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7/30/2019 process control experiment
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1-THEORY
1.1-PROCESS CONTROL
Process control refers to the methods that are used to control process variables when
manufacturing a product. For example, factors such as the proportion of one ingredient to
another, the temperature of the materials, how well the ingredients are mixed, and the
pressure under which the materials are held can significantly impact the quality of an end
product. Manufacturers control the production process for three reasons(1):
Reduce variability
Increase efficiency
Ensure safety1
In controlling a process there exist two type of classes of variables(2).
1. Input VariableThis variable shows the effect of the surroundings on the process. It
normally refers to those factors that influence the process. An example of this would be the
flow rate of the steam through a heat exchanger that would change the amount of energy put
into the process. There are effects of the surrounding that are controllable and some that are
not. These are broken down into two types of inputs.
a.Manipulated inputs: variable in the surroundings can be control by an operator or the
control system in place.
b.Disturbances: inputs that can not be controlled by an operator or control system. There
exist both measurable and immeasurable disturbances.
2. Output variable- Also known as the control variable These are the variables that are
process outputs that effect the surroundings. An example of this would be the amount of CO2
gas that comes out of a combustion reaction. These variables may or may not be measured.
As we consider a controls problem. We are able to look at two major control structures.
1. Single input-Single Output (SISO)- for one control(output) varible there exist one
manipulate (input) variable that is used to affect the process
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2. Multiple input-multiple output(MIMO)- There are several control (output) variable that are
affected by several manipulated (input) variables used in a given process (2).
1.1.1- Transfer Functions
A Transfer Function is the ratio of the output of a system to the input of a system, in the
Laplace domain considering its initial conditions and equilibrium point to be zero. If we have
an input function ofX(s), and an output function Y(s), we define the transfer functionH(s) to
be(3):
(1)
Figure 1.1 :Block diagram of Transfer functions
For comparison, we will consider the time-domain equivalent to the above input/output
relationship. In the time domain, we generally denote the input to a system asx(t), and the
output of the system asy(t). The relationship between the input and the output is denoted as
the impulse response, h(t).
We define the impulse response as being the relationship between the system output to its
input. We can use the following equation to define the impulse response:
(2)
Impulse Function
It would be handy at this point to define precisely what an "impulse" is. The Impulse
Function, denoted with (t) is a special function defined piece-wise as follows:
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(3)
The impulse function is also known as the delta function because it's denoted with the Greek
lower-case letter . The delta function is typically graphed as an arrow towards infinity, as
shown below:
Figure 1.2 : mpulse (delta) function
1.1.2- Step Response
Similarly to the impulse response, the step response of a system is the output of the system
when a unit step function is used as the input. The step response is a common analysis tool
used to determine certain metrics about a system. Typically, when a new system is designed,
the step response of the system is the first characteristic of the system to be
analyzed.However, the impulse response cannot be used to find the system output from the
system input in the same manner as the transfer function(3).
http://en.wikibooks.org/wiki/File:Delta_Function.svg7/30/2019 process control experiment
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1-2 DYNAMIC BEHAVIOURS OF FIRST ORDER AND SECOND ORDER
SYSTEMS
1.2.1 First-Order Systems
A one-degree-of-freedom first-order system is governed by the first-order ordinary
differential equation(4,5,6)
(4)
where y(t) is the response of the system (the output) to some forcing function F(t) (the input).
Eq. (4) may be rewritten as
(5)
where =a1/a0 has the dimension of time and is the time constant for the system and k =1/a0
is the gain.
Response of a First-order System to a Step Input
Consider a first-order system subjected to a constant force applied instantaneously at the
initial time t = 0 (4,5,6)
(6)
The initial condition is y(0) = 0. The solution to Eq. (5) with the step input Eq. (6) is then
(7)
The response approaches the final value y= kA exponentially. By using the boundary
conditions equation (7) then may be rewritten as
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(8)
The rate at which the response approaches the final value is determined by the time constant.
