MECH466 L16 BodeDiagram

(1)

2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

Lecture 16

Bode diagram

Course roadmap

Laplace transform Laplace transform

Transfer function Transfer function

Models for systems Models for systems •

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization Linearization

Modeling

Modeling AnalysisAnalysis DesignDesign

Time response Time response •

•TransientTransient •

•Steady stateSteady state

Frequency response Frequency response •

•Bode plotBode plot

Stability Stability •

•RouthRouth--HurwitzHurwitz •

•NyquistNyquist

Design specs Design specs

Root locus Root locus

Frequency domain Frequency domain

PID & Lead PID & Lead--laglag

Design examples Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Frequency response (review)

Steady state output

FrequencyFrequencyis same as the input frequencyis same as the input frequency

AmplitudeAmplitudeis that of input (A) multiplied byis that of input (A) multiplied by

PhasePhaseshiftsshifts

Frequency response function

(FRF):

G(jω

G(j

ω

)

Bode plot

: Graphical representation of

G(jω

G(j

ω

)

Gain

G(s

)

y(t y(t)) Stable

Stable

Bode plot of

Bode plot of G(j

G(j

ω

) (review)

Bode plot consists of

gain plot

&

phase plot

(2)

Sketching Bode plot

Basic functions

Constant gainConstant gain

Differentiator and integratorDifferentiator and integrator

Double integratorDouble integrator

First order system and its inverseFirst order system and its inverse

Second order systemSecond order system

Time delayTime delay

Product of basic functions

1.

1. Sketch Bode plot of each factor, andSketch Bode plot of each factor, and

2.

2. Add the Bode plots graphically.Add the Bode plots graphically.

10-2 10-1 100 101 102 19

19.5 20 20.5 21

10-2 10-1 100 101 102 -1

-0.5 0 0.5 1

Bode plot of a constant gain

TF

Sketching Bode plot

Basic functions

Product of basic functions

1.

2.

10-2 10-1 100 101 102 -40

-20 0 20 40

10-2 10-1 100 101 102 89

89.5 90 90.5 91

Bode plot of a differentiator

(3)

Bode plot of an integrator

TF

10-2 10-1 100 101 102 -40

-20 0 20 40

10-2 10-1 100 101 102 -91

-90.5 -90 -89.5 -89

Mirror image of

the Bode plot of

G(s

)=s with

respect to

respect to ω

ω

-

axis.

2008/09 MECH466 : Automatic Control 10 10-2 10-1 100 101 102

-100 -50 0 50 100

10-2 10-1 100 101 102 -181

-180.5 -180 -179.5 -179

Bode plot of a double integrator

TF

Sketching Bode plot

Basic functions

1.

2.

Bode plot of a 1st order system

TF

10-2 10-1 100 101 102 -50

-40 -30 -20 -10 0

10-2 10-1 100 101 102 -100

-80 -60 -40 -20 0

Corner frequency

Corner frequency

(4)

Bode plot of an inverse system

TF

10-2 10-1 100 101 102 0

10 20 30 40 50

10-2 10-1 100 101 102 0

20 40 60 80 100

Mirror image of

the original Bode

plot with respect

to

ω

-

axis.

Sketching Bode plot

Basic functions

1.

2.

Bode plot of a 2nd order system

10-1 100 101

-60 -40 -20 0 20

10-1 100 101

-200 -150 -100 -50 0

resonance resonance

Resonant freq. Resonant freq.

Peak gain Peak gain

1 0- 2

1 0- 1

1 00

1 01

1 02

-1 -0 . 5 0 0 . 5 1

1 0- 2

1 0- 1

1 00

1 01

1 02

-6 0 0 0 -4 0 0 0 -2 0 0 0 0

Bode plot of a time delay

Huge phase lag!

Huge phase lag!

The phase lag causes instability of the closed The phase lag causes instability of the closed--loop loop

(5)

Sketching Bode plot

Basic functions

Product of basic functions

1.

2.

Main advantage of Bode plot! Main advantage of Bode plot!

An advantage of Bode plot

Bode plot of a series connection G

11

(s)G

22

(s) is

the addition of each Bode plot of G

11

and G

22

.

GainGain

PhasePhase

We use this property to design

C(s

) so that

G(s)C(s

) has a

“

desired

”

shape of Bode plot.

Short proofs

Use polar representation

Then,

Therefore,

Sketch the Bode plot of a transfer function

1.

Decompose

G(s

) into a product form:

2.

Sketch a Bode plot for each component on the

same graph.

3.

Add them all on both gain and phase plots.

Example 1

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Example 1 (cont

Example 1 (cont’

’d)

d)

dB dB

deg deg

--2020

Example 2

dB dB

deg deg

--2020

Example 3

dB dB

deg deg

--2020

--4040

Example 4

dB dB

deg deg

+20

--2020

+45

+45 --4545

(7)

Example 5

dB dB

deg deg

+20

--4040

+45

+45 --4545

--9090

--2020

Remark

Use

Matlab

“

bode.m

”

to obtain precise shape.

ALWAYS check the correctness of

Low frequency gain (DC gain) Low frequency gain (DC gain)

High frequency gainHigh frequency gain

Example

Summary and exercises

Sketches of Bode plot

Basic functionsBasic functions

Products of basic functionsProducts of basic functions

Sketching Bode plot is just

…

.

to get a rough idea of the characteristic of a system.to get a rough idea of the characteristic of a system.

to interpret the result obtained from computer.to interpret the result obtained from computer.

to detect erroneous result from computer.to detect erroneous result from computer.

Exercises

Read Section 10.2. Solve Problem 10.4.Read Section 10.2. Solve Problem 10.4.

Next,

Nyquist

stability criterion

Exercises

(8)

Exercises

Sketch the Bode plot of:

Find a transfer function having the gain plot:

0dB 0dB 20dB 20dB

2

2 2020

--4040

+20

Ans.

Announcement

See the updated schedule for Lab 4

MECH466_LabSchedule_0809 (Mar 5).pdf.

If your group wants to perform Lab 4 on Tuesday

March 24 (4

-

6pm, or possibly 6

-

8pm), please

contact me ASAP.