Nonlinear adaptive observation of an exothermic stirred-tank reactor

Nonlinear adaptive observation of an exothermic stirred-tank reactor

Chemcal6ngm~rmg Scrence, “al. 46, No 3, pp. 793-805, KQ-2509/91 S3.D3+ 0.00 0 1991Pergamon Press plc 1991. Pnnted III Great Bntam. NONLINEAR A...

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Chemcal6ngm~rmg

Scrence,

“al.

46, No

3, pp. 793-805,

KQ-2509/91 S3.D3+ 0.00 0 1991Pergamon Press plc

1991.

Pnnted III Great Bntam.

NONLINEAR ADAPTIVE OBSERVATION OF AN EXOTHERMIC STIRRED-TANK REACTOR LULU LIMQUECO, JEFFREY C. KANTOR’ and STUART HARVEY Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46554, U.S.A. (Receiued 31 October

1989; accepted

3 April 1990)

Abstract-Nonlinear control of an exothermic stirred-tank reactor has been previously demonstrated using a number of techniques, including nonlinear static state feedback and output feedback using observers. This paper considers the task of adaptive observation. It is shown that an adaptive observer for concentration can be constructed for an arbitrary order reaction system when only temperature measurements are available. An adaptive observer is constructed to identify the pre-exponential Arrhenius constant and to provide on-line estimation of the unmeasured reactant concentration for a global nonlinear state-feedback controller. The reaction system 1s transformed into a time-varying adaptive observer canonical form (AOCF) which is linear in the unknown observer states and adaptive parameters. The AOCF was proposed by Bastin and Gevers, who have provided sufficient conditions for its stability. It is shown that the sufficient conditions on the structure of the system and on the signals for global stability of the observer are satisfied by the reaction system, thereby admitting the construction of the adaptive observer for an arbitrary order reaction. Proper choice of observer gains is crucial for an acceptable rate of convergence. Simulations show that the combined observer/controller provides satisfactory closed-loop behavior when the observer time constant is approximately 4 times smaller than the closed-loop time constant. It is also shown why certain transformations result in better adaptive observers.

I. INTRODUCTION The nonlinear control of an exothermic stirred-tank reactor has been previously described using a number of related techniques. Hoo and Kantor (1985) demon-

strated nonlinear control using nonlinear static state feedback, assuming that both concentration and temperature measurements are available. Limqueco and Kantor (1990) subsequently demonstrated output feedback control of the same reactor using a nonlinear observer to estimate concentration when only temperature measurements are available. Because of the stringent requirements for the existence of the nonlinear observer, the application was limited to the case of a first-order irreversible reaction. This paper now reports the construction of an adaptive observer suitable for arbitrary reaction order. The adaptive observer is used to identify the preexponential Arrhenius constant and to provide an online estimate of the unmeasured reactant concentration for a nonlinear state-feedback controller obtained by exact feedback linearization. The reaction system is transformed into a time-varying adaptive observer unknown

canonical observer

form (AOCF) which is Iinear in the states and adaptive parameters.

The AOCF was proposed by Bastin and Gevers (19881, who have provided sufficient conditions for stability and parameter convergence of the resulting observer. The sufficient conditions for the structure of the system and the signals for global stability of the observer are satisfied by the reaction system, thereby admitting the construction of the adaptive observer. Below, we discuss how to choose the gains of the adaptive observer to provide an acceptable rate of tAutho

to whom correspondence

should

be addressed.

convergence. We also demonstrate why a particular form of transformation results in an adaptive observer with better performance than other possible transformations. In the next section, we give an overview of the adaptive observer proposed by Bastin and Gevers (1988). Application to an nth-order irreversible model of the exothermic stirred-tank reactor is then discussed in Section 3. Readers interested primarily in the application of the observer may wish to read Section 3 first. Finally, we give some concluding remarks. 2. THEORY

The structure of an adaptive observer Consider the single-input single-output

2.1.

nonlinear

system S(r) =I(4

u, PI

y(r) = XI

(1)

where X(C)E% is the state vector, u( tj~ 3 is a measurable input, yea is a measurable output, and p(t) ES is the vector of unknown, bounded, possibly time-varying parameters. The nonlinear system can be transformed by a smooth time-invariant, nonlinear transformation

< [I =T(x, 0

from

c2,.

