CONTROL SYSTEM ANALYSIS & DESIGN
BY FREQUENCY RESPONSE
MAPUA INSTITUTE OF TECHNOLOGY
School of Chemical Engineering & Chemistry
Process Control & Instrumentation
LEARNING OBJECTIVES
• Be able to apply the FREQUENCY RESPONSE method in stability analysis and in the design of controllers
• Be familiar with the BODE Stability Criterion
• Have an understanding of the concept of GAIN and PHASE MARGINS
• Be able to determine initial PID settings
FREQUENCY RESPONSE• The response of an output variable to a sinusoidal input.• Key determinants are the AMPLITUDE and the FREQUENCY
of the sinusoid.
U(s)
Consider a 1st Order System:
Transfer Function = ratio of the Laplace Transform of
the output variable to that of the input variable
• Y(s) = Laplace Transform of the DEVIATION VARIABLE Y
Y(s)Y(s)
where:
• U(s) = Laplace Transform of the DEVIATION VARIABLE U
U(s)=
K
(s + 1)
(5-3)
Suppose a Sinusoidal Change in u is applied:
0 time, t
u
us
A u(t) = us + (A)sint
Y(t) = ?
A
A
where:• A = Amplitude = radian frequency
U(t) = (A)sintPeriod of Oscillation, T
(5-14)
• Radian Frequency = 2fwhere: f = cyclical frequency = 1/T
Parameters of the Sinusoidal Input:
• T = Period of Oscillation = the time it takes to complete one cycle
Sinusoidal Response of a 1st Order System:
0 time, t
y
ys
A
Y(t) = AKe-t/
(22 + 1)
AK
(22 + 1)+ sin(t + )
AK
(22 + 1)
(5-26)
• First Order Sinusoidal Response is attenuated– ratio of the amplitude of Y to that of U is less than 1
• First Order Sinusoidal Response lags the Sinusoidal Input by the phase angle = tan-1(-)• phase lag = ||/ 360f
Properties of 1st Order Sinusoidal Response:
(5-27)
0 time, t
u, y
us, ys
Input (u)
Output (y)
1st Order Frequency Response:• As the frequency increases, the phase angle Φ approaches 90o.
0.00
0.01
0.10
1.00
10.00
0.00
10.
0020.
004
0.00
80.
010.
020.
040.
08 0.1
0.2
0.4
0.8 1 2 4 8 10 20 40 80 10
0
-100.00
-90.00
-80.00
-70.00
-60.00
-50.00
-40.00
-30.00
-20.00
-10.00
0.00
Amplitude Ratio Phase Angle
BODE Plot of a 1st Order Frequency Response:•Log-log plot of Amplitude Ratio and Frequency•Semi-log plot of Phase Angle and Frequency
Lo
gar
ith
m o
f A
MP
LIT
UD
E R
AT
IOP
HA
SE
AN
GL
E
Logarithm of FREQUENCY
• Substitute j for s in the Transfer Function:• G(s) = G(j)
• Multiply the resulting equation by its complex conjugate• Determine the MAGNITUDE and ARGUMENT of the resulting
complex number:• G(j) = Real Part + Imaginary Part• MAGNITUDE = |G(j)| = √[Real Part2 + Imaginary Part2]• ARGUMENT = LG(j) = tan-1 [Imaginary Part/Real Part]
Alternative Way of Determining Frequency (i.e. Sinusoidal) Response from the Transfer Function:
AMPLITUDE RATIO = MAGNITUDEPHASE ANGLE = ARGUMENT
Illustrative Example #1:
Determine the frequency response of a 1st Order System.
