ECE 486 FREQUENCY RESPONSE Fall 08
Reading: FPE, Section 6.1
In this portion of the course, we will study control system design from a frequency response perspective.
This will be a dual perspective to the root locus design method, and will offer new insights into controller
design.
Suppose we consider a linear system given by the transfer function H(s). Recall that, if the system is stable,
the steady state output of the system will be A|H(jω)|cos(ωt +∠H(jω )) when the input is Acos(ω t):
This is called the frequency response of the system. The magnitude and phase of H(s) when s=jω
can therefore be determined by applying the input cos(ωt), and examining the output. As we will see, this
will tell us a great deal about the system. We will be working with graphs of the magnitude and phase of
the system as we sweep ωfrom 0 to ∞.
As an example, consider an underdamped second order linear system with transfer function
H(s) = ω2
n
s2+ 2ζωns+ω2
n
.
The magnitude of this function at s=jω is given by
|H(jω)|=ω2
n
| − ω2+ 2ζωnω j +ω2
n|=ω2
n
p(ω2
n−ω2)2+ 4ζ2ω2
nω2=1
q(1 −(ω
ωn)2)2+ 4ζ2(ω
ωn)2
,
and the phase is given by
∠H(jω) = ∠ω2
n
−ω2+ 2ζωnω j +ω2
n
=∠1
−(ω
ωn)2+ 2ζ(ω
ωn)j+ 1 =−tan−1
2ζω
ωn
1−ω
ωn2
.
Since these quantities are a function of ω
ωn, we can plot them vs ω
ωnfor various values of ζ:
