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Adaptive Enhanced Genetic Algorithm-BasedProportional Integral Controller Tuning for pH ProcessK. Valarmathi a; D. Devaraj a; T. K. Radhakrishnan aa Chemical Engineering Department, National Institute of Technology, Tiruchrappalli,India

Online Publication Date: 01 November 2007To cite this Article: Valarmathi, K., Devaraj, D. and Radhakrishnan, T. K. (2007)'Adaptive Enhanced Genetic Algorithm-Based Proportional Integral Controller Tuningfor pH Process', Instrumentation Science & Technology, 35:6, 619 - 635To link to this article: DOI: 10.1080/10739140701651607

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Adaptive Enhanced GeneticAlgorithm-Based Proportional Integral

Controller Tuning for pH Process

K. Valarmathi, D. Devaraj, and T. K. Radhakrishnan

Chemical Engineering Department, National Institute of Technology,

Tiruchrappalli, India

Abstract: The control of a pH process is a challenging task. For the pH process, Pro-

portional Integral (PI) control has been successfully used for many years. Tuning of the

PI controller is necessary for the satisfactory operation of the system. This paper

presents an Enhanced Genetic Algorithm (EGA) for the tuning of PI controllers. To

avoid the premature convergence and to reduce the computation time, advanced

genetic operators have been used in the proposed GA approach. The proposed GA

tuned PI controller is implemented in the experimental pH process. The performance

of the controller with Enhanced GA tuning is compared with traditional Ziegler

Nichols tuning and Internal Model Control for various set point and trajectory

responses of the pH process. The experimental results show that the proposed GA is

very much capable to adapt the controller to dynamic plant characteristics changes

in pH process.

Keywords: PI controller, Genetic algorithm, Nonlinear, pH process, IMC, ZN tuning

INTRODUCTION

The regulation and control of a pH process is a typical problem found in a

variety of industries, including pharmaceuticals, biotechnology, and chemical

processing. It is a non-trivial task arising from the non-linearity of the

titration process. High performance and robust pH control is often difficult to

achieve due to its nonlinear characteristics.

Address correspondence to T. K. Radhakrishnan, Chemical Engineering Depart-

ment, National Institute of Technology, Tiruchrappalli, India. E-mail: radha@nitt.edu

Instrumentation Science and Technology, 35: 619–635, 2007

Copyright # Taylor & Francis Group, LLC

ISSN 1073-9149 print/1525-6030 online

DOI: 10.1080/10739140701651607

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Numerous schemes have been employed to handle this non-linear

process, which includes the simple PID for regulation,[1] self-tuning

adaptive control,[2] linearising control and model-based control schemes.[3]

In the simple PID controller, the Ziegler-Nichols tuning procedure works

quite well, but this method is laborious and time consuming, particularly for

processes with large time constants or delays.[4] In linearising control, the

main idea is to transform the non-linear states to other states that are linear;

control of the pH is then achieved through these states. This method is

referred to as the strong base equivalent scheme. While it is possible to

achieve good control, this scheme requires full knowledge of the process.[5]

The non-linear control methods include the reference control structure and

the non-linear adaptive control proposed by Henson and Seborg.[6]

De Leon-Morales[7] proposed observer-based controllers using singular per-

turbations theory and differential algebra for position and speed control of a

single flexible joint robot manipulator. Assawinchaichote[8] described a

fuzzy observer based controller for a non-linear system. The observer-based

controller has a stabilization problem for linear singular systems; it needs a

modification of singular systems. In the non-linear adaptive control, input-

output linearization is attained but, in a practical approach, it requires

knowledge of the states’ variables.

