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Journal of Process Control 16 (2006) 913–921

Control design for suppressing transversal patternsin reaction–(convection)–diffusion systems

Yelena Smagina *, Olga Nekhamkina, Moshe Sheintuch

Department of Chemical Engineering, Technion, Haifa 32000, Israel

Received 13 February 2006; received in revised form 12 June 2006; accepted 14 June 2006

Abstract

The stabilization of planar stationary fronts solutions in a two-dimensional rectangular or cylinder domain, in which a diffusion–con-vection–reaction process occurs, is studied by reducing the original two-variable PDEs model to an approximate one-dimensional modelthat describes the behavior of the front line. We consider the control strategy based on sensors placed at the designed front line positionand actuators that are spatially-uniform or space dependent. We present a systematic control design that determines the number ofrequired sensors and actuators, their position and their form. The control used linear analysis of a lumped truncated model and conceptsof finite and infinite zeros of linear multidimensional systems.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Reaction–convection–diffusion processes; Planar front; Transversal patterns; Control; Root-locus method; System zeros

1. Introduction

Nonlinear parabolic partial differential equations(PDEs), which typically describe reaction–diffusion (R–D)systems, may admit spatially-dependent solutions like sta-tionary fronts as well as spatiotemporal patterns. The lattercan often be described as composed of slow-moving frontsseparated by domains of moderate changes. Propagatingfronts and patterned states may emerge in several technol-ogies including catalytic reactors [1], distillation processes[2], flame propagation and crystal growth [3] (see also [4]for references) as well as in physiological systems like theheart [5]. Our interest lies in catalytic reactors, in which sta-tionary or moving fronts and spatiotemporal patterns havebeen observed and simulated in various systems like flowthrough a catalyst [6], fixed-bed reactors [7,8], reactors withflow reversal [9] and loop reactors [10]. The instabilitiesemerge due to thermal effects in exothermic reactions,

0959-1524/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2006.06.003

* Corresponding author. Tel.: +11 972 482 92823; fax: +11 972 48230476.

E-mail address: cermssy@tx.technion.ac.il (Y. Smagina).

due to self-inhibition by a reactant and due to slow revers-ible modifications of the surface. The construction of acontroller that can stabilize a certain inhomogeneous solu-tion in an one-dimensional (1-D) R–D or a reaction–con-vection–diffusion (R–C–D) system is currently a subjectof intensive investigation [11–14]. Yet, most catalytic reac-tors, as well as physiological systems like the heart, exhibita behaviour that can be properly described by two- or eventhree-dimensional patterns.

In the present work we are interested in stabilizing sta-tionary (planar) fronts in a two-dimensional (2-D) domainin which a chemical R–D or R–C–D process occurs; i.e., wewant to suppress patterns that are transversal to the maindirection. The kinetic model can be described by coupledfast-activator and a slow-inhibitor. This terminology willbe elaborated on below. Previous studies of 1-D systemsdemonstrated that the simplest approach is by applying apoint-sensor control in which a single space-independentactuator responds to a sensor that is located at the frontposition [12]. However, a single point sensor control can-not stabilize planar fronts in a wide 2-D system. To over-come this problem we have recently studied control by

914 Y. Smagina et al. / Journal of Process Control 16 (2006) 913–921

several sensor/actuators situated at the front line [15]. Wehave also proposed there a new systematic strategy forinput–output selection, which assures that the control sys-tem obtained is a minimal-phase system1 (i.e., its finitezeros [16,17] lie in the left-half part of the complex plane).This property allows in turn to use the root-locus technique[18] for control design. The concepts of finite [16,17] andinfinite zeros [18] of a linear multivariable system areexplored to realize the above strategy.

This formal approach, however, lacks insight into thewave behavior, an insight that can be used to suggest effi-cient modes of control. Here we employ an approximatemodel reduction to a one-dimensional form that followsthe front position while approximating the front velocity.Such an approximate reduced model allows us to qualita-tively analyze various control strategies. We show that thisformalism accounts for a large class of problems and weapply it to R–D and R–C–D problems. Finally we comparethe results of these formal and approximate approaches.

The root-locus method, first introduced by Evans in1948, has been a powerful tool for the analysis and designof feedback system with single input/single output systemsas well as for multi-input/multi-output systems (e.g., see[18]). Unfortunately no control design applications thatuse the multivariable root-locus technique (with systemzeros analysis) have been reported to systems that aredescribed by partial differential equations. Such systemsare common in reaction–diffusion and in fluid-flow pro-cesses [14,19–22].

