Robust Finite-Time Stabilization of Temperature in Batch Reactors

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Ind. Eng. Chem. Res. 2002, 41, 2238-2247

Robust Finite-Time Stabilization of Temperature in Batch Reactors† Jose Alvarez-Ramirez,* Julio Solis-Daun,‡ and Rocio Solar Division de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F. 09340, Mexico

A robust finite-time control design for temperature tracking in batch reactors is presented. The proposed controller is composed by (a) a nominal inverse-dynamics feedback function that induces convergence in finite time and (b) a calorimetric balance estimator to compensate for uncertainties due to unmeasured concentrations, reaction kinetics, and heat-transfer parameters. It is shown that the proposed robust feedback controller can be seen as a type of nonlinear proportionalintegral controller with an antireset windup scheme. Numerical simulations are provided to illustrate the flexibility and the performance of the controller. 1. Introduction Batch reactors (BRs) are an important form of production in modern chemical industries for realizing frequently changing production plants that require products of many kinds, in variable quantities. In comparison to continuous reactors, BRs offer flexible and economically attractive options because the same equipment can be used for different products depending on market demand. Examples of processes where BRs are becoming widely used are food products, pharmaceuticals, paints, polymers, and speciality chemicals. This trend is expected to continue over the next decades as the modern industry pursues the manufacture of lowvolume, high-value-added chemicals. Optimal control theory has been widely used to derive operation policies for BRs.1 Because of the complexity of the reaction mixtures and the difficulty to perform on-line concentration measurements, optimal operation designs have focused on the computation of optimal temperature profiles,2,3 which provide useful information by giving information on the expected yield of the reaction system and show how the temperature should change during the reaction course to obtain the maximum yield or the desired final product characteristics. Optimally designed temperature trajectories are important in relation to quality control of the product. If a given trajectory cannot be realized properly, off-specification material may be produced. Hence, there is a necessity of designing robust control strategies to provide stable and safe BR operation with achievement of optimal production tasks. Because open-loop implementations of optimal temperature profiles are very sensitive to modeling errors and disturbances,4 the trends in BR control research focused on the development of efficient feedback control configurations. Within the control theory framework, the problem can be formulated as a tracking problem where the computed optimal temperature profile becomes the reference trajectory for a feedback controller.5 † This project was supported by CONACyT (Grant 4002005-34739-U). * Corresponding author. Also at Programa de Matematicas Aplicadas y Computacion, Instituto Mexicano del Petroleo. Fax: +52-5-8044900. Phone: +52-5-8044649. E-mail: jjar@ xanum.uam.mx. ‡ Departamento de Matematicas.

The flexibility offered by BR, however, gives rise to challenging control problems that are due to the nonstationary, nonlinear, and finite-time duration nature of the underlying dynamics. Because in a BR operation there is no steady state, there is no normal condition at which controllers can be tuned. Accordingly, satisfactory control responses have not, thus far, been obtained.4 Moreover, it is recognized that traditional proportionalintegral/proportional-integral-derivative (PI/PID) controllers, which perform quite well for continuous reactors, provide poor performance because of the nonstationary nature of BRs. To address the control problem for BRs, several design methodologies have been proposed.4-10 Marroquin and Luyben6 used a control configuration to demonstrate experimentally that nonlinear temperature control can give a better performance than conventional linear ones. Kravaris and Chung5 used input/output linearizing techniques to construct nonlinear controllers for BR control. Because the proposed controller relies on exact cancellation of nonlinearities, it suffers from robustness due to modeling errors and unmeasured disturbances. Rotstein and Lewin9 used robust stability tools11 to design linear controllers for the temperature control of an unstable BR. The resulting controllers are of the PI/PID type, which can tolerate uncertainties in process gain and nonmodeled dynamics. A disadvantage of this approach is that the identification of a (Laplace domain) linear model, on which the robust control design is based, is a difficult task because BR does not have a steady-state operation. Jutan and Uppal7 used a calorimetric balance approach12 to estimate the current amount of heat being released in the reactor at any moment in time. In conjunction with a feedback controller, the estimated signal was used to counterbalance the effects of reaction heat generation. The control performance reported by the authors was not satisfactory in the sense that the reactor is not smoothly delivered to the desired temperature. Cott and Macchietto8 improved Jutan and Uppal’s control design by using a model-based estimator for the heat released in the reactor. The feedback part of the controller was designed from the generic model control (GMC) algorithm,13 which can be seen as a particular case of the more general inverse-dynamics technique reported by Kravaris and Chung.5 Although the proposed controller was shown to be robust with respect to changes in process parameters and to model mismatch, its imple-

10.1021/ie010450f CCC: $22.00 © 2002 American Chemical Society Published on Web 04/02/2002

