Synchronization and convergence of piecewise smooth systems and networks

Abstract

In this thesis the study of convergence of piecewise smooth systems and discontinuous dynamical networks is considered. The contributions of the thesis are related both in the investigation of synchronization of networks with discontinuities in the agents's dynamics or in the inter-agents communication and in the investigation of incremental stability of nonsmooth systems. In particular, a framework for the study of synchronization when the nodes' dynamics may be both piecewise smooth and/or nonidentical across the network is presented. The analysis is performed using nonsmooth analysis tools, in particular set-valued Lyapunov functions. In this way, sufficient conditions are derived in order to guarantee global bounded synchronization, with analytical expression of the estimates of the synchronization bound and of the minimum coupling strength required to achieve synchronization. Different hypotheses on the heterogeneity of the agents' dynamics is considered, and the analysis is conducted both for linear and nonlinear coupling protocols. Differently from the few results available in the literature, boundedness of trajectories of the coupled systems is not required a priori, as well as any condition of synchrony among switching signals. Referring to the networks where discontinuities are present at the level of the communication among the agents, in the thesis novel event-triggered control strategies are developed. Such distributed algorithms are able to synchronize networks of general Lipschitz nonlinear systems. The results represent an innovative contribution on the existing literature for two main reasons. The existing literature considers only synchronization of integrators or linear systems and the control signals and communication signals are not both simultaneously piecewise constant, which is instead guaranteed in the developed strategies. Furthermore, a strategy able to guarantee a lower bound for the inter-event times between consecutive updates of the control law is proposed. More in detail, the considered strategy is model based and each node uses the dynamical model of its neighbours to predict their evolution and evaluate the triggering events. Both the cases of exact and not exact information of the neighbours' dynamical model are addressed and synchronization is analytically proven in both the cases. Furthermore, Zeno behaviour are excluded. The focus on the convergence properties and on the incremental stability of nonsmooth dynamical systems leads in the thesis to extend contraction theory, which is a tool used in the literature to address incremental exponential stability of smooth systems. In particular, the extension of such tool to the class of time switching non differentiable dynamical systems which satisfy Caratheodory conditions for the existence and uniqueness of a solution is considered. More in detail, it is proved that infinitesimal contraction of each mode of a switched system of interest gives a sufficient condition for global exponential convergence of trajectories towards each other. Such extension can be related with analogous results already present in the literature for convergent systems, which instead make use of Lyapunov conditions. The results are then used to develop smooth and piecewise smooth distributed protocols to coordinate and synchronize networks of piecewise smooth agents. Furthermore, the open problem of proving incremental stability of Filippov systems is introduced and preliminary results are given for the case of planar systems. All the results in the thesis are analytically proven and illustrated via a set of representative numerical examples.