- Email: [email protected]
State and Parameter Estimation of Bioprocess Models a Case Study
Copyright © IFAC Control of Industrial Systems. Belfort, France, 1997
STATE AND PARAMETER ESTIMATION OF BIOPROCESS MODELS A CASE STUDY
Vincent G. Ryckaert and lan F. Van Impe
BioTeC - Bioprocess Technology and Control Laboratory for Industrial Microbiology and Biochemistry K atholieke Universiteit Leuven Kardinaal Mercierlaan 92 B-3001 Leuven (Belgium) tel.: +32/(0)16/321585 - fax.: +32/(0)16/321997 email: [email protected] email: [email protected]
Abstract: The application of modern model based control techniques to biotechnological processes -such as biological wastewater treatment and fermentation processesis hampered due to the complex, nonlinear and time varying characteristics of these systems, and the lack of reliable, accurate and cheap on-line sensors. An approach to tackle the measurement problem is the use of estimation schemes. The estimation problem is analyzed for a typical bioprocess model. A direct link between the measurability of the process and the design of the plant is established. The analysis of the estimation problem is shown to be a powerful tool for determining an optimal investment policy for sensors. Keywords: Observers - Parameter estimation - Sensors - Biotechnology - Bio control
1. INTRODUCTION
process, and (ii) available on-line measurements. The following question can then be stated.
The application of modern model based control techniques to biotechnological processes -such as biological wastewater treatment and fermentation processes- is hampered due to (1) the complex, nonlinear and time varying characteristics of these systems, and (2) the lack of reliable, accurate and cheap on-line sensors.
Given a measurement set (hardware sensors) and additional knowledge about the system (model structure and some model parameters), which unmeasured state variables and model parameters can be extracted? This question has been solved for a simple model (Section 2), useful for instance, in the context of wastewater treatment (Ryckaert et al., 1994, 1996a, 1996b; Claes et al., 1996a, 1996b) for a class of estimation algorithms (Section 3). The results have been summarized and are discussed in Section 4.
By applying recent developments in nonlinear adaptive control theory the first problem can be dealt \vith. The second problem can be relaxed by using software sensors. A software sensor is a (parameter and/or state) estimation algorithm, based on (i) a model of the
1563
The analysis indicates a direct link between the measurability of the process and the design of the plant and indicates the impact of the system configuration on the estimation problem. Moreover it provides a powerful tool for determining an optimal investment policy for sensors.
CS at [mg02/L] and kLa [l/h] are the oxygen saturation concentration and oxygen transfer coefficient. Notice that in the following model only exogeneous metabolism and exogeneous respiration is assumed to be relevant and that dilution of the oxygen concentration is neglected, which could be motivated for the plant under consideration.
In the analysis the observability of the states and the identifiability of the parameters are analysed for various conditions. This analysis is however not sufficient. It is necessary to determine whether the measurements are sufficiently rich. Only when certain conditions of persistency of excitation are fulfilled the state and parameter estimators will converge.
GueousoUl1low
Llquldi"-
Notice that in this text the (parameter and state) estimation based on on-line measurements of the process is considered. Another approach, described by, e.g., Vanrolleghem (1994), is to measure in a separate bioreactor, filled with substrate and biomass of the process. This approach has as advantage that now the measurement experiment can be designed in an optimal way without a modification of the process operation. In such a way measurements with a high information content can be generated. The disadvantage is that one must assume that the dynamics in the bioreactor is representative for the dynamics in the plant. Also an extra investment in equipment is necessary.
•
Liquid 0UIII0w
Aeration
Stirred tank biological reactor
d~s d~x dgo
-k 1 . J1.Cx
+ D· (Cs in - Cs) J1.Cx + D·(CXin -Cx) -k 3 · J1.Cx + kLa· (CS at - Co) (1)
The problem can now be reformulated as follows.
The application of parameter estimators (and control) in wastewater treatment is described by Marsili-Libelli (1989) and Dochain et al. (1992).
Which states (Cs,Cx ,Co) and which parameters (k 1 , k 2 and J1.) can b~ estimated when a certain state, or a combination ofstates, and additional parameters are given.
