Internal Model based Cascade Controller implementation for a jacketed CSTR process

GRD Journals | Global Research and Development Journal for Engineering | International Conference on Innovations in Engineering and Technology (ICIET) - 2016 | July 2016

e-ISSN: 2455-5703

Internal Model based Cascade Controller Implementation for a Jacketed CSTR Process 1S.

Nagammai 2K. Dhanalakshmi 3S. Latha 1 Department of Electronics and Instrumentation Engineering 2,3Department of Electrical and Electronics Engineering 1 K.L.N. College of Engineering, Pottapalayam, Sivagangai 630 612 2,3Thiagarajar College of Engineering, Madurai 625015, India Abstract The continuous stirred tank reactor process (CSTR) process is uncertain and nonlinear in nature. It possess many challenging characteristics like multivariable interactions, unmeasured state variables, and time varying parameters. The three mode controller namely PID controllers are used in general and such simple feedback control will be very effective in compensating for changes in feed temperature and less effective in compensating for changes in jacket temperature. The response of the simple feedback control can be improved by measuring jacket temperature and taking control action before its effect has been felt by the reacting mixture. The single loop auto tuned PID controller is not best suitable for disturbance rejection. Cascade control scheme is mainly used to achieve fast rejection of disturbance before it propagates to the remaining parts of the plant. In this paper, Internal Model based cascade controller is designed to control the temperature of the reactor. Simulation shows that the Internal Model based cascade controller yields better results when compared with classical cascade control scheme.

Keyword- Jacketed CSTR, Cascade controller, Internal Model based cascade controller __________________________________________________________________________________________________

I. INTRODUCTION Many chemical industries have undergone significant changes in the past two decades. PID controllers are the most common controllers used in process industry. The most common tuning method is ZN method and the above mentioned approach frequently need manual tuning if the process has nonlinearities. For this reason a number of classical optimization techniques such as GA, PSO and EA methods are often used to find optimal values. In all of these, tuning is obtained for an operating point where the model can be considered linear. U.Sabura Banu and G.Uma proposed a gain scheduled genetic algorithm (GA)-based PID for a continuous stirred tank reactor (CSTR).

The paper is organized as follows. Section 2 gives details about the mathematical modelling of jacketed CSTR process.

Section 3 describes the conventional PID controller design. The conventional cascade controller implementation is given in section 4. The general approach of designing Internal model based PID controller is dealt in Section 5.The simulated results are given in section 6. Finally conclusion is given in section 7.

II. PROCESS DESCRIPTION A continuous stirred tank reactor (CSTR), is considered where in a first order exothermic reaction takes place at a temperature T with a cooling jacket. The chemical reaction is first order with Arrhenius temperature dependence. Perfect mixing is assumed.

In the jacketed CSTR the heat is added or removed by virtue of the temperature difference between a cooling jacket fluid and the reactor fluid. In the standard feedback control strategy the temperature of the reactor is measured and the jacket coolant flow rate is manipulated. If a disturbance in the jacket feed temperature occurs, which will affect the jacket temperature which in turn affects the reactor temperature.

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Internal Model based Cascade Controller Implementation for a Jacketed CSTR Process (GRDJE / CONFERENCE / ICIET - 2016 / 086)

Fig. 1: Schematic diagram of Jacketed CSTR process

In the schematic diagram shown in Fig.1. The temperature of the reactor is measured and compared with the desired temperature. The output of the reactor temperature controller acts as a set point to the jacket temperature controller. The jacket temperature controller manipulates the jacket cooling water flow rate. Here two measurements are made but only one manipulated variable is ultimately adjusted. In this cascade control strategy, the reactor temperature controller is the primary controller, whereas the jacket temperature controller is the secondary controller. This is effective because the jacket temperature dynamics are normally significantly faster than the reactor temperature dynamics. An inner loop disturbance, such as jacket feed temperature will be felt by the jacket temperature before it has a significant effect on the reactor temperature. The secondary controller adjusts the manipulated variable before a substantial effect on the primary process output has occurred.

