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Nonlinear self-tuning regulator for pH systems
Vol. 30, No. 10, pp. 1579-1586, 1994 Copyright(~) 1994 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0005-1098/94 $7.00+ 0.00
Automatica,
0005-1098(93) E0061-8
Pergamon
Nonlinear Self-tuning Regulator for pH Systems* SANG D E U K LEE,t JIETAE LEES and SUNWON P A R K t
A nonlinear adaptive control method, based on the rigorous model of p H dynamics, was presented and tested with simulations. The method was found to be effective for p H control systems. Key Words--pH control; adaptive control; neutralization; nonlinear system.
Abstract--A nonlinear self-tuning regulator for controlling
1972; Gustafsson, 1985; Choi and Rhinehart, 1987; Parrish and Brosilow, 1988). Among those an internal adaptive model controller developed by Choi and Rhinehart (1987) seems to be easily implementable and applicable to wastewater treatment processes. On the assumption that a wastewater stream is a hypothetical monoprotic acid, the controller can estimate unknown parameters of the internal model online and can calculate the manipulated variable. However, it has disadvantages of iterations in the estimation of unknown parameters. For severely nonlinear systems similar to the pH control system, the nonlinear adaptive control method which utilizes the structure of nonlinear dynamics of a specific process was reported to provide significant improvement over conventional controllers or linear adaptive methods (Agarwal and Seborg, 1987). Here we propose a new nonlinear adaptive control method for weak acid and strong base systems which applies the wen-known linear self-tuning regulator technique to the rigorous dynamic model of the pH process systematically. The nonlinear equations describing the neutralization process are transformed so that we can estimate the unknown parameters--the concentration and the dissociation constant of the influent weak acid. To the transformed model we apply the basic linear self-tuning regulation technique (Astrom and Wittenmark, 1973; Goodwin and Sin, 1984) such as the recursive least squares method with a variable forgetting factor (Fortescue et al., 1981) for parameter identification and the one-step-ahead control law for manipulated variable calculation.
the pH value of weak acid streams by a strong base is proposed. The nonlinear state equations describing the neutralization process are transformed so that we can estimate the unknown parameters, the concentration and the dissociation constant of the influent weak acid, via the recursive least squares method with a variable forgetting factor. Then we apply the one-step-ahead control law. Simulation results show that the regulator provides good performance for realistic situations such as multicomponent weak acid and strong base systems. 1. I N T R O D U C T I O N
THE CONTROLOF pH is important in wastewater treatment and in the chemical industry. But pH processes are difficult to control because of the inherent nonlinearity, the high sensitivity at the neutral point and the time-varying gain (Trevathan, 1978; Buchholt and Kummel, 1979; Jutila, 1983). For these reasons, pH control by conventional proportional-integral-derivative controllers or advanced controllers based on linear system theory is ineffective. Goodwin et al. (1982) presented a nonlinear self-tuning regulator which performs well for strong acid and strong base systems. But weak acid and strong base systems or weak base and strong acid systems reveal more complicated dynamic behaviours due to varying buffering effects of the weak acid (McAvoy et al., 1972) or of the weak base. Thus, their pH values are very difficult to control. Several works have been carried out to treat these systems (McAvoy, * Received 31 December 1990; revised 29 August 1991; revised 25 March 1993; received in final form 22 November 1993. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Torsten S6derstrt~m under the direction of Editor P. C. Parks. Corresponding author Professor Sunwon Park. Tel. +82 42 869 3901; Fax +82 42 869 3910. t Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon 305-701, Korea. ~: Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea.
2. PROCESS D E S C R I P T I O N
Consider the control system shown in Fig. 1. An n-component weak acid stream with the flow 1579
1580
SANG DEUK LEE et al.
I
Weak acid
®
I.. . . . . . . . .
]
'~_ ......
,
Ft
FcCu)
N~"I
vary in the reactions and they are linearly independent of each other. In this system, X corresponds to -CB in the reactor. Inserting the equilibrium constant into equations (2) and (3) gives
stream
I/Vi = ( C H / K a i + l + K b i / C H + KbiKdi/C2H)Cbi
i = 1, 2 . . . . . n X = CH -- K w / C H -- ~
(4)
(l + 2KBi/C H
i=1
FIG. 1. pH control system.
(5)
+ 3Kb,Ka,/C2)Cb,.
rate Ff and the concentration Cfi flows into the reactor of volume V and is neutralized by a strong base with the flow rate F~ and the concentration Co. The reactions can be represented as
dW,./dt = FfWfi/V
Kw
>H + + OH
H20 (
Assuming perfect mixing, material balances for the reaction invariant variables, W/ and X, give n + 1 independent equations. The state equations can be written as follows:
- (El "~- F c ) W i / V
dX/dt
Ka~
a~(
)H+ + b i
i=1,2
bi~
~H + + d i
i=1.2 ..... n
)H+ +gi
i=1,2 ..... n
.....
n
(1)
i = 1, 2 . . . .
,n
= F c X c / V - (Ff + F c ) X / V
(6) (7)
where Wfi and - X c corresponding to the concentration of the influent weak acid i, Cti, and the concentration of the strong base of the titrant stream, Co, respectively.
Kdi
di~
3. THE NONLINEAR pH CONTROLLER
BOH
>B + + O H -
where a . bi, di and gi denote an undissociated weak acid i, 1-, 2- and 3-acidic base. Ka. Kb~ and Ko~ are the first, second and third dissociation constants of the weak acid i, respectively, Kw is the ion product of water, and B O H is a strong base. The equilibrium constants for the reactions (1) are Kw
= CHCoH
K~i= CnCbi/C~i
i=1,2
.....
n
Kbi = C H C d i / f b i
i z 1, 2 . . . . .
n
gdi = C H C g i / C d i
i = 1, 2 . . . . .
n
w = (C./Ka +
n
(2)
X
= C H -
Coil
-- ~ i--I
(Cbi -{- 2Cdi + 3Cgi)
(3)
i = 1, 2 . . . . .
