Design of Optimal Controller for A Non-Linear Batch Process

Design of Optimal Controller for A Non-Linear Batch Process

Dwain Jude Vaz, Praseetha P P, Rahul Vijayan,

SELECT, VIT University, Vellore, Tamilnadu

1. Bagyaveereswaran,

   AssistantProfessor (Senior) SELECT, VIT University Tamilnadu

   Abstract This paper provides a theoretical framework for modelling and simulation for optimal control design of a nonlinear dynamic system. In this paper we have considered a Batch reaction as nonlinear dynamics. During the mixing of the chemical reactants, a sudden unpredictable amount of heat is released causing exothermic reaction. This may affect the product quality and may damage the system. An optimal control technique such as Linear Quadratic Regulator (LQR) and Proportional Integral Control(PID) method are used for control of the temperature of the chemical process and hence maintain the adequate conditions for the process to take place. The non- linear system states are fed to the LQR which is designed using linear state space model. The analysis of the simulation results revealed that LQR and two PID controllers together can give better performance than a simple LQR controller

   Key words Batch process , LQR, PID, Optimal control, Non- linear dynamic system .

   1. INTRODUCTION

      In an exothermic reactor, a large amount of heat liberated during the mixing of reactants can cause thermal runaway[1] if the generated heat exceeds the cooling capacity of the reactor tank. This may affect the product quality and pose safety problem to the plant. Hence it is necessary to have a precise temperature control[2] in such reactors. Here the control problem consists of obtaining the model of the reactor, and using this model to determine the

      control laws or strategies to achieve the desired system

      and modelling of the reactor. Section III presents the design of controller

   2. MATHEMATICAL MODELLING

      1. Linearization of the System

         Jacobian Linearization method is used to linearize the non linear system, about a specific operating point, called an equilibrium point.

         Consider a non-linear differential equation

         = ( , ()) (1)

         where f is a function mapping × . A Point

         s called an equilibrium point if there is a specific

         such that , = .

         Defining deviation variables to measure the difference

         = . (2)

         = . (3)

         The Relation between x(t) and u(t) are given by the differential equation

         = ( , ()) (4)

         Substituting the deviation variables in (4), we get

         = ( + (), + ()) (5) Using Taylor series expansion in equation (5)

         + . (6)

         response and performance.

         The Proportional-Integral-Derivative (PID) control is used to give efficient solution to various real-world control problems[3]. The transient and steady- state responses are taken care of with three-terms (i.e. P, I, and D). To make the performance of the system optimal LQR(Linear Quadratic Regulator ) optimization is used.

         As the input flow rates of the reactants increases the tank temperature as well as level also increases. Here we use LQR to control the temperature and later we use LQR +two PID controllers[6] for controlling temperature as well as level.

         The organisation of the paper is as follows. Section II discusses about the mathematical modelling of the mixing tank which includes linearization of the system equation

         The higher order terms are neglected.

         The above differential equation holds good for the deviation variables as long as the deviation variables are small. It is a linear, time-invariant[9], dierential equation, since the derivatives of x are linear combinations of the x variables and the deviation inputs, u. The matrices

         = = = , (7)

         = = = n×n (8)

         are constant matrices. The Linear system can now be defined as

         = + ()). (9)

         This is the Jacobian Linearization of the nonlinear system about the equilibrium point ( , ). For small values of

         the linear equation approximately governs the exact relationship between the deviation variables

         = 0 2 1 2

         1

         (16)

         .

         = 0 2 1 + 2

         (17)

         2

      2. Chemical Reactor

      Consider a mixing tank[7], with constant supply

      As ui represents flow rates into the tank, they are non- negative real values due to physical restrictions. This

      temperatures TC and TH and input flow rates qc(t) and qH(t). implies that 0 and TC TH..The differential

      The equations for the tank are: equation for TT , the tank temperature, implies that it is

      1

      =

      +

      0

      2

      (10)

      inversely proportional to the height of the tank . Hence, the dierential equationof a model is valid while h(t) > 0, so we

      = 1

      [

      ] + [

      ] (11)

      further restrict 1 > 0. Under those restrictions, the state ,

      ()

      is indeed an equilibrium point.

      where A is the area of the tank, TT is the temperature of the product inside the tank.