To explain the issues involved in suppression of trans-versal patterns let us review mechanisms that induce suchpatterns. Stationary fronts in one-variable one-dimensionalR–D systems are typically structurally unstable and existonly as a boundary between conditions that lead to expan-sion of one (say, hot) or the other (cold) state. Changing aparameter will monotonically change front velocity andcan cause a sign-change. Two-variable 1-D such systemswith a fast diffusing-activator and a slow non-diffusinginhibitor (like the FitzHugh-Nagumo model, see below)also cannot produce stationary fronts and fronts willeventually disappear out of the system. The change infront-velocity, upon changing a parameter, may be morecomplex and involve a hysteresis loop [23]. As stated abovefronts can be pinned to their position in 1-D systems bymeasuring the state variable at the front and using it toattenuate one of the parameters that affect front velocity.Such point-sensor control of planar fronts in 2-D systemswill lead to transversal patterns in a sufficiently wide sys-tem. That calls for control that uses spatial actuators, thatare discussed below. Stationary fronts and patterns mayemerge in 1- or 2-D activator–inhibitor systems when theinhibitor diffusivity is sufficiently large (the Turing mecha-nism), or due to global-coupling [24], to end effects or to

1 Let’s note that nonminimal-phase systems are always troublesome forcontrol.

inhomogeneities (see [25] for a recent review); these effectsare not discussed here.

The behavior of fronts in R–C–D systems is somewhatsimilar to R–D system after accounting for the effect ofconvection. Decades of investigation into front propaga-tion in catalytic reactors, using a thermo-kinetic model thataccounts for reactant concentration and reactor tempera-tures as its variables, have shown that for the physically-common case that reactant diffusivity is small or negligiblethe problem can be reduced to an one-variable presentationand stationary fronts exist only as a boundary betweenexpanding hot and cold zones [7,26,27]. Front velocity isextremely slow, due to the high heat capacity, and nearthe stationary front conditions it is linear with fluid veloc-ity. Thus, the simplest approach to control stationaryfronts in 1-D reactor is to use the flow rate or feed condi-tions as an actuator.

Transversal patterns may emerge in R–C–D systemswith oscillatory kinetics, in which the thermo-kineticsmodel is coupled with a slow non-diffusing inhibitor. Sym-metry breaking in the azimuthal direction, of a cylindri-cally-shaped thin catalytic reactor, was simulated whenthe perimeter is sufficiently large [27] in realistic reactorsmodels of high Le, ratio of solid- to fluid-phase heat capac-ities, and high Pe, ratio of convection to conduction num-bers. An approximate description of front position wasdeveloped there [27] and is employed here. Transversal pat-terns may also emerge in simple thermo-kinetic (i.e., non-oscillatory) model but apparently only with a sufficientlylarge reactant diffusivity (PeC/PeT < 1) due to Turing-likeinteraction between the temperature and the concentration[28,29]. However, in most situations PeT < PeC.

2. Problem statement

Consider the R–C–D problem, in the (z, s) rectangularor cylindrical shell domain of length L and width S, whichis described by a nonlinear parabolic PDEs of the form

yt þ ðV þ kV Þyz � yzz � yss ¼ Pðy; hÞ þ k; ð1Þht ¼ eQðy; hÞ ð2Þwhere y = y(z, s, t) is typically an activator (e.g., tempera-ture), undergoing diffusion and convection, and h =h(z, s, t) is the slow inhibitor that is a local (surface) prop-erty; k and kV are the control variables; e is the ratio of timescales (the diffusivity here is set to unity since it was scaledinto the length scale). No-flux or periodic boundary condi-tions are applied in the s-direction for a rectangular or acylindrical domain, respectively.

We have previously studied Eqs. (1) and (2) with thepolynomial source functions [15]

Pðy; hÞ ¼ �y3 þ y þ h; Qðy; hÞ ¼ �cy � h ð3Þsince the cubic P(y,h) adequately simulates the phenome-non of multiple steady states (bistable kinetics) due to ther-mal and autocatalytic effects [30], and since severalanalytical results are available. In Eq. (3) c < 1 is a positive

0 5 10 15 201

0

1

y,θ

z

a

yθ

0 10 20 30 40 500

1

2

3

1/eps

min

(abs

(k))

b

-

Fig. 1. A typical solution of the reaction–diffusion full system (Eqs. (1)–(3), kV = 0) (a) and the minimal gain (min(abs(k))) required to stabilize thefront in the full model with the control law (20) (b). Circle denotes theminimal gain value [abs(k)] calculated by using the approximate modelwhen e! 0. Other parameters: L = 20, S = 5, c = 0.45, Z* = L/2, s1 = S/2.

Y. Smagina et al. / Journal of Process Control 16 (2006) 913–921 915

constant. We have also studied the problem of a thermoki-netic first-order exothermic reaction [12], which may be re-duced to a form similar to Eq. (1) (see Section 5.2 below).

