Analytical functions, complex variable
Analytical functions are nice and useful applications of the group theory. Such functions play a fundamental role in the mathematical representation of the problems of systems theory. Analytic functions describe linear and linearly approximated processes with their definitive characteristic that within their domain of interpretation they can be differentiated at any point in the form of a Taylor series, i.e. they can be described as a power series. This description is the essence of providing a mathematical representation of the process; a model which is realistic under certain conditions and which can be handled with mathematical (in our case computational) methods.
Analytical functions behave similarly in the domain of complex variables, so our conclusions are valid to process representations with complex variables as well. The representation with complex variables reflects the image of the physical reality of processes which expresses the magnitude of effects in absolute terms, the amplification or weakening, while the rotation phase angles express the delay of the effect.
For this reason, in practice, changes are primarily represented with the concepts and tools of complex analysis (the theory of functions of a complex variable).
The application of functions of a complex variable provides a consistent framework for time and frequency domain representations.
After Gauss and Cauchy, the two-dimensional nature of the complex variable extends the interpretation of integration. In this interpretation, the definite integral along a line will be the space run through by the vector representing the variables, where the internal properties of the space are added up in the integral. As there are both positive and negative areas in this domain, which is closed from the perspective of each process cycle, the process only integrates the internal singularities, sources and sinks, so otherwise the process returning to the initial state results in zero. In the physical reality, this is described by Stokes’ law: all closed flow fields, whether fluid, gas or magnetic, can only provide the flow generated or absorbed in that field.
In process dynamics these sources and sinks are the poles and zeros; their mathematical forms in the two-dimensional space are the points where the process system displays a singular behaviour: it absorbs all effects and becomes hyperstable and uncontrollable, or it gives an infinite response to even the smallest effect and so becomes unstable. Thus, control design focuses on these state singularities and when designing the system it tries to avoid these points of singularity as much as possible or compensate them by properly setting up the sources and the sinks.
Let us take the suspension of a car as a simple example. If the suspension is too stiff, it will result in a bumpy ride even on a relatively smooth road, ultimately breaking the car but in any case shortening its lifecycle. Too soft suspension, on the other hand, is not only uncomfortable for the passengers but also causes serious swaying which can also break the car.
Classic control engineering deals with the representation of these variables, analysing stability with the Nyquist stability criteria.