      The necessary partial derivatives are given by :

      1

      1

      2

      1

      2 =

      2

      0

      21

      1 2 + 2 2

      0

      1 + 2

      1

      2

      1

      1

      2

      1

      1

      1

      1

      1

      2 =

      2

      2 2 2

      1

      2

      1

      1

      In order to linearize the given system it is required that the matrices of partial derivatives be evaluated at the equilibrium points.

      Fig.1 Batch Process – Mixing Tank

      Let the state vector X and input vector U be defined as:

      () = ,() = () (12)

   3. CONTROLLER DESIGN

      Optimal control is used to minimize the performance index. A control law is synthesized using optimal control technique, which results in best possible behaviour of the system .Linear quadratic regulator (LQR) is one of the

      ()

      , = 1

      1 1 1

      +

      ()

      2 0 21 (13)

      optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions. The control law is given by

      (, ) = 1 [

      ] + [

      ] (14)

      = (18)

      2 1

      2

      2 2

      where, X is the states of the system and K is feedback gain

      Where 1= (),

      2 = () and 1

      = , 2 = ()

      matrix[8] and it is derived from minimization of the cost function

      For any height > 0 and any tank temperature , satisfying < < should be a possible equilibrium point. With and chosen, the equation f( , )=0 can be written as

      = ( + ) (19)

      where, Q and R are positive semi-definite and positive definite symmetric constant matrices respectively. The LQR gain vector K is given by

      1 1

      1 = 0 2 1 (15)

      = 1 (20)

      2 2 2 0

      2

      The 2×2 matrix is invertible if TC not equal to TH. Hence as long as TC not equal to TH, there is a unique equilibrium input for any choice of ^. It is given by :

      where, P is the solution ofthe Algebraic Ricatti Equation[10] – (21)

      + 1 + = 0 (21)

      In the optimal control of mixing tank total temperature of the tank have been considered available for measurement which are directly fed to the LQR. The LQR is designed using the linear state-space model of the system. The optimal control value of LQR is given as a negative feedback along with the PID controller. The tuning of the PID controller and PID+LQR controller is done by Zeigler Nicholas method[11].

   4. SIMULATION & RESULT

      The MATLAB-SIMULINK models for the control of temperature and height of the mixing tank have been developed. The typical parameters of the reactor is selected as = 10, = 90, = 32 ,0 = 0.05m, constant =

      0.7. After linearization the system matrices used to design

      LQR are computed as below:

      A= 0.0258 0

      PIDControl schemes	Height	temperature
      KP	1.9438	1.45028
      KI	3.16	7.146180
      KD	-0.0049493	-0.145766

      Table 1. PID controller parameters

      0 0.0517

      B= 0.333 0.333

      21.67 5

      C= 0 1

      With the choice of

      Q= 1 0

      0 0

      R= 1 0

      Fig.5 Block diagram of Two PID With LQR

      The response of above model is as shown in fig. 6.

      0 1

      we obtain LQR gain vector as following:

      K= 0,9079 0.2295

      0.2338 0.9495

      the temperature response with LQR is shown in fig. 4 .

      Fig.4 Control of temperature with LQR

      SIMULINK model of the system using two PID controllers having parameters,

      Fig.6 Response of temperature with two PID and LQR

      Fig.7 Response of height with two PID and LQR

      TIME DOMAIN SPECIFICATION	2PID+LQR	LQR
      RISE TIME	6.3053	21.6764
      SETTLING TIME	17.7636	108.5134
      OVERSHOOT	0	9.6023
      UNDERSHOOT	0	0.8522
      PEAK	27.4006	12.5025
      PEAK TIME	57	63

   5. CONCLUSION

      Table 2. Time Domain Specification Comparison

      It is observed that the product temperature reaches the setpoint without overshoot and offset while using two PID and LQR.

      The setpoint tracking and disturbance rejection capability of the controller is verified by using the fig 7

      Fig 7. Block diagram of setpoint tracking and disturbance rejection of the system

      Its response is as shown in fig 8.

