Measurement-Based Optimization and Predictive Control for an Exothermic Tubular Reactor System

Measurement-Based Optimization and Predictive Control for an ExothermicTubular Reactor System

Wei Wu* and Chao-Wei ChenDepartment of Chemical Engineering, National Yunlin UniVersity of Science and Technology, Douliou,Yunlin 64002, Taiwan, Republic of China

In this article, the analytic optimization algorithm connected to the measurement-based predictive controlframework is implemented on an exothermic tubular reactor system. The two stages of design procedureinclude (i) the desired input/output references, which are determined by the steady-state optimization approach,and (ii) the output regulation design of nonlinear distributed parameter systems, which is addressed usingnondistributed predictive controls. Under the assumption of steady-state and plug-flow characteristics, thebang-bang type of extremal control is applied. Because of the fact that inlet perturbations could trigger thehot spots or thermal runaway, we propose two dependent manipulated variables to dominate the heat exchangefunction for cooling devices. With respect to specified state/input constraints, the output feedback architecture,which uses a few sensors for measurement of the reactor temperature at the prescribed axial position, issuccessfully demonstrated. It is a specific measurement-based feedback technique used to combat the effectsof heat or unknown disturbances. All tests show that the no-offset output tracking is achieved and the undesiredpeak temperature is removed while physical constraints and unknown disturbances are being consideredsimultaneously.

1. Introduction

The use of partial differential equation (PDE) models forsystem analysis and the design of estimation and controlalgorithms has become increasingly important recently.1-3

However, those control strategies for a class of nonlineardistributed parameter systems were too complex to implementin practice, because of the original distributed control frame-work. Although tubular reactors have been largely used inchemical process industry, they cannot be operated at excessivetemperatures, because of exothermic reactions and inlet pertur-bations, such that a undesired peak (hot-spot) temperature orthermal runaway4 would induce undesired side reactions andthermal decomposition of the products. Therefore, the problemof suppressing the magnitude of the hot-spot temperature oftubular reactors using a coolant device has been recentlyaddressed.5,6 Because the position and magnitude of the hot-spot temperature cannot be directly established via explicitnonlinear functions, the state feedback control framework orsteady-state optimization approach is often applied.

Recently, model predictive control (MPC) algorithms havebeen widely studied and applied in many chemical processes.7

Unfortunately, reports of MPC based on partial differentialequation (PDE) models are relatively scarce. Shang et al.8

proposed a modified MPC scheme that is capable of effectivelydriving a distributed process output to its setpoint, Dufour etal.9 provided a general MPC framework for a PDE model ofthe autoclave curing process, and Dufour and Toure10 developeda multiple-input/multiple-output (MIMO) MPC for PDE sys-tems. Notably, input/output constraints are handled in theoptimization task using a nonlinear programming method, andthe objective of approximating a PDE model is to reduce thecalculation time for the online MPC procedure. In addition toinput/output constraints and the characteristic of distributedparameter systems, the exothermic tubular reactor system shouldconsider the other crucial and practical problems on hot spots,coolant devices, and process optimization. Moreover, low-order

predictive control of a class of PDE systems has beendeveloped.11,12Note that the decomposition and transformationis used to dominate the model reduction such that closed-loopstability in the face of state/input constraint should be dependenton the state estimation design and the number of outputmeasurements.

Regarding optimal control and design, the terminal-costoptimization problem was usually implemented for batch processoptimization for the past several years.13-16 Notably, the pathand terminal constraints are considered simultaneously, and theexplicit synthesis of optimal control input was of the bang-bang type. Moreover, the optimal control design was appliedfor tubular reactor systems. Smet et al.17 described an optimaltemperature control problem of a steady-state plug-flow reactor(PFR), which has a tradeoff between process performance andheat loss. Birk et al.18 proposed a computation of optimal feedrates that can maximize the overall profit of the process. It isnoted that those feedforward-like control schemes could beseriously degraded, because of model errors or disturbances.Recently, the boundary feedback control for nonlinear PDEmodels was presented to improve closed-loop robustness.19,20

For the some control problems on the elimination of hot spots,boundary feedback design, and physical constraints, in thisarticle, the analytical optimization algorithm connected to themeasurement-based predictive control framework is imple-mented by two stages of design procedure. The optimaltemperature and concentration profiles of exothermic tubularreactors were determined under the assumption of steady-stateand plug-flow characteristics. For the first stage of steady-stateoptimization approach, Pontryagin’s minimum principle21 wasapplied to the terminal-cost optimization problem, in which theextended terminal cost criterion may enable a tradeoff betweenprocess performance and the overall heat effect. The bang-bang type of extremal control can facilitate the optimizationalgorithm for tubular systems. In the second stage, the on-linecomputation and feedback-based implementation scheme wasstudied according to the optimum operating policy and ap-proximate difference-based models. Moreover, nondistributedmodel predictive control (NMPC) strategies were developed,

* To whom correspondence should be addressed. Tel.: 886-5-5342601. Fax: 886-5-5312071. E-mail: [email protected].

