Temperature control in catalytic cracking reactors viaa robust PID controller

R. Aguilara,*, A. Poznyakb, R. Martınez-Guerrab, R. Maya-Yescasc

aDivision de Ciencias Basicas e Ingenierıa, Universidad Autonoma Metropolitana—Azcapotzalco, Apartado Postal 75-338.

07300, Mexico, D.F. MexicobDepartamento de Control Automatico, CINVESTAV-IPN, Mexico

cPrograma de Tratamiento de Crudo Maya, Instituto Mexicano del Petroleo, Mexico

Received 3 August 2000; received in revised form 24 May 2001; accepted 30 May 2001

Abstract

The stabilization of fluid catalytic cracking reactors is tackled in this paper. A robust PID control law is developed in order to

control the outlet reactor temperature. The suggested control is based on a reduced order model of the reactor given by a system ofordinary differential equations. The controller is synthesized using an input/output linearizing control law coupled to a propor-tional-derivative reduced order observer to infer on-line the unknown heat of reaction. The proposed control algorithm leads to a

classical PID structure. New tuning rules are given, based on the system structure, estimations and closed-loop time constants. Thiscontrol strategy turns out to be robust against model uncertainties, noisy temperature measurements and set point changes. Theperformance of the reaction temperature in a tubular riser reactor is numerically compared when the proposed control scheme and

standard PID controllers are applied. # 2002 Elsevier Science Ltd. All rights reserved.

Keywords: FCC; PID control; Robust temperature regulation; Tubular reactor; Tuning rules

1. Introduction

Tubular reactors, such as fixed, moving and riser beds,are of great importance in the chemical industry becauseof their efficient product extraction, large flow velocities,and high yields. These reactors are used, for example, inethanol production, anaerobic digestion, acetylenehydrotreating and catalytic cracking [1,2,3]. Since thiskind of reactors is adequately modeled as distributedparameter equation systems, a set of Partial DifferentialEquations is especially derived to describe such models.Control theory for these systems is still under develop-ment [4].

In order to study the control of distributed parametersystems, two different methodologies have been pro-posed [5]. In the first one, the control algorithm is basedon a distributed parameter system, which is reduced to afinite dimensional model. However, model uncertaintiesmake this approach inadequate in practice [6]. In the

second approach, the distributed parameter system isreduced to a set of ordinary differential equations [7,8].This reduced order model keeps the essential dynamics ofthe process and is useful for regulation purposes. More-over, analyses in terms of stability and performance areeasily done. In addition, the problem is tractable becausethe system is represented by a finite dimensional set ofordinary differential equations. Therefore, most controlresearch has followed this second approach.

Several papers deal with the control of tubular reactorsusing the input–output linearizing control technique [7].However, this approach needs precise knowledge of thenonlinear dynamics of the system, since realization pro-blems arise within this methodology.

If the information related to the model description isincomplete, the problem to solve is the control of uncer-tain process. This is a research field of a great importance,since, in reality, any mathematical model contains (withinits structure) some parameters, which can not be eval-uated exactly by an experimental way. To confront thesedrawbacks, several papers have been published devel-oping robust control strategies for some classes ofuncertain systems. Kolavennu et al. [9] develop an inter-esting study [with applications to continuous stirred tank

0959-1524/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0959-1524(01 )00034-8

Journal of Process Control 12 (2002) 695–705

www.elsevier.com/locate/jprocont

* Corresponding author. Fax: +52-5394-7378.

E-mail address: raguilar@correo.azc.uam.mx (R. Aguilar).

reactor (CSTR)], where the proposed control design isbased on a linear transformed model of the original sys-tem using I/O-linearization techniques with H2/H1

implementation. Other approach has been presented byLee and Sullivan [10] under the framework of optimalcontrol, where non-linear process models can be imbed-ded directely into the controller without resorting tolinaerization. Aguilar et al. [11] proposes ‘‘a full orderuncertainty observer’’ to obtain the on-line estimates ofthe heat of reaction in continuous reactors, which is cou-pled with a linearizing controller to regulate the reactortemperature.

Specific application of non-linear controllers to fluidcatalytic cracking (FCC), based on uncertainty observers,can be found in Alvarez-Ramırez et al. [12,13] and Agui-lar et al. [14], where the temperature regulation problemfor a regenerator-riser system is studied. These approa-ches produce practical controllers where the closed-looptemperature trajectories are forced to remain in aneighborhood close to the set point.

Another approach deals with the sliding-mode controltechnique [5]. Although it has been shown to exhibit goodperformance, the implementation by industrial plantoperators of such advanced control strategies seems to benot friendly enough. Hence, standard PID controllers arecommonly employed in commercial hardware forindustrial plants.

Typically, the control of tubular reactors has beendone using PIDs, despite the fact that these systems arenonlinear. Nevertheless, some PID tuning techniqueshave been more successful than others [e.g. internalmodel control (IMC) has been better than the Ziegler–Nichols method (ZN)] [15]. This situation is related tothe relationship between process dynamics and processcontrol. Process design can be viewed as a sequence ofdecision and evaluation steps. This iterative process triesto pursue a dual objective: some measure of profitabilityand the resilience of the plant is to be maximized. Theterm ‘‘resilience’’ is used in the sense proposed by Mor-ari [16]. For an industrial plant the dynamic resilience isclosely related to its physical design rather than a con-troller structure or type. In case of a plant alreadydesigned, it seems that a control which is closer to itsdynamics will present better performance. Followingthis idea and because of the popularity of PIDs, it seemsto be interesting to investigate the proper tuning of thiskind of controllers based on the intrinsic dynamics of aprocess, i.e. to obtain ad hoc controllers that ensuredynamic resilience for plants already in operation.

