Sliding mode control: an approach to regulate nonlinear chemical processes

Sliding mode control: an approach to regulate nonlinearchemical processes

Oscar Camacho a,*, Carlos A. Smith b

aDepartamento de Circuitos y Medias, Universidad de Los Andes, MeÂrida 5101, VenezuelabChemical Engineering Department, University of South Florida, Tampa, Florida, USA

Abstract

A new approach for the design of sliding mode controllers based on a ®rst-order-plus-deadtime model of the process,is developed. This approach results in a ®xed structure controller with a set of tuning equations as a function of thecharacteristic parameters of the model. The controller performance is judged by simulations on two nonlinear chemicalprocesses # 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Sliding mode control; Variable structure control; Nonlinear chemical processes

1. Introduction

Sliding mode control (SMC) is a robust andsimple procedure to synthesize controllers for lin-ear and nonlinear processes. To develop a slidingmode controller, SMCr, knowledge of the processmodel relating the controlled variable, Xc t� �, tothe manipulated variable, U t� �, is necessary.However, there are two problems with the use of amodel as far as chemical processes are concerned.First, the development of a complete model is dif-®cult due mainly to the complexity of the processitself, and to the lack of knowledge of some processparameters. Second, most process models relatingthe controlled and the manipulated variables areof higher-order. Generally, the SMC procedureproduces a complex controller, which could con-tain four or more parameters resulting in a di�culttuning job. Therefore, the use of the traditional

procedures of SMC presents disadvantages intheir application to chemical processes.An e�cient alternative modeling method for

process control is the use of empirical models,which use low order linear models with deadtime.Most times, ®rst-order-plus deadtime (FOPDT)models are adequate for process control analysisand design. But, these reduced order models pre-sent uncertainties arising from imperfect knowl-edge of the model, and the process nonlineare�ects contribute to performance degradation ofthe controllers.Conventional controllers, such as PID, lead-lag

or Smith predictors, are sometimes not su�cientlyversatile to compensate for these e�ects. Thus, aSMCr could be designed to control nonlinear sys-tems with the assumption that the robustness ofthe controller will compensate for modeling errorsarising from the linearization of the nonlinearmodel of the process.The aim of this paper is to design a SMCr based

on a ®rst-order-plus-deadtime (FOPDT) model ofthe actual process. The overall idea is to develop a

0019-0578/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PI I : S0019-0578(99 )00043-9

ISA Transactions 39 (2000) 205±218

www.elsevier.com/locate/isatrans

* Corresponding author. Tel.: +58-74-402-891; fax: +58-

74-402-890.

E-mail address: [email protected] (O. Camacho).

general SMCr, which can be used for self-regulat-ing chemical processes. The parameters of themodel, process gain, K, process time constant, �,and process deadtime, t0, are used to obtain theinitial estimates of the tuning terms in the SMCr.This article is organized as follows. Section 2

brie¯y presents the process model. Section 3, pre-sents some basic concepts of the SMC method.Section 4 shows the procedure to design a SMCrusing the FOPDT model. Tuning equations forthe controller are also given in this section. InSection 5 the simulation of the SMCr for twononlinear chemical processes is presented. Section6 concludes the paper.

2. Process model

The process reaction curve, Fig. 1, is an often-used method for identifying dynamic models 1. It

is simple to perform, and provides adequate modelsfor many applications. The curve is obtained byintroducing a step change in the output from thecontroller and recording the transmitter output.From the process curve shown in the ®gure, and

the procedure presented in the reference, thenumerical values of the terms in the FOPDTmodel given in Eq. 1 are obtained

X s� �U s� � �

Keÿt0s

�s� 1�1�

where X s� � is the Laplace transform of the con-trolled variable, the transmitter output, and U s� �is the Laplace transform of the manipulated vari-able, the controller output. Both X s� � and U s� � aredeviation variables. In this paper we use the unitof X s� � as fraction of the transmitter output, frac-tion TO; the unit of U s� � is fraction of the con-troller output, fraction CO. K, t0 and � werepreviously de®ned.

