What is ‘hybrid' about hybrid systems?
A hybrid model is adopted based on the simple hybrid system (SHS) model in which a finite control automaton interacts with a continuous–time plant at sample times, these two components being coupled by analog–to–digital and digital–to–analog converters. Three forms of uncertainty are introduced into this model, with particular interest given to transition dynamics that describes the system dynamics over some time subinterval of the sampling interval. The set of possible solutions generated by this instance of a simple hybrid system is characterized in two related ways to define the notion of hybrid trajectory used in this work. Using this notion of hybrid trajectory, an ordering is given to compare both hybrid solution segments as well as overall hybrid trajectories. This definition requires the characterization of the class of functions that are considered admissible in defining hybrid trajectories. The treatment of solutions given in this work provides a unified method of treating the constructive, approximate, and continuous properties of hybrid solutions generated by this simple hybrid system model. The relationships between the continuity of an operator defined relative to the qualitative property being designed for (e.g., viability), the existence of a fixed point of this operator, and the existence of a hybrid trajectory satisfying the desired qualitative property (corresponding to three control problems) are examined. The three–tank problem is introduced using this modeling formalism and modifications of the basic problem are made in order to illustrate various controlled and uncontrolled hybrid behaviour, in particular, various forms of transition dynamics models.
KeywordsSampling Interval Hybrid Model Transition Dynamic Trajectory Segment Input Alphabet
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