2. PROBLEM STATEMENT
In a wastewater treatment plant for carbon removal, the waste (substrate concentration Cs [mgCOD/L]) is degraded by micro-organisms (biomass concentration Cx [gMLSS/L)) which consume oxygen (oxygen concentration Co [mg02/L)). The model consists of three mass balances, describing biotransformation and transport. The biotransformation tank ofthe wastewater treatment plant is represented as a stirred tank continuous biological reactor. J1.(Cs) [l/h] is the specific biomass growth rate, which can be modelled, for instance, with the Monod law. k 1 [mgCOD/gMLSS] and k 3 [mg02/gMLSS] are the inverses of the yield of biomass on substrate and biomass on oxygen respectively. D [l/h] is the dilution rate and is defined as the influent flow rate F [m 3 /h] divided by the volume V [m 3 ] of the system. C Sin and C Xin are the influent substrate and biomass concentration respectively. The inflow of biomass in the tank models (i) the recycle stream often encountered in a classical treatment plant or (ii) just transport between a cascade of biodegradation tanks.
Notice that J1.( Cs) is considered to be a (time varying) parameter. This follows from a minimal modelling principle which states that parts of the system with high uncertainty (the kinetics) are not modelled but replaced by a parameter. 3. PROBLEM SOLUTIOl\: STATE OBSERVERS AND OBSERVER BASED PARAMETER ESTIMATORS A general model for biotechnological systems is described by Bastin et al. (1990). is the ndimensional state of the process. K is the n x r yield matrix while <;:,(e) is the r-dimensional vector of reaction rates. The transport terms are described by the n x n diagonal dilution matrix D and the n-dimensional input term f.
e
De 1564
+
f
(2)
model (2) and the parameter estimator equations (4). The tuning matrices nand r must be constructed such that the error svstem is stable. .
3.1. State observers
e
The state of the process can now be estimated from on-line measurements of a part of the state by the following algorithm.
e,
3.3. Transformations
~
De
K
+
f
The potential of the above described methods can be enlarged when transformations of the process model are allowed. In this study linear state transformations, introduction of new states and (invertible) reparametrizations are used.
(3)
+ n(e, - {,)
e is the
state estimate. Notice that this algorithm consists of a copy of the model plus a correction term, proportional with the difference and the estimate between the measurements of this part of the state {,. n is a n x p dimensional tuning matrix if is a p-dimensional vector.
e, e,
The dynamics of the state estimator can be understood by examining the estimation error model. The estimation error e is defined as the difference between the measurements and the estimate of this state The error model can be constructed by combination of the process model (2) and the state estimator equations (3). The tuning matrix n must be constructed such that the error system is stable.
e
e.
3.2. Observer based parameter estimators
dp
dI
De
KH(e)p
dt
+
n(e -
+
f
e)
ea is a r- dimensional vector of states e with r
the rank of the matrix K and eb is the (n - r)dimensional vector containing the remainder of the states Ca and Cb are respectively (n r) x rand (n-r) x (n-r) matrices. The process model can now be reformulated.
(4)
e.
Notice that the unknown reaction rate vector
e
dz dt
(Ca· K a + Cb . Kb) .
The dynamics of the parameter estimator can be understood by examining the estimation error model. Two errors can now be defined. The state estimation error el is defined as the difference between the measurements and the estimate of this state while the parameter estimation error e2 is defined as the difference between the real process parameter p and the estimate of this parameter p. The error model can be constructed by combination of the process
e
(5)
z
e)
(KH(e))T r(e -
e
Notice that the design of the tuning functions nand r is not discussed here. An extensive evaluation of design methods (exact linearisation, local linearisations in original or observer coordinates and high gain techniques) and design criteria (convergence area, noise sensitivity and robustness against parametric and structural uncertainty) can be found by Ryckaert et al. (1996b). The importance of the linear state transformations is now illustrated. A new state n - r dimensional state vector z is introduced by the following state transformation.
The unknown parameters p can be estimated bv the following algorithm. .
~
The aim of this study is to determine which states and which parameters can be estimated with the state observers and observer based parameter estimators when a certain state or combination of states are on-line measured and additional parameters are given for the model transformations mentioned above.
Z
+(C a ·fa+Cb .fb)
(6)
+(C b . D bb · Ci/ . Ca
e
-Ca· D aa ) .
ea
with K a , Kb' D aa , Dbb are the corresponding submatrices of the yield matrix K and the dilution matrix D respectively. By selecting Ca
1565
and Cb such that Ca' K a + Cb' Kb = 0 the dynamics of z is given by differential equations which are independent on the reaction rates. Recall that the kinetics equations are always the uncertain parts of the biological model. The dynamical system given by (6) and the differential equations for are equivalent to the original process model.
