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Chemical Engineerin# Science, Vol. 52, No. 14, pp. 2223-2230, 1997

PIh

S0009-2fi09(97)00060-2

© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009-2509/97 $17.00 I- 0.00

Robust stabilization of temperature in continuous-stirred-tank reactors Jos6 Alvarez-Ramirez,* Rodolfo Sufirez and Ricardo Femat Divisidn de Ciencias B~isicas e Ingenieria, Universidad Autdnoma Metropolitana-lztapalapa, Apdo. Postal 55-534, 09000 Mrxico D.F., Mrxico (Received 12 July 1995; accepted 26 February 1996)

Abstract--We study the stabilization problem of continuous-stirred-tank reactors controlled with the coolant temperature. It is assumed that reaction rates are unknown, such that uncertainties are associated with heat generation terms. We construct a dynamic feedback which is composed of a linearizing-like feedback and an uncertainty estimator, which is based on energy and mass balance equations. Both local and practical convergence of this control algorithm are proved when the uncertainty estimator satisfies a high-gain assumption. The performance of the controller is illustrated by means of a numerical example. © 1997 Elsevier Science Ltd. All rights reserved.

Keywords: Chemical reactors; nonlinear control; robust stabilization; uncertainty estimation. 1. INTRODUCTION Stabilization of uncertain systems is a central issue in control theory. In the past few years much interest has been devoted to the design of robust controls for dynamic systems with uncertainties. Attention has been given to both the problem of robust stabilization and robust performance of uncertain linear systems and a number of results covering these issues have been reported in the literature (Barmish, 1985; Leitmann, 1979; Petersen, 1987). On the other hand, despite the advances in the theory of nonlinear control systems (Isidori, 1989) to date the issue of designing robust feedback controls of nonlinear systems with uncertainties has not been fully investigated. Robust stabilization of a class of uncertain nonlinear systems satisfying the so-called matching condition has been analyzed in a number of papers (Barmish et al., 1983; Chen and Leitmann, 1987). However, the issue of robust stabilization and robust performance of nonlinear systems with uncertainties remains to be important and challenging. Continuous-stirred-tank reactors (CSTR) are ineluded in an important class of industrial processes where control has to deal with uncertainties. In CSTRs, uncertainties arise because of fluctuations in input flow conditions, noise and imperfections in stirring, and poor knowledge of the reaction mechanism. The latter being the most important source of uncertainties. Examples of processes where the reaction mechanism is poorly known are: (a) Fluid catalyzed crackin9 (FCC). This process is composed of two chemical reactors: the riser,

* Corresponding author.

where the oil is converted into gasoline, and the regenerator, where a burning reaction is realized to regenerate the catalyst activity. In both reactors, temperature must be controlled. Because of the presence of an intricate network of reactions, the reaction mechanism is poorly known and fairly approximated by a lumped mechanism of two or three reactions (Caldwell and Dearwater, 1991 ). (b) Mass polymerization. Here the reaction mechanism is well established; however there is an important uncertainty about the reaction parameters (enthalpies and reactions constants). Being the chemical reactions the generators of the dynamics in CSTRs, it is not hard to conclude that in order to have robust stabilization and robust perfbrmance, a control strategy must include certain knowledge of the reaction terms. Of course, the better the knowledge of such reaction functionalities, the better the control performance. Most robust control strategies for nonlinear systems (Corless and Leitmann, 1981; Chen and Leitmann, 1987) requires the knowledge of upper bounds for uncertainty terms to design a feedback that essentially is a high-gain one. Although robust stabilization can be guaranteed, the performance of the control input is poor (with large overshoot) and can, in some cases, be saturated. Applications of the above-discussed strategies to the control of chemical reactors is not a successful idea because establishing an upper bound for reaction rates terms is not easy. In this work, we take a different approach, namely, an approach that makes use of on-line estimation of uncertainties. We construct an uncertainty estimator (reaction rates) by taking the uncertainty as a new state