(x, p) to (<,8)

p, c*, .

parametrized

, c,) T(.)E9’“+m by n -

(21

I parameters

, c, into the AOCF 5’(t) = R<(t) +

Y(t)= r,(r)

QC4t)lW) + s(t) (3)

LULU LIMQUECO etni.

798

where c(t) E 9” is a state vector of the same dimension as x(t), Q(t) E W’ is a vector of unknown time-varying parameters which will be estimated on-line, o(t) E&Y is a vector of known functions of u(t) and v(t), a[(~,( t)] is an n x m matrix whose elements are all of the form nij[w(t)] = asw(r) for known constant, possibly zero, vectors gij E E, and g(t) E @ is a vector of known functions of time. R is a known constant n x n matrix of the following form:

I

kT

The observer mechanism is outlined in Fig. 1. The uppermost block is the process which gives the output p(t) when subject to input u(t). The block below the process serves as a measurement transformation that generates w(t) for the adaptive observer. The adaptive observer comprises three blocks: a measurement filter which generates V(t) and q(t) based on the measurement information contained in w(t); an adaptation subsystem which uses q(t) and y(t) to estimate the parameter 6(r); and an observer subsystem that uses inputs w(t), v(t), 8(r) and j(t) to produce the estimated output P(L) and the estimated transformed states f(t). If the inverse transformation exists for eq. (2). then the estimated states _? and parameters j can be solved from the algebraic

(4) where k,, . , k, are known constants and F(c,, , c,) is an (n - 1) x (n - 1) constant matrix whose eigenvalues can be freely assigned by a proper choice of the constant design parameters c2,. . , c,. Typically,F=diag(-cc,,..., -c,)withci>O. The AOCF (3) is linear in the unknown quantities t(t) and Q(t), while n[w(t)] is a known nonlinear or time-varying function of u(t) and V(I). The following adaptive observer can be applied to systems given in AOCF form. A state estimate is generated by the system i(t) = R&t)

+ [

D(1)=

+ n[w(r)]&t)

+ g(r)

f,(r),J(Z) =

B(t)= rq(t)j(t)

rp(f) = VT(t)k + n:[o(r)]

is introduced for analyzing stability. the following error system is obtained by using eqs (5)-(S):

t* [II

R*

(7) (8)

In terms of p, for the observer

VT

---I 0

--------

0

-I-q0

o...o -v 1,

I

s

(11)

where

(6)

V(O)= 0

An aux-

I;$1

F*=f-

(5)

where j(t) is the output error, that is, the difference between the actual and estimated output; and r is a positive definite matrix normally chosen such that I- = diag(y,, _ , y,) with yi > 0. q(t) is the so-called “regressor” vector, containing measurement information on process inputs and outputs. In addition, an auxiliary filter is introduced to generate the regressor vector q(t) from actual measurements. The filter equations are given by

e^are the state

and the parameter error vectors, respectively. iliary error vector

L

Y(f) - 3(f)

+ Q[o(r)],

af the adaptive ohseruer The variables F= < - Sand $ = B -

B =

where c1 is a positive constant, and c2,. . . ,c, are chosen such that the eigenvalues of F(c, , . . , cn) are in the open left-half plane. Variables marked by the circumflex denote estimated quantities. Parameter estimates are given by

3(t) = W(t)

2.2. Stahi&y

L

~b%]

equation.

Oi

i_------_1

..

1 I m,, . . . CJ

0

i

-cl ___ R*=

1

1

kT

(W

I

Bastin and Gevers (1988) presented the following theorem that provides sufficient conditions which guarantee that the homogeneous part of the error system in eq. (11) is exponentially asymptotically stable, that the error system is BIBO-stable, and that 1 and 6 are bounded if 6 is bounded. Theorem 1. If

where

a=

; [

1

(9)

V(t) is an (n - 1) x m matrix, and p(t) is an m-vector.

