Transcript Chapter 8
Chapter 11 Frequency Response Analysis
Overall Course Objectives • Develop the skills necessary to function as an industrial process control engineer.
– Skills • Tuning loops • Control loop design • • Control loop troubleshooting
Command of the terminology
– Fundamental understanding • • Process dynamics
Feedback control
Frequency Response Analysis • Is the response of a process to a sinusoidal input • Considers the effect of the time scale of the input.
• Important for understanding the propagation of variability through a process.
• Important for terminology of the process control field.
• But it is NOT normally used for tuning or design of industrial controllers.
Process Exposed to a Sinusoidal Input
c
(
t
)
G p
(s)
y s
(
t
)
Key Components of Frequency Response Analysis D
t p a c a y c
A r
a y a c
y s Time
D
t p
2 360 º
Effect of Frequency on
A r
and D
t p
D
t p c y s Time c Time
Bode Plot: A Convenient Means of Presenting
10
A r
and versus
1 0.1
0.01
0.01
0.1
1
10 100
Ways to Generate Bode Plot • Direct excitation of process.
• Combine transfer function of the process with sinusoidal input.
• Substitute
s
=
i
into
G p
(
s
) and convert into real and imaginary components which yield
A r
( ) and ( ).
• Apply a pulse test.
Developing a Bode Plot from the Transfer Function
G p
(
i
)
R
( )
i I A r
( )
R
2 ( )
I
2 ( ) ( ) tan 1
I
( )
R
( )
Derivation of the Bode Plot for a First Order Process
G p
(
s
)
K p p s
1
G p
(
i
)
i
K p p
1 After
G p
(
i
rationaliz ) 2
K p
2
p
ation 1
i
K
2
p
2
p
p
1
A r
( )
K
2
p
2 2
p K
2
p
2 1 2
p
K p
2 2
p
1 ( ) tan 1 (
p
)
Properties of Bode Plots Consider :
G p
(
s
)
G a
(
s
)
G b
(
s
)
G c
(
s
)
G d
(
s
)
A r
G a
(
s
)
G c
(
s
) ln[
A r
( )] ln
G b
(
s
)
G d
(
s
)
G a
(
i
) or ln
G b
(
i
) ln
G c
(
i
) ln
G d
(
i
) ( )
G a
(
i
)
G b
(
i
)
G c
(
i
)
G d
(
i
)
Bode Plot of Complex Transfer Functions • Break transfer function into a product of simple transfer functions.
• Identify A r ( ) and of each simple transfer function from Table 8.1.
• Combine to get A r ( ) and ( ) for complex transfer function according to properties.
• Plot results as a function of .
Example of a Bode Plot of a Complex Transfer Function
G p
(
s
) 2
p s
2
K p
e
2
p
s s
1
K c
1
I
1
s
A r
1 2
p s
2
K
2
p p
s
1 360 2
A r
1 2 2
p
2
K p
2
p
2 tan 1 2 1 2
p
2
p
Example Continued
K c
1
I
1
s
A r
K c
For overall process : 1 1 2
I
2 tan 1 1
I
A r
1 2 2
p K
2
p K c
2
p
2 360 2 1 1 2
I
2 tan 1 2 1 2
p
2
p
tan 1 1
I
Bode Stability Criterion
Y sp (s) G c (s) Y s (s) G a (s) G s (s) G p (s) Y(s) Y sp (s) G c (s) Y s (s) G a (s) G s (s) G p (s) Y(s)
Bode Stability Criterion • A system is stable if
A r
is less than 1.0 at the critical frequency (i.e., that corresponds to =-180º) • Closed loop stability of a system can be analyzed by applying the Bode Stability Criterion to the product of the transfer functions of the controller and the process, i.e.,
G c
(s)
G p
(s).
Gain Margin
10 1 0.1
0.01
0.01
0.1
M 10 1
100
Gain Margin • Gain Margin = 1/
A r
* – Where
A r
* is the amplitude ratio at the critical frequency.
– Controllers are typically designed with gain margins in the range of 1.4 to 1.8 which implies that
A r
at the critical frequency varies between 0.7 and 0.55, respectively.
10 1 0.1
0.01
0.001
0.0001
0.01
Phase Margin
0.1
co 1
10 100
Phase Margin • PM = * 180 – Where * is at the crossover frequency.
– Controllers are typically designed with a PM between 30º to 45º.
Tuning a Control from the Gain Margin Problem: Determine
K c
process (
K p
2,
p
3, for a gain margin equal 1.7 for a FOPDT
p
1.5).
Solution: From Table 8.1, one can determine A r and .
First, determine such that 180º 180 1 1.218 radians per unit time Rearranging the equation for A , r
K c
A * r 9 2 1 1.12
2
Tuning a Control from the Phase Margin • Phase margin determines the phase angle at the crossover frequency.
• The amplitude ratio at the crossover frequency is one; therefore, the controller gain can be calculated from the equation for the amplitude ratio.
Example of a Pulse Test
y s c Time
Developing a Process Transfer Function from a Pulse Test
A
( )
C
( )
R
( )
I
( ) 0
y
s
(
t
) cos
t dt B
( ) 0
c
(
t
)
A
( ) cos
C
( )
t C
2 ( )
dt B
( )
D
2 ( )
D
( )
D
( ) 0 0
y
s
(
t
) sin
t dt c
(
t
) sin
t dt A
( )
D
( )
C
2 ( )
B
( )
C
( )
D
2 ( )
A r
( ) ( )
R
2 ( ) tan 1
I
( )
R
( )
I
2 ( )
Limitations of Transfer Functions Developed from Pulse Tests • They require an open loop time constant to complete.
• Disturbances can corrupt the results.
• Bode plots developed from pulse tests tend to be noisy near the crossover frequency which affects GM and PM calculations.
Nyquist Diagram
A r
Real Axis
Nyquist Diagram (Complex Plane Plot)
I R
A r A r
cos sin Therefore, you can use the same equations used to develop a Bode plot to make a Nyquist diagram.
Nyquist Stabilty Criterion
2 unstable 0 -2 -2 .
(-1,0) stable 0 Real Axis 2
Closed Loop Frequency Response
D(s) Y sp (s) G c (s) G a (s) Y s (s) G s (s) G d (s) G p (s) + + Y(s)
Example of a Closed Loop Bode Plot
1.2
0.8
0.4
0 0.01
0.1
1
pf
10 100
Analysis of Closed Loop Bode Plot • At low frequencies, the controller has time to reject the disturbances, i.e., A r is small.
• At high frequencies, the process filters (averages) out the variations and A r is small.
• At intermediate frequencies, the controlled system is most sensitive to disturbances.
Peak Frequency of a Controller • The peak frequency indicates the frequency for which a controller is most sensitive.
Overview • Understanding how the frequency of inputs affects control performance and control loop stability is important.
• The analytical aspects of frequency response analysis are rarely used industrially.