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Multivariable Long-Range Predictive Control Algorithm Applied to a Continuous Flow Fermentation Process
Cop\'rig hl © I FAC Illh Triennial World Congress. Tallinn. Estonia. L·SSR. 1'l~11I
MULTIVARIABLE LONG-RANGE PREDICTIVE CONTROL ALGORITHM APPLIED TO A CONTINUOUS FLOW FERMENTATION PROCESS B. Dahhou*'**,
J.
Bordeneuve** and
J.
P. Babary*
Labomtoire d'Automatique et d'Analyse des Systemes du C.N.R.S., 7, avenue du Colonel Roche, 31077 Toulouse Ctidex, France *c. T.B.M .lUlliversitt! Paul Sabatier, avenue de Rangueil, 31077 Toulouse Ctidex, France **Greco-Sarta Systemes Adaptatifs
Abstract: The dynamics of biotechnological processes are highly complex and usually unknown . These processes are described by a set of non linear and non stationary equations derived from mass balance considerations. The complexity of these systems limits the application of classical control techniques and indicates that adaptive techniques may be of interest. This paper presents a simulation study of the multivariable generalized predictive control applied to a continuous flow fermentation process . The control design is based on predicting the process outputs over several steps and assumptions on future control actions. The control design is based on a linear discrete-time model with unknown and possibly varying parameters . The simulation is performed using a nonlinear and time-varying model of the process. These parameters are estimated by a constant trace adaptation a l gorithm with necessary features (data filtering and normalisation, UD factorization, dead zone . . ) that enhance the robustness of the resulting self-tuning algorithm. In the parametric model, the dilution rate and the influent substrate concentration are the control variables (Inputs) and the biomass and substrate concentrations stand for the controlled variables (Outputs). Numerical results illustrate the good performance and the excellent properties of the presented control scheme, particularly its ability to cope with varying dynamics and high level disturbances.
Keywords:
Adaptive
control;
bioreactor;
l ong
range
prediction;
robust
parameter
estimator.
I - I NTRODUCT I ON
solved using the Runge Kutta-Merson method. A linear discrete-time stochastic equation with time varying p a r ameters which are identified on-line to obtain the best r epresentation at every sampling period (in t h e squares sense) is used for the
Over the last years, a great number of studies have been devoted to the theoretical investigat i on and practical implementation of adaptive control. Most of these theoretical developments apply to linear systems only. However, the adaptive contro l of nonlinear systems is still a challenging issue, despite many successful attempts. The plants used in the biotechnological industry are highly interactive, interconnected systems best examplified by continuous flow fermentation processes because biological transformations are carried out in them. These processes involve highly complex dynamics which normally are not completly known. This complexity limits the application of standard optimal control techniques and indicates the need for adaptive techniques (Munack and Thomas, 1986). These techniques can track changes in the process dynamics and adjust the control signals in accordance with a spec i fied control objective. Over the last decade, many adaptive predictive control algorithms have been developed and successfully applied to different industries. The Generalized Predictive Control (GPC) algorithm developed by Clarke and co-authors (1987) is part of the long range predictive control methods, a nd has shown high performance in simulation a n d in practica l app l ications. In this paper, the multivariab l e GPC i s a pp l ied to the control of a continuous flow fermentati on p rocess. This process is described b y a s et o f nonlinear differential equations used f or simulation purposes. The model equ a t ion s are
bioreactor
d ynamics
representation.
To overcome
the prob l em associated with the on-line impleme n tation of the estimation part, such as the b l ow-up of the covariance matrix, the boundedness of the mode l error and the conditions concerning persistency of excitation, we make use of regularized UD constant trace algorithm with norma l ized data and of an information measure to detect the satisfactory process operation (in steady state). Practical aspects of the implementation as well as the commissioning of the control al gor ithm are also discussed. This paper is organ ized as follows: in part 11, a br ief desc r iption of the non linear model of the f erment a t i on p rocess used is given. In Section Ill , t h e multivar i able generalized predictive con t r o l method i s outlined. The identification algorithm is descr i bed in Section IV. To illu st r ate the u se of the multivariable GPC algo ri thm and to eva l uate its performa.)ce, simulation re sults are presented in Section V.
