- Email: [email protected]
Adaptive Linear Quadratic Control of an Extractor
Copyright © IFAC Adaptive Control of Chemical Processes, Copenhagen, Denmark, 1988
SEPARATION PROCESSES
ADAPTIVE LINEAR QUADRATIC CONTROL OF AN EXTRACTOR K. Najim*, E. Irving**, H. Youlal* and M. Najim*** *Ecole Natjollale Superieure d'lngenieurs de Genie Chilllique Ch em ill de la loge, 31078 Toulouse Cedex, Frallce **Electricite de France , Directioll des Etudes et Recherches. 1 avenue de Ghuiral de Gaulle. 92141 Clamart Cedex, France ***Ecoll' Natiollale Superieure d'Electrollique et de Radioeletririte de Bordeaux, 351 (Ours de la liberatioll, 33405 TalfllCf Cedex, FrailCl'
Abstract. The paper describes an application of an adaptive linear quadratic co ntrol algorithm to a pulsed liquid-liquid extraction column . The extractors are used in several industries as a separation technique. They perform a high degree of purification and are not heavy o n energy. The considered plant has the same dimensions as those cur rently used in fine chemical processes. The control objective is to optimize the column behaviour in spite of fluctuations in flow and physical properties of the solven t (extractor) and the solute (l iquid mixture) . The physical model developed for the column is too complex to use for control purpose. Therefore a single input-single o utput discrete time linear model is adopted. The selected control variables a r e the pulse frequency and the conductivity measured at the bottom of the column. The experie nces have been made with a mixture of water and toluene. The contro l algorithm is derived from a minimization of a quadratic cost function. The resulting Riccati eq uation is iterated until the c l osed-loop poles do belmng to a predefined s tabilit y domain included in the unit circle. The adaptive control algorithm involves a parameter identification procedure and a state-feedback control law which uses the estimated parameters. The estimation procedure uses normalised data and a forgetting factor which is adjusted to maintain the trace of the estimator gain matrix constant . Details on the practical implementation of the control algo rithm which is based on PDP-l 1 micro - computer are provided. The experimental results obtained show the ability of this algorithm to improve the efficien cy of the considered process. Keywords. Adaptive LQG control , Extraction columns, Process control, Robustness, Stabi lity
1. INTRODUCTION
cess lie in a predefined domain which is included in the classical stability domain (the unit circle). The usual Riccati equation associated with the LQG design is iterated until the constrained stability test is fulfilled. A transformation which maps the unit ci rcle into the restricted s tability domain is introduced t o ca rry out a more suitable state desc ription.
The necessit y of elaborating a robust control sys tem and the need of digital control in chemical engineering systems in order to improve quality and to optimize production efficiency is becoming more and more obvious. We present herein an application of an adaptive linear quadratic control algorithm to the control of a highly sensitive chemical process, a liquid-liquid extraction co lumn. The useof liquid-liquid extraction columns is gaining in creasing attention in several processes such as pharmaceutical, petrochemi cal , nuclear and hydrometal lurgical industries. The extraction is not heavy in energy compared to other separation techniques like the distillation, and leads to a high product purity. The closed loop dynamics are ge nerall y not, controlled by the LQG design. A controller using pure L.Q.G. does not provide the optimal performance under all conditions because the quadratic c riterion is the best compromise between conflicting demands. For example, in Astrom and Wittenmark (1984), design of LQG algorithm which guarantees exponential stability of the closed-loop system with all poles inside a circle with a given radius and centered at the origin has been described.
The structure of the paper is as follows : in next section the state repres enta tion of the system is ou tlined and the control objectives are stated. In section 3 the LQG basedJcontrol algorithm with a restricted closed loop stability requirement is developed. A robust parameter estimation algorithm and the adaptive form of the controller are presented in section 4. Practical features of the control algorithm as well as the implementation considerations are also discussed. In section 5 experimental results from the application of this control scheme to a liquid-liquid extraction column are presented. 2. PROCESS DESCRIPTION The liquid - liquid extraction co lumns have been inc reasingly used in several industries for the separation of constituents of a homogeneous liquid mixture. This separation technique is not energy consuming compared to the distillation. It leads to a high product purity. It consists in creating andintimate contact between the solution containi ng the component to be extracted (solute, continuous phase) with a solvent (extractor, dispersed phase) which play the role of the heat produced by the boiler in the case of separation by distillation.
