Adaptive Control Design For A MIMO Chemical Reactor

Adaptive Control Design For A MIMO Chemical Reactor

S. Sundari, Alamelu Nachiappan

Abstract

This paper deals with the operation, mathematical modeling and controller design for the jacketed continuous stirred tank reactor. Controller is designed to control the reactant mixture temperature within the reactor. The scheme is simulated using Matlab and the performance of adaptive controller is compared with the conventional PID controller. The simulation results show that the adaptive controller is best suited for CSTR.

Keywords-MIT rule, CSTR, Mathematical modelling, adaptation law, model reference adaptive control, adaptation gain.

Introduction

There are several types of stirred reactors used in chemical or biochemical industry. Continuous Stirred Tank Reactors (CSTR) are common used because of their technological paramers. This paper describes the existing techniques of the Continuous Stirred Tank Reactor (CSTR) with the mathematical model of the system, followed by the existing techniques for the implementation of Model reference adaptive controller (MRAC) using MIT rule for the CSTR temperature process .

The Continuous Stirred-Tank Reactor used in this work represents typical nonlinear plant described mathematically by the set of two nonlinear ordinary differential equations (ODE) and has two stable and one unstable steady-state which could lead to very unstable or unoptimal output responses with the use of conventional control methods[1]. One way to overcome this inconvenience is the use of the adaptive control (Åström and Wittenmark 1989) [2],which adopts parameters of the controller to the actual state of the system via recursive identification of the External Linear Model (ELM) as a linear representation of the originally nonlinear system .The results of the adaptive control on this concrete mathematical model can be found for example in [3]. The temperature control of reactor has quite remarkably improved by using the Model Predictive Controller [4]. RathikaraniDuraisamy proposed an adaptive optimization scheme for controlling air flow process with satisfactory transient performance[5]. The automatic tuning of PI controller has been

investigated using MRAC concept and AMIT rule. Rahul Upadhyay [6] proposed an analysis of CSTR temperature control with adaptive Controller using Lyapunov rule and PID Controller. Indirect adaptive control based pole placement and adaptive general predict control (GPC) was analyzed in [7]. In [8], a class of nonlinear PID controllers was presented using a nonlinear generalized predictive control

approach to a set nonlinear systems. Rajesh singla proposed an application of adaptive control with various types of command inputs in a process plant (CSTR)[9]. The multiple model and neural based adaptive multi- loop PID controller is proposed for a CSTR process[10]. The two controllers designed use the same control law and differ only in the calculation of controller parameters. The NN Adaptive-PID has good set point tracking and disturbance rejection is better than MM Adaptive- PID. An on-line adaptive control for non-linear processes under influence of external disturbance is proposed .An intelligent design of PID controller for a continuous stirred tank reactor was proposed[12] which presents various control methods to control the CSTR temperature control process such as Fuzzy logic PID controller.This paper includes the following parts: section 2 provides an overview of model reference adaptive control. In section 3 mathematical modelling of a chemical reactor is developed. Section 4 and section 5 describes in detail the design and simulation of both non-adaptive and adaptive controller and comparison of performance of adaptive controller and non-adaptive (conventional) controller.Section 6 represents the performance of expanded adaptive controller.

1. Model Reference Adaptive Control (MRAC)

   Model reference adaptive controller (MRAC) is a controller used to force the actual process to behave like a model process. MRAC systems adapt the parameters of a normal control system to achieve this match between model and process.

   Figure 1 : Basic model reference adaptive control structure The standard implementation of MRAC based systems is shown in figure 1. The reference model defines the desired performance characteristics of the process being controlled. The adaptation law uses the error between the process and the model output, the process output and input signal to vary the parameters of the control system. These parameters are varied so as to minimize the error between the

   The Arrhenius expression is used for the rate of reaction rA=K0*e(-E/RT) C

   * A

   * A

   Where K0 is the frequency factor, E is the activation energy, R is the ideal gas constant, and T is the reactor temperature.The reactor energy balance assume constant volume, heat capacity Cp and density and neglect the changes in the kinetic and potential energy is

   V Cp d/dt (T) =FCp (Tf T) + (H) V*rA -UA (T- Tj) (3)

   Where H is the heat of reaction, U is the heat transfer co- efficient, A is the heat transfer area, Tf is the reactant temperature, and Tj is the coolant temperature in the jacket . The steady state solution is obtained by equating the derivatives of reactant concentration ,reactor temperature set equal to zero, that is

   process and the reference model.

   d CA

   /dt = 0 = F/V(C

   Af CA) rA* CA

   (4)

2. MATHEMATICAL MODELING

   It is a simple exothermic reaction occurred in the reactor, this is cooled by a coolant that flows in the jacket around the reactor.CSTR process is shown in figure 2.