Y(s)
U(s)=
K
(s + 1)
• Substitute j for s in the Transfer Function• G(s) = G(j)
Illustrative Example #1:
G(s) =K
(s + 1)
G(j) =K
(j + 1)
• Multiply the resulting equation by its complex conjugate
Illustrative Example #1:
G(j) =K
(jτ + 1)x
(-jτ + 1)
(-jτ + 1)
G(j) =(2τ2 + 1)
=K(-jτ + 1) K(1 - τj)
(2τ2 + 1)
G(j) =(2τ2 + 1)
-K Kτ
(2τ2 + 1)j
√[ ]2
[ ]2
Illustrative Example #1:
|G(j)| = +(2τ2 + 1)
K -Kτ
(2τ2 + 1)
• Determine the MAGNITUDE and ARGUMENT of the resulting complex number:
• G(j) = Real Part + Imaginary Part• MAGNITUDE = |G(j)| = √[Real Part2 + Imaginary Part2]
AMPLITUDE RATIO = |G(j)| =K
√(2τ2 + 1)
Illustrative Example #1:
LG(j) = tan-1 / K
(2τ2 + 1)
PHASE ANGLE = tan-1(-τ)
• Determine the MAGNITUDE and ARGUMENT of the resulting complex number:
• G(j) = Real Part + Imaginary Part• ARGUMENT = LG(j) = tan-1 [Imaginary Part/Real Part]
(2τ2 + 1)
-Kτ[ ]
AMPLITUDE RATIO (AR) & PHASE ANGLE (Φ)for
Different Systems (1/3)
• 1st Order Systems in Series:• G(s) = K/(τs + 1)n
• AR = K[1/√(τ22 + 1)]n
• Φ = n[tan-1(-τ)]
• Dead Time:• G(s) = e-θs where: θ = Dead Time
• AR = 1
• Φ = -θ[360/(2π)]
AMPLITUDE RATIO (AR) & PHASE ANGLE (Φ)for
Different Systems (2/3)
• Proportional Controller:• G(s) = Kc
• AR = Kc
• Φ = 0
• Proportional-Integral Controller:• G(s) = Kc[1+1/(τIs)]
• AR = Kc√[1+1/(2τI2)]
• Φ = tan-1[-1/(τI)]
AMPLITUDE RATIO (AR) & PHASE ANGLE (Φ)for
Different Systems (3/3)
• Proportional-Derivative Controller:• G(s) = Kc[1+τDs]
• AR = Kc√[1+2τD2)]
• Φ = tan-1[τD]
• Proportional-Integral-Derivative Controller:
• G(s) = Kc[1+ 1/(τIs + τDs )]
• AR = Kc√[1+ {τD - 1/(τI)}2]
• Φ = tan-1 [τD - 1/(τI)]
Properties of BODE PLOTS:
• Overall AMPLITUDE RATIO (AR) for a Series of Transfer Functions
• Overall PHASE ANGLE (Φ) for a Series of Transfer Functions
= Product of Individual ARs
= Sum of Individual Φs
BODE STABILITY CRITERION:
A control system is stable if the OPEN-LOOP Frequency Response exhibits an AMPLITUDE RATIO (AR) of less than 1 at its CRITICAL FREQUENCY (c).
• CRITICAL FREQUENCY = Frequency () at which the PHASE ANGLE (Φ)
is -180o
Illustrative Example #2: (P-Control, 1st Order Process)
• Determine if the control system shown below will be stable using the Bode Stability Criterion. The Controller is a Proportional-only controller with the gain set at 1.6. The Process exhibits first order dynamics with a steady state gain of 1 and a time constant of 1. The Measuring Element and the Final Control Element have negligible dynamic lags.
R +Controller
FinalControlElement
Process
MeasuringElement
B
CMV
U+
+
-
e
0.0100
0.1000
1.0000
10.0000
0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100
-360
-315
-270
-225
-180
-135
-90
-45
0
45
90
Amplitude Ratio Phase Angle
BODE PLOT for Illustrative Example #2: (P-Control, 1st Order Process)
• Determine if the control system shown below will be stable using the Bode Stability Criterion. The Controller is a Proportional-Integral controller with the Gain set at 1.6 and the Integral Time set at 0.2. The Process exhibits first order dynamics with a steady state gain of 1 and a time constant of 1. The Measuring Element and the Final Control Element have negligible dynamic lags.
R +Controller
FinalControlElement
Process
MeasuringElement
B
CMV
U+
+
-
e
Illustrative Example #3: (PI-Control, 1st Order Process)
0.0100
0.1000
1.0000
10.0000
100.0000
1000.0000
10000.0000
0.00
10.
002
0.00
40.
008
0.01
0.02
0.04
0.08 0.
10.
20.
40.
