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458.308 Process Control & Design

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458.308 Process Control & Design

Jong Min Lee

School of Chemical and Biological Engineering Seoul National University

Lecture 8: Frequency Response

(2)

Preliminaries: sin(!

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t)

𝜔 t: [radian] 𝜔 : [rad/time] ( ) frequency

T: period → !T = 2⇡ , ! = 2⇡

T

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(2⇡ = 1 cycle)

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⌫ = 1 T

 cycles

time = 1

T [Hz]

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-1 -0.5 0 0.5

1 sin(2t)

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sin(t)

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higher freq.

(3)

Preliminaries:

• a vector in the complex plane with constant length one, making angle ωt with the positive real axis

• As time t increases the angle ωt increases, and so the vector rotates around the origin in the complex plane, in the counter-clockwise

sense. It makes one complete rotation when ωt = 2π radians. If that occurs when t = T, then ω = 2π/T, which has the units of radians per time.

• ω is called the angular frequency of the vector’s rotation.

• If we measure frequency in units of rotations (or cycles) per second, and if we denote that number by ν, then ν = 1/T and ω = 2πν.

• The only difference between e jωt and e −jωt is ωt, that is, e −jwt rotates in the opposite direction.

e

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j ! t

Real Imaginary

ω t ω t

e j ω t

e -j ω t

(4)

Example

T in

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T

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100 gal.

1 10s + 1

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well-insulated mixed tank

Assume inlet T in varies in a sinusoidal fashion between 20℃ - 30℃

(5)

0 100 200 300 400 500 600 700 800 20

21 22 23 24 25 26 27 28 29 30

Tin

Temp [℃] T

time [min]

T in = 5 sin(0.01t) + 25

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(6)

0 10 20 30 40 50 60 70 80 20

21 22 23 24 25 26 27 28 29 30

Temp [℃]

time [min]

T

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in = 5 sin(0.1t) + 25

T

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in

T

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(7)

0 1 2 3 4 5 6 7 8 20

21 22 23 24 25 26 27 28 29 30

Temp [℃]

time [min]

T in = 5 sin(t) + 25

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T in

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T

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At high frequencies, the inlet changes so rapidly

that the tank T does not change significantly

(8)

0 1 2 3 4 5 6 7 8 20

21 22 23 24 25 26 27 28 29 30

Temp [℃]

time [min]

T in = 5 sin(t) + 25

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A

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A ˆ

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T in

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T

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(9)

Definition of Frequency Response

Linear System:

Sinusoidal forcing → Sinusoidal response

System

A sin(ωt) A ˆ sin(ωt + φ) φ < 0

Defined by two quantities that depend on the frequency 𝜔 A ˆ

A (!)

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: Amplitude Ratio (“gain” for sinusoidal change)

(! )

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: Phase Angle (“delay” in terms of radians)

(10)

Bode Plots

Hendrik Wade Bode (1905-1982) wikipedia.org

Since Amplitude Ratio (A.R.) and Phase Angle (P.A.) depend on the frequency of ω, a continuous

parameter, it is convenient to display as plots

• ω vs. A.R. on a log-log scale

• ω vs. P.A. on a log-linear scale

(11)

Objective of This Lecture

• Understand the importance of “frequency response” for dynamic system (beyond system’s response to sinusoidal forcing).

• Why do we care about them even when pure sinusoidal changes are rare?

• Learn to recognize the characteristics of typical dynamic systems by looking at their Bode plots.

- “Making Friends” with Bode plots

• Learn how to calculate the frequency response and sketch Bode plots for a given input-output linear system

- Represented by a transfer function

(12)

Why Study Frequency Response?

For many electro-mechanical systems, frequency responses are informative and natural representation of system dynamics

(characteristics).

expensive speaker

20Hz 20KHz

AR

cheap speaker

electric signal sound wave signal

1

(13)

Continued

Ultrasonic Ringtones: The new ringtones that only kids can hear!

http://www.ultrasonic-ringtones.com

(14)

Continued

Yellow Filter Yellow Filter white light

(=red+orange+yellow+green +blue+violet)

yellow light

AR

1

Red Orange Yellow Green Blue Violet

Frequency

Gain (AR) is ~0 for all frequencies other than those for the yellow light.

Yellow Filter Yellow Filter white light

(=red+orange+yellow+green +blue+violet)

yellow light

AR

1

Red Orange Yellow Green Blue Violet

Frequency

Gain (AR) is ~0 for all frequencies other than those for the yellow light.

(15)

Why Study Frequency Response?

Any linear dynamical system is completely defined by its frequency response (for all 𝜔 )

2

Frequency

Response Information (Bode Diagram)

Any input (including non-sinusoidal)

Output response

• Frequency Response is NOT just for predicting output response to sinusoidal changes. It can be used to predict the output

response for any type of input change.

• This can be done in a numerically efficient way (called Fast

Fourier Transform).

(16)

Why Study Frequency Response?

Frequency response representation of system dynamics is very convenient for designing a feedback controller and analyzing a closed-loop system.

3

• Feedback system can cause process input to cycle in a sinusoidal manner and therefore cycle the process output.

+

+ Controller Process

y sp y

y

t y

t y

t y

t

Low K c Medium K c Critical K c > Critical K c

• Feedback induced cycling and instability happens because of the phase lag reaching −180°.

• Later: Bode stability criteria and Gain/Phase Margin

(17)

How to Compute Frequency Response

Quickly from TF

Y 0 (s)

U 0 (s) = G(s)

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s

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= j !

G(j ! ) = A ˆ A e j

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A ˆ

A (!) = q

Re 2 [G(j ! )] + Im 2 [G(j !)] = mod [G(j !)]

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(!) = tan 1

✓ Im[G(j !)]

Re[G(j !)]

= arg[G(j !)]

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Im

G ( jω )

A ˆ A

φ

(18)

Find frequency response of a first-order system, with Y 0 (s)

U 0 (s) = G(s) = K

⌧ s + 1

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G(j !) = K

⌧ (j !) + 1 = K

1 + ! 2 ⌧ 2 j K !⌧

1 + ! 2 ⌧ 2

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Referensi

Dokumen terkait

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