• Transformation with gen concentration.
ea equal to the oxy-
ea
dC o dt
For the model under consideration the rank of K is equal to one, so the state is atmost one state. Three possibilities can be considered. For the measurability analysis performed here, the model must be rewritten in terms of the state z (dynamics independent of the kinetics) and the measurable state This is illustrated below when the oxygen concentration Co is considered to be measurable on-line. By a detailed analysis of the several transformations and the resulting process models conclusions about the measurability of the system can be drawn under various assumptions concerning the knowledge of the parameters and availability of on-line measurements.
ea
dCo dt
k1 ·3
k1 +DCs in - k kLa . C. at 3
-Dz 2 1
3
4. CASE STUDY The results are summarized in three figures: Figure 1 describes the case where only on-line oxygen measurements are assumed to be available. Figure 2 summarizes the results for the case where both biomass and oxygen concentration are measured, while Figure 3 contains the results for the case where full state measurements are available.
1
(Co - Z2)) 3 +kLa . CS at - kLa . CSin 1
1
-Dz 2
k3
• Transformation with biomass concentration.
Each figure lists which yield coefficients, transport and input parameters are assumed to be known. There is also indicated whether explicit knowledge of the growth rate model is necessary or not and whether additional constraints (on the input) must be satisfied. In the figure can be found which states and/or parameters can be estimated. There is also indicated whether the convergence of the estimator is asymptotic (determined only by the process conditions, often the transport terms) or exponential (chosen by the designer of the algorithm).
+ DCXin
+ kLa . CS at
-y;;DCs in
+ (D -
ea
+ DCXin
+k"(kLa - D)Co 3 1 - k kLa . C Sat )
+k
-Dz1 - k DCSin
Co)
3
-Dz 1 + (kLa - D)-k Co
ea equal to the sub-
-k3l.L(Zl
1
+k
+kLa . C Sat - kLa . Co
em.
• Transformation with strate concentration.
-k3 J.L( Z2
kLa)Co
equal to the
-k1Cx
+ Cs
4.1. On-line measurements of oxygen concentra-
-k 3 Cx
+ Co
tion
Out of the oxygen concentration measurement the biomass and substrate concentration can be estimated even without knowledge of the growth rate. The convergence is however asymptotic and determined by the transport terms. In a wastewater treatment system the obtained convergence speed is often quite low due to the overdimensioning of the installations. Notice that here an explicit link between the design of the installation and the measurability of the process is made.
dCo dt +kLa' CS at - kLa . Co -Dz 1 + DCSin -k1DCXin -Dz 2
+ kLa . CS at
1566
Exponential estimation of the growth rate is possible when complete knowledge of the oxygen balance is available.
needed. The systematic analysis of the estimation problem can however be used to determine an optimal investment policy.
Estimation of the growth rate, independently of (some) yield coefficients, and even simultaneous estimation of the growth rate and some yield coefficients, is only possible in asymptotic way if some conditions on the inputs are fulfilled. These conditions must be examined for the particular system under study.
Further research will focus on more complex models, useful for instance for modelling of nitrogen removal systems, and other classes of estimation algorithms and model transformations. ACKN O\VLEDG EMENTS Vincent Ryckaert is a research assistant with the Flemish Institute for Scientific and Technological Research in Industry (IWT). Work supported in part by NFWO Project G.OI41.95, NFWO Project G.0286.96, and KULeuven Project OT/95/20. The scientific responsibility is assumed by its authors.
4.2. On-line measurements of oxygen and biomass concentration
Exponential estimation of the substrate concentration Cs is possible when the growth rate model is explicitly known. The substrate concentration Cs can still be estimated without the growth rate model but only in asymptotic manner.
REFERENCES
Exponential estimation of the growth rate, independently of (some) yield coefficients, is possible. The estimation of the yield k 3 from the available on-line measurements is trivial.
Bastin G. and D. Dochain (1990). On-line estimation and adaptive control of bioreactors,
Elsevier Science Publishing Corporation, Amsterdam.
4.3. On-line measurements of oxygen, biomass and substrate concentration
Claes J.E., V.G. Ryckaert, J.F. Van Impe, R. Gerards and L. Vriens (1996a). Mathematical modelling and identification of UNITANK® wastewater treatment systems. European Water Pollution Control, 6 (6), 34-44.
The growth rate can be exponentially estimated independently of the yield coefficients while the yield coefficients can be determined independently of the growth rate model. Also the simultaneous estimation of the yield coefficients and growth rate is possible.