2064 Ind. Eng. Chem. Res.2007,46, 2064-2076

10.1021/ie0611296 CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 03/06/2007

and two dependent manipulated variables were used to dominatethe heat exchange function for cooling devices. Particularly, theboundary feedback control, which uses a few sensors formeasurement of the reactor temperature at the prescribed axialposition, could accurately dominate the heat exchange of coolingdevices such that the undesired peak temperature disappearedthroughout the entire reactor. Compared to recent works fortubular reactor temperature control, our scheme is different fromthe design of multiple cooling zones22 or the cooling coilapparatus.23 Through theoretical analysis and closed-loop simu-lations, both NMPC techniques, with respect to state/inputconstraints, show that the no-offset output regulation of theillustrative phthalic anhydride reactor system can be achievedby tuning a single parameter.

2. Exothermic Tubular Reactor System

2.1. Process Description and Modeling.Several phthalicanhydride reactors have been presented by Froment24 and Chenand Sun.25 This reactor is packed with a V2O5 catalyst, the feedstream iso-xylene and air, and the outside of the tubular reactoris covered with a co-current cooling jacket, as depicted in Figure1.

Assume that the axial diffusion of reactants, the wall capacity,and the heat loss of the reactor are negligible. The reactionkinetics are denoted as pseudo-first-order, because of the largeexcess of oxygen. Under the mass balance and energy balance,four coupled nonlinear PDEs can be expressed:

with the boundary conditions

and

and the initial conditions

and

whereCA andCB are the concentrations ofo-xylene and phthalicanhydride, respectively,Le) [(1 - ε)Fscps + εFfcpf]/(Ffcpf), andr′ ) r0[(r1/r0)2 - 1]. The reaction rates are given by

and

whereCA,ss, CB,ss, Tss, andTc,ssrepresent steady-state variables.Most of the symbols have been specified in the Notation section,and the process parameters and operating conditions are givenin Table 1. In regard to the process design and controloptimization, the operating policies of this exothermic tubularsystem are stated as follows:

(1) The coolant flow rate (Vc) and coolant temperature (Tcf)in the feed are non-negative and bounded. To increase theeffectiveness of the heat transfer, bothVc andTcf are treated asdependent manipulated variables.

(2) The intitial steady-state variablesCA,ss, CB,ss, Tss, andTc,ss

are treated as the optimum operating conditions.(3) Physical constraints involve (i) bounded inputs and (ii)

the reactor temperature profile: bounded inputs are shown bythe equations

whereas the reactor temperature profile obeys

(4) The feed compositions (CAf) and the feed temperature ofthe cooling (Tcf) step could be affected by unknown distur-bances.

(5) The exit product concentration (CB,out) is not measurable;however, the exit reactor temperature (Tout), as the controlledvariable, is actually measurable.

Table 1. Process Parameters and Operating Conditions

L ) 4 m Ff ) 0.582 kg/m3

ε ) 0.35 Fc ) 1851.456 kg/m3

k01 ) 2.418× 109 s-1 Fs ) 2000 kg/m3

k02 ) 2.706× 109 s-1 cpf ) 1045 J kg-1 K-1

k03 ) 1.013× 109 s-1 cpc ) 483.559 J kg-1 K-1

E1 ) 1.129× 105 J/kmol cps ) 836.0 J kg-1 K-1

E2 ) 1.313× 105 J/kmol r0 ) 0.0125 mE3 ) 1.196× 105 J/kmol r1 ) 0.0225 m-∆H1 ) 1.285× 106 J/kmol U ) 96.02 J kg-1 K-1 s-1

-∆H2 ) 3.276× 106 J/kmol µ ) 1-∆H3 ) 4.561× 106 J/kmol V ) 2.06 m/sCAf ) 0.181 kmol/m3 Tf ) 628 KCBf ) 0 kmol/m3

R1 ) k01 exp(-E1

RgT)CA

R2 ) k02 exp(-E2

RgT)CB

R3 ) k03 exp(-E3

RgT)CA

RA ) -R1 - R3

RB ) R1 - R2

R } ∑i)1

3

(-∆Hi)Ri

Vc,mine Vc e Vc,max (4a)

Tcf,min e Tcf e Tcf,max (4b)

Tr(z,t) e Tr(L,t) ∀z∈ [0,L] (5)

ε∂CA(z,t)

∂t) -V

∂CA(z,t)

∂z+ µ(1 - ε)RA(CA,T) (1a)

ε∂CB(z,t)

∂t) -V

∂CB(z,t)

∂z+ µ(1 - ε)RB(CA,CB,T) (1b)

LeδT(z,t)

δt) -V

δT(z,t)δz

+µ(1 - ε)

FfcpfR(CA,CB,T) +

2ULr0Ffcpf

(Tc(z,t) - T(z,t)) (1c)

V∂Tc(z,t)

∂t) Vc

∂Tc(z,t)