1)Cb
X = C H - Kw/C H - C b
where C denotes a concentration. Cai, for example, is the concentration of the undissociated weak acid i in the reactor. To obtain the dynamic model for this system, we introduce reaction invariant variables (Gustafsson and Waller, 1983) as W i = Cai q- Cbi q- Cdi Jr Cg i
In real processes we do not know the numbers of weak acids in the influent stream and cannot measure the dissociation constants and the concentrations of 1-acidic bases in the reactor, so we cannot use equations (6) and (7) for designing a controller. Thus we assume the influent weak acid stream as a hypothetical monoprotic acid with the dissociation constant K~ and the concentration Ct. From the above assumption, equations (4), (5), (6) and (7) can be simplified as
(8)
(9)
dW/dt
= F f W f / V - (Ff + F c ) W / V
(10)
dX/dt
= F ~ X c / V - (Fe + F c ) X / V
(11)
where W corresponds to the of the hypothetical acid and the reactor. Eliminating the unknown equations (8) and (9), we equation (12):
total concentration its ionic species in variable Cb from obtain the cubic
C 3 + ( g a - X)C2H
where W~ is the total concentration of each conjugate acid-base system present, and X represents the electroneutrality condition. W~ and X are the state variables which do not
- (K,X
+ K ~ W + K w ) C H - K , K w = 0.
(12)
Here we design a nonlinear self-tuning regulator which estimates the unknown para-
Adaptive control of pH systems meter Ka and the unmeasured influent weak acid concentration Ct using equations (10)-(12), and calculates the strong base flow rate which compensates for the variations of the effluent pH value with the estimated unknowns. Since we cannot apply the self-tuning regulator technique directly to equations (10)-(12), we derive a state equation by differentiating equation (12): (3C~ + 2KaCH -
1581
3.1. P a r a m e t e r estimation
Equation (19) can be parametrized in terms of unknown parameters, Ka and KaCf, as equation (20): [(2dk+, - Ff - Uk)C3k - ( d k + , 2 ~ + u k C c ) C h k + Kw(Ff + uk)CHk] + Ka[(dk+, - Ff - Uk)CZk - u k C c C m + (d1,+1 + Ff + uk)Kw] + K.Cy[FfCHk] - 6k[dk+, CZk]
2 X C H - K a X - K a W - Kw) d C H / d t - ( C ~ + K~CH) d X / d t - K~CH d W / d t = O.
(13) From the definition of pH, C H = 1 0 -pH
or
pH=-log(Cu)
d C u / d t = - C H In (10) dpH/dt.
(14) (15)
Substituting equations (10), (11) and (15) into equation (13) results in In (10)[3C3H + 2(ga - X ) C 2 -- ( K a W + K a g
+ gw)CH]
dpH/dt
+ 1/V[(FcXc - FfX - F~X)C 2
-- epH{V In (10)[2C3k + (Ka --
Sk - ak)C2k + K a K w ] }
= 0
(20)
where dk+l = V In (10)(pHk+l -- pHk)/T, Xk is the approximated value of X k ()(k = X k - 6k), and 6k is the error at t = k T caused by the difference, 60, between the true value of X at t = 0 and the initial guess, 20. The error 6k goes to zero exponentially as time increases and the error epH also becomes negligible with a small sampling period T. Let Yk+l -~ [(2d,+, - F f - Uk)C3k
+ K a ( F c X ¢ - F f X - FcX)CH] + 1 / V [ K a ( F f W f - F f W - FcW)CH] = 0.
(16)
Using equation (12) to eliminate W and replacing Wf and X¢ with C, and - C c equation (16) can be rewritten as
-- ( d k + l S k + ukCc)C2k + Kw(Ff + Uk)CHk]/C2Hk
(21)
(22)
0 ~ (Ka, K a C t ) T
qJk+, ~- ( - [(d,+, - F f - u , ) C ~ k - UkC¢CHk
dpH/dt = {Ff[C3 + KaC~4 - (Kw + K~COCH - K~Kw]
+ (dk+l + Ff + u k ) K w ] / C ~ , ,
+ Fc[C 3 + (K a + C c ) C 2
- [F,C.kllC~,k) T
Kw)CH - K a K w ] } / {V In (10)[2C 3 + (Ka - X ) C 2
(23)
+ ( K a C c --
+ KaKw]}.
ek+l =- {6kdk+,C2k + epH V In (10)
(17)
Then from equations (11) and (17) we get the following discrete models in X and pH: X k + , = X k -- T / V [ u k C c + (Ff + uk)Xk] + ex (18)
(pHk+l - pH~)/T = {FfIC3k + KaC2Hk
- (Kw +
+ Kaff)fHk
Uk[C3Hk+
(Ka +
-
K , Kw]
Cc)C2Hk
{V In (10)[2C3k + (Ka - Xk)C~k epH
(19)
where T is a sampling period, Uk is the Fc at instance k, and ex and epn are the errors due to discretization by the Euler integration method.
( K a - S k --
Ok)C2Hk
+ K.Kw]I/C~k
(24)
where the division by C~ik has been used to avoid numerical problems due to handling very small numbers. Then we have Yk+l
+ (KaCc - Kw)CHk -- KaKw]}/ + KaKwl} +
x [2C3k +
=
T Y l k + l O + ek+l.
(25)
Equation (25) is derived by assuming perfect measurements. In the case of measurement noises, the errors caused by measurements can be included in ek+l. We estimate the unknown parameters Ka and Cf of equation (25) via the recursive least squares method with a variable forgetting factor (Fortescue et al., 1981; Cordero and Mayne,
SANG DEUK LEE et al.
1582 198l), that is
To prevent extreme manipulated variable, a implemented as
A + , = W~+,0,¢ Y Kk+, = PkWk+ ,/(1 + Wk+lPk~k+,)
if 0k + l < 0mi, then 0k + 1 = 0mi,
(26)
if Ok+~ > 0m,x then Ok+ ~ = 0m,~ = 1 -
(1
- Wk+,Kk+O(yk+IT
of the filter is
Uk+l = exp ( - T/r)Uk + [1 -- exp (-- T/T)Iuc,, (28)
Ok+l ~- Ok ~- gk+i(yk+, -- Yk+,)
Ak+,
fluctuation first order
where r is the time constant of the first-order filter.