11 - PROCESS MODEL schemat i c d i agr a m o f the continuous fl ow f ermen t at i on i s s h own in Fig. 1 and briefly descr i bed below . The g rowth me di um, con taining a carbon-rich A
393
substrate (glucose), flow continuously through a bioreactor, whose biological activity is maintained to accomplish the conversion from substrate to product (alcohol). In biotechnological processes, the evolution of the bacterial growth is usually described by a set of nonlinear equations derived from material balance considerations. The model equations are :
The controller design is based on the minimization of the following cost function
(1) [ ds/dt - - (l/R) .IJ. . X + u. (Sa -s) where x s u
is the biomass concentration is the substrate concentration is the dilution rate (i.e., influent flow rate/volume) Sa is the influent substrate concentration R is a yield coefficient lJ.(x , s) is the specific growth rate
y(t+j) - q-d A- 1 (q-1) B(q-1) l,l(t+j) + + A- 1 (q-1) D- 1 (q'1) ~(t+j) (5) The random term A- 1 (q-1) D- 1 (q-1) ~(t+j) contains terms which are independent of the observations. To separate these terms, the matrices E(q-1) and F(q-1) are defined such that: Im - D(q - ') E(q - 1) A(q - ') + q - j F (q - 1 ) (6)
In this representation, the specific growth rate lJ.(t) is known to be a complex function of several (biomass and substrate concentrations,
pH, ... ). Many analytical laws have been suggested in the literature for modelling this parameter (Monod, 1942; Andrews, 1968). The most popular expression is certainly Monod's law :
this equation is the first Diophantine equation. Then D- 1 (q-1 )A" (q-1
)~(t+j)
F(q
(2) where
IJ.max is
the maximum
growth rate and
Ks
_ E(q-1
)~(t+j
)+D-1 (q-1)
-1)A-1(q-1)~(t+j)
Multiplying equation (5) by D(q-1) E(q-1) substituting the term D(q-1 )A(q-1 )E(q-1) equation (6) yields :
is
the "Michaelis-Menton" parameter.
Note that this expression is far from being the only one as in the literature, more than thirty different models are detailed for the specific growth rate lJ.(t) . Therefore, the choice of an appropriate model for lJ.(x,s), is not easy
(7)
then from
y(t+j )_D(q-1 )E(q-1 )B(q-1 )l,l(t+j _d)+D(q-1 )~(t+j) +F(q-1 )y(t) or :
y(t+j )_E(q-1 )B(q-1 )cSl,l(t+j -d) + E(q-1 )~(t+j)
III - MULTIVARIABLE GENERALIZED PREDICTIVE CONTROL ALGORITHM
+F(q-1 )y(t)
(8)
the first term of the second member can be written
To regulate biomass and substrate concentrations, an input/output process model using the dilution rate and the influent substrate concentration as the control variables is considered. It is represented in a form suitable for digital control by means of the following equation : A(q-1) yet) - B l,l(t-d) + D- 1 (q-1) ~(t)
(4)
where ~(t) is the vector of reference sequence. The weighting parameter makes it possible to trade off decreased variance in control effort against an increased regulation error (y(t) - ~(t». Clarke et al. (1987) developed an algorithm to solve this problem. In this approach the prediction horizon Np is taken greater than the time delay. N is the minimum prediction horizon. To rewrite the criterion J in a more suitable form, consider the process output at time t+j :
dx/dt - (IJ. - u).x
parameters
[[y(t+j) - ~(t+j)l2 +~ [cSl,l(t+j-dll
J -
as
:
E(q-1 )B(q-1 )cSl,l(t+j -d)_R(q-1 )cSl,l(t+j -d) +S(q-1 )cSl,l(t-d)
(9)
with:
(3)
where A(q-1), B(q- 1 ), D(q-1) are matrices in the backward-shift operator defined by q- i yet) y(t-i).
