UNIVERSITI MALAYSIA PERLIS
SEMESTER 2 SIDANG AKADEMIK 2011/2012
ERT 210 – Kawalan dan Dinamik Proses / Process Control and Dynamics
PROBLEM BASED LEARNING (PBL)
QUESTIONS FOR GROUP 6, 7, 8, 9 AND 10
It is desired to develop the dynamic model of an absorber from its step response data. The
schematic diagram of the system is shown in Figure Q2. Before this gas is vented to
atmosphere, it is necessary to remove most of the NH3 from it by absorbing it with water.
The absorber has been designed so that the outlet NH3 concentration in the vapor is 50
ppm. During the design stage, a simulation test was conducted and the step response data
in Table Q2 was collected.
Air
NH3
H2 O
Air
NH3
H2O
NH3
Figure Q2
a) Propose a feedback control strategy to maintain the concentration of NH3 at a
desired level and show it using schematic diagram.
SEM2, 2011/2012
ANIS ATIKAH@ UniMAP
[15M]
b) Develop a first order plus time delay model from the step response data in TABLE Q2
below. Determine the time constant and steady state gain of the process.
TABLE Q2
Time
s
Water Flow
m3/hr
Outlet NH3
Concentration
0
0+
10
20
30
40
50
60
70
80
90
100
110
120
130
140
160
180
200
220
57
57
46
46
46
46
46
46
46
46
46
46
46
46
46
46
46
46
46
46
50
50
50
50
50.12
50.30
50.60
50.77
50.90
51.05
51.20
51.26
51.35
51.48
51.55
51.63
51.70
51.76
51.77
51.77
[15M]
c) Draw a block diagram of the feedback control system based on part (a).
[15M]
SEM2, 2011/2012
ANIS ATIKAH@ UniMAP
d) The behaviour of control valve can be approximated by a first order transfer function
with a time constant τ, and steady state gain as follows:
Group 6
Group 7
Group 8
Group 9
Group 10
K
2.0
2.1
2.2
2.3
2.4
τ
3.0
3.5
4.0
4.5
5.0
The transfer functions of the sensor, 𝐺𝑚 (𝑠) are estimated as follows:
Group 6
𝐺𝑚 (𝑠)
Group 7
1
2𝑠 + 1
1
2.5𝑠 + 1
Group 8
1
3𝑠 + 1
Group 9
Group 10
1
3.5𝑠 + 1
1
4𝑠 + 1
Derive the overall transfer function for a servo problem and discuss the response of
the closed-loop system with respect to stability if a proportional only controller is
used.
[20M]
e) Determine Cohen Coon and Ziegler-Nichols setting of a PI controller for the
proposed feedback controller.
[25M]
f) For both settings, determine the response of the closed loop system for a step
change of 55 in set point and calculate the offset.
[10M]
SEM2, 2011/2012
ANIS ATIKAH@ UniMAP
SOLUTION HINT
(a) Draw a feedback control structure by indicating transmitter, controller, transducer and
control valve at the appropriate location.
(b) Plot a graph of output vs time and draw a tangent line at the inflection point. Derive a firstorder-plus-time-delay model by estimating the value of K, τ and θ from the plot.
___________________________________________________________________
SEM2, 2011/2012
ANIS ATIKAH@ UniMAP
(c) Draw a block diagram of your control structure.