In this work, FCC riser reactors are chosen as the casestudy. Since FCC units produce around 40% of thegasoline pool, the choice of this particular process isbased on its importance in the refining industry. By thisreason and in view of the large production and the

Nomenclature

Ccat coke fraction on catalyst at the reactor outlet(dimensionless)

Crc coke fraction on regenerated catalyst atreactor inlet (dimensionless)

COR catalyst/feedstock mass ratio (dimensionless)Cp specific heat at constant pressure ([kJ/kg K)e error of temperature tracking (K)Ecf activation energy of coke formation (kJ/kg)Ff flow of feedstock entering the reactor (kg/s)FS flow of catalyst entering the reactor (kg/s)�Hf heat of cracking reaction (kJ/kg)K1 kinetic constant of feedstock cracking (s�1)K2 kinetic constant of gasoline cracking (kg/s)kC kinetic constant of coke formation in Kur-

ihara’s expression (kgcoke/kgcat s)Kp proportional controller gainn exponential of the coke generation kineticsR universal constant of gases (kJ/kmol K)t time (min)tc residence time of catalyst in the reactor (s)Tr reactor temperature (K)u control inputW catalyst hold-up (Ton)]x vector of states

y control outputyf feedstock mass fraction (dimensionless)yg gasoline mass fraction (dimensionless)z length (dimensionless)

Greek letters� crackeability factor for the feedstock

(dimensionless)� uncertain term� dimensional time�f density of feedstock (kg/m3)�g control gain (inverse of the closed-loop char-

acteristic time) (min�1)�1 observer gain (inverse of the estimation

characteristic time) (min�1)�2 observer gain (inverse of the estimation

characteristic time)�I characteristic time of integral action�D characteristic time of derivative action’ fraction of original activity of the catalyst

(dimensionless)

Superscipts� derivative with respect to timeU estimated value

696 R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705

importance of generated products, the development ofad-hoc control strategies can be clearly justified, sincesmall operating improvements turns out to be econom-ically very attractive.

FCC reactors have a number of interesting featuressuch as the strong correlation of reactor yields to theeffluent temperature [17]. This is in consequence of theexponential dependence of the reaction rates on tempera-ture. Furthermore, conversion and yields set the unitprofitability. Despite operational experience with thisclass of units has been satisfactory, the lack of the possi-bility of a direct control of yields and compositions is amatter of concern. Some feedstock characteristics are dif-ficult to estimate, especially the heat of the reaction thatsets the reaction temperature and, therefore, the yields andthe composition during the reaction time. In this case, thevariable to be controlled is the outlet reactor temperatureand the manipulated variable is the catalyst flow comingfrom the regenerator [17–19], as it is done in industrialpractice.

In this paper, a finite dimensional model is obtainedby spatial discretization of a distributed parametermodel for a FCC reactor. Then, a nominal input–outputlinearizing controller based on an uncertainty estimator isproposed to control the outlet temperature of the reactor.The uncertainty estimator consists of proportional andderivative contributions of the estimation error. The deri-vative action of the estimator generates fast estimates ofthe heat of reaction. When the estimator is coupled to theproposed control law, a controller structure, similar tothe PID, is obtained. The controller tuning procedure isreduced to a selection of the proportional, integral andderivative gains following the proposed tuning rules in astraightforward manner. The concept of ‘‘practical sta-bility’’ [20] is employed in order to show that the con-troller proposed leads to a globally state restraint andpractical stabilizability of the chemical reactor, providingthe global restraint of the state trajectories. The controlperformance is compared to ‘‘standard’’ PID controllerstuned by using ZN and IMC methodologies [21].

2. Mathematical model for a fluid catalytic cracking

reactor

Catalytic cracking reactors are tubular transportedbed reactors, also called risers. They crack heavy feedstocks coming from other process units (such as vacuumdistillation, coking and hydrotreating), in order toobtain lighter and valuable fuels (gasoline, LPG, C3 andC4 olefins). The efficient operation of the FCC unit hasan important influence, practically, on most refineryprocesses. Hence, there are very economic incentives tooperate these units at their optimum operating point thatcould lie close to the operating restrains [22]. The reac-tor model is obtained performing the classical mass and

energy balances to describe feedstock mass fraction yf(1), gasoline mass fraction yg (3) and temperature Tr (5)in the reactor. The kinetic scheme proposed by Week-man and Nace [23] is used and hydrodynamic plug flowbehavior is assumed. All variable definitions are given inthe nomenclature section.

2.1. Infinite dimensional model

The corresponding infinite dimensional model is givenbelow.

� Feedstock mass balance

W@yf@t

¼ Ff@yf@z

� WK1ðTrÞy2f COR½ ’=�f ð1Þ

Here, ’ is a specific parameter known as the catalystactivity decaying function, proposed by Voorhies [24]and satisfying

’ ¼ ’0 exp �� tc COR½ zð Þ ð2Þ

� Gasoline mass balance is expressed in a similar way:

W@yg@t

¼ Ff@yg@z

� W� 2K1ðTrÞy2f

� K2ðTrÞyg COR½ ’=�f ð3Þ

� Kurihara’s correlation [17] is used to describe the con-centration of coke produced Ccat:

Ccat ¼ kct

12c

Cnrc

exp �Ecf

RTro

� �ð4Þ

� An energy balance for the gas–solid mixture is also per-formed defining the relation between the reactor tem-perature Tr and the heat of the cracking reaction�Hf .