We consider the problem of stabilizing the planar frontin a 2-D R–C–D system (Eqs. (1)–(3)) at a position z = Z*

(a constant value) which, in an open-loop system, is sta-tionary but unstable (for a typical 1-D solution seeFig. 1a). Narrow systems behave like a 1-D system andadmit a stationary front solution that is unstable and typ-ically oscillates or travels out of the system [13]. Point-sen-sor control will stabilize this front. However, in asufficiently wide 2-D system, the front-line may undergosymmetry breaking so that in part of it the upper stateexpands while in other parts the lower state propagates.

We will investigate control with several actuatorslocated along the front position z = Z* at the points(z, s) = (Z*, sd), d = 1, . . .g and apply a general feedbackcontrol law of the form ðk; kV Þ ¼ ð�~k; 0Þ or ðk; kV Þ ¼ð0; ~kÞ with

~k ¼ kXg

d¼1

½yðZ�; sd ; tÞ � y�d �wdðz; sÞ ð4Þ

where k is a scalar gain coefficient, yðZ�; sd ; tÞ � y�d are devi-ations of the sensors reading from the set points,y�d ¼ y�ðZ�; sdÞ, and wd(z, s) are some space-dependent func-tions. We will seek control (4) with the simplest space-inde-pendent or space-dependent actuator functions wd(z, s) anda minimal number g of sensors. Such control structure imi-tates the unity negative output feedback with a scalar gaink < 0 in the related lumped model and can be found byusing the root-locus technique.

We consider a control design method based on the fol-lowing approximate model reduction.

3. Approximate model

Consider the following approximations (see [15] for adetailed derivation): When the activator (y) is fast whilethe inhibitor (h) is slow (i.e., e� 1) we can study, to a firstapproximation, the velocity of the activator front for fro-zen h profile. The opposite inclination of the activatorand inhibitor profiles (Fig. 1a) is the source of instability,

leading to front motion in the one-dimensional system. IfZ(s, t) is the front position, and c = c1(ho(Z(s)),k) is theapproximate velocity of a planar front (oZ/os = 0) in theabsence of convection (V = 0), so that c > 0 implies expan-sion of the y > 0 domain, which depends on the localsteady-state inhibitor (ho) and control variable (k), thenthe velocity of a low- curvature front in the presence ofconvection is c ffi c1 � V � Zss � kv. Then, the positionof an ascending front (dy/dz > 0), assuming a small pertur-bation from the stationary planar front (c1 � V = 0),approximately follows [15]

� dZdt¼ c ¼ c1ðhðZðs; tÞ; kÞÞ � V � kV �

o2Z

os2ð5Þ

The front propagation equation linearized aroundZ*(k = kV = 0) becomes

� dZdtffi oc1

oh

� �f

ohoz

� �f

Z þ oc1ok

k� kV �o2Zos2

where Z ¼ Zðs; tÞ � Z�, the subscript ‘f’ denotes that deriv-atives are estimated at Z*. Since Q(yo,ho) = �cyo � ho = 0at steady state then we obtain for small deviations of yo

and ho: (oh/oz)f = �c(oy/oz)f. Thus, the linearized frontpropagation equation becomes

dZdt¼ c

oc1oh

� �f

oyoz

� �f

Z � oc1ok

kþ kV þo2Zos2

ð6Þ

If the uncontrolled (k = kV = 0) system (6) is unstable (i.e.,when oc1/oh > 0) then a simple point-sensor control in theform

~k ¼ k½yðZ�; tÞ � ysðZ�Þ� � koyoz

� �f

ðZ � Z�Þ ð7Þ

is sufficient to maintain the front in its set position in anarrow system (Eqs. (1)–(3), yss = 0). Here k < 0 when~k ¼ kV ðk ¼ 0Þ or sign(k) = sign(oc1/ok) when ~k ¼ kðkV ¼ 0Þ. Using the solution of the 1-D one-variable frontvelocity c1(h), and the two-variable profile, we can evalu-ate the minimal absolute value of gain coefficient requiredfor control by k or kV.

In a wider system we need to account for small pertur-bations in the transversal directions of the front line. Thisstimulates us to extend the point-sensor control law (7)as follows:

~k ¼ koyoz

� �f

Xg

d¼1

Zðsd ; tÞwdðsÞ ð8Þ

where sd, d = 1,2, . . . ,g are the sensor positions along thefront line in s-direction, wd(s) are the actuator functions.For convenience we set wn(s) = /n(s) where /n(s) are theeigenfunctions of the linear operator of PDE (6)

/n ¼Cnffiffiffi

Sp cos

ðn� 1ÞpS

s ð9Þ

with appropriate boundary conditions (n = 1,2, . . . forplane domain or n = 1,3, . . . for cylinder), Cn = 1 if n = 1or Cn ¼

ffiffiffi2p

if n > 1.

