Simulation-based reachability analysis for nonlinear systems using componentwise contraction properties

A shortcoming of existing reachability approaches for nonlinear systems is the poor scalability with the number of continuous state variables. To mitigate this problem we present a simulation-based approach where we first sample a number of trajectories of the system and next establish bounds on the convergence or divergence between the samples and neighboring trajectories. We compute these bounds using contraction theory and reduce the conservatism by partitioning the state vector into several components and analyzing contraction properties separately in each direction. Among other benefits this allows us to analyze the effect of constant but uncertain parameters by treating them as state variables and partitioning them into a separate direction. We next present a numerical procedure to search for weighted norms that yield a prescribed contraction rate, which can be incorporated in the reachability algorithm to adjust the weights to minimize the growth of the reachable set

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Motion of water and sediment due to non-breaking waves in the swash zone

A long wave run-up theory is applied to the modelling of wave-induced flow velocities, sediment transport rates and bottom changes in a swash zone. First, the properties of the water tongue motion and the resulting lithodynamic response are analysed theoretically. Next, an attempt is made to run the model for the natural conditions encountered on the southern Baltic Sea coast. The Lagrangian swash velocities are used to determine the Eulerian phase-resolved bed shear stresses with a momentum integral method, after which the motion of sand is described by the use of a two-layer model, comprising bedload and nearbed suspended load. Seabed evolution is then found from the spatial variability of the net sediment transport rates. The results presented are limited to cases of the small-amplitude waves that seem to be responsible for accretion on beaches

Breach: A Toolbox for Verification and Parameter Synthesis of Hybrid Systems

Abstract. We describe Breach, a Matlab toolbox providing a coherent set of simulation-based techniques aimed at the analysis of deterministic models of hybrid dynamical systems. The primary feature of Breach is to facilitate the computation and the property investigation of large sets of trajectories. It relies on an efficient numerical solver of ordinary differential equations that can also provide information about sensitivity with respect to parameters variation. The latter is used to perform approximate reachability analysis and parameter synthesis. A major novel feature is the robust monitoring of metric interval temporal logic (MITL) formulas. The application domain of Breach ranges from embedded systems design using Simulink to the analysis of complex non-linear models from systems biology.

Coastal hydrodynamics beyond the surf zone of the South Baltic Sea

Summary: The paper presents experimental and theoretical investigations of hydrodynamic processes in a coastal region located close to the seaward boundary of the surf zone. The analysis is based on field data collected near Lubiatowo (Poland) by measuring equipment operated simultaneously by the Institute of Hydro-Engineering of the Polish Academy of Sciences (IBW PAN) and the Maritime Institute in Gdańsk (IMG). The data consist of wind velocity and direction measured at the IBW PAN Coastal Research Station (CRS) in Lubiatowo, deep-water wave buoy records, current profiles and sea bottom sediment parameters. Mean flow velocities measured in the entire water column have almost the same direction as wind. Nearbed flow velocities induced by waves and currents, as well as bed shear stresses, are modelled theoretically to determine sediment motion regimes in the area. It appears that the nonlinear wave–current interaction generates bed shear stresses greater than those that would result from the superposition of the impacts of waves and currents separately. The paper discusses the possibility of occasional intensive sediment transport and the occurrence of distinct seabed changes at greater coastal water depths adjacent to the surf zone. It was found that this can happen under the joint influence of waves and wind-driven currents. Keywords: Wind-driven current, Wave-induced nearbed oscillations, Bed shear stresses, Friction velocity, Baltic Sea, Moderate depth

Bounded and unbounded safety verification using bisimulation metrics

International audienceIn this paper, we propose an algorithm for bounded safety verification for a class of hybrid systems described by metric transition systems. The algorithm combines exploration of the system trajectories and state space reduction using merging based on a bisimulation metric. The main novelty compared to an algorithm presented recently by Lerda et.al. lies in the introduction of a tuning parameter that makes it possible to increase the performances drastically. The second significant contribution of this work is a procedure that allows us to derive, in some cases, a proof of unbounded safety from a proof of bounded safety via a refinement step. We demonstrate the efficiency of the approach via experimental results