Design of controller using variable transformations for a nonlinear process with dead time

1. ISA
TRANSACTIONS®
ISA Transactions 44 ͑2005͒ 81–91
Design of controller using variable transformations for a nonlinear
process with dead time
R. Anandanatarajan,a,* M. Chidambaram,b T. Jayasinghc
a
Pondicherry Engineering College, Pondicherry, India
b
IIT, Chennai, India
c
St. Xaviers College of Engineering, Nagercoil, India
͑Received 30 October 2003; accepted 5 June 2004͒
Abstract
In this work, a globally linearized controller ͑GLC͒ for a first-order nonlinear system with dead time is proposed.
This is similar to the GLC proposed by Ogunnalke ͓Ind. Eng. Chem. Process. Des. Dev. 25, 241–248 ͑1986͔͒ for
nonlinear systems without dead time. Two methods are proposed. One is based on the Smith prediction from the model
in the transformed domain and the other is based on Newton’s extrapolation method. The simulation study is made on
the conical tank level process and the results are compared with those obtained using a conventional PI controller and
the Smith PI controller based on the transfer function model about the operating point 39%. Finally, experimental
results on the laboratory conical tank level process are also given. © 2005 ISA—The Instrumentation, Systems, and
Automation Society.
Keywords: Nonlinear control; GLC; Conical tank
1. Introduction volves linearizing the modeling equation around a
steady state and applying linear control theory re-
Time delays occur in many control applications, sults ͓4͔. It is obvious that the controller perfor-
typically as a result of transporting material or en- mance in this case will deteriorate as the process
ergy. When the time delay is significantly greater moves further away from the steady state around
than the process dominant time constant, serious which the model was linearized.
degradation of the closed-loop performance can be Apart from the ‘‘local linearization’’ approach,
expected with traditional controllers. For instance, there are a few other ‘‘special purpose’’ design
the traditional PI controller results in very sluggish procedures ͓5͔ which may be applied directly to
closed-loop response ͓1͔. Faster reference tracking nonlinear systems. However, as noted by Ray ͓5͔,
and disturbance rejection can be achieved by the these usually have limited applicability and are of-
Smith predictor. However, its main weakness, ten based on accumulated experience with a spe-
compared to the PI controller, is a lack of robust- cial type of nonlinear system. A new modified
ness to variations in process parameters ͓2,3͔. Smith predictor is proposed by Vrecko ͓6͔ to im-
The traditional and easiest approach to the con- prove the robustness but again it is a linear con-
troller design problem for nonlinear systems in- troller.
The design of a controller based on the variable
*Corresponding author. E-mail address: transformation for a nonlinear system without
annandaគnatarajan@yahoo.com dead time is presented by Ogunnalke ͓7͔. In this
0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.