3 This assures the non singularity [16] of system (14).

916 Y. Smagina et al. / Journal of Process Control 16 (2006) 913–921

We determine now the sensors number (g) and theirpositions (sd). Expanding small perturbations of Zðs; tÞand Zðsd ; tÞ as

Zðs; tÞ ¼X

n

anðtÞ/nðsÞ; Zðsd ; tÞ ¼X

n

anðtÞ/nðsdÞ ð10Þ

we lump the linearized Eq. (6) with control (8) by Galerkinmethod to obtain the following discrete-version realization

_an ¼ coc1oh

� �f

oyoz

� �f

� qn

" #an

þ kckoyoz

� �f

Xg

d¼1

bdn

Xj

ajhdj; n ¼ 1; 2; . . . ð11Þ

where qn are the eigenvalues of the above linear operator(qn = (n � 1)2p2/S2, n = 1,2, . . . for plane domain orn = 1,3, . . . for cylinder), ck = oc1/ok when ~k ¼�kðkV ¼ 0Þ or ck = 1 when ~k ¼ kV ðk ¼ 0Þ and

hdj ¼ /jðsdÞ ð12Þ

bdn ¼Z S

0

/dðsÞ/nðsÞds ¼1; d ¼ n

0; d 6¼ n

�ð13Þ

In vector-matrix notation Eq. (11) is a linear infinite-dimensional dynamical system with a finite-dimensional in-put vector v and output vector w

_a ¼ KaþIg

O

� �v; w ¼ Ha ð14Þ

closed by the finite-dimensional linear output feedback

v ¼ kckoyoz

� �f

Igw ð15Þ

In (14) a = a(t) is the infinite dimensional vector; v and w

are g vectors; matricesIg

O

� �and H = [hdj], d = 1, . . .,g,

j = 1,2, . . ., have g infinite-dimensional columns and rows,respectively, and Ig is an g · g unit matrix. The diagonalelements of the matrix K = diag(r1,r2,� � �) are

rn ¼ coc1oh

� �f

oyoz

� �f

� ðn� 1Þ2p2

S2ð16Þ

From Eq. (14) it follows that the sensor positions stipulatethe structure of the output matrix H (see Eq. (12)) and theparameter g assigns the number of inputs and outputs ofthe system. Thus, the problem can be restated as follows:For the linearized infinite-dimensional ODEs (14) it is nec-essary to find the matrix H with the minimal number ofrows such that the finite-dimensional output feedback con-trol (15) stabilizes the closed-loop system 2.

Below we use a truncated (finite-dimensional) approxi-mation of Eq. (14) with truncated order N. An approxi-mate estimate of truncation order N may be obtained byconventional methods.

2 In the work [15], where the full-dimensional (2-D) problem wasstudied, we searched the input and output matrices.

4. Root-locus control design

To design control (15) we use the multivariable root-locus technique [18] that is based on analysis of finite zerosand infinite zeros [16,17] of the open-loop system. For real-ization of the root-locus technique we need at first to esti-mate the minimal number of sensors (g) that assuresstabilization of the closed-loop system (14), (15). This gcoincides with number of inputs and outputs of the open-loop system (Eq. (14)). Then we need to find matrix H offull rank3 which ensures that the finite zeros of (14) are sit-uated in the left-half part of the complex plane. In the finalstep we need to provide that all infinitely increasing eigen-values of the closed-loop system (Eqs. 14 and 15 withabs(k)!1)4 tend to infinity along asymptotes with a neg-

ative real angle. This is assured if rank HIg

O

� �� �¼ rank

H g ¼ g where Hg is the first g · g block of the matrix H

H g ¼

/1ðs1Þ /2ðs1Þ . . . /gðs1Þ/1ðs2Þ /2ðs2Þ . . . /gðs2Þ

..

. ...

. . . ...

/1ðsgÞ /2ðsgÞ . . . /gðsgÞ

2666664

3777775 ð17Þ

and the matrix (17) has only eigenvalues with positive realparts. Otherwise, we need to find an g · g matrix M thatensures above property to the matrix MHg.

To construct the matrix H (its elements depend on sen-sor positions in s-direction (see (12)) we need to seek theappropriate sensor positions. The following theorem givesconditions on the sensor number that assures ‘negative’finite zeros.

Theorem 1. The minimal g coincides with a number ofpositive eigenvalues r1, . . .,rg of the matrix K. This gguarantees that finite zeros of (14) lie in the left–half part

of the complex plane.

See Appendix for proof.

Remark 1. The conditions si 5 sj,i 5 j, i,j = 1,2, . . .,gassure that the matrix H is of full rank because functions/k(s), k = 1,2, . . . are orthonormal ones on [0,S]. More-over, we need to choose si to satisfy rank Hg = g.