3. Basic concepts about sliding mode control

Sliding Mode Control is a technique derivedfrom variable structure control (VSC) which wasoriginally studied by [2]. The controller designedusing the SMC method is particularly appealingdue to its ability to deal with nonlinear systemsand time-varying systems [3±5]. The robustness tothe uncertainties becomes an important aspect indesigning any control system.The idea behind SMC is to de®ne a surface

along which the process can slide to its desired®nal value; Fig. 2 depicts the SMC objective. Thestructure of the controller is intentionally alteredas its state crosses the surface in accordance with aprescribed control law. Thus, the ®rst step in SMCis to de®ne the sliding surface S t� �. S t� � is chosento represent a desired global behavior, for instancestability and tracking performance; The S t� �selected in this work, presented by [4], is an inte-gral-di�erential equation acting on the tracking-error expression.

S t� � � d

dt� l

� �n�t0

e t� �dt �2�Fig. 1. Process reaction curve.

206 O. Camacho, C.A. Smith / ISA Transactions 39 (2000) 205±218

where e t� � is the tracking error, that is, the di�er-ence between the reference value or set point, R t� �,and the output measurement, X t� �, or e t� � �R t� � ÿ X t� �. l is a tuning parameter, which helpsto de®ne S t� �; This term is selected by thedesigner, and determines the performance of thesystem on the sliding surface, n is the systemorder.The objective of control is to ensure that the

controlled variable be equal to its reference valueat all times, meaning that e�t� and its derivativesmust be zero. Once the reference value is reached,Eq. (2) indicates that S�t� reaches a constantvalue. To maintain S�t� at this constant value,meaning that e�t� is zero at all times; it is desiredto make

dS t� �dt� 0 �3�

Once the sliding surface has been selected,attention must be turned to design of the controllaw that drives the controlled variable to its refer-ence value and satis®es Eq. (3). The SMC controllaw, U�t�, consists of two additive parts; a con-tinuous part, UC�t�, and a discontinuous part,UD�t�, [6]. That isU t� � � UC t� � �UD t� � �4�

The continuous part is given by

UC t� � � f X t� �;R t� �� 5�

where f X t� �;R t� �� is a function of the controlledvariable, and the reference value.The discontinuous part, UD t� �, incorporates a

nonlinear element that includes the switching ele-ment of the control law. This part of the controlleris discontinuous across the sliding surface.

UD t� � � KDS t� �

S t� �� 6�

where KD is the tuning parameter responsible forthe reaching mode. � is a tuning parameter used toreduce the chattering problem. Chattering is ahigh-frequency oscillation around the desiredequilibrium point. It is undesirable in practice,because it involves high control activity and alsocan excite high-frequency dynamics ignored in themodeling of the system [3,4,6].In summary, the control law usually results in a

fast motion to bring the state onto the slidingsurface, and a slower motion to proceed until adesired state is reached.

4. SMCr synthesis from an FOPDT model ofthe process

This section presents the development of ageneral SMCr, for self-regulating processes, usinga ®rst-order-plus-deadtime (FOPDT) processmodel. The FOPDT model is an approximation tothe actual higher-order model. The developmentof this controller signi®cantly simpli®es the appli-cation of sliding mode control theory to chemicalprocesses.The literature reviewed does not reveal a simple

and practical method to apply SMC to processwith dead time [7±9]. In this chapter, a SMCrstructure based on the FOPDT model of theactual process is designed. Thus, the ®rst step is topropose a way to handle the deadtime term.The deadtime can be approximated in two dif-

ferent ways. A ®rst-order Taylor series approx-imation to the deadtime term produces.

eÿt0s � 1

t0s� 1�7�

Fig. 2. Graphical interpretation of SMC.

O. Camacho, C.A. Smith / ISA Transactions 39 (2000) 205±218 207

The above approximation can also be written as a®rst-order PadeÂ approximation

eÿt0s � 1ÿ 0:5t0s

1� 0:5t0s�8�

Fig. 3 shows a comparison among the deadtimeterm and the ®rst-order Taylor series and PadeÂapproximations. The ®gure shows that the PadeÂapproximation works very well between 0 and 1 butbeyond the approximation brakes down. On otherhand, the Taylor series approximation improvesas t0 increases.In [10] is shown that the ®rst-order Taylor

approximation or the PadeÂ approximation can beconsidered as good approximations for the dead-time term for chemical processes.The next section shows the development of a

SMCr using both approximations.

4.1. SMCr development based on a ®rst-orderTaylor series approximation

In this section a SMCr is developed based on the®rst-order Taylor series expansion. Additionally, arule to choose the tuning parameters will also bepresented.