When t = , y has reached 63.2% of its final value as illustrated in Figure 3. When t =5, y has
reached 99.3% of its final value.
Figure 1.3 : First Order systems
The time constant of a system can be determined from the measured response using a linear
regression. Taking the natural log Eq. (8) yields
(9)
The slope s of the natural log term plotted against t gives the time constant through the
relation s = -1/(4,5,6).
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Transient Response of a Thermocouple
The dynamic response of a sensor is often an important consideration in designing a
measurement system. The response of a temperature sensor known as a thermocouple (TC)
may be modeled as a first-order system. When the TC is subjected to a rapid temperature
change, it will take some time to respond. If the response time is slow in comparison with the
rate of change of the temperature that you are measuring, then the TC will not be able to
faithfully represent the dynamic response to the temperature fluctuations(6).
A model of the response of a TC is based on a simple heat transfer analysis. The rate at which
the sensor exchanges heat with its environment must equal the rate of change of the internal
energy of the sensor. If the dominant mechanism of heat exchange is convection (neglecting
conduction and radiation), as it is for a TC in a fluid, then this energy balance is
(10)
h is the convection coefficient, A is the surface area of the sensor, T is the temperature, m is
the TC mass, and c is the heat capacity. Writing Eq. (11) in the form of Eq. (5)
(11)
where the time constant is
(12)
1.2.2 Analysis Of Second-Order Systems
A second-order system is one whose output, y(t), is described by a second-order differential
equation. For example, the following equation describes a second-order linear system(7):
(13)
If ao 0, then Equation (13) yields
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(14)
Equation (14) is in the standard form of a second-order system, where
= natural period of oscillation of the system
= damping factor
K = steady state gain
The very large majority of the second- or higher-order systems encountered in a chemical
plant come from multicapacity processes, i.e. processes that consist of two or more first-order
systems in series, or the effect of process control systems. Laplace transformation of
Equation (14) yields
(15)
Case A: (over-damped response), when > 1, we have two distinct and real poles. In this
case the inversion of Equation (15) by partial fraction expansion yields
(16)
Where cosh(.) and sinh(.) are the hyperbolic trigonometric functions defined by
(17)
Case B: (critically damped response), when = 1, we have two equal poles (multiple pole).
In this case, the inversion of Equation (15) gives the result
(18)
Case C: (Under-damped response), when < 1, we have two complex conjugate poles. The
inversion of Equation (15) in this case yields
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(19)
Figure 4 : Underdamped Systems
- Overshoot: Is the ratio of a/b, where b is the ultimate value of the response and a is the
maximum amount by which the response exceeds its steady state value. It can be shown that
it is given by the following expression:
(20)
- Decay ratio: Is the ratio of the amount above the stead state value of two successive peaks,
c/a. it can be shown that it can be calculated by the following equation:
(21)
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- Rise time: tr is the the process output takes to first reach the new steady state value.
- Time to first peak: tp is the time required for the output to reach its first maximum value.
- Settling time: ts is defined as the time required for the process output to reach and remain
inside a band whose width is equal to 5 % of the total change in the output.
- Period: Equation (21) defines the radian frequency, to find the period of oscillation P (i.e.
the time elapsed between two successive peaks), use the well-known relationship = 2/P;
(22)
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2. EXPERIMANTAL METHOD
The experimental set-up consists of different U-manometers in different diameters and
that contains diffrent type of liquids via their properties such as water, glycerol and their
mixtures.
The pressure difference in the U-manometer was created by a vacuum generator.
2.1.DESCRIPTION OF APPARATUS
Figure 2.1. U tube manometer[8]
2.2. EXPERIMENTAL PROCEDURE
Pressure difference was applied on the U-manometer by vacuum generator and determine the
variation of the liquid level with time until the manometer balanced. The vacuum pump was
stoped when the constant liquid level was observed. This process was repeated for all
overdamp U-manometer, and determine again the variation of the liquid level with time.