2223

J. Alvarez-Ramirez et al.

2224

evolving in a manifold immersed in an extended state space. Then the uncertainty is estimated via an observer design approach. We study the control problem of a class of uncertain CSTR that are stabilizable via temperature regulation. Temperature control is the first requirement for the stable operation of industrial processes including CSTR, before applying secondary control (quality control) manipulations. We construct a dynamic feedback which is composed of a linearizing-like feedback (lsidori, 1989) and an uncertainty estimator. Both local and practical global convergence of this control algorithm are proved when the uncertainty estimator satisfies a high-gain assumption. Such uncertainty estimator is based on energy and mass balance equations. Balancing the heat in a defined volume leads to conservation equations of the following form: heat accumulation = heat transport + heat sources/losses.

2. PRELIMINARIES In this section, we provide the model of the CSTR under consideration and study its properties. Consider a CSTR where a set of m reactions among n chemical species is taking place. If the reactor is perfectly mixed, the dynamics are governed by the following equations (derived from mass and energy balances):

J'~-O(Tin(t)- T) + AH~:R(C, T ) + y ( u - T)

R(C, T )=R~(C)Rz(T )

(2)

where Rl(C)=diag[Rli(C)] E ~}~mxm and Rz(T) E ~m. This is not a restrictive assumption since many reaction rates satisfy it. Moreover, our results in the next section will hold for the nonseparable case under some mild assumptions. System (1) is assumed to satisfy the following assumptions:

Assumption1. R2(T ) is an Arrhenius-type vector (i.e. R2i(T)=klexp(-~i/T), where ~i is the activation energy for the ith reaction).

Assumption 2.

0 ~< IICin(t)l I ~ cimax, 0 ~ Tin(t) -~<~"inTmax and IdT/dt I <<,~

Assumption3.

This balance is used with easily available signals, like temperature, pressures, flows, to estimate instantaneous heat sources/losses. The design of such estimator is simple, as it is based on standard state'reconstruction techniques (Luenberger-like observers). One feature of the resulting feedback controller is that it carries out an almost cancellation of nonlinearities in the temperature dynamics. We illustrate the performance of the controller by a numerical example. The uncertainty estimation strategy here presented is closely related to the method of calorimetric balances (Schuler and Schmidt, 1992) for the estimation of reaction rates, conversion, and rate of production in chemical reactors. A number of experimental work (Schuler and Schmidt, 1992 and references therein) showed that calorimetric balances work quite well in a wide range of chemical reactors. The results in this work provide a theoretical framework to conceptualize calorimetric balances methods.

(~ = O(Cin(t) - C)+ER(C, T)

of generality, assume that R(C, T) is separable, i.e.

(1)

where C E ~ n is the vector of concentrations of chemical species, T E ~ is the temperature, EE{R nxm is a constant matrix (the stoichiometric matrix), R(C, T)E ~ " is the vector of reaction rates, A H E ~ " is the vector of reaction enthalpies, O=F/V is the mean residence time, 7 is the heat transfer coefficient, and u is the coolant temperature (the manipulated variable). For the sake of simplicity in presentation, and without loss

(a) AH E ~m is an unknown but a constant vector. (b) The heat transfer coefficient 7 is exactly known.

Assumption4.

The reaction rates vector R(C, T) is unknown. Additionally R(C, T) E C 2.

Assumption 5.

For a given T = Tr > 0, the isothermic

reactor

C= O(C, - C) + ER(C, Tr)

(3)

has a unique equilibrium point C*, which is a global attractor. Some comments regarding the above assumptions are in order. It is well established experimentally that most chemical reaction rates follow Arrenhius functionalities in temperature. Assumption 2 is satisfied in all practical situations. Assumption 3(a) is made for the sake of clarity in presentation; however, functionalities like A H ( T ) E cg~ can be taken without additional complications in the proof of the results in the next section. Assumption 3(b) is a restriction because in some cases (for instance, mass polymerization reactors) the heat transfer coefficient 7 changes with temperature and concentration variations. Since we are able to establish exponential convergence, a certain robustness margin to small variations of 7 is expected. Assumption 5 is a reasonable practical assumption. A large set of chemical reaction networks satisfies such assumption (Feinberg, 1990). The case of chemical reactors that do not satisfy Assumption 5, but are feedback linearizable, will be addressed in a forthcoming paper. This work is concerned with the problem of designing a feedback controller for system (1) such that the equilibrium point (C*, Tr) of the closed-loop system is globally uniformly (practically) asymptotically stable for all admissible uncertainties (satisfying Assumptions 2-4). In this case we will say that the closed-loop system is robustly stable. Before designing a feedback controller for the system (1), we need to state some properties.