2. 82 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91
present paper, it is shown that how transforma-
tions may be found which transform the nonlinear
system with dead time into one that is ideally lin-
ear. The approach is based on the hypothesis that a
system which is nonlinear with dead time in its
original variables is linear in some transformation
of the predicted future original variables. Con-
structive methods for finding the appropriate trans-
formations are presented. The design of a control-
ler for the linear transformed system may then be
carried out with a great deal of facility. The result-
Fig. 1. Conceptual configuration of the GLC controller.
ing controller will of course be nonlinear when
recast in terms of the original system variables. In
the present work, the specific problem of a conical
tank level process system having dead time is used
to illustrate the potentials of this approach. An ex- where a and b are constants, v is a state variable,
ample simulation and real time implementation of and g ( • ) is a function to be determined. Differen-
the controller on a conical tank laboratory level tiating Eq. ͑3͒ with respect to t,
process are presented.
dz dg dx
ϭ . ͑5͒
dt dx dt
From Eqs. ͑1͒ and ͑2͒
2. Design of globally linearized controller
„GLC… dx
ϭc 1 f 1 ͑ x ͒ ϩc 2 f 2 „x,u ͑ tϪT d͒ …. ͑6͒
dt
2.1. Variable transformation
Substituting Eq. ͑6͒ in Eq. ͑5͒,
A single-input, single-output nonlinear control
dz dg dg
system with dead time can be represented in gen- ϭc 1 f 1 ϩc 2 f 2 . ͑7͒
eral by dt dx dx
dx Let
ϭF„x,u ͑ tϪT d͒ …, ͑1͒
dt dg
c 1 f 1͑ x ͒ ϭa ͑8͒
where F ( • ) is an arbitrary nonlinear function of x, dx
the system state variable, and u, the control vari- and
able, and T d is the dead time in the process. The
function F„x,u ( tϪT d) … is split up as dg
c 2 f 2 „x,u ͑ tϪT d͒ … ϭb v . ͑9͒
F„x,u ͑ tϪT d͒ …ϭc 1 f 1 ͑ x ͒ ϩc 2 f 2 „x,u ͑ tϪT d͒ …, dx
͑2͒ The nonlinear system is thus mapped to
where f 1 ( x ) is a function of x alone and dz
f 2 „x,u ( tϪT d) … is a function of both x and ϭaϩb v . ͑10͒
u ( tϪT d) . Both f 1 and f 2 are taken to be nonlin- dt
ear. c 1 and c 2 are constants. A linear system may be effectively controlled with
zϭg ͑ x ͒ ͑3͒ a PI controller as in Fig. 1. From Eq. ͑8͒, the trans-
formation g ( x ) which transforms the given non-
is a transformation for mapping the nonlinear sys- linear system with dead time into a linear system
tem F ( • ) to a linear system is
dz
dt
ϭaϩb v , ͑4͒ g͑ x ͒ϭ
a
c1
• ͵ f dxx ͒ .
͑
1
͑11͒

3. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 83
2.2. Prediction of process variable and controller
output
A conventional PI controller gives an output
v ( t ) which effectively controls z. The future value
of f 2 is predicted either by using Newton’s method
or by using the transformed model of the system.
Since f 2 is a function of v , once f 2 is predicted, v
can be easily obtained. The nonlinear control law
can be obtained using Eq. ͑12͒.
Fig. 2. Block diagram representation of the nonlinear con-
troller with Newton’s predictor method ͑NEM͒. The PI con-
troller is designed for the linear system ͓Eq. ͑10͔͒. 2.3. Newton’s extrapolation method
Newton’s extrapolation formula, used to predict
the future value, is
Substituting the value of dg/dx from Eq. ͑8͒ into
Eq. ͑9͒ yields f ͑ pϩh ͒ ϭ f ͑ p ͒ ϩhٌ f ͑ p ͒ ϩhigher-order terms,
͑13͒
c 1 f 1 b v ͑ tϩT d͒
u͑ t ͒ϭ . ͑12͒
c 2 a f 2 „x ͑ tϩT d͒ … where f ( pϩh ) is the future value to be predicted
after an interval of h sec and p is the sampling
Let time at which the latest value is available. ٌ is the
difference operator. ٌ f ( p ) is the difference be-
u ͑ t ͒ ϭq ͑ • ͒ , tween the latest sampled value and the previous
sampled value. The number of terms that has to be
where v ( tϩT d) is the predicted manipulated vari- considered in the above formula depends on the
able and f 2 „x ( tϩT d) … is the predicted function of number of past data available and the designer’s
state variable. In deriving Eq. ͑12͒, the specific interest. In their work, only the first two terms are
case of f 2 ( x ) •u ( tϪt d) is considered than considered ͓8͔. This method may be called New-
f 2 „x,u ( tϪt d) …. In general case u has to be solved ton’s extrapolation method ͑NEM͒.
numerically.
Thus a nonlinear controller u ( t ) is designed
based on a variable transformation for the first- 2.4. Variable transformation predictor
order nonlinear process with dead time. The con-
troller performance is tested by simulation of the The nonlinearity in the process and hence in the
conical tank level process. The proposed control- model limits the prediction. The model which is
ler is expected to be highly robust when the oper- used to map the nonlinear process by variable
ating point of the process is shifted over the entire transformation is linear and this model is used
span of the tank. q ( • ) is the transformation which here to predict the future variable as shown in Fig.
transforms the linear controller output in the trans- 3. The configuration is a combination of Smith
formed domain into the nonlinear controller out- predictor and variable transformation and hence
put in the original domain. may be called the variable transformation predic-
A PI controller is designed and interfaced with tor ͑VTP͒. The transfer function model of the
the pseudolinear system shown in Fig. 2. This is transformed pseudolinear system ͑10͒ is
made possible by the use of the transformations z ( s ) / v ( s ) ϭb/s. The transfer function model b/s
g ( x ) and u ( t ) . x * is the set point and z * is the set will give the predicted transformed process vari-
point in the transformed domain. Thus the entire able z Ј . The model prediction error e m is the dif-
design procedure boils down to the determination ference between the predicted output and the
of g ( x ) to compute the corresponding u ( t ) by us- transformed process variable z. e m is added in the
ing Eq. ͑12͒. feedback path as an additional error.