Substituting Eq. (7) into Eq. (1) produces

X s� �U s� � �

K

�s� 1� � t0s� 1� � �9�

In di�erential equation form

t0�d2X t� �

dt2� t0 � �� dX t� �

dt� X t� � � KU t� � �10�

and since this is a second-order di�erential equa-tion, n � 2, from Eq. (2) S�t� becomes

S t� � � de t� �d t� � � l1e t� � � l0

�t0

e t� �dt �11�

Where l1 � 2l and l0 � l2

From Eq. (3)

dS t� �dt� d2e t� �

dt2� l1

de t� �dt� l0e t� � � 0 �12�

Substituting the de®nition of the error,e�t� � R�t� ÿ X�t�, into the ®rst two terms of theabove equation gives

Fig. 3. Comparison among eÿx (1), Taylor (2) and PadeÂ (3) approximations.


d2R t� �dt2

ÿ d2X t� �dt2

� l1dR t� �

dtÿ dX t� �

dt

� �� l0e t� � � 0 �13�

Solving for the highest derivative from Eq. (10),substituting it into the Eq. (13), and solving forU�t� provides the continuous part of the controller

UC t� � � t0�

K

� �" t0 � �t0�

l1

� �dX t� �

dt� X t� �

t0�

� l0e t� � � d2R t� �dt2

� l1dR t� �

dt

#�14�

This procedure, involving Eqs. (11) and (13), toobtain the expression for the continuous part ofthe controller is known in the SMC theory as theequivalent control procedure [2].In [11] is shown that the derivatives of the

reference value can be discarded, without anye�ect on the control performance, resulting in asimpler controller. Thus,

UC t� � � t0�

K

� �" t0 � �t0�ÿ l1

� �dX t� �

dt

� X t� �t0�� l0e t� �

#�15�

UC�t� can be simpli®ed by letting,

l1 � t0 � �t0�

�16�

It has been shown that this choice for l1 is thebest for the continuous part of the controller [11].To assure that the sliding surfaces behave as a

critical or overdamped system, l0 should be

l04l214

�17�

Then, the complete SMCr can be represented asfollows

U t� � � t0�

K

� � X t� �t0�� l0e t� �

� �� KD

S t� �S t� �� 18a�

with

S t� � �sign K� �

ÿ dX t� �dt� l1e t� � � l0

�t0

e t� �dt� � �18b�

Eqs. (18a) and (18b) constitute the controllerequations to be used. These equations presentadvantages from process control point of view,®rst they have a ®xed structure depending on thel's parameters and the characteristic parametersof the FOPDT model, and second the action ofthe controller is considered in the sliding surfaceequation, by including the term sign�K�, in Eq.18b. Note, that sign�K� only depends on the staticgain, therefore it never switches. From an indus-trial application perspective, Eq. (18b) represents aPID algorithm [12].To complete the SMCr, it is necessary to have a

set of tuning equations. For the tuning equationsas ®rst estimates, using the Nelder±Mead search-ing algorithm [13], the following equations wereobtained [11].

. For the continuous part of the controller andthe sliding surface

l1 � t0 � �t0�

�� time� �ÿ1 �19a�

l0 � 1

4

t0 � �t0�

� �2

�� time� �ÿ2 �19b�

. For the discontinuous part of the controller

KD � 0:51

Kj j�

t0

� �0:76

�� fraction CO� � �19c�

� �0:68� 0:12 Kj jKDl1 ��

fraction TO=time� ��19d�

Eqs. (19c) and (19d) are used when the signalsfrom the transmitter and controller are in frac-tions (0±1). Sometimes, the control systems workin percentages that is, the signals are in% (0±100)of range. In these cases the values of KD and � aremultiplied by 100.


4.2. SMCr development based on the PadeÂ

approximation

This section contains the development of thecontrol law when the deadtime term of the FOPDTprocess model is approximated by the PadeÂapproximation, Eq. (8). The procedure followed inthis section is similar to that one presented in theprevious part. Substituting Eq. (8) into Eq. (1),gives

XC s� �U s� � �

K 2ÿ t0s� ��s� 1� � 2� t0s� � �20�

Using a similar procedure as shown above, thecontinuous part of the controller, UC�s�, is

UC s� � � ÿ �K

s2 � l1sÿ �

R s� � �"

2� � t0t0�

ÿ l1

� �s

� 2

t0�

#XC s� � � l0e s� �

sÿ 2

t0

26666666666666664

37777777777777775�21�

Eq. (21) has a pole �2=t0� � on the right side ofthe complex plane. Thus, the continuous part ofthe controller contains an unstable term.Eq. (20) represents a nonminimun phase system.