Lam (1982) suggests to rescale the weighting factor in order to keep the closed-loop poles in the same position by taking into account any gain variation of the plant. While in Amerongen and co authors (1986), a method where by the control requirements are translated into the wieghting factor is proposed. In this paper, we present a method which can ensure that the closed loop poles of the controlled pro-
To improve the contact between the continuous phase
AC-N
189
190
K. Najim et al.
and the dispersed one. and consequently to enhance the performance of the plant. the extractor is intooduced counter currently in the column in the : form of drops through a distributor. and the column is equiped with a multiorifice plate s. The extraction efficiency is also increased by mechanical agitation. A schematic representation of the experimental setup used in this work is shown in figure 1. The co lumn is a cy lindrical glass ves sel. Its inside diameter is 5 cm while its height is 110 cm. The plate spacing is 5 cm. This column has dimensions of tho~ se commonly used in the fine chemistry. The mechanical agitation is performed by a pUlsing unit which is drived by a d.c. motor. The interface level. in the settling zone is controlled by means of a capacity probe. The dispersed and cont inuous phase flows are co ntrolled by pumps. The flow- rates are measured by means of electromagnetic flow-meters. The pulser is driven by a d.c. motor. A conductivity cell is set below the dispersed phase distrib utor and connected to a conductivit y sensor. This device affords a determination of the amount of dispersed phase contained in the co ntinuous one . The measured conductivity varies be tween two limits : the upper limit corresponds to the raffinate conduc tivity (water + residual solute) . The lower level corresponds to the conductivity of the dispers ed phase occupying practically all the base of the column. Previous hydrodynami c and mass transfer studies. Casamatta and Vogelphol (1985). have demonstrated that f or maintaining the co lumn at its optimal ope ratin g co nditions. it is necessa r y to maintain the operation in the vicinity of the flooding point by acting on the pulsing intensity. This operating zo ne is an unstable r egime . Increase in pulsation frequen cy can provoke movement toward the unstable flooding region : the co nduc tivit y the n attains the lower limit and v aries littl e . Return t o the desi red regime is possibl e but requires a long time and is much more difficult to obtain. Therefore. a robust adaptive cont ro l system with the cap ability to operate the co lumn in this diffi cul t mode. to recover normal operation and to s tabili ze it in the desired operating zone. is ne eded. Input-output model The plant und e r consideration is assumed t o be gove rn ed on the discrete-time set (t=1.2 •... ) by a linear input-output equation of the form : A(q-l)y(t) = B(q-l )u(t - k) + wet)
(1)
where ye t), u(t) and wet) are respe cti vely the system output. input and bound ed . disturbance, k is the system delay. A and B are polynomials in the backshift operator : -1
A(q)
-1
yet) y(t-1)
u(t-k-m+l HT
(1.0 ..... 0)
Let us now introduce the following variables y ' (t)
yet) - yc(t) 6(q-l)u(t) ; 6( q-l) = (1 _ q-l)
u' (t)
(5)
(6)
where yc(t) is the desired output at time t. The system (1) may be expressed as follows A"(q -1 )y ' (t) = B"(q -1 ) u' (t -k) + wet)
(7)
This representation ensures zero offset in steady state operation . From equations 1. 5. 6 and 7. it follows: A"(q-l) (1 _ q-l ) A(q-l ) B"(q-l)
B(q-l )
(8) (9)
with the notation: A"(q-l) = l+a"lq-l+ ... +a"n+lq-n-l
(10)
Equation (5) yields the following state-sp ace representation: X"(t+l) = <1>" X"(t) + [ u' (t)
( 11)
y' (t) = HT" X"(t) + wet)
( 12)
with X" (t) [
H"
= =
(y ' (t) , y' (t-l ) •..• y' (t-n-l ) • u' (t-k) , .•. , u' (t - k-m+ 1 ) }T
(b , 0, .... O. 1, .... O)T 1 (1, 0, ... , 0) T
( 13)
<1> " has the same structure as in (4) using the parameters of A"(q-l) and B"(q-l).
