On a practical criterion for optimal sensor configuration - application to a fixed-bed reactor

ON A PRACTICAL CRITERION FOR OPTIMAL SENSOR CONFIG...

14th World Congress of IFAC

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ON A PRACTICAL CRITERION FOR OPTIMAL SENSOR CONFIGURA TION - APPLICATION TO A FIXED-BED REACTOR

A.

'7aode

Wou,,~er, N.

Point, S. Porteman, M. Remy

Laboratoire d'Autoffuttique, Faculte Polytechnique de Mons, 31 Boulevard Dolez, 7000 Mons) Belgium E-fnail: [email protected]. be

Abstract: This paper considers the problem of sensor configuration for parameter and/or state estimation in distributed parameter systems. A practical criterion for assessing the influence of the sensor configuration is presented, which is based on a measure of independence of the sensor responses in the case of state estimation, and on a measure of independence of the- sensitivity functions in the case of parameter estimation, respectively. The procedure is illustrated with the optimal placement of temperature sensors in a catalytic fixed-bed reactor. Copyright © 1999lFAC Keywords: parameter parameter systems.

l~

estirnation~

slate

estimation~

INTRODUCTION

nonlinear observers, distrlbuted-

of the linearized, discretized:- model, the proposed criterion is independent of the model formulation, linearization or solution techniques. In fact, our optimality criterion is closely related to the one proposed in (Sadegh and Spa]], J998) in which no a

Nowadays~ complex mathematical models, i.e., mixed sets of differential-algebraic and partial differential equations, are increasingJy being applied

for process simulation and possibly~ for on-line state estimation and model-based predictive control.

priori knowledge of the system is assumed. In previous studies (Point~ et al. ~ 1996; Vande Wouwer, et al., 1998), several practical aspect~ of the distributed paraIlleter estimation problem are considered, including the comparison of several optimization algorithms and gradient computation techniques. Some results concerning the determination of experimental conditions, Le., input signal s and sensor configuration, are presented. Here, in addition to these results, attention is more focused on system observability and the on-line estimation of the temperature and concentrati on profiles in a catalytic fixed-bed reactor. In particular, the influence of sensor locations on the performance of a nonlineBr distributed parameter observer is highli ghted.

This study considers t"vu stages in the model development and exploitation, in which inference must be dra\vn on the system from experimental data: (a) estimation of sets of unknown parameters contained in the 010 del equations (b) on-line state estimation based on an appropriate observer or fitter. In both cases~ it is of interest to select an optimal sensor configuration~ i.e.~ number and location~ so as to maximize the information contained in the

experimental data. The optimality criterion developed in this paper is based on a test of independence of quantities of Le., the sensor responses in the context of state esti mation and tbe sensiti vity functions in the context of parameter estimation. As opposed to analysis procedures which make use of the model structure~ as for instance the observability criterion devised in (Dochain, et al., 1996) which is based on the conditioning number of the observability matrix interest~

This paper is organized as follows. The criterion for optima) sensor configuration is developed in Section 2. The procedure is illustrated with the optimal placement of temperature sensors in a catalytic fixedbed reactor. In Section 3, the reactor model and a

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

ON A PRACTICAL CRITERION FOR OPTIMAL SENSOR CONFIG...

14th World Congress of IFAC

the L 2 norm of the elements of S~ i.e~, their magnitude, \vhile the off-diagonal elements of M are the cross-scalar products of elements of S, i.e., represent a degree of dependence between these quantities. As the determinant of a positive semidefinite matrix increases with its diagonal elements and decreases with its off-diagonal elements, the maximization of det(l\:I) with respect to the sensor locations Xl, •.. , XM, allovls for the determination of the spatial regions where the elements of S take the largest values and are independent of one another. A vanishing Gram determinant indicates a linear dependence between the elements of S.

nonJinear distributed parameter observer for the online estimation of the temperature and concentration profiles are derived. Section 4 presents some numerical results and highlights the influence of the sensor locations on the performance of the observer. Some results are also given for the estimation of unknown model parameters. FinaHy, Section 5 is devoted to concluding remarks.

