This paper presents the design and performance analysis of Differential Evolution (DE) algorithm based Proportional-Integral-Derivative (PID) controller for temperature control of Continuous Stirred Tank Reactor (CSTR) plant in chemical industries. The proposed work deals about the design of Differential Evolution (DE) algorithm in order to improve the performance of CSTR. In this, the process is controlled by controlling the temperature of the liquid through manipulation of the coolant flow rate with the help of modified Model Reference Adaptive Controller (MRAC). The transient response of temperature process is improved by using PID Controller, Differential Evolution Algorithm based PID and fuzzy based DE controller. Finally, the temperature response is compared with experimental results of CSTR.

CSTR is highly non linear and the reaction that occurs inside the reactor may be exothermic or endothermic in nature. The reactor is generally combined with an outer jacket in order to control the temperature parameter. During exothermic reactor evolution of heat is removed by the use of coolant stream during exothermic reaction whereas on endothermic heating medium is passed through jacket or coil for maintaining the reaction temperature. The exothermic or endothermic reactions are involved in the reactor, the temperature of the mixture varies with respect to time. This can be overcome by using a suitable controller which adapts to nonlinearity present in the process. Since model reference adaptive controller has the capability to adapt to the change in variation due to disturbance or any environmental conditions [

The literature survey of various techniques employed for improving the performance of transient response of the CSTR process and the review is made on each process and their limitation is also discussed. Swarnkar et al. [

From the literature survey it is observed that MRAC controller is to be used to maintain non-linear systems with the ability to adapt to change in variation due to environment or by load changes. But use of MRAC can control the system with large oscillation and overshoots with increase of adaptation gain as 5. This is to be reduced depending upon the constraint [

CSTR is highly nonlinear feature in nature and usually has wide operating ranges and the real time setup is shown in

The CSTR runs at steady state and is mixed. Due to this process, the CSTR is modeled to have less variation in concentration, temperature in the vessel. Since the temperature and concentration are same all around in the reaction tank, they are the equal at the outlet point [

a) Correct mixing in the process reactor and jacket

b) Steady value of volume reactor and jacket

Mass balance equation is given as,

where, C_{A} is the outlet concentration of reactor A in the reactor and r_{A} is the reaction rate per unit volume. The Arrhenius equation is used for the reaction rate. The first- order reaction rate results in the corresponding equation,

where, ko is the constant rate of reaction, E is the activation energy, R is the constant of gas, T is the temperature of reactor (R, Rankine or K, Kelvin).

The energy balance equation is given as,

The _{A}. They are joined and it is impossible to solve single equation alone. In the following section, state representation models are obtained. [

The transfer function of plant and the model are obtained from above

Thus the model obtained should provide desired response for the process which tells about how the PID parameters are adjusted to the variation in the process due to change in set point by using MRAC.

The block diagram of adaptive PID controller for CSTR plant is shown in

Command Signal (Input) is the set value of Temperature, Output, ‘y’ is the process variable such as Actual Temperature of CSTR, ‘Ym’ is the reference transfer function, ‘U’ is the control signal or controller output to the CSTR process, ‘e’ is the error values to the adaptation mechanism and acts as an input to the controller, Controller parameters are Kp, Ti, Td present in the controller.

The basic idea in adjusting controller parameters in MRAC is commonly known as the gradient approach. Thus, the gradient method of MRAC is often called the MIT rule. The MIT rule performs well when the adaptation gain

The derivative

As MIT rule does not guarantee stability, it is imperative for people to modify it. As mentioned above, MIT rule let the users decide the critical parameter, which determines the rate of decrease of the cost function [

In the above equation, the existence of the parameter

The response of modified MRAC still has high oscillation, overshoot and long settling time. These can be reduced by tuning adaptive controller parameter using PID controller [

The concept of tuning MRAC by incorporating Fuzzy Inference System which improves performances better compare to conventional controller. The response of modified MRAC –PID has large overshoot with respect to increase of adaptation gain. This is to be reduced by use of some intelligent techniques. Therefore, the Mamdani type fuzzy inference system is used to tune the MRAC controller in order to reduce the transient characteristics performance of the CSTR plant to maintain the temperature of the reactor constant throughout the time. In the proposed work the input to the fuzzy is error and change in error and output is the manipulated variable that is inlet coolant flow rate. The number of membership function framed is 7 for each and the membership function used is triangular. The number of rules framed is 49. The range for error, change in error and coolant flow rate is (-15 10), (0 20) and (–1 1) is given to the system.