Hence, the equivalent control procedure applieddirectly over this kind of systems produce unstablecontrollers. An approach to solve the previousproblem, and that permit the use of SMC to non-minimun phase processes is presented in [10].In summary, up to now, the synthesis of a

SMCr has been shown from the linearization of anonlinear chemical process. The linear modelrepresenting the nonlinear chemical process is anFOPDT model. The characteristic parameters ofthe FOPDT model also are used in the tuningequations.

Thus, from the previous results, the controllerequation to be used is that obtained from theTaylor series approximation. The next part illus-trates the controller performance.

5. Simulation results

This section simulates the control performanceof the SMCr designed and given in Eqs. (18a) and(18b). The ®rst process, a mixing tank, comparesthe performance of the SMCr with respect to aPID controller. The second process, a chemicalreactor, presents further performance character-istics.

5.1. Mixing tank

Consider the mixing tank shown in Fig. 4. Thetank receives two streams, a hot stream, W1�t�,and a cold stream, W2�t�. The outlet temperatureis measured at a point 125 ft downstream from thetank. The following assumptions are accepted

. The liquid volume in the tank is consideredconstant

. The tank contents are well mixed

. The tank and the pipe are well insulated.

The temperature transmitter is calibrated for arange of 100 to 200�F. Table 1 shows the steady-state conditions and other operating information.The following equations constitute the process

model

Fig. 4. Mixing tank.


. Energy balance around mixing tank

W1 t� �Cp1 t� �T1 t� � �W2 t� �Cp2 t� �T2 t� �

ÿ W1 t� � �W2 t� �� Cp3 t� �T3 t� �

� V�Cv3dT3 t� �

dt�22�

. Pipe delay between the tank and the sensorlocation

T4 t� � � T3 tÿ t0� � �23�. Transportation lag or delay time

t0 � LA�

W1 t� � �W2 t� � �24�

. Temperature transmitter

dTO t� �dt

� 1

�T

T4 t� � ÿ 100

100ÿ TO t� �

� ��25�

. Valve position

dVp t� �dt� 1

�Vp

m t� � ÿ Vp t� �� 26�

. Valve equation

W2 � 500

60CVLVp t� �

��Gf�Pv

p�27�

. Sliding mode controller (SMCr)

U t� � � UC t� � �UD t� � �28�

where

W1�t�=mass ¯ow of hot stream, lb/minW2�t�=mass ¯ow of cold stream, lb/minCp =liquid heat capacity at constant pressure,

Btu/lb-�FCv =liquid heat capacity at constant volume,

Btu/lb-�FT1�t� =hot ¯ow temperature, �FT2�t� =cold ¯ow temperature, �FT3�t� =liquid temperature in the mixing tank, �FT4�t� =equal to T3�t� delayed by t0,

�Ft0 =deadtime or transportation lag, min� =density of the mixing tank contents,

lbm/ft3

V =liquid volume, ft3

TO�t�=transmitter output signal on a scale from0 to 1

VP�t� =valve position, from 0 (valve closed) to1 (valve open)

m�t� =fraction of controller output, from 0 to 1CVL =valve ¯ow coe�cient, gpm/psi1/2

Gf =speci®c gravity, dimensionless�Pv =pressure drop across the valve, psi�T =time constant of the temperature sensor,

min�Vp =time constant of the actuator, minA =pipe cross section, ft2

L =pipe length, ft

Following the procedure, presented in Section 2,to obtain the parameters of the FOPDT modelyields: K � ÿ0:78 fraction TO/fraction CO, � �2:32 min., and t0 � 2:97 min. Using these valuesthe tuning parameters for the SMCr are

l1 � 0:76minÿ1; KD � 0:54 fraction CO

l0 � 0:147minÿ2; � � 0:719 fractionTO=min

The tuning parameters for the PI controller areKC � ÿ0:5 and �1 � 2:32 min, using the tuningformulas for Dahlin synthesis, which producesmoother responses than Ziegler±Nichols tuningequations, working better for process with dead-time [1]. Note that the comparison is done usingthe initial tuning parameters for both controllers,to show the good performance obtained for the