∂z+ 2UL

r′Fccpc(T(z,t) - Tc(z,t)) (1d)

CA(0,t) ) CAf (2a)

CB(0,t) ) CBf (2b)

T(0,t) ) Tf (2c)

Tc(0,t) ) Tcf (2d)

CA(z,0) ) CA,ss(z) (3a)

CB(z,0) ) CB,ss(z) (3b)

T(z,0) ) Tss(z) (3c)

Tc(z,0) ) Tc,ss(z) (3d)

Ind. Eng. Chem. Res., Vol. 46, No. 7, 20072065

Remark 1: In regard to the operating policies of the illustratedexample,Vc andTcf are considered to be variables and are treatedas controls, the optimal operating points (initials) are not yetdetermined, the specified reactor temperature profile is restrictedto follow eq 5, and so on. All specifications will be investigatedin the article.

2.2. Steady-State System.Inspired by some articles for thesame illustrative example,6,26 these results show that theaforementioned exothermic reactions may cause the hot-spottemperature (Tr,max) in the reactor, because of the heat effect. Ifthe reactor has no coolant device, the wall temperature of thereactor (Tw) is used to dominate the heat-exchange effect. Asfor observing the characteristics of tubular systems, the steady-state model is easily investigated by

subject to

and

and initial conditions

and

Discussion:Inspired by the temperature policy for a uncon-vered steady-state PFR,6,27,28assume that the reactor has perfectcoolant function, which means that the overall wall temperatureis uniform, because of the fast dynamics of the coolant device.Figure 2 shows that the higher wall temperature (Tw > 645 K)will cause the undesired hot spot or thermal runaway phenomenaalong the trail of plug flow in the reactor. However, the lowwall temperature can reduce the appearance of the hot-spottemperature, but it also reduces the conversion of reactants atthe outlet of the reactor.

3. Optimal Temperature Control of the Steady-StateSystem

The higher or lower wall temperature could be inappropriatefor the steady-state operation of the tubular system. Generally,thestablereactor temperature associated with the high conver-sion rate of reactions at the outlet of reactor must be satisfiedsimultaneously under the feasible optimization strategy.

3.1. Optimal Design for Constrained Steady-State System.Referring to Figure 2, the physical constraints are bounded byTr,max e 680 K and Tcf e 633 K. The off-line computingalgorithm is added to obtain the distribution of wall temperatureTw(z), whereasCB is the amount of phthalic anhydride beingmaximized at the outlet of the reactor andTr,max is the peaktemperature being restricted under the upper limit of the reactortemperature (T < Tr,max). The usual independent variable (thetime t) for optimal control problem is replaced by the spatialvariablez. Moreover, the terminal-cost optimization of a steady-state tubular system, subject to physical constraints, is shownby

subject to

Figure 1. Illustration of the phthalic anhydride reactor system covered with a co-current cooling jacket.

minuss

J ) h(x(L)) + K1∫0

Lg(x(z)) dz

) -x2|z)L + K1∫0

L[uss- x3(z)]

2 dz (9)

dxdz

) f(x) + buss (10a)

x(0) ) x0 (10b)

dCA,ss

dz) (µV)(1 - ε)RA(CA,ss,Tss) (6a)

dCB,ss

dz) (µV)(1 - ε)RB(CA,ss,CB,ss,Tss) (6b)

dTss

dz)

µ(1 - ε)VFfcpf

R(CA,ss,CB,ss,Tss) + 2ULVr0Ffcpf

(Tw - Tss) (6c)

0 e z e L (7a)

CA,ss g 0 (7b)

CB,ssg 0 (7c)

Tss> 0 (7d)

CA,ss(0) ) CAf (8a)

CB,ss(0) ) CBf (8b)

Tss(0) ) Tf (8c)

2066 Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007

with x g 0 anduss e 633 K. The specified physical constraintis set by

i.e.,

where

and

Remark 2: Generally, the optimal control problem is to findan admissible distributionu* so that the trajectory of steady-state model (eq 10) can follow an admissible trajectory underthe specified constraint (eq 11). Through the aforementionedminimization algorithm, the objective function (eq 9) involvesboth objectives of the high conversion rate of reaction and thebounded integral cost due to the heat effect. In addition, theoverall temperature profile is influenced by the selection ofparameterK1.

3.2. Extremal Control Profile Determination. Accordingto Pontryagin’s minimum principle, the preceding minimizationproblem is equivalent to minimizing the following HamiltonianH:

where the adjoint vectorλ is the solution of

Obviously, the boundary conditions in eqs 8 and 11 are specifiedat z ) 0 or z ) L.