W@Tr@t

¼ ðFf þ FsÞ@Tr

@zþ

W�HfK1ðTrÞy2f COR½ ’

�fCpð5Þ

Adiabatic operation and no phase change areassumed. Details of the kinetics and model parameterscan be found in Weekman and Nace [23] as well as inBalchen et al [25].

2.2. Finite dimensional model

The model represented by Eqs. (1)–(5) can be reducedto a finite dimensional model by discretizing the spatialderivatives. In this work, the backward finite differencetechnique is used. It is easy to implement and conservethe basic dynamics when an appropriate number of

R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705 697

discretization points are used. The equations resultedfrom the application of this methodology are as follows:

� Feedstock mass balance:

Wdyf ;idt

¼ Ffyf ;i � yf ;i�1

h� WK1ðTr;iÞy

2f COR½ ’=�f

for i ¼ 1; ::::; nð6Þ

� Gasoline mass balance:

Wdyg;idt

¼ Ffyg;i � yg;i�1

h� W � 2K1ðTr;iÞy

2f ;i

� K2ðTr;iÞyg;i COR½ ’=�f for i ¼ 1; ::::; nð7Þ

� Energy balance:

WdTr;i

dt¼ ðFf þ FsÞ

Tr;i � Tr;i�1

h

þW�HfK1ðTr;iÞy

2f ;i COR½ ’

�fCpfor i ¼ 1; ::::; n

ð8Þ

Here, i represents the inner nodes of the discretizationscheme and h is the constant length between nodes.

3. Temperature control algorithm

3.1. Nominal control design

Nonlinear approaches to design control laws have beenused successfully in theoretical research. In particular, theinput–output linearizing technique shows attractive char-acteristics for the control of the nonlinear systems. In viewof this approach, a nonlinear system is represented as

x:1 ¼ f1ðx1; x3Þ

x:2 ¼ f2ðx1; x2; x3Þ

x:3 ¼ f3ðx1; x3Þ þ Bðx3Þu

y ¼ hðxÞ ¼ x3 ð9Þ

For this particular case, it is considered the finitedimensional model, Eqs. (6)–(8) evaluated in at thereactor outlet. With the system structure given by Eq.(9), the model can be written as:

f1 x1; x3ð Þ ¼ Ffyf ;n � yf ;n�1

h� WK1ðTr;nÞy

2f COR½ ’=�f

f2 x1; x2; x3ð Þ ¼ Ffyg;n � yg;n�1

h� W � 2K1ðTr;nÞy

2f ;n

� K2ðTr;nÞyg;n COR½ ’=�f

f3 x1; x3ð Þ ¼ FfTr;n � Tr;n�1

h

þW�HfK1ðTr;nÞy

2f ;n COR½ ’

�fCp

B x3ð Þ ¼Tr;n � Tr;n-1

h

The function u ¼ Fs is the scalar control input, x ¼

yf ;n; yg;n;Tr;n� �

are the system states and y ¼ Tr,n is thescalar system output. So, throughout of this paper x 2

R3; u; y 2 Rþ; f1; f3 : R2! R; f2 : R ! R are

smooth vector fields. The system (9) can be transformedinto canonical controllable form via a nonlinear changeof coordinates given by

ð"T; &TÞT ¼ hðxÞ;LfhðxÞ; :::;Lr�1f hðxÞ; &1; :::; &n-r

� �T

LB&i ¼ 0; i ¼ 1; 2; :::; n � r

Here, r is the relative degree of the system (9) andLfhðxÞ is the Lie derivative of hðxÞ along fðxÞ; both defi-nitions are used in the sense of Kravaris [26].

System in terms of the transformed variables is repre-sented by (10).

":i ¼ "iþ1 for i ¼ 1; 2; :::; r � 1

":r ¼ Lr

fhðxÞ þ LBLr�1f hðxÞu

&:¼ gð"1; "2; :::; "r; &Þ

y ¼ "1 ð10Þ

The conditions for the existence of this transforma-tion are related to a well-defined relative degree of thesystem [26].

The zero dynamics (when the output time derivative isequal to zero) are given by the ðn � rÞ-dynamic sub-system &

:¼ gð0; "2; :::; "r; &Þ. For this process, a plug

flow reactor and endothermic reactions, zero dynamicsare locally stable. Moreover, each spatial point of thereactor can be viewed as an isothermal reactor, whichdynamics are usually stable [27]. Consequently, the sta-tic feedback controller for LBLr�1

f hðxÞ 6¼ 0 and somebeing external input related to u by the equalityu ¼

�LrfhðxÞ

LBLr�1f

hðxÞ, can linearize the input/output map of the

system. In the case under study, the relative degree ofthe system is r ¼ 1. The study of such systems is veryimportant for many control applications, since a wideclass of chemical reactors has r ¼ 1. They are mathe-matically modeled as affine systems with respect to thecontrol input [9]. Systems that present relative degree onepossess some interesting features, such as the equivalentdissipativeness by means of a state or an output feedback.In general, it is easier to stabilize dissipative systems than

698 R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705

non dissipative ones [9,28]. For instance, bounded con-trollers can stabilize dissipative systems, but they are notable to do it with non-dissipative systems.