Table 1

Design parameters and steady-state values

Variable Value Variable Value

W1 250.00 lb/min V 15 ft3

W2 191.17 lb/min TO 0.5

Cp1 0.8 Btu/lb-�F Vp 0.478

Cp2 1.0 Btu/lb-�F CVL 12 gpm/psi1/2

Cp3;Cv3 0.9 Btu/lb-�F �Pv 16 psi

Set point 150�F �T 0.5 min

T1 250�F �vp 0.4 min

T2 50�F A 0.2006 ft2

T3 150�F L 125 ft

� 62.4 lb/ft3 m� 0.478 CO


SMCr initial tuning equations, but they can beadjusted, ®ne tuning, until acceptable control per-formance be obtained.Please note that the controller equations, Eqs.

(18a) and (18b), were developed using deviationvariables. The following changes the ``deviationvariables'' in the controller to ``actual variables''

U t� � � m t� � ÿm�

and

X t� � � TO t� � ÿ TO

e t� � � R t� � ÿ TO t� �

where m t� � is the controller output, in fractionCO, TO t� � is the transmitter output, in fraction,and R t� � is the reference value, or set point, frac-tion TO. The overbars indicate steady-statevalues.Since the process gain is negative, sign K� � is

negative, the controller equation to be used is

m t� � � m� ÿ t0�

K

1

t0�TO t� � ÿ TOÿ �� l0e t� �

� �� KDS t� �

S t� �� 18c�

with

S t� � � dTO t� �dtÿ l1e t� � ÿ l0

�t0

e t� �dt �18d�

Fig. 5 shows the response of the temperature,T4 t� �, when the ¯ow of hot water changes from250 lb/min to 200 lb/min, then to 175 lb/min, to150 lb/min, and ®nally to 125 lb/min. The curvesclearly show that as the operating conditionschange, the performance of the PID controllerdegrades, while that of the SMCr maintains itsperformance and stability. In this case, as the ¯owof hot water decreases, with a correspondingdecrease in cold water, the deadtime between thetank and the sensor increases. This increase indeadtime certainly adversely a�ects the perfor-

mance of the PID controller. To recover stability,new tunings are required for the PI controllerwhile none are required for the SMCr.In spite of the controller being synthesized using

a Taylor approximation and the tuning equations,Eqs. 19a±19d, are empirical, the proposed methodcan be successfully used in processes with a dead-time to time constant ratio larger than one. In ourexperience, they can be applied for t0=� around 3.

Fig. 5. Temperature response under SMCr and PID controller.


5.2. Chemical reactor

The reactor shown in Fig. 6 is a continuousstirred tank where the exothermic reaction A!Btakes place. To remove the heat of reaction thereactor is surrounded by a jacket through which acooling liquid ¯ows.The following assumptions are accepted

. heat losses from the jacket to the surround-ings are negligible

. densities and heat capacities of the reactantsand products are both equal and constant

. the heat of reaction is constant.

. level of liquid in the reactor tank is constant;that is, the ¯ow out is equal to the ¯ow in.

. the reactor and the jacket are perfectlymixed.

The temperature controller is calibrated for arange of 80±100�C. Table 2 shows the steady-stateand other operating information.The following equations constitute the process

model.

. Mole balance on reactant A

dCA t� �dt� F t� �

VCA t� � ÿ CA t� �� ÿ kC2

A t� �

�29�

. Energy balance on reactor contents

dT t� �dt� F t� �

VTi t� � ÿ T t� �� ÿ kC2

A

�HR

�Cp

ÿ UA

V�CpT t� � ÿ TC t� �� 30�

. Energy balance on jacket

dTC t� �dt� UA

VC�CCpcT t� � ÿ TC t� ��

ÿ FC t� �VC

TC t� � ÿ Tci t� �� 31�

. Reaction rate coe�cient

k � k0eE

R T�273� � �32�

. Temperature transmitter

dTO t� �dt

� 1

�T

T t� � ÿ 80

20ÿ TO t� �

� ��33�

. Sliding mode controller (SMCr)

U t� � � UC t� � �UD t� � �34�

. Equal percentage control valve (air to close)

FC t� � � FC max�ÿm t� � �35�

Fig. 6. Scheme of continuous stirred tank reactor.

