PDF Archive

Easily share your PDF documents with your contacts, on the Web and Social Networks.

Share a file Manage my documents Convert Recover PDF Search Help Contact



51N13 IJAET0313499 revised .pdf



Original filename: 51N13-IJAET0313499 revised.pdf
Author: "Editor IJAET" <editor@ijaet.org>

This PDF 1.5 document has been generated by Microsoft® Word 2013, and has been sent on pdf-archive.com on 13/05/2013 at 13:51, from IP address 117.211.x.x. The current document download page has been viewed 724 times.
File size: 405 KB (7 pages).
Privacy: public file




Download original PDF file









Document preview


International Journal of Advances in Engineering &amp; Technology, Mar. 2013.
©IJAET
ISSN: 2231-1963

OPTIMAL DESIGN AND ANALYSIS OF A CHEMICAL PROCESS
CONTROL SYSTEM
Ashis Kumar Das
Faculty of Tech., Uttar Banga Krishi Viswavidyalaya, Pundibari, Coochbehar, W.B., India

ABSTRACT
The present work describes the design and analysis of a chemical process control system. The design is
accomplished to attain the optimality of control operation. The total system is supposed to consist of the suitable
controller operated in the closed loop manner with negative feedback path, affording the suitable output to the
input. The optimality of the performance for the system is considered to be attained with gain of the (PD) controller
[1], so chosen that the integral square error becomes a minimum. The overall system is found to be stable,
controllable, and observable. The system is also analyzed in sampled data control domain (z domain). The stability
in z domain is analyzed using Jury’s Stability test. MATLAB software is appropriately used in the entire analysis.

KEYWORDS: Stability, PD controller, Integral Square Error, Sample data control.

I.

INTRODUCTION

The study of the overall system with the use of control theory demands. Here input is the desired
temperature of the chemical system and the controller i.e. PD controller is used because it develops a
signal which is a linear combination of terms that are proportional to ,and the derivative of the incoming
signal f(t). In this system controller, actuator and proportional valve, and the chemical process are in
cascade form with the temperature sensor is connected in negative feedback path. The desired
temperature is coming from input and the temperature from the output is compared in the controller.
The controller act to reduce the error signal.
The total control system is analyzed using MATLAB 7.10 software. The system designed in the work
is found to be stable with appropriate gain margin &amp; phase margins. The controllability and
observability of the total control system are tested.
The system is found to be controllable and observable. The sample data analysis of the system is also
done in the present study and the system is also found to be stable in the sample data system as done by
Jury’s test. Thus the control system designed in the present work is one sufficiently realizable system.
R(s)
G1(s)

G2(s)

G3(s)

Y(s)

H(s)
Figure 1: Block diagram of chemical process control system.

513

Vol. 6, Issue 1, pp. 513-519

International Journal of Advances in Engineering &amp; Technology, Mar. 2013.
©IJAET
ISSN: 2231-1963
𝐺1 (𝑆) = 𝑘𝑝 + 𝑘𝑑 𝑠[𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑒𝑟];
4
[𝐴𝑐𝑡𝑢𝑎𝑡𝑜𝑟 &amp; 𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑉𝑎𝑙𝑣𝑒];
𝑠+1
0.5
[𝐶ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑃𝑟𝑜𝑐𝑒𝑠𝑠];
𝐺3 (𝑆) =
𝑠+3
0.5
[𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑆𝑒𝑛𝑠𝑜𝑟];
𝐻(𝑆) =
𝑠+6
𝐺2 (𝑆) =

𝑅(𝑆) = 𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒;
𝑌(𝑆) = 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒;

II.

CONVERSION FROM S-DOMAIN TO Z-DOMAIN

Z –transform helps in the analysis and design of sample data control system, as Laplace transform does
in the analysis and design of continuous data control system. The z transforms F (z) of a sample data
control signal f (KT) is defined by the relation:
−𝑘
𝐹(𝑧) = ∑∞
𝑘=0 𝐹(𝐾𝑇)𝑧

… (1)

The above relation is derived from the Laplace transform as applied to sample data control signal.
Assuming 𝑒 𝑠𝑇 = 𝑧 be the concerned transformation variable in Laplace Transformation, we have
𝑇

𝑠𝑇 = 𝑙𝑛𝑧 i.e. 𝑠 −1 = 𝑙𝑛𝑧
Using power series an expansion of 𝑙𝑛𝑧, the above equation becomes:
𝑠 −1 =