A number of classical optimization techniques, such as a gradient method

and a point estimation method[9] are often used to find optimal values. These

optimization methods obtain PID parameters by optimizing an Integral

Square Error (ISE), Integral Absolute Error (IAE), and Integral Time Square

Error (ITSE) performance index. In all of these, tuning is obtained around an

operating point where the model can be considered to be linear. This implies

that there is sub-optimal tuning when a process operates outside the validity

zone of the model. This situation is common when the reference is not a set

point, but a trajectory response. Recently, Evolutionary Computation techniques

have been proposed to tune the PID controller by taking into account all non-

linearities and additional process characteristics. Oliveira et al.[10] used a

standard GA to determine initial estimates for a variety of classes of linear

time-invariant (LTI) systems, encompassing minimum phase, non-minimum

phase, and unstable systems. In an independent enquiry, Porter and Jones[11]

proposed a GA-based technique as a simple, generic method of tuning digital

PID controllers. Wang and Kwok[12] tailored a GA using inversion and pre-

selection of ‘micro-operators’ to PID controller tuning. Yeong-Koo et al.[13]

proposed the simple GA coupled with cubic spline interpolation method for

the control of a pH process. Dionisio Pereira[14] proposed system model identi-

fication and PID controller tuning for first order plant. This simple GA concept

led to premature convergence and, also, it takes a long time to reach the

solution. In this paper, the PI controller parameters are first obtained from the

ZN tuning method and these preliminary controller parameters are employed

in the pH process. The optimal controller parameters are computed with the

Enhanced Genetic Algorithm approach and applied to the PI controller.

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EXPERIMENTAL

System Description and Model Development

The pH is the measurement of the acidity or alkalinity of a solution. The pH

process consists of neutralization of two monoprotic reagents of a weak acid

(acetic acid) and a strong base (sodium hydroxide). The method implement

mass balances on components that are called reaction invariants of the

Continuous Stirred Tank Reactor (CSTR) solution. The CSTR has two inlet

streams: the influent process stream, and the titrating stream, with one

effluent stream at the output. The model of the pH neutralization process

used in this work follows that proposed by McAvoy et al.[15] and is shown

in Fig. 1. Assumption of perfect mixing is general in the modeling of pH

processes. Material balances in the reactor can be given by

Vdxa

dt¼ FaCa � Fa þ Fbð ÞXa ð1Þ

Vdxb

dt¼ FbCb � Fa þ Fbð ÞXb ð2Þ

Where Fa is the flow rate of the influent stream, Fb is the flow rate of the

titrating stream, Ca is the concentration of the influent stream, Cb is the con-

centration of the titrating stream, xa is the concentration of the acid solution, xbis the concentration of the basic solution and V is the volume of the mixture in

the CSTR.

The above mathematical equations describe how the concentration of the

acidic and basic components xa and xb change dynamically with time, subject

to the input streams Fa and Fb. Invoking the electro-neutrality condition, the

sum of the ionic charges in the solution must be zero.

½Naþ� þ ½Hþ� ¼ ½AC�� þ ½OH�� ð3Þ

Figure 1. pH neutralization process.

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The [X ] denotes the concentration of the X ion. The equilibrium relations also

hold for the water and the acetic acid:

Ka ¼½AC

�

�½Hþ�

HAC½ �ð4Þ

Defining, Xa ¼ [HAC]þ [AC2], Xb ¼ [Naþ] and use of Eqs. (3) and (4),

½Hþ�3þ ½Hþ�

2fKa þ Xbg þ ½Hþ�fKaðXb � XaÞ � Kwg � KwKa ¼ 0 ð5Þ

Let pH ¼ 2log10 [Hþ] and pKa ¼ 2log10 Ka. The titration curve is given by

Xb þ 10�pH � 10�pH�14

�Xa

1þ 10 pKa�pH¼ 0 ð6Þ

Where Ka and Kw are dissociation constant of acetic acid at 258C(Ka ¼ 1.778 � 1025 & Kw ¼ 10214). From Eq. (1), it can be seen that

the CSTR is only mildly non-linear. If the condition Fa � Fb holds, the

dynamics are essentially linear. This assumption is not entirely unreason-

able because, in most situations, the reagents used in the titrating stream

can be adjusted for concentration in such a way that the relationship

holds. If this assumption holds, the CSTR is then essentially governed by

Eq. (l), which reduces to a linear system. Eq. (6), which is strictly static

non-linear relationship between the states xa, xb and the output pH

variable, manifests itself as the familiar titration curve of the neutralization

process.

