ROBUST EXPONENTIAL NONLINEAR INTERVAL OBSERVERS FOR ...
Alcaraz-Gonzalez V., A. Genovesi, J. Harmand, A. V. Gonzalez, A. Rapaport and J. P. Steyer (1999) : "Robust Exponential Nonlinear Observers for a Class of Lumped Models Useful in Chemical and Biochemical Engineering - Application to a Wastewater Treatment Process", International Workshop on Application of Interval Analysis to Systems and Control, MISC'99, Girona, Espagne, February 24-26.
ROBUST EXPONENTIAL NONLINEAR INTERVAL OBSERVERS FOR A CLASS OF LUMPED MODELS USEFUL IN CHEMICAL AND BIOCHEMICAL ENGINEERING. APPLICATION TO A WASTEWATER TREATMENT PROCESS. Alcaraz-González V.1, Genovesi A.1, Harmand J.1, González A.V.3, Rapaport A.2 and Steyer J.P. 1 1
INRA-LBE, Avenue des Etangs, 11100 Narbonne - France. Fax : 33-468-425-160 [email protected], [email protected], [email protected], [email protected] 2 INRA-Biométrie, 2, Place Viala, 34060 Montpellier - France. Fax : 33-499-521-427. [email protected] 3 Departamento de Ingeniería Química de la Universidad de Guadalajara. Blvd. Marcelino Garcia Barragán y Calzada Olímpica, 44860 Guadalajara Jalisco - México. Fax : 523-619-4028. [email protected]
Abstract :
In this paper, two new robust observers are detailed for a class of lumped models useful in chemical and biochemical engineering. The first observer is an exponential nonlinear observer which is robust against uncertainties on the kinetics of the process. To cope with additional disturbances on the inputs, thus leading to a non detectable system, a second observer is designed, that is a nonlinear interval observer. With an a priori knowledge of the bounds concerning the unknown initial conditions, this interval observer also provides guaranteed intervals for the unmeasured variables. Simulation results are presented using a model of an anaerobic digestion process for the treatment of an industrial wine distillery wastewater. Finally, the observers are experimentally tested using data obtained from a 1 m3 continuous fixed bed bioreactor.
Keywords : Robust nonlinear state observers, nonlinear mass balance models, interval, wastewater treatment processes, anaerobic digestion.
I - Introduction Working on the control of biological processes – and especially of biological wastewater treatment processes - raises a number of very challenging problems. Indeed, if a number of on-line sensors providing state information are today available at the industrial scale, they are still very expensive and their maintenance is usually time consuming. Furthermore, biological processes are highly nonlinear time varying systems in which kinetic parameters are badly or poorly known. To overcome these difficulties, the notion of software sensors has been introduced in the early eighties. In fact, these software sensors simply consist in observers designed to estimate unmeasured states from the available on-line measurements and they have been demonstrated to be of a great help for biological processes (see for example [DOCHAIN and PAUSS, 1988], [BASTIN and DOCHAIN, 1990] [GAUTHIER and KUPKA, 1994], [BEN et al., 1995], [BUSAWON, 1996] or [FARZA et al., 1998]). However, in all previously cited schemes, the knowledge of all the inputs of the process, including for example the substrate input concentration, is needed. Unfortunately, if this knowledge can be available for biological processes used in the food or in the pharmaceutical industries, this is normally not the case in the wastewater treatment field. As a consequence, part of the process input vector is considered as unmeasured input disturbances and classical observer schemes cannot be used since the system can
be not observable nor detectable depending on the available on-line measurements. In this paper, conditions for designing robust nonlinear observers for a class of nonlinear models are derived. Based upon the work of [RAPAPORT, 1998], [RAPAPORT and HARMAND, 1998a-b], existence conditions of these observers are derived assuming that part of the input vector is not measured. As a consequence, it is possible that the considered system is not detectable. In other terms, it is then not possible to estimate the unmeasured states from the available on-line measurements. The new observers derived in this paper are called set-observers since they allow the user to reconstruct a guaranteed interval on the unmeasured states instead of reconstructing their precise numerical values. The only requirement is to know an interval in which the unmeasured inputs of the process evolve. The paper is organized as follows. First, the classical nonlinear asymptotic observer for unknown kinetics proposed in [BASTIN and DOCHAIN, 1990] is extended in some sense without any restriction on the inputs. Then, the equations of this observer are used to build a new nonlinear robust interval observer assuming that input disturbances are unknown but that they belong to a prescribed bounded set. This methodology allows one to compute guaranteed intervals for the unmeasured states when bounds on the unknown initial conditions are given. Simulation and experimental results are then provided and finally, conclusions and perspectives are drawn.