The infinite zeros of the system (14) coincide with eigen-values of the matrix Hg (17). If several eigenvalues of Hg

are not positive values we need to find the matrix M thatassures that all eigenvalues of MHg are positive. Introduc-ing the nonsingular g · g precompensator M transforms5

control (15) to the form v ¼ kckoyoz

� fMw and control (4)

(with wd(z, s) = /d(s)) becomes in the final form

4 Infinite zeros of system (14).5 Such operation does not change the finite system zeros of (14), (15)

because they are invariant to any nonsingular transformation of an output[16,17].

Y. Smagina et al. / Journal of Process Control 16 (2006) 913–921 917

~k ¼ kXg

d¼1

Xg

j¼1

mdj�yðZ�; sd ; tÞ" #

/dðsÞ ð18Þ

where mdj are elements of the matrix M, �yðZ�; sd ; tÞ ¼yðZ�; sd ; tÞ � yoðZ�; sdÞ.

Therefore, the general strategy of the method is asfollows:

1. Find actuator shapes that are eigenfunctions of linearoperator of PDE (6).

2. Calculate the minimal number g of actuator/sensors,that is the number of positive eigenvalues of the dynam-ics matrix K.

3. Assign sensor positions in s-direction satisfying the con-ditions si 5 sj (i 5 j, i, j = 1,2, . . .,g) and rank Hg = g.

4. If necessary find the precompensator M that assurespositiveness of real parts of all eigenvalues of the matrixMHg.

Resume: Such the strategy assures that control (18) stabi-lizes the front position in s-direction for a sufficiently largegain k according the root-locus method.

Remark 2. Using the approximate 1-D model considerablysimplifies the controller design (i.e., the spatial form of theactuators, the minimal number of sensors and theirpositions) when compared with the full 2-D system analysisconsidered in [15].

Remark 3. The first term in control (18) (with space-inde-pendent actuator) arrests also the front in z-direction.

Remark 4. Let us note that the linear feedback (18) assureslocal asymptotic stability of the original nonlinear systemin accordance with Lyapunov’s linearization theorem[31]. Thus, this control well stabilizes the process for smallperturbations from the steady state solution. At the sametime applying the root-locus technique for design ensuresrobustness of the closed-loop system with respect to smalluncertainties in parameters. Thus, stabilization is carriedout even when sensors are not exactly located at the frontline.

6 Note, that for the present system (oc1/oh) = (oc1/ok).

5. Applications

5.1. Polynomial model (Cubic source-function, Eq. (3))

Let us apply this procedure to design additive controlk(kV = 0) that will stabilize the (analytical) steady-statesolution of R–D system (Eqs. (1)–(3), V = 0) with e! 0in the rectangular domain. The steady state yo and c1 atthe front position are calculated as yo ¼

ffiffiffiffiffiffiffiffiffiffiffi1� cp

tanh

ððz� Zf Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� cÞ=2

pÞ, c1f ffi 3ðho þ kÞ=

ffiffiffi2p

[15] with oc1oh

� f

ffi 3=ffiffiffi2p

and oyoz

� f¼ ð1� cÞ=

ffiffiffi2p

(see details in [32]). We willcompare our results with previous ones [15] that were basedon the zero structure of exact 2-D model (i.e., for any e).

The space-dependent actuators take the form (9) withn = 1,2, . . .. Now, we need to calculate a number (g) ofpositive eigenvalues (rn) of the matrix K, which are givenby the relation

rn ¼ 0:82c� ððn� 1Þp=SÞ2; n ¼ 1; 2; . . . ð19ÞThis g coincides with number of actuator/sensors in control(18) for an assigned width S. Then we assign sensor posi-tions in the s-direction, si, i = 1, . . .,g, and calculate the ma-trix Hg (17). If several (or one) eigenvalues of Hg are not‘positive’ we need to find the matrix M that assures thatall eigenvalues of MHg are ‘positive’.

For a certain set of parameters [c = 0.45,L = 20], thatwas also examined in [15] with e = 0.1, we find for theapproximate model the following domains:

(i) S < 5.3, one space-independent actuator is sufficient

k ¼ k�yðZ�; s1; tÞ ð20Þ(ii) 5.4 6 S 6 10, two actuators/sensors are required with

k ¼ kX2

d¼1

X2

j¼1

mdj�yðZ�; sj; tÞ cosððd � 1Þps=SÞ" #

ð21Þ

(iii) 10 < S 6 16, three actuators/sensors are requiredwith

k ¼ kX3

d¼1

X3

j¼1

mdj�yðZ�; sj; tÞ cosððd � 1Þps=SÞ" #

ð22Þ

In [15] it was shown that control with the one space-inde-pendent actuator is effective for S 6 5.5. For wider systems(e.g., S = 7) we found the required control to be of theform (21) with m11 = m22 = 0, m12 = m21 = 1. This demon-strates good corroboration between these two approachesbut the selection process here is considerably simplified.