Robust stabilization of temperature Property 1. There is a subset M C ~,+1 where the dynamical system ( 1) is defined: M = [0, C~ ax] x ... x [0, CJ]'ax] × (-cx~, +c~). In fact, M is invariant under the solutions o f system (1). Here 0 <~ C max < cx~ is the maximal possible concentration o f the ith species. This" property is consequence o f Assumption 2 and standard mass balance arguments (Gavalas, 1968). The subset Mp=[O, Cr~ax] × - . . X [0, Cnmax] x [0,4-00] is the natural domain o f definition o f system (1) because it contains only zero and positive temperature. In this way, Mp i~" called to as the physical manifold o f ( 1 ). However, Mp is' not invariant under the dynamics of(i ). The following property shows that chemical reaction rates are upper bounded. This is a consequence of the finite mass contained in the reactor volume (Gavalas, 1968 ). Property 2.

NR(C,T)Jl < / h ,

Jor aft (C, T) E Mp.

Proof." Since Rt(C) is a continuous function and C is bounded, it is sufficient to show that Rz(T) is bounded for all 0 ~< T ~< + cx~. In fact, R2.i(T)=ki exp(-~i/T), so that 0 ~< R2,i(T) <<,ki. [~ Note that R2(T) is not bounded for T < 0. However, we can redefine a functionality R2(T) compatible with Property 2 such that Re(T) be bounded for all T E ~: R2(T)= { A s i n A s s u m p t i ° n

1

ifT~<0 if T < 0.

It must be pointed out that R2(T) was redefined only for technical reasons (we will need it in Section 4). Property 3. dR2(T)/dt is" continuous (i.e. R2(T) E ~1 ). Proofi Continuity follows from the fact that R=(0) = 0. On the other hand, dR2.i( T )/dt = c~ikiexp(-c(i/T )/T 2, for T ~>0, and e(-:% T )

lim - r ~ ~0 T 2 consequently, Re( T ) E ~l.

=0;

D

3. EVOLUTION IN AN EXTENDED STATE SPACE In this section we pose the original reactor dynamics (1) into an extended state space by considering the heat sources/losses (uncertainty terms) as a new state. Such extension is made only to help technical argumentations (proof of results) and it is not necessary to design the final control feedback. If the reactor rates R(C, T ) were known, Assumption 5 would imply that the state feedback

2225

1990; Kravaris and Palanki, 1988). However, because

R ( C , T ) is poorly known, A H T R ( C , T ) in (4) must be replaced by its estimate. The state feedback (4) is called to as Ideal Feedback because it cancels exactly the nonlinearities of the system. Note that concentration must be available for measurement in order to realize the control (4). Let q(C, T ) = AHTR(C, T ) and 0(C, T ) be an estimate of q(C, T ). The following result states that the chemical reactor (I) can be globally practically stabilized by using the estimate 0(C, T ) in a linearizing-like feedback. Practical stability implies that any initial condition (C0, T0) can be taken arbitrarily close to the desired set point by means of a parametrized control law (Barmish, 1985). Proposition 1 (Alvarez-Ramirez, 1994). The cmmol

Jeedback

T,.)]

