Chapter 5 Classical Feedback Analysis
© National Instruments Corporation 5-5 Xmath Control Design Module
As the gain varies, small ✱’s appear on the locus indicating the closed-loop
pole location for that choice of gain. The locus shown in Figure 5-2 shows
that for small gain values the closed-loop system is stable, with all of its
roots in the left half of the complex plane.
Frequency Response and Dynamic Response
The frequency response of a dynamic system is the output, or response, of
a system given unit-amplitude, zero-phase sinusoidal input. A sinusoidal
input with unit amplitude and zero phase, and frequency ω produces the
following sinusoidal output:
where A is the magnitude of the response as a function of ω, and φ is the
phase. The magnitude and phase of the system output will vary depending
on the values of the system poles, zeros, and gain. In many practical
engineering applications, the system poles and zeros are not precisely
known. Because the frequency response can be determined experimentally,
undesirable parts of the system’s frequency response then can be improved
by adding known compensation to the system.

freq( )

H=freq(Sys,F,{Fmin,Fmax,npts,track,delta})
The freq( ) function calculates the frequency response of a system in
several different ways, depending on the system representation. For
continuous-time transfer functions, the frequency response H(ω) at a given
frequency ω is obtained by substituting the complex frequency value jω for
qin the following equation. For discrete-time transfer functions, the value
ejwT, with T the system sampling interval, is substituted for q instead.
For continuous-time state-space systems, the basic method for finding
frequency response is to substitute different frequency values, represented
by ω, into the following equation:
Hjω() Aω()ejφω()
=
Sys q() qz
1
+()…qz
m
+()
qp
1
+()…qp
n
+()
---------------------------------------------=
Hjw() CjwI A–()
1–BD+=