For underdamped U manometer the vacuum generator was opened and then oscilation wasobserved . The liquid level and their times was determined for step and impulse function.
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3.0 RESULTS AND DISCUSSION
3.1 U TUBE MANOMETERS
Table 3.1 U tube manometers properties
properties Manometer
1
Manometer
2
Manometer
3
Manometer 4 Manometer5 Manometer 6
(g/cm3) 0,885 0,997 1,261 0,885 1,058 1,261
(Cp) 137,6 0,894 902,85 137,6 1,362 902,85
D (cm) 0.6 1,1 0.6 1,10 1,10 1,10
L( cm) 88 95 102 98 85 116
(s) 0.212 0.220 0.228 0.224 0.208 0.243
14,64 0,026 72,52 4,6 0,0354 23,03
According to Table 3.1 the viscosity of liquid in manometer 2 and 5 were realy smaller than
other and their diameter were same or bigger. This conditions effected the damping factor to
be smaller than 1. As a result manometer 2 and 5 could not absorb the effect of disturbition
like others so that their response will to be underdamped conditions. To determine the
response time we must look their time constant. The time constant was proportional with
square root of their lenght.As a result manometer 4 and 6 had a fast response time.
3.2. RESULTS for OVERDAMPED U-MANOMETERS
Table 3.2.1 Experimental Responses of Overdamped U-manometers to step change
Manometer 1 Manometer 3 Manometer 4 Manometer 6
t(S) t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp
0 0,000 0,000 1,000 0,000 0,000 1,000 0,000 0,000 1,000 0,000 0,000 1,000
3 14,151 0,415 0,510 13,158 0,430 0,589 13,393 0,420 0,594 12,346 0,571 0,457
6 28,302 0,701 0,238 26,316 0,645 0,336 26,786 0,623 0,319 24,691 0,771 0,229
9 42,453 0,844 0,143 39,474 0,766 0,206 40,179 0,754 0,145 37,037 0,886 0,114
12 56,604 0,918 0,068 52,632 0,850 0,131 53,571 0,841 0,072 49,383 0,943 0,057
15 70,755 0,952 0,041 65,789 0,916 0,075 66,964 0,884 0,029 61,728 0,971 0,029
18 84,906 0,980 0,007 78,947 0,944 0,047 80,357 0,928 0,014 74,074 1,000 0,000
21 99,057 0,993 0,000 92,105 0,972 0,019 93,750 1,000 0,000
24 113,208 1,000 105,263 0,991 0,000
27 118,421 1,000
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According to table 3.2.1 as predicted at table 3.1.1 fast response occured in manometer 4 and
6. Because of the tube lenght and diameter of the tube was bigger than other tubes so that
manometer 6 can easily absorp the effect of distirubition and give us fast response. But
manometer 4 must had a fast response time because its viscosity was smaller than manometer
6s liquid maybe some personal mistake in the experiment.
Figure 3.2.1. Experimental hr/kp versus t/ values
According to Figure 3.2.1 we can determine the response time . Kp values were the ultimate
values. Manometer 6 was reach their ultimate values faster than others when fluid was rising.
0,000
0,200
0,400
0,600
0,800
1,000
1,200
0,000 20,000 40,000 60,000 80,000 100,000 120,000 140,000
hr/Kp
t/to
M1
M3
M4
M6
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Figure 3.2.2. Experimental hf/Kp versus t/ values
When the fluid was falling again the manometer 6 had a fast response time others .