   Figure 2 Jacketed Continuous Stirred Tank Reactor

   The dynamics of the reacting mixture depends on the mass of the reactants and energy of reactants within the reactor. The reactor material balance equation is d/dt(V ) =Fin -Fout [1]

   Where V is the reactors volume, is the density of reactants,Fin is flow rate of reactant and Fout is the flow rate of product. The flow rates are assumed constant. Consider a simple reaction A B. The balance on component A is

   Vd/dt(CA ) = FC AF – FCA – V*rA (2)

   dT/dt=0=F/V(TfT)+(-H/CP)K0exp(-E/RT)CA) – UA/VCP(T Tj) (5)

   To solve these equations, all parameters and variables except for two (CA and T) must be Specified

   Table 1: Reactor Parameters value

   Reactor parameters	Values
   F/V,hr-1	1
   Ko,hr-1	16.96*1012
   (- H),kcal/kmol	5960
   E,kcal/kmol	11843
   Cp,kcal/m3	500
   Tjf,c Caf,kmol/m3	25
   UA/V,kcal/m3c.hr	150
   TjC	25

   1. Linearization of Dynamic Equation

      Let the state and input variables be defined in deviation variable form:

      The stability of the non-linear equation can be determined by finding the following state space form: X = AX + BU

      And determine the eigen values of the A (state space) matrix. Let the state and input variables be defined in deviation variable form

      Where CA

      is the concentration of reactant within the X =

      reactor and rA is the rate of reaction per unit volume.

      U=

      A=

   2. Stability Analysis

      The stability of particular operating point is determined by finding the A-matrix for that particular operating oint and finding the Eigen values of the A-matrix.

      Substituting the reactor parameter values ,we get

      A=

      =eig(A)=-0.8882,-0.5800

      Both of the Eigen values are negative, indicating that the point is stable.

      B=

      C D=

   3. Derivation of transfer function

      Transfer functions for concentration of the reactant to flow rate and jacket temperature and the temperature of the reactant to flow rate and jacket temperature are determined using matlab and are given by the equations 6,7 8 and 9 respectively.

      Transfer function relating CA to F

      H11 = (6)

      Transfer function relating CA to Tj

      H12 = (7)

      Transfer function relating T to F

      H21 = (8)

      Transfer function relating T to Tj

      H22 = (9)

   4. Design of Decouplers

      Let

      Y1 = CA, Y2 = T,

      m1=F, feed flow rate,

      m2 =Tj, jacket temperature

      Input output relation for Concentration control system and Temperature control system are given by equations (10)& (11) respectively.

      Y1 = m1 +

      m2 (10)

      Y2= m1+

      m2 (11)

      Coupling requires finding of the Relative Gain Array matrix:

      (y1/m1) at m2 =const

      11 = ————————

      (y1/m1) at y2 =const

      =3.0635/1.33=2.3

      Relative Gain Array matrix

      =

      The diagonal elements are negative, so the system is unstable. In order to eliminate the interaction, a decoupler D1(s) & D2(s) must be designed for the two systems and are given by the equations (12)&

      1. respectively.

         To cancel the effect of jacket temperature (m2) on the outlet concentration (Y1)

         D1(s) = (12)

         To cancel the effect of feed flow rate (m1) on the reactor temperature (Y2):

         D2(s) = (13)

         Where H11 & H12 are given by equations (6)&(7) respectively and H21& H22 are given by equations (8)&(9) respectively. The complete block diagram of the process with two decouplers is shown in figure 3&its equivalent representation of the process for Concentration and temperature control is shown in figure 4&5 respectively.

