A ROBUST ASYMPTOTIC OBSERVER FOR CHEMICAL AND ...
A ROBUST ASYMPTOTIC OBSERVER FOR CHEMICAL AND BIOCHEMICAL REACTORS 1
1
3
Alcaraz-González V. , Harmand J. , Dochain D. , Rapaport A.2, Steyer J.P.1, Pelayo Ortiz C.4, and González-Alvarez V.4 1
INRA-LBE, Avenue des Etangs, 11100 Narbonne - France. Fax : (+33)-468-425-160 2 INRA-LASB,2 Place Viala, 34060 Montpellier - France. 3 CESAME, Université Catholique de Louvain, 4-6 avenue Georges Lemaître B-1348, Louvain la Neuve - Belgium. Fax : (+32)-10-472-180 4 Universidad de Guadalajara. CUCEI. Calz. Gral M. García Barragan 1451. Guadalajara, México.
Abstract : A simple state observer is proposed for a class of lumped nonlinear time-varying systems useful in chemical and biochemical engineering. It is shown that this asymptotic nonlinear observer is stable in the presence of time-varying elements and robust in the face of initial conditions uncertainty and a total lack of knowledge on the nonlinearities of the system. Experimental results are presented using a model of an anaerobic digestion process for the treatment of industrial wastewater from a wine distillery and tested using real data obtained from 3 a 1 m continuous fixed bed pilot bioreactor. Copyright 2003 IFAC
Keywords: Robust estimation, Stability, Time-varying systems, Biotechnology.
1. INTRODUCTION One of the most frequent and important challenges in the control of chemical and biochemical processes is finding adequate and reliable sensors to measure all the important state variables of the plant. However, if a number of on-line sensors providing state information are available today at the industrial scale, they are still very expensive and their maintenance is usually time consuming. This is especially true in the field of chemical and biological processes. Furthermore, even when the information of the important variables is readily available, controlling the biological processes is a very hard task since these processes are highly nonlinear and their kinetic parameters are usually badly or inadequately known. To overcome the difficulties imposed by the lack of reliable or/and available sensors and the poor knowledge of the nonlinearities, several approaches have been proposed in the past. In addition to the well known Kalman filters and Luenberger observers, some of these approaches have included statistically
linearized filters, global linearization methods, pseudo-linearization methods and extended linearization methods (Misawa and Hedrick, 1989). Other approaches include the so-called high-gain observers and adaptive observers (Gauthier and Kupka, 1994). However, most of these approaches have shown to have limited performance results and stability properties only locally valid. In recent years the asymptotic observers have been proposed for relatively simple systems (Bastin and Dochain, 1990). Advantages of asymptotic observers compared to the previously cited approaches are their easy implementation in practice, the minimum number of easily verifiable conditions and the fact they permit the exact cancellation of the nonlinear terms. In this paper, the asymptotic observer originally proposed by Chen (1992) to reconstruct state variables of complex systems, is applied to a real anaerobic digestion process. In addition, using a property called cooperativity, an alternative issue for stability is proposed when the matrix that explicits the linear dependence between the state variables is time-dependent.
2. CONSIDERED SYSTEMS A very large number of chemical and biochemical processes can be described by the following class of nonlinear ordinary differential equations systems: x& (t ) = C f (x(t ),t ) + A(t )x(t ) + b(t ) n
(1) n×r
where x(t) ∈ ℜ is the state vector, C ∈ ℜ represents a matrix of coefficients (e.g. stoichiometric or yield coefficients) and f(x(t),t) ∈ r ℜ denotes the vector of nonlinearities (including reaction rates) which, in this paper, is supposed to be fully unknown. The time-varying matrix A(t) ∈ n×n n ℜ is the state matrix while b(t) ∈ ℜ belongs to a vector gathering the inputs (i.e. the mass feeding rate vector) and/or other functions possibly timevarying (e.g. the gaseous outflow rate vector, if any). The following hypotheses about the model (1) are introduced: Hypotheses H1: a) A(t) is known ∀ t ≥ 0 b) b(t) is known ∀ t ≥ 0 . c) C is constant and known. d) m states are measured on-line. Then the state space can be split in such a way that (1) can be rewritten as: x&1 (t ) = C1 f (x(t ),t ) + A11 (t )x1 (t ) + A12 (t )x2 (t ) + b1 (t ) x& 2 (t ) = C2 f (x(t ),t ) + A21 (t )x1 (t ) + A22 (t )x2 (t ) + b2 (t ) where the m measured states have been grouped in the x2(t) vector (dim x2(t) = m) and the variables that have to be estimated are represented by x1(t) s×s (dim x1(t) = s= n - m). Matrices A11(t) ∈ ℜ , A12(t) s×m m×s m×m s×r ∈ ℜ , A21(t) ∈ ℜ , A22(t) ∈ ℜ , C1 ∈ ℜ , C2 m×r s m ∈ ℜ b1 ∈ ℜ and b2 ∈ ℜ are the corresponding partitions of A(t), C and b(t) respectively. e) A(t) is bounded; that is, there exist constant − + A and A such as matrices A− ≤ A(t ) ≤ A+ ∀ t ≥ 0 , where the operator ≤ should be understood as a collection of inequalities between components. 3. THE ASYMPTOTIC OBSERVER Design of the asymptotic observer. Given that the nonlinearities f(x(t),t) are unknown, the asymptotic observer is designed in such a way that it enables the reconstruction of the unmeasured states from the measured ones, whatever the unknown nonlinearities are (Chen, 1992). This can be done by finding a suitable linear combination of the state s×n variables w(t) = Nx(t) with N ∈ ℜ , such that:
NC = 0.