Table 3.2.2 Theoretical responses of Overdampded U-manometers to step change
Manometer 1 Manometer 3 Manometer 4 Manometer 6
t(s) t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp t/ hr/Kp hf/Kp
0 0,000 0,000 1,000 0,000 0,000 1,000 0,000 0,000 1,0000 0,000 0,000 1,000
3 14,151 0,384 0,616 13,158 0,087 0,913 13,393 0,771 0,2290 12,346 0,238 0,762
6 28,302 0,620 0,380 26,316 0,166 0,834 26,786 0,948 0,0524 24,691 0,419 0,581
9 42,453 0,811 0,189 39,474 0,238 0,762 40,179 0,988 0,0120 37,037 0,558 0,442
12 56,604 0,891 0,109 52,632 0,304 0,696 53,571 0,997 0,0027 49,383 0,663 0,337
15 70,755 0,938 0,062 65,789 0,365 0,635 66,964 0,999 0,0006 61,728 0,743 0,257
18 84,906 0,964 0,036 78,947 0,420 0,580 80,357 1,000 0,0001 74,074 0,804 0,196
21 99,057 0,979 0,021 92,105 0,470 0,530 93,750 1,000 0,0000
24 113,208 0,988 0,012 105,263 0,516 0,484
27 118,421 0,558 0,442
This table show the theoretical responses of overdamped u manometers t/ values must be
same with the experiment . hr/Kp values were different with experimental because of the
persanol mistakes.
0,000
0,200
0,400
0,600
0,800
1,000
1,200
0,000 20,000 40,000 60,000 80,000 100,000 120,000
hf/kp
t/to
M1
M3
M4
M6
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Figure 3.2.3. Theoretical hr/Kp versus t/ values
M1 and M4 included same fluid and their viscoty values were smaller so that their response
times must be faster than others and also M6s lenght was bigger than M3 so that M6 must
gives us fast response time.
Figure 3.2.4. Theoretical hf/Kp versus t/ values
Same approach with the Figure 3.2.3 when the fluid was falling
0
0,2
0,4
0,6
0,8
1
1,2
0 50 100 150
hr/kp
t/to
M1
M3
M4
M6
0
0,2
0,4
0,6
0,8
1
1,2
0 50 100 150
hf/Kp
t/to
M1
M3
M4
M6
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3.3 RESULTS FOR UNDERDAMPED U-MANOMETERS (TO STEP CHANGE)
Table 3.3.1 Period of Oscillation and Radian Frequency of Underdamped U-Manometers
Manometer
2
Manometer
5
Period of Oscillation T(s) 1,383 1,33
Radian Frequency W(s) 4,544 4,802
Period of oscilation of manometer 2 and 5 were nearlly close together but manometer 2 little
bit long. The reason maybe the viscoty of liquid in manometer 2 was small so it rised more
than manometer 5 and that effected the raidan frequency .
Table 3.3.2 Experimental Responses of Underdamped U-manometers to Step Change
Manometer 2 Manometer 5
texp(s) t/ h/Kp texp(s) t/ h/Kp
0 0,000 0,000 0,000 0,000 0,000 0,000
1 1,510 6,864 1,000 1,430 6,875 1,000
2 1,780 8,091 0,522 1,890 9,087 0,500
3 2,590 11,773 0,882 2,470 11,875 0,890
4 3,230 14,682 0,676 3,460 16,635 0,646
5 3,960 18,000 0,809 4,220 20,288 0,768
6 4,900 22,273 0,728 5,190 24,952 0,720
7 5,810 26,409 0,699 6,110 29,375 0,720
8 6,890 31,318 0,743 7,160 34,423 0,744
As we expected the h /Kp values shows us the oscillation was occured because of the their
damping factor . And also the input was step function so that the osicalliton was reach one
point
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Table 3.3.3 Theoretical Responses of Underdamped U-manometers to Step Change
Manometer 2 Manometer 5
ttheo(s) t/ h/Kp ttheo(s) t/ h/Kp
0 0,327 1,486 0,138 0,417 2,005 0,154
1 1,018 4,627 1,795 1,081 5,197 1,740
2 1,708 7,764 0,265 1,745 8,389 0,354
3 2,398 10,900 1,680 2,409 11,582 1,562
4 3,088 14,036 0,372 3,073 14,774 0,513
5 3,778 17,173 1,581 3,737 17,966 1,421
6 4,468 20,309 0,463 4,401 21,159 0,637
7 5,158 23,445 1,496 5,065 24,351 1,311
8 5,848 26,582 0,541 5,729 27,543 0,734
9 6,538 29,718 1,424 6,393 30,736 1,226
10 7,228 32,855 0,608
The cause of reading mistakes the experimental values was not close with the experimental
values.
