# A calculus for hybrid sampled data systems

## Abstract

This paper addresses the problem of formally describing hybrid sampled data systems. Using the techniques proposed in Duration Calculus, we first develop a formal calculus, called *Accumulation Calculus*, for specifying real valued *step functions* of time. The key idea is that such functions are described using their integrals in bounded closed intervals. The semantics of Accumulation Calculus formulae as well as a relatively complete axiom system are presented.

Sampled data are a particular type of step functions which change at regular periods only. Suitable axioms are introduced to capture this. Special formulae and modalities are introduced to describe such data. Some proof rules are proposed for these modalities and shown to be sound. Resulting system is called a calculus for hybrid sampled data systems. Well established representations of sampled data systems from control theory such as difference equations can be directly translated into such formulae. At the same time the calculus allows reasoning about asynchronous events and their effect on sampled data. A detailed case study of a heating system is given to illustrate the applicability of the calculus to hybrid sampled data systems.

## Keywords

Difference Equation Hybrid System Boolean Function Heat System Predicate Symbol## Preview

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