(2)
The following additional hypothesis is recalled: Hypothesis H2: rank C = rank C2 = c . This implies the following important properties: a) m and s are fixed as m ≥ c, and thus, s ≤ n-c. Then, this hypothesis establishes how the partition of the hypothesis H1d features. b) C1 can be written as a linear combination of C2 s×m (i.e., C1 = K C2 with K ∈ ℜ ). c) The non-trivial solution of (2) admits at least s columns of N to be arbitrary chosen. Then, a non-trivial solution of (2) can be established as follows: Let N be rewritten as N = [N1 M N 2 ] with N1 ∈ ℜ , s×s
s×m
and (2) be equivalently rewritten as N2 ∈ ℜ N1C1 + N 2C2 = 0 . Then, thanks to the properties (b) and (c), the simplest solution of (3) is that N1 can be arbitrarily chosen to compute N2 as § § N 2 = − N1C1C2 where C 2 is a generalized pseudoinverse of C2 with the property C2C2 C2 = C2 . Thus, under hypotheses H1a-d and H2, the following system (Chen, 1992): §
wˆ& (t ) = W (t )wˆ (t ) + X (t )x2 (t ) + Nb(t ) wˆ (0) = Nxˆ (0) x (t ) = N −1 (w(t ) − N x (t )) ˆ 1 2 2 ˆ1 with
(3)
W (t ) = (N1 A11 (t ) + N 2 A21 (t ))N1−1
X (t ) = N1 A12 (t ) + N 2 A22 (t ) − W (t )N 2 is a nonlinearities independent asymptotic observer for the model (1).
Remarks R1: a) It should be noted that a choice of s columns of N other than N1 is also possible. However, −1 whatever the choice, the existence of N1 is necessary for rebuilding the unmeasured states. § b) Notice that C 2 is not necessarily unique. Moreover, in order to eliminate the dependence of f(x(t),t), Chen (1992), distinguishes between two cases: a) c ≤ r, and b) c < r proposing different solutions of (2) for each case. In this paper, the use of C 2§ in the solution of (2) covers these two cases. Stability of the asymptotic observer. Bastin and Dochain, (1990) provide sufficient conditions for the stability of (3) when A(t) is proportional to the identity matrix. When A is not proportional to the identity matrix and it has a defined structure but it is constant Chen, (1992), provide also sufficient conditions for stability of (3). Then, in this section an alternative issue for the stability of (3) is proposed when both A(t) is not proportional to the identity matrix and it is time-dependent. For doing
that, a property called cooperativity will be invoked. Thus, the following lemma is recalled: Lemma 1 (Smith, 1995): Let ζ& = f(ζ , t) . This system is said to be a cooperative system if ∂f i (ζ , t ) ≥ 0,∀i ≠ j . It implies that if ζ (0) ≥ 0 ∂ζ j
such a case, the stability properties of (3) are strictly governed by the operating conditions in the plant. This situation occurs also if A21 ≠ 0, but − C 2§ = C2 1 . b) Notice however that, in general, as has been § stated above, C 2 could be not unique. Then, the stability properties of (3) are a function of both the § operational limits and a suitable choice of C 2 .
Until now, this property had been used in observer schemes only for robustness purposes (see for example, (Alcaraz et al., 1999)). In this paper, instead, it will be used to establish sufficient conditions for stability. Now, consider the matrix
c) Besides the role that C 2 plays in the stability of the asymptotic observer, notice that the structure of the matrix associated with the error estimation § We (t ) suggests that C 2 could play on the tuning of
then ζ (t ) ≥ 0, ∀t ≥ 0 .
We (t ) = N1 1W (t )N1 = A11 (t ) − C1C2§ A21 (t ) −
(4)
which is associated with the dynamics of observation error of (3). Notice that under hypothesis H1e it is possible to compute − + and We such constant matrices We
the the two that
We ≤ We (t ) ≤ We ∀t ≥ 0. −
+
§ § Proposition: Whatever C 2 is (i.e., C 2 is a generalized pseudo-inverse (no unique solution) or § C2 is a full rank square matrix and then C 2 −1
becomes C2 (unique solution)), the observer (3) is asymptotically stable if the following conditions hold: a) We−, ij ≥ 0, ∀i ≠ j − + b) We and We are Hurwitz stable
Proof: Firstly, with reference to the previous lemma, the first condition of this proposition − simply states that the matrix We and thus, the
matrices We (t ) and We are cooperative. Secondly, let e(t ) = xˆ1 (t ) − x1 (t ) if xˆ1 (0 ) − x1 (0 ) ≥ 0 or e(t ) = x1 (t ) − xˆ1 (t ) if xˆ1 (0 ) − x1 (0 ) ≤ 0 be the observation error associated to (3). It is easy to verify that e has the dynamics: e&(t ) = We (t )e(t ) . +
− Then, thanks to the cooperativity of We (t ) , We and
We+ , e(t) is bounded by e1(t), solution of e&1 (t ) = We e1 (t ) , e1(0) = e(0), and e2(t), solution of −
e& 2 (t ) = We e 2 (t ) , +
e2(0) = e(0)
(i.e.,
e1 (t ) ≤ e(t ) ≤ e2 (t ), ∀t ≥ 0 ). In addition, if We and −
We− are Hurwitz stable, then e(t) necessarily converges asymptotically to e = 0 as long as t → ∞ for any e1(0). Therefore, xˆ1 (t ) also converges asymptotically towards x1 (t ) as long as t → ∞ for any initial conditions. Remarks R2: a) A particular case may arrive when A21 = 0. In
§
§
the asymptotic observer. Indeed, if C 2 is not unique and if the sub-matrix A21≠0, then C 2 could be used for generating more than one pair of stable − + matrices We and We placing desired poles in these matrices. However, this problem is beyond the framework of this paper. §