         1. Temperature control system

            The simplified transfer function model of the process is given by

            Figure. 3. Block Diagram of the Process with Two Decouplers

            Gp2(s)=

            (15)

            Figuer4. Equivalent Representation for concentration control system

            Figuer 5 . Equivalent Representation for temperature control system

3. Non-Adaptive Control Analysis

   Simulink model is designed using PID controller for concentration and temperature control system and its transfer functions are given by the equations 14&15 respectively. The conventional PID controller is implemented based on the auto tuning method.

   (i)Concentration control system

   The simplified transfer function model of the process is given by

   Gp1(s)=

   Figure 5: Step Input Response of Concentration Control

   Figure 6: Step Input Response of Temperature Control System

   The step input response of Concentration control system and temperature control system is shown in figure 5&6respectively.It was observed that the response has large settling time and overshoot for both Concentration and temperature control system,which is not desirable.

4. Adaptive control design and simulation

   In this section the model reference adaptive control is designed and implemented in simulink. The performance of an adaptive control to a conventional controller is compared. The modification of model reference adaptive controller is done to adapt faster and better accommodate variations in the parameters. The model that the plant is designed to have the following characteristics.

   1. Concentration control

      For the concentration control a maximum overshoot (Mp) of 5% and a settling time (Ts) of less than 4 seconds are selected. The following equation is used to determine the required damping ratio and natural frequency of the system.

      = *

      =0.68 and n=2.1986 rad/s. The transfer function for the model is therefore

      Gm(s) = (16)

   2. Temperature control

   A reference model (second order) with a maximum overshoot (Mp) of 2.5% and settling time(Ts) of 1 second is chosen.

   The transfer function for the model is therefore Gm(s) = (17)

   This has improved the overshoot to below 10% and the settling time is now less than 2 seconds. Conventional controller has large overshoot and settling time.By increasing the adaptation gain the system becomes unstable .

5. Expanded adaptive control

   The controller parameters are initialized to a value closer to their final value ,the performance of the adaptive controller will be improved. Table 2 shows the comparative overview of different transient domain parameters like overshoot, rise time and settling time of PID and adaptive controller. Comparing the Figure 7 and Figure 8,the adaptive controller has less settling time with no overshoot and rise time compared to conventional controller.

   Table 2: Transient domain parameters of controllers

   criterion	Concentration control		Temperature control
   	PID contr oller	Adaptive controller	PID controller	Adaptive controller
   Rise time (sec)	1.03	0	1.04	0
   Settling time (sec)	10.6	3.5	9.33	1.5
   Overshoot( %)	8.51	0	8.28	0

   Figure 7: Comparison of Adaptive controller and PID controller with a step input for concentration control

   Figure 8: Comparison of Adaptive controller and PID controller with a step input for temperature control

6. Conclusion

   This paper describes the behavior of a system controlled by model reference adaptive control scheme using MIT rule. The effect of adaptation gain is viewed on the time response characteristic of the second order system. The paper demonstrated that while the adaptive controller exhibits superior performance and the PID controller has the convergence time of typically large (greater than 10 seconds) and there is large overshoot. Increasing the adaptation rate improves the performance of the adaptive controller at the cost of increased oscillation.It is possible to significantly improve the performance of an adaptive controller simply by initializing the controller parameters to a value close to their final value.Proposed adaptive controller has good results of control and fulfilled the maximum control requirements such as stability,reference signal tracking and disturbance rejection.

7. References

1. Vojtesek, J.; P. Dostal. 2010 Adaptive Control of Continuous-stirred Tank Reactor in Two Stable Steady- States, In Proceedings of the IFAC Workshop Adaptation and Learning in Control and SignalProcessing 2010, Antalya, Turkey 2010, ISBN 978-3-902661-85-2

2. KJ Astrom and B Wittenmark, Adaptive Control, Addison Wesley, 1989.

3. Vojtesek, J, J. Novak P. Dostal. 2011. Effect of External Linear Models Order on Adaptive Control of CSTR, In Proceeding of The 19th IASTED International Conference on Applied Simulation and Modelling (ASM2011), p. 82-87. ISBN 978-0-88986- 884 -7.

4. Rahideh, M.H.Shaheed , Constrained output feedback model predictive control for nonlinear systems, Control Engineering Practice 20 (2012), pp 431443.

5. Rathikarani Duraisamy and Sivakumar Dakshinamurthy, An adaptive optimisation scheme for controlling air flow process with satisfactory transient performance, Maejo International Journal of Science and Technology, p.221-234, 2010.

