-
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: [email protected] Telephone:
9408572742
Markana Anilkumar is with Systems and Control Engineering Indian
Insti-tute of Tech.-Bombay,Mumbai-400076, India Email:
[email protected]
Nishant Parikh is with School of Petroleum Technology Pandit
Deen-dayal Petroleum University Gandhinagar, Gujarat-382007,
India.Email:[email protected] 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.