Deterministic systems and signals - Transforms and abacuses

Writing down the differential equations governing the operation of a physical system (set of elements) is a fundamental operation, an indispensable preliminary to any quantitative study. The importance of this cannot be overstated: on the one hand, experience shows that mistakes made in design offices are more often the result of incorrectly written equations than of incorrectly solved equations; on the other hand, solving differential equations can be entrusted to a computer, but no machine can write the equations of a physical system.

Drawing the functional diagram of a physical system is a highly instructive exercise for the engineer: starting with a description of the system, the aim is to understand how it works, then to express it in the form of a functional diagram, showing the variables of interest and the relationships between the component parts.

This approach brings into play the four Aristotelian causes: material, formal, efficient and final.

– To understand how it works, we first ask what the final cause is, i.e. the purpose or function of the device. This allows us to identify the input (command) and the output (the device serves to maintain the latter equal to the former).– This reveals formal causality, i.e. the system's operating structure, its internal organization.– All that remains is to express the intervention of the last two causes, by depicting the rectangles representing the system's components (material causality) and linking them with arrows, which express efficient causality.

The analysis of a system therefore consists in evaluating the behavior of one or more quantities involved in the system under the effect of one or more signals applied to it.

In this article, we present the various tools available to the engineer, and then characterize systems by an algebraic operator that contains all their properties: their transfer function.