4. EXAMPLE: APPLICATION TO A WASTEWATER TREATMENT PROCESS The anaerobic digester model. Anaerobic digestion is a multistep biological process in which organic matter is degraded into a gas mixture of methane (CH4) and carbon dioxide (CO2). It thus reduces the Chemical Oxygen Demand (COD) of the influent while producing valuable energy (i.e., methane). The biological scheme involves several multisubstrate multi-organism reactions that are performed both in series and in parallel (see for example Henze and Harremoes, (1983)). In the following, a model of an anaerobic digestion process carried out in a continuous fixed bed reactor for the treatment of industrial wine distillery vinasses, developed and experimentally validated by Bernard et al., (1998), is considered: T x& (t ) = [X& 1 X& 2 Z& C& TI S&1 S& 2 ] = Cf (ξ (t ), t ) + A(t )ξ + b(t )
(5)
with f (x(t )) = [µ 1 x1 µ 2 x 2 ] , T
[
]
T
− k1 k 2 1 0 0 k4 = M , 0 1 0 k 0 − k 3 5 A11 (t )M A12 (t ) A(t ) = K . K A21 (t )M A22 (t ) 0 0 0 M 0 − αD(t ) 0 0 − αD(t ) 0 0 0 M 0 0 0 − D(t ) 0 0 M 0 = 0 ( ( ) ) 0 0 k − D t + k − k 7 M 7 7 . K K K K K K 0 0 0 M− D(t ) 0 0 0 0 0 M 0 − D(t ) 0 T T
C = C1 MC 2 T
[
]
N = [N 1 M N 2 ]
b(t ) = b1T (t )M b2T (t )
T
kkP = D(t ) x1i (t ) x2i (t ) x3i (t ) x4i (t ) + 7 8 CO2 M x5i (t ) x6i (t ) D(t ) which is identical to model (1), where X1, X2, S1, S2, CTI and Z are respectively the concentrations of acidogenic bacteria, methanogenic bacteria, COD, Volatile Fatty Acids (VFA), total inorganic carbon and strong ions. For these variables, the upper index “i” indicates “influent concentration”. The parameter α is associated with the dilution rate for bacteria. The CO2 partial pressure is represented by PCO2 The variable D(t) is the dilution rate and it is supposed
∫
t +δ
t0
to
be
D(τ ) dτ > 0 ∀t
a
persisting
input,
i.e.
for some positive δ. The
microbial growth rates; µ1 and µ2, are highly nonlinear functions given respectively by the Monod and Haldane kinetics (Henze and Harremoes, 1983). Parameters k1 to k7 are yield coefficients while k8 is the Henry’s constant. The values of all parameters used for experimental runs are listed in Table 1. Table 1: Model parameters, (Bernard et al., 1998) Parameter
Value and units
k1 k2 k3 k4 k5 k6 k7 k8 α
6.6 Kg COD/Kg X1 7.8 mol VFA/Kg X1 611.2 mol VFA/Kg X2 7.8 mol CO2/Kg X1 977.6 mol CO2/Kg X2 1139.2 mol CH4/Kg X2 50 d-1 0.1579 mol CO2/ m3-KPa 0.5 (adimentional)
Observer design. In agreement with the Hypoteses H1a-b, D(t) as well as inputs were considered as known. Thus, A(t), and b(t) are known ∀ t ≥ 0 . Thereafter, in agreement with the hypotheses H1e, D(t) varies as 0.01 < D(t) < 1.2 d −1 , (see Figure 1) and thus, A(t) is bounded ∀ t ≥ 0 . Afterwards, in agreement with the hypotheses H1c, C was considered constant and known using the parameters reported in Table 1. From (6), it is clear that rank C = 2. Thus, in agreement with the hypotheses H1d, only a minimum of two measurements is required to reconstruct the state space. In fact, if the two substrate concentrations S1 and S2 are used to estimate on-line X1, X2, Z and CTI. Then, partitions of the hypothesis H1d are as shown in (5). Without loss of generality, N1 has been arbitrary chosen as N1 = I4. Therefore, N takes the form:
k 1 k 3 0 0 0 M k3 0 1 0 k1 k 3 0 0 M k k 2 1 = 0 0 k1 k 3 0 0 k1 k 3 0 M M 0 0 0 k 1 k 3 (k 3 k 4 + k 2 k 5 )k 1 k 5 It is easy to verify that W(t), We(t) and X(t) are described by the following matrices: 0 0 − αD(t ) 0 0 − αD(t ) 0 0 W (t ) = We (t ) = 0 0 − D(t ) 0 0 0 k 7 − (D(t ) + k 7 ) 0 (α − 1)k 3 D(t ) 1 (α − 1)k 2 D(t ) (α − 1)k1 D(t ) X (t ) = 0 0 k1 k3 ( ) ( ) k k k k k k k k k + − 2 5 7 1 7 5 3 3 4 + − and We and We take the values:
0 0 0 − 0.005 0 0 . 005 0 0 , − We = 0 0 0 − 0.01 0 0 50 − 50.01 0 0 − 0.6 0 − 0 0 . 6 0 0 , − We = 0 0 − 1.2 0 0 0 50 − 51.2 +
with eigenvalues respectively: T + λ We = [− 0.005 − 0.005 − 0.01 − 50.01] and
( ) λ (W ) = [− 0.6
− 0.6 − 1.2 − 51.2] which meets the conditions of the Proposition P1. Thus, the estimation is asymptotically stable. −
T
e
Simulation results, applied on this model can be found in Alcaraz et al., (1999). In this paper, rather than showing simulation results, it has been preferred to present experimental results directly, where stability properties are stood out. Experimental runs. The experimental runs were 3 carried out using a 1 m upflow anaerobic fixed bed reactor for the treatment of industrial wine distillery vinasses obtained from local distilleries in the Narbonne area (France). These experimental runs were carried out over a 35 day period. The measurements of the dilution rate as well as the measurements of S1, S2 and the partial CO2 pressure were performed on-line. They are represented in i i i Figures 1 to 4. The measurements of S1 , S2 , Z and i CTI were obtained from off-line data and they were assumed constant between measurements. They are represented in Figures 5 to 8. The influent biomass i i concentrations, X1 and X2 were assumed negligible.
