Decoupling the variables of that system will improve the control of that ... To avoid loop interactions, MIMO systems can be decoupled into separate loops known ...
ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014
Design and Decoupling of Control System for a Continuous Stirred Tank Reactor (CSTR) (1-2)
Georgeous, N.B*1 and Gasmalseed, G.A, Abdalla, B.K University of Science and Technology, Sudan Department of Chemical Engineering.
Abstract: To apply the methods of stability analysis and tuning, it is necessary first to develop a control strategy for possible noninteracting and interacting loops. Then the transfer functions were identified by mathematical modeling and the overall gains were cited from the literature. Pairing of the loops were undertaken and the interact in loops were coupled according to the RGA (Relative Gain Array), thus the loops with minimal interaction were selected and put in block diagrams. The characteristic equations were obtained for both closed and open loops. An exothermic reaction in a continuous stirred tank reactor (CSTR) was selected as a case study. The loops were subjected to tuning, stability and offset investigation. Summary of the adjustable parameters, offsets and response behavior were tabulated for comparison between the methods. It is clear that all the methods are in agreement, but the Bode criteria showed a superior consistency over other methods. It is recommended that the method of Bode has to be preferentially selected for tuning and stability analysis. Index Terms: Methods of tuning, Stability, Transfer function identification, Offset investigation.
I. INTRODUCTION A control system is composed of interacting loops and that the number of feasible alternative configurations needed to be configured are very large. It must be recognized that for a process with n controlled variables and n manipulated variables there are n different ways to form control the loops [1]. The question is which one to selected? The answer is to consider the interaction between the loops for all n loops and then the RGA is applied to select a loop when the interaction is minimal. The RGA provides such a methodology by pairing the input and output that give minimum interaction when together coupled, RGA was first proposed by Bristol and today it is a very popular tool for selection of control loops giving minimal interaction [1].The methods of pairings and the RGA were applied to an exothermic reaction in a jacketed CSTR, the process is a 2 2 controlled and manipulated variables. The method of stability and tuning were applied using Routh-Hurwitz, direct substitution, root-Locus, and Bode and Nyquist criteria.
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II. OBJECTIVES To select the loops with minimal interaction in CSTR. To study the dynamics of a CSTR. To investigate the methods of tuning and stability analysis. To compare the accuracy of these methods with respect to stability, adjustable parameters and offset.
III. LITERATURE REVIEW Multiple input, multiple output (MIMO) systems describe processes with more than one input and more than one output which require multiple control loops. Examples of MIMO systems include heat exchangers, chemical reactors, and distillation columns. These systems can be complicated through loop interactions that result in variables with unexpected effects. Decoupling the variables of that system will improve the control of that process [2]. An example of a MIMO system is a jacketed CSTR in which the formation of the product is dependent upon the reactor temperature and feed flow rate. The process is controlled by two loops, a composition control loop and a temperature control loop. Changes to the feed rate are used to control the product composition and changes to the reactor temperature are made by increasing or decreasing the temperature of the jacket. However, changes made to the feed would change the reaction mass, and hence the temperature, and changes made to temperature would change the reaction rate, and hence influence the composition. This is an example of loop interactions. Loop interactions need to be avoided because changes in one loop might cause destabilizing changes in another loop [2]. To avoid loop interactions, MIMO systems can be decoupled into separate loops known as single
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ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 input, single output (SISO) systems. Decoupling may be done using several different techniques, including restructuring the pairing of variables, minimizing interactions by detuning conflicting control loops, opening loops and putting them in manual control, and using linear combinations of manipulated and/or controlled variables. If the system can’t be decoupled, then other methods such as neural networks or model predictive control should be used to characterize the system [2,3,4]. There are two ways to see if a system can be decoupled. One way is with mathematical models and the other way is a more intuitive educated guessing method. Mathematical methods for simplifying MIMO control schemes include the relative gain array (RGA)[2]. The RGA provides a