PI Controller for the pH Process

This section presents tuning of the PI controllers in an industrial chemical

process using Ziegler Nichols and Internal Model Control. It is the type of

feedback controller that makes a plant less sensitive to changes in the sur-

rounding environment and small changes in the plant. A system consisting

of a plant and a controller with unity feedback is shown in Fig. 2. The

Figure 2. Block diagram of a plant with PI controllers.

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controlled output is given by

CðtÞ ¼ KceðtÞ þKc

ti

ðeðtÞdt ð7Þ

Where Kc is the proportional gain and ti is the integral time constant.

Ziegler Nichols Tuning

The PI controller parameters are initially determined by using a conventional

Ziegler-Nichols method. The tuning technique of Ziegler-Nichols is used to

provide a first guess for the PI controller parameters (Kc and ti), which are

then adjusted by trial and error. The Ziegler-Nichols technique is a closed

loop tuning method which is implemented as follows:

1. The feed disturbance flow is fixed at the mean of its range;

2. With the process in open loop, the manipulated variable is manually

adjusted to bring the process to the desired steady-state operating point;

3. A square wave is used to apply small set point changes (typically 0.1 pH

units);

4. The proportional gain is gradually increased until the process oscillates

continuously. This proportional gain is the ultimate gain, KU, and the

period of sustained oscillation is ultimate period Tu;

5. The PI tuning parameters (Kc and ti) are obtained by applying ZN rules.

Internal Model Control (IMC) Method

IMC, which depends on an accurate process model of the process, leads to a

design of a control system that is stable and robust. A process reaction curve

can be obtained by introducing a step change in the acid flow rate. It is one of

the widely followed process identification techniques. This method is used for

identifying the pH model. The process is divided into four zones and, in each

of these zones, a step change is given in both positive and negative directions

and reaction curves are recorded. Skogested[16] proposed an IMC based PI

controller for a first order process. The filter used for the design is

FðsÞ ¼1

ðtcsþ 1Þr; ð8Þ

where tc is the desired closed loop response time. The reaction curves indicate

the process as First Order Plus Time Delay (FOPDT) model with varying

process gains (Kp), time constants (tp), and delays (ud). Skogested approxi-

mates time delay as a first order pade approximation and rearranges it in the

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standard form of a PI controller. The PI controller settings, in this case, are

KcKp ¼tp

ðtc þ uÞand ti ¼ tp: ð9Þ

The PI controller parameters tuned by ZN and IMC are not satisfactory for

the pH process. The ZN tuning method needs frequent tuning for tracking

the pH performance, and IMC is based on approximating the pH process

into a FOPDT model. To avoid this difficulty, GA is capable of locating the

controller settings in the complex pH process.

Overview of the Genetic Algorithm

GA’s search algorithms are based on natural selection and natural genetics.

They draw their search power from the natural ‘law’ of survival of the

fittest. They operate on a string structure (chromosomes), typically, a conca-

tenated list of binary digits representing a coding of the control parameters

(phenotype) of a given problem. The real value of the control parameter,

encoded in a gene, is called an allele.[17] GA consists of three fundamental

operators, i.e., reproduction, crossover, and mutation. GA can be implemented

by using the following sequence of steps:

1. A chromosomal representation of solution to the problem.

2. Creation of an initial population of solutions.

3. Evaluation of a function that plays the role of the environment, rating

solution in terms of their fitness.

4. Choice of a set of operators used to manipulate the genetic composition of

the population.

5. Determination of parameter values used in GA (population size, probabil-

ities of applying genetic operators).