_ yk+I)2/Zo
4. S I M U L A T I O N R E S U L T S
if Ak+~ <
Amin
then hk+~
=
Amin
P,+, = (Pk - Kk+,WT'+,Pk)l,~k +, ^
where denotes an estimate, A is the variable forgetting factor and Z0 is a scalar quantity to speed up adaptation for sudden changes of a set point or load. 3.2. Control law With estimated values of Ka and Cf, we calculate the manipulated variable by the one-step-ahead control law from equation (19) (Goodwin and Sin, 1984) as uc,, = {V In (10)[2C3k+t
+ (/~a --
f(k+l)CHk+l
+ / ( . K w ] ( p H s - phk+,)/T - Ff[c-L,,+, + Rac~,~+, - (Kw + RaCf)Cnk+l -
R.Kw]}I
[ c L + , + ( S a + Cc)CL+~ + (R.Cc - Kw)CHk+i - K . K w ]
(27)
if Uca~< Umi,, then Ucal = Umm if U~aj> Umax then u~a~= Umax where pHs is the set point of p H at instance k+2. The denominator of equation (27) can easily be shown to be always greater than zero.
4.1. Single component weak acid and strong base system Simulations have been carried out for a process where the objective is to neutralize a single component weak acid stream by a strong base. To simulate the process, we calculated the reaction invariant variables, W and X, by integrating the model equations (10) and (11) analytically and obtained numerical solutions of CH and C b from equations (8) and (9) using a nonlinear equation solver. The data used for the simulation study are given in Table 1. The unknown parameters, Ka and Cf, of the influent weak acid stream are initially set to 5.0 × 10 _6 mol/l and 0.005 mol/1, respectively. At time t = 1000s, those are suddenly changed to 1.8× 10 -Smol/l and 0.01 mol]l in order to disturb the system. Simulations have been performed for the cases with perfect measurement and with measurement noises. A zero-mean uniform random noise with +0.21 pH units was used as a measure noise. In the case with perfect measurement, a random perturbation signal was added to the control input for parameter convergence. Figures 2 and 3 show the control input and output, and the parameter estimates in the case with perfect
TABLE [. DATA FOR THE SIMULATION STUDY IN THE CASE OF A SINGLE COMPONENT WEAK ACID AND STRONG BASE SYSTEM
Symbol V F~. b~
Value in process model
Kw T Po h,~,n
4001 -2.0 l/s 0.05 moll1 0.005 mol/I for 0 -< t -< 1000 s 0.01 mol/l for 1 0 0 0 < t - < 2 0 0 0 s 5.0 × 10 ~ mol/1 for 0 -< t -< 1000 s 1.8× 10 5mol/1 for 1 0 0 0 < t - < 2 0 0 0 s 1.0 × l 0 14 mo12/l 2 ls ---
Yo
--
C,, C~ K,
Value in controller 4001 limits: [0, 1.0 l/s] 2.0 l/s 0.05 tool/1 initial: 0.012 mol/l limits: [10 4, 1.0mol/l] initial: 1.0 × 10 5 mol/I limits: [10 s 10 2mol/I] 1.0 × 10 14 mo12/l 2 ls 103 I 0.1 (for perfect m e a s u r e m e n t ) 0.8 (for m e a s u r e m e n t noises) 10 -7 (for perfect m e a s u r e m e n t ) 5.0 (for m e a s u r e m e n t noises)
Adaptive control of pH systems
1583
l.fBO
F© 1,116 I ( I/see )
F~ ( 1/sec )
1
11.50
0 .~58
o.ee
,t,l,,,,I,,,,I,,,,I 8
588
1080
1588
2880
8.88 8
588
1888
1580
2888
1O
pH
I
10-
6
I
4
pH
2
e
•.............
4 '--
,,,I,,,,I,,,,I,,,,I
8
6 ~ .............
5BB IB80 tile(sec)
1588
2888
2 -
FIG. 2. Input and output in the case of a single component weak acid and strong base system with perfect measurement.
B
,,,,I,,,tl,,,,I,,,,I fl 588 I808
1580
2880
ti=e(se¢)
measurement, respectively. From these figures, we can see that the nonlinear pH controller gives good control performances. In the case with measurement noise in pH, the results are presented in Figs 4 and 5. In Fig. 5 we observe that the estimated values of Cf are very close to the true values, but those of K~ are highly oscillatory over a wide range and do not converge to their true values. This kind of
KoxlO s
(=olii)
c : , : .
8.0 8
508
i
1888
158o
,888
0.020 Cr
(=ol/l) 0.810
O.BeO
i,, 8
I,, 588
,
,,,,I
1088
....
1580
I 2888
tile(sec) n true -
value
estimated
value
FIG. 3. Parameter estimates in the case of a single component weak acid and strong base system with perfect measurement.
F1G. 4. Input and output in the case of a single component weak acid and strong base system with measurement noises.
problem may be solved by using other types of parameter estimation schemes such as a least squares method with dead zone (Middleton et al., 1988) or with covariance resetting (Ydstie, 1992). However, in spite of poor estimates of Ka and measurement noises the regulator works quite well as shown in Fig. 4. It appears that pH is less sensitive to Ka than to Cf, and the estimated values of a Ka-Cf pair and the true K,-Cf values give the same base flow rate, F~, at the operation point ( p H = 7 ) . Figure 6 shows that the estimated titration curves match well with the process titration curves near the neutral region, especially at pH = 7. In Fig. 6, the estimated titration curves were plotted using the estimates of Cf and the worst estimates of K a which were obtained at 680 and 1230 s as shown in Fig. 5. On the basis of reviewing Fig. 6, an alternative based on fixing K~ to some reasonable value and estimating only Cf instead of estimating both Ka and Cf may work well too. It will be studied in future research. According to the results of simulation studies, which are not presented in this paper, the control performance remains satisfactory even with a change in the pH set point from 5 to 7 and with the pH measurement noise. 4.2. Multi-component weak acid and strong base system In order to investigate the possibility of applying this regulator to wastewater treatment
1584
SANG DEUK LEE et al. TABLE
2.
