Studies in Informatics and Control, xx(x) 1-3, Month Year ISSN: 1220-1766 eISSN: 1841-429X [631222]

Studies in Informatics and Control, xx(x) 1-3, Month Year ISSN: 1220-1766 eISSN: 1841-429X
https://doi.org/10.24846/x
ICI Bucharest © Copyright 2012-2017. All righ ts reserved http://www.sic.ici.ro 1
1. Introduction
As it is known, due to the simplicity of the PID
structure and its capability to control almost all
processes, the PID controllers are the oldest
and the most widely used in industrial
applications despite the recent advances in
information technology and computer science
(Marlin, 1995; Brosilow and Joseph, 2002; Liu
et al, 2005; Chia and Lefkowitz, 2010; Visioli
and Zhong, 2011, Jin and Liu, 2014). For many
complex processes (especially with time delay,
with overshoot or of nonminimum phase), the
PID controllers cannot achieve a very good control performance; in addition, there is not a
simple and intuitive method of controller
tuning (Ziegler and Nichols, 1942; Garcia and
Morari, 1982; Duma et al, 2011; Nicolau, 2013;
Singh et al, 2014; Tabatabaei, 2016). The
proposed control algorithm is inspired from the internal model control (IMC) philosophy, which states that an accurate control can be
achieved only if the control system
encapsulates some representation of the
process to be controlled (Francis and Wonham,
1976; Bengtsson, 1977; Garcia and Morari, 1982; Rivera et al, 1986; Horn et al, 1996).

In any IMC structure, the process model is
connected in parallel to the process and the
difference between their outputs comes back to the internal controller. The overall controller of
an IMC structure is parameterized as follows
()()
1( )( )C
MQsGs
QsG s=
−, (1)
where ()MsG is the transfer function of the
process model and ()sQ – the transfer function
of the internal controll er. In the IMC strategy,
()sQ is an approximate inverse of the model
transfer function, which includes a suitable
filter to guarantee the controller properness.
Note that the lag time constant of the filter is
just the controller tuning parameter.
Despite its advantages, the IMC strategy did
not become a strong practical alternative to the PID strategy (Normey and Camacho, 2007;
Saxena and Hote, 2012; Vanavil et al, 2014;
Cirtoaje, 2017) because of the multitude of
model variants that depend on the process type
(linear or nonlinear, with or without overshoot,
with or without oscillations, of minimum or
nonminimum phase, of proportional or integral
type, stable or unstable etc.). The proposed On a Model Based Practical Control Algorithm
Vasile CIRTOAJE, Alina BAIESU
Department of Control Engineering and Computers,
Petroleum and Gas University of Ploiesti,
Bdul Bucuresti, 39, 100680, Ploiesti, Romania
[anonimizat] (corresponding author ), [anonimizat]

Abstract : The design of the proposed algorithm relies on three basic ideas: (1) finding a model based controller such tha t
for any stable process of proportional type, the closed-loop controller output to a step reference removes the steady-state
error and has a step shape (or close to this form); (2) refining the controller structure such that the initial value of the
controller output to a step reference is K times its final value, where K is a tuning parameter wi th standard value 1; (3)
extending the controller structure to address integral processes and some unstable processes by turning them into stablecompensated processes of proportional type. The overall controller is a series connection P-IMC of two systems: one o
f
pure proportional type and another one of IMC type. There is a simple procedure to verify online if the model parameters
(steady-state gain, time delay and transient time) have suitable values and to adjust them to improve the model. As the
PID control algorithm, the proposed algorithm is quasi-universa l and practical, but it is superior by its control performance
and the simplicity of the tuning procedure (which allows persons with poor preparation to easily operate the control
system). Also, it is more practical than the classical IMC algo rithm because its equations have a unique form for processes
of all types (as PID algorithm), and has as tuning parameter a control gain in stead of a filter time constant. Several
applications are given to show the effectiveness of the algorithm for processes of various types.
Keywords: practical and quasi-universal algorithm, model based al gorithm, online tuning, transient time, time delay,
proportional process, integral process, unstable process, compensated process.