In order to stabilize the system defined by Eq. 9; reg-ulating x3, the following nominal input-output lineariz-ing feedback control is proposed [Eq. (11)].

u ¼ B�1ðx3Þ �ge � f3 x1; x3ð Þ� �

ð11Þ

As usual, e ¼ y � ysp and ysp are tracking error andset point, respectively. The controller defined by Eq. (11)guarantees asymptotic stability of nonlinear systemswith no uncertainties and perfect measurements [29,30].It imposes a linear behavior to the system canceling thenonlinearities.

3.2. Non-ideal control design

As can be noticed, the synthesis of the nominal con-trol law needs accurate knowledge of the mathematicalmodel of the process to be realizable, in order to cancelcompletely the nonlinear characteristics of the system.However, a perfect model is impossible to be obtainedand, consequently, for uncertain systems a conventionalI/O controller design is not adequate.

In this work, it is supposed that the reactor outlettemperature TrðtÞ5 0 is bounded for all t > 0. Conse-quently, the concentrations inside the reactor are boun-ded input to bounded output state [31]. It is assumedtoo, that the uncertain term remains bounded, i.e. � 2�min; �max½ and the relative degree r has well definednormal form for this interval [9].

With the above assumptions, the basis of the non-ideal controller design is the nominal control law [Eq.(11)] presented in Subsection 3.1. In order to design thenon-ideal control law, consider the uncertain term in thestate equation to be regulated, for this, let us proposethe following nonlinear dynamic system representation:

x:1 ¼ f1ðx1; x3Þ

x:2 ¼ f2ðx1; x2Þ

x:3 ¼ f3ðx1; x3Þ þ ðB& ðxÞ þ�BÞu

y ¼ hðxÞ ¼ x3 ð12Þ

The functions f3ðx1; x3Þ and �B are model uncertain-ties related to the nonlinear system. According to theabove definitions f3(x1, x3) is related to the heat ofreaction (which is supposed unknown) and �B is theerror of the energy convective flow approximation. B& ðxÞis a nominal value of the control input coefficient. Now,it is possible to define the following change of variables:

�ðx; uÞ ¼ f3ðx1; x3Þ þ�Bu ð13Þ

By substitution of Eq. (13) into Eq. (12), a newrepresentation of the system is obtained.

x:1 ¼ f1ðx1; x3Þ

x:2 ¼ f2ðx1; x2Þ

x:3 ¼ �ðx; uÞ þ B& ðxÞu

y ¼ hðxÞ ¼ x3 ð14Þ

Since the uncertainty term, �ðx; uÞ, is an unknownfunction of the states and the control input, (x and u),the nominal control law for the regulation of x3 is notcausal and, therefore, not realizable.

Nevertheless, there is another way to develop aninput–output linearizing like controller that is robustagainst uncertainties. The procedure shown belowdefines, firstly, a method to estimate the uncertainty term�ðx; uÞ. This approach is based on a reduced order linearestimator where the uncertainty is only a function theestimation error, ". Defining the following dynamicsubsystem [Eq. 15].

x3 ¼ �þ B&u

�:¼ 'ðx; uÞ

y ¼ hðxÞ ¼ x3 ð15Þ

The uncertain term, �, is considered as a new state and'ðx; uÞ is a nonlinear unknown function that describes�’s dynamics. Firstly, an algebraic observability condi-tion to prove that the pair �; x3f g is observable shouldbe done. So, consider a differential algebraic approachdescribed by the following definitions [32].

Definition 1. There is a dynamic system which is afinitely generate differential algebraic extension @=k uh i.This means that any element of @ satisfies a differentialalgebraic equation with coefficients, which are rationalfunctions over k in the components of u and a finitenumber of their time derivatives.

Definition 2. Let u; y� �

be a subset of @ in the dynamicsystem @=k uh i. An element in @ is said to be observablewith respect to u; y

� �if is algebraic over k u; y

. There-

fore a state x is said to be observable if, and only if, it isobservable with respect to u; y

� �.

Now, according to Definition 1, Eq. (15) defines analgebraic differential dynamic system. From this sub-system, the following differential-algebraic equationscan be obtained:

x3 � y ¼ 0 ð16Þ

y:� B&u � � ¼ 0 ð17Þ

Hence, the following definition is introduced in order todefine an algebraically observable uncertainty condition.

R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705 699

Definition 3. An element Xi in @ is said to be analgebraically observable uncertainty if Xi satisfies a dif-ferential algebraic equation with coefficients over k u; y

.

From Definitions 2 and 3 and Eqs. (16) and (17) thepair �; x3f g is universally observable in the Diop-Fliesssense.

From the system given by Eq. (15), it can be seen thata standard observer structure design, which consist in acopy of the system plus correction of the measurementerror, is not possible since the term ' is an unknownfunction of the system output x3. Let us propose thefollowing uncertainty observer:

�:¼ �1ð�� �Þ þ �2ð�

:� �:Þ ð18Þ

This uncertainty estimator is a reduced order obser-ver, which infers the uncertain term (heat of reaction)from temperature measurements, where the uncertaintyobserved, � is obtained from the energy balance equation.Note that the observer contains proportional and deriva-tive actions, the aim of the derivative term is to improvethe speed of the estimation algorithm, because it enhancesanticipatory and stabilizing effects of derivative actionsand can diminish enough the peaking phenomena pre-sent in controllers based on observers [33].