S(q-1) _ So + S, q-1 +
-n
+ Sn s q
s
equation (9) can be rewritten as :
(10) which is the second Diophantine equation. D(q-1) - Im (1_q-1) and
The output y(t+j) is given by :
D(q-1 )y(t) - cSy(t) - yet) -y(t-l)
y(t+j) - R(q - 1) cSl,l(t+j -d) + S(q-1) cSl,l(t-d)
The matrix integrator D- 1 (q-1) is introduced to eliminate the steady-state error. t denotes discrete time index (number of sampling period), d is the process time delay in integer number of sample intervals. yet) and l,l(t) denote the system output vector (biomass and substrate concentrations) and input vector (dilution rate and influent substrate concentration) respectively . ~(t) is an uncorrelated random sequence of zero mean and finite variance.
+F(q- 1 )y(t) +E(q-1 )~(t+j) Define
ff(t+j)
(11)
by :
ff(t+j) - S(q-1) cSl,l(t-d) + F(q-1) yet)
(12)
In this vector, we have gathered the available data (input-output measurements). The output vector is written as :
394
(13)
parameter-estimates behaviour and reduces their variations. The latter property is quite consistent with stability requirements in adaptive control (M'Saad, 1987). Normalization of input-output data is useful to prevent the effects of unbounded modelling errors and to ensure the boundedness of the signals before processing by the algorithm (Praly, 1986). Then, the output, the regressor and the noise are
with
'1.-
[yT(t+N)
11-
[ogT (t+N-1)
ff -
[.e (t+N)
i(t+Np)P OgT(t+Np_d)]T
.e (t+Np)]T
....
[~ ~T (t+N) .. the dimension of Hp - Np-N and
EN
p-1
divided by the special norm ~~(t), given by :
~T (t+N p )] T
these vectors
~ (t) - ~ ~ (t - 1) + (1- ~) max (W: ( t) W f (t), ~o) (20)
is (m*H p ) , with
where 0 ~ ~ ~ 1
~o
> 0
To ensure numerical efficiency, it is advisable to use the U-D factorization to update the estimator gain, i.e.,
o G-
pet) - U(t) D(t) UT (t) RNp-1···· · ··· ·· ·· · ···RN-1
the reference vector is [HT (t+N) . .. HT (t+N p ) ]T
(22)
The cost function will be written as J - E[(G11+ff+~-H)T (14) The
prediction
(G11+ff+~-H) + ~ 11T111
error
is
where d0 is a regularizing constant. It is therefore advisable to compute an information measure at every sampling interval using the input-output data, and to implement a decision rule to take into consideration the way in which the current information can improve the estimation process. The following function can be used to measure the extent to which the incoming information differs from the preceding one:
with the
uncorrelated
available data.
H) + 2 [(GTG +
~11
I - 0
~ Im) 11 + GT (ff - H)]
o
(15)
set) -
where Im is the identity matrix.
For a prediction j > d, the variations of control signals og(t+l), ... ,o\O(t+j -d) are unknown. these variations are equal to zero,
the dimension of control horizon) .
G matrix
then
is (m*N u ). (Nu is the
IV - PARAMETER ESTIMATION The
plant model
can be
w)
(23)
estimates.
written in the following V - SIMULATION RESULTS
form DYf (t) - aT W f (t-l) + DW f (t) where
(t) P(t-l) W f (t)
If set) is less than a certain value, the parameters can be frozen at their previous values. These improvements are used together with the recursive least-squares algorithm, in which the trace of the covariance matrix is kept constant . Robustness of this parameter adaptation algorithm has been provided through appropriate data treatment. Control law implementation is relatively simple, and some simulation results are shown in the following section. This control law (16) can be implemented in a self-tuning context, by replacing these unknown matrices by their
The control law is therefore given by :
Assuming
(21)
where the factor U is an unitary upper triangular matrix and the factor D is a positive diagonal matrix (Bierman, 1977). It is straightforward to ensure a lower bound on the estimator gain matrix by monitoring the elements of D(t) - diag (d; (t»), as suggested in Ljung and Soderstrom (1983), i.e.,
the dimension of G matrix are (m*Hp,m*H p )
H -
and
a
is
the
parameter
The simulation experiments described here were carried out using the Runge Kutta's fourth order algorithm to integrate the nonlinear physical model of the bioreactor, described by the set of equations (1) used as the true system with the specific growth rate following Monod's law:
(17)
matrix, and
~
is the
regressor vector :
~T (t-l) -
~(s)
[-y(t-l) ... -y(t-na) g(t-d) . .. g(t-d-nb) (19)
~max S - ---
Ks + S
with ~max
Of"
denotes signal filtering by G'/F', G' and F' being two asymptotically stable polynomials. Low - pass filtering of the measurements may be used in the parameterestimation procedure in order to reduce the high frequency modes due to both noise and unmodelled dynamics. This ensures smooth
-
0.3 h- 1 , Ks - 4.5 g/£ and R - 0.5
the initial state values were
3~5
; 1"
12.---------------------------------------------, "
and the sampling period T - 1 h. the control and the prediction horizons were chosen respectively to be equal to Nu - 1 (control horizon) and Np - 6 (prediction horizon).