𝑇 𝟏
4
[ − 𝑢
2 𝒖
3



4
45

𝑢3 −

44 5
𝑢 ]
945

… (2)
… (3)

𝟏−𝒛−𝟏

Where 𝑢 = 𝟏+𝒛−𝟏

… (4)

In general, for any positive integral value of n
𝑇 𝑛 1

4

4

44

𝑠 𝑛 = (2) [𝑢 − 3 𝑢 − 45 𝑢3 − 945 𝑢5 ]

𝑛

… (5)

By using binomial expansion in the above equation for various values of n, we may have the
transformation from s to z domain instead of the time domain calculation of J (integral square error),
the complex frequency domain can be used. According to a theorem in mathematics by Parseval


𝐽 = 𝐼𝑆𝐸 = ∫0 𝑒 2 (𝑡) 𝑑𝑡 =

+𝑗∞
1
𝐸(𝑠)𝐸(−𝑠) 𝑑𝑠

2𝜋𝑗 −𝑗∞

… (6)

The determinant Bn is found by first calculating

Bn =

𝑏2𝑛−2

……

𝑏2

𝑏0

𝐷𝑛

𝐷𝑛−2

……

……

0

𝐷𝑛−1

𝐷𝑛−3

……

𝐷𝑛

𝐷𝑛−2

… …]

[ 0

Where E(s) can be expressed as follows:
E(s) =

𝑁𝑛−1 𝑠𝑛−1 +⋯+𝑁1 𝑠+𝑁0

𝐷𝑛𝑠𝑛+𝐷𝑛−1 𝑠𝑛−1 +⋯+𝐷1 𝑠+𝐷0

… (7)

J follows from complex variable theory .To clarify the effect of system order, the subscript for J will be
the system order. For an nth -order system
𝐽𝑛 = (−1)𝑛−1

514

𝐵𝑛
2𝐷𝑛 𝐻𝑛

… (8)

Vol. 6, Issue 1, pp. 513-519

International Journal of Advances in Engineering &amp; Technology, Mar. 2013.
©IJAET
ISSN: 2231-1963
Where Hn and Bn are determinants. Hn is the determinant of the n×n Hurwitz matrix. The first two rows
of the Hurwitz matrix are formed with the coefficients of D(s), while the remaining rows consist of right
shifted versions of the first two rows until the n×n matrix is formed. Thus we write
𝐷𝑛−1

𝐷𝑛−3

⋯⋯

⋯⋯

𝐷𝑛

𝐷𝑛−2

⋯⋯

⋯⋯

0

𝐷𝑛−1

𝐷𝑛−3

⋯⋯

0

𝐷𝑛

𝐷𝑛−2

……

[⋯ ⋯

⋯⋯

⋯⋯

⋯ ⋯]

Hn =

… (9)

The determinant Bn is found by first calculating
N(s) N (-s) = b2n-2s2n-2 +…..+b2s2 +b0

... (10)

Then first row of the Hurwitz matrix is replaced by the coefficients of N(s) N (-s) while the Remaining
rows are unchanged [2].

III.

SYSTEM DESIGN

The overall control system under study consists of one appropriate compensator in cascade with the
actuator and proportional valve and chemical process with temperature sensor in feedback path. Under
present situation, overall transfer function of the system is given by the relation:
𝑌(𝑠)
𝑅(𝑠)

2𝑘 𝑠2 +2𝑠(1+6𝑘𝑑 )+12
𝑑 +27)+19

𝑑
= 𝑇(𝑠) = 𝑠3 +10𝑠
2 +𝑠(𝑘

... (11)

Thus the entire system characteristics equation is 𝑠 3 + 10𝑠 2 + 𝑠(𝑘𝑑 + 27) + 19 = 0. For the
characteristics equation to make the design problem with stability, the Routh array is constructed as
below:
𝑠3

1

𝑘𝑑 + 27

𝑠2

10

19

𝑠1

(27 + 𝑘𝑑 )10 − 19
10

𝑠0

19

For stability, 𝑘𝑑 ≥ −25.1.
Now again
𝑌(𝑠)
2𝑘𝑑 𝑠 2 + 2𝑠(1 + 6𝑘𝑑 ) + 12
𝑇(𝑠) =
=
𝑅(𝑠) 𝑠 3 + 10𝑠 2 + 𝑠(𝑘𝑑 + 27) + 19
1−𝑇(𝑠)
𝑠

