Non-linear systems - First harmonic method

In automation, as in mechanics, electrical engineering and electronics, analysis and synthesis problems were initially posed on the assumption of linearity. Theories and methods were developed which enabled significant progress to be made in the fields of servo-control and regulation. However, engineers soon realized that this approach was not suitable for studying the behavior of many real systems. As a result, from the 1950s onwards, a great deal of study and research was carried out in the field of non-linear systems.

Initially, non-linearities were seen mainly as imperfections, but engineers soon became aware of the advantages they could derive from non-linearities in the design of more efficient systems. An example of this is plus/minus control, which, if properly designed, applies the bang-bang principle to obtain responses in the shortest possible time.

While the principles of proportionality and superposition lead to very general formulations and methods of analysis and synthesis for linear systems, the situation is quite different for non-linear systems. Indeed, by definition, the term non-linear systems encompasses systems of very different natures, requiring very different approaches.

One consequence of the eminently negative nature of this definition is that a unified theory is impossible in nonlinear automation: the engineer currently has at his or her disposal a range of methods that differ greatly from one another. The simplest method is to linearize the nonlinear system, and in particular to carry out this linearization in the frequency domain. Other linearization methods do exist, but they will not be presented in this article. Among these are the study of the singular points of the second-order system and the first Ljapunov method.

This harmonic linearization is most often referred to as first harmonic approximation or describing function in the Anglo-Saxon literature.