Claes J.E., V.G. Ryckaert and J.F. Van Impe (1996b). Modelling and identification of the specific growth rate and the oxygen uptake rate on a full scale wastewater treatment system. In: P. Borne, R. Soenen, Y. Sallez and S. El Khattabi (Eds.), Compu-
5. CONCLUSIONS • The analysis illustrates that quite often the convergence speed is asymptotic and determined by the transport terms. This clearly indicates the influence of the design of the plant on the measurability of the process. Often wastewater treatment systems are oversized which reduces the measurability of the system.
tational Engineering in Systems Applications 1996, Volume 1: Modelling, A nalysis and Simulation, GERF EC Lille - Cite
Scientifique, Villeneuve d'Ascq, 142-14i. [Multiconjerence Co.mputational Engineering in Systems Applications CESA '96 - Symposium on Modelling, Analysis and Simulation, Lille (France), July 9-12 1996]
• The estimation of states and parameters is often only possible when certain conditions on the input parameters are fulfilled. This depends strongly on the system under study. The condition on the biomass concentration of the input stream is for instance not fulfilled in a classical wastewater treatment system due to the biomass recycle flow.
Dochain D. and M. Perrier (1992). Adaptive linearizing control of the activated sludge process M ededelingen Fac. Landbouww. Fniv. Gent, 57 (4b), 2229-2248. [Proceedings of the 1992 Forum for Applied Biotechnology, Brugge (Belgium), September 24-25, 1992] Marsili-Libelli S. (1989). Modelling, identification and control of the activated sludge process. Advances in Biochemical Engineering Biotechnology, 38, 89- 148.
• This study indicates that an important investment in on-line measurements is
1567
Ryckaert V.G., J .F. Van Impe, J. Curinckx and K. Van Gool (1994). State space analysis of periodic biological wastewater treatment plants. M ededelingen Fac. Landbouww. Univ. Gent, 59 (4a), 2047-2055. [Proceedings of the 1994 Forum for Applied Biotechnology, Brugge (Belgium), September 28-30, 1994]
Online measurement or oxygen Co Yield
k3
I
. ~!~.:?
--
G~~th
Asympt. Cxin=O Indep. Kj ._ ----.- ._-------. I Growth Csin=O raJ3 Asympt.
I
Figure 1. Estimation based on oxygen concentration measurements Online measurements or oxygen Co and biomass Cx Yield kl.k3
Ryckaert V.G. and J .F. Van Impe (1996b). Design methodologies for nonlinear observers - application to biological models. In: P. Borne, R. Soenen, Y. Sallez and S. El Khattabi (Eds.), Computational Engineering in Systems Applications 1996, Volume 1: and
' j
Csin ' --...........kLa.csat _-_ _- ----D kLa.Csat, Cxin
Ryckaert V.G., C.H. Herremans, J .E. Claes, J.F. Van Impe, R. Gerards and L. Vriens (1996a). Modelling and dynamical analysis of UNITANK® wastewater treatment systems. Mathematical Modelling of Systems, 21 p. (in press)
Analysis
kLa.Csat
:::~:::. ::::~::::C~;;;::!
Ryckaert V.G., J.E. Claes, C.H. Herremans, J .F. Van Impe, R. Gerards and 1. Vriens (1995). Observer based estimation of oxygen uptake rates in cyclically operated wastewater treatment plants. Mededelingen Fac. Landbouww. Univ. Gent, 60 (4b), 2377- 2384. [Proceedings of the 1995 Forum for Applied Biotechnology, Gent (Belgium), September 27-29, 1995]
Modelling,
Converg.
I Transp. D
Inputs Kineties Constr. Estimate Converg. Cxin
Expon.
kLa. Csat
Figure 2. Estimation based on oxygen and biomass concentration measurements Online measurements or oxygen Co, biomass ex and substrate Cs
Simulation,
Yield
Transp. Inputs
Kineties Constr. Estimate Converg.
GERF EC Lille - Cite Scientifique, Villeneuve d'Ascq, 1052-1057. [Multiconference Computational Engineering in Systems Applications CESA '96 - Symposium on Modelling, Analysis and Simulation, Lille (France), July 9-12 1996]
.---------
-_ .. __ .. _-
Growth rate Indep.K
Expon.
kl.k3 Indep.1!
RLS
_.
__
Growth k'i".lZ3
Expon.