3.3. Extremal Control Approach. According to the objectivein eq 6, the bang-bang type of extremal controlu*, with respectto the minimization ofH, is characterized as

where z* represents the position away from the inlet of thereactor. Becauseψ remains identically zero over a finite distanceinterval [z*,L], the singular controlusing is obtained by repeatedlydifferentiating the functionψ until u appears explicitly. In thefirst step, the spatial derivative ofψ ) λTb is shown as

If the controlu does not appear in eq 15, then the second spatialderivative is written as

Moreover, the singular control law is obtained by settingd2ψ/dz2 ) 0:

Remark 3: Equations 10 and 13 show two point-boundary-value problems (TPBVPs). However, the extremal control thatis connected to singular control (eqs 14 and 17) performs theanalytical optimal control approach. Two parameters (K1, z*)

Figure 2. Open-loop steady-state profiles of a tubular system with a uniformwall temperature: (a) concentration of A response, (b) concentration of Bresponse, and (c) reactor temperature response.

x3 e x3|z)L (11)

Tr,max ) x3|z)L

x ) [CA,ss,CB,ss,Tss]T

K1 g 0

uss) Tw

f(x) ) [(µV)(1 - ε)RA(x)

(µV)(1 - ε)RB(x)

µ(1 - ε)VFfcpf

R(x) - 2ULVr0Ffcpf

x3]

b ) 2ULVr0Ffcpf

H(x,u*,λ) ) λTf(x) + (λTb)u*

} φ + ψu* (12)

λT ) - δHδx

(13a)

λ(L) ) ∂h∂x

|z)L (13b)

u*(z) ) {umax if ψ < 0using(z) if ψ ) 0, for z∈ [z*,L] } (14)

dψdz

) -λT∂f(x)∂x

b (15)

d2ψdz2

) λT∂f(x)∂x (∂f(x)

∂xb) - λT∂

2f(x)

∂x2b(f(x) + bu*) (16)

using )λT∂f(x)

∂x (∂f(x)∂x

b) - λT∂2f(x)

∂x2bf(x)

λT∂2f(x)

∂x2b2

(17)


can be tuned according to the specified optimization algorithmfor the high conversion rate of reactions, as well as thesafereactor temperature that is being achieved simultaneously. The

explicit form of using for the steady-state model was obtainedfrom the symbolic software of Maple, which is a general-purpose computer algebra system.

Figure 3. Optimal state and input profiles for the steady-state tubular system by adjustingK1 ) 1 × 10-5 (bold solid line),K1 ) 5 × 10-5 (solid line), andK1 ) 1 × 10-4 (dashed line): (a) concentration of A distribution, (b) concentration of B distribution, (c) reactor temperature distribution, and (d) walltemperature distribution.

Figure 4. Open-loop dynamic response of a tubular system with a co-current cooling jacket: (a) the steady-state coolant temperature profile is determinedby the minimization algorithm, (b) concentration of B response, (c) reactor temperature response, and (d) coolant temperature response.


Furthermore, the optimal temperature distribution of the wallis established using the following steps:

Step 1:For the first interval [0,z*], the wall temperatureTw

) umax ) 633 K is fixed at the upper limit in order to triggerthe large conversion rate of reactions.

Step 2:For the second interval [z*,L], the singular controlTw ) using(z) is applied to satisfy the dual objectives, i.e., thelargest amount ofCB at the outlet and no peak temperature inthis interval.

Step 3:Both tuning parameters (K1, z*) are detuned to searchthe wall temperature profiles appropriately while the reactortemperature distribution (eq 11) is being satisfied.

Remark 4: The extremal control with two parameters (K1,z*) aims to find out the appropriately optimal trajectory of thissteady-state system. Under the extremal control with fixed tuningparameters, Figure 3 shows the steady-state optimal profiles forprocess variables corresponding to computed value for walltemperature. When the uniform wall temperature (K1 ) 1 ×10-5) shown in Figure 3d is considered, the bold solid line ofFigure 3c shows that the peak temperature is observed in thereactor. Note that the optimal control is failed and saturated,because of an inappropriate parameterK1. If a K1 value of 5×10-5 is tuned, the solid line in Figure 3c shows that the peaktemperature is suppressed by constrained optimal design (eqs9 and 11). In addition, we find thatK1 g 5 × 10-5 induces thepoint z* f 0. WhenK1 ) 1 × 10-4 andz* ) 0 are given, wefind that the smooth distribution of the wall temperature (thedashed line) shown in Figure 3c is acceptable in a real operatingenvironment. Afterward, both parameters (K1, z*) are fixed inour control system, with respect to physical constraints.Moreover, the optimal state distribution as initial conditions areshown as

and

with CAf ) CA,ss(0), CBf ) CB,ss(0), andTf ) Tss(0).

4. Predictive Control Strategies

Referring to the aforementioned optimization strategy for anuncovered tubular reactor, the desired wall temperature distribu-tion is obtained via the off-line computation. According to theillustrative example, the exothermic tubular system is coveredwith a co-current cooling jacket. Two manipulated inputs (Vc,Tcf) have strong interactive effect on the dynamics of the coolantdevice. Thus, the steady-state coolant temperature is expectedto follow the profile of optimal wall temperature by solvingthe following minimization algorithm:

subject to

where V*c and T*cf, as initials, are obtained using iterativecomputation through the minimization algorithm (eq 19).

Moreover, the initial conditions of the distributed coolant system(eq 1d) are obtained and written as

and

Remark 5: Under the analytic optimal algorithm for steady-state systems, the initial conditions of the tubular system witha co-current cooling jacket (eq 1) have been determined. Figure4a shows that the steady-state coolant temperature profile isdetermined by the minimization algorithm between the coolantdevice and the reactor’s surface, and Figure 4b-d demonstratesthe dynamical responses of the illustrated tubular system.Notably, the base values ofTcf andVc are confirmed, becauseof smooth, no-peak dynamic responses.

Assuming that unmeasurable disturbances may appear in thefeed of the tubular reactor, the feedback-based scheme isnaturally required for the closed-loop performance and stability.From the process control viewpoint, the MPC scheme is anefficiently optimization-based feedback control strategy. More-over, the aforementioned optimal operating policy should beintegrated into the NMPC framework for the output regulationof the distributed reactor system while state/input constraintsand unknown inlet disturbances are being considered simulta-neously.

In our study, the one-step-ahead predictive horizon isimplemented to minimize the following objective function withno penalty on the control increments:

Let zi represents the spatial position along the reactor length,andzi ) (L/N)i, for 0 e i e N. N is the number of spatial points,ym,i represents the model output at the location ofzi, yi representsthe process output at locationzi, andyi

sp represents the referenceoutput at the location ofzi.

Regarding the previous steady-state optimization design, thedesired reference output is set byyi

sp ) Tss/ (zi). Following that

piecewise formulation, the originally distributed tubular systemmust be reduced to a lumped difference model:

with initial conditions

where the process states are

and

Consider that the space discretization is based on the finitedifference approximation; e.g., the five-point central differenceformula29 is shown for

C*A,ss(z) ) CA(z,0) (18a)

C*B,ss(z) ) CB(z,0) (18b)

T*ss(z) ) T(z,0) (18c)

minV*c, T*cf

∫0

L(Tw - Tc,ss)

2 dz (19)

dTc,ss

dz) 2UL

V*cr′Fccpc(Tc,ss- Tss) (20a)

Tc,ss(0) ) T*cf (20b)

T*c,ss(0) ) T*cf (21a)

T*c,ss(z) ) Tc(z,0) (21b)

Ji ) [ym,i(k + 1) - ym,i(k) + yi(k) - yisp(k)]2

(for i ) 1, 2, ...,N) (22)

xi(k + 1) ) F(xi(k),u(k)) (for i ) 1, 2, ...,N) (23)

xi(0) ) xss(zi) (for i ) 1, 2, ...,N)

xi(k) } [CA(zi,k),CB(zi,k),T(zi,k),Tc(zi,k)]T

xss(k) } [CA,ss(zi),CB,ss(zi),Tss(zi),Tc,ss(zi)]T


such that the difference-based nonlinear model described by eq1 is shown at locationzi:

Let Vc ) u1 and Tc(z0,t) ) Tcf ) u2. Moreover, the timediscretization of eq 25 using an implicit Euler method with asample time of∆t will yield the discrete-time model shown ineq 23.

Remark 6: The finite-difference method is a very simpletechnique, such that a large number of node points are used toimprove the approximate accuracy and convergence properties.In the case of the predictive control approach by eq 22, afeedback controller is described by a set of difference equations,

to minimize the lumped objective function (∑j)1N Jj). The

control law (eq 26) is an implicit function of the measured outputat locationzi, and the model stateêi corresponds to the positionsof zi. In the present framework, the closed-loop system is formedby augmenting the open-loop system with an output feedbackcontroller (eq 26).

Remark 7: If the lumped difference model (eq 23) is an open-loop stable system, i.e., a region of attraction is bounded by apositive constantBd,

Furthermore, the asymptotic output regulation is achieved, i.e.,

Using the previous minimization algorithm for∑j)1N Jj, the no-

offset tracking performance is obtained:

In fact, the control law described by eq 26 is difficult to obtain,

because of the solvability of the minimization of the lumpedobjective functions. In addition, the numerous outputs (distrib-uted output) feedback design may obviously increase the costof the overall devices, and the no-offset tracking performancefor multiple outputs are hardly achieved at the finite time.

4.1. Nondistributed Model Predictive Control (NMPC).According to the previous steady-state minimization approachfor cooling jacket design, the base values of the two manipulatedinputs (initials) are set by

and

and the discretized reference outputyisp (for i ) 1, 2, ...,N) are

determined by the previous steady-state Pontryagin’s minimiza-tion approach. Figure 4b-d has demonstrated the smooth, no-peak dynamic responses for the illustrated tubular system. Forthe reduction of the previous controller (eq 26), the one-step-ahead prediction horizon is proposed for the specified singleoutput at the outlet, with respect to amplitude and velocity

∂f(xi)

∂z≈ -f(xi+2) + 8f(xi+1) - 8f(xi-1) + f(xi-2)

12∆z

)∂f(xi)

∂z(24)

CA(zi,t) ) - Vε

∂CA(zi,t)

∂z+ µ

(1 - ε)ε

RA(CA(zi,t),T(zi,t))(25a)

CB(zi,t) ) - Vε

∂CB(z,t)

∂z+ µ

(1 - ε)ε

RB(CA(zi,t),CB(zi,t),T(zi,t))

(25b)

T(zi,t) ) - VLe

∂T(zi,t)

∂z+ µ

(1 - ε)FfcpfLe

R(CA(zi,t),CB(zi,t),T

(zi,t)) + 2ULr0FfcpfLe

(Tc(zi,t) - T(zi,t)) (25c)

Tc(zi,t) )Vc

V∂Tc(z,t)

∂z+ 2UL

r′FccpcV(T(zi,t) - Tc(zi,t))

(25d)

êi(k + 1) ) F(êi(k),u(k)) (for i ) 1, 2, ...,N) (26a)

u(k) ) Ψ(ê1(k), ...,êN(k); y1(k), ...,yN(k)) (26b)

∑i)1

N

||xi(k) - xss(zi)||eBd (ask f ∞) (27)

limkf∞

||ym,i(k + 1) - ym,i(k)|| ) 0 (for i ) 1, 2, ...,N) (28)

limkf∞

||yi(k + 1) - yisp(k)|| ) 0 (for i ) 1, 2, ...,N) (29)

Figure 5. Proposed nondistributed model predictive control (NMPC)scheme: (a) the first control scheme, without sensing state information,and (b) the second control scheme, with sensing state information at theprescribed location.

u1(0) ) V*c (30a)

u2(0) ) T*cf (30b)


constraints on both manipulated inputs. The NMPC algorithmis reduced by

subject to (i) a nominal discrete-time model as the dynamicmodel constraint,

(ii) amplitude constraints on the manipulated input,

and (iii) velocity constraints on the manipulated input,

where∆ym,N(k + 1) ) ym,N(k + 1) - ym,N(k), ∆uj(k) ) uj(k) -uj(k - 1), CN ) [0 0 1 0], kmpc is the input weight factor, and∆uj represents the movement of the manipulated input incre-ment.

Remark 7:The proposed predictive control algorithm is anondistributed output feedback controller. It manipulates adistributed reactor system using the steady-state optimization

Figure 6. Disturbances rejection using the first NMPC scheme: (a) response of the exit concentration of B, (b) response of the exit reactor temperature,(c) the corresponding manipulated coolant flow rate, (d) the corresponding manipulated inlet coolant temperature, (e) three-dimensional plot of the reactortemperature response during a+20% change in the reactant concentration in the feed, and (f) three-dimensional plot of the reactor temperature responseduring a-20% change in the reactant concentration in the feed.

minu(k)

JN ) [∆ym,N(k + 1) + yN(k) - yNsp(k)]2 +

kmpc∑j)1

2

∆uj2(k) (31)

êi(k + 1) ) F(êi(k),u(k)) (for i ) 1, 2, ...,N) (32a)

ym,N(k) ) CNêN(k) (32b)

uj,min e uj(k) e uj,max (for j ) 1, 2) (33)

∆uj,min e ∆uj(k) e ∆uj,max (for j ) 1, 2) (34)


approach and an open-loop observer. Moreover, the proposedmodel-based control scheme is depicted in Figure 5a. Note thatthe lumped difference model as the output prediction, withrespect to the input constraints, is treated as the feedback-basedimplementation scheme. The step disturbance rejection isachieved due to the error between the process output and themodel prediction, i.e.,yN(k) - ym,N(k), added in the prescribedobjective function (eq 31).

When the inlet disturbances are added, the previous steady-state approach cannot seize the regular dynamic behavior, e.g.,the location of the peak temperature. In addition, the stateconstraint by eq 11 for the exothermic reactor system is strictlyrequired. Obviously, the distributed state response will degradethe former NMPC scheme, because of the appearance of amoving peak temperature and the inaccurate state information

by the tubular systems. Thus, a measurement-based NMPCalgorithm is proposed and written as

subject to (i) the lumped difference process model,

Figure 7. Disturbances rejection using the first NMPC scheme: (a) response of the exit concentration of B, (b) response of the exit reactor temperature,(c) the corresponding manipulated coolant flow rate, (d) the corresponding manipulated inlet coolant temperature, (e) three-dimensional plot of the reactortemperature response during+1% changes in the coolant temperature in the feed, and (f) three-dimensional plot of the reactor temperature response duringa -1% change in the coolant temperature in the feed.

minu(k)

JM ) ∑l)1

M

∆ym,l2(k + 1) +

[yN(k) - yNsp(k)]2 + kmpc∑

j)1

2

∆uj2(k) (35)

xi(k + 1) ) F(xi(k),u(k)) + η(d(k)) (for i ) 1, 2, ...,N)(36a)

ym,l(k) ) Clxl(k) (for l ) 1, 2, ...,M) (36b)


(ii) the state constraint,

and (iii) the same conditions for amplitude and velocityconstraints on the manipulated input. In eqs 35-37, ∆ym,l(k +1) ) ym,l(k + 1) - ym,l(k), ym,l represents the measurable outputusing sensor measurements at the prescribed positionzl, ηrepresents the effect of unknown disturbancesd, andM is thenumber of spatial points that settle on measurement devices.