A standard block diagram of feedback controller (closed loop):
Replace Ysp, Ỹsp, Ym, Y with appropriate symbols (based on control structure in part (a)
(d)
i) Derive an overall transfer function for servo problem:
For servo problem, D=0
Thus, the overall transfer function is:
𝐾𝑚 𝐺𝑐 𝐺𝑣 𝐺𝑝
𝑌
=
𝑌𝑠𝑝 1+𝐺𝑐 𝐺𝑣 𝐺𝑝 𝐺𝑚
(Replace Yspand Y with your input & output variable)
Substitute the value of Km, Gm, Gv, Gp, Gc
Proportional controller is used; so, Gc=Kc
Gm, Gv,is given
Gp is derived in part (b)
Km can be obtained from Gm =(since Km is the transmitter gain)
𝐼𝑓 𝐺𝑚 =
1
, 𝑠𝑜 𝑲𝒎 = 𝟏
2𝑠 + 1
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ANIS ATIKAH@ UniMAP
ii) Discuss the response with respect to the stability
1) Use frequency response approach (Bode Diagram-Chapter 14)
[use this approach instead of using Routh stability criterion due to the existence of
time delay function]
OR
2) Use MATLAB to observe the response from small value of Kc to big value of Kc
Steps to use frequency response approach:
** Closed-loop system is stable if AR<1
1) Find GOL(s) using 𝐺𝑐 = 𝐾𝑐
𝐺𝑂𝐿 (𝑠) = 𝐺𝑐 𝐺𝑣 𝐺𝑝 𝐺𝑚
2) Substitute s by wj
3) Find critical frequency (frequency at which the phase angle equals to –π radian)
4) Find Kc at AR<1
Example:
𝐺𝑐 = 𝐾𝑐
𝐺𝑚 =
𝐺𝑂𝐿 (𝑠) = 𝐾𝑐 ∙
𝟏) 𝑮𝑶𝑳 (𝒔) =
1
2𝑠 + 1
𝐺𝑝 =
−0.1609𝑒 −21𝑠
86𝑠 + 1
𝐺𝑣 =
2
3𝑠 + 1
2
−0.1609𝑒 −21𝑠
1
∙
∙
3𝑠 + 1
86𝑠 + 1
2𝑠 + 1
−𝟎. 𝟑𝟐𝟏𝟖𝑲𝒄 𝒆−𝟐𝟏𝒔
(𝟑𝒔 + 𝟏)(𝟖𝟔𝒔 + 𝟏)(𝟐𝒔 + 𝟏)
2) 𝑺𝒖𝒃𝒔𝒕𝒊𝒕𝒖𝒕𝒆 𝒔 𝒃𝒚 𝒘𝒋
𝐺𝑂𝐿 (𝑤𝑗) =
−0.3218𝐾𝑐 𝑒 −21𝑤𝑗
(3𝑤𝑗 + 1)(86𝑤𝑗 + 1)(2𝑤𝑗 + 1)
3) Find critical frequency,𝒘𝒄 at 𝜙 = -π
𝜙 = −21𝑤 − 𝑡𝑎𝑛−1 (3𝑤) − 𝑡𝑎𝑛−1 (86𝑤) − 𝑡𝑎𝑛−1 (2𝑤)
−𝜋 = −21𝑤𝑐 − 𝑡𝑎𝑛−1 (3𝑤𝑐 ) − 𝑡𝑎𝑛−1 (86𝑤𝑐 ) − 𝑡𝑎𝑛−1 (2𝑤𝑐 )
𝜋 − 21𝑤𝑐 − 𝑡𝑎𝑛−1 (3𝑤𝑐 ) − 𝑡𝑎𝑛−1 (86𝑤𝑐 ) − 𝑡𝑎𝑛−1 (2𝑤𝑐 ) = 0
𝑤𝑐 = 0.067𝑟𝑎𝑑/𝑠𝑒𝑐
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4)Find 𝑲𝒄 at AR < 1 and using 𝒘 = 𝒘𝒄
−0.3218𝐾𝑐
𝐴𝑅 = |𝐺(𝑤𝑗)| =
√(3𝑤)2 + 12 ∙ √(86𝑤)2 + 12 ∙ √(2𝑤)2 + 12
−0.3218𝐾𝑐
√(3𝑤𝑐 )2 + 12 ∙ √(86𝑤𝑐 )2 + 12 ∙ √(2𝑤𝑐 )2 + 12
< 1
−0.3218𝐾𝑐
√(3 × 0.067)2 + 12 ∙ √(86 × 0.067)2 + 12 ∙ √(2 × 0.067)2 + 12
< 1
𝐾𝑐 < −18.702
Therefore, the closed loop response is stable when 𝑲𝒄 < −𝟏𝟖. 𝟕𝟎𝟐
______________________________________________________________
(e) Determine Cohen Coon and Ziegler-Nichols setting of a PI controller.