The controller design is so that the zeros of the closed loop characteristic equation lie within a ci rcl e C(8 . a) of radius a and ce nter ed at a point of coordinate 8 on the real axis. Let us introduce the followin g transformation of the matrix <1>" and o f the vectors [ ' and H' : <1> '
a (" - 8 I)
( 14 )
['
a ["
( 15)
H'
H"
( 16)
3. CONTROL ALGORITHM
-n
1 + a q + ... + anq 1 - 1 -1 -m B(q ) = b + b q + ... + b + q (2) 2 m 1 1 For the L.Q.G .• the following state-space representation has been used :
(4)
X(tp y(t-n+l) u(t-k)
The control design is based on the minimieation of the following quadratic performance criterio n: J
N(t) 2 2 L {y ' (t+i) + AU' (t+i-l )} i=l
( 17)
where A is the control weighting factor and N(t) is a time horizon which will be defined in the sequel. The control law for L.Q.G. design is given by : u' (t)
- L (N(t»
X' (t)
(18)
with : L(N(t)) S(N(t» S*(N(t»
,TS*(N(t»'+H'H,T = S(N(t)-1)-S(N(t)-1) [,[,TS(N(t)-l)(I+['TS(N(t) -l)[, )-l
5(0) = H'H,T
(19)
The closed loop system is governed by X' (t+l) = (' - ['L(N(t»X' (t)
(20)
Adaptive Linear Quadratic Control of an Extractor The parameter N(t), i.e. number of iterations, is such that the eigenvalues of the matrix /T{~' - r' L (N(t))} - S 7 are less than a in absolute values. This conditIon when it is verified ensures the robustness of the control algorithm. The contr~l gain L(N(t)) is of dimension (n+m+l). and will be written as : L(N(t)) -
L- 11.1 2 .···.1 n+m+l_7 T
(21)
The closed-loop system is governed by the following characteristic equation : A(q-l)G(q-l)+q-k B(q-l)F(q-l) _ 0 (22) where F(q-l) G(q-l)
11+12 q
-1
+ .. + lna+l q
1 + lna+2 q
-1
-na
+ .. + 1na+m+ 1 q
(23) -m
Stability requirements can be achieved by substituting in the characteristic polynomial the operator q-l by (S+aq-l). Consequently. any stability criterion can be used. 4. ADAPTIVE L.Q.G. Parameters estimation The development in the previous sections was based on the assumption that the process parameters were known. An adaptive strategy results when the parameters in the control law are substituted by estimates generated by an on line parameter estimation procedure. A constant trace algorithm (Irving. 1979) associated to data filtering and normalization (Praly. 1986) procedures has been used to estimate the process parameter. The U/D factorization method has been also used to avoid numerical rounding errors (Bierman, 1977). Software and hardware The column has been interfaced with a PDP-l 1 microcomputer using the multitasking microRSX operating system for programs development and execution supervision. The overall software involves three concurent tasks for data acquisition. algorithm computations and control signals updating. A supervision mode for the former tasks and for operator communication has been also implemented. It provides other facilities such as plant operation supervision. data loggings. as well as on-line facilities for the adjustement of some critical design parameters like for example the sampling rate. 5. RESULTS In practice the dynamical model order of the column is obviously unknown a priori. Good results have been obtained with the following choices : - na-3. nb-2 and k-l. with a sampling oeriod of 10 seconds. These values have been obtained from transient analysis - the stability domain is the circle of radius 0.9 centered at the origin. The variable yet) is measured at time t and the control variable u(t) is evaluated and applied to the actuator at time t+1. where 1 is the computation time related to the control algorithm. Figure 2 shows the variation of the conductivity. This signal has not filtered. The measured conductivity under adaptive control is very close to the desired value (0.45 mS/cm). The variation of the control signal (pulse frequency) is depicted on figure 3. This signal is not smooth because the output measurement is highly disturbed. Figures 4-a and 4-b show respectively the variation of continuous and dispersed flow rates. These
191