Vanrolleghem P. (1994). On-line modelling of activated sludge processes: Development of an adaptive sensor. PhD thesis, Faculty
Figure 3. Estimation based on full state measurements
of Agricultural and Applied Biological Sciences, Rijksuniversiteit Gent (Belgium)
1568
STATE AND PARAMETER ESTIMATION OF BIOPROCESS MODELS A CASE STUDY
Vincent G. Ryckaert and lan F. Van Impe
BioTeC - Bioprocess Technology and Control Laboratory for Industrial Microbiology and Biochemistry K atholieke Universiteit Leuven Kardinaal Mercierlaan 92 B-3001 Leuven (Belgium) tel.: +32/(0)16/321585 - fax.: +32/(0)16/321997 email: [email protected] email: [email protected]
Abstract: The application of modern model based control techniques to biotechnological processes -such as biological wastewater treatment and fermentation processesis hampered due to the complex, nonlinear and time varying characteristics of these systems, and the lack of reliable, accurate and cheap on-line sensors. An approach to tackle the measurement problem is the use of estimation schemes. The estimation problem is analyzed for a typical bioprocess model. A direct link between the measurability of the process and the design of the plant is established. The analysis of the estimation problem is shown to be a powerful tool for determining an optimal investment policy for sensors. Keywords: Observers - Parameter estimation - Sensors - Biotechnology - Bio control
1. INTRODUCTION
process, and (ii) available on-line measurements. The following question can then be stated.
The application of modern model based control techniques to biotechnological processes -such as biological wastewater treatment and fermentation processes- is hampered due to (1) the complex, nonlinear and time varying characteristics of these systems, and (2) the lack of reliable, accurate and cheap on-line sensors.
Given a measurement set (hardware sensors) and additional knowledge about the system (model structure and some model parameters), which unmeasured state variables and model parameters can be extracted? This question has been solved for a simple model (Section 2), useful for instance, in the context of wastewater treatment (Ryckaert et al., 1994, 1996a, 1996b; Claes et al., 1996a, 1996b) for a class of estimation algorithms (Section 3). The results have been summarized and are discussed in Section 4.
By applying recent developments in nonlinear adaptive control theory the first problem can be dealt \vith. The second problem can be relaxed by using software sensors. A software sensor is a (parameter and/or state) estimation algorithm, based on (i) a model of the
1563
The analysis indicates a direct link between the measurability of the process and the design of the plant and indicates the impact of the system configuration on the estimation problem. Moreover it provides a powerful tool for determining an optimal investment policy for sensors.
CS at [mg02/L] and kLa [l/h] are the oxygen saturation concentration and oxygen transfer coefficient. Notice that in the following model only exogeneous metabolism and exogeneous respiration is assumed to be relevant and that dilution of the oxygen concentration is neglected, which could be motivated for the plant under consideration.
In the analysis the observability of the states and the identifiability of the parameters are analysed for various conditions. This analysis is however not sufficient. It is necessary to determine whether the measurements are sufficiently rich. Only when certain conditions of persistency of excitation are fulfilled the state and parameter estimators will converge.
GueousoUl1low
Llquldi"-
Notice that in this text the (parameter and state) estimation based on on-line measurements of the process is considered. Another approach, described by, e.g., Vanrolleghem (1994), is to measure in a separate bioreactor, filled with substrate and biomass of the process. This approach has as advantage that now the measurement experiment can be designed in an optimal way without a modification of the process operation. In such a way measurements with a high information content can be generated. The disadvantage is that one must assume that the dynamics in the bioreactor is representative for the dynamics in the plant. Also an extra investment in equipment is necessary.
•
Liquid 0UIII0w
Aeration
Stirred tank biological reactor
d~s d~x dgo
-k 1 . J1.Cx
+ D· (Cs in - Cs) J1.Cx + D·(CXin -Cx) -k 3 · J1.Cx + kLa· (CS at - Co) (1)
The problem can now be reformulated as follows.
The application of parameter estimators (and control) in wastewater treatment is described by Marsili-Libelli (1989) and Dochain et al. (1992).
Which states (Cs,Cx ,Co) and which parameters (k 1 , k 2 and J1.) can b~ estimated when a certain state, or a combination ofstates, and additional parameters are given.