** Use Cohen Coon & Ziegler Nichols table for PI controller to get the value of 𝑲𝒄 & 𝝉𝑰
(Steps in determining PID parameter using Ziegler Nichol’s method have been uploaded
in the portal)
** Thus, we will get 2 set of transfer functions for PI controller to be:
1
1) 𝐺𝑐 = 𝐾𝑐 (1 + 𝜏 𝑆)  For Cohen Coon
𝐼
2)
1
𝐺𝑐 = 𝐾𝑐 (1 + 𝜏 𝑆) For Ziegler Nichols
𝐼
(f) Determine the response of the closed loop system for a step change of 55 in set point
and calculate the offset.
** Calculate the response of the closed-loop system for a step change of 55 (using both
Cohen Coon and Ziegler Nichols PID parameter):
Example:
Using Cohen Coon PID parameter:
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ANIS ATIKAH@ UniMAP
𝑌=
𝐾𝑚 𝐺𝑐 𝐺𝑣 𝐺𝑝
∙𝑌
1+𝐺𝑐 𝐺𝑣 𝐺𝑝 𝐺𝑚 𝑠𝑝
𝑌=
𝐾𝑚 𝐺𝑐 𝐺𝑣 𝐺𝑝 55
∙
1+𝐺𝑐 𝐺𝑣 𝐺𝑝 𝐺𝑚 𝑠
Substitute the value of 𝐺𝑐 𝐺𝑣 𝐺𝑝 𝐺𝑚 in the above equation using Gc obtained from part (e).
𝐺𝑐 = 𝐾𝑐 (1 +
1
1
) = −11.71 (1 +
)
𝜏𝐼 𝑠
46.48𝑠
(−11.71)(46.48𝑠) − 11.71
2
−0.1609𝑒 −21𝑠
1(
)
(
)
(
55
46.48𝑠
3𝑠 + 1
86𝑠 + 1 )
𝑌=
∙
−21𝑠
(−11.71)(46.48𝑠) − 11.71
2
−0.1609𝑒
1
𝑠
1+(
)
(
)
(
)
(
46.48𝑠
3𝑠 + 1
86𝑠 + 1
2𝑠 + 1)
𝑌=
(−544.28𝑠 − 11.71)(2)(−0.1609𝑒 −21𝑠 )
55
(46.48𝑠)(3𝑠 + 1)(86𝑠 + 1)(2𝑠 + 1) + ((−11.71)(46.48𝑠) − 11.71)(2)(−0.1609𝑒 −21𝑠 ) 𝑠
∙
**Calculate the offset for a step change of 55 in set point:
𝑂𝑓𝑓𝑠𝑒𝑡 = 𝑆𝑉 − 𝑃𝑉
= 𝑌𝑠𝑝 (∞) − 𝑌(∞)
𝑌𝑠𝑝 (∞) = 55
𝑌(∞) = lim [𝑠𝑌(𝑠)]
𝑠→𝑂
𝑌(∞) = lim [𝑠 ×
𝑠→𝑂
(−544.28𝑠 − 11.71)(2)(−0.1609𝑒 −21𝑠 )
(46.48𝑠)(3𝑠 + 1)(86𝑠 + 1)(2𝑠 + 1) + ((−11.71)(46.48𝑠) −
11.71)(2)(−0.1609𝑒 −21𝑠 )
∙
55
]
𝑠
𝑌(∞) = 55
Therefore;
𝑂𝑓𝑓𝑠𝑒𝑡 = 55 − 55 = 0
It is proved that an integral action in PI controller eliminate offset.
*Repeat the same steps to know the response of the closed-loop response using Ziegler
Nichols PID parameter.
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“Hope this hint will help you to complete the PBL report..”=)
SEM2, 2011/2012
ANIS ATIKAH@ UniMAP