quantitative approach to the analysis of the interactions between the controls and the output, and thus provides a method of pairing manipulated and controlled variables to generate a control scheme. The RGA is a normalized form of the gain matrix that describes the impact of each control variable on the output, relative to each control variable's impact on other variables. The process interaction of open-loop and closed-loop control systems are measured for all possible input-output variable pairings [5]. A ratio of this open-loop gain to this closed-loop gain is determined and the results are displayed in a matrix. The array will be a matrix with one column for each input variable and one row for each output variable in the MIMO system[5]. The best pairing is discovered by taking the maximum value of RGA Matrix for each row. A digital computer can be used to control simultaneously several outputs, the control program is composed of several subprograms, each one used to control a different loop. Furthermore, the control program should be able to coordinate the execution of the various subprograms so that each loop and all together function properly [6]. Although a controller may be stable when implemented as an analog controller, it could be unstable when implemented as a digital controller due to a large sampling interval. During sampling the aliasing modifies the cutoff parameters. Thus the sample rate characterizes the transient response and stability of the compensated system, and must update the values at the controller input often enough so as to not cause instability. When substituting the frequency into the z operator, regular stability criteria still apply to discrete control systems. Nyquist criteria apply to z-domain transfer functions as well as being general for complex valued functions. Bode stability criteria apply similarly. Jury criterion determines the discrete system stability about its characteristic polynomial [6]. The system stability can be tested by considering its response to a finite input signal. This means the analysis of system dynamics in the actual time domain which is usually cumbersome and time consuming. Several methods have been developed to deduce the system stability from its characteristic equation. They are short-cut methods for providing information without finding out the actual response of the system. They give information from the s-domain without going back to the actual time-domain. All these methods are based on the criterion that a sufficient condition for stability of a control loop is to have a characteristic equation with only negative real roots and or complex roots with negative real parts. The short-cut methods for assessment of the stability of a system include the direct method, the Routh–Hurwitz stability criterion and graphical methods of investigating the behavior of the roots of the characteristic equation, i.e. Root Locus method and Nyquist stability criterion. Bode plots are common graphical method. It depends only upon the open loop transfer function (OLTF). The OLTF relates the feedback or measured variable to the set point, when the feedback loop is disconnected from the comparator when the loop is broken or opened. IV. RESULTS AND DISCUSSION Interaction, coupling and control of CSTR Consider a process with two controlled outputs and two manipulated inputs (Figure 1.). The transfer functions are: 1.GC1(s)= KC1,
….(1)
2.GC2(s)= KC2,
….(2)
3.Gf1(s)=
,
….(3)
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ISSN: 2319-5967 ISO 9001:2008 Certified 4.Gf2(s)=
International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 , ….(4)
5.Gm1(s)=
….(5)
,
6.Gm2(s)= 7. H11(s)= 8.H12(s)=
….(6)
,
….(7)
,
….(8)
,
9.H21(s)=
….(9)
,
10.H22(s)=
….(10)
. Loop 1
Gm1(s)
-
Gf1(s)
GC1(s)
H11
,sp +
+
+
H21 H12
,sp
+
-
GC2(s)
Gf2(s)
+
+H22
Gm2(s) Loop 2 Fig 1. Block diagram of CSTR with two controlled outputs and two manipulative variables.
The Routh–Hurwitz stability criterion is used for determination the stability of a system, the characteristic equation of closed loop is used. Also Bode plots, root-Locus method and Nyquist stability criterion are used for determination the stability of a system, the OLTF is used. The amplitude ratio (AR) is used to get the ultimate gain(KU) and Ziegler-Nichols tuning is brought to edge of stability under proportional control and it is used for tuning the adjustable parameters, which are substituted in overall gains for study the response. Loop 1 is closed and loop 2 is open, the transfer function is: ….(11)
….(12)
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ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014
….(13)
The characteristic equation of closed loop is: 0.15 s4 +1.45s3 + 3.85 s2 +3.6 s + (1+ KC1) = 0
….(14)
Putting the characteristic equation in Routh Array: Table 1: Routh-Hurwitz Array
S4 S3 S2 S1 S0
0.15 1.45 3.4776 (3.183-0.417 KC1) (1+ KC1)
3.85 3.6 (1+ KC1) 0 0
(1+ KC1) 0 0 0 0
0 0 0 0 0
For the system to be critically stable: (3.183-0.417 KC1) = 0, KC1= KU1 = 7.633, The system become stable for all values of KC1 =
KU1 7.633. The auxiliary equation is1.45s3 +3.6 s = 0, s= iω; ωco = 1.5757 rad / sec; Pu =
= 3.99 sec.