2. PROBLEM STATEMENT
In a wastewater treatment plant for carbon removal, the waste (substrate concentration Cs [mgCOD/L]) is degraded by micro-organisms (biomass concentration Cx [gMLSS/L)) which consume oxygen (oxygen concentration Co [mg02/L)). The model consists of three mass balances, describing biotransformation and transport. The biotransformation tank ofthe wastewater treatment plant is represented as a stirred tank continuous biological reactor. J1.(Cs) [l/h] is the specific biomass growth rate, which can be modelled, for instance, with the Monod law. k 1 [mgCOD/gMLSS] and k 3 [mg02/gMLSS] are the inverses of the yield of biomass on substrate and biomass on oxygen respectively. D [l/h] is the dilution rate and is defined as the influent flow rate F [m 3 /h] divided by the volume V [m 3 ] of the system. C Sin and C Xin are the influent substrate and biomass concentration respectively. The inflow of biomass in the tank models (i) the recycle stream often encountered in a classical treatment plant or (ii) just transport between a cascade of biodegradation tanks.
Notice that J1.( Cs) is considered to be a (time varying) parameter. This follows from a minimal modelling principle which states that parts of the system with high uncertainty (the kinetics) are not modelled but replaced by a parameter. 3. PROBLEM SOLUTIOl\: STATE OBSERVERS AND OBSERVER BASED PARAMETER ESTIMATORS A general model for biotechnological systems is described by Bastin et al. (1990). is the ndimensional state of the process. K is the n x r yield matrix while <;:,(e) is the r-dimensional vector of reaction rates. The transport terms are described by the n x n diagonal dilution matrix D and the n-dimensional input term f.
e
De 1564
+
f
(2)
model (2) and the parameter estimator equations (4). The tuning matrices nand r must be constructed such that the error svstem is stable. .
3.1. State observers
e
The state of the process can now be estimated from on-line measurements of a part of the state by the following algorithm.
e,
3.3. Transformations
~
De
K
+
f
The potential of the above described methods can be enlarged when transformations of the process model are allowed. In this study linear state transformations, introduction of new states and (invertible) reparametrizations are used.
(3)
+ n(e, - {,)
e is the
state estimate. Notice that this algorithm consists of a copy of the model plus a correction term, proportional with the difference and the estimate between the measurements of this part of the state {,. n is a n x p dimensional tuning matrix if is a p-dimensional vector.
e, e,
The dynamics of the state estimator can be understood by examining the estimation error model. The estimation error e is defined as the difference between the measurements and the estimate of this state The error model can be constructed by combination of the process model (2) and the state estimator equations (3). The tuning matrix n must be constructed such that the error system is stable.
e
e.
3.2. Observer based parameter estimators
dp
dI
De
KH(e)p
dt
+
n(e -
+
f
e)
ea is a r- dimensional vector of states e with r
the rank of the matrix K and eb is the (n - r)dimensional vector containing the remainder of the states Ca and Cb are respectively (n r) x rand (n-r) x (n-r) matrices. The process model can now be reformulated.
(4)
e.
Notice that the unknown reaction rate vector
e
dz dt
(Ca· K a + Cb . Kb) .
The dynamics of the parameter estimator can be understood by examining the estimation error model. Two errors can now be defined. The state estimation error el is defined as the difference between the measurements and the estimate of this state while the parameter estimation error e2 is defined as the difference between the real process parameter p and the estimate of this parameter p. The error model can be constructed by combination of the process
e
(5)
z
e)
(KH(e))T r(e -
e
Notice that the design of the tuning functions nand r is not discussed here. An extensive evaluation of design methods (exact linearisation, local linearisations in original or observer coordinates and high gain techniques) and design criteria (convergence area, noise sensitivity and robustness against parametric and structural uncertainty) can be found by Ryckaert et al. (1996b). The importance of the linear state transformations is now illustrated. A new state n - r dimensional state vector z is introduced by the following state transformation.
The unknown parameters p can be estimated bv the following algorithm. .
~
The aim of this study is to determine which states and which parameters can be estimated with the state observers and observer based parameter estimators when a certain state or combination of states are on-line measured and additional parameters are given for the model transformations mentioned above.
Z
+(C a ·fa+Cb .fb)
(6)
+(C b . D bb · Ci/ . Ca
e
-Ca· D aa ) .
ea
with K a , Kb' D aa , Dbb are the corresponding submatrices of the yield matrix K and the dilution matrix D respectively. By selecting Ca
1565
and Cb such that Ca' K a + Cb' Kb = 0 the dynamics of z is given by differential equations which are independent on the reaction rates. Recall that the kinetics equations are always the uncertain parts of the biological model. The dynamical system given by (6) and the differential equations for are equivalent to the original process model.
