Planar Hybrid Systems Research Articles

Limit cycles and chaos in planar hybrid systems

In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic dynamics.

ParaPlan: A Tool for Parallel Reachability Analysis of Planar Polygonal Differential Inclusion Systems

We present the ParaPlan tool which provides the reachability analysis of planar hybrid systems defined by differential inclusions (SPDI). It uses the parallelized and optimized version of the algorithm underlying the SPeeDI tool. The performance comparison demonstrates the speed-up of up to 83 times with respect to the sequential implementation on various benchmarks. Some of the benchmarks we used are randomly generated with the novel approach based on the partitioning of the plane with Voronoi diagrams.

Hybrid control design for limit cycle stabilisation of planar switched systems

ABSTRACTThis paper addresses the control problem of an important class of hybrid dynamical systems where the desired state does not belong to the set of subsystem equilibria. Beyond the practical stabilisation where the system trajectory has to be driven to a neighbourhood of a desired state, a control law is designed to solve the limit cycle stabilisation problem for planar hybrid systems. Using the hybrid Poincaré map, two hybrid controllers are developed, which guarantee a local asymptotic/finite-time stability of the desired limit cycle. Illustrative examples are provided to highlight the effectiveness of the derived results.

Sensitivity engineering in direct contact palladium-gold nano-sandwich hydrogen sensors [Invited

In recent years, plasmonic hydrogen sensing schemes using complex hybrid Pd@Au nanostructures have attracted significant attention. However, so far, most studies have focused on investigating the sensing performance of nanosensor geometries where the constituent materials are laterally coupled. In contrast to such planar hybrid systems, which often require complex multi-step fabrication approaches, sensing devices where the materials are stacked directly on top of each other can be fabricated in a single lithography step, enabling straightforward high-throughput processing. Here, we demonstrate a novel hydrogen sensing scheme which incorporates complex hybrid plasmonic nanostructures consisting of stacked gold and palladium nanodisks. In particular, we study the influence of stacking order and geometry, experimentally and numerically, to find an optimal arrangement for a hydrogen sensor device. With an optimized sensing geometry – a stack of gold as lower and palladium as upper disk – we obtain spectral shifts as large as 30 nm at 4 vol.% H2, which is a strong improvement compared to previous indirect sensing designs. Our samples yield large absorption and scattering signals and are fabricated by low-cost hole-mask colloidal lithography and therefore yield sample sizes over areas of 1 cm2.

Reachability analysis of complex planar hybrid systems

Hybrid systems are systems that exhibit both discrete and continuous behavior. Reachability, the question of whether a system in one state can reach some other state, is undecidable for hybrid systems in general. In this paper we are concerned with GSPDIs, 2-dimensional systems generalizing SPDIs (planar hybrid systems based on “simple polygonal differential inclusions”), for which reachability have been shown to be decidable. GSPDIs are useful to approximate 2-dimensional control systems, allowing the verification of safety properties of such systems.In this paper we present the following two contributions: (i) an optimized algorithm that answers reachability questions for GSPDIs, where all cycles in the reachability graph are accelerated. (ii) An algorithm by which more complex planar hybrid automata are over-approximated by GSPDIs subject to two measures of precision. We prove soundness, completeness, and termination of both algorithms, and discuss their implementation.

Melnikov theory for a class of planar hybrid systems

In this paper, we extend the well-known Melnikov theory for smooth systems to a class of two-zonal planar hybrid piecewise smooth systems. In this class, each zone of differentiability is separated by a straight line and in these zones, the dynamics are governed by a smooth system. When an orbit reaches the separation line then a reset map applies before entering the orbit in the other zone. The family of non-generic systems of this class which has a continuum of periodic orbits is considered. Then, we study the periodic orbits of the continuum that persist under a perturbation. To achieve this objective, first we build an appropriate Poincaré map, after that, we introduce a function of one variable whose zeros provide us the periodic orbits that remain after the perturbation. This function will be called the Melnikov function because it is very close to the so-called Melnikov function for smooth systems. Finally, we apply the obtained results to some piecewise linear systems, continuous as well as discontinuous ones.

HYBRID LIMIT CYCLES AND HYBRID POINCARÉ-BENDIXSON

We present two results about regular hybrid systems with no branching (Simic et al., 2000a). The first one provides a condition for asymptotic stability of hybrid closed orbits in terms of contraction-expansion rates of resets and flows in a hybrid system. The second one is a generalization of the Poincaré-Bendixson theorem to planar hybrid systems.