4. 84 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91
Fig. 3. Block diagram representation of the variable trans-
formation predictor. The PI controller is designed for the
linear system ͓Eq. ͑10͔͒. Fig. 4. Block diagram of the Smith PI controller.
but practically it is found that selection of large
3. Application of GLC to conical tank level
values of a and b end up with overflow error in the
process
transformed variable.
The transfer function of Eq. ͑15͒ is
The mathematical model of the conical tank liq-
uid level system considered for the study is ex- G ͑ s ͒ ϭb/s. ͑16͒
pressed as
The forward path transfer function of the closed-
dh ͓ u ͑ tϪT d͒ ϩu d͑ tϪT d͒ Ϫc ͱh ͔ loop system with a PI controller is
ͫ ͬ
ϭ , ͑14͒
dt R2 2 G f͑ s ͒ ϭbK c͑ 1ϩT is ͒ /T is 2 . ͑17͒
␲ 2h
H T i and K c can be obtained by assuming a closed-
where H, R, c are total height, top radius, and out- loop time constant and damping factor ͓9͔. Inter-
flow valve coefficient of the conical tank, respec- estingly, the tuning parameters are independent of
tively. h, u, and u d are liquid level, inflow rate, and local time constant and local gain of the process.
disturbance of the conical tank level process, re- The transformation which transforms the present
spectively. The transformation g ( h ) transforms process in to a linear system is
the nonlinear system represented by Eq. ͑14͒ into
Ϫ2a⌸R 2
a linear system. The transformed system is a g͑ h ͒ϭ h 5/2. ͑18͒
pseudolinear system, whose mathematical model 5H 2 c
is
4. Tuning of controllers
dz
ϭaϩb v . ͑15͒
dt The responses are compared with a conventional
The values of a and b are assumed as Ϫ0.7657 PI controller tuned about nominal operating point
and 0.0177, respectively. Mathematically there is of 39%. The time constant and gain of linearized
no restriction on the selection of values of a and b model are 76 sec and 1.2, respectively. The pro-
cess dead time is 32 sec. The Ziegler and Nichols
Table 1 Table 2
ISE of Regulatory responses for 15% decrease in load. ISE of Regulatory responses for 15% increase in load.
Operating ZN Smith Operating ZN Smith
point PI PI GLC point PI PI GLC
39% 1285 3269 928 39% 1236 3367 807
24% 2324 2762 1492 24% 1780 2991 2162
54% 1224 3740 518 54% 1221 3779 448

5. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 85
Table 3
ISE of servo responses for 20% increase in set point.
Operating ZN Smith
point PI PI GLC
39% 23670 35480 16090
24% 19260 27530 15780
54% 32030 47660 16260
͑ZN͒ tuning parameters for the linearized model
are K cϭ0.95 and T iϭ0.020. The tuning param-
eters for the Smith PI controller as shown in Fig. 4
are based on Haalman’s tuning rule. They are K c
ϭ0.5747 and T iϭ0.022 ͓10͔. Similarly for both
GLC’s are K cϭ2.87 and T iϭ0.025.
The simulation and experimentation are carried
out by taking 39% as nominal level. 24% and 54%
are other operating points used to test the robust- Fig. 5. Regulatory responses for a 15% decrease in load at
ness of the controller tuned at 39% nominal level. 39% nominal operating point.
The integral square error ͑ISE͒ values are pre-
sented in the Tables 1– 4. Regulatory responses for
a 15% decrease in load at nominal operating point a better response ͑26% lesser ISE͒ than the con-
39% ͑refer to Fig. 5͒ show that the proposed con- ventional Smith predictor. The ZN-PI controller
troller gives an improved response ͑28% lesser gives an oscillatory response. Regulatory re-
ISE͒ while the conventional Smith predictor sponses for a 15% decrease in load at 54% oper-
͑150% higher ISE͒ gives a poorer performance ating point but tuned at 39% ͑refer to Fig. 9͒ show
than the ZN-PI controller. that the proposed controller gives an improved
Regulatory responses for a 15% increase in load performance ͑58% lesser ISE͒ while the conven-
at nominal operating point 39% ͑refer to Fig. 6͒ tional Smith predictor ͑205% higher ISE͒ provides
show that the proposed controller improves the re- a poorer performance than the ZN-PI controller.
sponse ͑35% lesser ISE͒ while the conventional
Smith predictor ͑172% higher ISE͒ gives a poorer
performance than the ZN-PI controller.
Regulatory responses for a 15% decrease in load
at 24% operating point but tuned at 39% ͑refer to
Fig. 7͒ show that the proposed controller provides
a better response ͑36% lesser ISE͒ than the con-
ventional Smith predictor ͑19% higher ISE͒. The
ZN-PI controller gives an oscillatory response.
Regulatory responses for a 15% increase in load
at 24% operating point but tuned at 39% ͑refer to
Fig. 8͒ show that the proposed controller provides
Table 4
ISE of servo responses for 20% decrease in set point.
Operating ZN Smith
point PI PI GLC
39% 46920 28900 20030
24% 18918 14260 10930
Fig. 6. Regulatory responses for a 15% increase in load at
54% 28980 36270 18420
39% nominal operating point.

6. 86 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91
Fig. 7. Regulatory responses for a 15% decrease in load at Fig. 9. Regulatory responses for a 15% decrease in load at
24% operating point. 54% operating point.
Regulatory responses for a 15% increase in load proved response ͑31% lesser ISE͒ than the con-
at 54% operating point but tuned at 39% ͑refer to ventional Smith predictor. The ZN-PI controller
Fig. 10͒ show that the proposed controller im- provides an oscillatory response. Servo responses
proves the performance ͑63% lesser ISE͒ while for a 20% increase in the set point at nominal op-
the conventional Smith predictor ͑209% higher erating point 39% ͑refer to Fig. 12͒ show that the
ISE͒ gives a poorer response than the ZN-PI con- proposed controller improves the response ͑32%
troller. lesser ISE͒ while the conventional Smith predictor
Servo responses for a 20% decrease in the set ͑50% higher ISE͒ gives a poorer performance than
point at nominal operating point 39% ͑refer to Fig. the ZN-PI controller.
11͒ show that the proposed controller gives an im-
Fig. 8. Regulatory responses for a 15% increase in load at Fig. 10. Regulatory responses for a 15% increase in load at
24% operating point. 54% operating point.

7. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 87
Fig. 11. Servo responses for a 20% decrease in set point at
39% nominal operating point. Fig. 13. Servo responses for a 20% decrease in set point at
24% operating point.
Servo responses for a 14% decrease in the set domain are lesser than the tolerance that can be
point at 24% operating point but tuned at 39% handled by the computer.
͑refer to Fig. 13͒ show that the proposed controller Servo responses for a 20% increase in set point
improves the response ͑33% lesser ISE͒ more than at 24% operating point but tuned at 39% ͑refer to
the conventional Smith predictor. The ZN-PI con- Fig. 14͒ show that the proposed controller im-
troller gives an oscillatory response. At this oper- proves the response ͑18% lesser ISE͒ while the
ating point it is not possible to decrease the set conventional Smith predictor ͑43% higher ISE͒
point greater than 14% because of numerical gives a poorer performance than the ZN-PI con-
round-off error. Some values in the transformed troller.
Fig. 12. Servo responses for a 20% increase in set point at Fig. 14. Servo responses for a 20% increase in set point at
39% nominal operating point. 24% operating point.