The nonlinear equations that model the CSTR behaviour are given as [1] The component material balance on the reactant gives

C A = f1  C A , T , T j  = •

-E 

  F  C A0 - C A  -  e  RT C A V

The energy balance in the reactor system is T = f 2 CA ,T ,Tj   •

 -H F T0 - T  +  V  ρr C pr

-E    Au RT  e  C A T - T j  VρC p 

The energy balance in the jacket is T j = f3  CA ,T ,T j   •

Fj Vj

T

jf

- Tj  +

Au T - T j  V j ρ j C pj

(1)

The four nonlinear differential equations expressed in equations 1.a to 1.d cannot be solved analytically. The approximate model is derived about the steady-state operating point of the reactor given in Table.2. The variables C A , T & T j are considered as state variables and Tjf & T0 are the disturbance variables. The manipulated variable is the cooling water flow rate

 F  .The controlled variable is the reactor temperature  T  .The linearized state space model of the plant is obtained using local j

point linearization.

The state equation and output equation of the CSTR process is given in equation [2].

•  C A   a11 a12  •  = a a  T   21 22  •   a31 a32  T j 

a13  C A   a14   a23   T  +  a24 a33  T j   a34

a15  F  a25    Fj a35   

 y1  1 0 0  C A   y  = 0 1 0   T  (2)  2     y3  0 0 1  T j  Substituting the numerical values given in Table 1 & 2, the constants are evaluated. The state space model of the system thus obtained is given in equation [3].

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Internal Model based Cascade Controller Implementation for a Jacketed CSTR Process (GRDJE / CONFERENCE / ICIET - 2016 / 086)

•  0  C A  -1.67 -0.255 0  CA   2.92 F   •  =  23.4 -28.8 20.9   T  + -28.6 0       T   Fj  •   0 -200.3 -217  Tj   0 -415.3    T j 

(3)

The various transfer functions of the plant are, CA (s) 2.918 S 2+724.2 S +32006.1 = 3 F(s) S +247.4S 2 +10842.7S +18706.3 CA (s) 2213.3 = 3 2 Fj (s) S +247.4S +10842.7S +18706.3

T(s) -28.6S 2 - 6182.5 S +4450.4 = 3 F(s) S +247.4S 2 +10842.7S +18706.3 -8679.7  S +1.7  T(s) = 3 Fj (s) S + 247.4S 2 +10842.7S +18706.3 Tj (s) F(s) T j (s)

=

-5720S - 4103.8 S + 247.4S 2 +10842.7S +18706.3 3

-415.3S 2 - 12654.2S - 22452.4 Fj (s) S 3 + 247.4S 2 +10842.7S +18706.3 The transfer function of the primary process (outer loop) is, 0.65 0.587S +1 T(s) g1 = = T j (s) 0.018S 2 +0.555S +1 =

(4)

The transfer function of the secondary process (inner loop) is, -415.3  S 2 + 30.4 S + 54.8  T j (s) g2 = = 3 Fj (s) S + 247.4S 2 +10842.7S +18706.3 variable V Vj

Description Reactor volume (m3) Jacket volume (m3) Arrhenius exponential factor (hr -1) Activation energy (KJ/K mol) Heat transmission coefficient (K J/hr m2 K ) Heat transmission surface( m2) Perfect gas constant (K J/K mol K)

Value 1.36 0.085 7.08 x 1010 69815

Reaction heat (KJ/K mol)

69815

Cp

Thermal capacity (K J/Kg K)

3.13

C pj 

Water Thermal capacity (K J/Kg K)

4.18

Product density(Kg/ m3)

800



E U A R  H 

j variable

CAos

(5)

3065 23.22 8.314

3

Water density(Kg/ m ) 1000 Table 1: CSTR variables and parameter values Description value Steady state Feed concentration 8 (K mol /m3)

T0s

Steady state Feed temperature (K)

294.7

Ts T jfs

Steady state Reactor temperature (K)

333.6

Steady state cooling water input temperature (K)

294.7

T js

Steady state jacket temperature (K)

331.4

CAs

Steady state Reactor concentration (K mol /m3)

4.031

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Internal Model based Cascade Controller Implementation for a Jacketed CSTR Process (GRDJE / CONFERENCE / ICIET - 2016 / 086)

Fs

Steady state Feed flow rate (m3/hr)

1.13

F js

Steady state cooling water flow rate (m3/hr)

1.41

Table 2: Reactor steady state parameter values

III. CONVENTIONAL PID CONTROLLER DESIGN It is clear that, the reactor temperature T will respond much faster to changes in Ti than to changes in T j Therefore, the simple feedback control will be very effective in compensating for changes in Ti and less effective in compensating for changes in T j The process transfer function relating reactor temperature T and the coolant flow rate Fi is given in equation [7]