Substituting in the nominal controller defined by Eq.(11), the estimate of the uncertain term, the followingnon-ideal controller is obtained [Eq. (19)].

u ¼ B&�1 �ge � �� �

ð19Þ

Now, the regulation error is defined by e ¼ Y�

Ysp ¼ x3 � x3;sp. For this case x3;sp ¼ constant, conse-quently e

:¼ x:3. From the energy balance equation

� ¼ x:3 � B& ðx3Þu; substituting into it Eq. (19) and taking

the time derivative of the resulting equation the follow-ing expression is obtained: �

:¼ e€ � �ge þ �

:. This is

introduced in Eq. (18) to obtain:

�:¼ �2e€ þ ð�1 � �2�gÞe

:� �g�1e ð20Þ

Since the proposed controller uses estimated values ofthe uncertainty, it cannot cancel the system nonlinearities,completely. Thus, the system trajectory remains inside aneighborhood close to the set point. Practical stability isachieved as long as the uncertainty estimation error isbounded [10,31]. The restraint of the heat of reaction(uncertain term) is common for a wide class of chemicalreactions and is consequence of characteristics of themathematical modeling commonly employed; chemicalreactions are usually Lipschitz with respect to temperature.It is not hard to see that global Lipschitz of �HfR yf :Trð Þ

property is found if the functionality R yf ;Trð Þ with respectto temperature is of the Arrhenius type [27].

The final expression for the input-output linearizingcontroller with uncertainty estimation can be obtained

integrating the estimator [Eq. (15)] and substituting itinto the non-ideal controller [Eq. (14)], to obtain:

u ¼ B&�1ðxÞ

� ð�g � ð�1 � �2�gÞÞe þ �g�1

ðt

o

eð�Þd� � �2e:

� ð21Þ

Note that this controller [Eq. (21)] has a PID struc-ture and is equivalent to the linearizing controller basedon proportional-derivative uncertainty observer [Eqs.(13) and (14)].

Furthermore, for the resulting PID controller, thetuning procedure is simplified to select the controllergains in a straightforward manner. The controller gainsare given in terms of the system structure B& ðxÞ, �g, �1and �2, which have strong physical meaning for theplant operators. The characteristic closed-loop time (�g)can be considered as the inverse of the prescribed meansettling time. It is selected from the order of the dominanttime constant corresponding to the open-loop dynamicsof the riser reactor. Hence, �1 can be chosen greater thanthe inverse temperature sampling time. �2 will be con-sidered later. Therefore, empirical tuning rules for tun-ing both the controller and the estimator are avoided.This parameterization is given by Eqs. (22)–(24).

Kp ¼ B&�1ðxÞ �g � ð�1 � �2�gÞ� �

ð22Þ

�I ¼ B&�1ðxÞ�1�g ð23Þ

�D ¼ B&�1ðxÞ�2 ð24Þ

where Kp is the proportional gain, �I is the integral timeand �D is the derivative time.

When these rules are compared to other robust tuningrules [34] for linear PID controllers, which have fixedcontrol’s gains, it is noted that the controller proposedperforms like a PID one with time varying gains,because the gains contain the nominal term of the con-troller coefficient B& ðxÞ.

Note that the aids of the mathematical developmentpresented for the input–output linearizing controllerbased on proportional-derivative reduced order uncer-tainty observer is to lead a robust PID control structure.It is more tractable, generally speaking, for the plantoperators and would be easily implemented in commercialhardware (i.e. programming logic controllers).

Another problem that arises when distributed systemsare studied is related to the time delay between thecontrol input and the measured output system, whichleads to non minimum-phase behavior with strong con-trol design implications.

In order to avoid the time delay and the non-mini-mum phase characteristics of tubular reactors, some

700 R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705

authors have been using the temperature of the reactorjacket as control input [35]. Nonetheless, in some cases,as the one treated here, these reactors are operating asadiabatic systems, so the control input must be relatedto the inlet flow to the reactor. If the flow can be con-sidered incompressible, the density along riser remainsconstant. This assumption might suppose an immediateresponse of the reactor outlet to changes in the inputflow, consequently the control design is done consider-ing a minimum phase system, the time delay has beendiminished enough [7].

Kravaris and Daoutidis [36] proposed a nonlinearstate control design for second order non-minimumphase systems, which has a structure similar to system(15). It was successfully applied to CSTR, resulting in aclosed-loop response of a nonlinear first order all pass inseries with a linear first order lag. The control law pro-posed by these authors is given by:

u ¼ �LB ð Þ�1 v � � �Lf ð Þ ð25Þ

Here, �; x3ð Þ is a function that satisfies:

@

@�¼ �2

@'

@�@'

@x3

and@

@x3¼ �1

on the line ' x; uð Þ ¼ 0 and

@

@x36¼ 0 8 �; x3ð Þ:

If this approach has to consider uncertainties in themodel, the terms �; x3ð Þ, LB and Lf will beunknown. Now, taking the controller proposed here, itis easy to notice that if LB � B& ðx3Þ and �; x3ð Þ þ

Lf ¼ � are considered, it leads to the similar structureof Eq. 25. This approach would be extended to nonminimum-phase systems in future works.

3.3. Stability remarks

The stability issue has been addressed in two differentways. The first is related to consider the effect of theuncertainties as a perturbation to the input–output linearmodel and to use linear robust control techniques toaccount for the uncertain terms [9]. The second approachconsiders the effect of the uncertainties in the originalnonlinear model and uses the characteristics of the closed-loop behavior of the state equations to show stabilityproperties [11,31], this methodology has strong physicalmeaning, therefore it is employed in this work.