8. 88 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91
Fig. 15. Servo responses for a 20% decrease in set point at
54% operating point.
Fig. 17. Servo responses of conical tank level process for
Ϫ8% step change at nominal operating point 39% using
GLC ͑VTP͒ and GLC ͑NEM͒ with the same PI settings.
Servo responses for a 20% decrease in set point
at 54% operating point but tuned at 39% ͑refer to
Fig. 15͒ show that the proposed controller im- ler and the conventional Smith predictor ͑49%
proves the response ͑36% lesser ISE͒ while the higher ISE͒ provides a very poor response.
conventional Smith predictor ͑25% higher ISE͒ The servo responses of the conical tank level
gives a poorer response than the ZN-PI controller. process for a 8% decrease in set point at the nomi-
Servo responses for a 20% increase in set point at nal operating point of 39% using GLC
54% operating point but tuned at 39% as in Fig. controllers—one with NEM prediction and the
16 show that the proposed controller improves the other with proposed VTP prediction but both with
response ͑51% lesser ISE͒ than the ZN-PI control- same tuning parameters—are shown in Fig. 17.
The response given by NEM has slight oscilla-
tions. The magnitude of oscillations increases as
the step magnitude increases. This indicates that
the prediction by NEM is very poor compared
with the proposed GLC with VTP.
Figure 18 shows the experimental servo re-
sponses for various set point changes at 39%
nominal level. The responses are slightly oscilla-
tory. Figures 19 and 20 show the experimental
regulatory responses for various magnitudes of
load changes at the nominal operating point. The
regulatory responses are found to have small os-
cillations as compared to the servo responses. Fig-
ure 21 shows a block diagram of the experimental
setup.
5. Conclusion
Fig. 16. Servo responses for a 20% increase in set point at A nonlinear controller is designed based on the
54% operating point. variable transformation for the first-order nonlin-

9. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 89
Fig. 18. Experimental servo responses of conical tank level process for set point changes at nominal operating point 39%
using GLC ͑VTP͒.
ear process with dead time. The performances of worse when the operating point is shifted to 54%.
proposed GLC ͑VTP͒ are tested by simulation and On the other hand, the proposed controller gives
compared with ZN PI and Smith PI controllers. better dynamics.
The ZN PI controller gives an oscillatory response Consider the regulatory responses at 24% oper-
for decrease in set point even at a nominal operat- ating point for decrease in load as shown in Fig. 7.
ing point. The situation becomes worse when the The ZN-PI controller gives a highly oscillatory re-
operating point is shifted to 24%. For any increase sponse. On the other hand, Smith PI and GLC give
in the set point the Smith PI controller gives a an oscillation free response. Similarly, consider
very poor response due to a very small dead time the regulatory responses at 24% operating point
to time constant ratio. This situation becomes for an increase in load as shown in Fig. 8. Here
Fig. 19. Experimental regulatory responses of conical tank level process for increase in load changes at nominal operating
point 39% using GLC ͑VTP͒.