u(C,T)=I[-L(TI-O(C,T)-K(T

(5)

is a practical stabilizer jor the system (1). That is, given an3' set ~ C M containing (C*, T~), there evists a gain K > 0 jbr which the control (5) renders this set the stable attractor o f the system (1) within ato' neighborhood 6(Z) C M o[ ~, and moreover any vystem trajecto D, beginning in 5(~) converges to ~. Note that the only restriction on 0(C, T ) is that of being bounded, so that even the controller (5) with O(C, T ) = 0 is a practical stabilizer. Practical stabilizability means that the controller (5) can drive the trajectories of system (1) as closer to (C*, 7;,.t as desired. However, such convergence is attained at the expenses of a high-gain feedback (large values of K). From the practical point of view, a high-gain controller is not desirable because induces large control actions (saturation) and high sensivity to external noise. Of course, the better estimation of q, the better the control performance. As a conclusion, a way of estimating r/ must be provided to the controller. In what follows, a procedure to estimate the uncertainty term q( C, T ) = AHT R( C, T ) via temperature measurements will be provided. The idea is to take r/(C, T ) as a new state variable and extend the system (1) defined on M C ~"+~ to a system on an (n + 1 )-dimensional manifold immersed in a !}?,+2 space. For the sake of simplicity in the presentation, and without loss of generality, assume constant input flow conditions (Cin(t) and Tin(t) are constant). Let ~/~(C, 7", q) = q - AHTR(C, T ). The set M ' = {(C, T, q) E ~I?"~=: c~(C,T,q)-O} is a (n + l)-dimensional manifold ~mmersed in ~"+:. Consider the system

C=F~(C,T)=t)(C~ -C)+ER(C.T) T=F2(C,T,u)=OIT,~-T)+rl+

7(u-'I)

(6a)

(6b)

tl = A H i diag[R2,i(T )]JRI(C)Ft(C. T ) ui(C, T ) --- I-[-L(T ) - AH'rRI(C)R2(T ) ), -K(T-

Tr)]

+AHT Rt(C)JR2(T )F2(C, T,u) (4)

where L(/" ) = 0(T~. - r ) - 7T, stabilizes system (1) in an equilibrium point (C*, T~) (Suarez and Alvarez,

(6c)

where F t ( C , T ) E ~ " , and F2(C,T;u)~!£. JRI(C)C sR~x" is the Jacobian matrix of RI(C) and JR2(T ) = [dR2j(T )/dT] 6 ?£".

J. Alvarez-Ramirez et al.

2226

Proposition 2. The set {(C, T, q) E ~n+2 : (D (C, T~ ~1) = constant} is invariant under the vector field defined by system (6).

Proof: It suffices to show that d~b (C, T, r/)/dt = 0 along the trajectories of system (6), or equivalently O~b. ( ) + OrqS. ~k+ O,~b. ~ = 0. This is automatically satisfied because c ~ q S = - l ; r}=c~b.tff+c~r~b./~. [] Let n : (C, T, r/) --~ (C, T ) be the canonical projection. We have the following result.

Proposition 3. For all input u E ~R, the system (6) has the same solutions as the system (1) module n, if r / ( t = O ) = A H V g l ( C ( t : O ) ) R 2 ( T ( t = O ) ) . That is, if tPt(Co, To, rlo) is a solution to system (6), then no~pt ( Co, To, qo ) is a solution to system (1).

Proof: Integrate the r/equation in system (6) (use the

form (Gauthier and Bonard, 1981 ). So that, system (6) satisfies the uniform observability condition. [] From the above results we conclude that, in principle, q can be reconstructed by means of a state observer. Such observer would be structured as a copy of the system (7) corrected by an error of observation (Luenberger observer). However, this classical observer structure is not possible since the term F3(C, T; u) is not known. Therefore, a problem of state reconstruction with uncertainties must be confronted. This is an important conclusion: the original problem of controlling an uncertain system has been transformed into a problem of controlling a system with nonmeasured states. The following controller is proposed [which is analogous to eq. (5)]:

u(T, O) = I [ - L ( T ) - Sat(i) - K ( T - Tr)]

(8)

