Modeling variable structure systems as hybrid automata: From PWL Lorenz to PWL Chen

We analyze and simulate variable structure systems (VSS) in the framework of hybrid automata (HA). VSS combine different vector fields with logical decisions to fully describe the dynamics of a given system. On the other hand, HA are computational models that permit the definition of a set of discrete locations, each with its own difference/differential description, which are related thought transitions given by a set of symbolic logic conditions. Therefore, is relatively straight forward to interpret VSS as HA. Taking this computational view of VSS, allows one to pose analysis, control and simulation problems in a different context, and provide alternative novel solutions. As an illustration of the potential applicability of HA to the analysis of VSS, in this contribution we synthesize a HA to represent a parameterized PWL chaotic system that bridges the gap between the PWL Lorenz and PWL Chen systems. Our main contribution, consists on the symbolic analysis and classification of different attractors produced by the PWL chaotic system. Our results, further illustrates the relation between the different components of the so-called generalized PWL Lorenz family of chaotic attractors, and can prove to be significant in the study of chaos in VSS. The resulting HA are simulated by means of the open source program Ptolemy II ©.