NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1, NO.2, JUL-DEC 2010 51

Performance analysis of IMC based PID controllertuning on approximated process model

Ankit K. Shah, Markana Anilkumar and Nishant Parikh

AbstractClassical Proportional Integral Derivative(PID) con-troller remains the most popular approach for industrial processcontrol. Poor tuning of PID controller can lead to mechanicalwear associated with excessive control activity, poor controlperformance and even poor quality products. In this paper, wedesign procedure for the internal model control(IMC) approachfor tuning of conventional PID controller with proper tuningrules. Furthermore, with help of analytical rule of step testobtaining the effective first order time delay model of the process.A simulation example of continuous stirred tank reactor is usedin which the IMC based PID tuning method implemented andthe step response of the closed loop system is compared withclassical tuning methods like Ziegler-Nichols and Cohen-Coon.

Index TermsNormalized cutset, discrete wavelet transform,high boost filter

I. INTRODUCTION

NEVERTHELESS , PID controllers are still widely usedin industrial applications including for process control.The reason is that PID controller has a simple structure whichis easy to be understood by the engineers who design it. Thisis not only due to the simple structure which is conceptuallyeasy to understand and, which makes manual tuning possible,but also to the fact that the algorithm provides adequateperformance in the vast majority of applications. It is widelyused in process industries because of its simple structureand robustness to the modeling error. Sophisticated controlalgorithms, such as model predictive control, are built onthe basis of the PID algorithm. Even in non-linear controldevelopment, PID control has been used as comparisonreference [1].

According to a survey conducted by Japan electricMeasuring Instruments Manufacturers Association in 1989,90 percent of the control loops in industries are of PID type[1] and only small portion of the control loop works well.Also survey by ender [2] indicates 30 percent of the controllerare operated in manual mode and 20 percent of the loopsuse factory tuning. It means that PID controller is widelyused but poorly tuned. Poor tuning can lead to mechanicalwear associated with excessive control activity, poor controlperformance and even poor quality products. The present

Ankit K. Shah is with Instrumentation and Control DepartmentSardar Vallabhbhai Patel,Inst. of Tech.,Vasad-388306,India.Email: ak-snishaa@gmail.com Telephone: 9408572742

Markana Anilkumar is with Systems and Control Engineering Indian Insti-tute of Tech.-Bombay,Mumbai-400076, India Email: anil.markana@iitb.ac.in

Nishant Parikh is with School of Petroleum Technology Pandit Deen-dayal Petroleum University Gandhinagar, Gujarat-382007, India.Email:nish23481@gmail.com Telephone: (+91)79-232-75025

work is aimed to provide PID controller tuning guidelinesusing Internal Model Control(IMC) approach. Recently muchresearch effort has been focused on the automatic tuning ofPID controllers, which was first proposed by Astrom andHagglund (1984) [2]. They have introduced novel relay tuningmethod for finding the critical gain and critical frequencyof closed loop process [2] and proposed several tuning rulesfor PID controllers based on this information. The PIDcontroller tuning is method of computing the three controlparameters Proportional gain, Derivative time and Integraltime, such that the controller meets desired performancespecification. Since the exact dynamics of the plant isgenerally unknown, the basic function of autotuners is someexperimental procedure through which plant informationis obtained in order to compute the controller parameters.Tuning techniques can therefore be classified according tothis experimental procedure. This is particularly true for theoptimal PID controller tuning for time delay processes sincethe stability check for a given time-delay closed-loop systemis not a trivial task.

In the second section, the design steps for Internal ModelControl(IMC) will be describe. In the same section tuningformulas for the conventional PID controller closed loopsystem will be given. In the third section simulation resultson continuous flow stirred-tank reactor(CSTR) will be givenwith comparison study of closed loop PID controller responseusing Ziegler-Nichols, Cohen-coon and IMC tuning methods.The fourth section concludes the our approach of IMC basedPID controller parameter tuning.

II. BASIC DESIGN OF INTERNAL MODELCONTROL(IMC)

In this section, we will develop the IMC approach forPID controller tuning. The name comes from the fact thatthe controller has explicit model of the plant as its part [3].The premise of IMC is that in reality, we only have anapproximation of the actual process. Even if we have thecorrect model, we may not have accurate measurements ofthe process parameters. Thus the imperfect model should befactored as part of the controller design.