      Fig 8. Response for setpoint tracking and disturbance rejection

      PID with LQR controller, is used to control the effective temperature of a batch reactor. In order to compare the results initially system with only LQR is implemented and later on system with two PIDs with LQR is implemented. The MATLAB-SIMULINK models have been developed for the simulation of both control schemes. The simulation results justify that performance of two PID+LQR control scheme is better than LQR control scheme. Also it is verified that the system tracks the setpoint and rejects the disturbance in an effective manner. The performance investigation of this approach with fuzzy controller may be done as a future scope of this work.

   6. REFERENCES

Centre Autom. et Syst., Ecole Nat. Superieure des Mines de Paris,

Fontainebleau, France .DEC,30,1991

1. Optimizing and tracking of temperature control trajectory of batch process Shuqian Lin ; Jing Wang ; Liulin Cao ; Qibing Jin Control and Decision Conference (CCDC), 2010 Chinese

   Digital Object Identifier: 10.1109/CCDC.2010.5498871 Publication Year: 2010 , Page(s): 2086 – 2091

2. A neural methodology for batch process optimizing control Enbo Feng ; Yihui Jin Control Applications, 1993., Second IEEE Conference on Digital Object Identifier 10.1109/CCA.1993.348321 Publication Year: 1993 , Page(s): 703 – 707 vol.2

3. Robust LQR tracking control for a class of affine nonlinear uncertain systems Hai-Ping Pang ; Qing Yang

   Control and Decision Conference (CCDC), 2012 24th Chinese Digital Object Identifier: 10.1109/CCDC.2012.6244191 Publication Year: 2012 , Page(s): 1197 120

4. Controller performance assessment of batchprocesses based on dynamic time warpingYijun Cai ; Mengfei Zhou ; Tao Zou ; Luyue Xia Intelligent Control and Automation (WCICA), 2010 8th World Congress on Digital Object Identifier: 10.1109/WCICA.2010.5553994

   Publication Year: 2010 , Page(s): 4165 4168

5. Modelling and Simulation for Optimal Control of Nonlinear Inverted Pendulum Dynamical System Using PID Controller and LQRPrasad,

   L.B. ; Tyagi, B. ; Gupta, H.O. Modelling Symposium (AMS), 2012 Sixth Asia

   Digital Object Identifier: 10.1109/AMS.2012.21 Publication Year: 2012 , Page(s): 138 143

6. PID adaptive control of exothermic stirred tankreactors M'Saad,

   M. ; Bouslimani, M. ; Latifi, M.A.

   Control Applications, 1993., Second IEEE Conference on Digital Object Identifier: 10.1109/CCA.1993.348296 Publication Year: 1993 , Page(s): 113 – 117 vol.1

   Cited by: Papers (1)

7. Optimal state-feedback design for non-linear feedback-linearisable systemsEsfahani, P.M. ; Farokhi, F. ; Karimi-Ghartemani, M. Control Theory & Applications, IET

   Volume: 5 , Issue: 2 Digital Object Identifier: 10.1049/iet- cta.2009.0242 Publication Year: 2011 , Page(s): 323 333

8. Estimating nonlinear systems in a neighborhood ofLTI- approximantsEnqvist, M. ; Ljung, L. Decision and Control, 2002, Proceedings of the 41st IEEE Conference on Volume: 1 Digital Object Identifier: 10.1109/CDC.2002.1184641

   Publication Year: 2002 , Page(s): 1005 – 1010 vol.1 Cited by: Papers (2)

9. The exact slow-fast decomposition of the algebraicRicatti equation of singularly perturbed systemsSu, W.C. ; Gajic, Z. ; Shen,

   X.M. Automatic Control, IEEE Transactions on Volume: 37 , Issue: 9 Digital Object Identifier: 10.1109/9.159592 Publication Year: 1992 , Page(s): 1456 – 1459 Cited by: Papers (15)

10. Tuning of PID controller using Ziegler-Nichols methodfor speed control of DC motor Meshram, P.M. ; Kanojiya, R.G.

Advances in Engineering, Science and Management (ICAESM), 2012 International Conference on Publication Year: 2012 , Page(s): 117 – 122 Cited by: Papers (3)