7

invariance of ~b(C, T, q)):

q( t ) = AHT RI( C)R2( T ) + 11 where 11 is a constant of integration. The condition r/o = AHTRI(Co)Rz(To) implies that I1 = 0 . When r/(t) is back-substituted in system (6), we obtain the solution to system (1). The solution to system (6) becomes qJt(Co, To, r/o), which is the solution to the upper subsystem in system (6) when r/ is back-substituted. Thus, n o tpt(Co, To, r/0) = t~l,t(Co, To), where ~bl.t(Co, To) is a solution to system (1) with initial condition

where ~ is an estimate of the state r/. The function Sat(t~) is defined as follows: Sat(q)= ~, if ]ql "%

(Co, ~o). ~" = L ( T ) + ~ + 7u(T,~l) - gl(e)(V - T) The above result is important because guarantees that if the initial condition satisfies O(Co, T0, r/0)=0, the solution to the extended system (1) correspond essentially to solution to a system (1). The use of system (6) to design a control law has some advantages over using system (1). The main advantage is that the uncertainty term AHVR(C, T ) appears now as a state variable q, such that techniques of for designing observers for nonlinear systems can be used. Of course, the righthand side (RHS) of the r/-equation is unknown.

(9)

= - g 2 ( g ) ( T - T) where g l ( e ) = - 2 e and g2(e)~-----/32, with e > 0 . Together eqs (8) and (9) define a dynamic feedback controller. The system (9) will be called to as uncertainty reconstructor (UR). The following result will be useful to study the stability properties of the system (1) under the controller (8), (9).

Proposition 5. F3(C, T; u) in eq. (7) is a c6 function of its arguments, and is bounded for all ( C, T ) E M.

4. MAIN RESULTS We state now our main results. Specifically, we are going to provide a dynamic feedback to control the uncertain system (1). The following result establishes the reconstructibility property of the uncertain variable r/ via temperature measurements.

Proposition 4. The subsystem (6a), (6b) is uniformly observable for measured temperature T.

Proof: The subsystem (6b), (6c) can be written in the following form:

T=L(T,t)+q+ ~=F3(C,T;u)

Tu

Proof: Continuity of F3(C, T; u) follows from the continuity properties of FI(C, T ) E ~ " and F2(C, T; u) E ~. On the other hand, IF3(C, T; u)[ < II ~

11[/33max {ki} +/34] < cxD

where NJR1(C)F1(C,T)I[ ~

(7)

where F3(C, T; u) is the RHS of the last equation in system (6). System (7) is written as an observability

The above property was expected to hold since q(C, T ) is bounded (Property 2) and differentiable. For the sake of simplicity in notation, let us consider the

Robust stabilization of temperature P(2)=~~ +(K + 25+~(5-

following change of coordinates: xc = C E N", x~+~ -T - T,. and x,+2 = q. System (1) and controller (8), (9) can be written as follows: x,. = O(Ci, - x~) + ER(x,,,x,+l ) 2n~l

--Xn+l)q-Xn+2 -~-~(U- - X n + l )

-- 0(Tin(T)

.;,,.~2 -

K~(x,,x,,<;u) (lO)

-~.-, = L(-~.I) + x . . 2 4- )'U(Xn+l,Xn~-2) - - , q l ( 8 ) ( X n + l -- Xn+l )

.~.,_. = -.q2(5)(x,,, i

-'~,,+~)

where u(x,,+l ,-r,,+z) - [ - L ( x , < ) - 2 , + 2 + KX,+l]/7. Define the errors el = X,+l -2,,+1 and ez = (x,+e - x,+e)//5. Then kl = 5[-2el + e2]

(ll) & = r,[F3(x, e)/52 - el] where e = (el, e2)V and x = (x,,,x,+j)T. The closed-loop temperature dynamics become -fn+l = ~,e2 -- Kxn+l

2227 1))22

+ (2~:K + c2 + ('(5°)) - K52 + ("(81.