ICI Bucha r2
control a
using a
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In the
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last years,
d various t u
PID controll e
s (Brosilow ,
2010; Jin a n
anavil et al, 2
housiya et al,
performance
ID algorithm s
obtained by u
IMC type.
neral structur e
system is sho w
ocess transfe r
function, Y-
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t), E- c o
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1. Block diagr a
sy
nsfer functio n
s R and V ha
CP
CPGG
GG+, G
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ws the follo w
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ng to (4), in o
ance of the cl
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ht 2012-2017. Amoves this dis
odel structu r
he process to
proportional t y
s (if the p r
ble).
many publi c
uning techni q
ers for vari o
, 2002; Na g
nd Liu, 2014 ;
2014; Santos h
2017). In ou r
derived by
s is generall y
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e of a linea r
wn in Figure
r function, G
– controlled
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ontrol erro r,
am of a close d
ystem.
ns between
ave the expre
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wing relation
nit step refe r
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order to analy
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p reference.
Vasile Cirt o
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re which i s
be controlle d
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rocess is o f
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geswara an d
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uture researc h
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out minimu m
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sfer function
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wing practica l
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unction of a
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ights reserved (9 )
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ntrol algorith m
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where Pτ
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where T
integer v
time del a
M
MlTτ⎡=⎢⎣
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()u∞ of
to a unit
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control r
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where 1(
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a perfe c
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MK
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ion, the resp o
/MKK on the
ng/decreasin g
human oper a
ronger/weak e
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response ()ut
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PK, then the i n
ue ()u∞ are e
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nction of ma g
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odel steady- st
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initial value
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final value).
ent of the m o
1
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mpling time
ratio betwe e
mpling time;

(0)+ and th e
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e are
1)
PK= .
onse ()ut kee
time interva l
g the tuning
ator can ma k
er.
tial and final
) to a unit s t
()1 /PK∞= . In
nitial value u
equal. For a p
PK= ), the re s
gnitude 1/PK

it step functi o
perfect, the r
eference cha n
ion. Assume
tate gain MK
state gain K
nse ()ut to a s
(0 )u+ diffe r
()u∞; mo rVasile Cirt o
All rights reserve dlay, and 1t is
e response y
odel (10) ha s
/,( 1 3 )MTT
and Ml- the
en the mode l
that is,
(14)
e final valu e
response ()ut
(15)
eps its initia l
l [0, ]Mτ. By
gain K, the
ke the contro l
values of th e
tep referenc e
n addition, i f
(0 )u+ and th e
perfect mode l
sponse ()ut is
P, i.e.
(16)
on. Since th e
response ()ut
nge rΔ is no t
further tha t
M differs fro m
PK, then th e
step referenc e
rent from it s
re precisely ,
oaje, Alina Bai e
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time trT
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negativ e
In conc
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to be su i

