Use of The Complex Plane in Control Theory
In control theory, one use of the complex plane is known as the 's-plane'. It is used to visualise the roots of the equation describing a system's behaviour (the characteristic equation) graphically. The equation is normally expressed as a polynomial in the parameter 's' of the Laplace transform, hence the name 's' plane.
Another related use of the complex plane is with the Nyquist stability criterion. This is a geometric principle which allows the stability of a closed-loop feedback system to be determined by inspecting a Nyquist plot of its open-loop magnitude and phase response as a function of frequency (or loop transfer function) in the complex plane.
The 'z-plane' is a discrete-time version of the s-plane, where z-transforms are used instead of the Laplace transformation.
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