That is, P(2) is the characteristic polynomial of N perturbed by terms of at the most, of the order of 5. For 5 > 0 sufficiently large K + 2~: + (r(5 ~ ) > 0 , and [K + 2c + (~(5-I )][25K + 5";2 4- (' (.F0)] >" Kc 2 4- (!'(c). Therefore, the Routh Hurwitz criteria implies that there exists a 5* > 0 such that for all c > c*, the matrix has all its eigenvalues in the open left-half complex plane. This prove the local stability of eqs ( I 1 ). ( 121 for large enough values of the gain 5. 4.2. Global stabili O' We will not be able to show the global asymptotic stability of the closed-loop system. Instead, we will prove the global practical stability of the closed-loop system. That is, the trajectories of the closed-loop system can be taken arbitrarily as close to the equilibrium point (C*,0,0,0) as desired by tuning the control parameter 5. Integration of eq. i 1 1 ) yields e(t ) -- exp(cN~t)e(O) + exp (5Nit)

( 12 )

×

Note that eqs (12), (11) is a linear system with associated matrix

/[

exp(-5Nls)F(x(s),e(s))ds

where N =

0 0

-25 --5

il

N1 =

which is stable with eigenvalues { - K , - 5 , - 5 } , perturbed by a bounded term F3(x,e). As a consequence of this observation, we have the following conclusion.

+ ,

f,

al/5

--5+a2/~

a3/~

1

where al =dF3/dx,+l(O), a2 =dF3/del(O) and a3 =dF3/ de2(0). The characteristic polynomial of N- is given by P(2) - 23 + (K + 25 - a2/5122 + (2oK - 3a2 - a2K/5 - m + 5)2 + (3ka2 + 25ai - ks)

F3(x. e ) ~:

II exp(sNl(t - s)F(x(s),e(s)l[I ds.

0

Since N~ is a stable matrix, there exists constants 1~, 2>0 such that [pexp(cN, t ) e l l ~ e x p ( - 5 2 t ) l l e l l . Therefore, Ilelp ~< ~ expO:),t)pMo)[I c).,!Z'exp(-,~Z*)[[F(x(s).e(s))[[-- ds.

But F ( x , e ) is bounded (Proposition 5) by a constant so that

fis,

][e]l ~<~ exp(c2t)He(O)[] +

I:°

E°I

Ilell ~ rl exp(sN~t)e(O)[I

+Yl exp( 4. I. Local stability The local stability of eqs (11), (12) in the equilibrium point (0,0,0) is given by the eigenvalues of the matrix

F(x, e)

Using the Triangle and Schwartz inequalities, we get

Proposition 6. The controller (8),(9) is internalh' stable. That is, all the states (x,+t, el, e2) are bounded The proof of the above result is straightforward and is a direct consequence of the fact that all the trajectories of an asymptotically stable linear system with bounded perturbations are bounded (Jordan and Smith, 19771.

f 1 -2 - 1 +1 0

x

['

fl~ exp ( -~:2t ) E

exp ( - 5 2 s ) ds

Ilell-~ lexpO,/ot)Ile(0)l[-~-j

(13)

+ ,:_~.

The above inequality shows that the error e(t) is uniformly bounded• On the other hand, eq. (12) can be written as follows:

J. Alvarez-Ramirez et aL

2228

where C = (0, 1). As above, integration of this equation yields Xn+l ~-

exp (-Kt)x,+l (0)

Remark 1. From Theorem 1, we conclude that since

x~+l(t) is uniformly bounded then the controller (8), (9)

+ e exp ( - K t ) --~o"t exp (Ks )Ce(s ) ds

yields an internally stable closed-loop system. Besides, if the temperature converges exponentially to a ball of finite radius r, Assumption 5 implies that the concentrations should be bounded and converge to a region around the desired operating point. By defining the deviation vector ~ = C - C*, the concentration equation in eq. (1) can be written as

from where [Ix,+~ II ~< exp(-gt)lX,+l(O)l-b/3fl6 e x p ( - K t ) ×

are practically stable according to the notion of stability given in Barmish et al. (1982).

exp (gs)lle(s)l[ ds.

= -0~, + ER(~, T(t)) - ER(C*, T* ).