D A ' I A F O R I H E S I M U L A ] ' I O N S I ' U I ) ' ~ IN ]HI-: ( ' A S [ O F A M t r l j F I C O M P O N E N I
Symbol
WEAK ACID AND SIRON(i
Value in process model
V
Value in controller
100 I
F,
2.0 l/s
C,. Cn, Cre
0.1 mol/l 0.01 mol/l, 0.02 mol/1 for 0 -< t <- 1000 s 0.005 mol/l, 0.01 mol/l for 1000 < t < 2000 s 1.8× 10-s mol/I, 1,0x 10- 7 mol/l for 0_
Kal Ka2
Kb~, Kt~2 Kdl, Kd2 Kw T P0
100 1 limits: [0, 1.0 l/s] 2.0 I/s 0.1 mol/l initial: 0.012 mol/l (as monoprotic weak acid) limits: [10 4, 1.0 mol/1] (as monoprotic weak acid) initial: 1.0 x 10 5 mol/l (as monoprotic weak acid) limits: [10 s, 10 2 mol/l] (as monoprotic weak acid) 1.0 × 10 I,~ mo12/i 2 ls 10-~1 0.1 (for perfect measurement) 0.8 (for measurement noises) 10 _3 (for perfect measurement) 5.0 (for measurement noises)
.)tmi n
Zo
systems, we have applied the regulator to a system which has the influent weak acid stream consisting of a mixture of a monobasic and a dibasic weak acid. The data for the simulations are summarized in Table 2. The results are shown in Figs 7 and 8 for the case with perfect measurement and in Figs 9 and 10 for the case with measurement noises. From these figures we can see that the regulator
10,0
m
KoxlO 4
BASE S'rSI [M
performs as well for multicomponent systems as for single component systems.
5. C O N C L U S I O N
We propose a nonlinear self-tuning regulator for controlling pH values of weak acid streams by a strong base. A standard recursive least square identification method and one-step-ahead control law are used in the regulator. Since a rigorous model describing the real pH neutralization process of a weak acid and strong base system is used, unknown parameters are estimated so that the estimated K a - C , pair and
(=oi/1 ) 5.0 14
0,0 0
500
1000
1500
12
2000
18
pR
0.820
8 6
Cr
(mol/1) 4
0,010
L~ 0,000
.... fl
2
~ ~ ~ , _ , - ~ I ....
I ....
500 1800 time(see)
I .... 1580
8
I
I
I
8,fl
2080
I
1
[
I
11
8.5
I
I 1.8
Fc(l/sec)
0 p _ror~___ t i t r a t i o o curve ~ u - i r ~ 0-1000 sectmds. • Estimated t i t r a t i o n curve at 650 ~ .
[] true v a l u e - estimated
Process t i t r l t i ~
value
FIG. 5. Parameter estimates in the case of a single component weak acid and strong base system with measurement noises.
•
curve d u r i r ~ 1000--2000 s e c t o r .
E s t i l a ~ e d t i t r a t i o n carve at 1280 s e c m ~ .
FIG. 6. Titration curves in the case of a single component weak acid and strong base system with measurement noises.
Adaptive control of pH systems 1.00
--
1.00
F©
Fc
(l/see)
(I/see)
0.511
-
8.58
itilllll
8.88 8
10
iliiliilll
588
1888
1508
8.88
2880
0
m
6 4
580
1888
1588
2888
18-
8L
8
pH
1585
I
I
pH
6
2 8
8
588
1888
1588
-
4"28 ,,,,I,,,,I,,,,I,,,,I
m
2888
8
588
time(see)
1888
1588
2888
time(see)
FIG. 7. Input and output in the case of a multicomponent weak acid and strong base system with perfect measurement.
FIG. 9. Input and output in the case of a multicomponent weak acid and strong base system with measurement noises.
2.0
3.8
K,xl0 e (,,o1/I)
2.0
IC,xlO~
1.0
(moll
,,,I,,,,
8.8 8
588
,,,,I,,,,I 1008
1588
)
) 1 8.8
2888
0
8.848
588
1088
1588
2888
8.048
Cr
Cf
(eolll)
(,,ol/l)
t
8.828
0.880
I
....
8
I ....
588
I ....
1888
8,828 I ....
1588
I
2888
time(sec)
FIG. 8. Parameter estimates in the case of a multicomponent weak acid and strong base system with perfect measurement.
t h e t r u e Ka-Cf v a l u e s give t h e s a m e b a s e flow r a t e at t h e o p e r a t i o n p o i n t . S i m u l a t i o n results s h o w t h a t the s e l f - t u n i n g r e g u l a t o r p r o v i d e s g o o d c o n t r o l p e r f o r m a n c e s in spite o f m e a s u r e m e n t n o i s e a n d v a r i a t i o n s o f influent w e a k acid c o n c e n t r a t i o n a n d d i s s o c i a t i o n c o n s t a n t o f the w e a k acid. F o r w i d e a p p l i c a t i o n s to r e a l p H p r o c e s s e s , f u r t h e r s t u d i e s such as t h e o r e t i c a l s t a b i l i t y
0,080
,,,
8
t .......
588
1888
I ....
1588
I
2888
time(sec)
FIG. 10. Parameter estimates in the case of a multicomponent weak acid and strong base system with measurement noises.
analysis, c o n s i d e r a t i o n o f n o i s e d y n a m i c s a n d e x p e r i m e n t a l v e r i f i c a t i o n to n e u t r a l i z a t i o n p r o cesses o f m u l t i c o m p o n e n t influent w e a k acid streams would be required.
Acknowledgements--Financial support from the Korea Ministry of Science and Technology and the Automation Research Center at POSTECH is gratefully appreciated.
SANe; D~t:K LE[~ et al.