Figure 3.3.1. Comparison of Experimental and Theoretical Responses for M-2
Experimental values were not correctly readed .
0,000
0,200
0,400
0,600
0,800
1,000
1,200
1,400
1,600
1,800
2,000
0,000 5,000 10,000 15,000 20,000 25,000 30,000 35,000
h/kp
t/to
M2-exp
M2-theo
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3.4. RESULTS FOR UNDERDAMPED U-MANOMETERS (TO IMPULSE CHANGE)
Table 3.4.1 Experimental Responses of Underdamped U-manometers to Impulse Change
texp(s) t/ h/Kp texp(s) t/ h/Kp
0 0,000 0,000 0 0,000 0,000
1 0,650 2,955 1,000 1,1 5,288 1,000
2 1,280 5,818 -0,693 1,79 8,606 -0,791
3 1,890 8,591 0,511 2,5 12,019 0,674
4 2,550 11,591 -0,341 3,12 15,000 -0,372
5 3,270 14,864 0,295 4 19,231 0,186
6 4,170 18,955 -0,239 4,89 23,510 -0,140
7 5,010 22,773 0,295 5,75 27,644 0,070
8 5,770 26,227 -0,239 6,59 31,683 -0,023
9 6,670 30,318 0,193 7,71 37,067 0,012
10 7,870 35,773 -0,114
8,690 39,500 0,091
The input was the impulse function so that the h/Kp values changes positive to negative. The
lenight of oscicallation should reach 0.
Table 3.4.2 Theoretical Responses of Underdamped U-manometers to Impulse Change
ttheo(s) t/ h/Kp theo(s) t/ h/Kp
0 -0,346 -1,573 -1,042 -0,333 -1,599 -1,057
1 0,346 1,573 0,960 0,333 1,599 0,947
2 1,038 4,718 -0,885 0,998 4,796 -0,845
3 1,730 7,864 0,815 1,663 7,993 0,753
4 2,422 11,009 -0,751 2,328 11,190 -0,6685 3,114 14,155 0,692 2,993 14,387 0,591
6 3,806 17,300 -0,638 3,658 17,584 -0,522
7 4,498 20,445 0,588 4,323 20,781 0,459
8 5,190 23,591 -0,541 4,988 23,978 -0,403
9 5,882 26,736 0,499 5,653 27,175 0,352
10 6,574 29,882 -0,460 6,318 30,373 -0,306
11 7,266 33,027 0,423 6,983 33,570 0,266
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12 7,958 36,173 -0,390
13 8,650 39,318 0,359
14 9,342 42,464 -0,331
15 10,034 45,609 0,305
16 10,726 48,755 -0,281
17 11,418 51,900 0,259
Figure 3.4.1. Theoretical and experimental values for M-2
According to Figure 3.4.1 the experimental and theoretical curve was close early but than
some of the mistakes maybe reading mistakes was effectted the phase of the oscillation. But
both of them was aproach to zero because of the impulse function.
-1,5
-1
-0,5
0
0,5
1
1,5
0,000 10,000 20,000 30,000 40,000 50,000 60,000h/kp
t/to
M2 exp
M2-theo
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Figure 3.4.2. Theoretical and experimental values for M-5
According to Figure 3.4.2 the experimental and theoretical curve was close early but than
some of the mistakes maybe reading mistakes was effectted the phase of the oscillation. But
both of them was aproach to zero because of the impulse function.