Ind. Eng. Chem. Res., Vol. 41, No. 9, 2002 2239

mentation requires the measurement or estimation of certain compositions. On the other hand, the GMC algorithm introduces a classical integral action to compensate for uncertainties in heat-transfer parameters and unmeasured disturbances. We will show in this paper that the incorporation of the classical integral action is redundant and overparametrizes the resulting controller. Recently, Chen and Peng10 proposed a simple learning control strategy for temperature trajectory tracking. Contrary to previous papers, the proposed control strategy is not a model-based one, so that it does not exploit the underlying dynamics of BRs and leads to a conservative closed-loop performance. By considering that the process gain and the “time-constant” of a BR vary with time, Lakshmanan and Arkun4 used linear models which describe the batch dynamics locally along the optimal trajectory. As a result, a linear parameter-varying model obtained by interpolation between these multiple models was used to design a model predictive control for regulation and targeting of the product specification at the end of the batch. The major criticism to this approach is that linear models are obtained along the optimal trajectory. In this way, if the BR model contains strong uncertainties and the process is subjected to disturbances, as is the case in practice, the corresponding optimal trajectory will deviate strongly from the computed optimal trajectory. In turn, this deviation can introduce strong uncertainties in the linear models. Despite its proven robust functioning capability, the temperature control design for BRs still lacks understanding and systematization in the sense of the recent robustness-oriented constructive control approaches. The systematization is required to simplify the construction, the tuning, and the maintenance of the control schemes, and the understanding issue is important to assess the basic capabilities and limitations in control designs intended for high performance with adequate robustness levels and safety margins and, in principle, to extend the idea to other batch chemical processes. The results reported in the literature constitute interesting advances because they show how the temperature control design can be improved by employing linear or nonlinear advanced control techniques. However, for the development of a control design methodology with a systematic construction-tuning procedure and a rigorous theoretical backup along the recently developed constructive nonlinear control ideas, the following issues must be addressed: the closed-loop robust stability condition, its relationship with the gain tuning scheme, the identification and characterization of the robustness-performance tradeoff that underlies the highly robust functioning, the design with input and state saturation, the extent of the robust stability region, the performance limitation by the presence of noise, and the interplay between process and control designs. This motivates and justifies the present work, where some of the aforementioned issues are studied. In this work a constructive control approach is employed to study the temperature control problem for BRs. First, an exact inverse-dynamics temperature controller is constructed to induce finite-time convergence of the reactor temperature to the reference temperature trajectory. Finite-time stabilization is a desirable property in BRs because these processes have a finite-time operation policy. In this way, it is important to dispose of feedback controllers with guaranteed

finite-time convergence. In a second step, motivated by calorimetric estimation ideas,7,12 a robustly convergent estimator is built to on-line infer the rates of heat generation and exchange from temperature measurements and dynamic heat balances. Following the constructive ideas of the estimator-based and modelingerror compensation14 nonlinear control approaches, the combination of an approximation of the exact inversedynamics controller with the calorimetric balance estimator yields the proposed measurement-driven temperature controller. With respect to existing results in the literature, our contribution can be summarized as follows: (i) A systematic control design with a transparent construction-tuning procedure is provided. In particular, the use of calorimetric balance approaches7,12 is systematized by posing the control design problem within a constructive control framework.15 (ii) A feedback controller with finite-time convergence capability is obtained. In this way, the convergence time of the reactor temperature can be prescribed by adjusting a sole control parameter. (iii) The resulting controller can be interpreted as a nonlinear PI controller, with nonlinear integral action that generalizes conventional PI control configurations. (iv) In the limit of high rates of estimation, the proposed feedback controller recovers the behavior of the exact input/output inverse-dynamics controller.5 This property guarantees good performance of the BR temperature controller despite modeling errors and unmeasured disturbances. A specific BR example is presented to illustrate these facts and to demonstrate the versatility and generality of the proposed control scheme. 2. BR Model We consider BRs in which m reactions take place involving n (n > m) chemical species. Mass and energy balances on the chemical reaction system lead to the following set of differential equations:

c˘ ) Er(c,T)

(1)

T˙ ) HTr(c,T) + γ(c,T,t) [u - T] with initial conditions c(0) ) c0 and T(0) ) T0. In this system, (i) c ∈ Rn is the vector of concentrations of the involved chemical species; (ii) T ∈ R is the reactor temperature; (iii) r(c,T) ∈ Rm is the vector of reaction kinetics; (iv) E ∈ Rn × Rm is the stoichiometric matrix; (v) H ∈ Rm is the vector of constant reaction enthalpies; (vi) γ(c,T,t) is the heat-transfer parameter; and (vii) u is the jacket temperature, which is taken as the control input. A feasible operation of the BR requires that γ(c,T,t) > 0, so that in the rest of the paper γ(c,T,t) * 0, for all c ∈ Rm, T ∈ R, and t g 0, will be assumed. From the principle of mass conservation,16,17 there exists a positive vector ω ∈ Rn, such that ωTE ) 0. The equality ωTE ) 0 states that what is produced by the reaction system is equal to what is consumed. As a consequence, it can be shown that the polyhedral set C ) {c ∈ Rn: ωTc g 0, ci g 0, 1 e i e n} is positively invariant under the dynamics of the BR.16,17 Hence, throughout the rest of the paper the vector of concentrations c will be restricted to the bounded and closed set C.

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Figure 1. Schematic diagram of a BR.