inputs are not constant. In spite of these flow rates variations, the column is maintained in its optimal behavior zone. 6. CONCLUSION We have presented in this paper an adaptive LQG control algorithm and its application to a liquidliquid extraction column. The control scheme is based on a reduced order model which parameters are identified using a robust estimation procedure. The control law is derived from the minimization of a quadratic cost function with a stability constraints on the closed-loop system. To achieve the restricted stability requirements. a simple transformation on the backward shift operator is introduced and the Riccati difference equation is iterated until the resulting closed-loop characteristic polynomial has all its roots in the prescribed stability domain. Experimental results show the ability of this control algorithm to improve the efficiency of this highly sensitive chemical plant. REFERENCES Amerongen J. Van. Nauta Lemke H.R. and Klu~t P.G.M. Van der "Adaptive adjustment of the weighting factors in a criterion", Journal A. Vol. 27. n03. pp. 163-168 (1986) Astrom. K.J. and B. Wittenmark. "Computer controlled systems, Theory and design". Prentice-Hall Inc, Englewood Cliffs. N.J. (1984) Bierman. G.J .• "Factorization methods for discrete sequential estimation". Academic Press. New York ( 1977) Casamatta, G. and A. Vogelphol. "Modeling of fluid dynamics and mass transfer in extraction COlumns",
German Chemical Engineering. Vol. 8. 96 (1985) Irving, E. et al.. "Improving power network stability and unit stress with adaptive generator control". Automatica. Vol. 15, 31 (1979) Lam. K.P •• "Design of stochastic discrete time linear optimal regulators". Int. J. Systems. Vol. 13, n09. pp. 979-1011 (1982) Praly. 1., "Robustesse des algorithmes de commande adaptative", in Commande adaptative : aspects theoriques et pratiques, I.D. Landau et L. Dugard. Eds, Masson (1986)
K. Najim et at.
192
13.
Cont. i nuolJs phMF!
=t~==:::!J
12.36
11. 72 -
Figure 4-a - Continoous phase flow in (l/h)
Di spersed phose
-A
-W -'"fO&~...'"
Pulse fr tlQuerrcy 14.928 .. 1•. 836 -
Conductlvfty
1•• 744
Figure 1 .- Liquid-liquid extraction column
14.652
.S
Figure 4-h
.14 .41
.32
.U
o.
o.
50.
100.
150.
200.
250.
Figure 2 - Behaviour of the cond uctivity signal in (mS /cm)
z. I. '"
I.. 1.28
1.04 .8
o.
Figure 3 - Behaviour of the pulsing frequency signal in (Hz)
250.
ryispersed phase flow in (l/h)
SEPARATION PROCESSES
ADAPTIVE LINEAR QUADRATIC CONTROL OF AN EXTRACTOR K. Najim*, E. Irving**, H. Youlal* and M. Najim*** *Ecole Natjollale Superieure d'lngenieurs de Genie Chilllique Ch em ill de la loge, 31078 Toulouse Cedex, Frallce **Electricite de France , Directioll des Etudes et Recherches. 1 avenue de Ghuiral de Gaulle. 92141 Clamart Cedex, France ***Ecoll' Natiollale Superieure d'Electrollique et de Radioeletririte de Bordeaux, 351 (Ours de la liberatioll, 33405 TalfllCf Cedex, FrailCl'
Abstract. The paper describes an application of an adaptive linear quadratic co ntrol algorithm to a pulsed liquid-liquid extraction column . The extractors are used in several industries as a separation technique. They perform a high degree of purification and are not heavy o n energy. The considered plant has the same dimensions as those cur rently used in fine chemical processes. The control objective is to optimize the column behaviour in spite of fluctuations in flow and physical properties of the solven t (extractor) and the solute (l iquid mixture) . The physical model developed for the column is too complex to use for control purpose. Therefore a single input-single o utput discrete time linear model is adopted. The selected control variables a r e the pulse frequency and the conductivity measured at the bottom of the column. The experie nces have been made with a mixture of water and toluene. The contro l algorithm is derived from a minimization of a quadratic cost function. The resulting Riccati eq uation is iterated until the c l osed-loop poles do belmng to a predefined s tabilit y domain included in the unit circle. The adaptive control algorithm involves a parameter identification procedure and a state-feedback control law which uses the estimated parameters. The estimation procedure uses normalised data and a forgetting factor which is adjusted to maintain the trace of the estimator gain matrix constant . Details on the practical implementation of the control algo rithm which is based on PDP-l 1 micro - computer are provided. The experimental results obtained show the ability of this algorithm to improve the efficien cy of the considered process. Keywords. Adaptive LQG control , Extraction columns, Process control, Robustness, Stabi lity
1. INTRODUCTION
cess lie in a predefined domain which is included in the classical stability domain (the unit circle). The usual Riccati equation associated with the LQG design is iterated until the constrained stability test is fulfilled. A transformation which maps the unit ci rcle into the restricted s tability domain is introduced t o ca rry out a more suitable state desc ription.