Figure 1: Experimental on-line measurements of D.
Figure 6: Experimental influent S2 concentration.
Figure 2: Experimental on-line effluent S1 concentration.
Figure 7: Experimental influent Z concentration.
Figure 3: Experimental on-line effluent S2 concentration.
Figure 8: Experimental influent CTI concentration.
Figure 4: Experimental on-linePCO2 measurements.
Experimental results. Estimation results for the unmeasured states are presented in Figures 9 to 12. Notice that these experimental runs were perturbed with real experimental noise in the input concentrations, the dilution rate and the measured states. Despite that, the observer (3) presented excellent properties of convergence and stability. It was able to estimate, with a reasonably good accuracy, the unmeasured states exhibiting a nondivergent behavior. Also, in relation to stability, the asymptotic observer (3) was tested with two sets of initial conditions (see Figures 9 to 12) showing excellent stability properties in all cases. Furthermore, as it can be seen in these figures, the observer exhibits fast convergence properties even in the non-steady state. Nevertheless, it must be noticed that, for this example, the convergence rate cannot be tuned. Effectively, as A21=0, the stability only depends on the operating conditions at the plant.
Figure 5: Experimental influent S1 concentration.
Figure 9: Estimate of the acidogenic bacteria concentration.
such an observer. For time-varying systems, sufficient conditions for stability were established. This observer was satisfactorily tested and validated experimentally using real data obtained 3 from a 1 m fixed bed reactor for the treatment of industrial distillery wine vinasses. The asymptotic observer scheme showed a satisfactory performance and excellent stability properties even in the presence of noise due to operating conditions. A logical extension of this approach, now under study, is the introduction of tuning parameters into the observer. Because of the major interest of these observers at the experimental scale, their use in robust nonlinear control schemes with application to continuous bioreactors is also under study. REFERENCES
Figure 10: Estimate of the methanogenic bacteria concentration.
Figure 11: Estimate of the concentration of Z (-- : estimated state, __ : experimental on-line measurements).
Figure 12: Estimate of the total inorganic carbon concentration (-- : estimated state, __ : experimental on-line measurements).
5. CONCLUSIONS AND FUTURE WORK In this paper, the robust asymptotic observer initially proposed by Chen, (1992) was reviewed. Easily verifiable conditions on the general structure of the model have been established for designing
Alcaraz-González V., Genovesi A., Harmand J., González-Alvarez V., Rapaport A. and Steyer, J.P. (1999) Robust exponential nonlinear interval observers for a class of lumped models useful in chemical and biochemical engineering. Application to a wastewater process. Applications of Interval Analysis to Systems and Control, MISC’99. pp. 225-235, Gerona, Spain,. Bastin G. & Dochain D. (1990). On-line estimation and adaptive control of bioreactors. Elsevier. Bernard O., Dochain D., Genovesi A., Puñal A., Perez Alvarino D., Steyer J.P. & Lema, J. (1998). Software sensor design for an anaerobic wastewater treatment plant. International Workshop on "Decision and Control in Waste Bio-Processing", WASTE-DECISION'98, 8 pages (CD-ROM), Narbonne, France. Chen, L. (1992). Modelling, Identifiability and Control of Complex Biotechnological Systems. PhD Thesis, Université Catholique de Louvain, Louvain, Belgium. Gauthier J.P. & Kupka I. (1994). Observability and Observers for Nonlinear Systems. SIAM J. Control and Optim., vol. 34, n°4, pp. 975-994. Henze M. and Harremoes P. (1983). Anaerobic Treatment of Wastewater in Fixed Film Reactors- A Literature Review. Water Science and Technology., vol. 15, n°1, pp. 1-101. Misawa, E. A., & Hedrick, J. K. (1989). Nonlinear Observers. A State-of-the-Art Survey. Trans., of the ASME. Vol. 111, pp. 344-352. Smith, H.L. (1995) Monotone Dynamical Systems. An introduction to the Theory of Competitive and Cooperative Systems, AMS Mathematical Surveys and Monographs, vol. 41, (1995) pp. 31-53. Acknowledgements. The authors gratefully acknowledge the ECOS-Nord program for French-Mexican Scientific Cooperation (M97-B01 project) as well as ANUIES (Mexican National Association of Universities and High Education Institutes), the mexican program PROMEP, the French ADEME/AGRICE program (N° DVNAC 9901028 INRA/ADEME/AGRICE Contract) and the TELEMAC project (IST 2000-28156)) for the financial support that made this study possible.
