems are aimed at a better performance. At the same time the majority of mathematical publications on HDS is devoted to stability of equilibria for systems without inputs. If a HDS with input is convergent all solutions will tend to the limit solution that does not depend on initial conditions and therefore the performance of such systems can be rigorously defined. It should be noted that our approach to the analysis of performance is completely different from traditional Lp-gain approaches. -- synchronization in complex networks. Our previous studies have demonstrated that convergency of the subsystems that form a network is a crucial property to guarantee some synchronous regimes in the network. We plan to extend these results to the case of nonsmooth subsystems and towards a situation when the interconnections between the subsystems are rapidly changing, e.g. in ``small world' networks. The synchronization problems arise in studies of locomotion of biomechanical systems, neural activity and is a subject of starting collaboration between the Departments of Mechanical Engineering and Biomedical Technology of the Eindhoven University. A similar problem is to design/analyse the motion of a group of vehicles in two or three dimensional space.