Strong nonlinearities in model (1) are introduced by the reaction kinetics r(c,T). Commonly, the functionality of the reaction kinetics on c is of polynomial or rational (Langmuir) type. Besides, the functionality on T is of the Arrhenius type (i.e., exp(-EA/RgT), where EA > 0 is an activation energy and Rg > 0 is a universal gas constant). The class of BRs modeled by (1) includes important industrial processes, such as polymerization reactors18 and crystallizers.19 A typical operation of a BR can be described as follows. The reactants and catalysts are added into the reactor (see Figure 1), and the reactor temperature T is made to track a prescribed (commonly optimal) trajectory Tr(t) by manipulating the jacket temperature u. At first, the reactor temperature starts from the initial condition T0, such that a control policy must ensure temperature tracking in the sense that T(t) f Tr(t) as close and soon as possible. In this way, the controlled result of the temperature in the reactor has a major effect on the levels of the products in the reactor. For instance, an optimal temperature policy for minimum time, desired conversion, and molecular chain length is obtained at different initiator concentrations.20 3. Robust Control Design From a control viewpoint, major sources of uncertainties are the reaction kinetics r(c,T) and the heattransfer parameter γ(c,T,t). On the one hand, the function r(c,T) is poorly known in industrial applications. Even if r(c,T) is exactly known, the composition vector c is rarely available from measurements. In this way, one can consider that the reaction kinetics vector r(c,T) is not available for feedback control design. On the other hand, the heat-transfer parameter γ(c,T,t) can display a complex functionality with respect to concentration, temperature, and time due to changes in heattransfer area, viscosity, density, etc. Besides, complete concentration measurements are rarely made in industry because of expensive measurement devices or large time delays. In this way, a temperature controller for BR must confront robustness issues against functional uncertainties and unmeasured states. In the sequel, it will be assumed that only the reactor and jacket temperatures are available for feedback control design. 3.1. Exact Feedback Control Design. In this section, the temperature control of BRs with complete knowledge of the system dynamics will be addressed. That is, it will be assumed that the reaction kinetics r(c,T) and the heat-transfer parameter γ(c,T,t) are known and that all entries of the concentration vector

c are measured. Of course, the resulting controller cannot be implemented because of the lack of complete information; however, it will be used as an intermediate step toward the practical controller to be designed in the next section. It should be noted that, in general, the n + 1 differential equations in (1) are not linearly dependent. However, this is not a drawback to the inverse-dynamics controller design to be described in the following. Assume that the temperature reference Tr(t) and its first time derivative T˙ r are uniformly bounded for all t g 0. Let e(t) ) T(t) - Tr(t) be the temperature tracking error. Because BR are finite duration processes, with duration time TBR > 0, it seems to be natural to require an error dynamics with finite-time convergence. These kinds of controllers have gained much attention this past decade21 for one main reason: some robustness properties occur, even if sometimes the finite-time convergence property is lost.22 The problem of designing a robust finite-time controller for general nonlinear systems of the form x˘ ) f(x,u) is not an easy task.21,22 In the following, it will be shown that for temperature stabilization this problem can be addressed by means of an inverse-dynamics control approach endowed with a certain type of nonparametric estimation. As in GMC13 approaches,8 consider the following desired dynamics for the temperature regulation error e(t):

e˘ ) F(e), e(0) ) e0

(2)

where the function F(e) is chosen such that, for any initial condition e0 ∈ R, the solution e(t;e0) converges to the origin in finite time, say TFT < ∞. To achieve finitetime stabilization, the eigenvalue of the “linearized” part of F(e) tends to minus infinity as the state tends to zero. As a consequence of this “infinite eigenvalue assignation”, F(e) becomes non-Lipschitz while the error e reaches zero. The function can be either continuous or noncontinuous. Examples of finite-time stabilization functions are the following: (i) Noncontinuous dynamics: F(e) ) -KFT sign(e), where KFT is a prescribed positive constant. The solution of (2) is

{

-K t + e e(t) ) +KFTt + e0 FT 0

if e0 > 0 if e0 < 0

(3)

For a given parameter KFT, the convergence time TFT is given by

tFT ) |e0|/KFT

(4)

For initial conditions in a given compact set Ωe ⊂ R, the maximum convergence time, denoted by tmax FT , is given by

tmax FT ) max (|e0|/KFT) e0∈Ωe

(5)

Besides, tmax FT depends inversely on KFT; namely, the larger the value of KFT, the smaller the value of tmax FT . In this way, the tracking error dynamics (2) ensures convergence of the reactor temperature T(t) to the reference temperature Tr(t) in a time not larger than tmax FT .


(ii) Continuous dynamics: F(e) ) -KFT|e|-1/n, n > 1 and n is an odd integer. For instance, the solutions of e˘ ) -KFT|e|-1/3 are e(t) ) sign(e0)[|x0| - KFTt/3]3, if 0 < t < 3|x0|/KFT, and e(t) ) 0, if t g 3|x0|1/3/KFT. Thus, they converge in finite time to the origin, with tFT ) 3|x0|1/3/ KFT. To obtain the control input that provides the error dynamics (2), notice that the desired temperature dynamics is written as

T˙ ) F(e) + T˙ r

(6)

By matching the actual reactor dynamics (1) to the desired one (6), we get the exact feedback control function φex(c,T,t) given by (see Kravaris and Chung5)

u ) φex(c,T,t) def

) T + [T˙ r - HTr(c,T) + F(e)]/γ(c,T,t)

(7)

When u ) φex(c,T,t) is used in (1), the corresponding closed-loop dynamics of the BR is

c˘ ) Er(c,T) T˙ ) F(e) + T˙ r Stability of the error dynamics (2) ensures that T(t) f Tr(t). Once temperature convergence has been achieved, the concentration dynamics are governed by the system

When the dynamics (10) is matched to the desired temperature dynamics (6), the corresponding inversedynamics control input is5

u ) φ(η(c,T,u,t),T,t) def

) T + [T˙ r - η(c,T,u,t) + F(e)]/γ j (T,t)

(11)

Notice that the control input (11) is implicitly defined in the sense that u depends on itself. Because γ(c,T,t) is bounded away from zero, the equality u ) φ(η(c,T,u,t),T,t) can be easily solved for u to obtain that u ) φex(c,T,t). That is, the equality

φex(c,T,t) ) φ(η(c,T,φex(c,T,t),t),T,t)