8 1 2 4 8 10 20 40 80 100
-360
-315
-270
-225
-180
-135
-90
-45
0
45
90
Amplitude Ratio Phase Angle
BODE PLOT for Illustrative Example #3: (PI-Control, 1st Order Process)
Illustrative Example #4:
• Consider the feedback control system shown below. The Controller is a Proportional-only controller. The Process, the Measuring Element, and the Final Control Element exhibit first order dynamics. The Process has a gain of 1 and a time constant of 5 seconds. The Measuring Element has a gain of 1 and a time constant of 1 second. The Final Control Element has a gain of 1 and a time constant of 2 seconds. Determine if the loop will be stable for the following Controller Gain settings: (a) Kc = 1, (b) Kc = 12.8, and (c) Kc = 20 . (Adapted from Ex. 11.4/11.10-Seborg, 2ed)
R +Controller
FinalControlElement
Process
MeasuringElement
B
CMV
U+
+
-
e
BODE PLOT for Illustrative Example #4:(a) Kc = 1
0.0000
0.0000
0.0000
0.0000
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
100.0000
0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100
-360
-315
-270
-225
-180
-135
-90
-45
0
45
90
Amplitude Ratio Phase Angle
BODE PLOT for Illustrative Example #4:(b) Kc = 12.8
0.0000
0.0000
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
100.0000
0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100
-360
-315
-270
-225
-180
-135
-90
-45
0
45
90
Amplitude Ratio Phase Angle
BODE PLOT for Illustrative Example #4:(c) Kc = 20
0.0000
0.0000
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
100.0000
0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100
-360
-315
-270
-225
-180
-135
-90
-45
0
45
90
Amplitude Ratio Phase Angle
CONTROL SYSTEM DESIGNBASED ON FREQUENCY RESPONSE
• GAIN MARGIN > 1.7• GAIN MARGIN = 1/ARc
• ARc = AR at the Critical Frequency, c
• 30O < PHASE MARGIN < 45O
• PHASE MARGIN = 180O - Φc
• Φc = Φ at which AR is equal to 1
Source: James B. Riggs, TexasTech University
Illustrative Example #5:
• Determine the GAIN MARGIN and the PHASE MARGIN of the Control System in Illustrative Example #4 for the following Controller Gain settings:
(a) Kc = 1
(b) Kc = 3.085
(c) Kc = 10
(d) Kc = 12.77. (Adapted from Ex. 11.4/11.10-Seborg, 2ed)
0.0000
0.0000
0.0000
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
100.0000
0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100
-360
-315
-270
-225
-180
-135
-90
-45
0
45
90
Amplitude Ratio Phase Angle
BODE Plot for Illustrative Example #5:(b) Kc = 3.085
Illustrative Example #6:
• Consider the feedback control system shown below. The Controller is a Proportional-only controller. The Process is a first order system with dead time. The process steady state gain is 1, the process time constant is 1 min, and the process dead time is 1.02 min. The Measuring Element and the Final Control Element have negligible dynamic lags. Determine the Controller Gain that will result to a GAIN MARGIN of 1.7.
• (Adapted from Ex. 17.3-Coughanowr, 2ed)
R +Controller
FinalControlElement
Process
MeasuringElement
B
CMV
U+
+
-
e
0.0010
0.0100
0.1000
1.0000
10.0000
100.0000
0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100
-360
-315
-270
-225
-180
-135
-90
-45
0
45
90
Amplitude Ratio Phase Angle
BODE PLOT for Illustrative Example #6:
CONTROLLER TUNINGBASED ON FREQUENCY RESPONSE
• Prepare a BODE PLOT using a P-only Controller with a Kc of 1.
• Determine the c & ARc
• Compute the ULTIMATE GAIN (Kcu)• ULTIMATE GAIN (Kcu) = Kc at which a Proportional-only
control loop is on the verge of instability.
• Kcu = 1/ARc
• Compute the ULTIMATE PERIOD (Pu)• Pu = 2π/c
CONTROLLER TUNINGBASED ON FREQUENCY RESPONSE
• Determine the PID setting from the ZIEGLER-NICHOLS Tuning Correlation:
Kc τI D
P-Controller
0.5 * Kcu n.a. n.a.
PI Controller
0.45 * Kcu Pu/1.2 n.a.
PID Controller
0.6 * Kcu Pu/2 Pu/8
Illustrative Example #7:
• Consider the feedback control system shown below. The Controller is a Proportional-Integral controller. The Process is a first order system with dead time. The process steady state gain is 1, the process time constant is 1 min, and the process dead time is 1.02 min. The Measuring Element and the Final Control Element have negligible dynamic lags. Determine the Controller Gain and Integral Time using the Ziegler-Nichols correlation.
• (Adapted from Ex. 17.3-Coughanowr, 2ed)
R +Controller
FinalControlElement
Process
MeasuringElement
B
CMV
U+
+
-
e
Illustrative Example #8:
• Determine an initial Controller Gain (Kc) setting for the Control System in Illustrative Example #4 using Frequency Response and the Ziegler-Nichols correlation.
. (Adapted from Ex. 11.4/11.10-Seborg, 2ed)