10
!
Simulation experiments are shown for biomass and substrate concentrations tracking. performed with a square wave reference signal varying between 5 g/e and 10 g/e for the first output (biomass concentration) and between 3 g / e and 6 g/e for the second output (substrate concentration). Realistic measurements are reproduced by adding white noise signal to the outputs (biomass and substrate concentrations). Figures 2a - 2b show the evolution of the inputs variables (di lution rate and influent substrate concentration). The evolutions of outputs variables (biomass and substrate concentrations) are given in Figs 2c 2d.
,
I
,
time (h), lOO
Fig . 2c
200
JOO
Evoluti o n o f bi omass c o ncentration
l J~ " 'I.J,Jllj :~ f I
Another simulation was performed in the same conditions described above. except for reference signals; in this second simulation. these reference signals are not synchronous. The evolution of inputs and outputs variables is shown in Figs 3a - 3b - 3c - 3d.
l'
, 'I ,
Feeding
x.s
time (h) lOO
Drawing
u
200
JOO
x,s
Fig . 2d Fig . 1
I
i.
I
Bioreactor
u Sa
I
,
Evoluti o n o f substrate concentration
Schematic representation of the continuous fermentation process.
0.' 0.20
!I
0.3 ...,.-,- - - - - - - - - - - - - - - - - - -
1,
0.20
M'l
0.20
:: l 0.2
0.24
0.22
l
Cl.>
0.111 -!
0.11 0. 14
1 1
0.<>4
0-'"
time (h) l OO
200
t ime (h 100
JOO
Fig . 2a : Ev o luti on o f diluti on rate
200
JOO
Fig. 3a : Evolution o f diluti o n rate
'2.-------------------------------------~~------_,
'2,-------------~------------------------------,
JO
JO :lA
21
24 22 20 11 10
14
time (h) 100
Fig. 2b
time
200
100
Evolution of influent substrate conce ntrati o n
Fig . 3b
396
200
JOO
Evo luti o n o f influent substrate conce ntrati o n
13,------------------------------------------------,
Ljung, L. and T. S6derstr6m (1984), Theory and practice of recursive identification, MIT Press, Cambridge, MA. Monod, J. (1942), Recherche sur la croissance des cultures bacteriennes , Ed. Herman, Paris . M'Saad, M. (1987). Sur l'applicabilite de la commande adaptative. These d'Etat, Institut National Poly technique de Grenoble. Praly, L. (1986), Robustesse des algorithmes de commande adaptative. In 1.0. Landau et L. Dugard. Ed. Commande adaptative : Aspects theoriques et pratiques. Mass on, Paris .
12 11
I.
time (h 100
Fig . 3c
Fig. 3d
2DO
Ev o luti o n o f biomass concentration
Ev o lution o f substrate concent r ation
v -
CONCLUSION
In this paper a multivariable generalized predictive control algorithm combined with a constant trace identification method is used for the control of a nonlinear fermentation process. This is a difficult process due to complex relationships between the variables of interest in the physical model. The simulation results show the good performance of this adaptive predictive algorithm, mainly in the case where the process is subject to high disturbances.
REFERENCES Andrew, J. (1968), A mathematical model for the continuous culture of micro-organisms utilising inhibitory substrate . Biotechnology and Bioengineering, vol. 10 Bierman, G.J. (1977), Factorization methods for discrete sequential estimation. Academic Press, New York. Clarke, D.W., C. Mohtadi and P . S. Tuffs (1987), Generalized predictive control. Part. I . The basic Algorithm and Part 11. Extensions and interpretations. Automatica vol 23 n° 2, 137-160. Kucera, V. (1979), Discrete linear control: The polynomial equation approach. John Wi1ey & Sons.