The GA repeats the above steps until predetermined criteria are met. The

criteria could be a certain number of generations and/or a value for the

maximum fitness. When the solution is obtained, the binary string is

decoded to its corresponding real value. The GA searches a population of

points instead of a single solution. The GA is not easily sidetracked to

obtain a local optimal solution instead of a global optimal solution. It

doesn’t need information about the system, except for the fitness function.

Enhanced Genetic Algorithm (EGA)

The most important issues in the genetic evolution are the effective rearrange-

ment of the genotype information. In Simple Genetic Algorithm (SGA), cross

over is the main genetic operator for the exploitation of information, while

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mutation brings new non-existent bit structures. This SGA scheme is capable

of locating the neighborhood of the optimal or near optimal solutions but, in

general, requires a large number of generations to converge. This problem

becomes more intense for large-scale optimization problems with difficult

search spaces and lengthy chromosomes, where the possibility for the SGA

to become trapped in local optima increases and the convergence speed of

the SGA decreases. At this point, a suitable combination of the basic,

advanced, and problem specific genetic operators must be introduced in

order to enhance the performance of the GA. Advanced and problem-

specific operators usually combine local search techniques and expertise

derived from the nature of the problem. All problem specific operators

introduce random modification to all chromosomes; this proves to have

better fitness. It replaces the original one with the new. All problem specific

operators are applied with a probability of 0.2. The following problem

specific operators have been proposed in Reference [18].

1. Gene cross swap operator (GCSO): This operator randomly selects two

different chromosomes from the population and two genes, one from every

selected chromosome, and swaps their values, as shown in Fig. 3. While

crossover exchanges information between high-fit chromosomes, the GCSO

searches for alternative alleles, exploiting information stored even in low fit

strings.

2. Gene inverse operator (GIO): This operator acts like a sophisticated

mutation operator. It randomly selects one gene in a chromosome and

inverts its bits values from one to zero and vice versa, as shown in Fig. 4.

The GIO searches for bit-structures of improved performance exploits new

areas of the search spaces far away from the current solution, and retains

the diversity of the population.

In the enhanced GA shown in Fig. 5, after the application of the basic

genetic operators, the advanced and problem specific operators are applied to

produce the new generation. All chromosomes in the initial population are

Figure 4. Gene inverse operator.

Figure 3. Gene cross swap operator.

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created at random. Due to the decoding process, the corresponding control

variables of the initial population satisfy their upper-lower bound. Population

statistics are then used to adaptively change the crossover and mutation prob-

abilities. If premature convergence is detected, the mutation probability is

increased and the crossover probability is decreased. This happens in the case

of high population diversity.

Genetic Algorithm Implementation

Use of GA is a rapidly expanding area in control systems design. A genetic

tuning algorithm usually starts with no knowledge of the correct solution

and depends on the responses from its environment to give an acceptable

result. It has been shown that genetic algorithms are capable of locating

optimal regions in complex domains, avoiding the difficulties, or even

erroneous results in some cases, associated with the gradient descent

methods and with high-order systems. The enhanced GA can be performed

with an elitist approach, which means that a solution cannot degrade from

one generation to the next, but that the best individual of a generation is

copied to the next generation without any changes being made to it. A

binary GA is implemented with the matching binary crossover and binary

Figure 5. Enhanced genetic algorithm.

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mutation functions. Individuals of the first generation are generated at random.

Boundaries are set for the generated PI constants. These boundaries are found

experimentally from Z-N method. To obtain the PI tuning parameters, one

usually has to minimize a performance index. The performance index MSE

is used as the basis of the fitness function. It maximizes the fitness function,

whereas the MSE needs to be minimized.

Each individual in the population represents the parameters of the PI con-

trollers. The elements of that solution consist of all the control variables in the

system. For the pH process, the controller parameters are Kp and Ki, where

Ki ¼ Kc/ti. The PI controller parameters are transformed to a finite string

using a binary coding. For example, Kp and Ki all have 10 bits and form a

binary string as:

S ¼ 101 01010111|fflfflfflfflfflffl{zfflfflfflfflfflffl}Kp

01 0101000|fflfflfflfflffl{zfflfflfflfflffl}Ki

The boundaries are 0 , Kp , 1 and 0 , Ki , 5.