Table 2

Design parameters and steady-state values

Variable Value Variable Value

CA 1.133 kgmol/m3 Vc 1.82 m3

Cai 2.88 kgmol/m3 F�t� 0.45 m3/min

T 88�C Fc max 1.2 m3/s

Ti 66�C CPc 4184 J/kg-�CTci 27�C � 50

Set point 88�C �T 0.33 min

�HR ÿ9.6e7 J/kgmol Ko 0.0744 m3/s-kgmol

CP 1.815e5 J/kgmol ÿ�C E 1.182e7 J/kgmol

U 3550.0 J/s ÿm2 ÿ�C Tc 50.5�C�c 1000 kg/m3 m� 0.254 fraction CO

A 5.4 m2 V 7.08 m3

� 19.2 kgmol/m3


where

CA�t� = concentration of the reactant in thereactor, kgmol/m3

CAi�t� = concentration of the reactant in thefeed, kgmol/m3

T�t� = temperature in the reactor, �CTi�t� = temperature of the feed, �CTc�t� = jacket temperature, �CTci�t� = coolant inlet temperature, �CTO�t� = transmitter signal on a scale from 0 to 1

(fraction TO)F�t� = process feed rate, m3/sV = reactor volume, m3

k = reaction rate coe�cient, m3/kgmol-s�HR = heat of reaction, assumed constant,

J/kgmol� = density of the reactor contents,

kgmol/m3

Cp = heat capacity of the reactants andproducts, J/ kgmol- �C

U = overall heat-transfer coe�cient,J/s-m2- �C

A = heat transfer area, m2

Vc = the jacket volume, m3

�C = density of the coolant, kg/m3

Cpc = speci®c heat of the coolant, J/kg- �CFc�t� = coolant rate, m

3/s�T = time constant of the temperature

sensor, sU�t� = SMCr output signal on a scale from

0 to 1 (fraction CO)FCmax = maximum ¯ow through the control

valve, m3/s� = valve rangeability parameterk0 = Arrhenius frequency parameter,

m3/s-kgmolE = activation energy of the reaction,

J/kgmolR = ideal gas law constant, 8314.39

J/kgmol-Km�t� = valve position on a scale from 0 to 1

Fig. 7 shows the open loop response of thereactor; from this ®gure process parameters, are:K � 1:6 fraction TO/fraction CO; � � 13:0 min.;t0 � 3:0 min. For this process, because the processgain is positive, the SMCr is

m t� � � m� � t0�

K

1

t0�TO t� � ÿ TOÿ �� l0e t� �

� �� KDS t� �

S t� �� 18c�

with

S t� � � ÿ dTO t� �dt� l1e t� � � l0

�t0

e t� �dt �18b�

With the values of K, �, and t0, the continuouspart of the SMCr can be tuned using the lexpressions, Eqs. (19a) and (19b),

l1 � 0:410minÿ1

l0 � 0:0421minÿ2

Fig. 7. Process reaction curve for the reactor.


Also , from Eqs. (19c) and (19d)

KD � 0:96 fraction CO

� � 0:76 fraction TO=min

Fig. 8 shows the system response when a +10%change in inlet ¯ow occurs. The ®gure shows that,because the temperature of the inlet ¯ow is coolerthan the temperature in the reactor, the reactortemperature ®rst decreases somewhat. However,after a short while the temperature in the reactorincreases since more reactant is added to the reactor.Fig. 8 shows the control performance when the

modeling error between the real process and theFOPDT model is small. However, the model is

never perfect. Martin [14] considers that modelingerror of 25% in its parameters is a ``reasonableerror''. Let us consider two cases. The ®rst case isfor ÿ10% model error and the second one is for100% in model error. The second case could beconsidered an ``unreasonable error,'' but ourintent is to judge the controller. The error used isthe same in every parameter, that is, the sameÿ10% error in K, � and t0.Fig. 9 shows the open loop responses for the

actual process and for the model with a ÿ10 and100% error.Figs. 10 and 11 show the process response when

the inlet ¯ow changes by 10% and the modelingerror used is ÿ10 and 100%, respectively. A com-parison of Figs. 8 and 6, when no error in the

Fig. 8. System responses for 10% in inlet ¯ow.


Fig. 9. E�ect of modeling error.

Fig. 10. System responses for 10% change in inlet ¯ow for ÿ10% error in modeling.


model is present shows little di�erence in the pro-cess response. Fig. 9 shows that with 100% errorin the model, the control performance degradessomewhat. The most signi®cant di�erence is that ittakes longer to return the process to the set point.However, even with such a large error in themodel, the control is still stable.