• Transformation with gen concentration.
ea equal to the oxy-
ea
dC o dt
For the model under consideration the rank of K is equal to one, so the state is atmost one state. Three possibilities can be considered. For the measurability analysis performed here, the model must be rewritten in terms of the state z (dynamics independent of the kinetics) and the measurable state This is illustrated below when the oxygen concentration Co is considered to be measurable on-line. By a detailed analysis of the several transformations and the resulting process models conclusions about the measurability of the system can be drawn under various assumptions concerning the knowledge of the parameters and availability of on-line measurements.
ea
dCo dt
k1 ·3
k1 +DCs in - k kLa . C. at 3
-Dz 2 1
3
4. CASE STUDY The results are summarized in three figures: Figure 1 describes the case where only on-line oxygen measurements are assumed to be available. Figure 2 summarizes the results for the case where both biomass and oxygen concentration are measured, while Figure 3 contains the results for the case where full state measurements are available.
1
(Co - Z2)) 3 +kLa . CS at - kLa . CSin 1
1
-Dz 2
k3
• Transformation with biomass concentration.
Each figure lists which yield coefficients, transport and input parameters are assumed to be known. There is also indicated whether explicit knowledge of the growth rate model is necessary or not and whether additional constraints (on the input) must be satisfied. In the figure can be found which states and/or parameters can be estimated. There is also indicated whether the convergence of the estimator is asymptotic (determined only by the process conditions, often the transport terms) or exponential (chosen by the designer of the algorithm).
+ DCXin
+ kLa . CS at
-y;;DCs in
+ (D -
ea
+ DCXin
+k"(kLa - D)Co 3 1 - k kLa . C Sat )
+k
-Dz1 - k DCSin
Co)
3
-Dz 1 + (kLa - D)-k Co
ea equal to the sub-
-k3l.L(Zl
1
+k
+kLa . C Sat - kLa . Co
em.
• Transformation with strate concentration.
-k3 J.L( Z2
kLa)Co
equal to the
-k1Cx
+ Cs
4.1. On-line measurements of oxygen concentra-
-k 3 Cx
+ Co
tion
Out of the oxygen concentration measurement the biomass and substrate concentration can be estimated even without knowledge of the growth rate. The convergence is however asymptotic and determined by the transport terms. In a wastewater treatment system the obtained convergence speed is often quite low due to the overdimensioning of the installations. Notice that here an explicit link between the design of the installation and the measurability of the process is made.
dCo dt +kLa' CS at - kLa . Co -Dz 1 + DCSin -k1DCXin -Dz 2
+ kLa . CS at
1566
Exponential estimation of the growth rate is possible when complete knowledge of the oxygen balance is available.
needed. The systematic analysis of the estimation problem can however be used to determine an optimal investment policy.
Estimation of the growth rate, independently of (some) yield coefficients, and even simultaneous estimation of the growth rate and some yield coefficients, is only possible in asymptotic way if some conditions on the inputs are fulfilled. These conditions must be examined for the particular system under study.
Further research will focus on more complex models, useful for instance for modelling of nitrogen removal systems, and other classes of estimation algorithms and model transformations. ACKN O\VLEDG EMENTS Vincent Ryckaert is a research assistant with the Flemish Institute for Scientific and Technological Research in Industry (IWT). Work supported in part by NFWO Project G.OI41.95, NFWO Project G.0286.96, and KULeuven Project OT/95/20. The scientific responsibility is assumed by its authors.
4.2. On-line measurements of oxygen and biomass concentration
Exponential estimation of the substrate concentration Cs is possible when the growth rate model is explicitly known. The substrate concentration Cs can still be estimated without the growth rate model but only in asymptotic manner.
REFERENCES
Exponential estimation of the growth rate, independently of (some) yield coefficients, is possible. The estimation of the yield k 3 from the available on-line measurements is trivial.
Bastin G. and D. Dochain (1990). On-line estimation and adaptive control of bioreactors,
Elsevier Science Publishing Corporation, Amsterdam.
4.3. On-line measurements of oxygen, biomass and substrate concentration
Claes J.E., V.G. Ryckaert, J.F. Van Impe, R. Gerards and L. Vriens (1996a). Mathematical modelling and identification of UNITANK® wastewater treatment systems. European Water Pollution Control, 6 (6), 34-44.
The growth rate can be exponentially estimated independently of the yield coefficients while the yield coefficients can be determined independently of the growth rate model. Also the simultaneous estimation of the yield coefficients and growth rate is possible.