(12)

is satisfied. In this way, the inverse-dynamics control input u ) φ(η(c,T,u,t),T,t) is an implicit version of the exact feedback control function u ) φex(c,T,t). This implicit representation of the exact inverse-dynamics function (7) will be the departing point for the robust control construction. Because the modeling error function η(c,T,u,t) is unknown, a practical version of the implicit feedback function (11), and hence of the exact feedback function j (t) of η(c,T,u,t) u ) φex(c,T,t), is obtained if an estimate η can be provided. Hence, the practical feedback control, j ,T,t), becomes denoted by φpr(η

j ,T,t) u ) φpr(η def

c˘ ) Er(c,Tr)

(8)

By assumption, the trajectory Tr(t) is uniformly bounded for all t g 0. Because r(c,T) is a smooth function of its arguments, this implies that Er(c(t),Tr(t)) is uniformly bounded for all t g 0. From the above arguments, it is concluded that the exact feedback control u ) φex(c,T,t) guarantees finite-time tracking of the reference temperature trajectory Tr(t) with complete stability of the controlled BR (i.e., all of the states of the controlled BR remain uniformly bounded). 3.2. Robust Feedback Control Design. Unfortunately, the exact feedback control input u ) φex(c,T,t) cannot be implemented as it stands because of the lack of concentration measurements and uncertainties in the functions r(c,T) and γ(c,T,t). To overcome this problem, a robust feedback controller resembling the structure of the exact feedback controller u ) φex(c,T,t) will be designed below. Let γ j (T,t) > 0 be an estimate of the heat-transfer parameter γ(c,T,t). Notice that the estimate γ j (T,t) depends only on temperature and time. In particular, γ j (T,t) can be a positive constant reflecting a mean-intime value of the actual heat-transfer parameter function γ(c,T,t). Because c is not available for measurements and r(c,T) is an uncertain function, as a worstcase control design, assume that the reaction kinetics r(c,T) are unknown. Introduce the modeling error function η(c,T,u,t) as follows: def

η(c,T,u,t) ) HTr(c,T) + ∆γ(c,T,t) (u - T)

(9)

) T + [T˙ r - η j + F(e)]/γ j (T,t)

(13)

Given the temperature measurements {T(t), u(t)}, the modeling error signal η(t) ) η(c(t),T(t),u(t),t) can be obtained via a calorimetric balance7,12 in the reactor to give the following expression:

η(t) ) T˙ - γ j (T(t),t) (u(t) - T(t))

(14)

Exact temperature time derivatives T˙ cannot be obtained because its computation requires forward information (i.e., it is a noncausal operation). In this way, the estimate η j (t) can be obtained if a practical (i.e., causal) estimate of the time derivative can be provided. To this end, the following filter-based estimator is used:

j) η j˘ ) τe-1(η - η

(15)

where τe > 0 is an estimation time constant. Notice that, if η j (0) ) 0, the estimator (15) can be written as

η j (s) )

1 η(s) τ es + 1

(16)

where s is the Laplace variable. That is, the estimate η j (t) is a (low-pass) filtered version of the actual signal η(t). To implement the estimator (15), the calorimetric balance (14) can be used in (15) to obtain

η j˘ ) τe-1[T˙ - γ j (T,t) (u - T) - η j]

(17)

def

where ∆γ(c,T,t) ) γ(c,T,t) - γ j (T,t). Then, the reactor temperature dynamics can be described as

j - T. ConseNow introduce the variable w ) τeη quently, the estimator (17) is equivalent to the following system:

T˙ ) η(c,T,u,t) + γ j (T,t) (u - T)

w˘ ) - γ j (T,t) (u - T) - τe-1(w + T), w(0) ) w0 (18)

(10)

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where the estimated signal is computed in the following way:

η j ) τe-1(w + T)

(19)

In summary, the proposed feedback controller is composed of the approximate inverse-dynamics feedback j ,T,t) given by (13) and the modeling function u ) φpr(η error signal estimator (18) and (19), namely,

j + F(e)]/γ j (T,t) u ) T + [T˙ r - η w ) -γ j (T,t)(u - T) - τe-1(w + T),

w(0) ) w0

η j ) τe-1(w + T) From (16), we know that η j (s) f η(s) as τe f 0. That is, the smaller the value of the estimation time constant, the faster the convergence of the estimated modeling error signal η j (t) to the actual one η(t). As η j (t) converges j ,T,t) also to η(t), the practical feedback function φpr(η converges to the inverse-dynamics one φ(η(c,T,u,t),T,t). On the other hand, because φ(η(c,T,u,t),T,t) is an implicit version of the exact feedback function φex(c,T,t), it is concluded that the practical feedback controller given by (13), (18), and (19) converges and recovers the performance of the exact feedback controller φex(c,T,t) as τe f 0. This is an interesting fact that will be exploited to provide easy-to-use tuning guidelines. Finally, some comments on the implementation of the proposed controller are in order: (a) If the heat-transfer coefficient γ(c,T,t) is exactly known, the modeling error function η(c,T,u,t) is exactly the heat released by reaction HTr(c,T). In this case, the modeling error estimator of (18) and (19) can be seen as a continuous version of the discrete-time estimator proposed by Cott and Macchietto.8 (b) Discontinuous feedback functions F(e) such as KFT sin(e) may induce undesirable effects in the control loop, such as control input chattering and high sensitivity to measurement noise. These problems can be reduced if a continuous approximation of the function sign is used, such as sign(T - Tr) ≈ tanh(R(T - Tr)), R > 0. In the limit as R f ∞, the function tanh(R(T - Tr)) converges uniformly to the function sin. Besides, the parameter R modulates the sensitivity of the feedback control to measurement noise. In fact, the smaller the value of R, the lower the sensitivity of the feedback control loop to measurement noise. (c) From (6), we know that e˘ ) F(e). If e(t) f 0 in finite time, the function F(e) is necessarily non-Lipschitz at the origin, i.e., |dF(e)/de| f ∞ as e f 0. In this way, the dynamics e˘ ) F(e) has an infinite pole at the origin. Although infinite poles induce good robustness properties against uncertain parameters and functions, excessive sensitivity to measurement noise can be present. Assume that F(e) is a symmetric function. To reduce the adverse effects of the infinite pole, a boundary-layer (Lipschitz) modification FL(e;) of the non-Lipschitz function F(e) can be proposed as follows:

FL(e;) )

{

F(e) -Ke

if |e| > if |e| e

where K ) |dF(e)/de| and > 0 is the size of the boundary layer about the zero tracking error e ) 0. Notice that FL(e;) is a continuous function. It is noted

that FL(e;) does not induce finite-time convergence of the tracking error but rather asymptotic convergence only. However, finite-time convergence of the tracking error to a neighborhood of e ) 0 is preserved. That is, the trajectory e(t) converges in finite time, say tFT, to the interval [-, +] and remains there for all t > tFT. (d) Industrial BRs are equipped with jackets/coils, such that the actual manipulated variable is the flow rate of the cooling/heating fluid. Under this situation, the temperature controller can be implemented in a cascade configuration where the proposed controller of (13), (18), and (19) becomes the master controller, which provides the jacket temperature reference to a slave controller.23,24 In this way, the slave controller manipulates the flow rate of the cooling/heating fluid to achieve the performance asked by the (proposed in this paper) master controller. (e) Temperature measurements commonly contain noisy dynamics. Because the calorimetric balance estimator of (18) and (19) may be of high-gain nature, discontinuous feedback functions such as sign(T - Tr) can produce chattering and hence large (high-frequency) control actions. This adverse effect can be reduced if there is a prefiltering of temperature measurements to remove high-frequency components. Another approach can be the use of a low-pass filter on the controller output as follows:

uf )

1 u τfs + 1

(20)

where uf is the filtered control input and τf is the filter time constant. Where τf is large, abrupt changes in u can be prevented. However, if τf is too small, then the difference between u and uf may become too large and the deviation of the system from the finite-time convergence behavior becomes more pronounced. As a heuristic rule, it suggested to choose τf to be not larger than the prescribed convergence time TFT. (f) In this paper, for the sake of simplicity in presentation, we have restricted ourselves to BRs. However, the proposed robust control design methodology can be easily extended along the same lines to semibatch chemical reactors. (g) This paper focused on the development of efficient tracking control configurations. That is, the reference Tr(t) was assumed to be given from an optimization procedure. However, from a practical viewpoint, one wonders what the reference Tr(t) might be in the presence of uncertainties. Model errors can be the cause of the poor performance of the reactor operation when Tr(t) based on a model is implemented on the actual plant. Recently, Gattu and Zafiriou25 proposed a methodology for on-line setpoint modification in the presence of modeling errors. The underlying idea is to recompute the temperature profile after new measurements are available. This results in robustness with respect to model errors and allows improvements even with infrequent product property measurements. Because the temperature controller proposed before requires the least knowledge (namely, the estimate γ j ) of the BR dynamics, the reactor temperature can track the setpoint Tr(t) every time it is modified. 3.3. Control Structure. The proposed feedback controller of (13), (18), and (19) displays an interesting structure, which can be interpreted as a type of nonlinear PI controller equipped with an antireset windup


(ARW) scheme. Because magnitude constraints in control inputs are always present in a practical situation, j ,T,t) be the computed control input and let let uc ) φpr(η u ) Sat(uc;umin,umax) be the actual control input, i.e.,

{

umin u ) uc umax

if uc e umin if umin < uc < umax if uc g umax

where umin and umax are respectively the minimum and maximum control input limits. Equation 17 can be rewritten as

η j˘ ) τe-1[T˙ - γ j (T,t) (uc - T) - η j-γ j (T,t) (u - uc)] j ,T,t) is used, becomes which, when uc ) φpr(η

η j˘ ) τe-1[T˙ - T˙ r - F(e) - γ j (T,t) (u - uc)] When it is recalled that e ) T - Tr, a direct integration of the above equation gives

η j (t) ) τe-1[e - e0 -

∫0tF(e(σ)) dσ ∫0tγj (T(σ),σ)[u(σ) - uc(σ)] dσ]

(21)

where σ is a dummy integration variable. Equation 21 can be used in (13) to obtain the following expression for the computed control input:

where the feedforward term is given by

ψ(T,Tr,t) ) [γ j (T,t) T + T˙ r - τe-1e0]/γ j (T,t)

(23)

and the linear proportional KLP, nonlinear proportional KNP, and the nonlinear integral KNI gains are given by

KLP ) γ j (T,t)-1τe-1 KNP ) - γ j (T,t)-1 KNI ) - γ j (T,t)-1τe-1

(24)

The ARW gain KARW is given by

KARW ) γ j (T,t)-1τe-1

(25)

If the estimated heat-transfer parameter γ j (T,t) is constant, say γ j , the ARW scheme reduces to the simpler one:

KARW

∫0t[u(σ) - uc(σ)] dσ

with

KARW ) τe-1 The above computations show that the proposed feedback controller is a type of nonlinear PI control endowed with a natural ARW scheme of feedback

nature. In this way, when the control input is saturated, j (T(σ),σ) [u(σ) - uc(σ)] dσ the feedback signal KARW ∫t0 γ drives the error u - uc to zero by recomputing uc such that the controller output uc attains exactly the saturation limit. This prevents the controller from winding up.26 When the control input is not saturated, u ) uc and the ARW action has no effect on the control loop. Notice that the integral action ∫t0F(e(σ)) dσ is induced by the (calorimetric-balance-based) modeling error estimator of (18) and (19). This nonlinear integral action introduces the robustness capabilities of the proposed feedback against unknown reaction kinetics and uncertain heat-transfer parameters. In this way, another integral action, as is made by Cott and Macchietto,8 is redundant in the sense that it does not add new structure into the feedback controller. This can be seen if an asymptotic convergence model such as e˘ ) KATe is used instead of the finite convergence one e˘ ) F(e). It can be shown that the corresponding controller is equivalent to a linear PI controller,27 where the traditional integral action is induced by the corresponding calorimetric balance estimator. 3.4. Tuning Guidelines. The finite-time convergence property of the exact feedback controller u ) φex(c,T,t), and the calorimetric balance estimation configuration suggests that the tuning of a BR temperature control system proceeds according to the following steps: (a) Determine the value function F(e) up to a point where a satisfactory nominal response is obtained. For instance, if F(e) ) sign(e) and given a nominal set Ω of initial conditions, choose the GMC gain KFT such that the maximum convergence time tmax CT (see (5)) is not larger than a prescribed one. (b) Choose a sufficiently small value of the estimation j (t) f η(t) time constant τe such that the convergence η is faster than the nominal convergence time tmax CT . The rule τe < 0.2tmax CT is suggested. An important advantage of the proposed control configuration is that its tuning can be made in two sequential steps: in the first step, the nominal input/ output closed-loop behavior is specified via the selection of F(e); in the second step, control tuning depends only on one parameter, τe. Accordingly, the smaller the value of τe, the closer the controlled temperature behavior is to the one induced by the exact inverse-dynamics feedback controller (7). While these tuning guidelines are clearly reminiscent of well-accepted practice, the following remarks are in order. (1) Contrary to what happens in standard PI configurations, with the adoption of the proposed PI configuration, the improvement of the performance (via the selection of the function F(e)) may now be viewed as decoupled from modifications required for sensitivity reduction (via the tuning of the estimation time constant τe) and vice versa. (2) The tuning of the estimation time constant τe is particularly easy to carry out in view of the fact that, up to the point where the influence of nonmodeled dynamics and measurement noise is no longer negligible, the performance is improved monotonically with τe-1. 4. Numerical Simulations To illustrate the performance of the proposed robust feedback controller, let us consider the consecutive exothermic chemical reaction A f B f C in a BR, as

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was considered by Marroquin and Luyben.6 The reaction in each step is assumed to be first-order:

r(c,T) )

(

)(

r1(c,T) -k c1 ) k c1 k2c2 r2(c,T) 1 1

)

where c1 and c2 are respectively the concentrations (mol/ L) of the reactant A and the desired product B. The rate constants are given by

k1 ) 5.35 × 1010 exp(-9000/T) min-1 and

k2 ) 4.61 × 1017 exp(-15000/T) min-1 For the energy balance, the parameters are H1 ) 18.9 m3‚K/mol and H2 ) 12.65 m3‚K/mol. Besides, the heattransfer parameter γ(c,T,t) was chosen as

γ(c,T,t) ) 200.0 + 40 tanh[0.1(T - 335.0)] min-1 That is, the heat-transfer parameter γ(c,T,t) has a (20% variation in the operation region. Such variability in γ(c,T,t) can be due to, e.g., changes in the viscosity and density of the reaction mixture. The estimated heattransfer parameter γ j for control implementation was chosen as γ j ) 200 min-1. Finally, the batch time is specified as tf ) 30 min. Although the structure of the reaction system is simple, the corresponding BR retains the main dynamical characteristics of industrial systems, namely, nonstationary operation and limited operation period. (a) Constant-temperature policy. To demonstrate the versatility and generality of the proposed control scheme, let us consider the case of a constant-temperature policy with Tr ) 330 K. The initial conditions are c1(0) ) 0.95, c2(0) ) 0.05, and T(0) ) 310 K. The desired convergence time is tCT ) 5 min. The discontinuous error function F(e) ) KFT sign(e) was implemented via the approximation tanh(Re) with R ) 0.5. Because |e(0)| ) 20, KFT ) 4 K/min is chosen (see (4)). The control input bounds are umin ) 300 K and umax ) 400 K. Figure 2 presents the control performance for three different values of the estimation time constant τe. For comparison, the performance of the exact (inverse-dynamics) feedback controller u ) φex(c,T,t) is also shown. As expected, the smaller the value of τe, the closer the control behavior is to the one of the exact inverse-dynamics feedback controller (7). In this way, the prescribed convergence time tFT ) 5 min is achieved as τe goes to zero. Notice that, even that the modeling error estimator is of asymptotic convergence nature, the overall controller does not lose the finite-time convergence property. In this way, the proposed finite-time controller displays a good robust stability margin against modeling errors. The good performance of the proposed controller is provided by the on-line modeling error estimator because it can predict the speed at which combined reaction heat and uncertain transferred heat are released in the reactor. (b) Measurement noise. To illustrate the effects of measurement noise and disturbances on the measured temperature, consider that the measured temperature Tm is contaminated with σ2 ) 2 Gaussian disturbance ν(t). Figure 3 shows the controller output for two different values of the estimation time constant τe. Notice that the control output contains high-frequency

Figure 2. Effect of the estimation time constant τe in the performance of the control system. Notice that the trajectory T(t) arrives to the reference in finite time tf < 5 min.