In the block diagram shown in Fig. 1 [4] implementationof IMC on the process transfer function Gp is given. In thatGp is the approximate transfer function of the process Gpand Gc is the model controller. In Fig. 2 block diagram ofthe conventional feedback controller shown. By comparison ofFig. 1 and Fig. 2, conventional feedback controller Gc consistsof Gc and Gp. We first need to derive the closed-loop functions

52 NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1, NO.2, JUL-DEC 2010

Fig. 1. Block diagram of internal model control structure

Fig. 2. Block diagram of conventional PID structure

for the IMC system. Based on the block diagram of the IMC,the error is E = R-(C -C) where C = Gp. We can rearrangedthe equations and get the controller output P as

P =Gc

1GcGp(R C) (1)

This step is to shows the relationship between the conven-tional feedback controller transfer function Gc shown in Fig.2 and IMC structure in Fig. 1:

Gc =Gc

1GcGp(2)

The poles of Gc are inherited from the zeros of Gp. If Gphas positive zeros, it will lead to a Gc function with positivepoles. To avoid that, we split the approximate function as aproduct of two parts:

Gp = Gp+ + Gp (3)

with Gp+ containing all the positive zeros, if present. Thecontroller will be designed on the basis of Gp.. only. Wenow define the model controller transfer function Gc as

Gc =Gf

Gp(4)

where Gf is first order low pass filter [5] used to avoidmodel mismatch. The filter transfer function Gf defined as

Gf =1

fs+ 1(5)

where f is the filter time constant.

III. PID TUNING USING IMC

For conventional PID controller transfer function is givenby

Gc(s) = Kp[1 +1

is+ ds] (6)

where Kp is the proportional gain, i is the integral timeand d is the derivative time or lead time.To model our process by fitting the open-loop step test dataas a first order function with time delay, our measured orapproximate model transfer function Gp given by

Gp =Ketds

ps+ 1(7)

where K represents the dc gain, p and td the process timeconstant and time delay, respectively. Replacing the time delayof by a first order Pade rational approximation expressed by[5]

etds 1td2 s

1 + td2 s(8)

The approximate model transfer function represent in (7)can be factorized in to invertible and noninvertible factors as

Gp =K

(ps+ 1)(1 +td2 s)

(9)

Gp+ = (1 td2s) (10)

From (9) and (4), IMC controller transfer function can bederive as

Gc =(ps+ 1)(1 +

td2 s)

K(fs+ 1)(11)

Comparing equation (11) with the ideal PID controllerequation represented by (6), which will lead to the tuningparameters of an ideal PID controller based on IMC approach.Controller parameters are given as

Kp =2ptd

+ 1

K(2ftd

+ 1)(12)

=td2

+ p (13)

d =p

2ptd

+ 1(14)

IV. SIMULATION RESULTS ON CONTINUOUSSTIRRED TANK REACTOR(CSTR)

Fig. 3 shows the continuous stirred tank reactor for reactantA. In this example, we have a stirred tank with a volumeV of 4m3 being operated with an inlet continuous flow rateQ of 0.4m3/sec and which contains an inlet reactant A at aconcentration Cin.

The model equation for continuous flow stirred-tank reactorwith chemical reaction of the reactant A given as

Vd

dtCA = Q(Cin CA) V CA (15)

where CA is the molar concentration of reactant A and isthe first order reaction rate constant of value 0.1sec1. Aboveequation can be written in simplified manner by

NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1,NO.2, JUL-DEC 2010 53

Fig. 3. Continuous stirred tank reactor

V

Q

d

dtCA + (1 +

KV

Q)CA = Cin (16)

By taking laplace transformation of above equation thetransfer function of CSTR can be written as

Gps =CACin

=

11+KVQV

Q+KV s+ 1(17)

The analog output of the concentration detector is trans-mitted to a controller, which in turn sends a signal to theinjection regulating valve at input stream. Photodetector isused to monitor the concentration of reactant A. The magicphotodetector is extremely fast and the response is linearover a large concentration range. Unite of concentration CAin terms of gmol/m3 now it converted into mV using themeasurement gain 2.6mVm3/gmol and transport lag td =0.7sec. The regulating valve is especially designed so that inletconcentration of the reactant A in gmol/m3 varies linearlywith the valve position. The regulating valve is thus first orderwith a time constant of 0.2 sec and a steady state gain of0.6 gmol/m3mV . Now complete system open loop blockdiagram show in Fig. 4, in that Gm is the measurement transferfunction, Ga is the valve transfer function and Gps is the actualtransfer function. Transfer function of Gm , Ga and Gps aregiven by

Fig. 4. Complete open loop block diagram of CSTR

Gps =0.67

6.67s+ 1(18)

Gm = 2.6e0.7s (19)

Ga =0.6

0.2s+ 1(20)

Complete open loop transfer function of the CSTR given as

Gp =(0.67)(0.6)(2.6)e0.7s

(6.67s+ 1)(0.2s+ 1)(21)