As discussed earlier the final form of control (18)depends on mdj values. For small g (i.e., g = 2) we can find

analytically that M ¼ 0 11 0

� �. Indeed, calculating the

eigenvalues (vi) of the 2 · 2 matrix Hg (17) we find fromthe equation (1 � v)(cos(ps2/S) � v) � cos(ps1/S) = 0 thatRe(v1,2) > 0 if cos(ps1/S) < cos(ps2/S) or if s2 < s1. Toassure this condition we need to rearrange rows of thematrix Hg. This operation is equivalent to left multiplica-tion of Hg by M. Thus, in the range 5.4 < S 6 10 the finalcontrol (21) (with m11 = m22 = 0, m12 = m21 = 1) coincideswith one obtained by the exact method.

For approximate model we may evaluate the minimal(absolute) value of the required gain �k6 and compare it withthat predicted by the full exact model (with any e). Forexample, �k ¼ 1:0 for the system of width S = 5 with control(20). This �k corresponds to that for the full model withe = 0.3 (see Fig. 1b).

918 Y. Smagina et al. / Journal of Process Control 16 (2006) 913–921

The effectiveness of control (20), (21) with one and twoactuators in the thin and wider system is demonstrated inFig. 2a–c. Control (20) with the one space-independentactuator effectively suppresses transversal perturbations ina system of width S = 5 (Fig. 2a). This control does notwork when the width is S = 7(Fig. 2b) and a control withtwo actuator/sensors is required for such a system (Fig. 2c).

5.2. Thermokinetic model

In order to demonstrate that the know-how gained inthis work can be applied to realistic problems we considera two-dimensional model of a thin cylindrically-shapedadiabatic reactor. A cylindrical shell is a simple two-dimen-sional geometry that captures some of the properties of afull three-dimensional reactor. We analyze pattern forma-tion for the generic pseudo-homogeneous model of a fixedbed reactor catalyzing a first order reaction of Arrhenius-kinetics (i.e., thermokinetic model). The reaction is coupledwith slow changes of catalytic activity (i.e., oscillatorykinetics). The model consists of balance equations for threestate variables, reactant concentration (C), temperature (T)and catalytic activity (h), and may be written in the follow-ing dimensionless form:

oxosþ V

oxon� 1

Pec

o2x

on2þ 1eR2

o2x

os2

� �¼ Dahð1� xÞ exp

cycþ y

� �¼ ð1� xÞhhðyÞ ð23aÞ

Leoyosþ V

oyon� 1

PeT

o2y

on2þ 1eR2

o2yos2

� �¼ Bð1� xÞhhðyÞ ð23bÞ

n ¼ 0;1

PeC

oxon¼ Vx;

1

PeT

oyon¼ Vy; n ¼ 1;

oxon¼ oy

on¼ 0

ð23cÞyðn; 0Þ ¼ yðn; SÞ; xðn; 0Þ ¼ xðn; SÞ ð23dÞ

Khdhds¼ ah � bhh� y ¼ qðy; hÞ;Kh Le 1 ð23eÞ

Fig. 2. Testing the effectiveness of the number of actuator/sensors according tline) y profiles in s-direction (column1, inserts are initial and final profiles in z

z = L/2 (columns 2, 3) (parameters as in Fig. 1, e = 0.1, k = �20); (a) S = 5, usiactuator. (c) S = 7, using one space-independent and one space-dependent�0.01sin(ps/S) in (c). Sensors positions are at Z* = L/2 and s1 = S/2 in (a,b)

with conventional notations

x ¼ 1� CCin

; y ¼ cT � T in

T in; n ¼ z

L; eR ¼ S

L;

s ¼ tuo

L; V ¼ u

uo; c ¼ E

RT inB ¼ c

ð�DHÞCin

ðqcpÞf T in;

Da ¼ Að1� eÞLuo

e�c; Le ¼ ðqcpÞeðqcpÞf

;

PeT ¼ðqcpÞf Luo

ke; PeC ¼

Luo

eDfð24Þ

(see [27] for details). Note that we choose some averagevalue uo as the velocity scale, in order to study separatelythe effects of the velocity control (V). Other parametersthat may be used for actuators are feed conditions (Cin,Tin), which can be approximately described by manipulat-ing B.

For further analysis we employ the physically reason-able assumption that mass-axial mixing is negligible(PeC!1). Thus, reduced Eq. (23a) becomes

oxosþ V

oxon¼ ð1� xÞhhðyÞ ð23fÞ

The control of the one-dimensional version of this model(yss = 0) was studied in [12]. The properties of global con-trol and point-sensor control aimed to stabilize a frontsolution were studied by manipulating various reactorparameters, including fluid flow (V) and feed conditions.Patterns in the shell cylindrical reactors (two dimensional)were shown to undergo symmetry breaking in the azi-muthal direction when the perimeter is sufficiently large[27]. The generic regular patterns simulated then are rotat-ing multi-wave patterns of constant rotation-speed andoscillatory-’firing’ ones, and theirs selection is highly sensi-tive to governing parameters and initial conditions.