(i) c1 > 0 and c2, _ . , c, are chosen such that Ret,&(F)] < 0 Vi (ii) q( t)~ S, , where S, is any set of signals r(f) such that

Nonlinear adaptive observation of an exothermic stirred-tank reactor Process

w )

799

y(t) )

i(t) = f(x) + ” g(x: YW=Nx) L

I

I

I

Measwement

Transformation )

w(t) = functions of 4 u(t) ad ~0) W(t)

W

MC3SUIWIE~t Fiiter ir(t) = F V(t) + [email protected]&,, cp(t) = VT(t)k + fl:(o$t))

VW

v v Adnptmion Subsystem

r

Observer Subsystem h

e(t) = R &I + Wtit))

60)

&I) = r q?(l) F(I)

Inverse Transfomdon

Adam&e Observer

Fig. 1. Observer mechanism.

(1) r(c) is bounded Vr > 0; (2) i(t) is bounded Vt > 0 except possibly at a countable number of points (ti} such that minlti - tjl > A > 0 for some arbitrary fixed A. (iii) there exist a positive that Vt > 0

constants

a and T such

f+T

O-cal<

d~)cp=(~)d~

i (iv) there exists a positive

constant

(13)

In order to transfer the stability conditions to the system, they gave another theorem which provides a structural condition on the AOCF that guarantees the output reachability of the auxiliary filter. This ensures that cp(t) is persistently exciting when CD(~)is sufficiently rich. Theorem 2. The auxiliary filter (7) and (8) is output reachable from w(t) if and only if the m x m matrix

SCdt)l Sl

M, such that

Vt > 0

S[w(t)]

IY(

6 M, < @I

(I) le(t)l G K1140)l

Vt 2 0

+ KZ

(2) lim sup le(t)l < K, M,

(Ifi)

i-m

where

e(t)

1

= at, 6(t) [

:

(17)

I

(18)

where s1 = Q 1’ ,y. I = kTFj-Z n-

(15)

[w(t)1

[ s,Cdt)l

(14)

K, , K, and K, such

then there exist finite constants that

=

j=2,.

. ,m

(19)

has full column rank over W, i.e. if there exists no constant m-vector p # 0 such that S[w(t)]B = 0. Theorems 1 and 2 are used to obtain the following conditions on the system, the transformation T, and the signals which guarantee the global stability of the adaptive observer. Therefore, if these conditions are satisfied, then a stable adaptive observer exists for the system as stated in Theorem 3.

LULUhdQUEC0 et cd.

800

Structural conditions (on the system and the transformation T): S.l: The system (1) is BIB0 stable. S.2: The system and the transformation Tare such that s.2.1: the elements of w( t) are bounded functions of u(t) and v(t) s.2.2: R and n in (3) make S[w(t)] in (18) of full column rank over W (i.e. the auxiliary filter (7) and (8) is output reachable). S.3: The parameter variation d(t) formation Tare such that

and the trans-

(20)

Conditions on the signals: SI.l: t((t)cS& for some A > 0. SI.2: There exist positive constants that

y and T such

r#~ =;

2 yl > 0

Vt > 0

.=$t

x2

=

-

cr

x1

=

T - T,-o

-Y

Tfo

2 = Bqbx”zK(x,)

(21)

. w”“(Z)]

(24)

where u is the measured process input, x, is the dimensionless temperature, and x2 is the dimensionless reactant concentration. The resulting reduced model is given by

dr

[coT(?)cu

Q

e,

u=Y[6(Tc-Tf,)+q(T,-Tf,)] Tfo

dx

b)l

q=-

k,C;-‘e-Y

-2 = - $x;K(x,)

WT(r) = cs +

PC,Q,

E ’ = RTrO

where Q, and T,, are nominal or characteristic values of the volumetric tlowrate and feed temperature. The corresponding dimensionless variables are defined by

with I

UA

0

I+T W(z)Wr(r)dr

6=;

PCJ,,

c

for all t 2 0.