The schematic diagram of DE algorithm is shown in

Command Signal (Input) is the set value of Temperature, Output, ‘y’ is the process variable such as Actual Temperature of CSTR, ‘Ym’ is the reference transfer function, ‘U’ is the control signal or controller output to the CSTR process, ‘e’ is the error values to the adaptation mechanism and acts as an input to the controller, Controller parameters are Kp, Ti, Td present in the controller.

The steps involved in the tuning of proposed method are given below.

Step1: Setting DE optimization parameter

Step2: Initialize population size

Step3: Perform mutation and cross over operations. DE derived its name from the mutation operator it applies to mutate its individual. It generates the mutated individual for the principal parent. Then to complement the differential mutation search strategy, DE then uses a crossover operation, in which the mutated individual is mated with the principal parent and generates the offspring

Step4: Verify boundary constraints of the system

Step5: Selection process is made and is to ensure that the individuals given to the next generation are exactly with the best fitness values in the population

Step6: Repeat the steps 3 to 5 until new population completed

Step7: Repeat step 6 until fitness value converges and also met best solution till the end of generation.

The DE algorithm is used to get a optimal value for PID parameters. In the proposed work the number of population, crossover, mutation and generations used are 8, 0.7, 0.5 and 10. Total of 25 generations with a population size of 8 generated sufficient simulations. So, the initial population size chosen is 8. Crossover assists exploit and enhance the convergence. From empirical results and theoretical studies, all suggest a relatively higher probability pc for crossover in the range of 0.6 to 0.95, whereas the mutation probability pm is typically very low, around 0.1 to 0.5. First initialize P, I, D values. Normally, the PID values are initiated by applying Ziegler –Nichols tuning methods. The tuning values are 5.583, 0.4466 and 0.7274. Next is to reduce the transient characteristics since it calls the model response to read the overshoot and settling time value of current system and the start to evaluate the best member of the system up to 10th iteration since it is second order system. The fitness value is considered as the sum of overshoot and settling time. Next is the mutation here it replaces the low fitness population with the new vector and produce a new population with best fitness value. This process continues till the fitness values converges and met best solution till 10^{th} iteration to get optimize values of P, I, D parameters.

The response obtained after tuning MRAC controller using has large overshoot with increase of adaptation gain 5. This is reduced by use of some optimization tool; therefore DE is used to optimize the value of fuzzy membership function structure. Due to excess running time the rules is reduced to 9 and the DE is used for 9 rules. Assume number of population, generation, mutation and crossover as 9,5,0,3 and 0.5. Initialize the value randomly by giving the default starting point as [–15 10 0 20 –1 1]. The search space range is given to search around that region to get best value, since lower and upper range for search space is [–20 5 –10 10 –1 0.5] and [–5 20 5 30 –0.5 1]. Next it reads the fuzzy membership function of current generation and get information of overshoot and settling time from the response using this it find best value. Then it goes for mutation and crossover from this it replaces the new vector population of best fitness to the lower fitness value. This process continues till it reaches 5^{th} iteration. In each iteration, it finds best fitness value by using overshoot and settling time of that iteration. This makes the cost function to converge to make the global best. The result obtained using DE makes the fuzzy structure optimize and produces response with no overshoot and less settling time for increase of adaptation gain 5 compare to fuzzy at adaptation gain 5.

The result obtained using MRAC and incorporating different techniques like PID, PID+DE, Fuzzy and Fuzzy+DE with MRAC for improving the transient characteristics of CSTR plant are discussed here.