We consider two types of control aimed to suppresstransversal symmetry breaking (in the s-direction):

o control law (18). The figures present initial (dashed line) and final (solid-direction) and gray-scale plots in the planes (t,z) at s = S/2 or in (t,s) atng one space-independent actuator. (b) S = 7, using one space-independentactuators. Initial perturbation in s-direction is �0.01cos(ps/S) (a,b) andand s1 = S, s2 = S/2 in (c).

Fig. 3. Typical snapshots of the unfolded cylinder for the open-loop (a)and closed-loop system with type (i) control of the feed-concentration(actually of (B)) (b) or type (ii) control (velocity control) (c) for ahomogeneous reactor thermokinetic model (Eqs. (23a)–(23e)). Each row iscomposed of three (s,n) snapshots of temperature within the periodiccycle. Le = 100, V = 1, B = 30, c = 20, Da = 0.06, PeT = 65, PeC/PeT = 0.1, Kh = 1000, ah = 40, bh = 50, S = 0.20, k = 1.

Y. Smagina et al. / Journal of Process Control 16 (2006) 913–921 919

(i) Parameter control by manipulating feed concentra-tion (B! Bo + k, B is proportional to reactant feedconcentration).

(ii) Velocity control by manipulating flow rate (V!Vo + kV).

We will seek to design the control variables kV or k inthe form of Eq. (4) with the front position at Z* = n*L

and with y�dðZ�; sd ; sÞ as the set-points (temperature values)at the sensor positions (Z*, sd), d = 1, . . .,g, si 2 [0,S],n 2 [0,1]. The sensors are placed at the desired stationaryplanar front position.

For control design we apply an approximate descriptionof front position that was obtained by assuming that h var-ies slowly and that the activily profile, ho(n), is frozen. Thederivation of the front propagation velocity is based onassumptions of Frank-Kamenetsky [26] and Kiselev [7]and will be presented elsewhere. Then the position Z(s,s)of an ascending front (dy/dn > 0) around the steady state(c� V/Le) can be shown to be described by a nonlinearone-dimensional PDE

�LedZds¼ c ffi c�ðhoðZÞ; k; kV Þ �

1~PeT

o2Zos2

ð25Þ

where c* is the approximate planar front velocity in thevicinity of the steady state and

c� ¼ 2ð1þ B=cÞ2

BV ð1� V

V �Þ;

V � ¼ ð1þ B=cÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDahoðZÞ

BPeT

eB

1þB=c

sð26Þ

In (25) parameter ~PeT ¼ PeT=L does not depend on the sys-tem length. For small perturbations Z ¼ Z � Z� from thestationary planar front Z* we obtain the linearized model

LedZds¼ � oc�

oh

� �fo

ohon

� �fo

Z � oc�

ok

� �o

k

"

� oc�

okV

� �o

kV þ1

~PeT

o2Z

os2

#ð27Þ

where the subscript ‘fo‘ denotes that derivatives are esti-mated at Z*, Bo, Voand subscript ‘o’ denotes that deriva-tives are estimated at Bo,Vo.

We use the above considered strategy for design of con-trol of form (8) where wd(s) = /d(s) are eigenfunctions ofthe problem /ss(s) = �q/(s) with periodic boundary condi-tions and a number of sensors (g) calculated as the numberof positive terms rn in the expression

rn ¼1

bh

oc�

oh

� �fo

oyon

� �fo

� 4ðn� 1Þ2p2

~PeTS2; n ¼ 1; 3; . . . ð28Þ

In (28) we have used the specific form of Eq. (23e) to eval-uate (oh/on)f = �(oy/on)f/bh. The final form of controlcoincides with expression (18) where /d � cos(p(d � 1)s/S), d = 1,3, . . ..

The elaborated control was applied to stabilize the sta-tionary planar front for conditions (see Figs. 3 and 4, thecaptions list the parameters), in which a front is knownto exist at Z* = 0.45L for the one-dimensional problem.In a 2-D open-loop system the planar front is stable for acylinder of perimeter S < 0.137 and looses stability atS = 0.137 through an infinite-period bifurcation to compli-cated two-dimensional patterns [27]. We choose to studythe control at S = 0.20, when the open-loop system yieldsa ‘‘firing’’ pattern (see Figs. 3 and 4a) with a sharp frontnear the reactor inlet (the transient process leading fromthe unstable steady state to the ‘firing’ state is illustratedby Fig. 4a(1)). This pattern is characterized by a periodicalexpansion and shrinking of a cold spot that emerges alter-natively at the opposite sides of the cylindrical shell.

