Course roadmap

ME451: Control Systems Modeling Analysis Design

Laplace transform Time response

Design specs
• Transient
Lecture 23 Transfer function • Steady state
Root locus
Bode diagram of simple systems
Models for systems Frequency response
Frequency domain
• electrical • Bode plot
• mechanical
PID & Lead-
Lead-lag
• electromechanical Stability
Dr. Jongeun Choi Block diagrams • Routh-
Routh-Hurwitz
Design examples
Department of Mechanical Engineering Linearization • Nyquist

Michigan State University

(Matlab simulations &) laboratories

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Frequency response (review) Bode plot of G(jω) (review)

ƒ Steady state output ƒ Bode plot consists of gain plot & phase plot
ƒ Frequency is same as the input frequency
ƒ Amplitude is that of input (A) multiplied by
ƒ Phase shifts Gain
y(t)
y(t)
Stable

G(s) Log-
Log-scale

ƒ Frequency response function (FRF): G(jω)

ƒ Bode plot: Graphical representation of G(jω)

3 4
 Sketching Bode plot Bode plot of a constant gain
ƒ Basic functions (Today) ƒ TF
ƒ Constant gain 21

ƒ Differentiator and integrator 20.5

ƒ Double integrator 20

ƒ First order system and its inverse 19.5

ƒ Second order system 19

10
-2 -1
10 10
0 1
10
2
10

ƒ Time delay 1

ƒ Product of basic functions (Next lecture) 0.5

1. Sketch Bode plot of each factor, and 0

2. Add the Bode plots graphically. -0.5

Main advantage of Bode plot! -1

10
-2 -1
10 10
0 1
10
2
10
5 6

Bode plot of a differentiator Bode plot of an integrator

ƒ TF ƒ TF
40 40

20 20

0 0

-20 Mirror image of -20

-40
10
-2 -1
10
0
10
1
10
2
10
the bode plot of -40
10
-2 -1
10
0
10 10
1 2
10

91
1/s with respect -89

90.5
to ω-axis. -89.5

90 -90

89.5 -90.5

89 -91
-2 -1 0 1 2 -2 -1 0 1 2
10 10 10 10 10 10 10 10 10 10

7 8
 Bode plot of a double integrator Bode plot of a 1st order system
Corner frequency
ƒ TF ƒ TF 0

100 -10

-20
50
-30
0
-40
-50
-50
-2 -1 0 1 2
-100 10 10 10 10 10
-2 -1 0 1 2
10 10 10 10 10
0
-179
-20
-179.5
-40
-180
-60
-180.5 -80
Straight line
-181 approximation -100
-2 -1 0 1 2 -2 -1 0 1 2
10 10 10 10 10 10 10 10 10 10

9 10

Bode plot of an inverse system Bode plot of a 2nd order system

resonance
ƒ TF ƒ TF
20

50 0

40
-20
30

20
-40

Mirror image of
10
ƒ Resonant freq -6010 -1 0
10
1
10
0
-2 -1 0 1 2
10 10 10 10 10
the original bode 0
100
plot with respect 80 ƒ Peak gain -50

to ω-axis. 60
-100
40

20 -150

0
-2 -1 0 1 2 -200
10 10 10 10 10 -1 0 1
10 10 10
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 Bode plot of a time delay Remark
ƒ TF 1
ƒ Use Matlab “bode.m” to obtain precise shape.
0.5 ƒ ALWAYS check the correctness of
0
ƒ Low frequency gain (DC gain)
-0 . 5

-1
ƒ High frequency gain
-2 -1 0 1 2

ƒ Example
10 10 10 10 10
Huge phase lag!
0

-2 0 0 0

-4 0 0 0

-6 0 0 0
-2 -1 0 1 2
10 10 10 10 10

As can be explained with Nyquist stability criterion,

this phase lag causes instability of the closed-
closed-loop system,
and hence, the difficulty in control.
13 14

Exercises Summary
ƒ Sketch bode plot. ƒ Bode plot of various simple transfer functions.
ƒ Constant gain
ƒ Differentiator, integrator
ƒ 1st order and 2nd order systems
ƒ Time delay
ƒ Sketching Bode plot is just ….
ƒ to get a rough idea of the characteristic of a system.
ƒ to interpret the result obtained from computer.
ƒ to detect erroneous result from computer.
ƒ Next, Bode plot of connected systems

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