In order to show the stability properties of the closed-loop system, firstly a convergence analysis of the uncer-tainty observer should to be done.

Proposition 1. Let � and � be the uncertainty term andits estimate. The following dynamic system: �

:¼

�1ð�� �Þ þ �2ð�:� �:Þ, is a like-exponential reduced order

observer for the system (10).

Proof 1. Let us define the estimation error as:

" ¼ �� � ð26Þ

Now, the dynamic scalar equation of the estimationerror, according to Eq. (10) and (13) is given by:

":¼ �

�11 þ �2

"þ'ðx; uÞ

1 þ �2ð27Þ

Integration of Eq. (27) renders:

" ¼ "0exp ��1

1 þ �2t

� �

þ

ðt

0

exp�1

1 þ �2ðt � sÞ

� �'ðx; uÞ

1 þ �2ds ð28Þ

Now, consider the following assumption: A1. 'ðx; uÞis bounded, i.e. 'k k4*

Taking norms from Eq. (28):

"k k4 exp ��1

1 þ �2t

� �"0k k �

*

�1

� �þ*

�1ð29Þ

In the limit when t!1:

"k k4*

�1ð30Þ

Remark 1. It is important to analyze the structure ofEq. (29) in order to obtain some characteristics of theproposed observer. As can be observed, taking the limitwhen t ! 1, the estimation error remain around closedball with radius proportional to *

�1, this radius can be

made as small as desired by taking �1 large enough.Besides, in order to improve the speed of the uncer-

tainty observer convergence to the steady-state estima-tion error, two actions have to be taken. The first one is toconsider the parameter �1 large enough, which is necessaryto obtain a small steady-state error, as mentioned above,and the second one is faced with the influence of theparameter �2 related to the derivative action of the uncer-tainty observer. If �2 ! �1, then exponential term of theright hand side of Eq. (29) can be accelerated enough andconsequently, the convergence of the uncertainty obser-ver has better performance.

Remark 2. Note that if the measurements of the sys-tem are corrupted by additive noise, i.e. y ¼ x3 þ �, andthis noise is considered bounded, �k k4+, a methodol-

R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705 701

ogy similar to the one used to analyze the estimationerror " can be applied in order to prove that the steadystate estimation error becomes "k k4 Cþ+

�1. This con-

firms robustness against noisy measurements.Now, it is possible to implement a non-ideal con-

troller using the uncertainty estimated, to produce apractical control law, which can lead the trajectories ofreactor’s temperature inside a neighborhood close to theset point.

Proposition 2. The controller u ¼ B&�1 �ge � �� �

pro-duces globally state restraint and practical stabilizability,given that the temperature trajectories remains inside of aclose ball with radio proportional to ej j4 *þ+

�1�g.

Proof 2. In order to prove closed-loop stability of thetemperature of the reactor, it is necessary to analyze theclosed-loop equation of the energy balance when the non-ideal control law is introduced, which has the structureshown in Eq. 31.

x:3 ¼ e ¼ ��ge þ �� �ð Þ ð31Þ

If � ! �, the difference of the right hand side of Eq.31 is close to zero, then stability properties of this con-trol law can be recovered, as expected, to be analogousto the ideal control.

If the difference � � �ð Þ remains bounded, then the esti-mation error is "k k4 *þ+

�1. Then it is possible to solve the

linear ordinary differential Eq. (31) for the second state-ment, where the following inequation is obtained:

ej j4*þ+

�1�gþ e0j jexpð��gtÞ ð32Þ

It is easy to note that the limit when t ! 1 is:

ej j4*þ+

�1�gð33Þ

Remark 3. Note that the regulation error e can be madeas small as desired in two ways. The first one is to considerthe inverse of mean settling time �g large enough, but thisapproach has several problems like great control effort,input saturation and possible instability. The other wayconsists of good converge of the uncertainty estimator,which lead to " be small enough; this situation can bereached with a high gain of the estimator proposedrather than a high gain of the linearizing control.

4. Results and discussion

Numerical simulations were carried out in order tocompare the performance of the proposed controller withZiegler–Nichols (ZN) as reinterpreted by Smith and Cor-ripio and IMC-PID controllers, when considering set

point changes and typical disturbances to FCC reactors(Table 1). ZN and IMC controllers were tuned using astep perturbation for the identification of the riser reac-tor [37] and industrial operation conditions. The fol-lowing dynamic parameters for the open-loop reactorwere obtained: the time delay is 4 s, the open-loopcharacteristic time is 7.5 seconds and the correspondingsteady-state gain is 0.58 K s/kg.

Figs. 1 and 2 show the closed loop behavior of theriser outlet temperature (control output) and the cata-lyst flow rate (control input) when a ZN-PID controlleris used. It can be seen that the performance is very poorat short times, because the controller can not regulateadequately the outlet temperature of the riser when thedisturbances arrive. Even worse, when set point changesare imposed, there is temperature fall down resulting inthe quenching of the process.

Figs. 3 and 4 show the closed loop behavior when aIMC-PID controller is used. It is easy to note that theoutput behaves better than when a ZN controller is used(Fig. 3). However, the control effort is excessive. It is

Table 1

Disturbances expected in industrial FCC reactors [18]

Disturbance Time of arriving

5 K decrease in the feedstock

temperature, Tf

180 min after the start

2.5% increase in the combustion

coke rate, KC

250 min after the start

4 kg/min decrease in the

feedstock flow, Ff

420 min after the start

Fig. 1. Control output considering Ziegler–Nichols PID controller.

