Zeros, poles. Design considerations
One way to represent system changes mathematically is through transfer functions which describe the transformation of signals between the input and the output. This input-output relation is a fractional function whose numerator and denominator generate the states detected at the input and the output, respectively. The values of the two states which are critical or extreme from the perspective of operation are given by the roots of the numerator (the zeros) and the roots of the denominator (the poles).
If the system is modelled with linear differential equations, the zeros and poles are the extreme values of the solution of the differential equation: at the input, this is the state where even the biggest effect fails to trigger displacement and the system exhibits infinite inertia; while at the output, this is the state where even the smallest effect will trigger an unstoppable displacement causing instability.
If the model uses the Euclidean infinite complex plane, the limits of stability are shown by the left (negative) half-plane considering the poles of the system. In the solution of the differential equation described by the transfer function, the exponential function with a negative exponent trends toward a position of rest, while the positive exponent trends toward infinity where stability is lost. Today, the term referring to this exponential explosion is present in the vocabulary of all scientific disciplines using systems theory. It is a design and later a closed-loop control task to keep the system in this real (stable) domain.
In order to make processes transparent and predictable, this infinite operational plane is transformed with its inverse into a unit circle (or disk) where infinity is projected into its inverse, the starting point, and the phenomena into the inside and circumference of the circle.
In more complex cases, where phenomena can no longer be followed by simple review and corresponding calculation procedures, further transformations are used, such as hyperbolic spaces instead of the Euclidean planes, constantly searching where the complexity of the reality can be better represented and predicted. This is why we demonstrated a few space transformations earlier and why we call attention to the fact that even apparently natural, routine representations are nothing more than “models” which means both freedom of thinking and a dangerous departure from reality. This consideration is behind the philosophical references in the introductory part.