397
MULTIVARIABLE LONG-RANGE PREDICTIVE CONTROL ALGORITHM APPLIED TO A CONTINUOUS FLOW FERMENTATION PROCESS B. Dahhou*'**,
J.
Bordeneuve** and
J.
P. Babary*
Labomtoire d'Automatique et d'Analyse des Systemes du C.N.R.S., 7, avenue du Colonel Roche, 31077 Toulouse Ctidex, France *c. T.B.M .lUlliversitt! Paul Sabatier, avenue de Rangueil, 31077 Toulouse Ctidex, France **Greco-Sarta Systemes Adaptatifs
Abstract: The dynamics of biotechnological processes are highly complex and usually unknown . These processes are described by a set of non linear and non stationary equations derived from mass balance considerations. The complexity of these systems limits the application of classical control techniques and indicates that adaptive techniques may be of interest. This paper presents a simulation study of the multivariable generalized predictive control applied to a continuous flow fermentation process . The control design is based on predicting the process outputs over several steps and assumptions on future control actions. The control design is based on a linear discrete-time model with unknown and possibly varying parameters . The simulation is performed using a nonlinear and time-varying model of the process. These parameters are estimated by a constant trace adaptation a l gorithm with necessary features (data filtering and normalisation, UD factorization, dead zone . . ) that enhance the robustness of the resulting self-tuning algorithm. In the parametric model, the dilution rate and the influent substrate concentration are the control variables (Inputs) and the biomass and substrate concentrations stand for the controlled variables (Outputs). Numerical results illustrate the good performance and the excellent properties of the presented control scheme, particularly its ability to cope with varying dynamics and high level disturbances.
Keywords:
Adaptive
control;
bioreactor;
l ong
range
prediction;
robust
parameter
estimator.
I - I NTRODUCT I ON
solved using the Runge Kutta-Merson method. A linear discrete-time stochastic equation with time varying p a r ameters which are identified on-line to obtain the best r epresentation at every sampling period (in t h e squares sense) is used for the
Over the last years, a great number of studies have been devoted to the theoretical investigat i on and practical implementation of adaptive control. Most of these theoretical developments apply to linear systems only. However, the adaptive contro l of nonlinear systems is still a challenging issue, despite many successful attempts. The plants used in the biotechnological industry are highly interactive, interconnected systems best examplified by continuous flow fermentation processes because biological transformations are carried out in them. These processes involve highly complex dynamics which normally are not completly known. This complexity limits the application of standard optimal control techniques and indicates the need for adaptive techniques (Munack and Thomas, 1986). These techniques can track changes in the process dynamics and adjust the control signals in accordance with a spec i fied control objective. Over the last decade, many adaptive predictive control algorithms have been developed and successfully applied to different industries. The Generalized Predictive Control (GPC) algorithm developed by Clarke and co-authors (1987) is part of the long range predictive control methods, a nd has shown high performance in simulation a n d in practica l app l ications. In this paper, the multivariab l e GPC i s a pp l ied to the control of a continuous flow fermentati on p rocess. This process is described b y a s et o f nonlinear differential equations used f or simulation purposes. The model equ a t ion s are
bioreactor
d ynamics
representation.
To overcome
the prob l em associated with the on-line impleme n tation of the estimation part, such as the b l ow-up of the covariance matrix, the boundedness of the mode l error and the conditions concerning persistency of excitation, we make use of regularized UD constant trace algorithm with norma l ized data and of an information measure to detect the satisfactory process operation (in steady state). Practical aspects of the implementation as well as the commissioning of the control al gor ithm are also discussed. This paper is organ ized as follows: in part 11, a br ief desc r iption of the non linear model of the f erment a t i on p rocess used is given. In Section Ill , t h e multivar i able generalized predictive con t r o l method i s outlined. The identification algorithm is descr i bed in Section IV. To illu st r ate the u se of the multivariable GPC algo ri thm and to eva l uate its performa.)ce, simulation re sults are presented in Section V.