Tuning: Applying Ziegler- Nichols method ( Z-N method): Table 2: Z-N adjustable parameters for loop 1 Controller mode P PI PID
τI ( sec) 3.325 1.995
KC1 3.8165 3.4349 4.5798
τD ( sec) 0.4988
Substituting the value of KC1 = 3.8165 of proportional controller (P- action) in characteristic equation,0.15 s4 +1.45s3 + 3.85 s2 +3.6 s + 4.8165= 0 and in first column of Routh array. All elements of the first column were positive and there is no change of sign, therefore the system is stable. Response: Substituting the value of KC1 in the transfer function of loop1 (G(s)) and introducing an impulse forcing function: Impulse Response 1
System: sys Peak amplitude: 0.849 A t time (seconds): 1.12
Amplitude
0.5
System: sys Settling time (seconds): 16.9 0
-0.5
0
5
10
15
20
Time (seconds)
Fig 2. Impulse response when loop1 is closed.
12
25
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ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 Bode analysis and tuning: Open loop transfer function (OLTF) of loop 1is: ….(15)
Bode Diagram 0
Magnitude (dB)
-50 -100 -150 -200 -250 0 System: sys Frequency (rad/s): 1.58 Phase (deg): -180
Phase (deg)
-90 -180 -270 -360 -2
-1
10
0
10
1
10
2
10
Frequency
10
3
10
(rad/s)
Fig 3. Bode diagram when loop1 is open
From Bode diagram, ωco =1.58 rad /sec and the amplitude ratio (AR)= 1, Pu =
= 3.98 sec, the amplitude
ratio (AR) is used to get KC1. ….(16)
1=
Substituting the value of ωco = 1.58 rad / sec in the equation to get KC1,KC1= KU1= 7.9962 Response: For Kc1 = 3.9981
Impulse Response 1
System: sys Peak amplitude: 0.886 At time (seconds): 1.11
0.8
0.6
Amplitude
0.4
0.2 System: sys Settling time (seconds): 18.9 0
-0.2
-0.4
-0.6
0
5
10
15
20
25
Time (seconds)
Fig 4.Impulse response when loop1 is closed.
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ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 Root Locus: Root Locus 15
Imaginary Axis (seconds-1)
10
5
0
System: sys Gain: 7.52 Pole: -0.00429 + 1.57i Damping: 0.00273 Overshoot (%): 99.1 Frequency (rad/s): 1.57
-5
-10
-15 -20
-15
-10
-5
0
5
10
15
Real A xis (seconds -1)
Fig 5.Root-Locus diagram when loop1 is open
From root-Locus diagram, ωco = 1.57 rad / sec and the amplitude ratio (AR) = 1, KC1= KU1= 7.8939 Response: For Kc1 = 3.9469 Impulse Response 1
0.8
System: sys Peak amplitude: 0.875 A t time (seconds): 1.11
0.6
Amplitude
0.4
0.2 System: sys Settling time (seconds): 18.8 0
-0.2
-0.4
-0.6
0
5
10
15
20
25
30
Time (seconds)
Fig 6. Impulse response when loop1 is closed.