2. CRITERION FOR OPTIMAL SENSOR LOCATION Consider the general equation for the state v(t,x)

av

dV

a2 v

- = f(t, X, v ' - " - 2 ; 8), t > O,XE at ax ax

(O,L)

This criterion is closely related to the identifiability criterion proposed by Qureshi et al. (1980) based on the determinant of the Fisher information matrix. More recently, Sadegh and Span (1998) also developed a criterion for optimal sensor configuration based on maximizing the overall sensor response while minimizing the correlation among the sensor outputs- An attractive feature of this family of criteria is that they do not depend on the model formulation, complexity, or solution technique.

(1)

which represents a \vide variety of model equations incl uding nonlinear differential -algebraic and partial differential equations. In this expression, e denotes a P-vector of unkno\1\o'o parameters. Measurements are available at M locations Xi Yi (t) = h(v(t, xi), i = 1,

'U;

M

(2)

In order to estimate non-measurable state variables or to infer the values of unknown model parameters from the measurements Yj(t)~ it is of interest to maximize the amount of information - about the state or the parameters - in the collected data by selecting appropriate sensor locations.

Let S(Xl " .. , x M ~ t) denotes a set of quantItIes of interest in the several measurement points, i.e., either a set 0 f sensor responses

3. CASE STUDY: A CATALYTIC FIXED-BED REACTOR 3.1. A Model ofa Catalytic Fixed-Bed Reactor Consider a catalytic fixed-bed reactor in which the hydrogenation of small amount of CO'2 to methane is performed. The model of the transient behavior (Van Doesburg and De Jong~ 1974) consists of the following equations

ac at

-:::::: D or a set of sensitivity functions with respect to the unknown parameters

S ( Xl' ... , X M ~ t) :;;:: (s 0 (x 1 ~ t) ...s Op (x 1 , t) ...

aT at

...Se\ (x M ~ t).~.Sep (x ~1 ' t)]

}"eff

-:=

J

T

(4)

e

a2 c

ac

dX 2

dX

ff ---V~-T(C

T) t > 0 ~

,

X E

(O,L)

(6)

a2 T - ; £pgcPl! aT v'"

pep dX 2

pep dX

(7)

+ 2k w (Tw - T) + (-M-I) r(c, T) Rtpc p

The optimality criterion is based on a measure of independence between these quantities of interest, i.e the sensor responses in state estimation or the sensitivity functions in parameter estimation. In this study, the Gram determinant is used g(xl ~ ..., xM) ==

(5)

pep

in which c(tx) is the concentration of CO 2 and T(t,x) is the temperature~ The reaction rate is given by

-E

q

de{[S(XI •...• XM.t)ST(XI •...• XM.t)dt]

'

cexp(~)

RT l+kcc

(8)

These equations are suppJemented by boundary and initial conditions

ac

v

ox

D eff

-=--(c-cin)

where r is the time span of interest. In this expression, the diagonal elements of the r matrix M = S(XI.· ... XM .t)ST (Xl,,,,,XM.t)dt are

(9)

J o

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

ON A PRACTICAL CRITERION FOR OPTIMAL SENSOR CONFIG...

14th World Congress of IFAC

Table 1. Model parameters

ac =0 ax aT

-=0 dx

t > 0, x

~

c:= Co (x)

t == 0,

T=To(x) ~

=L

X E

[O~LJ

(10)

D eff

(11)

The temperature profiles can be measured by thermocouples inserted in the bed at several locations along the reactor axis Yi (t) ;:: T(t,Xi)' i = 1, ... , M

(12)

The model equations (6~11) are solved using a finite difference approxiInation of the spatial derivatives and an appropriate ODE solver. The parameter values are given in Table 1.

:=

5xlO- 4 m 2/s

Acff==

3.5xl 0. 4 kcal/m s K

pep ;;:: 364 kcaUm3 K

v

== 0.5 m/s

2.29 kcaVkg K

c

=0.6

;;:: 0.0775 kg/m3

kw

= 5x10-4 kcaUm2 s K

Tw

== 300 K

R t =0.01

L

= 0.2 ID

k r = O.971xl0 13 m- 3

c pg

=:

Pg

ID

S-l

25.211 kcaUmole kc == 12.7

E Hct

;;::

-~H

;:;: 6.006 kcal

R

:=

1.98 cal/rnole K

T(K)

050

Figure 1 shows temperature and concentration transients at reactor start-up, Le., due to step changes in the feed temperature and concentration Cin(t) == O----i-2.5 mole%, and Tin(t) == 300~500 K for 12:0. These results correspond to a gas velocity v = 0.5 m/so The transient temperature profiles for a higher flow velocity, e.g., v = 1.5 m/s, are depicted in Fig. 2. Very steep gradients In the concentration and temperature profiles develop in the region of high reaction rate.