The necessit y of elaborating a robust control sys tem and the need of digital control in chemical engineering systems in order to improve quality and to optimize production efficiency is becoming more and more obvious. We present herein an application of an adaptive linear quadratic control algorithm to the control of a highly sensitive chemical process, a liquid-liquid extraction co lumn. The useof liquid-liquid extraction columns is gaining in creasing attention in several processes such as pharmaceutical, petrochemi cal , nuclear and hydrometal lurgical industries. The extraction is not heavy in energy compared to other separation techniques like the distillation, and leads to a high product purity. The closed loop dynamics are ge nerall y not, controlled by the LQG design. A controller using pure L.Q.G. does not provide the optimal performance under all conditions because the quadratic c riterion is the best compromise between conflicting demands. For example, in Astrom and Wittenmark (1984), design of LQG algorithm which guarantees exponential stability of the closed-loop system with all poles inside a circle with a given radius and centered at the origin has been described.
The structure of the paper is as follows : in next section the state repres enta tion of the system is ou tlined and the control objectives are stated. In section 3 the LQG basedJcontrol algorithm with a restricted closed loop stability requirement is developed. A robust parameter estimation algorithm and the adaptive form of the controller are presented in section 4. Practical features of the control algorithm as well as the implementation considerations are also discussed. In section 5 experimental results from the application of this control scheme to a liquid-liquid extraction column are presented. 2. PROCESS DESCRIPTION The liquid - liquid extraction co lumns have been inc reasingly used in several industries for the separation of constituents of a homogeneous liquid mixture. This separation technique is not energy consuming compared to the distillation. It leads to a high product purity. It consists in creating andintimate contact between the solution containi ng the component to be extracted (solute, continuous phase) with a solvent (extractor, dispersed phase) which play the role of the heat produced by the boiler in the case of separation by distillation.
Lam (1982) suggests to rescale the weighting factor in order to keep the closed-loop poles in the same position by taking into account any gain variation of the plant. While in Amerongen and co authors (1986), a method where by the control requirements are translated into the wieghting factor is proposed. In this paper, we present a method which can ensure that the closed loop poles of the controlled pro-