The Enhanced Genetic Algorithm searches for the optimal solution by

maximizing or minimizing the function and, therefore, an evaluation

function which provides a measure of the quality of the problem solution.

As the objective functions in the optimization, integral square error (ISE),

integral time absolute error (ITAE), mean square error (MSE), settling time,

and peak overshoot are widely used. In the pH process, frequent change of

the magnitude of errors with respect to time make the control problem very

complicated. The MSE is chosen as the objective function to achieve the

minimal control errors. The MSE squares the error to remove negative error

components. MSE discriminates between over-damped and under damped

systems.

fMSE ¼1

n

Xnt¼1

ðeðtÞÞ2 ð10Þ

The fitness function is:

f ¼ fMSE þ fosþ fst: ð11Þ

During the GA run, GA searches for a solution with a maximum fitness-

function value. Hence, the minimization objective function, which is given by

Eq. (9) is transformed as:

Fitness ¼K

1þ fð12Þ

K is a constant. In the denominator, a value of 1 is added to f in order to avoid

division by zero.

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RESULTS AND DISCUSSION

This section presents the details of the design of PI controller for the pH

process. First, the proposed GA-based algorithm was applied to design the

PI controller for a simulated pH process and then the same algorithm was

applied to a real-time system. The GA code was written in MATLAB and

executed on a PC with a Pentium IV processor. The design details and the per-

formance of the controller are given below.

Simulation Results

The pH process was simulated based on Eqs. (1) and (2), using MATLAB

simulink. The model parameters for the experimental system are given in

Table 1. The pH model, simulated using MATLAB Simulink, is shown in

Fig. 6. A step signal is used as the input to the system. Uniformly distributed

noise is added to the step input through a uniform random generator. This is

Table 1. Model parameters for the pH process

Symbols Description Value

V Volume of the continuous stirred

tank reactor

7.4l L

Fa Flow rate of the influent stream 0.24 L min2l

Fb Flow rate of the titrating stream 0–0–0.8 L min21

Ca Concentration of the influent stream 0.2 g mol L21

Cb Concentration of the titrating stream 0.1 g mol L21

Figure 6. Simulation of pH model.

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done to cause no-linear distortion due to the dead zone non-linearity in the pH

process. A PI controller was used to control the pH process.

The GA-based algorithm is applied to find the optimal parameters of the

controller. The objective function in this pH process is minimization of error,

peak overshoot, rise time, and settling time. The optimization variables are

represented as binary numbers in the GA population. The initial population

is randomly generated between the variables’ lower and upper limits. Tourna-

ment selection is applied to select the members of the new population.

Advanced genetic operators are applied after the application of the

crossover and mutation operators on the selected individuals. The perform-

ance of GA for various values of crossover and mutation probabilities in the

ranges 0.6–1.0 and 0.001–0.1, respectively, was evaluated.

The best results of the proposed GA are obtained with the following

control parameters:

Number of generations ¼ 30

Population size ¼ 10

Crossover probability ¼ 0.8

Mutation probability ¼ 0.08

The EGA took 12 s to complete the 30 generations. After 30 generations,

it was found that all the individuals had reached almost the same fitness value.

This shows that the proposed GA has reached the optimal solution. Figure 7

shows the convergence of proposed GA algorithm. It is observed the

Figure 7. Convergence of proposed GA.

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variation of the fitness during the GA run for the best case and shows the gen-

eration of optimal variables. It can be seen that the fitness value increases

rapidly in the first 2 generations of the GA. During this stage, the GA concen-

trates mainly on finding feasible solutions to the problem. Then, the value

increases slowly, and settles down near the optimum value with most of the

Figure 8. Tracking performance of proposed GA, ZN, and IMC tuned PI.