Using the estimate in eq. (13),

Since the last two terms on the right-hand side are bounded (Property 2), it is obvious that the concentrations converge exponentially to neighborhood of the operating point (C*, T*) with a rate of the order of 0 (the inverse of the residence time).

[IXn+l I1 ~< exp (-gt)lx~+,(O)l + ~ eft6 exp ( - Kt)

exp (Ks)

x{llexp(-e2(s)(lleol]-~-~2 ) 5. AN ILLUSTRATIVE EXAMPLE

+851 ~22 j ds ILx.+~ II ~< exp

(-gt)lx.+l(O)l +

~1~6

(

le(0)ll -

x [exp ( - e 2 t ) - exp ( - K t ) ] + L~lfl6fls [1 - exp ( - K t ) ] . e2K

(14)

Inequality (14) shows that xn+l(t) is uniformly bounded. So that, we conclude that the controller (8), (9) yields an internally stable closed-loop system. Also, xn+l(t) converges exponentially to a ball B(r) with radius r=(~lflsfl6)/(~;)LK). ~l,fl6 and 2 are around 1.0, so that r ~ fls/eK, r can be interpreted as steady-state error, which is proportional to the bound f15 of the uncertainty and to the inverse product of control gain K and uncertainty reconstructor gain ~. r can be made arbitrarily small by taking large values of ~K. Increasing the control gain K leads to control degradation such as saturation. However, the alternative of increasing e can be taken without serious problem of input degradation. The above discussion and results can be summarized in the next theorem. Theorem 1. The parametrized control (with parameter e.) (8), (9) is a global practical stabilizer for system (1). That is', control (8), (9) takes the closed-loop

trajectories arbitrarily close to the set point ( C*, Tr). In certain sense, the above-established result states a Separation Principle for the stabilization of system (1) via the uncertainty reconstructor (9). One merely construct an uncertainty reconstructor like system (9) with large enough gain and then connect it to the system. Theorem 1 assures that such connected systems

We consider a CSTR with a simple chemical reaction which was also considered in Alvarez-Ramirez (1994), where it has been shown that robust control as in Corless and Leitmann (1981) can be applied to this system if an upper bound of the uncertainty is available. We assume here that no upper bound of the reaction rate is available and make use of the feedback controller with uncertainty estimator (8), (9). The model of the chemical reactor is as in eq. (1) with reaction rate

R( C, T) = Ck exp (-c~/T ) with k = exp(25) and ~ = 10,000. Moreover, 0 = 1.0, ~,= 1.0, A H = 200.0, Cin = 1.0 and T~o = 350.0. Under these parameters, system (1) satisfies Assumptions 15, so that a control like eqs (8),(9) can stabilize the chemical reactor (in the sense of practical stability). The system has three equilibrium points, one of them being unstable and located in (0.5, 400.0). For numerical simulation, we take K ~ ~ (no saturation function in eq. (8)), and the reference temperature Tr = 400.0. For moderated and small values of e, the trajectories of the closed-loop system converges globally to a limit cycle. The 'size' of such limit cycle can be seen as a measurement of the steady-state error. According to eq. (14), the size of the limit cycle must decrease as the parameter e becomes larger. This situation is illustrated in Fig. 1 where the limit cycle is shown for various values of e. Numerical simulations show that for ~> 15.0, the trajectories of the closed-loop system converge to the equilibrium point (0.5,400.0, 100.0). Figure 2 shows the system performance for e = 50.0 and K = 1.0. It can be observed that the reactor temperature converges monotonically to the target value as a first-order linear system. This behavior is due to the almost cancellation of nonlinearities carried out by the feedback controller with uncertainty estimator (8), (9). The control histories are shown in Fig. 3. The actual

Robust stabilization of temperature

2229

600 ,

~

500

1

=

....

0.0

:-..~,.

:/:." ~::.:.

-..,

I

I

I

0.3

0.5

0.8

1.0

Concentration, C Fig. 1. Globally attracting cycle for various values of ~:.