However, to use the empirical tuning relations, we needto fit the data to a first order transfer function with deadtime. Thus at this stage, we probably would have obtainedthe approximation of the CSTR by giving step signal to theinput injection regulating valve. From the Fig. 5, first orderapproximate transfer function of complete CSTR system isgiven by

Gp =1.04e0.85

7.1s+ 1(22)

From equation (12), (13), (14) and (22) values of PIDcontroller parameters are

Kp = 6.87, = 7.52sec, d = 0.4sec (23)

For the Ziegler-Nichols [6] and Cohen-Coon [7] methodsconventional PID parameters are evaluated using tuning rulesgiven in Table 1. The values of PID controller parametersfor IMC, Ziegler-Nichols and Cohen-Coon tuning rules aregiven in Table 2. From Table 2, IMC tune PID controller haveproportional gain less and integral time more as compare toother two methods. The unit step response for the close loopcontrol of CSTR shown in Fig. 6 using above setting

54 NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1, NO.2, JUL-DEC 2010

Fig. 5. Open loop step response of CSTR Structure

Fig. 6. Closed loop response using different PID tuning methods

of PID controller parameters for given three methods. Thecomparison between the Ziegler-Nichols, Cohen-Coon andIMC PID tuning rules closed loop step response given in Table3.

V. CONCLUSION

Internal model control methodology elementary notionswere reviewed, with particular relevance being given to theconversion from the IMC structure to a conventional PIDcontroller configuration. From these we can able tune thegains PID controller by using IMC parameters. From Fig. 6and Table 3, IMC based PID controller unit step responserequired less settling time to reach desired concentration andgive less overshoot in the response compared to other twomethods. We can conclude that the IMC based tuning of PIDcontroller outperformed the Ziegler-Nichols and Cohen-Coontuning methods.

REFERENCES[1] K. Astrom and B. Wittenmark, Computer Controlled Systems: Theory

and Design, NJ: Englewood Ciffs, Prentice-Hall, 1984.[2] K. J. Astrom and T. Hagglund, Automatic tuning of simple regulators

with specifications on phase and amplitude margins, Automatica, Vol.20, 645-693, 1984.

[3] K. Moudgalya, Digital Control, John Wiley and Sons Ltd., England, 2007.[4] P. B. Oliveira and J. B. Cunha, Teaching PID controller tuning through

internal model control, Control Eng. Practica, Vol.6, 155-165, 1998.[5] M. Sharusuzzoha, M. Lee and J. Lee, IMC-PID controller tuning for

improved disturbance rejection of unstable time delay processes, Theoriesand application of Chemical Engineering, Vol. 11, 2005.

[6] C.C. Yu, Autotuning of PID controllers, Springer, 1999.[7] M. A. Garcia-Alvarado, I. I. R. Lopez and T. T. Ramos Tuning of

muultivariate PID controller based on characteristic matrix eigenvalues,Lyapunov functions and robustness creteria, Chemical Engineering Sci-ence, Vol. 60, 897-905, 2005.

Anil Markana received his B.E. degree in In-strumentation and Control engineering from GECGandhinagar, Gujarat University, India, in 2000,M.Tech degree in Systems and Control engineeringfrom Indian Institute of Technology, Bombay, India,in 2008, and currently pursuing Ph.D. degree inSystems and Control engineering from the IndianInstitute of Technology, Bombay, Mumbai. He iscurrently a lecturer in the School of PetroleumTechnology, Pandit Deendayal Petroleum University,Gandhinagar, Gujarat, India. His current research

interests include advance Process Control, Optimal Control strategies likeModel Predictive Control, Linear Quadratic Gaussian Control, GPC, MinimumVariance Control, Industrial Automation, Distributed Control Systems, andModelling etc.

Ankit Shah received his B.E. degree in Instru-mentation and Control engineering from L. D. Col-lege of Engineering, Ahmedabad, Gujarat, India, in2002 and M.Tech degree in Systems and Controlengineering from Indian Institute of Technology,Bombay, India, in 2010. He is currently a lecturerin the Instrumentation and control dept. at SardarVallabhbhai Inst. Of Tech., Vasad, Gujarat, India.His current research interests include areas of thehybrid system identification, advance process controland digital control.

Nishant Parikh received his B.E. degree in Instru-mentation and Control engineering from ShantilalShah College of Engineering and Technology, Bhav-nagar, India, in 2002, M.Tech degree in Systemsand Control engineering from Indian Institute ofTechnology, Bombay, India, in 2008, and currentlypursuing Ph.D. degree in Chemical engineering fromthe Indian Institute of Technology, Bombay, Mum-bai. He is currently a lecturer in the School ofPetroleum Technology, Pandit Deendayal PetroleumUniversity, Gandhinagar, Gujarat, India. His current

research interests include areas of system identification, advance processcontrol and digital control.