Table 2. Comparison of different tuning methods

Tuning MSE OS Ts Tp Tr SP

ZN 0.1996 3.97 25 1 1 5

IMC 0.1510 4.16 20 1 1

GA 0.0141 3.95 5 1 1

EGA 0.0136 3.95 5 1 1

ZN 2.987 4.13 289 101 1 7

IMC 0.4985 14 68 14 1

GA 0.0762 3.33 372 184 1

EGA 0.076 1.78 14 25 1

ZN 14.47 3.44 513 195 33 9

IMC 0.2914 4.5 535 293 33

GA 0.207 2.70 453 189 17

EGA 0.1468 2.31 527 173 18

ZN 48.62 15.6 210 92 86 11

IMC 3.2960 10.256 70 25 20

GA 0.899 5.42 311 24 22

EGA 0.4186 5.20 5.51 22 19

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individuals in the population reaching that point. Next, for comparison, ZN

tuning, IMC, simple GA, and the proposed GA tune the PI controller for the

pH process with 20% changes in acid flow rate. The performance indices

obtained from various tuning methods are shown in Table 2. Figure 8

shows the tracking performance of the proposed GA, ZN, and IMC tuned PI

controller for various set point levels. The proposed GA has minimum

settling time and minimum peak overshoot, compared to the other tuning

methods. The ZN method shows satisfactory response, but it could not

settle at the set point 9. IMC tuning of PI shows more peak overshoot at the

set point 7 and set point 9.

Figure 9. Experimental setup of pH process.

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Experimental Results

This section presents the details of the experimental work carried out in lab-

oratory grade pH process. Figure 9 shows the arrangement of the experimental

setup used in this work. In this setup, acetic acid is fed to the reactor at a

constant flow rate and sodium hydroxide is added to the reactor through the

pump which controlled by the PI controller. Perfect mixing is ensured by

using a stirrer to maintain uniform concentration throughout the reactor.

The tank is provided with two nozzles to feed the inlet streams; the

capacity of the tank is 7.4 liters. The contents of the tank are agitated by a

four-paddle stirrer and the pH value is measured by a pH electrode.

The pH process is interfaced with the Pentium IV PC using a dSPACE

1104 real time data acquisition card. The card supports 8 ADC and 8 DAC

channels and the conversion speed of the card is 2 ms for ADC and 10 ms

for DAC. The MATLAB program is converted to C-code through the

dSPACE card and the control desk software is provided with the data acqui-

sition card. The real time coding is executed with a sampling rate of 10

seconds.

The results obtained using the proposed GA and IMC tuned PI controller

with set point level 11 is shown in Fig. 10. From the figure, it is found that the

IMC tuned PI controller has a greater rise time. The response of the process to

load disturbance, which is introduced in the set point 5, is shown in Fig. 11.

Figures 12 and 13 represent the behavior of the feed flow rate of base and

pH response of the pH process using different tuning methods at set point 7.

It shows that the control performance using IMC tuning has more peak

overshoot, i.e., similar to the results obtained using simulation. Overall the

performance of the controller is found to be good when it is tuned with the

proposed GA.

Figure 10. Comparison of EGA, GA, and IMC tuned PI (set point 11).

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Figure 11. Results of EGA with load disturbances (set point 5).

Figure 12. Feed flow rate of pH process using different tuning methods.

Figure 13. pH value of pH process using different tuning methods.

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CONCLUSION

A tuning technique based on the enhanced genetic algorithm is developed and

applied to the real-time control of a pH process. The pH control is quite

difficult, due to non-linearity of the neutralization process. This process

requires good control to overcome the load disturbances. The optimal par-

ameters of the PI controller at each pH region are computed by using the

enhanced genetic algorithm. From the simulations, this enhanced GA tuned

PI controller has minimum settling time and minimum MSE, compared to

the ZN and IMC methods. Experimental results show that the proposed con-

troller is able to tune quickly and reduce the tracking error to zero, in a manner

similar to the simulation results.

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Received May 17, 2007

Accepted July 30, 2007

Manuscript 1650

Proportional Integral Controller Tuning 635