1
3
Alcaraz-González V. , Harmand J. , Dochain D. , Rapaport A.2, Steyer J.P.1, Pelayo Ortiz C.4, and González-Alvarez V.4 1
INRA-LBE, Avenue des Etangs, 11100 Narbonne - France. Fax : (+33)-468-425-160 2 INRA-LASB,2 Place Viala, 34060 Montpellier - France. 3 CESAME, Université Catholique de Louvain, 4-6 avenue Georges Lemaître B-1348, Louvain la Neuve - Belgium. Fax : (+32)-10-472-180 4 Universidad de Guadalajara. CUCEI. Calz. Gral M. García Barragan 1451. Guadalajara, México.
Abstract : A simple state observer is proposed for a class of lumped nonlinear time-varying systems useful in chemical and biochemical engineering. It is shown that this asymptotic nonlinear observer is stable in the presence of time-varying elements and robust in the face of initial conditions uncertainty and a total lack of knowledge on the nonlinearities of the system. Experimental results are presented using a model of an anaerobic digestion process for the treatment of industrial wastewater from a wine distillery and tested using real data obtained from 3 a 1 m continuous fixed bed pilot bioreactor. Copyright 2003 IFAC
Keywords: Robust estimation, Stability, Time-varying systems, Biotechnology.
1. INTRODUCTION One of the most frequent and important challenges in the control of chemical and biochemical processes is finding adequate and reliable sensors to measure all the important state variables of the plant. However, if a number of on-line sensors providing state information are available today at the industrial scale, they are still very expensive and their maintenance is usually time consuming. This is especially true in the field of chemical and biological processes. Furthermore, even when the information of the important variables is readily available, controlling the biological processes is a very hard task since these processes are highly nonlinear and their kinetic parameters are usually badly or inadequately known. To overcome the difficulties imposed by the lack of reliable or/and available sensors and the poor knowledge of the nonlinearities, several approaches have been proposed in the past. In addition to the well known Kalman filters and Luenberger observers, some of these approaches have included statistically
linearized filters, global linearization methods, pseudo-linearization methods and extended linearization methods (Misawa and Hedrick, 1989). Other approaches include the so-called high-gain observers and adaptive observers (Gauthier and Kupka, 1994). However, most of these approaches have shown to have limited performance results and stability properties only locally valid. In recent years the asymptotic observers have been proposed for relatively simple systems (Bastin and Dochain, 1990). Advantages of asymptotic observers compared to the previously cited approaches are their easy implementation in practice, the minimum number of easily verifiable conditions and the fact they permit the exact cancellation of the nonlinear terms. In this paper, the asymptotic observer originally proposed by Chen (1992) to reconstruct state variables of complex systems, is applied to a real anaerobic digestion process. In addition, using a property called cooperativity, an alternative issue for stability is proposed when the matrix that explicits the linear dependence between the state variables is time-dependent.
2. CONSIDERED SYSTEMS A very large number of chemical and biochemical processes can be described by the following class of nonlinear ordinary differential equations systems: x& (t ) = C f (x(t ),t ) + A(t )x(t ) + b(t ) n
(1) n×r
where x(t) ∈ ℜ is the state vector, C ∈ ℜ represents a matrix of coefficients (e.g. stoichiometric or yield coefficients) and f(x(t),t) ∈ r ℜ denotes the vector of nonlinearities (including reaction rates) which, in this paper, is supposed to be fully unknown. The time-varying matrix A(t) ∈ n×n n ℜ is the state matrix while b(t) ∈ ℜ belongs to a vector gathering the inputs (i.e. the mass feeding rate vector) and/or other functions possibly timevarying (e.g. the gaseous outflow rate vector, if any). The following hypotheses about the model (1) are introduced: Hypotheses H1: a) A(t) is known ∀ t ≥ 0 b) b(t) is known ∀ t ≥ 0 . c) C is constant and known. d) m states are measured on-line. Then the state space can be split in such a way that (1) can be rewritten as: x&1 (t ) = C1 f (x(t ),t ) + A11 (t )x1 (t ) + A12 (t )x2 (t ) + b1 (t ) x& 2 (t ) = C2 f (x(t ),t ) + A21 (t )x1 (t ) + A22 (t )x2 (t ) + b2 (t ) where the m measured states have been grouped in the x2(t) vector (dim x2(t) = m) and the variables that have to be estimated are represented by x1(t) s×s (dim x1(t) = s= n - m). Matrices A11(t) ∈ ℜ , A12(t) s×m m×s m×m s×r ∈ ℜ , A21(t) ∈ ℜ , A22(t) ∈ ℜ , C1 ∈ ℜ , C2 m×r s m ∈ ℜ b1 ∈ ℜ and b2 ∈ ℜ are the corresponding partitions of A(t), C and b(t) respectively. e) A(t) is bounded; that is, there exist constant − + A and A such as matrices A− ≤ A(t ) ≤ A+ ∀ t ≥ 0 , where the operator ≤ should be understood as a collection of inequalities between components. 3. THE ASYMPTOTIC OBSERVER Design of the asymptotic observer. Given that the nonlinearities f(x(t),t) are unknown, the asymptotic observer is designed in such a way that it enables the reconstruction of the unmeasured states from the measured ones, whatever the unknown nonlinearities are (Chen, 1992). This can be done by finding a suitable linear combination of the state s×n variables w(t) = Nx(t) with N ∈ ℜ , such that:
NC = 0.