300 L-_----=~__=:::::...--=::=__..;;;;==_=;;=;.~~=_~;;..;;;

o

0.1

0.2

x(m)

3.2. Design of a nonlinear distributed paranleter observer From the model equations (6-11) and the available temperature measurements (12)~ a nonlinear distributed observer is developed

ac at

ac 2

- =D

ff

e

ac

~ ~

dX

'

- - - v - - r ( c T)

ox 2

1.5

~

0.5

t > O~ x E (O,L) (13)

aT

ot

Aeff (321' 2

pep

ax

PgC pg

at

pep

ax

0.1 xlm)

£v~---

Fig. 1. Evolution of the temperature and concentra-

~(Tw-f) ~ (-~H)r(c;r) Flpc p

+ k T (2~ T)(y -

0.2

tion profiles at t v=O.5 m/s

(14)

pep

=

0, 100, 200, ... , 5000 s -

y)

450

02 == 0

dx

aT

ax

400~

t> 0, x:=: L

350l

( 16)

=0

2~co(x)

T=To(x)

300

o

_ _ ._---=-_-..---l._---==----_~--"'::=-----'

_~

0.1

0.2

x(m)

t

= O~ X E

[O~LJ

(1 7)

Fig. 2. Evolution of the temperature profiles at t == lOO~ 200~ 4'.~ 5000 s - v=1.5 mJs

O~

4267

Copyright 1999 IFAC

ISBN: 0 08 043248 4

14th World Congress ofIFAC

ON A PRACTICAL CRITERION FOR OPTIMAL SENSOR CONFIG...

The design of the c.orrection term k T (c" T)(y ~ y)

The estimation errors appear as the solution of a convection-diffusion-reaction system. The correctlon

follows a method devised in (Zeitz, 1977), \\-'hich is based on a physical interpretation of the equations defining the estimation errors e(x, t)==T(t, x)~T(t, x)

term k T (c, T)(y - y) is selected so as to compensate potentially destabilizing source terms in equation ( 19) and ensure the observer con vergence (Zeitz, 1977), i.e.,

and £(t" x)~c(t, x)-c(t, x) , i.e.,

de

d2€

at

ox

a£

ar,. '"'

ar

A.

~

- = D eff ---v--(-(c,T)c+--Cc., T)e) 2 QX QC

t> 0,

ae

Aeff

ut

pep

~

ale

X E

(O,L)

pep

EV

ox.L-

In this way, a transfer term in the form -ae(t, x), i.e., equivalent to a heat loss.. is introduced in (19). The tuning parameter ex ~ 0 can be viewed as a heat transfer coefficient, whose magnitude determines the observer convergence,

de pep dx

(-aH)

ar

.~

'"'

ar

~

'"

- - -we + - - ( - ( c , T)E+-(c,T)e) (19) Rpc p pep de aT

+ k T (c, T)( y -

aE v -::::::;--E ax D eff ae

EVPgC pg

ax

Aeff

..

y)

to

t> 0'1

X

=0

The initial temperature profile (x) as well as the error profile e(t~x), \s,,'hich appears in the expression (23) of the correctjon term~ are evaluated by interpolating known values at the measurement

(20)

e

points.

4. INFLUENCE OF THE SENSOR CONFIGURATION

dE =0

ox

t> 0, x = L

ae ==0 ' ax

(21) 4~1.

E ~ Co (x)

e=eo(x)

(c, T»e(t, x) (23) T

(18)

PgC pg

~

2k

k T (2, T)(y - y)=:o-(a + (-Ml) :

aT

t = 0, X E [O,L]

(22)

'I

Note that, using the assumption of smaJ) estimation the reactions terms in the equations (18-19) have been linearized.

en~ors,

)( 10 15

State Estimation

rrhe optimality criterion (5) based on the sensor responses (12) during the reactor start-up (Figs. 1-2) is used to deterrnine the optimal location of one or t\\-'o sensors. In the one sensor case, the optimal sensor location simply corresponds to the reactor hot~ . spot. Figure 3 shows the evolution of the Gram determinant for two sensors~ and Table 2 summarizes the results.

