Poles and Zeros

Poles and Zeros of a transfer function are the frequencies for which the value of the transfer function becomes infinity or zero respectively. The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system. Let’s say we have a transfer function defined as a ratio of two polynomials:

H(s)=N(s)/D(s)

Where N(s) and D(s) are simple polynomials. Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s) = 0 and solving for s.Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting D(s) = 0 and solving for s. [1]

The poles and zeros are properties of the transfer function, and therefore of the differential equation describing the input-output system dynamics. Together with the gain constant K they completely characterize the differential equation, and provide a complete description of the system. A system is characterized by its poles and zeros in the sense that they allow reconstruction of the input/output differential equation. In general the system dynamics may be represented graphically by plotting their locations on the complex s-plane, whose axes represent the real and imaginary parts of the complex variable s (pole-zero plots). For the stability of a linear system, all of its poles must have negative real parts,that is they must all lie within the left-half of the s-plane. A system having one or more poles lying on the imaginary axis of the s-plane has non-decaying oscillatory components in its homogeneous response, and is defined to be marginally stable. [2]