T(s) -8679.7S - 14755.5 = 3 Fj (s) S +247.4S 2 +10842.7S +18706.3

(6)

The auto tuned PID controller parameters for the desired open loop bandwidth is 0.2 rad/s is obtained as, K Kc  0.000125, i  0.0005 hr; Ki  c  0.25

i

IV. CONVENTIONAL CASCADE CONTROLLER DESIGN The inner loop (secondary) controller GC2 is first designed and then the outer loop (primary) controller GC1 is designed with the inner loop closed. In this example, the inner loop bandwidth is selected as 2 rad/s, which is ten times higher than the desired outer loop bandwidth. In order to have an effective cascade control system, it is essential that the inner loop responds much faster than the outer loop. The auto tuned PID controller parameters of secondary controller & primary controller for the desired open loop bandwidth of 2 rad/s is obtained using pseudo Code. Secondary controller parameters are

Kc  0.000084, i  0.00005 hr; Ki 

Kc

i

 1.68

Primary controller parameters are

Kc  0.000154, i  0.0005 hr ; Ki 

Kc  0.31 i

V. INTERNAL MODEL BASED CASCADE CONTROLLER DESIGN The response of the simple feedback control can be improved to changes in the coolant temperature by measuring T j and taking control action before its effect has been felt by the reacting mixture. Thus, if T j goes up, the flow rate of the coolant is increased to remove the same amount of heat and decreased when T j decreases.

Thus two control loops using two different measurements, T and T j with single manipulated variable, Fj is used and which is shown in Fig.2. The loop that measures T (controlled variable) is the primary or master control loop and uses a set point supplied by the designer. Whereas the loop that measures T j , uses the output of the primary controller as its set point and is called the secondary or slave loop.

The control action for the inner loop or secondary controller is proportional integral (PI) with the gain set to a high value. But the action of the outer loop or primary controller is internal model (IMC) based PID controller.

Fig. 2: Block diagram of IMC Based Cascade Control Scheme

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536

Internal Model based Cascade Controller Implementation for a Jacketed CSTR Process (GRDJE / CONFERENCE / ICIET - 2016 / 086)

The Routh’s Stability analysis is applied to the secondary process given in equation (5) to obtain the secondary controller parameters. The value of ultimate gain

k cu

and frequency of sustained oscillation

c 0 is obtained as,

kcu  0.885, c 0  148.4 rad / sec Pu 

2

c 0



2  0.042sec 148.4

The ZN-PI tuning parameters are,

Kc  0.45kcu  0.4, Ki 

Kc

i

 11.37

The primary controller is designed based on Internal Model structure and the design procedure is given below:  The process model is factorized into good and bad elements. ~

~

~

g p  g p  (s) g p  (s) 

The ideal internal model controller is the inverse of the good portion of the process model.

~

~

1

q(s)  g p  (s) 

A filter factor is added to make the controller realizable.. ~

~

1

q( s)  q( s) f ( s)  g p  ( s) f ( s)



  1 f (s)   n   ( S  1)  and ‘n’ is In order to track step set point changes, the filter transfer function usually has the form

chosen to make the controller proper.

The filter tuning parameter is adjusted to vary the speed of response of the closed loop system. A small value of  results in fast closed loop system response whereas large value of  is “large”, the closed-loop system is more robust.

Consider the plant transfer function of the form,

g p (s) 

k p   s  1

( p1s  1)( p 2 s  1)

Where  is a positive real number, indicating a positive zero (yielding inverse response) in the process transfer function. For a dynamic open loop controller,

q( s)  g p1 ( s) f ( s) 

( p1s  1)( p 2 s  1) k p   s  1

f ( s)

And if we let f ( S )  1/   S  1 , we obtain

 1 q( s )    kp 

 ( p1s  1)( p 2 s  1)   ( s  1)(  s  1)

Now when the model inverse is used for control system design, the zeros of the process model become the poles of the controller. This creates an unstable controller and the possibility of unbounded, manipulated variable action. In order to make the controller proper a filter factor is added. The IMC filter parameter is tuned to improve robustness.

The transfer function of the primary process (outer loop) is,

g1 =

0.65 0.587S +1 T(s) = Tj (s) 0.018S 2 +0.555S +1

The internal model controller transfer function for the primary loop in case of set point tracking is,

q(s)=

1.54 0.018S 2 +0.555S +1 (λS +1)(0.587S +1)

The filter parameter

 is chosen as 0.04.