(2)
The following additional hypothesis is recalled: Hypothesis H2: rank C = rank C2 = c . This implies the following important properties: a) m and s are fixed as m ≥ c, and thus, s ≤ n-c. Then, this hypothesis establishes how the partition of the hypothesis H1d features. b) C1 can be written as a linear combination of C2 s×m (i.e., C1 = K C2 with K ∈ ℜ ). c) The non-trivial solution of (2) admits at least s columns of N to be arbitrary chosen. Then, a non-trivial solution of (2) can be established as follows: Let N be rewritten as N = [N1 M N 2 ] with N1 ∈ ℜ , s×s
s×m
and (2) be equivalently rewritten as N2 ∈ ℜ N1C1 + N 2C2 = 0 . Then, thanks to the properties (b) and (c), the simplest solution of (3) is that N1 can be arbitrarily chosen to compute N2 as § § N 2 = − N1C1C2 where C 2 is a generalized pseudoinverse of C2 with the property C2C2 C2 = C2 . Thus, under hypotheses H1a-d and H2, the following system (Chen, 1992): §
wˆ& (t ) = W (t )wˆ (t ) + X (t )x2 (t ) + Nb(t ) wˆ (0) = Nxˆ (0) x (t ) = N −1 (w(t ) − N x (t )) ˆ 1 2 2 ˆ1 with
(3)
W (t ) = (N1 A11 (t ) + N 2 A21 (t ))N1−1
X (t ) = N1 A12 (t ) + N 2 A22 (t ) − W (t )N 2 is a nonlinearities independent asymptotic observer for the model (1).
Remarks R1: a) It should be noted that a choice of s columns of N other than N1 is also possible. However, −1 whatever the choice, the existence of N1 is necessary for rebuilding the unmeasured states. § b) Notice that C 2 is not necessarily unique. Moreover, in order to eliminate the dependence of f(x(t),t), Chen (1992), distinguishes between two cases: a) c ≤ r, and b) c < r proposing different solutions of (2) for each case. In this paper, the use of C 2§ in the solution of (2) covers these two cases. Stability of the asymptotic observer. Bastin and Dochain, (1990) provide sufficient conditions for the stability of (3) when A(t) is proportional to the identity matrix. When A is not proportional to the identity matrix and it has a defined structure but it is constant Chen, (1992), provide also sufficient conditions for stability of (3). Then, in this section an alternative issue for the stability of (3) is proposed when both A(t) is not proportional to the identity matrix and it is time-dependent. For doing
that, a property called cooperativity will be invoked. Thus, the following lemma is recalled: Lemma 1 (Smith, 1995): Let ζ& = f(ζ , t) . This system is said to be a cooperative system if ∂f i (ζ , t ) ≥ 0,∀i ≠ j . It implies that if ζ (0) ≥ 0 ∂ζ j
such a case, the stability properties of (3) are strictly governed by the operating conditions in the plant. This situation occurs also if A21 ≠ 0, but − C 2§ = C2 1 . b) Notice however that, in general, as has been § stated above, C 2 could be not unique. Then, the stability properties of (3) are a function of both the § operational limits and a suitable choice of C 2 .
Until now, this property had been used in observer schemes only for robustness purposes (see for example, (Alcaraz et al., 1999)). In this paper, instead, it will be used to establish sufficient conditions for stability. Now, consider the matrix
c) Besides the role that C 2 plays in the stability of the asymptotic observer, notice that the structure of the matrix associated with the error estimation § We (t ) suggests that C 2 could play on the tuning of
then ζ (t ) ≥ 0, ∀t ≥ 0 .
We (t ) = N1 1W (t )N1 = A11 (t ) − C1C2§ A21 (t ) −
(4)
which is associated with the dynamics of observation error of (3). Notice that under hypothesis H1e it is possible to compute − + and We such constant matrices We
the the two that
We ≤ We (t ) ≤ We ∀t ≥ 0. −
+
§ § Proposition: Whatever C 2 is (i.e., C 2 is a generalized pseudo-inverse (no unique solution) or § C2 is a full rank square matrix and then C 2 −1
becomes C2 (unique solution)), the observer (3) is asymptotically stable if the following conditions hold: a) We−, ij ≥ 0, ∀i ≠ j − + b) We and We are Hurwitz stable
Proof: Firstly, with reference to the previous lemma, the first condition of this proposition − simply states that the matrix We and thus, the
matrices We (t ) and We are cooperative. Secondly, let e(t ) = xˆ1 (t ) − x1 (t ) if xˆ1 (0 ) − x1 (0 ) ≥ 0 or e(t ) = x1 (t ) − xˆ1 (t ) if xˆ1 (0 ) − x1 (0 ) ≤ 0 be the observation error associated to (3). It is easy to verify that e has the dynamics: e&(t ) = We (t )e(t ) . +
− Then, thanks to the cooperativity of We (t ) , We and
We+ , e(t) is bounded by e1(t), solution of e&1 (t ) = We e1 (t ) , e1(0) = e(0), and e2(t), solution of −
e& 2 (t ) = We e 2 (t ) , +
e2(0) = e(0)
(i.e.,
e1 (t ) ≤ e(t ) ≤ e2 (t ), ∀t ≥ 0 ). In addition, if We and −
We− are Hurwitz stable, then e(t) necessarily converges asymptotically to e = 0 as long as t → ∞ for any e1(0). Therefore, xˆ1 (t ) also converges asymptotically towards x1 (t ) as long as t → ∞ for any initial conditions. Remarks R2: a) A particular case may arrive when A21 = 0. In
§
§
the asymptotic observer. Indeed, if C 2 is not unique and if the sub-matrix A21≠0, then C 2 could be used for generating more than one pair of stable − + matrices We and We placing desired poles in these matrices. However, this problem is beyond the framework of this paper. §