1586 REFERENCES
Agarwal, M. and D. E. Seborg (1987). Self-tuning controllers for nonlinear systems. Automatica, 23, 209-214. Astrom, K. J. and B. Wittenmark (1973). On self-tuning regulators. Automatica, 9, 185-199. Buchholt, F. and M. Kummel (1979). Self-tuning control of a pH-neutralization process. Automatica, 15, 665-671. Choi, J. Y. and R. R. Rhinehart (1987). Internal adaptive-model control of wastewater pH. 1987 ACC, Boston, pp. 2084-2089. Cordero, A. O. and D. Q. Mayne (1981). Deterministic convergence of a self-tuning regulator with variable forgetting factor, l E E Proc. D, 128, 19-23. Fortescue, T. R., L. S. Kershenbaum and B. E. Ydstie (1981). Implementation of self-tuning regulators with variable forgetting factors. Automatica, 17, 831-835. Goodwin, G. C., B. Mclnnis and R. S. Long (1982). Adaptive control algorithms for waste water treatment and pH neutralization. Opt. Control Appl. Meth. 3, 443-459. Goodwin, G. C. and K. S. Sin (1984). Adaptive Filtering Prediction and ()mtrol. Prentice-Hall, Englewood Cliffs, NJ. Gustafsson, T. K. (1985). An experimental study of a class of
algorithms R~r adaptive pH control. ('hem. £n.~..~~ "4tk 827-837. Gustafsson, T. K. and K. V. Waller (1983). Dynamic modeling and reaction invariant control of pH. ('hem. Eng. Sci. 38, 389-398. Jutila, P. (1983). An application of adaptive pH-control algorithms based on physico-chemical modeling in a chemical wastewater treatment plant, btt. J. Control. 38, 639-655. McAvoy, T. J., E. Hsu and S. Lowenthal (1972). Dynamics of pH in controlled stirred tank reactors. Ind. Eng. Chem. Process Des. Devel. 11, 68-70. McAvoy, T. J. (1972). Time optimal and Ziegler-Nichols control. Ind. Eng. Chem. Process Des. Devel. 11, 71-78. Middleton, R. H., G. C. Goodwim D. J. Hill and D. Q. Mayne (1988). Design issues in adaptive control. IEEE, 33, 50-58. Parrish, J, R. and C. B, Brosilow (1988). Nonlinear inferential control. AIChE J., 34, 633-644. Trevathan, V. L. (1978). Advanced control of pH. ISA Annual Con.ference, Paper No. 839, Philadelphim PA, pp. 69-82. Ydstie, B. E. (1992). Transient performance and robustness of direct control. IEEE Trans. Autom. Control, AC-39, 1091 1105.
Automatica,
0005-1098(93) E0061-8
Pergamon
Nonlinear Self-tuning Regulator for pH Systems* SANG D E U K LEE,t JIETAE LEES and SUNWON P A R K t
A nonlinear adaptive control method, based on the rigorous model of p H dynamics, was presented and tested with simulations. The method was found to be effective for p H control systems. Key Words--pH control; adaptive control; neutralization; nonlinear system.
Abstract--A nonlinear self-tuning regulator for controlling
1972; Gustafsson, 1985; Choi and Rhinehart, 1987; Parrish and Brosilow, 1988). Among those an internal adaptive model controller developed by Choi and Rhinehart (1987) seems to be easily implementable and applicable to wastewater treatment processes. On the assumption that a wastewater stream is a hypothetical monoprotic acid, the controller can estimate unknown parameters of the internal model online and can calculate the manipulated variable. However, it has disadvantages of iterations in the estimation of unknown parameters. For severely nonlinear systems similar to the pH control system, the nonlinear adaptive control method which utilizes the structure of nonlinear dynamics of a specific process was reported to provide significant improvement over conventional controllers or linear adaptive methods (Agarwal and Seborg, 1987). Here we propose a new nonlinear adaptive control method for weak acid and strong base systems which applies the wen-known linear self-tuning regulator technique to the rigorous dynamic model of the pH process systematically. The nonlinear equations describing the neutralization process are transformed so that we can estimate the unknown parameters--the concentration and the dissociation constant of the influent weak acid. To the transformed model we apply the basic linear self-tuning regulation technique (Astrom and Wittenmark, 1973; Goodwin and Sin, 1984) such as the recursive least squares method with a variable forgetting factor (Fortescue et al., 1981) for parameter identification and the one-step-ahead control law for manipulated variable calculation.
the pH value of weak acid streams by a strong base is proposed. The nonlinear state equations describing the neutralization process are transformed so that we can estimate the unknown parameters, the concentration and the dissociation constant of the influent weak acid, via the recursive least squares method with a variable forgetting factor. Then we apply the one-step-ahead control law. Simulation results show that the regulator provides good performance for realistic situations such as multicomponent weak acid and strong base systems. 1. I N T R O D U C T I O N
THE CONTROLOF pH is important in wastewater treatment and in the chemical industry. But pH processes are difficult to control because of the inherent nonlinearity, the high sensitivity at the neutral point and the time-varying gain (Trevathan, 1978; Buchholt and Kummel, 1979; Jutila, 1983). For these reasons, pH control by conventional proportional-integral-derivative controllers or advanced controllers based on linear system theory is ineffective. Goodwin et al. (1982) presented a nonlinear self-tuning regulator which performs well for strong acid and strong base systems. But weak acid and strong base systems or weak base and strong acid systems reveal more complicated dynamic behaviours due to varying buffering effects of the weak acid (McAvoy et al., 1972) or of the weak base. Thus, their pH values are very difficult to control. Several works have been carried out to treat these systems (McAvoy, * Received 31 December 1990; revised 29 August 1991; revised 25 March 1993; received in final form 22 November 1993. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Torsten S6derstrt~m under the direction of Editor P. C. Parks. Corresponding author Professor Sunwon Park. Tel. +82 42 869 3901; Fax +82 42 869 3910. t Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon 305-701, Korea. ~: Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea.
2. PROCESS D E S C R I P T I O N
Consider the control system shown in Fig. 1. An n-component weak acid stream with the flow 1579
1580
SANG DEUK LEE et al.
I
Weak acid
®
I.. . . . . . . . .
]
'~_ ......
,
Ft
FcCu)
N~"I
vary in the reactions and they are linearly independent of each other. In this system, X corresponds to -CB in the reactor. Inserting the equilibrium constant into equations (2) and (3) gives
stream
I/Vi = ( C H / K a i + l + K b i / C H + KbiKdi/C2H)Cbi
i = 1, 2 . . . . . n X = CH -- K w / C H -- ~
(4)
(l + 2KBi/C H
i=1
FIG. 1. pH control system.
(5)
+ 3Kb,Ka,/C2)Cb,.
rate Ff and the concentration Cfi flows into the reactor of volume V and is neutralized by a strong base with the flow rate F~ and the concentration Co. The reactions can be represented as
dW,./dt = FfWfi/V
Kw
>H + + OH
H20 (
Assuming perfect mixing, material balances for the reaction invariant variables, W/ and X, give n + 1 independent equations. The state equations can be written as follows:
- (El "~- F c ) W i / V
dX/dt
Ka~
a~(
)H+ + b i
i=1,2
bi~
~H + + d i
i=1.2 ..... n
)H+ +gi
i=1,2 ..... n
.....
n
(1)
i = 1, 2 . . . .
,n
= F c X c / V - (Ff + F c ) X / V
(6) (7)
where Wfi and - X c corresponding to the concentration of the influent weak acid i, Cti, and the concentration of the strong base of the titrant stream, Co, respectively.