Figure 3
referenc e
Figure 4
referenc e
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()u∞ for MK
PK> ( Figu r
model time d e
time dela y
n from the s t
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n is positive
Pτ> (Figure 4
odel transie n
Pτ≈ differs
Pr, then ()ut
rm in a tim
n is positi v
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lusion, if th e
a step refer e
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respective p a
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elay Mτ dif
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tep form im m
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for M Pττ<,
4).
nt time trMT
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e zone wit h
ve for trMT
trPT ( Figure 5
e control sy s
ence is large r
of magnitud e
arameter of t h
sed/decrease d
tem responses
values of MK
tem response s
values of Mτ.
ici.ro (0 ) ( ) uu+<∞
ffers from th e
()ut has a
mediately aft e
,{ }PMττ; thi
, and negativ e
for MPKK≈
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5).
stem respons e
r/smaller tha n
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he model nee d
d.
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In practi c
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MK, Mτ a
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ponse ()ut ha
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del time del a
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u from the
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process m o
a good mode l
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fK is larger /
u is also lar g
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u to a ref e
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have the bes t
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()ut to a ste p
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e initial valu e
close to eac h
ay Mτ and ,
trMT – suc h
step form i n
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good contro l
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ence step fo r
gain K are
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he control r e
a step shap e
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values of K.
Consider the
shoot:
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1) (1 5 1)se
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step input.
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unit step r
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system r
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paramet e
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0.7K= ; 1; 1.
system respo n
0.7K= ; 1; 1.
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3; 2.
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3; 2.
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0M> and fo r
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(17 )
(18 )
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p reference f o
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13 and 14
responses (yt
parameters Mτ
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.5; (B): RK=
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Consider the
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31)
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2.
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P,
0tPτ− .
formulae
mating the p r
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tion 2.3. C
1.5(
(4 1)(5s s++
he unit step p
gets
5, 9Pτ=,
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0 32Ptτ−=−
9. Process res p
20 and 21
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show the c o
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(20 )
(21 )
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step input.
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0.4; 0.63 ; 1.
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23, 6Mτ=
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The pro
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feedbac k
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respons e
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feedbac k
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process.
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compens a
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preferably w
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k gain fK, an
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sated proces s
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the possibi l
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26. Closed-l o
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agram of the
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extended t o
and som e
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MATIC mod e
21
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a bumpless t
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step refer e
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27). For fK=
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7. Compensat e
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sed-loop sys t
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28, 29 and
he response
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8. Control sys t
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30. Control sy s
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suitable K=
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step or ram p
output.
31. Control sys
p disturbance.
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1.3= , Figure
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a unit st e
0.7fK=
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Figure 3 2
the unit s t
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step refer e
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process whos e
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49 5= .
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0.8; 1; 1.2.
Vasile Cirt o
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path wit h
turns into a
e response t o
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are:
sponses ()yt,
nce are show n
0.8= ; 1; 1.2.
tem remain s
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oaje, Alina Bai e
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Due to
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m
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p
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34. Control sy s
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5. Control sys t
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nclusions a
its capabilit y
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algorithm is
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dynamic pe r
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ty of th e
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t.
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0.8; 1; 1.2.
tem responses
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On a Model Based Practical Control Algorithm

http://www.sic.ici.ro ICI Bucharest © Copyright 2012-2017. All rights reserved 13
algorithm (used by the process human operator
to increase or decrease the control action), and three model parameters that can be easily determined by experimental way – the model
steady-state gain
MK, the model time delay
Mτ and the model transient time trMT . In
addition, there is a simple procedure to verify
and correct online the model parameters for
1K= starting from the fact that if the model
has a high accuracy, then the controller output
to a step reference is close to a step form. This
feature ensures and guaranties the stability of
the closed-loop system on an upper bounded
interval of K. Analyzing the deviation of the
controller response to a step reference from the ideal step form, the process operator can adjust
online the model parameters to improve the
model accuracy.
In the extended variant, the control algorithm
has one more parameter, a process feedback
gain
fK, which is used to control integral-type
processes and some unstable processes by
turning them into stable proportional processes (compensated processes). The proposed algorithm has been implemented in real time, in laboratory and industrial
applications, with excellent results.
In our opinion, the presented control algorithm
can be still improved to reduce the weight of
the tuning gain
K on the control action,
especially for the processes with large time
delay, where the controller response ( ) ut to a
unit step reference keeps its initial value
/M KK on the whole time interval [0, ]Mτ.
This can be achieved in future research by
improving the IMC component of the actual
P-IMC algorithm after the procedure shown in
[6], which allows to get a strictly monotone
response ( ) ut on [0, ]Mτ.
REFERENCES
1. Bengtsson G. (1977). Output regulation
and internal models – a frequency domain
approach, Automatica , 13(4), 333-345.
2. Brosilow C., Joseph B. (2002). Techniques
of Model-Based Control , Upper Saddle
River (NJ), Prentice Hall. 3. Chia TL., Lefkowitz I. (2010). Internal
model-based control for integrating
processes, ISA Transactions , 49(4), 519-
527.
4. Cirtoaje V. (2006). Process compensation
based control, Buletinul Universitatii
"Petrol-Gaze" din Ploiesti , Seria Tehnica,
LVIII(1), 48-53.
5. Cirtoaje V., Baiesu A. (2016). Comparative
study of three practical IMC algorithms with inner controller of first and second
order, JEEE Control and Computer
Science , 2(2), 21-28.
6.
Cirtoaje V. (2017). A quasi-universal
practical IMC algorithm, Automatika,
58(2), 168-181.
7. Duma R., Trusca M., Dobra P. (2011).
Tuning and implementation of PID
controllers using rapid control prototyping, J. of Control Engineering and Applied
Informatics , 13(4), 64-73.
8.
Francis B., Wonham W. (1976). The
internal model principle of control theory,
Automatica , 12(5), 457-465.
9. Garcia C., Morari M. (1982). Internal
model control – A unifying review and
some new results, Ind. Eng. Chem. Proc.
Des. Dev ., 21(2), 308-323.
10. Ghousiya BK., Seshagiri RA., Radha-
krishnan TK. (2017). Enhanced IMC based PID controller design for non-minimum
phase integrating processes with time
delay, ISA Transactions , 68, 223-234.
11.
Horn I., Arulandu J., Gombas C.,
VanAntwerp J., Braatz R. (1996).
Improved filter design in internal model
control, Ind. Eng. Chem. Res ., 35, 3437-
3441.
12. Jin B., Liu Q. (2014). Analytical IMC-PID
design in terms of performance/robustness
trade-off for integrating processes, J.
Process Control , 24(12), 22-32.
13. Liu T., Zhang W., Gu D. (2005). Analytical
design of two-degree-of-freedom control
scheme for open-loop unstable processes
with time delay, J. Process Control , 15(5),
559-572.

