Housekeeping

DERIVATION:

Beginning with a differential equation describing the plant to be controlled, the Laplace Transform may be used to move the equation from the time-domain into the s-domain. Because transfer functions do not account for initial conditions, this process allows us to ”separate” the input-output dynamics from the inputs and outputs themselves. A transfer function is represents the relationship of the system. Roots of the numerator are called ”zeros” (they drive the transfer function to 0) and roots of the denominator are called ”poles” (they drive the transfer function to ∞).

For example, consider the equation for a simple spring-mass-damper:

Staying in the complex plane we remind ourselves that the poles can be expressed as p = σ + jω where σ is the real component located on the ”x-axis” and jω is the complex [frequency] component located on the ”y-axis”.

POLE-ZERO MAP:

The pole-zero map is simply a map showing the locations of poles and zeros in the complex plane. It is useful for a quick check of stability and general behavior (half-plane location of poles and zeros) and also serves as a basis for further analysis tools such as the root locus and Nyquist plots.

The pole zero map is a 2D plot wherein the "x-axis" is the real axis and the "y-axis" is the imaginary axis. Any complex number can be represented as a point on the complex plane, with values on the real or imaginary axis being purely real or imaginary respectively.

Because of the nature of the Laplace Transform (consider and Euler's Formula) values in the left-half plane will have negative real components and will decay exponentially with values in the right-half-plane growing exponentially. Values on the real axis will exhibit purely exponential decay, while values on the imaginary axis will oscillate with no decay or growth. Values in either plane that do not lie on the real axis will demonstrate either oscillatory decay or oscillatory growth.

Note: Poles are named poles because the system response will have an infinite response at the pole locations and a negative infinite [peak] response at the zero locations.

PERSPECTIVE 1 (cartesian):

One perspective to look at points on the pole-zero map is in terms of exponential decay [or growth] rate and frequency. Points closer to the imaginary axis [smaller real components] will decay [or grow] more slowly with points further away decaying [or growing] more quickly. Points closer to the real axis [smaller imaginary components] will have lower oscillation frequencies with points further away oscillating more quickly.

PERSPECTIVE 2 (polar):

Another perspective is to consider the point locations in terms of the polar components of magnitude and frequency.