6. Rahul Upadhyay and Rajesh Singla, Application of adaptive control in a process control,International Conference on Education Technology and Computer, pp.323-327, 2010.

7. B. Jia, G. Ren, G. Long, Design and Stability Analysis of Fuzzy Switching PID Controller, 6th World Congress on Intelligent Control and Automation, Dalian, China, pp. 3934 3938, 2006.

8. J. Prakash, K. Srinivasan. Design of nonlinear PID Controller and nonlinear model predictive controller for a continuous stirred tank reactor, ISA Transactions, vol. 48, no. 3, 2009, pp. 273 -282.

9. Rajesh Singla and Rahul Upadhyay, Analysis of CSTR Temperature Control with Adaptive and PID Controller, International Journal of Engineering and Technology, Vol.2, No., 5.pp.453-458, 2010

10. Vinodha. R, Abraham Lincoln.S and Prakash.J, Multiple Model and Neural based Adaptive Multi- loop PID Controller for a CSTR Process, International Journal of Electrical and Computer Engineering.pp.251-2.56, 2010

11. Avinashi Kapoor, Saxena.T.K, Udaibi Singh and NishaJha, Online Adaptive Control for Non-Linear Processes Under Influence of External Disturbance International Journal ofArtificial Intelligence and Expert System,pp.36-46,2011

12. GlanDevadhas.G and Pushpakumar.S, An Intelligent Design of PID Controller for a Continuous Stirred Tank Reactor,World Applied Sciences Journal 14(5) pp.698- 703, 2011.

13. Vojtesek, J.; P. Dostal. 2010 Adaptive Control of Continuous-stirred Tank Reactor in Two Stable Steady- States, In Proceedings of the IFAC Workshop Adaptation and Learning in Control and SignalProcessing 2010, Antalya, Turkey 2010, ISBN 978-3-902661-85-2

14. KJ Astrom and B Wittenmark, Adaptive Control, Addison Wesley, 1989.

15. Vojtesek, J.; J. Novak; P. Dostal. 2011. Effect of External Linear Models Order on Adaptive Control of CSTR, In Proceeding of The 19th IASTED International Conference on Applied Simulation and Modelling (ASM2011), p. 82-87. ISBN 978-0-88986- 884 -7.

16. Rahideh, M.H.Shaheed , Constrained output feedback model predictive control for nonlinear systems, Control Engineering Practice 20 (2012) 431443.

17. Rathikarani Duraisamy and Sivakumar Dakshinamurthy, An adaptive optimisation scheme for controlling air flow process with satisfactory transient performance, Maejo International Journal of Science and Technology, pp.221-234, 2010.

18. Rahul Upadhyay and Rajesh Singla, Application of adaptive control in a process control,International Conference on Education Technology and Computer, pp.323-327, 2010.

19. B. Jia, G. Ren, G. Long, Design and Stability Analysis of Fuzzy Switching PID Controller, 6th World Congress on Intelligent Control and Automation, Dalian, China, pp. 3934 3938, 2006.

20. J. Prakash, K. Srinivasan. Design of nonlinear PID Controller and nonlinear model predictive controller for a continuous stirred tank reactor, ISA Transactions, vol. 48, no. 3, 2009, pp. 273 -282.

21. Rajesh Singla and Rahul Upadhyay, Analysis of CSTR Temperature Control with Adaptive and PID Controller , International Journal of Engineering and Technology, Vol.2, No., 5.pp.453-458, 2010

22. Vinodha. R, Abraham Lincoln.S and Prakash.J, Multiple Model and Neural based Adaptive Multi- loop PID Controller for a CSTR Process, International Journal of Electrical and Computer Engineering.pp.251-2.56, 2010

23. Avinashi Kapoor, Saxena.T.K, Udaibi Singh and NishaJha, Online Adaptive Control for Non-Linear Processes Under Influence of External Disturbance International Journal ofArtificial Intelligence and Expert System,pp.36-46,2011

24. GlanDevadhas.G and Pushpakumar.S, An Intelligent Design of PID Controller for a Continuous Stirred Tank Reactor,World Applied Sciences Journal 14(5)

pp.698- 703, 2011.