Figure 3. Control performance when the measured temperature is affected with a Gaussian measurement noise.

components with excessively large amplitude. Of course, such a dynamic behavior of the jacket temperature cannot be realized in practice. Intuitively, the effective control input that provides temperature tracking is the


Figure 4. Control performance when a low-pass filter is used to remove the high-frequency components.

Figure 5. Performance of finite-time and infinite-time PI controllers with τe ) 0.5 min, tFT ) 5 min, and τc ) 5 min.

mean dynamics of such a high-frequency signal, which can be obtained with a low-pass filter. Figure 4 shows the filtered jacket temperature uf obtained with the lowpass filter (20) with τf ) 4 min. In this case, most highfrequency components induced by the chattering of the feedback sin(T - Tr) are removed. (c) Comparison with an infinite-time (asymptotic) PI controller. Traditional PI controllers are commonly used in industry to track temperature in batch and semi-BRs. Drawbacks of traditional PI controllers u ) u j + Kpe + KI ∫t0e(σ) dσ when used in BRs are (i) there is not a nominal control input u j and (ii) there is a lack of reliable tuning procedures to choose the gains KP and KI. In fact, batch and semi-BRs do not have a nominal operating point to carry out traditional (e.g., Ziegler-Nichols) tuning procedures. The control design described before allows one to obtain an asymptotic PI controller if F(e) ) -τc-1e, where τc is a prescribed time constant of the first-order dynamics 3e ) -τc-1e. Figure 5 shows the performance of finite-time and infinite-time PI controllers with τe ) 0.5 min, tFT ) 5 min, and τc ) 5 min. On the one hand, although both controllers drive the temperature trajectory T(t) to the setpoint, the finitetime design has the advantage that the exact convergence time can be given a priori. On the other hand, both PI controllers have enormous advantages over the traditional representation of the PI control in that the tuning can be made quite easily. In fact, the tuning depends only on one parameter, τe. (d) Optimal temperature policy. In the consecutive reaction scheme, it is required to maximize the production of the desired component B. In this way, the performance index to be maximized is the concentration of component B at the specified final time, i.e., I ) c2(Tf). The standard optimal control problem is then to find the temperature profile, such that the performance

index I ) c2(Tf) is maximized. To find the optimal temperature profile, the time interval is first divided into P subintervals of equal length (0, t1), (t1, t2), ..., (tP-1, tP), where tP ) tf, so that the length of each subinterval is L ) tf/P. A piecewise linear continuous control policy for the temperature profile is seeked, so that in the time interval (tk, tk+1) the temperature is given by

T(t) ) T(k) +

(

)

T(k+1) - T(k) (t - tk) L

where T(k) is the temperature at time tk and so on. The optimal control problem is to find the P + 1 values of T(k), k ) 0, 1, ..., P, such that the performance index I ) c2(Tf) is maximized.3 For P ) 50, the optimal trajectory calculated by Luss and Okongwu3 can be correlated by

Tr(t) ) 345.9 - 5.6t + 0.69t2 - 0.0417t3 + 1.18 × 10-3t4 - 1.26 × 10-5t5 To resemble the industrial operation of BRs, a preheating period of 5 min is considered to take the mixture temperature at Tr ) 345.9 K. Once the reactor temperature has achieved the initial optimal temperature, the BR undergoes the reaction period by means of the addition of, e.g., a catalyst. For τe ) 0.2 min (see Tuning Guidelines), Figure 6 shows the response of the controlled reactor. Notice the small overshoot at t ) 5 min induced by the startup of the reaction. The reactor temperature converges to the reference trajectory in about 3 min thanks to the finite-time convergence feedback KFT sign(T - Tr) and to the calorimetric balance estimator of (18) and (19), which provides an on-line estimate of the modeling error signal η(t). Figures 7 and 8 show the performance of the finite-time

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Figure 6. Dynamics of the controlled BR under tracking of a optimal temperature trajectory.

Figure 8. Dynamics of the controlled BR when a low-pass filter is used to reduce high-frequency components.

surement noise. Such large-amplitude components in the control input are removed by means of a simple lowpass filter with tf ) 2 min. (e) Effects of disturbances. Failures in the reaction operation can appear because of the complexity of the reaction schemes. Let us show that the proposed controller is able to provide protection against moderate failures. Figure 9 presents the control performance when a -20% step change in the kinetics parameter k2 at t ) 10 min and a -25% step change in the heattransfer parameter γ(c,T,t) at t ) 15 min are present. The controller is able to counteract the adverse effects of these step changes within a time period of no longer than 2 min. 5. Conclusions

Figure 7. Performance of the finite-time PI controller when the measured temperature is contaminated with a σ2 ) 2 Gaussian noise.