Table 3.4.3 Comparison of Theoretical and Experimental Overshoot, Decay Ratio and
Response time to Impulse change
Monometer 2 Monometer 5
Experimental Theoretical Experimental Theoretical
Overshoot 0,511 0,922 0,674 0,897
Decay ratio 0,577 0,85 0,276 0,805
Response
Time
8,6 11,42 7,71 6,98
-1,500
-1,000
-0,500
0,000
0,500
1,000
1,500
0,000 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000h/kp
t/to
M5-exp
M5-theo
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4. CONCLUSIONS
In this experiment ,to determine the effects of liquid properties and shape of U-tube
manometers on response time by using step and impulse input, U-manometer systems, which
are manometer-1 with engine oil, manometer-2 with water, manometer-3 with glycerol,
manometer-4 with engine oil, manometer-5 with 15% glycerol solution and manometer-6
with glycerol were used.
In the overdamped systems (m-1,m-3,m-4,m-6), the damping factor was calculated and it was
observed that their damping factors were greater than 1. These systems can easily absorb the
energy of disturbiton and the reason of this viscosity of liquids that contained in these
manometers were high enough according to their diameter and length.Furtheremore, to
compare their response time, it was observed that higher length and higher diameter cause the
response time to get low for same liquid.
In the underdamped systems (m-2 ,m-5), the damping factor was calculated again and it was
observed that their damping factor were smaller than 1. As we expected they relased their
energy with doing oscillation step by step. Our experimental values was different from the
theoretical values.The reason of this the oscillation was realy fast so that the reading mistakes
was done. Howewer, according to theoretical and experimental response time, we could
observed that the impulse system had a higher response time than step system. The reason of
this, while they relasing their energy which comes from disturbition from vacuum generator,
the potential energy differences at step function was small than impulse function.
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5. NOMENCULATURE
A, B Constants in the transfer function
At Surface area of bulb for heat transfer (m2)
g Acceleration of gravity (m/s2)
Kp Static gain or gain (m)
L Total length of the liquid in U-manometer (m)
m Mass of liquid in the monometer (kg)
r Liquid lever difference at any time in U-manometer (m)
t Time (s)
tr Rise time (s)
T period of oscillation (s/cycle)
Q Volumetric flow rate of the liquid (m3/s)
p Time constant (s)
Viscosity of the liquid (Pa.s)
Density of the liquid (kg/m3)
Radian frequency (radian/s)
Damping factor
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6.REFERENCES
1- http://www.pacontrol.com/download/Process%20Control%20Fundamentals.pdf
2-
https://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminolo
gy
3-. http://en.wikibooks.org/wiki/Control_Systems/Transfer_Functions
4- J.P. Holman, Experimental Methods for Engineers, 7th Ed., McGraw-hill, New York,
2001: First-order systems, p. 19-23; Thermocouples p. 368-377; Linear regression p. 91-
94; Signal conditioning (RC Circuits) p. 183-190.
5-R.S. Figliola and D.E. Beasley, Theory and Design for Mechanical Measurements, Wiley,
New York, 1991, p. 63, 73.
6- Omega Technologies Handbook, Thermocouple Reference Tables, Omega Engineering
Inc., 1993, p. B172.
7http://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second
%20Order%20Systems.pdf
8.http://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmecha
nicsgeneral
http://www.pacontrol.com/download/Process%20Control%20Fundamentals.pdfhttps://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminologyhttps://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminologyhttp://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second%20Order%20Systems.pdfhttp://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second%20Order%20Systems.pdfhttp://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmechanicsgeneralhttp://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmechanicsgeneralhttp://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmechanicsgeneralhttp://www.edibon.com/products/?area=fluidmechanicsaerodynamics&subarea=fluidmechanicsgeneralhttp://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second%20Order%20Systems.pdfhttp://faculty.ksu.edu.sa/alhajali/Publications/Dynamic%20Behavior%20of%20First_Second%20Order%20Systems.pdfhttps://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminologyhttps://controls.engin.umich.edu/wiki/index.php/Process_Control_Definitions_and_Terminologyhttp://www.pacontrol.com/download/Process%20Control%20Fundamentals.pdf7/30/2019 process control experiment
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7. APPENDIX