4. EXAMPLE: APPLICATION TO A WASTEWATER TREATMENT PROCESS The anaerobic digester model. Anaerobic digestion is a multistep biological process in which organic matter is degraded into a gas mixture of methane (CH4) and carbon dioxide (CO2). It thus reduces the Chemical Oxygen Demand (COD) of the influent while producing valuable energy (i.e., methane). The biological scheme involves several multisubstrate multi-organism reactions that are performed both in series and in parallel (see for example Henze and Harremoes, (1983)). In the following, a model of an anaerobic digestion process carried out in a continuous fixed bed reactor for the treatment of industrial wine distillery vinasses, developed and experimentally validated by Bernard et al., (1998), is considered: T x& (t ) = [X& 1 X& 2 Z& C& TI S&1 S& 2 ] = Cf (ξ (t ), t ) + A(t )ξ + b(t )
(5)
with f (x(t )) = [µ 1 x1 µ 2 x 2 ] , T
[
]
T
− k1 k 2 1 0 0 k4 = M , 0 1 0 k 0 − k 3 5 A11 (t )M A12 (t ) A(t ) = K . K A21 (t )M A22 (t ) 0 0 0 M 0 − αD(t ) 0 0 − αD(t ) 0 0 0 M 0 0 0 − D(t ) 0 0 M 0 = 0 ( ( ) ) 0 0 k − D t + k − k 7 M 7 7 . K K K K K K 0 0 0 M− D(t ) 0 0 0 0 0 M 0 − D(t ) 0 T T
C = C1 MC 2 T
[
]
N = [N 1 M N 2 ]
b(t ) = b1T (t )M b2T (t )
T
kkP = D(t ) x1i (t ) x2i (t ) x3i (t ) x4i (t ) + 7 8 CO2 M x5i (t ) x6i (t ) D(t ) which is identical to model (1), where X1, X2, S1, S2, CTI and Z are respectively the concentrations of acidogenic bacteria, methanogenic bacteria, COD, Volatile Fatty Acids (VFA), total inorganic carbon and strong ions. For these variables, the upper index “i” indicates “influent concentration”. The parameter α is associated with the dilution rate for bacteria. The CO2 partial pressure is represented by PCO2 The variable D(t) is the dilution rate and it is supposed
∫
t +δ
t0
to
be
D(τ ) dτ > 0 ∀t
a
persisting
input,
i.e.
for some positive δ. The
microbial growth rates; µ1 and µ2, are highly nonlinear functions given respectively by the Monod and Haldane kinetics (Henze and Harremoes, 1983). Parameters k1 to k7 are yield coefficients while k8 is the Henry’s constant. The values of all parameters used for experimental runs are listed in Table 1. Table 1: Model parameters, (Bernard et al., 1998) Parameter
Value and units
k1 k2 k3 k4 k5 k6 k7 k8 α
6.6 Kg COD/Kg X1 7.8 mol VFA/Kg X1 611.2 mol VFA/Kg X2 7.8 mol CO2/Kg X1 977.6 mol CO2/Kg X2 1139.2 mol CH4/Kg X2 50 d-1 0.1579 mol CO2/ m3-KPa 0.5 (adimentional)
Observer design. In agreement with the Hypoteses H1a-b, D(t) as well as inputs were considered as known. Thus, A(t), and b(t) are known ∀ t ≥ 0 . Thereafter, in agreement with the hypotheses H1e, D(t) varies as 0.01 < D(t) < 1.2 d −1 , (see Figure 1) and thus, A(t) is bounded ∀ t ≥ 0 . Afterwards, in agreement with the hypotheses H1c, C was considered constant and known using the parameters reported in Table 1. From (6), it is clear that rank C = 2. Thus, in agreement with the hypotheses H1d, only a minimum of two measurements is required to reconstruct the state space. In fact, if the two substrate concentrations S1 and S2 are used to estimate on-line X1, X2, Z and CTI. Then, partitions of the hypothesis H1d are as shown in (5). Without loss of generality, N1 has been arbitrary chosen as N1 = I4. Therefore, N takes the form:
k 1 k 3 0 0 0 M k3 0 1 0 k1 k 3 0 0 M k k 2 1 = 0 0 k1 k 3 0 0 k1 k 3 0 M M 0 0 0 k 1 k 3 (k 3 k 4 + k 2 k 5 )k 1 k 5 It is easy to verify that W(t), We(t) and X(t) are described by the following matrices: 0 0 − αD(t ) 0 0 − αD(t ) 0 0 W (t ) = We (t ) = 0 0 − D(t ) 0 0 0 k 7 − (D(t ) + k 7 ) 0 (α − 1)k 3 D(t ) 1 (α − 1)k 2 D(t ) (α − 1)k1 D(t ) X (t ) = 0 0 k1 k3 ( ) ( ) k k k k k k k k k + − 2 5 7 1 7 5 3 3 4 + − and We and We take the values:
0 0 0 − 0.005 0 0 . 005 0 0 , − We = 0 0 0 − 0.01 0 0 50 − 50.01 0 0 − 0.6 0 − 0 0 . 6 0 0 , − We = 0 0 − 1.2 0 0 0 50 − 51.2 +
with eigenvalues respectively: T + λ We = [− 0.005 − 0.005 − 0.01 − 50.01] and
( ) λ (W ) = [− 0.6
− 0.6 − 1.2 − 51.2] which meets the conditions of the Proposition P1. Thus, the estimation is asymptotically stable. −
T
e
Simulation results, applied on this model can be found in Alcaraz et al., (1999). In this paper, rather than showing simulation results, it has been preferred to present experimental results directly, where stability properties are stood out. Experimental runs. The experimental runs were 3 carried out using a 1 m upflow anaerobic fixed bed reactor for the treatment of industrial wine distillery vinasses obtained from local distilleries in the Narbonne area (France). These experimental runs were carried out over a 35 day period. The measurements of the dilution rate as well as the measurements of S1, S2 and the partial CO2 pressure were performed on-line. They are represented in i i i Figures 1 to 4. The measurements of S1 , S2 , Z and i CTI were obtained from off-line data and they were assumed constant between measurements. They are represented in Figures 5 to 8. The influent biomass i i concentrations, X1 and X2 were assumed negligible.