500-

od 2 400 F--

300 0.00

I

I

I

0.25

0.50

0.75

Concentrotion, C Fig. 2. System performance for e = 50.0 and K = 1.0.

450

'Y"

400

g

350

-~

300

-E (.~

250 200 0

1

m 3

2

Time Fig. 3. Control (8), (9) histories.

1.00

J. Alvarez-Ramirez et al.

2230 125

100-

75^

50-

25-

i.. 0

I 1

I 2

I 3

I 4

Time Fig. 4. Actual uncertainty r/ and estimated uncertainty ~ histories.

uncertainty xn+2 and the estimated uncertainty J~+2 are compared in Fig. 4. After a short time, ~,+2 converges practically to xn+2. 6. CONCLUSIONS A robust control has been constructed for a class of CSTR with unknown reaction rates (time-varying uncertainty). The control is designed in a deterministic manner in the sense that it does not rely on any statistical property of the uncertainty. Moreover, the bound on the uncertainty is unknown. For control design, no a priori knowledge of the reaction rate is required. A numerical example illustrating how the control scheme works has been provided. REFERENCES

Alvarez-Ramirez, J. (1994) Stability of a class of uncertain continuous stirred chemical reactors with a nonlinear feedback. Chem. Engng Sci. 49, 1743-1748. Barmish, B. R. (1985) Necessary and sufficient conditions for quadratic stabilizability of an uncertain system. J. Optim. Theory Appl. 46, 399-408. Barmish, B. R., Corless, M. and Leitmann, G. (1983) A new class of stabilizing controllers for uncertain dynamical systems. SIAM J. Control Optim. 21, 246-255. Caldwell, J. M and Dearwater, J. G. (1991) Model predictive control applied to FCC units. In Chemical Process C o n t r o l - CPCIV, eds Y. Arkun and W. H. Ray (CACHE-AICHE, 1991), pp. 319-334. Chen, Y. H. and Leitmann, G. (1987) Robustness of uncertain systems in the absence of matching assumptions. Int. J. Control 45, 1526-1542. Corless, M. and Leitmann, G. (1981) Continuous state feedback Guaranteeign uniform ultimate boundedness for uncertian dynamical systems. IEEE Trans. Automat. Control 26, 1139-1144.

Esfandiari, F. and Khalil, H. K. (1992) Output feedback stabilization of fully linearizable systems. Int. J. Control 56, 1007-1037. Feinberg, M. (1990) Chemical reaction network structure and the stability of complex isothermal reactors - - I. The deficiency zero and the deficiency one theorems. Chem. Engng Sci. 42, 2229-2268. Gauthier, J. P. and Bonard, G. (1981) Observability for any u(t) of a class of nonlinear systems. IEEE Trans. Automat. Control 26, 922-926. Gavalas, G. R. (1968) Nonlinear Differential Equations of Chemical Reaction Systems. Springer, New York, U.S.A. Isidori, A. (1989) Nonlinear Control Systems. Springer, New York. Jordan, D. W. and Smith, P. (1977) Nonlinear Ordinary Differential Equations. Oxford University Press, Oxford, U.K. Kravaris, C. and Palanki, S. (1988) Robust nonlinear state feedback under unstructured uncertainty. A.LCh.E.J. 7, 1119-1127. Leitmann, G. (1979) Guaranteed symptotic stability for some linear systems with bounded uncertainties. Trans. ASME J. Dyn. Sytems Measurements Control 101, 212-216. Petersen, I. R. (1987) Disturbance attenuation and H ~ optimization: a design method based on the algebraic Riccati equation. IEEE Trans. Automat. Control 32, 427-429. Schuler, H. and Schmidt, C.-U. (1992) Calorimetricstate estimators for chemical reactor diagnosis and control: review de methods and applications. Chem. Engng Sci. 47, 899-915. Suarez, R. and Alvarez, J. (1990) Nonlinear control of isothermical chemical reactors. An approach based on a geometric characterization which incorporates stoichiometric and kinetics. Proceedings of the American Control Conference, pp. 593-598.