II - The considered model In this paper, chemical and biochemical processes that can be described by the following general nonlinear model are considered : x& = C R(x(t ),t ) + A(t )x (t ) + b(t )
(1)
where x ∈ ℜn is the state vector, R ∈ ℜm denotes the reaction rate vector and C ∈ ℜn×m represents the constant stoichiometric coefficients matrix. The matrix A ∈ ℜn×n explicits the linear dependence between the state variables while b ∈ ℜn is a vector gathering all the terms of the model that are not a function of the state. Let us now assume that the state space can be split in such a way that (1) can be rewritten as : x&1 = C1 R(x (t ),t ) + A11 (t )x1 (t ) + A12 (t )x2 (t ) + b1 (t ) x&2 = C2 R( x(t ),t ) + A21 (t )x1 (t ) + A22 (t )x2 (t ) + b2 (t )
(2)
where the measured variables have been grouped in the x2 vector and the variables that has to be estimated are represented by x1. We note dim x1 = s and we define r = dim x2 = n - s. Matrices C1 ∈ ℜs×m, C2 ∈ ℜr×m are the corresponding partitions of C and A11 ∈ ℜs×s, A12 ∈ ℜs×r, A21 ∈ ℜr×s and A22 ∈ ℜr×r are the corresponding partitions of A. The following hypotheses are furthermore introduced : Hypotheses H1 : a) R is unknown. b) A and C are known. c) Measurements on x2 are continuously available. d) Guaranteed bounds on the initial conditions of the state vector are known with x(0)≤ x(0) ≤ x(0)+. e) There may also exist uncertainties on some parameters involved in b(t) (but not contained in A) and only some bounds of these are known such that b-(t) ≤ b(t) ≤ b+(t).
Note : The operator ≤ applied between vectors should be understood as a collection of inequalities between components.
III - An exponential observer In this section, only the hypotheses H1a-H1c will be necessary. Thus, it is assumed that the vector b(t) is perfectly known. In addition, m < n and C has a rank p with p ≤ m < n. Then, the following additional hypotheses are introduced : Hypotheses H2 : a) rank C2 = rank C b) r = dim x2 ≥ rank C Under hypotheses H2, it is possible to make a change of variables to design a reduced order observer regarding the following procedure. Let N ∈ ℜs×n be a matrix representing a linear combination of the state variables, so that an auxiliary variable w can be defined as : w(t)
=
Nx(t)
(3) Introducing this change of variable into (1), it is straightforward to establish that the new system will be kinetic independent if and only if N satisfies : NC = 0
(4)
Under hypotheses H2, the solution of the algebraic system (4) allows the choice of at least s rows of NT. In other words, if N1 ∈ ℜ s×s , N 2 ∈ ℜ s×r are the corresponding partitions of N, then we can arbitrary
choose N 1T to compute N 2T = −(C2T ) C1T N1T . Assumptions H2 ensure that (C 2T ) , the right pseudo−1
−1
inverse of C2T (in a general sense) exists. The kinetic independence property of the observer is guaranteed by (4). Thus, the last condition to be satisfied is that N 1−1 exists in order to recover the original state. The following additional hypotheses are then introduced : Hypotheses H3 : a) N1 is invertible. b) The dynamic of the system z& = A11 (t )z is exponentially stable. The first condition is satisfied if N1 = kIs is chosen where k is a real and positive arbitrary parameter. The second condition is satisfied if the real part of any eigenvalues of A11(t) are strictly negative for each t ≥ 0 . Under hypothesis H3a the derivative of the linear transformation (3) can be done and the original model (1) rewritten as a function of w and x2 as follows : w& = W (t )w + X (t )x 2 + Nb(t )
(5)
where W (t ) = N1 A11 ( t ) N1−1 , X (t ) = N1 A12 ( t ) + N 2 A22 (t ) − W (t ) N 2 . System (5) is then a linear and kinetics independent model which is only a function of w and x2.
Proposition 1 : Under hypotheses H1a-H1c, H2 and H3, when the vector b(t) is perfectly known, the following system:
wˆ& = W (t )wˆ (t ) + X (t )x 2 (t ) + Nb(t ) wˆ (0) = Nxˆ (0) xