A Simple Feedback Control Approach for Output Modulation of

Article pubs.acs.org/IECR

A Simple Feedback Control Approach for Output Modulation of Spatiotemporal Patterns in a Class of Tubular Reactors Hector Puebla,*,† Eliseo Hernandez-Martinez,‡ Rogelio Hernandez-Suarez,§ Jorge Ramirez-Muñoz,† and José Alvarez-Ramirez∇ †

Departamento de Energía, Universidad Autónoma Metropolitana-Azcapotzalco, México DF, México Facultad de Ciencias Químicas, Universidad Veracruzana Campus Xalapa, Veracruz, México § Programa de Procesos de Transformación, Instituto Mexicano del Petróleo, México DF, México ∇ Departamento de Ingeniería de Procesos e Hidráulica, Universidad Autónoma Metropolitana-Iztapalapa, México DF, México ‡

ABSTRACT: Tubular chemical reactors can support spatiotemporal patterns. Pattern modulation in reactive systems may be of major importance for adequate operation of some industrial applications. In this work, we introduce a robust control approach for the output regulation and tracking of spatiotemporal patterns in a class of tubular reactors at a desired position. The control design includes a state estimator of lumped uncertain terms, which is coupled with an inverse model-based feedback control that assigns a desired output closed-loop behavior. The result is an efficient controller that is robust against feed disturbances while introducing good output regulation and tracking properties. The controller is implemented via numerical simulations in three benchmark tubular reactors displaying spatiotemporal patterns: (i) spatiotemporal periodic oscillation in a tubular reactor with axial dispersion, (ii) chaotic spatiotemporal patterns in a tubular reactor with recycle, and (iii) spatiotemporal periodic oscillations in the Belousov−Zhabotinseky reaction−diffusion system.

1. INTRODUCTION In 1952, Turing showed that a uniform steady state in reaction−diffusion (RD) systems, comprising a species that tends to grow autocatalytically (an activator) and an species that counteracts this tendency of growth (an inhibitor), may become unstable when the inhibitor diffuses sufficiently faster than the activator.1 For reaction−diffusion−convection (RDC) systems, the combined action of flow and diffusion with a nonlinear kinetic term may create spatiotemporal patterns.2−4 Turing and flow-diffusion instabilities conditions seem to be fulfilled in several examples of chemical distributed parameter systems (DPS), in particular in tubular reactors.4−7 Indeed, spatiotemporal patterns in tubular reactors are generated by interactions of chemical reactions with the transport and storage of mass and heat.3−11 In these units, heat and mass released by chemical reaction are activator variables and the extent of reaction is an inhibitor variable. Patterns, once formed, are sustained when local mixing is slower, compared to reaction. Two typical patterns in a one-dimensional (1D) system are homogeneous oscillations and a family of moving waves propagating with constant velocities. For instance, combustion processes exhibit both stationary and moving patterns,8 the Belousov−Zhabotinsky (BZ) reaction (catalytic oxidation of malonic acid), exhibits target and spiral patterns,9,10 and frontal polymerization displays propagating reaction waves.11 Pattern formation in reactive systems is interesting from a fundamental point of view, and with respect to potential applications ranging from material science, petroleum engineering, as well as biochemical processes. Indeed, reactors with spatiotemporal patterns have shown to provide significant efficiency enhancement to many catalytic processes.5−8,11−15 © 2013 American Chemical Society

Benefits include increase process productivity and selectivity, capital equipment savings, and operating cost reductions.5−8,11−15 Some applications reported in the literature include volatile organic compounds (VOC) oxidation,5,6,14 treatment of waste air,14 and propane combustion.8 Thus, the proper understanding and control of the pattern formation in tubular reactors may be of major importance for adequate and safe design, operation, and control of processes displaying spatiotemporal patterns. The basic idea in controlling spatiotemporal patterns is that one can force the system to display a desired output by imposing an appropriate external perturbation.16−20 For example, for photosensitive systems, one might vary the intensity or the frequency of the imposed illumination.10 Temperature is also a suitable control parameter, since it affects both reaction rates and transport properties.16,18 Indeed, the use of temperature, as a control parameter, dates back at least to the first experiments on Turing patterns in the CIMA reaction.2,4 Control of distributed parameter systems is a challenging problem because of several factors, including model complexity, the spatial dependence of process variables, complex interactions, and limited available online measurements.21 For several applications, the essential aim of the control of spatiotemporal patterns in tubular reactors is the output regulation or tracking of the outlet conditions, i.e., the Received: Revised: Accepted: Published: 17517

April 29, 2013 November 15, 2013 November 20, 2013 November 20, 2013 dx.doi.org/10.1021/ie4013562 | Ind. Eng. Chem. Res. 2013, 52, 17517−17528

Industrial & Engineering Chemistry Research

Article

∂y(ξ , t ) ∂y(ξ , t ) ∂ 2y(ξ , t ) − b1v + f1 (y(ξ , t ), z(ξ , t )) =D ∂ξ ∂t ∂ξ 2

modulation of the reactor’s exit controlled variable at will and keeping internal conditions below critical conditions.22−25 From a practical point of view, the heart of the control problem addressed here is either the operation of tubular chemical reactors at some fixed-point (regulation) or periodic oscillations (tracking) at the reactor exit via a simple and practical feedback control scheme. The regulation case is significant for suppression oscillatory behavior when it is it is unwanted or undesirable. The tracking case would be beneficial to convert uncontrolled oscillations into desired regular ones with periodic time dependence, in order to improve the process productivity.5−8,11−15 Control of chemical distributed parameter systems, including spatiotemporal patterns in RD and RDC systems has been addressed with both nonlinear model-based21−25 and linear16−18 controllers. The industrial application of nonlinear model-based controllers is limited, because of their complexity and dependency on an exact mathematical model of RD and RDC systems. For instance, high uncertainties in the rate and transport constants, kinetic parameters, and delay transport, in chemical and biochemical tubular reactors, limits the applicability of nonlinear control schemes, because of the assumption of a perfect model for control design purposes. On the other hand, the use of linear techniques is very limiting, because of the complex nature of spatiotemporal patterns. In this work, we introduce a practical and robust control approach for output regulation and tracking of spatiotemporal patterns in tubular reactors. We derive a feedback control law for the control input variable, which guarantees that the spatiotemporal pattern at the output axial position is suppressed or forced toward a desired behavior, despite model and parameter uncertainties and typical expected disturbances. The control design is addressed using a simple robust control approach that has two interesting properties for practical application of the resulting control law: (i) a systematic consideration of uncertainty that leads to a controller with a good robustness property, and (ii) a linear control structure that can be implemented in practice. Results showed that the proposed control approach is able to modulate, at will, the output of a general class of tubular reactors displaying spatiotemporal patterns. Three benchmark examples are used to illustrate the closed-loop performance: (i) a tubular reactor with axial dispersion,5 (ii) a tubular reactor with recycle,26 and (iii) a tubular reactor with the BZ reaction.27 This work is organized as follows. Section 2 presents the class of tubular reactor models supporting spatiotemporal dynamics. Section 3 describes the design of the proposed control approach. In section 4, numerical experiments on three case studies showed the control performance. Finally, in section 5, the main conclusions of the paper are drawn.

+ g1(y(ξ , t ), z(ξ , t ))u(ξ , t )

(1)

∂z(ξ , t ) ∂ 2z(ξ , t ) ∂z(ξ , t ) = B1 − B2 v + B3f2 (y(ξ , t ), z(ξ , t )) ∂t ∂ξ ∂ξ 2 (2)

where y(ξ,t) is the controlled state of the system, z(ξ,t) are the uncontrolled states, v is the fluid velocity, u(ξ,t) denotes the manipulated variable, D is the total dispersion coefficient of the controlled state, f1(y(ξ,t), z(ξ,t)), f 2(y(ξ,t), z(ξ,t)), and g1(y(ξ,t), z(ξ,t)) are nonlinear functions, and ∂/∂ξ and ∂2/ ∂ξ2 denotes convection and diffusion transport processes, respectively. B1, B2, and B3 are matrices of suitable dimension. The scalar b1 and the entries of matrix B2 are equal to either zero or one, where a zero and one entry denotes respectively that the ith process variable is fixed or “transported” into the process, such that taking zeros for b1 and B2 leads to the RD case. 2.2. Case Studies. A variety of tubular reactors have been reported that exhibit the formation of spatiotemporal patterns.5−16,26−32 We have selected, as case studies, three benchmark models of tubular reactors displaying pattern formation: (i) a tubular reactor with axial dispersion,5 (ii) a tubular plug-flow reactor with recycle,26 and (iii) a tubular reactor with the BZ reaction.27 2.2.1. Tubular Reactor with Axial Dispersion. The axial dispersion model is commonly used to describe nonideal flow in plug-flow reactors. In this model, it is assumed that, superimposed onto the main flow, there exists an axial mixing of material that is governed by an analogue of Fick’s law of diffusion. We consider a pseudo-homogeneous axial dispersion model as described by Jensen and Ray,5 where main assumptions include (i) a homogeneous reaction, (ii) properties of the reaction mixture are characterized by average values, (iii) axial mixing is described by a single parameter model, and (iv) first-order kinetics.5 The mathematical model is given by eq 1 with D = Pey−1Le−1, v = 1, b1 = Le−1, B1 = Pez−1, B2 = B3 = 1, g1(y(ξ,t),z(ξ,t)) = Le−1β, u = yj, and f1 (y(ξ , t ), z(ξ , t )) = Le−1BDa(1 − z(ξ , t )) ⎛ y(ξ , t ) ⎞ × exp⎜ ⎟ − Le−1βy(ξ , t ) ⎝ 1 + y(ξ , t )/λ ⎠

f2 (y(ξ , t ), z(ξ , t )) ⎛ y(ξ , t ) ⎞ = Da(1 − z(ξ , t )) exp⎜ ⎟ ⎝ 1 + y(ξ , t )/λ ⎠

where y and z are dimensionless temperature and conversion, respectively. Da is the Damköhler number, which is the ratio of convection or mixing time to reaction time and is defined, for this case, as Da = [l(1 − εp)k0 exp(−λ)/v]. Fast reactions have smaller reaction times and, therefore, large Da values. Slow reactions, on the other hand, have small Da values. For fast forward reactions, as in the case of combustion reactions, Da > 1 can be expected. Pey and Pez are the Peclet numbers, representing the ratio of advection to dispersion, for heat and mass transfer, and are defined as Pey = vl/Dl, Pez = vlρfCpf/kl, respectively. v is the fluid velocity, εp is the bed porosity, l is the reactor length, ρf is the fluid density, Cpf is the fluid heat capacity, and kl, and Dl are the longitudinal heat and mass

2. THE CLASS OF TUBULAR REACTORS In this section, we introduce the class of tubular reactors supporting pattern formation. Case studies of three benchmark examples of tubular reactors that display complex spatiotemporal patterns are also presented. 2.1. The General Model of Tubular Reactors Systems Displaying Spatiotemporal Dynamics. We consider the following single-input−single-output (SISO) mathematical model of a tubular reactor, in one spatial dimension with suitable boundary conditions (e.g., Robin, Danckwerts, Dirichlet, or Newmann boundary conditions), that supports spatiotemporal dynamics: 17518

dx.doi.org/10.1021/ie4013562 | Ind. Eng. Chem. Res. 2013, 52, 17517−17528


Article

degree for the reactant A (z = (C0 − C)/C0), where T0 and C0 are the feed temperature and concentration, respectively. r is the recycle fraction (i.e., the mass flow of the recycle/mass flow of the reactor), yj is the jacket reactor temperature, Da = VkC0n/FC0 is the Damköhler number, where V is the reactor volume, k is the reaction rate constant, F is the volumetric flow rate. t is a dimensionless time. δ = Akq/(ρCpFβ) is a dimensionless heat-transfer coefficient, where ρ is the fluid density, Cp is the fluid heat capacity, A is the heat exchange area, and kq is the heat exchange coefficient. γ is a dimensionless activation energy, and β is a dimensionless number related to adiabatic increase of temperature., The dynamic behavior of tubular reactors with recycle has been extensively studied in the literature.5,6,26,28 In particular, it is well-known that the recycle of either mass or heat, or both, can lead to oscillatory behavior, including chaos. The significance of the analysis and control of tubular chemical reactors with recycle stem from the fact that such systems are commonly used in industrial processes. We consider a PFR with recycle, where the recycle fraction varies from zero (0) to one (1), with 1 corresponding to total recycle and 0 corresponding to no recycle, reducing to that of a conventional PFR, for which the stationary solution always is unique. Thus, since we are addressing the control of oscillatory behavior in tubular reactors, we consider here recycle fraction values in the range of 0 < r ≤ 1. Chaotic spatiotemporal patterns are found for the following parameter values:26 β = 2, Da = 0.15, γ = 15, δ = 3, yj = −0.030, and r = 0.5. 2.2.3. Tubular Reactor with the BZ Reaction. The BZ reaction, which consists of oxidation and bromination in sulfuric acid of a malonic acid, or other organic substrate, in the presence of a redox-active catalyst (such as a Ce ion or ferroin, is a classic example of a chemical system that shows spatial, periodic, and wave properties.4,9,10,27 BZ reaction has been studied extensively in the context of various fields relevant to many chemical and biological problems.4,33 The mechanism of pattern formation in the BZ reaction consists in a positive feedback effect of the bromous acid that stimulates its own production (the activator), which is inhibited by the produced CO2 (the inhibitor). We consider the ferrioncatalyzed BZ with the addition of methanol as an activating reactant.27 A modified Oregonator model considering the reaction of methanol with sodium bromate and acid sulfuric was proposed by Zhang et al.27 The proposed model considers the following species: sodium bromate (A), malonic acid (B), methanol (M), H+ form the sulfuric acid (H), bromous acid (U), and the oxidized catalyst (V). Methanol reacts with sodium bromate and sulfuric acid, with a rate constant of km, producing bromous acid, which is the activator of BZ reaction system. The model,27 after a suitable rescaling, is written as eq 1 with the following values: D = Dy, b1 = 0, B1 = 0, v = 0, B2 = 0, B3 = 1, g1(y(ξ,t),z(ξ,t)) = 1, u = c, f1(y(ξ,t),z(ξ,t)) = ε−1(y − y2 − fz(y − q)/(y + q)), and f 2(y(ξ,t),z(ξ,t)) = (y − z). Here, y is the dimensionless bromous acid (U), and z is the dimensionless oxidized catalyst (V). ε = k5B/k3A, q = 2k4k1/k2k3, and f are parameters related to the original kinetic constants, which are kept fixed in the numerical simulations. In particular, f is a stoichiometry factor, representing the number of Br− ions produced in the bromination of the malonic acid. Du is the diffusion coefficient of variable y. In this forced model, c = 2k4kmHM/(k3k5B) represents the production of bromous acid related to the methanol reaction with sodium bromate and

dispersion coefficients, respectively. Le is the Lewis number, which measures the ratio of the heat capacitance term and the mass capacitance term. β is a dimensionless heat-transfer coefficient. B is a dimensionless adiabatic temperature rise, and λ is the dimensionless activation energy. It is noted the presence of the length of the reactor and flow rate, in the definition of Peclet numbers. Thus, small Peclet numbers are expected in laboratory reactors and some microfluidic devices. Depending on the model dimensionless parameters, different dynamical regimes are observed. Jensen and Ray5 have performed extensive bifurcation studies of this dispersion axial model, showing that the system exhibits very rich dynamic behavior for different regions in the bifurcation parameter space. In particular, the influence of Damköhler and Peclet numbers was investigated.5 At high Pe values (as Pe → ∞), plug flow occurs in the absence of oscillatory behavior. At intermediate Pe values (e.g., Pe = 5), the dispersion axial model can have multiple equilibria that can be either stable or unstable. As Pe → 0, complete back-mixing occurs, reaching the limit case of a CSTR. Jensen and Ray5 have shown that oscillations due to interaction of dispersion and reaction effects should not exist in fixed-bed reactors, and only can occur in very short empty tubular reactors, for instance, in thermal cracking and solution polymerization. Even though the presence of oscillatory behavior is possible only for reduced cases in tubular reactor with axial dispersion, there are practical situations where one wishes to avoid or control these oscillations to improve the performance of this class of tubular reactors.5−8,11−15 For the particular case with Pey = Pez = 5, B = 14, β = 3, Le = 1, λ = 20, Da = 0.185, and yj = 0.0, the operating steady state of the open-loop system is unstable, where the temperature and conversion profiles moves to an asymptotically stable periodic (limit cycle) spatially nonuniform steady state.5 2.2.2. Tubular Reactor with Recycle. Tubular chemical reactors with mass and heat recycle are commonly applied in industrial processes. Because of feedback caused by the recycle, a complex dynamic response may occur in this type of reactor. Indeed, periodic, quasi-periodic, and chaotic spatiotemporal patterns have been reported in tubular reactors with recycle.26,28 We consider a plug-flow chemical reactor (PFR) with a recycle of mass and heat, as reported by Berezowski.26 A fraction of the reactor exit is recycled and mixed with the fresh fed. Main assumptions of the model include (i) plug flow in the reactor, (ii) negligible recycle delay, (iii) instantaneous mixing of the feed and recycle streams, (iv) an irreversible exothermic first-order reaction (A → B), and (v) a cooling jacket is used to remove the heat from the reactor.26 In this case, model (1) is obtained with the following values: D = 0, b1 = 1, v = 1, B1 = 0, B2 = 1, B3 = 1, g1(y(ξ,t),z(ξ,t)) = (1 − r)δ, u(t) = yj, and f1 (y(ξ , t ), z(ξ , t )) ⎛ γβ(ξ , t ) ⎞ = (1 − r )Da(1 − z(ξ , t ))n exp⎜ ⎟ ⎝ 1 + βy(ξ , t ) ⎠ − (1 − r )δy(ξ , t ) f2 (y(ξ , t ), z(ξ , t )) ⎛ γβ(ξ , t ) ⎞ = (1 − r )Da(1 − z(ξ , t ))n exp⎜ ⎟ ⎝ 1 + βy(ξ , t ) ⎠

For this case study, y is the dimensionless temperature (y = (T − T0)/βT0), z is the dimensionless concentration or conversion 17519



Article

sulfuric acid.27 For numerical simulations, the following parameter values are used:27 Du = 1, ε = 0.1, q = 0.002, f = 1.25, and c = 0.3. Under the above conditions, the BZ system displays a stable periodic spatiotemporal pattern. With regard to the spatiotemporal dynamics supported by the above case studies, the following comments are in order: • In continuous tubular chemical reactors, the development of nonlinear spatiotemporal structures depends both on the local temporal kinetics and on diffusion transport.4,34 Indeed, the coupling between temporal oscillations arising of competing kinetic processes involving nonlinear feedback (e.g., chemical or thermal feedback) and diffusion transport can explain the occurrence of spatiotemporal patterns, including chaotic behavior.4,34−36 • Two interesting spatiotemporal patterns observed in tubular reactors are hot spots and thermal moving fronts.7,8,16,29 In particular, in a PFR, there exists a strong dependence of temperature and composition profiles on the inlet conditions, allowing the possible appearance of a maximum in the temperature profile (i.e., the hot spot) and the possibility of temperature runaway.37 On the other hand, thermal moving fronts are generated as follows. An initially cold catalyst bed is fed from one side at low temperature. If part of the catalyst bed is heated above an ignition temperature because of exothermic reactions, a traveling hot zone is established. If the flow velocity is periodically reversed, the hot temperature zone can be trapped inside the catalyst bed. This idea has been implemented to improve the performance of traditional tubular reactors for reversible and irreversible exothermic catalytic processes.8,14,17 • Uncontrolled hot spots and thermal fronts can have detrimental consequences on the operation of the reactor (for instance, catalyst deactivation and sintering and unwanted side products). Thus, control strategies for both the hot spot temperature and thermal fronts are of upmost importance. However, these patterns have a dynamic dependence with the reactor’s axial position, and its control is beyond of the scope of this paper. On the one hand, the hot spot location depends on the catalyst arrangement, the length of the reactor, and the nature of chemical reactions in the reactor.37 On the other hand, thermal fronts depend on the propagation velocity of the thermal front.17 • The BZ reaction in the tubular reactor is described by RD equations. RD equations can describe numerous phenomena in biology, ecology, and chemistry4,9,10 (for instance, prey−predator models, biomedical models, and various models in epidemiology33). A variety of spatiotemporal patterns arise from the competition between reaction and diffusion.1,4 For instance, unequal rates of diffusion and the nonlinearity of the reaction may result in the formation of complex spatiotemporal patterns, such circular and spiral waves, traveling spots, and highly disordered structures that resemble turbulence.1,2,4,9,10,20,33

ideas38−40 is presented. Both suppression (regulation) and periodic enforcing (tracking) of the reactor’s controlled output are framed as output feedback control design problems. The MEC approach is derived for the class of DPS model of tubular reactors described by eq 1. 3.1. Control Problem. Our goal is controlling the spatiotemporal dynamics at the reactor’s exit axial position for the class of tubular reactors described by eq 1. Although control of spatiotemporal patterns is one of the central problems of nonlinear dynamics, the study and control of pattern formation in systems where chemical reactions, diffusion, and convection arise simultaneously (such as in tubular reactors) remains largely unstudied. The control problem is either suppression or enforcing of spatiotemporal dynamics at the output of the reactor’s axial position ξ, via the manipulation of a suitable affine control input. Thus, since we are controlling at the reactoŕs output axial position ξ = L, both state variables and the control input are dependent only on time, i.e., y(t), z(t), and u(t). The control problem description is completed by the following assumptions: (A1) The measurement of the controlled variable at the output of the axial position of the reactor is available for control design purposes. (A2) Nonlinear functions f1(y(t), z(t)), and f1(y(t), z(t)) are smooth functions. (A3) The nonlinear function g1(y(t),z(t)) is bounded away from zero, globally bounded, and its sign (the directionality of the control action) is known. (A4) Nonlinear function f1(y(t),z(t)) at a desired position includes uncertain terms and terms including uncertain parameters that, for control design purposes, are considered completely unknown. (A5) There exists a rough estimate of the nonlinear function g1(y(t),z(t)) at a desired position given by g1̅ (y(t ), z(t )). (A6) Approximate diffusive and convective terms at the desired controlled position are considered as constant external perturbations χ(t) and completely unknown for control design purposes. The following comments are in order: • Usually, in chemical reactors, only temperatures and flow rates are easily measurable and, although advanced methods for accurate direct or indirect measurement of the concentrations have been developed, these have not been used frequently in industrial plants.37,41 Even in the absence of such measurements, state estimators can be designed, if the required unmeasured states of the chemical reactor are observable.41 • Assumption A2 is satisfied by most industrial chemical reactions. Assumption A3 is satisfied for the considered case studies and also for most tubular reactors chemical displaying spatiotemporal dynamics. Assumption A3 guarantees that the inverse of g1(y(t),z(t)) at a desired position exists and also is satisfied for most controlled chemical reactors. Assumptions A2 and A3 are considered for stability analysis purposes.40 • Assumptions A4−A6 considers model uncertainties, or, in the worst case, entire terms are unknown. Indeed, in chemical reactors, physicochemical parameters (diffusion coefficient, kinetics, etc.) have some degree of uncertainty, since these parameter values commonly are estimated from experimental data, which contain errors

3. ROBUST LINEAR OUTPUT FEEDBACK CONTROL DESIGN In this section, the control problem is stated and the control design based on modeling error compensation (MEC) 17520



Article

where τe is an estimation time constant and is the only design parameter of the reduced-order observer. (3) Assign a desired closed-loop behavior via an inverse feedback function u(t). Let e(t) = y(t) − yref(t) be the regulation or tracking error at the output of the reactor. We consider a desired closed-loop asymptotic behavior, i.e.,

that are due to both the estimation procedure adopted to fit data and the experimental errors of the data themselves.37,41 Moreover, in order to derive feedback control laws with minimum system information, both assumptions are appropriate. 3.2. Control Design. Robust control design based on MEC ideas is a control technique that has been applied to many complex systems.19,42−45 (for instance, batch and CSTR reactors,40,42 distillation columns,40 biochemical systems,19,44 and mechanical oscillators45). Sun et al.38 proposed a robust controller design method for single-input−single-output (SISO) minimum-phase linear systems. The central idea was to compensate the error due to uncertainty by determining the modeling error via plant input and output signals and using this information in the design. In addition to a nominal feedback, another feedback loop is introduced using the modeling error and this feedback action is explicitly proportional to the parametric error, which is the source of uncertainty. The MEC approach was extended for a class of linear time-varying and nonlinear linearizable lumped parameter systems with uncertain and unknown terms by Alvarez-Ramirez,40 where, instead of designing a robust state feedback to dominate the uncertain term, the uncertain term is viewed as an extra state that is estimated using a high-gain observer. The estimation of the uncertain term gives the control system some degree of adaptability. The control design is based on an equivalent model that lumps uncertain terms in a single state, which is estimated with a simple reduced-order observer, such that the assumption of a perfect model for control design purposes is not required. On the other hand, the control law forces a desired closed dynamics, including linear asymptotic40 and finite time behavior.42,43 Motivated by these features, in this paper, we have considered the robust control design of tubular reactors via MEC ideas. The control design is as follows: (1) Identify and lump all uncertain terms in a single state η. From the model described by eq 1 and assumptions A4 and A6, we have η (t ) = D

d2y(t ) dξ

− b1v

2 ξ=L

dy(t ) dξ

e(t )|ξ= L → 0

such that y(t) → yref(t) asymptotically. A suitable control law to this end is given by u(t ) =

+ f1 (y(t ), z(t ))

= η(t ) + g1̅ (y(t ), z(t ))u(t ) ξ= L

(3)

(4)

and the proposed reduced observer is d η ̅ (t ) dt

= τe−1(η(t ) − η ̅ (t )) (5)

ξ= L

using ω(t) = τeη̅ − y(t), the reduced-order observer can be written as follows: dω(t ) dt

= g1̅ (y(t ), z(t ))u(t ) − η ̅ (t ) ξ= L

⎛ d y (t ) ⎜ z(t )) τc−1e(t ) + η ̅ (t ) − ref ⎜ dt ⎝

⎞ ⎟ ⎟ ξ=L ⎠

where τc is a closed-loop time constant and is the only control law design parameter. The MEC controller is given by the feedback function (eq 8) and the modeling error estimator (eq 6). Notice that the resulting feedback controlled depends only on the measure of y(t), the estimated value of lumped uncertain and the rough estimate g1̅ (y(t ), z(t )). terms η(t), ̅ Some observations on the nature of the controlled systems are given as follows.40,42−45 • Lumped control approaches: In control systems involving DPS, an important question is whether the control and measurements are distributed, or concentrated, on the spatial domain. In many practical systems, only inlet and exit conditions of the spatial domain or selected points in the interior are accessible for measurement and control. A natural approach for control DPS is to have distributed actions over the entire spatial.21 However, applying distributed control can be difficult, because it becomes very complicated for both design and implementation, because of information delays, spatial complexity, long feedback loops, etc. In such cases, lumped approaches may have significant advantages over distributed controls (for instance, reduced complexity for both design and implementation). • Control of hot spots: The proposed control approach for the class of tubular reactors displaying spatiotemporal dynamics employs only one spatial location: the reactor’s exit axial position. Introducing appropriate sensors dispersed in the spatial domain, and if information about the localization of the hot spot is available (for instance, via sensitivity analysis37 or state observers41), then the case of hot spot control can also be addressed with the proposed control approach in conjunction with cascade control schemes and employing several heating/ cooling sections. 3.3. Stability and Robustness Issues. Stability analysis of nonlinear DPS is very complex and is beyond of the scope of this paper. However, since we are addressing the control of DPS at the reactor’s output axial position, the distributed nature is not considered for the stability analysis of the closed-loop system, such that previous results of the stability of the MEC approach for nonlinear lumped parameter systems39,40 are valid for the analysis of the closed-loop behavior. For the sake of completeness in presentation, a sketch of main ideas of the stability results for the MEC approach are provided as follows.

The function η(t) contains all of the uncertain terms of the system (1) at a desired position. (2) Design a reduced-order observer to get an estimate state η̅(t) of the real uncertain term η(t). The model described by eq 1 with lumped uncertain terms given by eq 3 is dy(t ) dt

−g1̅ −1(y(t ),

(8)

ξ=L

+ [g1(y(t ), z(t )) − g1̅ (y(t ), z(t ))]u(t )

(7)

t →∞

(6) 17521



Article

Given the regulation or the tracking error e(t) and defining then the closed-loop the estimation error as ϕ = η − η(t), ̅ system becomes de(t ) dt dϕ(t ) dt

show the controller robustness against model uncertainties, the control design is based on a rough estimate of function g1(y(t), z(t)), with a parameter mismatch of ∼15% of the nominal parameters used in the simulations. Moreover, typical disturbances for each case study are considered. 4.1. Control of Spatiotemporal Dynamics in a Tubular Reactor with Axial Dispersion. It has been reported that oscillatory behavior in tubular reactors can improve the mean productivity.7,8,11−14 Oscillatory behavior may also be of significant commercial for a class of oscillatory reactions in pollution abatement processes, polymerization processes, and in several partial oxidation processes.11−14 For instance, the time-averaged ammonia concentration (mean productivity of some species per cycle) observed in the non-steady-state operation was exceeded by 5%−27% of the ammonia concentration obtained under similar but under steady-state conditions.7 Thus, we have selected, as the control objective, the operation of the tubular reactor in a stable limit cycle. The manipulatable variable is the jacket cooling temperature, i.e., u(t) = yj, and the controlled variable is the reactor temperature, i.e., y(t). Given the distributed nature of the system, one would expect that the corresponding control input also varies along the reactor. It should be stressed that this is not taken into account in this case study, as it assumes that the jacket temperature is uniform along the reactor. This is a realistic assumption for systems with well-mixed jackets and for cases with high coolant flow rate. For cases with nonuniform jacket temperature, a jacket model should be incorporated. Control action u is then introduced in the energy balance as follows:

= − τc−1e(t ) + ϕ(t ) ξ=L

g (y(t ), z(t )) e(t ) + Ψ(e(t ), ϕ(t )) g ̅ (y(t ), z(t )) dχ (t ) + dt ξ = L

= − τe−1 ξ=L

(9)

where Ψ(e(t), ϕ(t)) represents the time derivative of lumped uncertain terms. By using assumptions A2−A6, it can be shown that such a time derivative is a continuous function of its arguments. Thus, there exists two positive constants ι1 and ι2 both independent of τe such that |Ψ(e(t ), ϕ(t ))| ≤ ι1|e(t )| + ι2 |ϕ(t )|

(10)

The closed-loop system (eq 9) can be seen as a nonlinear singularly perturbed system with e(t) and ϕ(t) as the slow and fast variables, respectively, and τe the small perturbation parameter. Hence, stability results from singularly perturbed systems implies the existence of a maximum estimation time constant τe*, such that, for all τe < τe*, the regulation or tracking error e(t) goes asymptotically to zero.39,40,42−45 In other words, the stability analysis establishes that the controlled variable y(t) can be regulated to a desired reference value yref(t) despite model uncertainties, as long as small values of the estimation time constant τe can be chosen. The maximum estimation time constant can be taken as a measure of the robustness of the controlled plant. Larger values of τe* leads to better robustness capabilities of the closed-loop system. Smaller values of τe lead to faster reconstruction of uncertain terms; however, excessively small values of τe may induce high sensitivity of the controller to high-frequency measurement noise.40 Thus, very small values of τe must be avoided in order to get a good tradeoff between stability margin and robust stability. A rigorous robustness analysis is beyond of the scope of this paper, however, in the next section, the closedloop system is subject to typical expected disturbances and model uncertainties in order to shown via numerical simulations the robustness of the proposed controller. From the analysis of the closed-loop system, the following simple tuning rules for the state estimator and the inverse feedback function, can be stated:40,42−45 (i) Starting with the residence time of the tubular reactor, which is the main characteristic time constant of the tubular reactor, as the upper base value for τc, determine a value of τc up to a point where a satisfactory nominal response is attained. (ii) The estimation time constant τe, which determines the smoothness of the modeling error, can be chosen as τe < 0.5 τc.

∂y(t ) ∂t

= Pe−1Le−1

∂ 2y(t )

ξ=1

∂ξ

− Le−1

2 ξ=1

∂y(t ) ∂ξ

ξ=1

⎛ y(t ) ⎞ + Le−1BDa(1 − z(t )) exp⎜ ⎟ ⎝ 1 + y(t )/λ ⎠ − Le−1βy(t ) + Le−1βu(t )

(11)

We consider that there exist great uncertainties in transport constants and kinetic parameters, such that the modeling error function is defined as follows: η ̅ (t ) = Pe−1Le−1

d2y(t ) dξ

− Le−1

2 ξ=1

dy(t ) dξ

ξ=1

⎛ y(t ) ⎞ + Le−1BDa(1 − z(t )) exp⎜ ⎟ ⎝ 1 + y(t )/λ ⎠ − Le−1βy(t ) + (Le−1β − Le−1β ̅ )u(t )

(12)

and the control law derived from the MEC approach is given by ⎛ dy (t ) u(t ) = − Le−1β ̅ ⎜τc−1e(t ) + τe−1(ω(t ) + y(t )) − ref ⎜ dt ⎝

4. NUMERICAL SIMULATIONS In this section, the control methodology described above is applied to the case studies. Numerical simulation shows the capabilities of the proposed control approach for both suppression of spatiotemporal dynamics and modulation of a sinusoidal signal, i.e., enforcing to a new spatiotemporal behavior, at the reactor’s output axial position. In order to

dω(t ) dt

⎞ ⎟ ⎟ ξ=1 ⎠ (13)

= Le−1β ̅ u(t ) − τe−1(ω(t ) + y(t )) ξ= 1

(14)

It can be seen that the feedback control law resulting from the MEC approach only uses the available measurement of 17522



Article

Figure 1. Controlled spatiotemporal dynamics in an axial dispersion tubular reactor: (a) spatiotemporal temperature map, (b) spatiotemporal concentration map, and (c) temperature time profiles at three reactor’s axial position. Control action is turned on at t = 25. Perturbations of feed conditions are activated at t = 50.

Figure 2. (a) Controlled reactor temperature for three sets of controller parameters, τc = [0.5, 0.05, 0.001], and τe = [0.1, 0.01, 0.002], plotted with red, green, and blue colors, respectively. (b) Corresponding control inputs.

17523



Article

Figure 3. Controlled spatiotemporal dynamics in a tubular reactor with recycle: (a) spatiotemporal temperature map, (b) spatiotemporal concentration map, and (c) temperature time profiles at three reactor axial positions. Control action is turned on at t = 10. A perturbation on the recycle ratio is activated at t = 20.

the suppression of chaotic behavior under the outlet conditions by manipulating the cooling jacket temperature. The energy balance of the reactor can then be written as

temperature (y(t)) and a rough estimation of two parameters (Le and β). We consider the control task of the tracking of a sinusoidal reference, yref (t ) = 2.5 + 2 sin(2.5t )

∂y(t ) ∂t

(represented by the dotted black line in Figure 1c). Control law is turned on at t = 25 time units. Three sets of control parameters are selected: τc = [0.5, 0.05, 0.001], and τe = [0.1, 0.01, 0.002]. Figure 1 shows the spatiotemporal evolution of the reactor’s variables, and the time evolution at three axial reactor positions, for τc = 0.001 and τe = 0.002. Figure 2 shows the controlled temperature and the control input for the three set of control parameters. The reactor temperature can be forced into a new sustained oscillatory behavior by the periodic variation of the cooling jacket temperature. Numerical simulation supported the role of cooling jacket temperature on the arising and suppression of oscillatory behavior in nonisothermal tubular reactors. As can be seen from Figures 1c and 2b, both the controlled variable and control input remain in a plausible operating region, and the controller tracks the output reactor temperature at the desired reference within a very good performance. Numerical simulations on Figures 1 and 2 also shows the control performance when the reactor is subject to disturbances in the feed conditions. 4.2. Suppression of Chaotic Spatiotemporal Patterns in a Tubular Reactor with Recycle. Chaotic oscillations under industrial conditions are commonly undesirable. In order to avoid the unfavorable range of occurrence of chaos, the control of this behavior is necessary. Thus, for this case study, the chaotic behavior for the operation of the PFR with recycle is assumed as undesirable dynamics. The control objective is

=− ξ=1

∂y(t ) ∂ξ

+ (1 − r )Da(1 − z(t ))n ξ=1

⎛ γβy(t ) ⎞ × exp⎜ ⎟ − (1 − r )δy(t ) ⎝ 1 + βy(t ) ⎠ + (1 − r )δu(t )

(15)

and at the desired output reactor position, we define the regulation error as e(t ) = y(t ) − yref

Under assumptions A4−A6, which consider uncertainties in the convective term, transport constants, and kinetic parameters, the modeling error function is defined as follows: η (t ) = −

dy(t ) dξ

+ (1 − r )Da(1 − z(t ))n ξ= 1

⎛ γβy(t ) ⎞ × exp⎜ ⎟ − (1 − r )δy(t ) ⎝ 1 + βy(t ) ⎠ + ((1 − r )δ − (1 − r ̅ )δ ̅ )u(t )

(16)

such that the MEC approach leads to the controller, given by 17524



Article

Figure 4. (a) Controlled reactor temperature for three sets of controller parameters, τc = [10, 5, 1], and τe = [2, 1, 0.2], plotted in green, blue, and red colors, respectively. (b) Corresponding control inputs.

⎛

u(t ) = −[(1 −

r ̅ )δ ̅ ]−1 ⎜τc−1e(t ) ⎜ ⎝

−

dω(t ) dt

dyref (t ) dt

literature on forced patterns in the BZ system (for instance, modifying the concentrations of added BZ reactants, the modulation of light on a light-sensitive variation of the BZ reaction, modulation of electric current, addition of activating reactants such that hydroquinone and methanol, have been used to control BZ-type oscillations).9,10,20,27,46 In this work, we consider the suppression of a periodic spatiotemporal pattern via the modulation of a parameter related to the addition of methanol, as described in section 2.2.3. Thus, the controlled system is written as follows:

−1

+ τe (ω(t ) + y(t ))

⎞ ⎟ ⎟ ξ=1 ⎠

(17)

= (1 − r ̅ )δ ̅u(t ) − τe−1(ω(t ) + y(t )) ξ= 1

(18)

∂y(t ) ∂t

The feedback control law only uses the available outlet reactor temperature measurement and a rough estimation of the term (1 − r)δ. For this case study, suppression of the spatiotemporal dynamics, relative to the reference temperature yref = 0.2, is considered. Control action is turned on at t = 10 times units, and a 10% perturbation for the recycle ratio r is implemented at t = 20 time units. Figures 3 and 4 show the closed-loop performance for the suppression of the chaotic spatiotemporal dynamics at the reactor exit position. Controller parameters are τc = [10, 5, 1], and τe = [2, 1, 0.2]. Figure 4 shows that the proposed control scheme is able to suppress the spatiotemporal pattern to the proposed reference temperature, via a control input with a quasi-periodic form, and the limits of the controlled variable are in reasonable values. Figure 4 also shows that an arbitrary convergence to the desired reference value can be prescribed, in the sense that the smaller the values of the controller parameters, the faster the convergence. On the other hand, despite the significant perturbation in a stabilitydependent parameter, the recycle ratio, the controller is able to handle it without a considerable control effort. 4.3. Suppression of Spatiotemporal Dynamics in a Tubular Reactor with BZ Reaction. There is significant

= Dy

∂ 2y(t )

ξ=1

∂ξ

2 ξ=1

⎛ y(t ) − q ⎞ + ε−1⎜y(t ) − y(t )2 − fz(t ) ⎟ y(t ) + q ⎠ ⎝

+ u(t )

(19)

In this case, the modeling error function is defined as follows: η ̅ (t ) = Dy

d2y(t ) dξ

2 ξ= 1

⎛ y(t ) − q ⎞ + ε−1⎜y(t ) − y(t )2 − fz(t ) ⎟ y(t ) + q ⎠ ⎝ (20)

and the controller is given by ⎛ d y (t ) u(t ) = −⎜τc−1e(t ) + τe−1(ω(t ) + y(t )) − ref ⎜ dt ⎝

⎞ ⎟ ⎟ ξ=1 ⎠ (21)

dω(t ) dt

= u(t ) − τe−1(ω(t ) + y(t )) ξ= 1

(22)

The feedback control law only uses the measurement of the HBrO2. The control task is the suppression of the open-loop periodic spatiotemporal pattern to the reference value of yref = 0.5. The control is connected at t = 20 time units. A 10% 17525



Article

Figure 5. Controlled spatiotemporal dynamics in the Belouzov−Zhabotinsky (BZ) reaction system: (a) spatiotemporal activator concentration map, (b) spatiotemporal inhibitor concentration map, and (c) activator concentration time profiles at three reactor axial positions. Control action is turned on at t = 20. A 10% perturbation in parameter f is implemented at t = 30.

Figure 6. (a) Controlled activator concentration for three sets of controller parameters, τc = [2, 0.5, 0.1], and τe = [0.5, 0.1, 0.02], plotted with green, blue, and red colors, respectively. (b) Corresponding control inputs.

perturbation on parameter f, related to the reaction stoichiometry, is implemented at t = 30 time units. Figure 5 shows the spatiotemporal map for controlled and uncontrolled variables, and the time evolution of the controlled variable at

three axial reactor positions. Figure 6 show the time profiles at the output axial reactor position for three sets of controller parameters: τc = [2, 0.5, 0.1], and τe = [0.5, 0.1, 0.02]. From Figure 6, it can be observed that after the control input displays 17526



Article

other is the design of an inverse feedback function to both drive the plant to behave in the desired manner and compensate the estimated uncertain term. The control approach is able to induce a desired closed-loop behavior in a class of distributed parameter models of tubular reactors displaying pattern formation. The negative feedback induced by the proposed controller, with the existing positive and negative loops in tubular reactors, results in the modulation or suppression of spatiotemporal patterns. Numerical simulations on three benchmark case studies show how spatiotemporal patterns on tubular reactors can be controlled at a desired reactor position via a simple robust model-based control design with good closed-loop performance.

a slight overshoot, a smooth transition to a steady-state value is followed. Figure 6 also shows that the controller provides good rejection capabilities of the implemented perturbation without excessive transient variations in the control input. Some additional comments on the closed-loop performance of the proposed controller are given as follows: • Spatiotemporal patterns in tubular reactors are mainly defined by the internal properties of these media, including negative and positive loops (for instance, chemical and thermal loops, or physical recycle). Thus, the feedback loop induced by the feedback controller allows the modification of the reactor dynamics via the introduction of a negative feedback. • The basic idea in controlling the spatiotemporal patterns is that, if one understands the underlying dynamics, one can force the system to display a desired pattern via a suitable external forcing. In tubular reactors, this is accomplished by externally forcing a control input variable, affecting either concentrations of the chemical species or the reactor temperature, or both. For case studies 1 and 2, the manipulation of the jacket reactor temperature, and for case study 3, the manipulation of an additive reactant, are suitable for the control of spatiotemporal patterns at the desired output reactor position, since these variables influence both heat transfer and mass transfer in the reactor. • The effect of the controller parameters is that for lower values, the controlled variable reaches faster the desired reference value, with a high control effort, but keeping the control input and the controlled variable in plausible process operating regions. • For all case studies, it can be seen that the proposed feedback control scheme is able to modulate the controlled variable at the output reactor position, despite the worst scenario of model uncertainties (i.e., complete unknowledge of local nonlinearities and diffusive and convective terms. Moreover, the controller was also able to handle typical expected perturbations for each case study, with good closed-loop performance and good robustness capabilities.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Turing, A. The Chemical Basis for Morphogenesis. Philos. Trans. R. Soc. London, Ser. B 1952, 237, 37−72. (2) Castets, V.; Dulos, E.; Boissonade, J.; De Kepper, P. Experimental Evidence of a Sustained Standing Turing-type Nonequilibrium Chemical Pattern. Phys. Rev. Lett. 1990, 64, 2953−2956. (3) Yakhnin, V. Z.; Menzinger, M. Convective Instability and Its Suppression in Packed-Bed and Monolith Reactors. Chem. Eng. Sci. 1999, 54, 4547−4557. (4) Yoshida, N. Characterization of Patterns, Oscillations and Chaos in Chemical Systems. J. Chem. Phys. 2001, 115, 8327−8330. (5) Jensen, K. F.; Ray, W. H. The Bifurcation Behavior of Tubular Reactors. Chem. Eng. Sci. 1982, 37, 199−222. (6) Subramanian, S.; Balakotaiah, V. Classification of Steady-State and Dynamic Behavior of Distributed Reactor Models. Chem. Eng. Sci. 1996, 51, 401−421. (7) Egyhazy, T.; Kovacs, J.; Scholtz, J. Experimental Study of Flow Reversal Ammonia Synthesis. Chem. Eng. Technol. 1998, 21, 967−974. (8) Salinger, A.; Eigenberger, G. The Direct Calculation of Periodic States of the Reverse Flow ReactorI. Methodology and Propane Combustion Results. Chem. Eng. Sci. 1996, 51, 4903−4913. (9) Vanag, V. K.; Zhabotinsky, A. M.; Epstein, I. R. Oscillatory Clusters in the Periodically Illuminated, Spatially Extended Belousov− Zhabotinsky Reaction. Phys. Rev. Lett. 2001, 86, 552−555. (10) Vanag, V. K.; Epstein, I. R. Pattern Formation in a Tunable Reaction−Diffusion Medium: The BZ Reaction in an Aerosol OT microemulsion. Phys. Rev. Lett. 2001, 87, 228301. (11) Chekanov, Y. A.; Pojman, J. A. Preparation of Functionally Gradient Materials via Frontal Polymerization. J. Appl. Polym. Sci. 2000, 78, 2398−2404. (12) Kulkarni, M.; Dudukovic, M. Periodic Operation of Asymmetric Bidirectional Fixed-Bed Reactors: Energy Efficiency. Chem. Eng. Sci. 1997, 52, 1777−1788. (13) Sapundzhiev, C.; Grosev, G.; Elenkov, D. Non-Steady-State Catalytic Decontamination of Waste Gases. Chem. Eng. Technol. 1991, 14, 209−212. (14) Bunimovich, G.; Vernikovskaya, N.; Strots, V.; Balzhinimaev, B.; Matros, Y. SO2 Oxidation in a Reverse Flow Reactor: Influence of a Vanadium Catalyst Dynamic Properties. Chem. Eng. Sci. 1995, 50, 565−580. (15) Reis, N.; Pereira, R. N.; Vicente, A. A.; Teixeira, J. A. Enhanced Gas−Liquid Mass Transfer of an Oscillatory Constricted-Tubular Reactor. Ind. Eng. Chem. Res. 2008, 47, 7190−7201. (16) Singh, S.; Lal, S.; Kaistha, N. Case Study on Tubular Reactor Hot-Spot Temperature Control for Throughput Maximization. Ind. Eng. Chem. Res. 2008, 47, 7257−7263.

