What are the applications of Complex numbers in Control theory?

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behaviour of dynamical systems with inputs, and how their behaviour is modified by feedback.

What I am looking for is the application of complex numbers in Control theory. Where do complex numbers come into play when dealing with control theory?

My research so far-

1. Complex numbers allow us to describe the properties of dynamic systems from the point of view of frequency. This analysis can be done by using the idea of the spectral transfer function, which is obtained from the transfer function by replacing the complex variable “s” by “jω”. Ref.
2. Control Theory: In control theory, systems are transformed from the time domain to the frequency domain using Laplace transformations. The poles and zeros of the system are analyzed in the complex plane. Ref. Ref.
3. Complex numbers are used in the Nyquist stability criterion. Ref.

Thanks in advanced!
Any resources which will help me find applications of complex numbers in mathematics are always welcome!

I have graduated as a control engineer, but now I am continuing my education in the department of mathematics. So, I may be able to help you. I will explain the role of complex analysis only in linear control but not further.

First of all, it is better to say the role of complex analysis but not the role of complex numbers.

All stability criteria which are thought in this course are related to complex analysis. Nyquist stability criterion is derived from Cauchy's argument principle which is a subject in complex analysis.

Analysis of the breakpoints, the asymptotes, the behaviour of the branches in root-locus all come from complex analysis.

Routh-Hurwitz stability criterion comes from an old subject in complex analysis: Analytic Polynomials. It started with the fundamental theorem of algebra. Also, since eigenvalues of a matrix are roots of a special polynomial, many things in linear algebra and its applications in control theory come from complex analysis. The book 'Analytic Theory of Polynomial' by Q.L. Rahman and G.Schmeisser can help you to understand this.