[email protected])l < M, < ~0

8=I:-Aff)Cfy

- (4 + 6)x, + u + q(1 -x2)

where K(x,) is the dimensionless

(26)

function

where 6 > 0 but otherwise arbitrary, and q is the number of elements of F’(t) which are not identically

zero. Theorem 3. If conditions S-S.3 and SI.i and SI.2 are satisfied, and if the design parameters c, , . . , c, are chosen such that c, > 0 and Re{l,[F(c,, I . . , c,)]} < 0 Vi, then there exist positive constants K, and K, such that (15) is satisfied. 3. APPLICATION

TO AN EXOTHERMIC

3.2. Consrruction of an cldaptirle nhertrer

We have found that applying the following transformation Tl to the dimensionless reaction system (26) generalized for an arbitrary reactor order:

STIRRED-TANK

REACTOR

of an exothermic stirred-tank reactor A model for an exothermic irreversible reaction is a stirred-tank reactor given by 3.1. Model

dC dt= dT

z-

reduces the reactor model to a dimensionless in AOCF form:

- k,C”exp

system

(-AH) ------k,C”exp pc, + g$-

(29)

T) P

where C and Tare reactant concentration and reactor temperature, respectively. The exponent n is the order of the reaction; the remaining notations are defined in the nomenclature section. The feed concentration C, is assumed to be known and constant. The concentration C and pre-exponential Arrhenius constant k, are both assumed to be unknown. It is assumed that only temperature measurements are available. Following the analysis of Uppal et al. (1974), the model is made dimensionless by introducing the parameters

Y =

51.

Note that estimates of x1 and 4 can be recovered from on-line estimates of t2 and 0 since the transformation (28) is invertible as follows:

(30)

Nonlinear

adaptive observation of an exotherrnic stirred-tank reactor

Thus the state and parameter estimates can be found explicitly from estimates of transformed states and parameters without resorting to numerical procedures. We next show that the reaction system satisfies the structural conditions for the stability of the adaptive observer. The system (26) will be operated near a BIBO-stable operating point. The system and the transformation T, are such that S[o(t)] has full coiumn rank and the elements of o(t) are bounded functions of y(t). Note that in this case where only one parameter is being identified, S[w(t)] is a column vector, thereby always satisfying the full column rank condition. The parameter 4 varies only as the reactor catalyst concentration changes; therefore lb(t)1 d M, < 00 for t 2 0. The conditions on the signals are also met since u(~)E S, for some A z 0. Instead of proving condition (21), we directly prove that condition (13) is satisfied. The following expression is obtained from eqs (7) and (8): cp = e-cl,

(4 - %)BY(t)dt

+ BY(~).

(311

1 It satisfies eq. (13) for fj9# 0. Since the structural conditions on the reaction system and the transformation, and the conditions on the signals are all satisfied, then the following adaptive observer can be used for the system

u - (4 + 4Y Bs+U fq-cc,)@---dyl_ 1

&, = rcp(t)jqt)

(32)

where P(t) = 4, (t) F(t) = Y(t) - P(t) P(t) = - cz V(t) + (4 - C2)BY cp(t) = v(t) + PY.

(33)

We choose the gains cl, c2 and r so that the adaptive observer converges at a reasonable rate. To accomplish that, we look at the eigenvalues of the Jacobian of the adaptive observer and the auxiliary filter. Solving for the Jacobian of eqs (32) and (33) and its eigenvalues results in

For the observer subsystem to be critically is desirable to choose the gains so that G Critical damping zero when * fi dynamics

- 2qpy,.

> 2q/?y,, or a highly

4-

2

close to

underdamped

< 2yj3y5.