Vasile Cirtoaje, Alina Baiesu
ICI Bucharest © Copyright 2012-2017. All righ ts reserved http://www.sic.ici.ro 14
14. Marlin T. (1995). Process Control –
Designing processes and Control Systems
for Dynamic Performance , New York,
McGraw Hill, Inc.
15. Nageswara Rao CV., Padma Sree R.
(2010). IMC based controllers design for
integrating system with time delay, Indian
Chem Eng , 52(3), 194-218.
16. Nguyen TH., Kaneko O., Yamamoto S.
(2013), Fictitious reference iterative tuning to modified IMC for unstable plants,
Journal of Control, Measurement, and
System Integration , 6(5), 345-352.
17.
Nicolau V. (2013). On PID controller
design by combining pole placement
technique with symmetrical optimum
criterion, Mathematical Problems in
Engineering , 1-8.
18. Normey-Rico JE., Camacho EF. (2007).
Control of Dead-time Processes , Spriger-
Verlag London Ltd.
19. Rivera D., Morari M., Skogestad S. (1986).
Internal model control – PID controller
design, Ind. Eng. Chem. Process Des.
Dev., 25, 252-265.
20. Santosh Kumar DB., Padma Sree R.
(2016). Tuning OF IMC based PID
controllers for integrating systems with
time delay, ISA Transactions , 63, 242-255.

21.
Saxena, S., Hote, Y. (2012). Advances in
Internal Model Control Technique: A
Review and Future Prospects, IETE
Technical Review , 29(6), 461-472.
22. Singh, R., Bala, R., Bhatia, B. (2014).
Internal Model Control and IMC Based
PID Controller, International Journal of
Advanced Research in Computer Science
and Software Engineering , 4(6), 915-922.
23. Tabatabaei, M. (2016). PID controller
design based on Laguerre orthonormal
functions, J. of Control Engineering and
Applied Informatics , 18(4), 1-12.
24. Vanavil B., Anusha AV., Perumalsamy M.,
Rao AS. (2014). Enhanced IMC-PID
controller design with lead-lag filter for
unstable and integrating process with time
delay, Chemical Engineering Communi-
cation , 201(11), 1468-1496.
25. Visioli A., Zhong QC. (2011). Control of
Integral Processes with Dead Time ,
Springer Verlag London Ltd.
26. Yamada, K., Hagiwara, T. (2006). A design
method of modified PID controller for
unstable plant, Intelligent Engineering
System Through Artificial Neural
Networks , 16, 753-7586.
27. Ziegler JG., Nichols NB. (1942). Optimum
setting for automatic controllers, ASME
Trans , 759-768.