702 R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705

possible to notice that the catalyst flow is moved from300 to 525 kg/s (see Fig. 4), situation that will take theindustrial unit to an input-saturation and will lost con-trol. Therefore, IMC controller cannot deal adequatelywith this control problem.

For the controller proposed, the control gain �g wasadjusted with the above considerations: the regulationerror for the nominal controller, behave as e=e(0) exp(-�g t), such that �g

�1 can be interpreted as the mean

convergence time of the controlled temperature. Thevalue �g

�1=9 s was employed. The value of �g�1 was

determined considering the catalyst residence time in thereactor, which allows to the controller to attenuate thedisturbances without excessive control efforts. The valueof �1 was choosed of the order of the inverse of themeasurement characteristic time, for temperature mea-surements the measurement frequency is around 0.01s�1. The observer parameter �2, in accordance with theremark 2 in Section 3.3 was chosen as 0.95.

Figs. 5 and 6 show the reactor performance when thecontroller proposed is used. Noisy measurements oftemperature of �2 K around the current temperatureare considered. Firstly, when a decrease of 5 K occurs infeedstock temperature Tf at 180 min, the convectiveenergy flow to the reactor inlet is diminished. The con-trol action is to increase the catalyst flow FS in order tomaintain the outlet temperature at the set point. As canbe seen, the system reaches its new steady state in lessthan 35 min following an acceptable trajectory. It isimportant to notice that, also, the control effort is veryreasonable.

Secondly, when a 2.5% increase occurs in the com-bustion rate of coke at 270 min, more coke is produced,gasoline production decreases due to catalyst deactiva-tion. The control action is to decrease the catalyst flowin order to recover the temperature, and also the pro-duction rate. Despite the complexity of the problem, thecontrol is able to take back the system to the set point inless than 35 min, following an acceptable trajectory.

Finally, a 4 kg/min decrease change in the feedstocksupply Ff occurs at 420 min, the flow of convective

Fig. 2. Control input considering Ziegler–Nichols PID controller.

Fig. 3. Control output considering IMC PID controller.

Fig. 4. Control input considering IMC PID controller.

R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705 703

energy is diminished. The system is forced to change theflow of energy in the process. In order to reject thisperturbation, the controller increase the regeneratedcatalyst flow rate, Fs (Fig. 4) and is able to maintain thesystem at the set point.

Given the time varying structure of the gains, the con-troller proposed showed good robustness and stabilityagainst disturbances and set-point changes. In contrast,two standard PIDs with fixed gains, exhibit unacceptableperformance, they could not deal with the disturbancessupplied and the nonlinear nature of the reactor. As it canbe noticed, this PID control design has shown increasedperformance during typical industrial operating situa-tions. It is important to notice that under the assump-tions done, the reactor analyzed does not present nonminimum-phase behavior.

5. Conclusions

A proportional-derivative reduced order observer hasbeen developed in order to produce on-line estimatevalues of the uncertain term present in the system, rela-ted to the heat generation by chemical reaction. Thisobserver is coupled to a linearizing control law in orderto obtain practical closed-loop stability. When the pro-posed control law is analyzed, it is important to observethat it presents the structure of a PID controller, withtime varying gains. The output temperature of a fluidcatalytic cracking reactor was controlled manipulatingthe catalyst flow, following industrial practice, using thePID structure proposed. This novel approach showedenhanced closed-loop performance in comparison toZN and IMC PID controllers.

New tuning rules for the controller’s gains were givenin terms of the system structure, closed-loop and thesampling characteristic times, which have importantphysical meaning, avoiding the application of identifica-tion methodologies and others methods which propose afixed controller’s gains (e.g. root locus, Ziegler–Nichols,IMC, etc.).

Acknowledgements

This work is supported under project FIES-98-111-IIof Instituto Mexicano del Petroleo. The authors aregrateful to the anonymous referees for their valuablecomments.

References

[1] D. Poncelet, Analyse Foundamentale des List Fluidises Biologi-

ques, PhD Thesis, Univ. Liege, Belgium, 1986.

[2] G. Bastin, D. Dochain, On-line Estimation and Adaptive Control

of Bioreactors, Elsevier, Amsterdam, 1990.

[3] P.D. Chistofides, P. Daoutiris, Robust control of hyperbolic

PDE systems, Chem. Eng. Sci. 53 (1) (1998) 85–105.

[4] A. Arbel, Z. Huang, I.H. Rinard, R. Sinnar, V. Sapre, Dynamic and

control of fluidized catalytic crackers. 1. Modeling the current gen-

eration of FCCs, Ind. Eng. Chem. Res. 34 (1995) 1228–1243.

Fig. 5. Control output considering the proposed controller.

Fig. 6. Control input considering the proposed controller.

704 R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705

[5] E.M. Hanczyc, A. Palazoglu, sliding mode control of nonlinear

distributed parameter chemical processes, Ind. Eng. Chem. Res.

34 (1995) 559–566.

[6] J. Gonzalez, R. Aguilar, M.A. Barron, J. Alvarez-Ramırez,

Nonlinear recursive feedback control for a tubular bioreactor, J.

Chem. Technol. & Biotechnol. 74 (1999) 766–770.

[7] D. Dochain, J.P. Babary, N. Tali-Maamar, Modelling and adaptive

control of nonlinear distribuied parameter bioreactors via orthogo-

nal collocation, Automatica 28 (1992) 873–883.

