Course roadmap
ME451: Control Systems
Modeling
Analysis
Laplace transform
Lecture 22
Frequency response
Transfer function
Models for systems
• electrical
• mechanical
• electromechanical
Block diagrams
Linearization
Dr. Jongeun Choi
Department of Mechanical Engineering
Michigan State University
Design
Time response
• Transient
• Steady state
1 Design specs
Root locus
Frequency response
3 Frequency domain
3 • Bode plot
Stability
• RouthRouth-Hurwitz
3 • Nyquist
1 PID & LeadLead-lag
2 Design examples
(Matlab simulations &) laboratories
1
What is frequency response?
2
A simple example
ƒ RC circuit
System
u(t)
u(t)
ƒ We would like to analyze a system property by
applying a test sinusoidal input u(t) and
observing a response y(t).
ƒ Steady state response yss(t) (after transient dies
out) of a system to sinusoidal inputs is called
frequency response.
R
C
y(t)
y(t)
ƒ Input a sinusoidal voltage u(t)
ƒ What is the output voltage y(t)?
3
4
An example (cont’d)
ƒ TF (R=C=1)
An example (cont’d)
1
ƒ Derivation of y(t)
r
y
0.8
0.6
0.4
0.2
ƒ u(t)=sin(t)
ƒ Inverse Laplace
0
Partial fraction expansion
-0.2
-0.4
-0.6
0 as t goes to infinity.
-0.8
-1
0
5
10
15
20
25
30
35
40
45
50
At steadysteady-state, u(t)
u(t) and y(t)
y(t) has same frequency,
but different amplitude and phase!
(Derivation for general G(s)
G(s) is given at the end of lecture slide.)
5
6
Response to sinusoidal input
Frequency response function
ƒ How is the steady state output of a linear system
when the input is sinusoidal?
ƒ For a stable system G(s), G(jω) (ω is positive) is
called frequency response function (FRF).
ƒ FRF is a complex number, and thus, has an
amplitude and a phase.
ƒ First order example
Im
y(t)
y(t)
G(s)
ƒ Steady state output
ƒ Frequency is same as the input frequency
Re
ƒ Amplitude is that of input (A) multiplied by
ƒ Phase shifts
Gain
7
8
Another example of FRF
First order example revisited
ƒ Second order system
ƒ FRF
Im
ƒ Two graphs representing FRF
ƒ Bode diagram (Bode plot) (Today)
Re
ƒ Nyquist diagram (Nyquist
(Nyquist plot)
9
Bode diagram (Bode plot) of G(jω)
ƒ Bode diagram consists of gain plot & phase plot
10
Bode plot of a 1st order system
ƒ TF
Corner frequency
0
-10
-20
-30
-40
-50
-2
10
LogLog-scale
-1
10
0
10
1
10
2
10
0
-20
-40
-60
-80
-100
-2
10
11
-1
10
0
10
1
10
2
10
12
Exercises of sketching Bode plot
ƒ First order system
Remarks on Bode diagram
ƒ Bode diagram shows amplification and phase
shift of a system output for sinusoidal inputs with
various frequencies.
ƒ It is very useful and important in analysis and
design of control systems.
ƒ The shape of Bode plot contains information of
stability, time responses, and much more!
ƒ It can also be used for system identification.
(Given FRF experimental data, obtain a transfer
function that matches the data.)
13
System identification
Summary and exercises
ƒ Sweep frequencies of sinusoidal signals and
obtain FRF data (i.e., gain and phase).
ƒ Select G(s) so that G(jω) fits the FRF data.
ƒ Frequency response is a steady state response
of systems to a sinusoidal input.
ƒ For a linear system, sinusoidal input generates
sinusoidal output with same frequency but
different amplitude and phase.
ƒ Bode plot is a graphical representation of
frequency response function. (“bode.m”)
ƒ Next, Bode diagram of simple transfer functions
ƒ Exercise: Read Section 8.
Agilent Technologies: FFT Dynamic Signal Analyzer
Generate sin signals
Collect FRF data
Sweep frequencies
Select G(s)
G(s)
14
Unknown
system
15
16
Derivation of frequency response
Term having denominator of G(s)
G(s)
0 as t goes to infinity.
17