To improve the contact between the continuous phase
AC-N
189
190
K. Najim et al.
and the dispersed one. and consequently to enhance the performance of the plant. the extractor is intooduced counter currently in the column in the : form of drops through a distributor. and the column is equiped with a multiorifice plate s. The extraction efficiency is also increased by mechanical agitation. A schematic representation of the experimental setup used in this work is shown in figure 1. The co lumn is a cy lindrical glass ves sel. Its inside diameter is 5 cm while its height is 110 cm. The plate spacing is 5 cm. This column has dimensions of tho~ se commonly used in the fine chemistry. The mechanical agitation is performed by a pUlsing unit which is drived by a d.c. motor. The interface level. in the settling zone is controlled by means of a capacity probe. The dispersed and cont inuous phase flows are co ntrolled by pumps. The flow- rates are measured by means of electromagnetic flow-meters. The pulser is driven by a d.c. motor. A conductivity cell is set below the dispersed phase distrib utor and connected to a conductivit y sensor. This device affords a determination of the amount of dispersed phase contained in the co ntinuous one . The measured conductivity varies be tween two limits : the upper limit corresponds to the raffinate conduc tivity (water + residual solute) . The lower level corresponds to the conductivity of the dispers ed phase occupying practically all the base of the column. Previous hydrodynami c and mass transfer studies. Casamatta and Vogelphol (1985). have demonstrated that f or maintaining the co lumn at its optimal ope ratin g co nditions. it is necessa r y to maintain the operation in the vicinity of the flooding point by acting on the pulsing intensity. This operating zo ne is an unstable r egime . Increase in pulsation frequen cy can provoke movement toward the unstable flooding region : the co nduc tivit y the n attains the lower limit and v aries littl e . Return t o the desi red regime is possibl e but requires a long time and is much more difficult to obtain. Therefore. a robust adaptive cont ro l system with the cap ability to operate the co lumn in this diffi cul t mode. to recover normal operation and to s tabili ze it in the desired operating zone. is ne eded. Input-output model The plant und e r consideration is assumed t o be gove rn ed on the discrete-time set (t=1.2 •... ) by a linear input-output equation of the form : A(q-l)y(t) = B(q-l )u(t - k) + wet)
(1)
where ye t), u(t) and wet) are respe cti vely the system output. input and bound ed . disturbance, k is the system delay. A and B are polynomials in the backshift operator : -1
A(q)
-1
yet) y(t-1)
u(t-k-m+l HT
(1.0 ..... 0)
Let us now introduce the following variables y ' (t)
yet) - yc(t) 6(q-l)u(t) ; 6( q-l) = (1 _ q-l)
u' (t)
(5)
(6)
where yc(t) is the desired output at time t. The system (1) may be expressed as follows A"(q -1 )y ' (t) = B"(q -1 ) u' (t -k) + wet)
(7)
This representation ensures zero offset in steady state operation . From equations 1. 5. 6 and 7. it follows: A"(q-l) (1 _ q-l ) A(q-l ) B"(q-l)
B(q-l )
(8) (9)
with the notation: A"(q-l) = l+a"lq-l+ ... +a"n+lq-n-l
(10)
Equation (5) yields the following state-sp ace representation: X"(t+l) = <1>" X"(t) + [ u' (t)
( 11)
y' (t) = HT" X"(t) + wet)
( 12)
with X" (t) [
H"
= =
(y ' (t) , y' (t-l ) •..• y' (t-n-l ) • u' (t-k) , .•. , u' (t - k-m+ 1 ) }T
(b , 0, .... O. 1, .... O)T 1 (1, 0, ... , 0) T
( 13)
<1> " has the same structure as in (4) using the parameters of A"(q-l) and B"(q-l).
The controller design is so that the zeros of the closed loop characteristic equation lie within a ci rcl e C(8 . a) of radius a and ce nter ed at a point of coordinate 8 on the real axis. Let us introduce the followin g transformation of the matrix <1>" and o f the vectors [ ' and H' : <1> '
a (" - 8 I)
( 14 )
['
a ["
( 15)
H'
H"
( 16)
3. CONTROL ALGORITHM
-n
1 + a q + ... + anq 1 - 1 -1 -m B(q ) = b + b q + ... + b + q (2) 2 m 1 1 For the L.Q.G .• the following state-space representation has been used :
(4)
X(tp y(t-n+l) u(t-k)
The control design is based on the minimieation of the following quadratic performance criterio n: J
N(t) 2 2 L {y ' (t+i) + AU' (t+i-l )} i=l
( 17)
where A is the control weighting factor and N(t) is a time horizon which will be defined in the sequel. The control law for L.Q.G. design is given by : u' (t)
- L (N(t»
X' (t)
(18)
with : L(N(t)) S(N(t» S*(N(t»
,TS*(N(t»'+H'H,T = S(N(t)-1)-S(N(t)-1) [,[,TS(N(t)-l)(I+['TS(N(t) -l)[, )-l
5(0) = H'H,T
(19)
The closed loop system is governed by X' (t+l) = (' - ['L(N(t»X' (t)
(20)