L,h = 0 L&h

# 0.

(36)

The choice of h = x2 satisfies eqs (36) and yields the following transformed state variables for n = 2: 2L=h(x,.x,)=x, =

-

#x:

control

K(q)

+ q(1 - x*)

(37)

law defined by

k,,(r - 2,) - k,,?, V,h

(34j

(35)

3.3. Nonlinear output feedback controller To achieve nonlinear output feedback control, we combine the adaptive observer and the nonlinear state-feedback controller constructed using exact linearization which Hoo and Kantor (1985) have successfully applied to a similar reactor model. Regarding u in eq. (23) as a control variable, for exact linearization there must exist a scalar function h(x,, x2) such that (Su, 1982)

t4=

-Jzqg

it

A typical simulation of the adaptive observer for a second-order reaction system with the parameters q=I,y=20,~=0.11,/?=7.0and6=0.5isshown in Fig. 2. For the gains r = 24.5083, c1 = 25 and ca = 15, the eigenvalues of the observer are ~ 15, - 15 and - 12.5 for xzs = 0.3. Initial estimates x2 = 0.8 and 4 = 0.2 are used when the actual values are x2 = 0.1 and q5 = 0.11. Both the concentration and the parameter 4 approach the actual values asymptotically and without bias. We know that if there is no initial error in the estimates, i.e. 5, = 5, at t,, then the observer should yield [ = 5 for all t > tO. This is not, however, true in the case of the adaptive observer because of the parameter adaptation. o^is obtained using eq. (32) which depends on the error between the actual and the estim_ated output, while 0 is obtained using eq. (28). For <, = c, # <,-and Q0 = 0, # @,at t,, the next cstimate of &is still 8, while 0 # 0, since the system is not at steady state. This is illustrated in Fig. 3 when a step disturbance is introduced to the input after the system has reached steady state, thereby changing r, and 0, for the new input.

with a feedback

_ -c*

damped,

avoids having an eigenvalue

when s

a2 = L,h

i

801

-

Ljh

(38)

where r is a setpoint for P, . The constants k,, and k,, are chosen for desirable closed-loop dynamics subject to input rate and limit constraints.

802

LULU

et al.

LIMQUECO

outlettemperature

Reactor 6.0

(a)

55

/

5.0 4.5 40 3.5

3.0 2.s 2.0 1.5 I .o

L

0

1

12

1

3

4

I1

5

Dimensionless

Actual and estimated

6

7

8

2

91

1

I

I

1

I

I

i

4

6

8

10

12

14

16

Dimensionless

time

concentration

18

Actual and estimated concentration

(b) i

(b)

I

0

2

3

L-U 4

5

6

7

:3

tnne

1 8

9

10

Estimated phi

Estimated phi

2.5

0

I

2

3

4

5

6

7

8

9

10

0

2

4

6

Dimensionless time

8

IO

12

14

16

18

20

Dimensmnless tome

Fig. 2. (a) Performance of the adaptive observer for u = 3.2159 with I- = 24.5083, c1 = 25 and c2 = 15. (b)Actual (solid) and estimated (dashed) concentration. The adaptive observer starts with an estimate of0.8 when it is actually 0.1. (cl Adapted parameter 4 when the actual value 1s 0.11. It starts with an initial guess of 0.2

Fig. 3. (a) Performance of the adaptive observer when a step change in the input temperature to u = 4.2159 was introduced at t = 10. The observer parameters are the same as those in the previous example. (b) Actual (solid) and estimated (dashed) concentration. (c) Adapted parameter d when the actual value is 0.11.

Unlike the linear case, there does aration principle for the nonlinear

controller dynamics

not exist a sepsystem. Conse-

quently, the designs of the observer and the controller are coupled. It is necessary that the observer and the

gains are chosen such that the observer are faster than the controller dynamics. For

our reactor example, yield time constants

the control gains are chosen to approximately 4 times longer

Nonlinear

adaptive

observation

of an exothermic

stirred-tank

reactor

803

feedback controller can provide effective feedback control over a wide range of operating conditions. A typical result is shown in Fig. 4 for bounded control action ( - 2 < u < 6). The controller gains are kc, = 9 and k,, = 6, and the observer gains are the same as before. A lower bound of 0.03 is also imposed on the parameter 4 to prevent L,L,h from becoming zero when 4 is very close to zero.