Remark 8: With the aid of a few sensors for measurement,the measurement-based predictive control algorithm controllerdesign can induce the stable and no-offset output regulation atthe outlet of the tubular reactor. Moreover, the proposed controlscheme is depicted in Figure 5b. In addition, the number ofsensorsM should be less than the number of spatial pointsNfor lumped difference equations. If the number of sensors is

sufficient, the prescribed state constraint (eq 37) for suppressingan undesired peak temperature could be satisfied.

Remark 9: Assume that the nominal solutionu(k) )Ψ(ê1(k), ...,êN(k); y1(k), ...,yN(k)) to the minimization of∑j)1

N Jj is obtained from the system without disturbances, e.g.,xi(k + 1) ) F(xi(k),u(k)) (for i ) 1, 2, ...,N). Apparently, thefeasibility of the aforementioned optimization algorithm in theface of physical constraints would be dependent on the controllerparameterkmpc. It implies that the feedback controlu(ym,N, yN,kmpc) can be determined by solving the aforementioned opti-mization problem (eq 31). Moreover, if the optimization problem(eq 31) is feasible, eq 35 also has a feasible solution, becauseof the aid of measurements, i.e., minJM e min JN.

4.2. Demonstration.The presented simulation tests are basedon the following conditions:

(a) The number of spatial pointsN for lumped differenceequations is 100.

æ(xi-1(k)) e æ(xi(k)) (for i ) 2, 3, ...,N) (37)

Figure 8. Disturbances rejection using the second NMPC scheme: (a) response of the exit concentration of B, (b) response of the exit reactor temperature,(c) the corresponding manipulated coolant flow rate, (d) the corresponding manipulated inlet coolant temperature, (e) three-dimensional plot of the reactortemperature response during a+20% change in the reactant concentration in the feed, and (f) three-dimensional plot of the reactor temperature responseduring a-20% change in the reactant concentration in the feed.


(b) The number of spatial points for sensor measurements is20.

(c) The bounds for the input constraints include (i) anamplitude constraint,

and (ii) a velocity constraint,

(d) The optimal input and output (steady-state optimizationapproach) includes (i) the initials of input,

and

and (ii) the desired setpoint as the reference output,

(e) The optimal state distribution (Figure 4) for the difference-based model (eq 25) is given as

and

Figure 9. Disturbances rejection using the second NMPC scheme: (a) response of the exit concentration of B, (b) response of the exit reactor temperature,(c) the corresponding manipulated coolant flow rate, (d) the corresponding manipulated inlet coolant temperature, (e) three-dimensional plot of the reactortemperature response during a+1% change in the coolant temperature in the feed, and (f) three-dimensional plot of the reactor temperature response duringa -1% change in the coolant temperature in the feed.

u2(0) ) T*cf ) 614.3 K (40b)

T*ss(zN) ) yNsp ) 630.5 K (41)

CA(zi,0) ) C*A,ss(zi) (42a)

CB(zi,0) ) C*B,ss(zi) (42b)

T(zi,0) ) T*ss(zi) (42c)

0.01 m/se Vc e 0.05 m/s (38a)

605 K e Tcf e 625 K (38b)

0 m/se ∆Vc e 10-4 m/s (39a)

0 K e ∆Tcf e 0.1 K (39b)

u1(0) ) V*c ) 0.02 m/s (40a)


(f) The undesired operating temperature is>640 K.

(g) The sample time∆t is 1 s.

(h) The tuning parameterkmpc is fixed (kmpc ) 0.01).

According to the first NMPC scheme shown in Figure 5a, itwill be validated by the illustrated reactor system while the state/input constraints and inlet disturbances are being consideredsimultaneously. For inlet disturbances, changes of(20% in thereactant concentration in the feed, and(1% in the coolanttemperature in the feed are considered in the process. Figure6b shows that the NMPC with only output information at theoutlet temperatureTout can ensure the stable and no-offset outputregulation in the presence of(20%CAf. Notably, Figures 6cand 6d show both actuatorsVc and Tcf being restricted underphysical constraints, and the three-dimensional figures shownin Figures 6e and 6f show the complete state distribution.However, the perturbation by+20%CAf will induce the peaktemperature close to the inlet bound. Under the same NMPCscheme for(1%Tcf, Figure 7b shows that the stable and no-offset output regulation can be achieved. The corresponding twocontroller actions are depicted in Figures 7c and 7d, and thethree-dimensional temperature responses are shown in Figures7e and 7f. Unfortunately, the perturbation by+1%Tcf willobviously induce the undesired peak temperature (>640 K) closeto the inlet bound. In our opinion, this (internal) model-basedcontrol scheme can be easily realized through the fast onlinecomputation; however, the nondistributed and output feedbackdesign cannot effectively dominate the entire temperature effectinside the reactor.