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Internal Model based Cascade Controller Implementation for a Jacketed CSTR Process (GRDJE / CONFERENCE / ICIET - 2016 / 086)

VI. SIMULATION RESULTS In order to analyse the performances of the proposed controllers, the system is simulated using MATLAB/SIMULINK. The controller parameter values for simulation are given Table 3.

The open loop response of the designed system is shown in Fig. 3.and which indicates that, the open loop system is stable but set point tracking is not obtainable.

Type of Controller Conventional PI IMC based Cascade controller Conventional Cascade controller

Kc Ki -0.000125 -0.0005 Primary controller IMC based PID Kc Ki Kd 28.5 51.3 0.92 Primary PI Controller Kc

Ki

0.000154

0.31

Kd Secondary PI Controller Kc Ki -0.4 -11.4 Secondary PI Controller Kc Ki -0.000084

-1.68

Table 3: Controller Parameter Settings

Fig. 3: Open loop step response of jacketed CSTR

The performance of conventional PI controller in comparison with cascade controller and Internal Model based cascade controller is shown in Fig. 4 for step change in cooling water flow rate.

Fig. 4: Servo response for change in cooling water flow rate

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Internal Model based Cascade Controller Implementation for a Jacketed CSTR Process (GRDJE / CONFERENCE / ICIET - 2016 / 086)

The set point variation has been introduced to assess the tracking capability of the proposed controllers for nominal and shifted operating point. The servo tracking response is shown in Fig 5.

Fig. 5: Servo tracking response for set point variations

Further, it is evident that the coolant flow rate (MV) variation is found to be smooth in all controllers which is shown in Fig.6.The variation in cooling water flow rate In order to demonstrate the disturbance rejection capability of various control schemes, simulation studies have been carried by changing the feed temperature as disturbance variable and which is shown in Fig.7.

Fig. 6: Variation of manipulated variable (MV)

Fig. 7: Regulatory response for change in feed temperature

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Internal Model based Cascade Controller Implementation for a Jacketed CSTR Process (GRDJE / CONFERENCE / ICIET - 2016 / 086)

VII.

CONCLUSION

The effectiveness of internal model based cascade controller algorithm is demonstrated for a modelled jacketed CSTR process. It has been shown that, the Internal Model based cascade controller tracks the set point with minimum error values at a lesser settling time. The IM based Cascade controller offers better transient response specifications than single loop and cascade controllers. The performance summary is given in Table 4.

Conventional IMC based Cascade controller Cascade controller Peak time tp 50 50 0.95 Rise time tr 11.14 9.85 0.248 Settling time ts 23.4 21 0.512 %Mp 0 0.03 0.12 ISE(servo) 282000 6370 4858 IAE(servo) 1692 251 37 Table 4: Comparison of Time domain Specifications and Performance indices Parameter

Classical PI Controller

REFERENCES [1] R. Vinodha, S. A. Lincoln, and J. Prakash, “Design and implementation of simple adaptive control schemes on simulated model of CSTR process,” International Journal of Modelling, Identification and Control, vol. 14, no. 3, pp. 159–169, 2011.

[2] Bequette,B.W., Process control: modelling, design, and simulation ,Prentice Hall, New York(1998) [3] Sigurd Skogestad, and Ian Postletwaite, “Multivariable Feedback control Analysis and design” John Wiley & sons,Ltd , pp.

344 – 347.

[4] George Stephanopoulos “Chemical Process Control”-.An Introduction to theory and Practice PHI 2004.Process dynamics and control By Dale E Seborg, Mellichamp and Edgar.

[5] U.Sabura Banu and G.Uma “Fuzzy Gain Scheduled CSTR With GA-Based PID” Taylor & Francis Volume 195, Issue 10, 2008.P.Albertos and A.Sala., “Multivariable control systems” Springer International Edition,pp.19-32 [6] H. Man and C. Shao, “Nonlinear predictive adaptive controller for CSTR process,” Journal of Computational Information Systems, vol. 8, no. 22, pp. 9473–9479, 2012.

[7] J. Prakash and R. Senthil, “Design of observer based nonlinear model predictive controller for a continuous stirred tank reactor,Journal of Process Control, vol. 18, no. 5, pp. 504–514, 2008.

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