We test two types of control for PDE (23a–e): parametercontrol (i) and velocity control (ii). In both cases we useproportional feedback control (18). The axial sensor waschosen to be positioned (n* = 0.445) close to the stationaryfront position, and the transversal coordinates sd were cho-sen to be equally spaced along the perimeter. The simula-tion revealed that both control strategies enabled tostabilize the front position around its steady state, withsmall fluctuations, using a single space independent actua-tor. The front shape, however, is not constant and patternsin the closed-loop system are essentially two-dimensional innature: they exhibit complex quasi-periodic oscillationscomposed of varying ‘‘firing’’ spots coupled with angularrotation (Figs. 3 and 4). The transversal-direction variation

Fig. 4. Patterns of the homogeneous reactor model (Eqs. (23a)–(23e)) inthe case of the open-loop (a) and the closed-loop B-control system usingone (b) or two (c) actuators/sensors. Row 1 presents the temperatureevolution in (s,n) plane at certain s (=0) showing (a) a transition processwhen the one-dimensional stationary planar front is used as initialconditions, and (b, c) quasi-periodic patterns in the closed-loop system.Row 2 presents typical snapshots of the spatial temperature profiles:s1 = 0, s2 = S/2. Other parameters are as in Fig. 3.

920 Y. Smagina et al. / Journal of Process Control 16 (2006) 913–921

of the front position, defined as a point with 50% conver-sion (x = 0.5), is about 9% with control type (i) and 25%with control type (ii). Increasing the number of actuatorsat the front-line can significantly reduce the amplitude ofthe front oscillations (compare Fig. 4b(1) with one sensorand 4c(1), with two equally spaced sensors). At the sametime the system is still not stable and large amplitude tem-perature oscillation (�8%) are evident in the downstreamsection (compare Fig. 4b(2) and 4c(2)).

6. Conclusions

We have developed a new methodology to stabilizeplanar stationary front solutions in a two-dimensionalrectangular and cylinder domain, in which a diffusion–convection–reaction systems occurs. To that end we usean approximate 1-D model that describes the behavior ofa front line. This reduction considerably simplified thesearch of a control structure in comparison with the exactprocedure applied to the original 2-D model. We use afinite-dimensional point-sensor feedback control that isdesigned by multivariable root-locus method. This controlshould be applied only for minimal-phase systems (havingonly negative finite zeros) and that stimulated us to developthe strategy of searching input-output sets to assure theabove property to the system. The method exploited theconcepts of finite and infinite zeros of linearized multivari-able finite-dimensional realization of the approximatedsystem.

The main drawback of the point sensor control is its sen-sitivity to sensors location, i.e., the reduced effective gainwhen sensors are not exactly located at a front position.This may be a major drawback as front position tend tovary due to changing conditions and deactivation. The sug-gested control calculated by using the root-locus techniqueassures robustness with respect to parameter uncertaintyand that helps to overcome the typical shortcoming ofthe point sensor control.

The analysis of the learning model with polynomialsource functions is quite simple. Its extension to the morerealistic thermokinetic model revealed that the main appli-cation can be conducted with little preparation but certainproblems remain: The concentration and temperature can-not always be lumped into a single variable, as they showcertain nonlinear interaction (see Fig. 4). The cylindricalsystem shape can lead to rotating patterns (which areabsent in the rectangular geometry). Most importantly, acontrol designed to supress a certain mode (say, movingor transversal patterns) results, beyond a bifurcation, inother (transversal or rotating, respectively) patterns.Future work will address modes of supressing rotatingpatterns.

Acknowledgements

Work supported by the US-Israel Binational ScienceFoundation. MS is a member of the Minerva Center forNonlinear Physics of Complex Systems. ON acknowledgespartial support by the Center for Absorption in Science,Ministry of Immigrant Absorption, State of Israel.

Appendix

Proof of Theorem 1. Let us build the (N + g) · (N + g)system matrix [33] P(l) for Eq. (14): PðlÞ ¼

lI � K �BH O

� �where B ¼ Ig

O1

� �and H is g · N matrix.

Finite zeros of system (14) are determined as the set ofcomplex l = li for which the normal rank [34] of thesystem matrix P(l) is reduced [33]. Partitioning the matrixK and H as K = diag(Pg,PN�g), H = [Hg,HN�g] wherePg = diag(r1, . . . ,rg) and PN�g = diag(rg+1, . . . ,rN) wewrite the serious of the rank equalities

rank P ðlÞ ¼ rank

½lIg � P g� O ...�Ig

O ½lIN�g � P N�g� ...

O

. . . . . . ...

. . .

H g H N�g...

O

2666666664

3777777775

¼ rank

O ...½lIN�g � P N�g�

. . . . . . . . .

H g...

HN�g

2666437775

Y. Smagina et al. / Journal of Process Control 16 (2006) 913–921 921

from which follows that the normal rank of the matrix P(l)loses if and only if the complex l coincides with any rj,j = g + 1, . . .,N. Thus, these negative rj are finite zeros ofsystem (14). h

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