Kdi
di~
3. THE NONLINEAR pH CONTROLLER
BOH
>B + + O H -
where a . bi, di and gi denote an undissociated weak acid i, 1-, 2- and 3-acidic base. Ka. Kb~ and Ko~ are the first, second and third dissociation constants of the weak acid i, respectively, Kw is the ion product of water, and B O H is a strong base. The equilibrium constants for the reactions (1) are Kw
= CHCoH
K~i= CnCbi/C~i
i=1,2
.....
n
Kbi = C H C d i / f b i
i z 1, 2 . . . . .
n
gdi = C H C g i / C d i
i = 1, 2 . . . . .
n
w = (C./Ka +
n
(2)
X
= C H -
Coil
-- ~ i--I
(Cbi -{- 2Cdi + 3Cgi)
(3)
i = 1, 2 . . . . .
1)Cb
X = C H - Kw/C H - C b
where C denotes a concentration. Cai, for example, is the concentration of the undissociated weak acid i in the reactor. To obtain the dynamic model for this system, we introduce reaction invariant variables (Gustafsson and Waller, 1983) as W i = Cai q- Cbi q- Cdi Jr Cg i
In real processes we do not know the numbers of weak acids in the influent stream and cannot measure the dissociation constants and the concentrations of 1-acidic bases in the reactor, so we cannot use equations (6) and (7) for designing a controller. Thus we assume the influent weak acid stream as a hypothetical monoprotic acid with the dissociation constant K~ and the concentration Ct. From the above assumption, equations (4), (5), (6) and (7) can be simplified as
(8)
(9)
dW/dt
= F f W f / V - (Ff + F c ) W / V
(10)
dX/dt
= F ~ X c / V - (Fe + F c ) X / V
(11)
where W corresponds to the of the hypothetical acid and the reactor. Eliminating the unknown equations (8) and (9), we equation (12):
total concentration its ionic species in variable Cb from obtain the cubic
C 3 + ( g a - X)C2H
where W~ is the total concentration of each conjugate acid-base system present, and X represents the electroneutrality condition. W~ and X are the state variables which do not
- (K,X
+ K ~ W + K w ) C H - K , K w = 0.
(12)
Here we design a nonlinear self-tuning regulator which estimates the unknown para-
Adaptive control of pH systems meter Ka and the unmeasured influent weak acid concentration Ct using equations (10)-(12), and calculates the strong base flow rate which compensates for the variations of the effluent pH value with the estimated unknowns. Since we cannot apply the self-tuning regulator technique directly to equations (10)-(12), we derive a state equation by differentiating equation (12): (3C~ + 2KaCH -
1581
3.1. P a r a m e t e r estimation
Equation (19) can be parametrized in terms of unknown parameters, Ka and KaCf, as equation (20): [(2dk+, - Ff - Uk)C3k - ( d k + , 2 ~ + u k C c ) C h k + Kw(Ff + uk)CHk] + Ka[(dk+, - Ff - Uk)CZk - u k C c C m + (d1,+1 + Ff + uk)Kw] + K.Cy[FfCHk] - 6k[dk+, CZk]
2 X C H - K a X - K a W - Kw) d C H / d t - ( C ~ + K~CH) d X / d t - K~CH d W / d t = O.
(13) From the definition of pH, C H = 1 0 -pH
or
pH=-log(Cu)
d C u / d t = - C H In (10) dpH/dt.
(14) (15)
Substituting equations (10), (11) and (15) into equation (13) results in In (10)[3C3H + 2(ga - X ) C 2 -- ( K a W + K a g
+ gw)CH]
dpH/dt
+ 1/V[(FcXc - FfX - F~X)C 2
-- epH{V In (10)[2C3k + (Ka --
Sk - ak)C2k + K a K w ] }
= 0
(20)
where dk+l = V In (10)(pHk+l -- pHk)/T, Xk is the approximated value of X k ()(k = X k - 6k), and 6k is the error at t = k T caused by the difference, 60, between the true value of X at t = 0 and the initial guess, 20. The error 6k goes to zero exponentially as time increases and the error epH also becomes negligible with a small sampling period T. Let Yk+l -~ [(2d,+, - F f - Uk)C3k
+ K a ( F c X ¢ - F f X - FcX)CH] + 1 / V [ K a ( F f W f - F f W - FcW)CH] = 0.
(16)
Using equation (12) to eliminate W and replacing Wf and X¢ with C, and - C c equation (16) can be rewritten as
-- ( d k + l S k + ukCc)C2k + Kw(Ff + Uk)CHk]/C2Hk
(21)
(22)
0 ~ (Ka, K a C t ) T
qJk+, ~- ( - [(d,+, - F f - u , ) C ~ k - UkC¢CHk
dpH/dt = {Ff[C3 + KaC~4 - (Kw + K~COCH - K~Kw]
+ (dk+l + Ff + u k ) K w ] / C ~ , ,
+ Fc[C 3 + (K a + C c ) C 2
- [F,C.kllC~,k) T
Kw)CH - K a K w ] } / {V In (10)[2C 3 + (Ka - X ) C 2
(23)
+ ( K a C c --
+ KaKw]}.
ek+l =- {6kdk+,C2k + epH V In (10)
(17)
Then from equations (11) and (17) we get the following discrete models in X and pH: X k + , = X k -- T / V [ u k C c + (Ff + uk)Xk] + ex (18)
(pHk+l - pH~)/T = {FfIC3k + KaC2Hk
- (Kw +
+ Kaff)fHk
Uk[C3Hk+
(Ka +
-
K , Kw]
Cc)C2Hk
{V In (10)[2C3k + (Ka - Xk)C~k epH
(19)
where T is a sampling period, Uk is the Fc at instance k, and ex and epn are the errors due to discretization by the Euler integration method.
( K a - S k --
Ok)C2Hk
+ K.Kw]I/C~k
(24)
where the division by C~ik has been used to avoid numerical problems due to handling very small numbers. Then we have Yk+l
+ (KaCc - Kw)CHk -- KaKw]}/ + KaKwl} +
x [2C3k +
=
T Y l k + l O + ek+l.
(25)
Equation (25) is derived by assuming perfect measurements. In the case of measurement noises, the errors caused by measurements can be included in ek+l. We estimate the unknown parameters Ka and Cf of equation (25) via the recursive least squares method with a variable forgetting factor (Fortescue et al., 1981; Cordero and Mayne,
SANG DEUK LEE et al.