6. Conclusions

This paper has shown the synthesis of a slidingmode controller based on an FOPDT model of theactual process. The controller obtained is of ®xedstructure. A set of equations obtains the ®rst esti-mates for the tuning parameters. The examplespresented indicate that the SMCr performance isstable and quite satisfactory in spite of non-linearities over a wide range of operating condi-tions. The relations given in Eq. (19) provided agood starting set of tunings.

The controller law, Eqs. (18a) and (18b) shouldbe rather easy to implement in any computer sys-tem (DCS) [12].

References

[1] C.A. Smith, A.B. Corripio, Principles and Practice of

Automatic Process Control, John Wiley & Sons, New

York, 1997.

[2] V.I., Utkin, Variable structure systems with sliding modes,

Transactions of IEEE on Automatic Control, AC ± 22

(1997) 212±222.

[3] H. Sira-Ramirez, O. Llanes-Santiago, Dynamical dis-

continuous feedback strategies in the regulation of non-

linear chemical processes, IEEE Transactions on Control

Systems Technology 2 (1) (1994) 11±21.

[4] J.J. Slotine, W. Li, Applied Nonlinear Control, Prentice-

Hall, New Jersey, 1991.

[5] M.C. Colantino, A.C. Desages, J.A. Romagnoli, A. Pala-

zoglu, Nonlinear control of a CSTR: disturbance rejec-

tion using sliding mode control, Industrial & Engineering

Chemistry Research 34 (1995) 2383±2392.

[6] A.S.I. Zinober, Variable Structure and Liapunov Control,

Springer±Verlag, London, 1994.

Fig. 11. System responses for 10% change in inlet ¯ow for 100% error in modeling.


[7] K.D. Young, V.I. Utkin, UÈ . OÈ zgumer, A control engi-

neer's guide to sliding mode control, IEEE Transactions

on Control Systems Technology 7 (3) (1999) 328±342.

[8] J.Y. Hung, W. Gao, J.C. Hung, Variable structure con-

trol: a survey, IEEE Transactions on Industrial Electro-

nics 40 (1) (1993) 2±21.

[9] G.E. Young, S. Rao, Robust sliding-mode of a nonlinear

process with uncertainty and delay, Journal of Dynamical

Systems, Measurement, and Control 109 (1987) 202±208.

[10] O. Camacho, R. Rojas, W. Garcia, Variable structure

control applied to chemical processes with inverse

response, ISA Transactions 38 (1999) 55±72.

[11] O.E. Camacho, A new approach to design and tune slid-

ing mode controllers for chemical processes, Ph.D. Dis-

sertation, 1996, University of South Florida, Tampa,

Florida.

[12] O. Camacho, C. Smith, E. ChacoÂ n, (1997). Toward an

implementation of sliding mode control to chemical pro-

cesses, in: Proceedings of ISIE'97, Guimaraes-Portugal,

1997, pp. 1101±1105.

[13] D.M. Himmelblau, Applied Nonlinear Programming,

McGraw-Hill, New York, 1972.

[14] T.E. Marlin, Process Control, McGraw-Hill, New York,

1995.

Oscar Camacho received the Electrical Engi-neering, and M.S. in Control Engineering degreesfrom Universidad de Los Andes (ULA)m MeÂ rida,Venezuela, in 1984 and 1992, respectively, and theM.E. and Ph.D. in Chemical Engineering at Uni-versity of South Florida (USF), Tampa, FL, in1994 and 1996, respectively.He has held teaching and research positions at

ULA, CIED-PDVSA since 1985. He has been theChairman of the Electrical Engineering School atULA since 1998. His current research interestincludes sliding mode control, hybrid systems andpassivity-based control. He is the author of morethan 30 publications in journals and conferenceproceedings.

Carlos A. Smith, is Professor of Chemical Engi-neering and Associate Dean for Academic A�airsat the University of South Florida. Before joiningUSF he worked for Dow Chemical USA from1967 through 1972. Professor Smith is a co-authorof two editions of Principles and Practice of Auto-matic Process Control, 1986 and 1975, publishedby John Wiley & Sons. Professor Smith has beenin consultancy work, as well as lecturing on shortcourses, for many companies in the US, Canada,Latin America, and Europe. Professor Smith lovesteaching, working with graduate students, anddealing with industry.


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Sliding mode control: an approach to regulate nonlinear chemical processes