5. CONCLUSIONS Complex behavior including spatiotemporal patterns can be supported by continuous tubular chemical reactors. For certain chemical reactions, with potential industrial applications, periodic oscillations at the reactor’s output axial position might enhance the performance, and forcing the system to behave periodically would be not detrimental. However, if the periodic behavior deteriorates the reactor performance, this behavior should be suppressed. Therefore, the control of spatiotemporal patterns in chemical reactors is of much practical interest. In this paper, we have presented a simple and practical control design for the suppression and control of tubular reactors that display a variety of spatiotemporal patterns. Although suppression of the complex dynamics through control at a single position does not seem to be feasible for many systems, the proposed control approach based on modeling error compensation ideas works for a class of tubular reactors displaying spatiotemporal dynamics. There are two important processes involved in the proposed control approach. One is the lumping of uncertain terms in a single state, which is estimated with a reduced-order observer. The 17527



Article

(40) Alvarez-Ramirez, J. Adaptive Control of Feedback Linearizable Systems: A Modelling Error Compensation Approach. Int. J. Robust Nonlinear Control 1999, 9, 361−377. (41) Dochain, D. State and Parameter Estimation in Chemical and Biochemical Processes: A Tutorial. J. Process Control 2003, 13, 801− 818. (42) Alvarez-Ramirez, J.; Solis-Daun, J.; Solar, R. Robust Finite-Time Stabilization of Temperature in Batch Reactors. Ind. Eng. Chem. Res. 2002, 41, 2238−2247. (43) Alvarez-Ramirez, J.; Monroy-Loperena, R. Output-Feedback Control of Reactive Batch Distillation Columns. Ind. Eng. Chem. Res. 2000, 39, 378−386. (44) Puebla, H.; Alvarez-Ramirez, J. A Cascade Feedback Control Approach for Hypnosis. Annals Biomed. Eng. 2005, 33, 1449−1463. (45) Puebla, H.; Alvarez-Ramirez, J. Suppression of Stick-Slip in Drillstrings: A Control Approach Based on Modeling Error Compensation. J. Sound Vib. 2008, 310, 881−901. (46) Huh, D. S.; Choe, Y. M.; Park, D. Y.; Park, S. H.; Zhao, Y. S.; Kim, Y. J.; Yamaguchi, T. Controlling the Ru-Catalyzed Belousov− Zhabotinsky Reaction by the Addition of Hydroquinone. Chem. Phys. Lett. 2006, 417, 555−560.

(17) Panfilov, V.; Sheintuch, M. Control Strategies for Front Stabilization in a Tubular Reactor Model. AIChE J. 2001, 47, 187− 196. (18) Pellegrini, L.; Possio, C. T.; Albertoni, S.; Biardi, G. Different Scenarios in a Controlled Tubular Reactor with a Counter Current Coolant. Chaos, Solitons Fractals 1993, 3, 537−549. (19) Puebla, H.; Martin, R.; Alvarez-Ramirez, J.; Aguilar-Lopez, R. Controlling Nonlinear Waves in Excitable Media. Chaos, Solitons Fractals 2009, 39, 971−980. (20) Schmidt, B.; De Kepper, P.; Muller, S. C. Destabilization of Turing Structures by Electric Fields. Phys. Rev. Lett. 2003, 90, 118302. (21) Christofides, P. D. Nonlinear and Robust Control of PDE Systems; Birkhauser: Boston, 2001. (22) Wu, W. Finite Difference Output Feedback Control of a Class of Distributed Parameter Processes. Ind. Eng. Chem. Res. 2000, 39, 4250−4259. (23) Wu, W.; Chen, C.-W. Measurement-Based Optimization and Predictive Control for an Exothermic Tubular Reactor. Ind. Eng. Chem. Res. 2007, 46, 2064−2076. (24) Wu, W.; Huang, M. Nonlinear Inferential Control for an Exothermic Packed-Bed Reactor: Geometric Approaches. Chem. Eng. Sci. 2003, 58, 2023−2034. (25) Wu, W.; Ding, S.-Y. Model Predictive Control of Nonlinear Distributed Parameter Systems Using Spatial Neural-Network Architectures. Ind. Eng. Chem. Res. 2008, 47, 7264−7273. (26) Berezowski, M. Spatio-Temporal Chaos in Tubular Reactor with the Recycle of Mass. Chaos, Solitons Fractals 2000, 11, 1197−1204. (27) Zhang, C. X.; Liao, H. M.; Zhou, L. Q.; Ouyang, Q. Pattern Selection in the Belousov−Zhabotinsky Reaction with the Addition of an Activating Reactant. J. Phys. Chem. B 2004, 108, 16990−16994. (28) Berezowski, M. Effect of Delay Time on the Generation of Chaos in Continuous Systems. One-Dimensional Model. TwoDimensional Model − Tubular Chemical Reactor with Recycle. Chaos, Solitons Fractals 2001, 12, 83−89. (29) Burghardt, A.; Berezowski, M.; Jacobsen, E. W. Approximate Characteristics of a Moving Temperature Front in a Fixed-Bed Catalytic Reactor. Chem. Eng. Process. 1999, 38, 19−34. (30) Chaudhury, A.; Chakraborty, S. Dynamics of Mixing-Limited Pattern Formation in Nonisothermal Homogeneous Autocatalytic Reactors: A Low-Dimensional Computational Analysis. Ind. Eng. Chem. Res. 2011, 50, 4335−4344. (31) Ganesh, A.; Sheintuch, M.; Luss, D. Transversal Hot Zones Formation in Catalytic Packed-Bed Reactors. Ind. Eng. Chem. Res. 2008, 47, 7509−7523. (32) Gupta, A.; Chakraborty, S. Linear Stability Analysis of High-and Low-Dimensional Models for Describing Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactors. Chem. Eng. J. 2009, 145, 399−411. (33) Murray, J. D. Mathematical Biology; Springer−Verlag: Berlin, 1998. (34) Kapral, R., Showalter, K., Eds. Chemical Waves and Patterns; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1995. (35) Doherty, M. F.; Ottino, J. M. Chaos in Deterministic Systems: Strange Attractors, Turbulence, and Applications in Chemical Engineering. Chem. Eng. Sci. 1988, 43, 139−183. (36) Ciotti, L.; Budroni, M. A.; Masia, M.; Marchettini, N.; Rustici, M. Competition Between Transport Phenomena in a Reaction− Diffusion−Convection System. Chem. Phys. Lett. 2011, 512, 290−296. (37) Froment, G. F.; Bischoff, K. B.; De Wilde, J. Chemical Reactor Analysis and Design; John Wiley & Sons: Hoboken, NJ, 2010. (38) Sun, J.; Olbrot, A. W.; Polis, M. P. Robust Stabilization and Robust Performance Using Model Reference Control and Modeling Error Compensation. IEEE Trans. Automatic Control 1994, 39, 630− 635. (39) Alvarez-Ramirez, J.; Morales, A.; Cervantes, I. Robust Proportional−Integral Control. Ind. Eng. Chem. Res. 1998, 37, 4740−4747. 17528


A Simple Feedback Control Approach for Output Modulation of

Recommend Documents