11 - PROCESS MODEL schemat i c d i agr a m o f the continuous fl ow f ermen t at i on i s s h own in Fig. 1 and briefly descr i bed below . The g rowth me di um, con taining a carbon-rich A
393
substrate (glucose), flow continuously through a bioreactor, whose biological activity is maintained to accomplish the conversion from substrate to product (alcohol). In biotechnological processes, the evolution of the bacterial growth is usually described by a set of nonlinear equations derived from material balance considerations. The model equations are :
The controller design is based on the minimization of the following cost function
(1) [ ds/dt - - (l/R) .IJ. . X + u. (Sa -s) where x s u
is the biomass concentration is the substrate concentration is the dilution rate (i.e., influent flow rate/volume) Sa is the influent substrate concentration R is a yield coefficient lJ.(x , s) is the specific growth rate
y(t+j) - q-d A- 1 (q-1) B(q-1) l,l(t+j) + + A- 1 (q-1) D- 1 (q'1) ~(t+j) (5) The random term A- 1 (q-1) D- 1 (q-1) ~(t+j) contains terms which are independent of the observations. To separate these terms, the matrices E(q-1) and F(q-1) are defined such that: Im - D(q - ') E(q - 1) A(q - ') + q - j F (q - 1 ) (6)
In this representation, the specific growth rate lJ.(t) is known to be a complex function of several (biomass and substrate concentrations,
pH, ... ). Many analytical laws have been suggested in the literature for modelling this parameter (Monod, 1942; Andrews, 1968). The most popular expression is certainly Monod's law :
this equation is the first Diophantine equation. Then D- 1 (q-1 )A" (q-1
)~(t+j)
F(q
(2) where
IJ.max is
the maximum
growth rate and
Ks
_ E(q-1
)~(t+j
)+D-1 (q-1)
-1)A-1(q-1)~(t+j)
Multiplying equation (5) by D(q-1) E(q-1) substituting the term D(q-1 )A(q-1 )E(q-1) equation (6) yields :
is
the "Michaelis-Menton" parameter.
Note that this expression is far from being the only one as in the literature, more than thirty different models are detailed for the specific growth rate lJ.(t) . Therefore, the choice of an appropriate model for lJ.(x,s), is not easy
(7)
then from
y(t+j )_D(q-1 )E(q-1 )B(q-1 )l,l(t+j _d)+D(q-1 )~(t+j) +F(q-1 )y(t) or :
y(t+j )_E(q-1 )B(q-1 )cSl,l(t+j -d) + E(q-1 )~(t+j)
III - MULTIVARIABLE GENERALIZED PREDICTIVE CONTROL ALGORITHM
+F(q-1 )y(t)
(8)
the first term of the second member can be written
To regulate biomass and substrate concentrations, an input/output process model using the dilution rate and the influent substrate concentration as the control variables is considered. It is represented in a form suitable for digital control by means of the following equation : A(q-1) yet) - B l,l(t-d) + D- 1 (q-1) ~(t)
(4)
where ~(t) is the vector of reference sequence. The weighting parameter makes it possible to trade off decreased variance in control effort against an increased regulation error (y(t) - ~(t». Clarke et al. (1987) developed an algorithm to solve this problem. In this approach the prediction horizon Np is taken greater than the time delay. N is the minimum prediction horizon. To rewrite the criterion J in a more suitable form, consider the process output at time t+j :
dx/dt - (IJ. - u).x
parameters
[[y(t+j) - ~(t+j)l2 +~ [cSl,l(t+j-dll
J -
as
:
E(q-1 )B(q-1 )cSl,l(t+j -d)_R(q-1 )cSl,l(t+j -d) +S(q-1 )cSl,l(t-d)
(9)
with:
(3)
where A(q-1), B(q- 1 ), D(q-1) are matrices in the backward-shift operator defined by q- i yet) y(t-i).
S(q-1) _ So + S, q-1 +
-n
+ Sn s q
s
equation (9) can be rewritten as :
(10) which is the second Diophantine equation. D(q-1) - Im (1_q-1) and
The output y(t+j) is given by :
D(q-1 )y(t) - cSy(t) - yet) -y(t-l)
y(t+j) - R(q - 1) cSl,l(t+j -d) + S(q-1) cSl,l(t-d)
The matrix integrator D- 1 (q-1) is introduced to eliminate the steady-state error. t denotes discrete time index (number of sampling period), d is the process time delay in integer number of sample intervals. yet) and l,l(t) denote the system output vector (biomass and substrate concentrations) and input vector (dilution rate and influent substrate concentration) respectively . ~(t) is an uncorrelated random sequence of zero mean and finite variance.
