OPTIMAL DESIGN AND ANALYSIS OF A CHEMICAL PROCESS CONTROL SYSTEM
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1 International Journal of Advances in Engineering & Technology, Mar. 23. 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 [], 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. software. The system designed in the work is found to be stable with appropriate gain margin & 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) G (s) G2(s) G 3(s) Y(s) H(s) Figure : Block diagram of chemical process control system. 53 Vol. 6, Issue, pp
2 International Journal of Advances in Engineering & Technology, Mar. 23. G (S) = k p + k d s[controller]; G 2 (S) = 4 [Actuator & Proportional Valve]; s + G 3 (S) =.5 [Chemical Process]; s + 3 H(S) =.5 [Temperature Sensor]; s + 6 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: F(z) = k= F(KT)z k () 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 st = lnz i.e. s = T lnz Using power series an expansion of lnz, the above equation becomes: (2) s = T 2 [ u 4 3 u 4 45 u u5 ] (3) Where u = z +z In general, for any positive integral value of n (4) s n = ( T 2 )n [ u 4 3 u 4 45 u 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 = I SE = e 2 (t) dt = 2πj j The determinant B n is found by first calculating B n = Where E(s) can be expressed as follows: E(s) = [ +j E(s)E( s) b 2n 2 b 2 b D n D n 2 D n D n 3 D n D n 2 ] ds (6) N n s n + +N s+n D n s n +D n s n + +D s+d (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 J n = ( ) n B n 2D n H n (8) 54 Vol. 6, Issue, pp
3 International Journal of Advances in Engineering & Technology, Mar. 23. Where H n and B n are determinants. H n 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 H n = D n D n 3 D n D n 2 D n D n 3 D n D n 2 [ ] The determinant B n is found by first calculating (9) N(s) N (-s) = b 2n-2s 2n b 2s 2 +b... () 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) = T(s) = 2k ds 2 +2s(+6k d )+2 s 3 +s 2 +s(k d +27)+9... () Thus the entire system characteristics equation is s 3 + s 2 + s(k d + 27) + 9 =. For the characteristics equation to make the design problem with stability, the Routh array is constructed as below: s 3 k d + 27 For stability, k d 25.. s 2 9 s (27 + k d ) 9 s 9 Now again T(s) = Y(s) R(s) = 2k ds 2 + 2s( + 6k d ) + 2 s 3 + s 2 + s(k d + 27) + 9 T E (s) = T(s) s = s2 +s( 2k d )+25+k d s 3 +s 2 +s(27+k d )+9. (2) N 2 =, N =-2k, N = 25+k d... (3) D 3=, D 2=, D = 27+k d, D = 9... (4) J n = B n 2D n H n... (5) H 3 = D 2 D D 3 D [ D 2 D ]... (6) 55 Vol. 6, Issue, pp
4 International Journal of Advances in Engineering & Technology, Mar. 23. N(s) = s 2 + s( 2k) k d... (7) N( s) = s 2 s( 2k) k d... (8) N(s)N( s) = s 4 s 2 (5 42k d + 4k d 2 ) + ( k d + k d 2 )... (9) J n = B n 2D n H n J 3 = 86k d 2 279k d k d Now, for maximum or minimum, dj 3 dk d =. Simplifying we get k d = 3.9, is the only positive value. It is tested that with this value of k = 3.9, d2 J 3 dk d 2 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 & 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: >> num=[6.8, 39.8, 2]; >> den=[,, 3.9, 9]; >> sys=tf(num,den) Transfer function: 6.8 s^ s S^3 + s^ s + 9 Finding Gain Margin &Phase Margin: >> margin(sys) Gm=Inf db ; W pc = 5.32 rad/sec. Pm= 3 ; W gc = 5.32 rad/sec. 56 Vol. 6, Issue, pp
5 International Journal of Advances in Engineering & Technology, Mar. 23. Bode Diagram Gm = Inf, Pm = 3 deg (at 5.32 rad/sec) 5 Magnitude (db) Phase (deg) Frequency (rad/sec) 2 Figure 2: Bode diagram and Phase Margin, Gain Margin. Finding Impulse response: >>impulse(sys) Impulse Response Amplitude Time (sec) Figure 3: Impulse response. Finding Root locus: >>rlocus(sys) Root Locus 2.5 Imaginary Axis Real Axis Figure 4: Root locus. 57 Vol. 6, Issue, pp
6 International Journal of Advances in Engineering & Technology, Mar. 23. Finding Step response: >>step(sys) Step Response.4.2 Amplitude Time 4 (sec) Sample data control: >> num=[6.8, 39.8, 2]; >> den=[,, 3.9, 9]; >>T=.; >>[numz,denz]=c2dm(num,den,t, zoh ); >>printsys(numz,denz, z ) num/den =.533 z^ z z^ z^ z >>dstep(numz,denz) Figure 5: Step response Step Response.4.2 Amplitude Time (sec) Finding Controllability and Observability: >>[A B C D]=ssdata(sys) Figure 6: Step response of discrete-time linear systems. 58 Vol. 6, Issue, pp
7 International Journal of Advances in Engineering & Technology, Mar A= 4. B= C= [ ] [.5 ] [ ] D = >>M=ctrb(A,B); >>rank_of_m = rank(m) rank_of_m = 3 >>system_order = length(a) System_order = 3 >>N = obsv(a,c); >>rank_of_n = rank(n) Rank_of_N = 3 VI. VII. RESULT Gain Margin= Inf db,very high. Phase Margin= 6 deg. 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 []. Stefani, Shahian, Savant, Hostetter, Design of feedback Control System 4 th ed, Oxford University Press, pp [2]. R.C. Dorf,Bishop, Modern Control System 8 th ed, Addison Wesley, 999 [3]. Achintya Das, Mrinmoy Chakraborty, Design and Analysis of an Artificial Control of Standing and Leg Articulation System, WCECS 28, pp [4]. Hans P. Geering Optimal Control with Engineering Applications, Springer [5]. Anderson,Moore, Linear Optimal Control, Prentice-hall, Inc. [6]. MATLAB 7. 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. 59 Vol. 6, Issue, pp
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