05-12-2012, 04:49 PM
Understanding Poles and Zeros
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All of the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros must
be either purely real, or appear in complex conjugate pairs. In general for the poles, either pi = σi,
or else pi, pi+1 = σi±jωi. The existence of a single complex pole without a corresponding conjugate
pole would generate complex coefficients in the polynomial D(s). Similarly, the system zeros are
either real or appear in complex conjugate pairs.
The Pole-Zero Plot
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 poles and zeros of a transfer function may be
complex, and 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.
Such plots are known as pole-zero plots. It is usual to mark a zero location by a circle (◦) and a
pole location a cross (×). The location of the poles and zeros provide qualitative insights into the
response characteristics of a system. Many computer programs are available to determine the poles
and zeros of a system from either the transfer function or the system state equations [8]. Figure 1
is an example of a pole-zero plot for a third-order system with a single real zero, a real pole and a
complex conjugate pole pair, that is;
System Stability
The stability of a linear system may be determined directly from its transfer function. An nth order
linear system is asymptotically stable only if all of the components in the homogeneous response
from a finite set of initial conditions decay to zero as time increases,