1582 198l), that is
To prevent extreme manipulated variable, a implemented as
A + , = W~+,0,¢ Y Kk+, = PkWk+ ,/(1 + Wk+lPk~k+,)
if 0k + l < 0mi, then 0k + 1 = 0mi,
(26)
if Ok+~ > 0m,x then Ok+ ~ = 0m,~ = 1 -
(1
- Wk+,Kk+O(yk+IT
of the filter is
Uk+l = exp ( - T/r)Uk + [1 -- exp (-- T/T)Iuc,, (28)
Ok+l ~- Ok ~- gk+i(yk+, -- Yk+,)
Ak+,
fluctuation first order
where r is the time constant of the first-order filter.
_ yk+I)2/Zo
4. S I M U L A T I O N R E S U L T S
if Ak+~ <
Amin
then hk+~
=
Amin
P,+, = (Pk - Kk+,WT'+,Pk)l,~k +, ^
where denotes an estimate, A is the variable forgetting factor and Z0 is a scalar quantity to speed up adaptation for sudden changes of a set point or load. 3.2. Control law With estimated values of Ka and Cf, we calculate the manipulated variable by the one-step-ahead control law from equation (19) (Goodwin and Sin, 1984) as uc,, = {V In (10)[2C3k+t
+ (/~a --
f(k+l)CHk+l
+ / ( . K w ] ( p H s - phk+,)/T - Ff[c-L,,+, + Rac~,~+, - (Kw + RaCf)Cnk+l -
R.Kw]}I
[ c L + , + ( S a + Cc)CL+~ + (R.Cc - Kw)CHk+i - K . K w ]
(27)
if Uca~< Umi,, then Ucal = Umm if U~aj> Umax then u~a~= Umax where pHs is the set point of p H at instance k+2. The denominator of equation (27) can easily be shown to be always greater than zero.
4.1. Single component weak acid and strong base system Simulations have been carried out for a process where the objective is to neutralize a single component weak acid stream by a strong base. To simulate the process, we calculated the reaction invariant variables, W and X, by integrating the model equations (10) and (11) analytically and obtained numerical solutions of CH and C b from equations (8) and (9) using a nonlinear equation solver. The data used for the simulation study are given in Table 1. The unknown parameters, Ka and Cf, of the influent weak acid stream are initially set to 5.0 × 10 _6 mol/l and 0.005 mol/1, respectively. At time t = 1000s, those are suddenly changed to 1.8× 10 -Smol/l and 0.01 mol]l in order to disturb the system. Simulations have been performed for the cases with perfect measurement and with measurement noises. A zero-mean uniform random noise with +0.21 pH units was used as a measure noise. In the case with perfect measurement, a random perturbation signal was added to the control input for parameter convergence. Figures 2 and 3 show the control input and output, and the parameter estimates in the case with perfect
TABLE [. DATA FOR THE SIMULATION STUDY IN THE CASE OF A SINGLE COMPONENT WEAK ACID AND STRONG BASE SYSTEM
Symbol V F~. b~
Value in process model
Kw T Po h,~,n
4001 -2.0 l/s 0.05 moll1 0.005 mol/I for 0 -< t -< 1000 s 0.01 mol/l for 1 0 0 0 < t - < 2 0 0 0 s 5.0 × 10 ~ mol/1 for 0 -< t -< 1000 s 1.8× 10 5mol/1 for 1 0 0 0 < t - < 2 0 0 0 s 1.0 × l 0 14 mo12/l 2 ls ---
Yo
--
C,, C~ K,
Value in controller 4001 limits: [0, 1.0 l/s] 2.0 l/s 0.05 tool/1 initial: 0.012 mol/l limits: [10 4, 1.0mol/l] initial: 1.0 × 10 5 mol/I limits: [10 s 10 2mol/I] 1.0 × 10 14 mo12/l 2 ls 103 I 0.1 (for perfect m e a s u r e m e n t ) 0.8 (for m e a s u r e m e n t noises) 10 -7 (for perfect m e a s u r e m e n t ) 5.0 (for m e a s u r e m e n t noises)
Adaptive control of pH systems
1583
l.fBO
F© 1,116 I ( I/see )
F~ ( 1/sec )
1
11.50
0 .~58
o.ee
,t,l,,,,I,,,,I,,,,I 8
588
1080
1588
2880
8.88 8
588
1888
1580
2888
1O
pH
I
10-
6
I
4
pH
2
e
•.............
4 '--
,,,I,,,,I,,,,I,,,,I
8
6 ~ .............
5BB IB80 tile(sec)
1588
2888
2 -
FIG. 2. Input and output in the case of a single component weak acid and strong base system with perfect measurement.
B
,,,,I,,,tl,,,,I,,,,I fl 588 I808
1580
2880
ti=e(se¢)
measurement, respectively. From these figures, we can see that the nonlinear pH controller gives good control performances. In the case with measurement noise in pH, the results are presented in Figs 4 and 5. In Fig. 5 we observe that the estimated values of Cf are very close to the true values, but those of K~ are highly oscillatory over a wide range and do not converge to their true values. This kind of
KoxlO s
(=olii)
c : , : .
8.0 8
508
i
1888
158o
,888
0.020 Cr
(=ol/l) 0.810
O.BeO
i,, 8
I,, 588
,
,,,,I
1088
....
1580
I 2888
tile(sec) n true -
value
estimated
value
FIG. 3. Parameter estimates in the case of a single component weak acid and strong base system with perfect measurement.
F1G. 4. Input and output in the case of a single component weak acid and strong base system with measurement noises.
problem may be solved by using other types of parameter estimation schemes such as a least squares method with dead zone (Middleton et al., 1988) or with covariance resetting (Ydstie, 1992). However, in spite of poor estimates of Ka and measurement noises the regulator works quite well as shown in Fig. 4. It appears that pH is less sensitive to Ka than to Cf, and the estimated values of a Ka-Cf pair and the true K,-Cf values give the same base flow rate, F~, at the operation point ( p H = 7 ) . Figure 6 shows that the estimated titration curves match well with the process titration curves near the neutral region, especially at pH = 7. In Fig. 6, the estimated titration curves were plotted using the estimates of Cf and the worst estimates of K a which were obtained at 680 and 1230 s as shown in Fig. 5. On the basis of reviewing Fig. 6, an alternative based on fixing K~ to some reasonable value and estimating only Cf instead of estimating both Ka and Cf may work well too. It will be studied in future research. According to the results of simulation studies, which are not presented in this paper, the control performance remains satisfactory even with a change in the pH set point from 5 to 7 and with the pH measurement noise. 4.2. Multi-component weak acid and strong base system In order to investigate the possibility of applying this regulator to wastewater treatment
1584
SANG DEUK LEE et al. TABLE
2.