+F(q- 1 )y(t) +E(q-1 )~(t+j) Define
ff(t+j)
(11)
by :
ff(t+j) - S(q-1) cSl,l(t-d) + F(q-1) yet)
(12)
In this vector, we have gathered the available data (input-output measurements). The output vector is written as :
394
(13)
parameter-estimates behaviour and reduces their variations. The latter property is quite consistent with stability requirements in adaptive control (M'Saad, 1987). Normalization of input-output data is useful to prevent the effects of unbounded modelling errors and to ensure the boundedness of the signals before processing by the algorithm (Praly, 1986). Then, the output, the regressor and the noise are
with
'1.-
[yT(t+N)
11-
[ogT (t+N-1)
ff -
[.e (t+N)
i(t+Np)P OgT(t+Np_d)]T
.e (t+Np)]T
....
[~ ~T (t+N) .. the dimension of Hp - Np-N and
EN
p-1
divided by the special norm ~~(t), given by :
~T (t+N p )] T
these vectors
~ (t) - ~ ~ (t - 1) + (1- ~) max (W: ( t) W f (t), ~o) (20)
is (m*H p ) , with
where 0 ~ ~ ~ 1
~o
> 0
To ensure numerical efficiency, it is advisable to use the U-D factorization to update the estimator gain, i.e.,
o G-
pet) - U(t) D(t) UT (t) RNp-1···· · ··· ·· ·· · ···RN-1
the reference vector is [HT (t+N) . .. HT (t+N p ) ]T
(22)
The cost function will be written as J - E[(G11+ff+~-H)T (14) The
prediction
(G11+ff+~-H) + ~ 11T111
error
is
where d0 is a regularizing constant. It is therefore advisable to compute an information measure at every sampling interval using the input-output data, and to implement a decision rule to take into consideration the way in which the current information can improve the estimation process. The following function can be used to measure the extent to which the incoming information differs from the preceding one:
with the
uncorrelated
available data.
H) + 2 [(GTG +
~11
I - 0
~ Im) 11 + GT (ff - H)]
o
(15)
set) -
where Im is the identity matrix.
For a prediction j > d, the variations of control signals og(t+l), ... ,o\O(t+j -d) are unknown. these variations are equal to zero,
the dimension of control horizon) .
G matrix
then
is (m*N u ). (Nu is the
IV - PARAMETER ESTIMATION The
plant model
can be
w)
(23)
estimates.
written in the following V - SIMULATION RESULTS
form DYf (t) - aT W f (t-l) + DW f (t) where
(t) P(t-l) W f (t)
If set) is less than a certain value, the parameters can be frozen at their previous values. These improvements are used together with the recursive least-squares algorithm, in which the trace of the covariance matrix is kept constant . Robustness of this parameter adaptation algorithm has been provided through appropriate data treatment. Control law implementation is relatively simple, and some simulation results are shown in the following section. This control law (16) can be implemented in a self-tuning context, by replacing these unknown matrices by their
The control law is therefore given by :
Assuming
(21)
where the factor U is an unitary upper triangular matrix and the factor D is a positive diagonal matrix (Bierman, 1977). It is straightforward to ensure a lower bound on the estimator gain matrix by monitoring the elements of D(t) - diag (d; (t»), as suggested in Ljung and Soderstrom (1983), i.e.,
the dimension of G matrix are (m*Hp,m*H p )
H -
and
a
is
the
parameter
The simulation experiments described here were carried out using the Runge Kutta's fourth order algorithm to integrate the nonlinear physical model of the bioreactor, described by the set of equations (1) used as the true system with the specific growth rate following Monod's law:
(17)
matrix, and
~
is the
regressor vector :
~T (t-l) -
~(s)
[-y(t-l) ... -y(t-na) g(t-d) . .. g(t-d-nb) (19)
~max S - ---
Ks + S
with ~max
Of"
denotes signal filtering by G'/F', G' and F' being two asymptotically stable polynomials. Low - pass filtering of the measurements may be used in the parameterestimation procedure in order to reduce the high frequency modes due to both noise and unmodelled dynamics. This ensures smooth
-
0.3 h- 1 , Ks - 4.5 g/£ and R - 0.5
the initial state values were
3~5
; 1"
12.---------------------------------------------, "
and the sampling period T - 1 h. the control and the prediction horizons were chosen respectively to be equal to Nu - 1 (control horizon) and Np - 6 (prediction horizon).