[8] P.H. Wallman, A.S. Foss, Experiences with dynamics estimators for

fixed-bed reactors, Ind. Eng. Chem. Fundam. 20 (1981) 234–239.

[9] Kolavennu, S. Palanki, Cockburn, J. Robust control of I/O line-

alizable systems via multi-model H2/H< synthesis. Chem. Eng.

Sci., (2000), 55, 1583-1589.

[10] Lee, P.L. and Sullivan, G.R. Generic model control (GMC).

Comp. Chem. Eng. 12, (1988) 6, 573-580.

[11] R. Aguilar, J. Gonzalez, J. Alvarez, M.A. Barron, Temperature

regulation of a class of continuous chemical reactor based on a

nonlinear luenberger-like observer, J. Chem. Tech. Biotech. 70

(1997) 209–216.

[12] J. Alvarez-Ramırez, Felipe lopez-isunza and ricardo aguilar. a

strategy to regulate temperature in fcc units with poor knowledge

of chemical kinetics, Comp. Chem. Engng. 13 (1996) S859–S864.

[13] J. Alvarez-Ramirez, R. Aguilar, F. Lopez-Isunza, Robust reg-

ulation of temperature in reactor-regenerator fluid catalytic

cracking units, Ind. Eng. Chem. Res. 35 (1996) 1652–1659.

[14] R. Aguilar, J. Gonzalez, M. Barron, J. Alvarez-Ramirez, Control

of a fluid catalytic cracking units based on proportional-integral

reduced order observers, The Chemical Engineering Journal 75

(1999) 77–85.

[15] C.G. Economou, M. Morari, Internal model control 5. extension

to nonlinear systems, Ind. Eng. Chem. PDD. 25 (1986) 403–411.

[16] M. Morari, Design of resilient plants-III. A general framework

for the assessment of dynamic resilience, Chem. Eng. Sci. 38

(1983) 1881–1891.

[17] A. Arbel, I.H. Rinard, R. Sinnar, V. Sapre, Dynamics and con-

trol of fluidized catalytic crackers. 4. The impact of design on

partial control, Ind. Eng. Chem. Res. 36 (1997) 747–759.

[18] M. Hovd, S. Skogestad, Procedure for regulatory control struc-

ture selection with application to the FCC process, AIChE J. 39

(1986) 1938–1953.

[19] R. Maya-Yescas, D. Bogle, F. Lopez-Isunza, Approach to the

analysis of the dynamics of industrial FCC units, J. Proc. Con-

trol. 8 (1998) 89–100.

[20] B. Barmish, M. Corless, G. Leitmann, A new class of stabilizing

controllers for uncertain dynamical systems, SIAM, J. Control

Optim. 21 (1983) 246–255.

[21] D.E. Rivera, M. Morari, S. Skogestad, Internal model control 4.

PID controller design, Ind. Eng. Chem. PDD. 25 (1986) 252–265.

[22] P. Venuto, E. Habib, Catalyst-feedstock-engineering Interactions

in fluid catalytic cracking, Cat. Rev.- Sci. Eng. 18 (1978) 1–50.

[23] V.W. Weekman, D.M. Nace, Kinetics of catalytic cracking

selectivity in fixed, moving and fluid bed reactors, AIChE J. 16

(1970) 397–404.

[24] A. Voorhies, Carbon formation in catalytic cracking, Ind. Eng.

Chem. 37 (1945) 4–8.

[25] J.G. Balchen, D. Lungquist, S. Strand, State-space predictive

control, Chem. Eng. Sci. 47 (1992) 787–798.

[26] C. Kavaris, Input/output linearization: a nonlinear analog of

placing poles at process zeros, AIChE J. 34 (1988) 1803–1812.

[27] G.R. Gavalas, Nonlinear differential equations of chemically

reacting systems, Springer-Verlag, New York, 1968.

[28] C. Byrnes, A. Isidori, J. Willems, Passivity feedback equivalence

and the global stabilization of minimum phase nonlinear systems,

IEEE Trans. Aut. Control 36 (1991) 1228–1240.

[29] J.J. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall,

Englewood Cliffs, NJ, 1991.

[30] A. Isidori, Nonlinear Control Theory, Springer-Verlag, New

York, 1985.

[31] J. Alvarez-Ramirez, Stability of a class of uncertain continuous

stirred chemical reactor with a nonlinear feedback, Chemical

Engineering Science 49 (11) (1994) 1743–1748.

[32] R. Martınez-Guerra, J. de Leon-Morales, Nonlinear estimators:

a differential algebraic approach, Appl. Math. Lett. 4 (1996) 21–

25.

[33] H.J. Sussman, P.V. Kokotovic, The peaking phenomenon and

the global stabilization of nonlinear systems, IEEE Trans.

Autom. Control 36 (1991) 424–440.

[34] M. Morari, E. Zafiriou, Robust Process Control, Prentice Hall,

NJ, USA, 363 p. 121, 1989.

[35] F. Doyle III, H. Budman, M. Morari, Linearizing controller

design for a packed-bed reactor using a low order wave propa-

gation model, Ind. Eng. Chem. Res. 35 (1996) 3567–3580.

[36] C. Kravaris, P. Doutiris, Nonlinear state feedback control of

second-order nonminimum phase systems, Computers Chem.

Engng. 14 (1990) 439–449.

[37] B.A. Ogunnaike, W.H. Ray, Process Dynamics, Modeling and

Control, Oxford University Press, UK, 1994.

R. Aguilar et al. / Journal of Process Control 12 (2002) 695–705 705