Figure 1: Experimental on-line measurements of D.
Figure 6: Experimental influent S2 concentration.
Figure 2: Experimental on-line effluent S1 concentration.
Figure 7: Experimental influent Z concentration.
Figure 3: Experimental on-line effluent S2 concentration.
Figure 8: Experimental influent CTI concentration.
Figure 4: Experimental on-linePCO2 measurements.
Experimental results. Estimation results for the unmeasured states are presented in Figures 9 to 12. Notice that these experimental runs were perturbed with real experimental noise in the input concentrations, the dilution rate and the measured states. Despite that, the observer (3) presented excellent properties of convergence and stability. It was able to estimate, with a reasonably good accuracy, the unmeasured states exhibiting a nondivergent behavior. Also, in relation to stability, the asymptotic observer (3) was tested with two sets of initial conditions (see Figures 9 to 12) showing excellent stability properties in all cases. Furthermore, as it can be seen in these figures, the observer exhibits fast convergence properties even in the non-steady state. Nevertheless, it must be noticed that, for this example, the convergence rate cannot be tuned. Effectively, as A21=0, the stability only depends on the operating conditions at the plant.
Figure 5: Experimental influent S1 concentration.
Figure 9: Estimate of the acidogenic bacteria concentration.
such an observer. For time-varying systems, sufficient conditions for stability were established. This observer was satisfactorily tested and validated experimentally using real data obtained 3 from a 1 m fixed bed reactor for the treatment of industrial distillery wine vinasses. The asymptotic observer scheme showed a satisfactory performance and excellent stability properties even in the presence of noise due to operating conditions. A logical extension of this approach, now under study, is the introduction of tuning parameters into the observer. Because of the major interest of these observers at the experimental scale, their use in robust nonlinear control schemes with application to continuous bioreactors is also under study. REFERENCES
Figure 10: Estimate of the methanogenic bacteria concentration.
Figure 11: Estimate of the concentration of Z (-- : estimated state, __ : experimental on-line measurements).
Figure 12: Estimate of the total inorganic carbon concentration (-- : estimated state, __ : experimental on-line measurements).
5. CONCLUSIONS AND FUTURE WORK In this paper, the robust asymptotic observer initially proposed by Chen, (1992) was reviewed. Easily verifiable conditions on the general structure of the model have been established for designing
Alcaraz-González V., Genovesi A., Harmand J., González-Alvarez V., Rapaport A. and Steyer, J.P. (1999) Robust exponential nonlinear interval observers for a class of lumped models useful in chemical and biochemical engineering. Application to a wastewater process. Applications of Interval Analysis to Systems and Control, MISC’99. pp. 225-235, Gerona, Spain,. Bastin G. & Dochain D. (1990). On-line estimation and adaptive control of bioreactors. Elsevier. Bernard O., Dochain D., Genovesi A., Puñal A., Perez Alvarino D., Steyer J.P. & Lema, J. (1998). Software sensor design for an anaerobic wastewater treatment plant. International Workshop on "Decision and Control in Waste Bio-Processing", WASTE-DECISION'98, 8 pages (CD-ROM), Narbonne, France. Chen, L. (1992). Modelling, Identifiability and Control of Complex Biotechnological Systems. PhD Thesis, Université Catholique de Louvain, Louvain, Belgium. Gauthier J.P. & Kupka I. (1994). Observability and Observers for Nonlinear Systems. SIAM J. Control and Optim., vol. 34, n°4, pp. 975-994. Henze M. and Harremoes P. (1983). Anaerobic Treatment of Wastewater in Fixed Film Reactors- A Literature Review. Water Science and Technology., vol. 15, n°1, pp. 1-101. Misawa, E. A., & Hedrick, J. K. (1989). Nonlinear Observers. A State-of-the-Art Survey. Trans., of the ASME. Vol. 111, pp. 344-352. Smith, H.L. (1995) Monotone Dynamical Systems. An introduction to the Theory of Competitive and Cooperative Systems, AMS Mathematical Surveys and Monographs, vol. 41, (1995) pp. 31-53. Acknowledgements. The authors gratefully acknowledge the ECOS-Nord program for French-Mexican Scientific Cooperation (M97-B01 project) as well as ANUIES (Mexican National Association of Universities and High Education Institutes), the mexican program PROMEP, the French ADEME/AGRICE program (N° DVNAC 9901028 INRA/ADEME/AGRICE Contract) and the TELEMAC project (IST 2000-28156)) for the financial support that made this study possible.