Claes J.E., V.G. Ryckaert and J.F. Van Impe (1996b). Modelling and identification of the specific growth rate and the oxygen uptake rate on a full scale wastewater treatment system. In: P. Borne, R. Soenen, Y. Sallez and S. El Khattabi (Eds.), Compu-
5. CONCLUSIONS • The analysis illustrates that quite often the convergence speed is asymptotic and determined by the transport terms. This clearly indicates the influence of the design of the plant on the measurability of the process. Often wastewater treatment systems are oversized which reduces the measurability of the system.
tational Engineering in Systems Applications 1996, Volume 1: Modelling, A nalysis and Simulation, GERF EC Lille - Cite
Scientifique, Villeneuve d'Ascq, 142-14i. [Multiconjerence Co.mputational Engineering in Systems Applications CESA '96 - Symposium on Modelling, Analysis and Simulation, Lille (France), July 9-12 1996]
• The estimation of states and parameters is often only possible when certain conditions on the input parameters are fulfilled. This depends strongly on the system under study. The condition on the biomass concentration of the input stream is for instance not fulfilled in a classical wastewater treatment system due to the biomass recycle flow.
Dochain D. and M. Perrier (1992). Adaptive linearizing control of the activated sludge process M ededelingen Fac. Landbouww. Fniv. Gent, 57 (4b), 2229-2248. [Proceedings of the 1992 Forum for Applied Biotechnology, Brugge (Belgium), September 24-25, 1992] Marsili-Libelli S. (1989). Modelling, identification and control of the activated sludge process. Advances in Biochemical Engineering Biotechnology, 38, 89- 148.
• This study indicates that an important investment in on-line measurements is
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Ryckaert V.G., J .F. Van Impe, J. Curinckx and K. Van Gool (1994). State space analysis of periodic biological wastewater treatment plants. M ededelingen Fac. Landbouww. Univ. Gent, 59 (4a), 2047-2055. [Proceedings of the 1994 Forum for Applied Biotechnology, Brugge (Belgium), September 28-30, 1994]
Online measurement or oxygen Co Yield
k3
I
. ~!~.:?
--
G~~th
Asympt. Cxin=O Indep. Kj ._ ----.- ._-------. I Growth Csin=O raJ3 Asympt.
I
Figure 1. Estimation based on oxygen concentration measurements Online measurements or oxygen Co and biomass Cx Yield kl.k3
Ryckaert V.G. and J .F. Van Impe (1996b). Design methodologies for nonlinear observers - application to biological models. In: P. Borne, R. Soenen, Y. Sallez and S. El Khattabi (Eds.), Computational Engineering in Systems Applications 1996, Volume 1: and
' j
Csin ' --...........kLa.csat _-_ _- ----D kLa.Csat, Cxin
Ryckaert V.G., C.H. Herremans, J .E. Claes, J.F. Van Impe, R. Gerards and L. Vriens (1996a). Modelling and dynamical analysis of UNITANK® wastewater treatment systems. Mathematical Modelling of Systems, 21 p. (in press)
Analysis
kLa.Csat
:::~:::. ::::~::::C~;;;::!
Ryckaert V.G., J.E. Claes, C.H. Herremans, J .F. Van Impe, R. Gerards and 1. Vriens (1995). Observer based estimation of oxygen uptake rates in cyclically operated wastewater treatment plants. Mededelingen Fac. Landbouww. Univ. Gent, 60 (4b), 2377- 2384. [Proceedings of the 1995 Forum for Applied Biotechnology, Gent (Belgium), September 27-29, 1995]
Modelling,
Converg.
I Transp. D
Inputs Kineties Constr. Estimate Converg. Cxin
Expon.
kLa. Csat
Figure 2. Estimation based on oxygen and biomass concentration measurements Online measurements or oxygen Co, biomass ex and substrate Cs
Simulation,
Yield
Transp. Inputs
Kineties Constr. Estimate Converg.
GERF EC Lille - Cite Scientifique, Villeneuve d'Ascq, 1052-1057. [Multiconference Computational Engineering in Systems Applications CESA '96 - Symposium on Modelling, Analysis and Simulation, Lille (France), July 9-12 1996]
.---------
-_ .. __ .. _-
Growth rate Indep.K
Expon.
kl.k3 Indep.1!
RLS
_.
__
Growth k'i".lZ3
Expon.
Vanrolleghem P. (1994). On-line modelling of activated sludge processes: Development of an adaptive sensor. PhD thesis, Faculty
Figure 3. Estimation based on full state measurements
of Agricultural and Applied Biological Sciences, Rijksuniversiteit Gent (Belgium)
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