Table 2. optimal sensor location for state estimation (cm)

0.5 mJs v == 1.5 mls v;:=

0.2

2.5

1.5

0.5

o

0.2

Pig. 3. Evolution of the Gram determinant g(Xl,X2) v=:O.5 mls (above) and v===1.5 mJs (below)

1 sensor 0.4 11

2 sensors 2.6,20 0.8,20

In order to illustrate the influence of the sensor placement on the observer performance, the following test is considered. The observer starts at t :: 500 s from interpolated initial conditjons~ and from this time~ uses temperature measurements from one or two sensors with a sampJing interval of 10 s. Figure 4 shows the observer convergence in the case of one sensor placed in the optimal location Xl == O~4 cm or at the reactor outlet Xl ;:::; 20 cm, respectively. The observer starts from an initial uniform temperature profile equal to the measured value~ Le., 500 K in the first case and 0 K in the second case. The observer gain a:= 1. In the second case, although the first teluperature profiles are more satisfactorily reproduced, the performance deteriorates significantly and~ at final times, the estimated temperature profiles oscillate.

4268

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ON A PRACTICAL CRITERION FOR OPTIMAL SENSOR CONFIG...

14th World Congress ofIFAC

sensitivity functions. At this stage~ the temperature sensitivities \vith respect to kw and kr are computed by direct differentiation (Point et al., 1996). The results in the two sensors case. are depicted in Fig. 7. Table 3 gives the optimal sensor locations.

650 T~K)

550

In previous studies (Point~ et al., 1996; Vande Wouwer, et al., 1998)l the authors have discussed the influence of the sensor placement on the parameter estimation procedure~ i.e., the minimization of an output-least square (OLS) c.riterion using an optimization al gorithm.

450

400 350

----'---_ _

~

o

_ _

~--....L,

O.1~

0.05

~-------J

0,2

For exanlple~ Figure. 8 illustrates the shape of the OLS criterion in the one sensor case when the reactor operates with a gas velocity v = L.5 mls. When the sensor is placed in the first half of the reactor, where the Gram determinant is vanishing, the OLS criterion is almost t1at~ which indicates that the sensitivity \vith respect to the parameters is [0\\.'. On the other hand, when the sensor is placed in the optimal location, the OLS criterion displays a much steeper profile, indicating much higher parameter sensitivities.

650 T{K)

600

900 r - - - - - - - - . - - - - - , - - - - - - - - , r - - - - - . , . . . . . - - - - - . 300 L_---=~~~-=::::::::;;::::::=..:;;;;:;;;;;;;;;;;iiiiii ;;;;;;;;IiIIiI;;;;;;;;;;;;;;;;;;:; o 005 01 015 0_2

T{K)

Kern)

800

Fig. 4. Estimated temperature profiles - one sensor located in Xl :=:: 0.4 cm (abo\le) or Xl ~ 20 cm

7DO

(bel0'"') 600

These oscillations can also be seen in Fig. 5, which shows the temporal evolution of the estimated temperature in x = 4 cm, for several ~ensor configurations (1, 2, and 3 sensors placed in several locations). Figure 5 also illustrates the influence of the observer gain on the convergence, i.e., the graphs correspond to (( == 0 and et == 0.1, respecti veJy_ Even \vith a very poor observer tuning, the temperature estimates corresponding to an optimal sensor configuration converge quickly. Of course, a better observer tuning improves the convergence in all cases.

500

400

100D

150D :(s)

2000

250D

3000

55tJr--------,-----~-r-~---.....-----...---~

T(K)6CO

,:'"'"

,~,~.,.<;:~-.-. -~.,~ :_~.;;~.~ j

550 .:

500

Finally ~ Figure 6 shows the evolution of the estimated temperature profiles in the case of two sensors in the optimal locations (see Table 2), \vhen the reactor operates \vith a gas velocity of 1.5 m/s. Al though the initial interpolated profile is a very crude approx.imation~the observer converges quickly.