Adaptive Linear Quadratic Control of an Extractor The parameter N(t), i.e. number of iterations, is such that the eigenvalues of the matrix /T{~' - r' L (N(t))} - S 7 are less than a in absolute values. This conditIon when it is verified ensures the robustness of the control algorithm. The contr~l gain L(N(t)) is of dimension (n+m+l). and will be written as : L(N(t)) -
L- 11.1 2 .···.1 n+m+l_7 T
(21)
The closed-loop system is governed by the following characteristic equation : A(q-l)G(q-l)+q-k B(q-l)F(q-l) _ 0 (22) where F(q-l) G(q-l)
11+12 q
-1
+ .. + lna+l q
1 + lna+2 q
-1
-na
+ .. + 1na+m+ 1 q
(23) -m
Stability requirements can be achieved by substituting in the characteristic polynomial the operator q-l by (S+aq-l). Consequently. any stability criterion can be used. 4. ADAPTIVE L.Q.G. Parameters estimation The development in the previous sections was based on the assumption that the process parameters were known. An adaptive strategy results when the parameters in the control law are substituted by estimates generated by an on line parameter estimation procedure. A constant trace algorithm (Irving. 1979) associated to data filtering and normalization (Praly. 1986) procedures has been used to estimate the process parameter. The U/D factorization method has been also used to avoid numerical rounding errors (Bierman, 1977). Software and hardware The column has been interfaced with a PDP-l 1 microcomputer using the multitasking microRSX operating system for programs development and execution supervision. The overall software involves three concurent tasks for data acquisition. algorithm computations and control signals updating. A supervision mode for the former tasks and for operator communication has been also implemented. It provides other facilities such as plant operation supervision. data loggings. as well as on-line facilities for the adjustement of some critical design parameters like for example the sampling rate. 5. RESULTS In practice the dynamical model order of the column is obviously unknown a priori. Good results have been obtained with the following choices : - na-3. nb-2 and k-l. with a sampling oeriod of 10 seconds. These values have been obtained from transient analysis - the stability domain is the circle of radius 0.9 centered at the origin. The variable yet) is measured at time t and the control variable u(t) is evaluated and applied to the actuator at time t+1. where 1 is the computation time related to the control algorithm. Figure 2 shows the variation of the conductivity. This signal has not filtered. The measured conductivity under adaptive control is very close to the desired value (0.45 mS/cm). The variation of the control signal (pulse frequency) is depicted on figure 3. This signal is not smooth because the output measurement is highly disturbed. Figures 4-a and 4-b show respectively the variation of continuous and dispersed flow rates. These
191
inputs are not constant. In spite of these flow rates variations, the column is maintained in its optimal behavior zone. 6. CONCLUSION We have presented in this paper an adaptive LQG control algorithm and its application to a liquidliquid extraction column. The control scheme is based on a reduced order model which parameters are identified using a robust estimation procedure. The control law is derived from the minimization of a quadratic cost function with a stability constraints on the closed-loop system. To achieve the restricted stability requirements. a simple transformation on the backward shift operator is introduced and the Riccati difference equation is iterated until the resulting closed-loop characteristic polynomial has all its roots in the prescribed stability domain. Experimental results show the ability of this control algorithm to improve the efficiency of this highly sensitive chemical plant. REFERENCES Amerongen J. Van. Nauta Lemke H.R. and Klu~t P.G.M. Van der "Adaptive adjustment of the weighting factors in a criterion", Journal A. Vol. 27. n03. pp. 163-168 (1986) Astrom. K.J. and B. Wittenmark. "Computer controlled systems, Theory and design". Prentice-Hall Inc, Englewood Cliffs. N.J. (1984) Bierman. G.J .• "Factorization methods for discrete sequential estimation". Academic Press. New York ( 1977) Casamatta, G. and A. Vogelphol. "Modeling of fluid dynamics and mass transfer in extraction COlumns",
German Chemical Engineering. Vol. 8. 96 (1985) Irving, E. et al.. "Improving power network stability and unit stress with adaptive generator control". Automatica. Vol. 15, 31 (1979) Lam. K.P •• "Design of stochastic discrete time linear optimal regulators". Int. J. Systems. Vol. 13, n09. pp. 979-1011 (1982) Praly. 1., "Robustesse des algorithmes de commande adaptative", in Commande adaptative : aspects theoriques et pratiques, I.D. Landau et L. Dugard. Eds, Masson (1986)
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