𝑇𝐸 (𝑠) =

𝑠2 +𝑠(10−2𝑘 )+25+𝑘𝑑
𝑑 )+19

𝑑
= 𝑠3 +10𝑠2 +𝑠(27+𝑘

N2 =1, N1=10-2k, N0= 25+kd
D3=1, D2=10, D1= 27+kd, D0= 19
𝐵𝑛
𝑛 𝐻𝑛

𝐽𝑛 = 2𝐷

…..... (13)
...…… (14)
…... (15)

𝐷2

𝐷0

0

𝐻3 = 𝐷3

𝐷1

0

[0

𝐷2

𝐷0 ]

515

……. (12)

……..... (16)

Vol. 6, Issue 1, pp. 513-519

International Journal of Advances in Engineering &amp; Technology, Mar. 2013.
©IJAET
ISSN: 2231-1963
𝑁(𝑠) = 𝑠 2 + 𝑠(10 − 2𝑘) + 25 + 𝑘𝑑
𝑁(−𝑠) = 𝑠 2 − 𝑠(10 − 2𝑘) + 25 + 𝑘𝑑
𝑁(𝑠)𝑁(−𝑠) = 𝑠 4 − 𝑠 2 (50 − 42𝑘𝑑 + 4𝑘𝑑2 ) + (625 + 50𝑘𝑑 + 𝑘𝑑2 )

…...… (17)
…...… (18)
........... (19)

𝐵𝑛
𝑛 𝐻𝑛

𝐽𝑛 = 2𝐷
𝐽3 =

2
86𝑘𝑑
−279𝑘𝑑 +7713
380𝑘𝑑 +9538

Now, for maximum or minimum,

𝑑𝐽3
𝑑𝑘𝑑

= 0. Simplifying we get kd = 3.09, is the only positive value. It

is tested that with this value of k = 3.09,

𝑑 2 𝐽3
2
𝑑𝑘𝑑

is positive implying the availability of minima or maxima.

Hence the design optimality gets satisfied [3].

IV.

METHODS AND MATERIALS

So long any control system is considered in continuous data control system (continuous time domain
Laplace domain), the system analysis and study get restricted for any change in the system parameter,
or input variation for easy and ready study. To circumvent this problem sample data (s.d.) control system
makes study &amp; analysis easy and ready available with variation in system parameter and also studied in
sample data control model. The stability of the present system is tested by the Jury’s stability test which
guarantees the stability of the overall system. Needless to mention, any stable system when operated in
s.d. mode, the system is not necessarily to be guaranteed to remain stable in the s.d. mode also, there
being the enhancement of the order of the system. As any control system deserves to reach its steady
state by which the system finally runs, and follows the input at that state, the designed parameter k is
accordingly decided, the other desirable characteristic performances being also available in the system
[4, 5].

V.

PROGRAMME IN MATLAB

Create Transform Function:
&gt;&gt; num=[6.18, 39.08, 12];
&gt;&gt; den=[1, 10, 30.09, 19];
&gt;&gt; sys=tf(num,den)
Transfer function:
6.18 s^2 + 39.08 s + 12
------------------------------------S^3 + 10 s^2 + 30.09 s + 19
Finding Gain Margin &amp;Phase Margin:
&gt;&gt; margin(sys)
Gm=Inf dB ; Wpc = 5.32 rad/sec.
Pm= 130°; Wgc = 5.32 rad/sec.

516

Vol. 6, Issue 1, pp. 513-519

International Journal of Advances in Engineering &amp; Technology, Mar. 2013.
©IJAET
ISSN: 2231-1963
Bode Diagram
Gm = Inf , Pm = 130 deg (at 5.32 rad/sec)
5

Magnitude (dB)

0
-5
-10
-15
-20

Phase (deg)

-25
45

0

-45

-90
-2

-1

10

0
Frequency 10
(rad/sec)

10

1

2

10

10

Figure 2: Bode diagram and Phase Margin, Gain Margin.

Finding Impulse response:
&gt;&gt;impulse(sys)
Impulse Response
7

6

5

Amplitude

4

3

2

1

0

-1

0

1

2

3

Time (sec)

4

5

6

7

Figure 3: Impulse response.