10. 90 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91
Fig. 20. Experimental regulatory responses of conical tank level process for decrease in load changes at nominal operating
point 39% using GLC ͑VTP͒.
also the ZN-PI controller gives a highly oscillatory controller gives oscillatory responses and the
response but the Smith PI gives an oscillation-free Smith PI gives sluggish responses, GLC gives os-
response. GLC gives an oscillatory response but cillation free responses with quick rise time. The
quickly settles unlike that of the ZN-PI controller. simulation results show the robustness of the
In this case the drawback with the GLC is the GLC.
large undershoot. But it has less ISE than that of The proposed controller outperforms the Smith
Smith PI. Consider the regulatory responses at PI and the ZN-PI controllers when the operating
54% operating point ͑refer to Figs. 9 and 10͒. The point of the process is shifted over the entire span
Smith PI controller gives a large undershoot and a of the tank. While comparing VTP and NEM, VTP
large overshoot compared to that of the ZN-PI gives better dynamics than NEM. As a future
controller. But GLC gives a very small overshoot work, the GLC can be tested on other nonlinear
and undershoot compared to that of the ZN-PI systems; the work in this direction is ongoing.
controller. Consider the servo responses at 39%
and 24% ͑refer to Figs. 11 and 13͒. The ZN-PI Acknowledgments
We are grateful to Professor K. Ethirajulu for his
constant encouragement and providing the facili-
ties. We thank Professor P. Dhananjayan for his
encouragement.
References
͓1͔ Meyer, C., Seborg, D. E., and Wood, R. K., A com-
parison of the Smith predictor and conventional feed-
back control. Chem. Eng. Sci. 31, 775–778 ͑1976͒.
͓2͔ Hagglund, T., A predictive PI controller for processes
with long dead times. IEEE Control Syst. Mag. 12 ͑1͒,
57– 60 ͑1992͒.
͓3͔ Tan, K. K., Lee, T. H., and Leu, F. M., Predictive PI
versus Smith control for dead-time compensation. ISA
Trans. 40, 17–29 ͑2001͒.
͓4͔ Schneider, D. M., Control of process with time delays.
IEEE Trans. Ind. Appl. 24 ͑2͒, 186 –191 ͑1988͒.
͓5͔ Ray, W. H., Advanced Process Control. McGraw-Hill,
Fig. 21. Block diagram of experimental setup. New York, 1981.

11. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 91
͓6͔ Vrecko, D., Vrancic, D., Juricic, D., and Strmcnik, S.,
A new modified Smith predictor: The concept, design M. Chidambaram obtained
and tuning. ISA Trans. 40, 111–121 ͑2001͒. his B.E. ͑Chemical͒ from An-
namalai University, M.E.
͓7͔ Ogunnalke, B. A., Controller design for nonlinear pro- ͑Chemical͒ and Ph.D. from In-
cess systems via variable transformations. Ind. Eng. dian Institute of Science in
Chem. Process Des. Dev. 25, 241–248 ͑1986͒. 1977 and 1984, respectively.
͓8͔ Chidambaram, M., Anandanatarajan, R., and Jayas- He was faculty member in IIT-
ingh, T., Controller design for nonlinear process with Bombay during 1984 to 1991.
dead time via variable transformations. Proceedings of Since September 1991 he has
been a faculty member in IIT-
the International Symposium on Process Systems En- Madras. He had been head of
gineering and Control ͑ISPSEC’03͒, IIT, Mumbai, the Department of Chemical
2003, pp. 223–228. Engineering during the period
͓9͔ Chidambaram, M., Applied Process Control. Allied October 2000 to October 2003.
Publishers, New Delhi, India, 1998. He has authored three books: Nonlinear Process Control. John Wiley,
͓10͔ Tian, Y. C. and Gao, F., Double-controller scheme for 1996; Applied Process Control. Allied Publishers, 1998; Computer
Control of Processes. Narosa Publishers, 2002. He has published 135
control of processes with dominant delay. IEE Proc.: research publications in journals and 45 papers in conferences pro-
Control Theory Appl. 145 ͑5͒, 479– 484 ͑1998͒. ceedings. His areas of interest in process control are PID control, relay
tuning, and nonlinear control.
R. Anandanatarajan ob-
tained his B.Sc. ͑Mathematics͒
from Madras University in
1984, Bachelor degree in Elec- T. Jayasingh obtained his
trical from the Institution of Ph.D. in Control and Instru-
Engineers ͑India͒ in 1989, mentation from IIT-Delhi. He
M.Sc. ͑Mathematics͒ and M.E. has more than two decades of
͑Process control and Instru- experience in teaching and re-
mentation͒ from Annamalai search at Anna University,
University in 1994 and 1998, Chennai, India. Presently he is
respectively, Ph.D. from Anna a visiting professor at St.
University in 2003. Presently Xaviers College of Engineer-
he is assistant professor of the ing, Nagercoil, India. His areas
Department of Instrumentation of interest include process con-
Engineering, Pondicherry Engineering College. He has authored two trol, computer based instru-
books titled Signals & Systems and Computer Peripherals and Inter- mentation, and sensor model-
facing. ing.