: 4

/ J

~

..

.6: 3

450 400

a= 0.1

350

3°~o6~

1000

1500 t{s)

4.2. Pararneter estifnation

Fig.

Consider the estimation problem of two unknown model parameters, e. g., the heat transfer coefficient with the cooling jacket kw and the pre-exponentiaI factor k r in the expression of the reaction rate. The optimality criterion (5)~ based on the measurements (12) during the reactor start-up~ is computed in order to evaluate the degree of dependence betv-reen the

2000

2500

3000

5 Temporal evolution of the estimated temperature in x ::= 4 cm for various sensor configurations (circled 1ine~ real temperature; dashed lines: estimation \vith 1 sensor - 1: 0.4 cm~ 2: 10 cm: 3: 20 cm - solid lines: estimation \vith 2 sensors - 4: 2.6 and 20 cm; 5: 0 and 5 cm; 6: 15 and 20 cm - crossed lines: estimation with 3 sensors - O~ 10 and 20 cm)

4269

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ON A PRACTICAL CRITERION FOR OPTIMAL SENSOR CONFIG...

14th World Congress ofIFAC

300

Table 3. optimal sensor location for parameter estimation (cm)

250

2CO

2 sensors 2.2,20 10.8 , 18.6

I sensor 2.2 10.8

v = 0.5 m/s v=1.5 m1s

'50 100

Generally t the selection of a sensor configuration will significantly influence the shape of the OLS criterion and the existence of a \\'ell-defined minimum., and in turn., the computational expense required by the optimization algorithm to estimate the unknown

1 KwlkwO

parameter values.

1.5

X 10!5

'iD

8.

5. CONCLUSIONS

In this paper, a criterion for optimal sensor configuration is developed, which is based on a measure of independence, Le.., the Gram determinant, of the s.ensor responses in the context of state estimation, or the parameter sensitivities in the context of parameter estimation. The illaximization of the Gram determinant allows the determination of the spati a] regions where the sensor should be located. The usefulne·ss of the criterion is illustrated with the optimal placement of temperature sensors in a fixed-bed reactor. Particularly, the influence of the sensor locations on the performance of a nonlinear distri buted observer is highlighted.

1.5

Fig. 8. 3D plots of the OLS criterion in the unknown parameter space - sensor in x=O (above) or x:;::10.8cm (below)

700

r

T(K)

REFERENCES

650r

Dochain D., N. Tali-Maamar and J. P. Babary (1996). Influence of the Sensor Location on the Practical Observability of a Fixed-Bed Bioreactor, Proc. 13th IFAC World Congress.,

600~ r

550

491-496. Point N., A. Vande Wouwer and M. Remy (1996). Practical Issues in Distributed Parameter Estimation: Gradient computation and Optimal Ex.periment Design, Control Eng. Practice, 4,

500

450

0.1

0.05

0.15

02

1553-1562. Qureshi Z.H., T. S. Ng and G. C. Goodwin (1980). Optimum Experimental Design for Identification of Distributed Parameter Systems~ Int J. Control, 31,21-29. Sadegh P. and J.C# SpaIl (1998). Optimal Sensor Configuration fOT Complex Systems~ Proc. An1,erican Control Conference, 3575-3579 \lande Wouwer A., N. Point, S. Porteman and M. Remy (1998). Numerical Experiments in Distributed Parameter Estimation~ Proc. lCOTA '98. Van Doesburg H. and W.A. De long (1974). Dynamic Behavior of an Adiabatic Fixed-Bed Methanator, Int. Sy1l1p. Chem. React. Eng., Evanston, Advances in Chenl. Series 133, 489503. Zeitz M. (1977). Nichtlineare Beobachter fUr chemische Reaktoren~ \TDI-Fortschritt-Berichte 8/27, VDI- \l er l ag.

x(m)

Fig. 6. Estimated temperature profiles in the case of two sensors in the optimal locations ~ v-=1.5m1s x 10~ 2.5

2

1.5

0.5

o

0.2

C2

o

0

Fig. 7. Evolution of the Gram determinant g(Xb X2) v=1.5 m/s

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Copyright 1999 IFAC

ISBN: 0 08 043248 4