10
!
Simulation experiments are shown for biomass and substrate concentrations tracking. performed with a square wave reference signal varying between 5 g/e and 10 g/e for the first output (biomass concentration) and between 3 g / e and 6 g/e for the second output (substrate concentration). Realistic measurements are reproduced by adding white noise signal to the outputs (biomass and substrate concentrations). Figures 2a - 2b show the evolution of the inputs variables (di lution rate and influent substrate concentration). The evolutions of outputs variables (biomass and substrate concentrations) are given in Figs 2c 2d.
,
I
,
time (h), lOO
Fig . 2c
200
JOO
Evoluti o n o f bi omass c o ncentration
l J~ " 'I.J,Jllj :~ f I
Another simulation was performed in the same conditions described above. except for reference signals; in this second simulation. these reference signals are not synchronous. The evolution of inputs and outputs variables is shown in Figs 3a - 3b - 3c - 3d.
l'
, 'I ,
Feeding
x.s
time (h) lOO
Drawing
u
200
JOO
x,s
Fig . 2d Fig . 1
I
i.
I
Bioreactor
u Sa
I
,
Evoluti o n o f substrate concentration
Schematic representation of the continuous fermentation process.
0.' 0.20
!I
0.3 ...,.-,- - - - - - - - - - - - - - - - - - -
1,
0.20
M'l
0.20
:: l 0.2
0.24
0.22
l
Cl.>
0.111 -!
0.11 0. 14
1 1
0.<>4
0-'"
time (h) l OO
200
t ime (h 100
JOO
Fig . 2a : Ev o luti on o f diluti on rate
200
JOO
Fig. 3a : Evolution o f diluti o n rate
'2.-------------------------------------~~------_,
'2,-------------~------------------------------,
JO
JO :lA
21
24 22 20 11 10
14
time (h) 100
Fig. 2b
time
200
100
Evolution of influent substrate conce ntrati o n
Fig . 3b
396
200
JOO
Evo luti o n o f influent substrate conce ntrati o n
13,------------------------------------------------,
Ljung, L. and T. S6derstr6m (1984), Theory and practice of recursive identification, MIT Press, Cambridge, MA. Monod, J. (1942), Recherche sur la croissance des cultures bacteriennes , Ed. Herman, Paris . M'Saad, M. (1987). Sur l'applicabilite de la commande adaptative. These d'Etat, Institut National Poly technique de Grenoble. Praly, L. (1986), Robustesse des algorithmes de commande adaptative. In 1.0. Landau et L. Dugard. Ed. Commande adaptative : Aspects theoriques et pratiques. Mass on, Paris .
12 11
I.
time (h 100
Fig . 3c
Fig. 3d
2DO
Ev o luti o n o f biomass concentration
Ev o lution o f substrate concent r ation
v -
CONCLUSION
In this paper a multivariable generalized predictive control algorithm combined with a constant trace identification method is used for the control of a nonlinear fermentation process. This is a difficult process due to complex relationships between the variables of interest in the physical model. The simulation results show the good performance of this adaptive predictive algorithm, mainly in the case where the process is subject to high disturbances.
REFERENCES Andrew, J. (1968), A mathematical model for the continuous culture of micro-organisms utilising inhibitory substrate . Biotechnology and Bioengineering, vol. 10 Bierman, G.J. (1977), Factorization methods for discrete sequential estimation. Academic Press, New York. Clarke, D.W., C. Mohtadi and P . S. Tuffs (1987), Generalized predictive control. Part. I . The basic Algorithm and Part 11. Extensions and interpretations. Automatica vol 23 n° 2, 137-160. Kucera, V. (1979), Discrete linear control: The polynomial equation approach. John Wi1ey & Sons.
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