The set point given for system is 100 degree F. The reference model will produce the desired response that should track by the plant in order to maintain the particular temperature inside the reactor. To get a constant temperature of feed at outlet of the reactor MRAC controller is used in the system. The adaptation gain chosen is 5 for controller. The results obtain is shown in below

The MRAC shows more oscillation and overshoots with respect to adaptation gain 5. The variation of manipulated variable and error signal response are shown in

It is from the responses that the response obtained using MRAC produces more overshoots and oscillation which needs to be reduced to prevent the system from damage therefore PID controller is incorporated with MRAC. The MRAC forces plant output to follow the desired response of reference model.

PID is to determine the control signal for controlling the temperature of CSTR process. The result obtained after incorporating PID with MRAC is shown in

Usage of PID makes the system to reach the steady state soon and overshoot gets reduced. The variation of manipulated variable and error after incorporating PID is shown in

Therefore, the PID controller is incorporated with MRAC and it is inferred that the transient characteristics of the plant response such as peak overshoot, settling time is improved when related to MRAC. The error-signal also gets reduced due to the incorporation of PID with MRAC and is shown in the

The values obtained with incorporation of PID in MRAC is optimized, may improve the response. Since DE algorithm is used to optimize the PID values the response obtained by use of DE for optimization of PID values are shown in

The convergence rate obtained after optimization is shown in

Performance criteria | Fuzzy | Fuzzy DE |
---|---|---|

MSE | 6.23 | 3.596 |

ITAE | 112.47 | 68.6 |

The tuning of MRAC with the help of fuzzy which makes the system to improve its transient characteristic performance with the adaptation gain 5. Here the number of rules framed to tune the MRAC is 49 and number of inputs and output are 2 and 1. Membership function framed is triangular method is used for defuzzification techniques. The

Membership function notation is represented by LN, MN, SN, Z, SP, MP and LP. [

Where, LN-large negative, MN-medium negative, SN-small negative, Z-zero, SP-small positive, MP-medium positive, LP-large positive.

The response obtained by the incorporation of Fuzzy in MRAC is shown in

The incorporation of Fuzzy in MRAC has an oscillation. The fuzzy range is fixed on trial and error basis, to get an optimal value have to use some optimization technique.

The result obtained by use of fuzzy has an oscillation and overshoot and it is to be reduced in order to meet the improved performance.

The fuzzy structure is to be optimized with the help of DE. Here the number of rules framed is reduced due to time consumption problem so the rules framed are 9.The response of the system after the use of DE makes the overshoot zero and settling time to be less and is shown in below ^{th} to 5^{th} due to time consumption problem and therefore the time taken to settle will get increases by 20% (2.5seconds). Hence, the

It is inferred from the

The PID values are obtained by Ziegler Nichols method shown in

Determination of PID values | P | I | D |
---|---|---|---|

Ziegler-Nichols method of tuning | 5.538 | 0.4466 | 0.7274 |

Optimization by DE | 9.7285 | 1.1582 | 0.01164 |

The comparative analysis of MRAC, incorporation of PID, PID+DE, Fuzzy and Fuzzy+DE in MRAC are given in the

Parameter specification | MRAC | PID | PID DE | FUZZY | FUZZY DE |
---|---|---|---|---|---|

Rise time (sec) | 1.1 | 1.4 | 1.3 | 2.5 | 5.8 |

Settling time (%) | 20 | 15.8 | 10.6 | 5 | 2.5 |

Overshoot (%) | 16.2 | 5.37 | 5 | 12.5 | 0.001 |

It is inferred from the

In chemical industries there is a problem of controlling the temperature of the liquid inside the chemical reactor due to the nature of exothermic and endothermic reaction. This makes the inside temperature of the plant to vary with time. This problem is reduced by designing the Modified Model Reference Adaptive Controller for CSTR process which controls the temperature of the plant by manipulating the coolant flow rate. This technique can be used for large gamma value as well as high set point using Normalization MRAC along with PID, FUZZY, PID+DE and FUZZY+DE which produces satisfactory result and the transient response characteristics of the CSTR plant is improved. From the result it is concluded that different techniques where used to improve the transient characteristics in which PID produce response with more overshoot and less time taken to settle, whereas fuzzy produce minimum overshoot and less time taken to settle. Depending upon our need either of the method can be used. For instance to make the system to reach the steady soon PID+DE can be used and in case of production from damage due to large overshoot Fuzzy +DE is used.