Finding Root locus:
&gt;&gt;rlocus(sys)
Root Locus
2

1.5

Imaginary Axis

1

0.5

0

-0.5

-1

-1.5

-2
-16

-14

-12

-10

-8
Real Axis -6

-4

-2

0

2

Figure 4: Root locus.

517

Vol. 6, Issue 1, pp. 513-519

International Journal of Advances in Engineering &amp; Technology, Mar. 2013.
©IJAET
ISSN: 2231-1963
Finding Step response:
&gt;&gt;step(sys)
Step Response
1.4

1.2

Amplitude

1

0.8

0.6

0.4

0.2

0

0

1

2

3

4 (sec) 5
Time

6

7

8

9

Figure 5: Step response

Sample data control:
&gt;&gt; num=[6.18, 39.08, 12];
&gt;&gt; den=[1, 10, 30.09, 19];
&gt;&gt;T=0.1;
&gt;&gt;[numz,denz]=c2dm(num,den,T,’zoh’);
&gt;&gt;printsys(numz,denz,’z’)
num/den =
0.51331 z^2 - 0.77821 z + 0.2723
-----------------------------------------------z^3 - 2.1755 z^2 + 1.5551 z - 0.36788
&gt;&gt;dstep(numz,denz)
Step Response
1.4

1.2

Amplitude

1

0.8

0.6

0.4

0.2

0

0

10

20

30

40
Time (sec)

50

60

70

80

Figure 6: Step response of discrete-time linear systems.

Finding Controllability and Observability:
&gt;&gt;[A B C D]=ssdata(sys)

518

Vol. 6, Issue 1, pp. 513-519

International Journal of Advances in Engineering &amp; Technology, Mar. 2013.
©IJAET
ISSN: 2231-1963
−7.0000
A= 4.0000
[

0

−3.0625

−0.5000

0

0

0.5000

0

2
B= 0
]

C= [0.3750

0.1406

2.7813]

[0]

D=0
&gt;&gt;M=ctrb(A,B);
&gt;&gt;rank_of_M = rank(M)
rank_of_M = 3
&gt;&gt;system_order = length(A)
System_order = 3
&gt;&gt;N = obsv(A,C);
&gt;&gt;rank_of_N = rank(N)
Rank_of_N = 3

VI.

RESULT
 Gain Margin= Inf dB,very high.
 Phase Margin= 60 deg.


VII.

Rank of the controllability and observability matrix = 3 as same as the order of the system and
hence the system is controllable an observable.

CONCLUSION

The system designed is found to be stable in both continuous and sample data control system,
controllable, observable. The system has also appropriate gain margin and phase margin. So the design
of a chemical process control system becomes feasible with optimal control operation.

VIII.

FUTURE WORK

The integral square error may be reduced further by incorporating different control elements in the
closed loop control path.

REFERENCES
[1]. Stefani, Shahian, Savant , Hostetter, “Design of feedback Control System” 4 th ed, Oxford University
Press, pp 208-209
[2]. R.C. Dorf ,Bishop,”Modern Control System” 8 th ed, Addison Wesley, 1999
[3]. Achintya Das, Mrinmoy Chakraborty, “Design and Analysis of an Artificial Control of Standing and Leg
Articulation System”, WCECS 2008, pp 933-937.
[4]. Hans P. Geering “Optimal Control with Engineering Applications”, Springer
[5]. Anderson,Moore,”Linear Optimal Control”, Prentice-hall, Inc.
[6]. MATLAB 7.10

AUTHORS
Ashis Kumar Das was born in West Bengal, India, received the B.E. in Electrical and Electronics
Engineering from Siliguri Institute of Technology affiliated to University of North Bengal, Siliguri
and M.Tech. in Electrical Engineering (Industrial Electrical System) from National Institute of
Technology (Deemed University), Durgapur, India. Currently, he is interested to research topics
include Power System, Control System. He is currently working as Assistant Professor of
Electrical Engineering Department, Faculty of Technology at Uttar Banga Krishi Viswavidyalaya,
Pundibari, Coochbehar, West Bengal, India.

519

Vol. 6, Issue 1, pp. 513-519


Related documents


PDF Document 51n13 ijaet0313499 revised
PDF Document ijeart03405
PDF Document ijetr011915
PDF Document 3i18 ijaet0118692 v6 iss6 2342 2353
PDF Document 33i17 ijaet1117374 v6 iss5 2253 2261
PDF Document ijetr2235


Related keywords