5

t

I

6

7

c-e10

3.4. Comments on transformation to AOCF The transformation of a given model to AOCF is not unique. The performance of the adaptive observer depends on the choice of transformation. We shall demonstrate this phenomenon for the exothermic reaction system and explain the practical significance. A transformation T2 which has the same
e=

Actual and estimated concentration W

results

Px* + (1 + 4 - cz)x, PKCY)

qh; -

in the AOCF

24-

+

(4 + 4Y

P9 + (1 + 9 -

cz)@

- dy)

1 (40)

y=51. The adaptive

0.11 0

1

1

I

1

I

J

I

I

2

3

4

5

6

7

observer

is then given by

8

Dimensionless time

+ Estimated

(39)

u - f9 + 4Y 84 + (1 + 9 - cz)(u - dy) 1

phi

0.7 ,

h = rcp(t)F(t)

(41)

where P(t) = El(t) F(t) = Y(t) - Q(t)

V(t) = Vt) + B K(Y).

-0.ll1 0

1

2

I

4

I

I

I

I

I

3

4

5

6

7

8

9

I IO

Dimensionless time Fig. 4. (a) Performance of the nonlinear output feedback controller for r = 24.5083, c1 = 25, c2 = 15, k,, = 9 and k,, = 6. (b) Actual (solid) and estimated (dashed) concentration. The adaptive observer starts with an estimate of 0.8 when it is actually 0.1. (c) Adapted parameter 4 when the actual value is 0.11. It starts with an initial guess of 0.2.

than the observer time constants. This allows the observer states to converge to the actual values before the system reaches steady state. Simulations demonstrate that the nonlinear output

142)

This adaptive observer is the same as the one described by eqs (32) and (33) except for fi[w( t)] and the auxiliary filter. A typical simulation of this adaptive observer with the same reactor parameters is shown in Fig. 5 for r = 50, c1 = 40 and cZ = 20. The eigenvalues of the Jacobian of the adaptive observer and the auxiliary filter are - 10, - 10, - 12.746 and - 27.254 for xZs = 0.65. The performance of the adaptive observer is acceptable. However, when it was used together with the controller for k,, = 9 and k,, = 6, bad estimates were obtained and it wduld not converge. Poor performance results from the use of this observer in feedback control (Fig. 6). As mentioned earlier, the only difference between the two AOCFs is the function Q[o( c)] which results

804

LULU EAMQUECO ez a/. Reactor

Reactor outlet temperature 2.8

outlettemperature

2.”

(a

(a)

2.6 -

I .6 1.4 1.2

0.8 owls

4

S

6

7

8

9

/

I .o 1

0

10

2

LA..-li 3 4

7

6

9

8

Dimensionless time

Dimensionless time

Actual and estimated

I

5

Actual and estimated concentration

concentration

1

(b)

‘.O1

6 (b) 5

Y

0

2

k

6

8

IO

12

Dimensionless

Estimated

14

16

18

20

0

I

I

1

2

I1

3

time

4

6

Dimensionless

phi

Estimared

I

I

1,

S

8

7

9

10

time

phi

0.2

Cc)

(c) 0.15 -

0.05 0

-0.05

i 2

4

6

8

10

Dimensionless

12

14

16

18

20

time

observer that resulted from transformation T2 for u = 0.9186 and I- = 50, c1 = 40 and c2 = 20. (b) Actual (solid) and estimated (dashed) con-

centration. The adaptive observer starts with an estimate of 0.9 when it is actually 0.2. (c) Adapted parameter I$ when the value is 0.11. It starts with an initial

I 12

34

, 5 Dimensionless

Fig. 5. (a) Performance of the adaptive

actual

-0.1 I 0

guess of 0.2.