PI controller when the measured temperature is contaminated with a (2 K Gaussian noise. As in the case of a constant-temperature setpoint, the unfiltered temperature signal induces a strong feedback of the mea-

A robust finite-time control strategy for temperature trajectory tracking in BRs has been presented. The robust control design uses a calorimetric balance algorithm to estimate modeling errors and counteract its effects via a finite-time stabilization feedback. The calorimetric balances introduce in a natural form a nonlinear integral action, which induces the robustness capabilities of the resulting controller. In this way, the resulting controller is able to cope with uncertainties due to reaction kinetics and heat-transfer parameters and with unmeasured disturbances. As a result of the systematic control construction, a transparent construction-tuning procedure is provided. The effectiveness and flexibility of the controller is illustrated via a specific example. We believe that the proposed constructive control approach has an important potential in becoming a very useful tool in the research area of control of batch processes. However, several important questions remain


Figure 9. Control performance when a -20% step change in the kinetics parameter k2 at t ) 10 min and a -25% step change in the heat-transfer parameter γ(c,T,t) at t ) 15 min.

open and need additional work. For example, a rigorous proof of the stability of the underlying closed-loop system and the effects of measurement noise are required to assess the basic capabilities and limitations of the proposed control scheme. Composition control is also a challenging problem because the inherent delays introduced by measurement devices limit seriously the performance of feedback controllers. This is an important control problem because, rather than the reactor temperature, reactor composition control is commonly the most important task. Literature Cited (1) Filippi-Bossy, C.; Bordet, J.; Villermaux, J.; MarchalBrassely, S.; Georgakis, C. Batch reactor optimization by use of tendency models. Comput. Chem. Eng. 1989, 13, 35-47. (2) Luss, R. Optimal control of batch reactors by iterative dynamic programming. J. Process Control 1994, 4, 218-226. (3) Luss, R.; Okongwu, O. N. Towards practical optimal control of batch reactors. Chem. Eng. J. 1999, 75, 1-9. (4) Lakshmanan, N. M.; Arkun, Y. Estimation and model predictive control of nonlinear processes using linear parameter varying models. Int. J. Control 1999, 72, 659-675. (5) Kravaris, C.; Chung, C. B. Nonlinear state feedback synthesis by global input/output linearization. AIChE J. 1987, 33, 592-603.

(6) Marroquin, G.; Luyben, W. L. Practical control studies of batch reactors using realistic mathematical models. Chem. Eng. Sci. 1973, 28, 993-1003. (7) Jutan, A.; Uppal, A. P. Combined feedforward-feedback servo control scheme for an exothermic batch reactor. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 597-602. (8) Cott, B. J.; Macchietto, S. Temperature control of exothermic batch reactors using generic model control. Ind. Eng. Chem. Res. 1989, 28, 1177-1184. (9) Rotstein, G. T.; Lewin, D. R. Control of an unstable batch chemical reactor. Comput. Chem. Eng. 1992, 16, 27-49. (10) Chen, Ch-T.; Peng, Sh-T. A simple adaptive control strategy for temperature trajectory tracking in batch processes. Can J. Chem. Eng. 1998, 76, 1118-1127. (11) Morari, M.; Zafiriou, E. Robust Process Control; PrenticeHall: New York, 1989. (12) Schuler, H.; Schmidt, Ch.-U. Calorimetric-state estimators for chemical reactor diagnosis and control: review of methods and applications. Chem. Eng. Sci. 1992, 47, 899-915. (13) Lee, P. L.; Sullivan, G. R. Generic model control. Comput. Chem. Eng. 1988, 12, 573-598. (14) Alvarez-Ramirez, J. Adaptive control of feedback linearizable systems: a modeling error compensation approach. Int. J. Robust Nonlinear Control 1999, 9, 361-377. (15) Sepulchre, R.; Jankovic, M.; Kokotovic, P. Constructive Nonlinear Control; Springer: London, 1997. (16) Feinberg, M. Chemical reaction network structure and the stability of isothermal reactorssI. The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 1987, 29, 2229-2268. (17) Gavalas, G. R. Nonlinear Differential Equations of Chemical Reacting Systems; Springer-Verlag: New York, 1968. (18) Kozub, D. J.; McGregor, J. F. State estimation for semibatch polymerization reactors. Chem. Eng. Sci. 1992, 47, 10471062. (19) Miller, S. M.; Rawlings, J. B. Model identification and control strategies for batch cooling crystalizers. AIChE J. 1994, 40, 1312-1326. (20) Gentric, C.; Pla, F.; Lafiti, M. A.; Corriou, J. P. Optimization and nonlinear control of a batch emulsion polymerization reactor. Chem. Eng. J. 1999, 75, 31-46. (21) Bath, S. P.; Bernstein, D. S. Continuous finite-time stabilization of translational and rotational double integrators. IEEE Trans. Autom. Control 1998, 43, 678-682. (22) Perruquetti, W.; Drakunov, S. Finite time stability and stabilization. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, 2000. (23) Russo, L. R.; Bequette, B. W. State space versus input/ output representations for cascade control of unstable systems. Ind. Eng. Chem. Res. 1997, 36, 2271-2278. (24) Tyner, D.; Soroush, M.; Grady, M. C. Adaptive temperature control of multiproduct jacketed reactors. Ind. Eng. Chem. Res. 1999, 38, 4337-4344. (25) Gattu, G.; Zafiriou, E. A methodology for on-line setpoint modification for batch reactor control in the presence of modeling error. Chem. Eng. J. 1999, 75, 21-29. (26) Kothare, M. V.; Campo, P. J.; Morari, M.; Nett, C. N. A unified framework for the study of anti-windup designs. Automatica 1994, 30, 1869-1883. (27) Alvarez-Ramirez, J.; Morales, A.; Cervantes, I. Robust proportional-integral control. Ind. Eng. Chem. Res. 1998, 37, 4740-4747.

Received for review May 18, 2001 Revised manuscript received September 28, 2001 Accepted December 27, 2001 IE010450F

Robust Finite-Time Stabilization of Temperature in Batch Reactors

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