Nyquist stability criterion: Nyquist Diagram 0.8
0.6
0.4 System: sys Real: -0.132 Imag: 0.00326 Frequency (rad/s): -1.58
Imaginary Axis
0.2
0
-0.2
-0.4
-0.6
-0.8 -1
-0.5
0
0.5
1
1.5
Real A xis
Fig 7. Nyquist diagram when loop1 is open
From Nyquist diagram, ωco = 1.58 rad / sec.The characteristic equation of closed loop is: 0.15 s4 +1.45s3 + 3.85 s2 +3.6 s + (1+ KC1) = 0 0.15 ω 4 -1.45 iω 3 - 3.85 ω 2 +3.6 iω + (1+ KC1) = 0
….(17)
14
….(14)
ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 0.15 ω 4 - 3.85 ω 2 + (1+ KC1) = 0 ….(18) Substituting the value of ωco = 1.58 rad / sec to get KC1, KC1= KU1 = 7.6763 Response: For Kc1 = 3.8382 Impulse Response 1
System: sys Peak amplitude: 0.854 At time (seconds): 1.12
Amplitude
0.5
System: sys Settling time (seconds): 16.9 0
-0.5
0
5
10
15
20
25
30
Time (seconds)
Fig 8. Impulse response when loop1 is closed Table 3: Comparison between the methods for continuous stirred tank reactor ( CSTR ) of loop1 Method
KU
PU (sec)
KC
Routh7.633 3.99 3.8165 -0.2076 Durwitz Bode 7.9962 3.98 3.9981 -0.2001 Root-Locus 7.8939 4.00 3.9469 -0.2021 Nyquist 7.6763 3.98 3.8382 -0.2067 Table 4: Comparison between the methods for continuous stirred tank reactor ( CSTR ) of loop1 Method
Overshoot
Routh-Durwitz Bode Root-Locus Nyquist
0.849 0.886 0.875 0.854
Rise time(sec) 1.12 1.11 1.11 1.12
Settling time(sec) 16.9 18.9 18.8 16.9
Loop 2 is closed and loop 1 is open, the transfer function is: ….(19)
The characteristic equation is: 0.4 S5 + 2.4 s4 +5.7s3 + 7.1 s2 +4.4 s + (1+ 2KC2) = 0
….(20)
The preceding procedure is applied. Table 5: Routh-Hurwitz Array
S5 S4 S3 S2
0.4 2.4 4.5167 (4.85+0.18 KC2)
5.7 7.1 (4.23-0.33KC2) (1+ 2KC2)
15
4.4 (1+ 2KC2) 0 0
0 0 0 0
ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 S1
(
S0
)
(1+2 KC2)
KU2 = 1.6029, ωco= 0.906 rad / sec, Pu =
0
0
0
0
0
0
= 6.94 sec, KC2 = 0.8015.
Response: For Kc2 = 0.8015 Impulse Response 0.5
0.4 System: sys Peak amplitude: 0.37 A t time (seconds): 1.75
Amplitude
0.3
0.2
0.1 System: sys Settling time (seconds): 22.9 0
-0.1
-0.2
0
5
10
15
20
25
30
35
Time (seconds)
Fig 9. Impulse response when loop2 is closed
Bode analysis and tuning: Open loop transfer function (OLTF) of loop 2is:
….(21)
Bode Diagram 50
Magnitude (dB)
0 -50 -100 -150 -200 0 System: sys Frequency (rad/s): 0.906 Phase (deg): -180
Phase (deg)
-90 -180 -270 -360 -450 -2
10
-1
10
0
10
1
10
Frequency (rad/s)
Fig 10. Bode diagram when loop2 is open
16
2
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ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 From Bode diagram, ωco = 0.906 rad / sec, Pu = = 6.9379 sec, and the amplitude ratio (AR) = ….(22)
KC2= KU2= 1.6054. Response : For Kc2 = 0.8027 Impulse Response 0.5
0.4 System: sys Peak amplitude: 0.371 At time (seconds): 1.75
Amplitude
0.3
0.2
0.1 System: sys Settling time (seconds): 23 0
-0.1
-0.2
0
5
10
15
20
25
30
35
Time (seconds)
Fig 11. Impulse response when loop2 is closed
Root Locus: Root Locus 6
System: sys Gain: 1.64 Pole: 0.00626 + 0.909i Damping: -0.00689 Overshoot (%): 102 Frequency (rad/s): 0.909
Imaginary Axis (seconds-1)
4
2
0
-2
-4
-6 -8
-6
-4
-2
0
2
4
Real A xis (seconds -1)
Fig 12. Root-Locus diagram when loop2 is open
From root-Locus diagram, ωco = 0.909 rad / sec and the amplitude ratio (AR) = 1, KC2= KU2= 1.6141. Response: For Kc2 = 0.8071 Impulse Response 0.5
0.4 System: sys Peak amplitude: 0.373 At time (seconds): 1.75
Amplitude
0.3
0.2
0.1 System: sys Settling time (seconds): 23 0
-0.1
-0.2
0
5
10
15
20
25
30