According to the second NMPC scheme shown in Figure 5b,it will be validated by the illustrated reactor system with thesame state/input constraints and inlet disturbances. Figure 8bshows that the NMPC with 20 sensors for measurement of thereactor temperature at the fixed interval can ensure the stableand no-offset output regulation in the presence of(20%CAf.The corresponding actuatorsVc and Tcf shown in Figures 8cand 8d are being restricted under physical constraints, and thecomplete state distributions are shown in Figures 8e and 8f.Compared to Figure 7b, the second NMPC scheme can providebetter tracking performance than the first scheme, but the more-oscillating responses are detected using the second scheme.Compared to Figure 7e, the second NMPC scheme can providea smooth temperature distribution, because of the multisensordesign. Under the same NMPC scheme for(1%Tcf, Figure 9bshows that the stable and no-offset output regulation can beachieved. The corresponding two controller actions are depictedin Figures 9c and 9d, and the three-dimensional temperatureresponses are shown in Figures 9e and 9f. Compared to Figure8b, it verifies again that the second NMPC scheme can providebetter tracking performance than the first scheme, and the secondNMPC scheme can almost reduce the peak temperature, asshown in Figure 9e.

Consequently, the proposed two NMPC schemes have beensuccessfully validated for controlling a distributed system inthe face of inlet disturbances or state/input constraints. Particu-larly, the second NMPC scheme almost captures the charac-teristic of distributed parameter systems, because of the numberof measurement devices. Based on the accurate steady-stateoptimization approach, the aforementioned two input controlschemes can ensure the stable/robust output regulation aroundprescribed operating conditions.

5. Conclusion

For the illustrative exothermic tubular reactor system, theproblems of hot spots, partial differential equation (PDE)models, boundary control, and physical constraints are chal-lenging to traditional feedback control design. Fortunately,Pontryagin’s minimum principle, in association with the extre-mal control, facilitates the steady-state optimization approach,so that the modified model predictive control (MPC) strategy,using several output measurements, is added to optimize theprocessing temperature profile and ensure the largest amountof products in a fixed space time. The two control actuators aredescribed as boundary control of distributed parameter systemsto dominate the heat exchange ability of the cooling device. Inregard to state/input constraints handling and the optimizationresolution, an exterior penalty method is used to account forthe controller parameter, with respect to closed-loop feasibilityduring on-line iterative procedures. All nondistributed modelpredictive control (NMPC) strategies are reduced to a one-step-ahead predictive horizon. The simplest tuning procedure in theoutput feedback framework is quite easy to implement inpractice. In regard to the control of hot spots, the proposedmeasurement-based NMPC is implemented to reduce theundesired peak temperature. Although the position and numbersof sensors aren’t explored by theoretical analyses, simulationtests indicate that the measurement sensors at the hot-spot zonecan effectively remove the hot-spot temperature.

Acknowledgment

This work is supported by the National Science Council ofthe Republic of China, under Grant No. NSC-94-2214-E-224-007.

Notation

CA ) concentration of A (kmol/m3)CAf ) feed concentration of A (kmol/m3)CB ) concentration of B (kmol/m3)CBf ) feed concentration of B (kmol/m3)CB,out ) exit product concentration (kmol/m3)cpc ) heat capacity of coolant (J kg-1 K-1)cpf ) heat capacity of fluid (J kg-1 K-1)cps ) heat capacity of catalyst (J kg-1 K-1)Ei ) activation energy of reactioni, for i ) 1, 2, 3 (J/kmol)-∆Hi ) heat of reactioni, for i ) 1, 2, 3 (J/kmol)L ) reactor length (m)r0 ) reactor radius (m)r1 ) coolant tube radius (m)Rg ) gas constantRi ) rate of reactioni, for i ) 1, 2, 3t ) time (s)T ) reactor temperature (K)Tf ) feed temperature (K)Tc ) coolant temperature (K)Tcf ) inlet coolant temperature (K)Tw ) wall temperature (K)Tout ) exit reactor temperature (K)Tr,max ) hot-spot temperature (K)∆t ) sample timeU ) overall heat transfer coefficients (J kg-1 K-1 s-1)V ) feed flow rate (m/s)Vc ) coolant flow rate (m/s)z ) axial distance (m)ε ) void volume fraction of reactorµ ) catalyst activity

Tc(zi,0) ) T*c,ss(zi) (42d)


Fc ) coolant density (kg/m3)Ff ) fluid density (kg/m3)Fs ) catalyst density (kg/m3)H ) Hamiltonian functionλ ) adjoint vector

Superscripts

* ) optimalsp ) set point

Subscripts

ss) steady state

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ReceiVed for reView August 28, 2006ReVised manuscript receiVed November 30, 2006

AcceptedFebruary 5, 2007

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Measurement-Based Optimization and Predictive Control for an Exothermic Tubular Reactor System

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