D A ' I A F O R I H E S I M U L A ] ' I O N S I ' U I ) ' ~ IN ]HI-: ( ' A S [ O F A M t r l j F I C O M P O N E N I
Symbol
WEAK ACID AND SIRON(i
Value in process model
V
Value in controller
100 I
F,
2.0 l/s
C,. Cn, Cre
0.1 mol/l 0.01 mol/l, 0.02 mol/1 for 0 -< t <- 1000 s 0.005 mol/l, 0.01 mol/l for 1000 < t < 2000 s 1.8× 10-s mol/I, 1,0x 10- 7 mol/l for 0_
Kal Ka2
Kb~, Kt~2 Kdl, Kd2 Kw T P0
100 1 limits: [0, 1.0 l/s] 2.0 I/s 0.1 mol/l initial: 0.012 mol/l (as monoprotic weak acid) limits: [10 4, 1.0 mol/1] (as monoprotic weak acid) initial: 1.0 x 10 5 mol/l (as monoprotic weak acid) limits: [10 s, 10 2 mol/l] (as monoprotic weak acid) 1.0 × 10 I,~ mo12/i 2 ls 10-~1 0.1 (for perfect measurement) 0.8 (for measurement noises) 10 _3 (for perfect measurement) 5.0 (for measurement noises)
.)tmi n
Zo
systems, we have applied the regulator to a system which has the influent weak acid stream consisting of a mixture of a monobasic and a dibasic weak acid. The data for the simulations are summarized in Table 2. The results are shown in Figs 7 and 8 for the case with perfect measurement and in Figs 9 and 10 for the case with measurement noises. From these figures we can see that the regulator
10,0
m
KoxlO 4
BASE S'rSI [M
performs as well for multicomponent systems as for single component systems.
5. C O N C L U S I O N
We propose a nonlinear self-tuning regulator for controlling pH values of weak acid streams by a strong base. A standard recursive least square identification method and one-step-ahead control law are used in the regulator. Since a rigorous model describing the real pH neutralization process of a weak acid and strong base system is used, unknown parameters are estimated so that the estimated K a - C , pair and
(=oi/1 ) 5.0 14
0,0 0
500
1000
1500
12
2000
18
pR
0.820
8 6
Cr
(mol/1) 4
0,010
L~ 0,000
.... fl
2
~ ~ ~ , _ , - ~ I ....
I ....
500 1800 time(see)
I .... 1580
8
I
I
I
8,fl
2080
I
1
[
I
11
8.5
I
I 1.8
Fc(l/sec)
0 p _ror~___ t i t r a t i o o curve ~ u - i r ~ 0-1000 sectmds. • Estimated t i t r a t i o n curve at 650 ~ .
[] true v a l u e - estimated
Process t i t r l t i ~
value
FIG. 5. Parameter estimates in the case of a single component weak acid and strong base system with measurement noises.
•
curve d u r i r ~ 1000--2000 s e c t o r .
E s t i l a ~ e d t i t r a t i o n carve at 1280 s e c m ~ .
FIG. 6. Titration curves in the case of a single component weak acid and strong base system with measurement noises.
Adaptive control of pH systems 1.00
--
1.00
F©
Fc
(l/see)
(I/see)
0.511
-
8.58
itilllll
8.88 8
10
iliiliilll
588
1888
1508
8.88
2880
0
m
6 4
580
1888
1588
2888
18-
8L
8
pH
1585
I
I
pH
6
2 8
8
588
1888
1588
-
4"28 ,,,,I,,,,I,,,,I,,,,I
m
2888
8
588
time(see)
1888
1588
2888
time(see)
FIG. 7. Input and output in the case of a multicomponent weak acid and strong base system with perfect measurement.
FIG. 9. Input and output in the case of a multicomponent weak acid and strong base system with measurement noises.
2.0
3.8
K,xl0 e (,,o1/I)
2.0
IC,xlO~
1.0
(moll
,,,I,,,,
8.8 8
588
,,,,I,,,,I 1008
1588
)
) 1 8.8
2888
0
8.848
588
1088
1588
2888
8.048
Cr
Cf
(eolll)
(,,ol/l)
t
8.828
0.880
I
....
8
I ....
588
I ....
1888
8,828 I ....
1588
I
2888
time(sec)
FIG. 8. Parameter estimates in the case of a multicomponent weak acid and strong base system with perfect measurement.
t h e t r u e Ka-Cf v a l u e s give t h e s a m e b a s e flow r a t e at t h e o p e r a t i o n p o i n t . S i m u l a t i o n results s h o w t h a t the s e l f - t u n i n g r e g u l a t o r p r o v i d e s g o o d c o n t r o l p e r f o r m a n c e s in spite o f m e a s u r e m e n t n o i s e a n d v a r i a t i o n s o f influent w e a k acid c o n c e n t r a t i o n a n d d i s s o c i a t i o n c o n s t a n t o f the w e a k acid. F o r w i d e a p p l i c a t i o n s to r e a l p H p r o c e s s e s , f u r t h e r s t u d i e s such as t h e o r e t i c a l s t a b i l i t y
0,080
,,,
8
t .......
588
1888
I ....
1588
I
2888
time(sec)
FIG. 10. Parameter estimates in the case of a multicomponent weak acid and strong base system with measurement noises.
analysis, c o n s i d e r a t i o n o f n o i s e d y n a m i c s a n d e x p e r i m e n t a l v e r i f i c a t i o n to n e u t r a l i z a t i o n p r o cesses o f m u l t i c o m p o n e n t influent w e a k acid streams would be required.
Acknowledgements--Financial support from the Korea Ministry of Science and Technology and the Automation Research Center at POSTECH is gratefully appreciated.
SANe; D~t:K LE[~ et al.
1586 REFERENCES
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