Time (seconds)
Fig 13. Impulse response when loop2 is closed
17
35
1=
ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 Nyquist stability criterion: Nyquist Diagram 2
1.5
1 System: sys Real: -0.62 Imag: 0.00572 Frequency (rad/s): -0.906
Imaginary Axis
0.5
0
-0.5
-1
-1.5
-2 -1
-0.5
0
0.5
1
1.5
2
2.5
Real Axis
Fig 14. Nyquist diagram when loop2 is open
From Nyquist diagram, ωco = 0.906rad / sec, The characteristic equation is: 0.4 S5 + 2.4 s4 +5.7s3 + 7.1 s2 +4.4 s + (1+2KC2) = 0, 2.4 ω4 - 7.1 ω 2 + (1+2KC2) = 0, Substituting the value of ωco = 0.906 rad / sec to get KC2.KC2= KU2 = 1.6054. Response : For Kc2 = 0.8027 Impulse Response 0.5
0.4 System: sys Peak amplitude: 0.371 A t time (seconds): 1.75
Amplitude
0.3
0.2
0.1 System: sys Settling time (seconds): 23 0
-0.1
-0.2
0
5
10
15
20
25
30
35
Time (seconds)
Fig 15. Impulse response when loop2 is closed Table 4. Comparison between the methods for continuous stirred tank reactor ( CSTR ) of loop2 Method
KU
PU (sec)
KC
Routh-Durwitz Bode Root-Locus Nyquist
1.6029 1.6054 1.6141 1.6054
6.94 6.94 6.91 6.94
0.8015 0.8027 0.8071 0.8027
-0.3842 -0.3838 -0.3825 -0.3838
Table 5. Comparison between the methods for continuous stirred tank reactor ( CSTR ) of loop2 Method
Overshoot
Rise time(sec)
Settling time(sec)
Routh-Durwitz
0.37
1.75
22.9
Bode
0.371
1.75
23
Root-Locus
0.373
1.75
23
Nyquist
0.371
1.75
23
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ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 Relative Gain Array (RGA): The ratio between the two loop gain is λ11, λ11 =
=
λ11 =
….(23)
….(24)
, λ11 = - 0.667, λ12 = 1+ 0.667= 1.667
s
.
=
λ11 = - 0.667< 0, , then m2 cause a strong effect on y1 and in the opposite direction from that caused by m1. In this interaction effect is very dangerous and must be avoid pairing m1 with y2. Nomenclature Symbols KU Ultimate gain PU Ultimate period(sec) KC Controller gain Offset Indices C Refer to controller f Refer to valve m Refer to measuring element 1 Refer to loop1
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ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 3, Issue 6, November 2014 2 Refer to loop2 V. CONCLUSION It is clear that all the methods are in agreement, but the Bode criterion has to be preferentially selected for tuning and stability analysis. For CSTR, λ11 = - 0.667, then m2 cause a strong effect on y1 and in the opposite direction from that caused by m1. In this interaction effect is very dangerous. ACKNOWLEDGMENT The authors wish to thank the Collage of Higher Studies and Research of Karary University for their help and for giving us opportunity for carrying out research in partial fulfillment for Ph.D in Chemical Engineering. REFERENCES [1] Stephanopoulos, G. (2005), Chemical Process Control, Prentice-Hall, India. [2] Tham, M.T. (1999). "Multivariable Control: An Introduction to Decoupling Control". Department of Chemical and Process Engineering, University of Newcastle upon Tyne. [3] McMillan, Gregory K. (1983) Tuning and Control Loop Performance. Instrument Society of America. ISBN 0-87664694-1. [4] Lee, Jay H., Choi, Jin Hoon, and Lee, Kwang Soon. (1997). "3.2 Interaction and I/O Pairing". Chemical Engineering Research Information Center. [5] Berber, Ridvan. (1994).Methods of Model Based Process Control, Kluwer Academic Publishers. [6] FRANKLIN, G.F.; POWELL, J.D. (1981). Digital control of dynamical systems. USA, California: Addison-Wesley. ISBN 0-201-82054-4.
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