in different V and cp since they are both functions of f2[w(t)]. It is not hard to see that the large magnitude of L?[w(t)] magnifies the error in 0, and, as a conse-

6

1 7

1 8

1 91

time

Fig. 6. (a) Performance of the nonlinear output feedback controller using transformation T2 for I- = 50, c, = 40, c2 = 20, k,, = 9 and k,, = 6. (b) Actual (solid) and estimated (dashed) concentration. The adaptive observer starts with an estimate of0.9 when it is actually 0.2. (c) Adapted parameter C#J when the actual value is 0.1 I. It starts with an initial guess of 0.2. quence, poor estimates of 5 are obtained. Besides, the magnitude of both Y and cp will also be large, which makes the problem even worse. R[w(t)lr, is a func-

Nonlinear

adaptive

observation of an exothermic stirred-tank reactor

805

tion of Y while fl[o(t)],, is a function of K(Y). The first observer performs better since Y ( K(y) makes it less sensitive to the error in $ Since the problem is related to the parameter ad-

t T T, T,

time, s reacTor temperature, assumed measurable, K coolant temperature, assumed measurable, K feed temperature, assumed measurable, K

aptation, the following equation (Sastry and Bodson, 1989) is used to slow down the rate of parameter

u U

dimensionless process input heat transfer coefficient, J/s/K/m2

V

reactor volume, assumed constant, m3 dimensionless temperature [y( T - T,, )/ T,-.] dimensionless reactant concentration (C/C,) measured variable: dimensionless temperature

adaptation:

B(t) =

x1

I-cp(t)Rt)

(43)

1 + cp(t)cp(t)’

Using this equation instead of (41) does not solve the problem. Slower parameter adaptation rate does not result in better state and parameter estimates fed to the state-feedback controller.

4. CONCLUSION

We have shown that an adaptive observer can be constructed for the nth-order reaction system and how the observer gains should be chosen for acceptable convergence rate. It is also demonstrated that the adaptive observer can be used with the nonlinear state feedback controller to form the nonlinear output feedback controller that performs well in a wide range of operating conditions. It should be noted that this method is applicable to a large class of nonlinear systems since the transformation can be functions of the unmeasured states. Finally, we have shown that by choosing a transformation where Q[w(t)] has elements with smaller magnitude results in an adaptive

observer

that performs

better.

NOTATION

A c Cf c, E AH k(T) k, 4 R

heat exchange surface area, m2 reactant concentration, mol/m3 feed concentration, mol/m3 specific heat capacity, J/kg/K activation energy, J/mol heat of reaction, assumed constant, Arrhenius rate law, l/s Arrhenius pre-exponential constant volumetric flowrate, m’/s gas constant, J/mol/K

x2

Y

J/mol

EY(T-

~,,)/~,,I

Greek Ietters dimensionless dimensionless dimensionless

heat of reaction heat transfer coefficient activation energy

B 6 :

Damkahler

V 0 P r W r

regressor vector transformed parameter reactant density, kg/m3 dimensionless time functions of u(t) and y(t) transformed state

number

Sub- and superscripts feed condition f initial condition 0 REFERENCES

Bastin, G. and Gevers, M. R., 1988, Stable adaptive observers for nonlinear time-varying systems. IEEE Trans. autom. Control 33, 650-657. Hoo, K. and Kantor, J. C., 1985, An exothermic continuous stirred tank reactor is feedback equivalent to a linear system. Chem. Engng Commun. l-10. Limqueco, L. and Kantor, J. C., 1990, Nonlinear output feedback control of an exothermic reactor. Comput. them. Engng (accepted). Sastry, S. and Bodson, M., 1989, Adaptive Conrrol, Stability, Comergenre, and Robustness. Prentice-Hall, Englewood Cliffs, NJ. Su, R., 1982, On the linear equivalents of nonlinear systems. System Control I&t. 2, 48 52. Uppal, A., Ray, W. H. and Poore, A. B., 1974, On the dynamic behavior of continuous stirred tank reactors. Chem. Engng Sci. 29, 967-985.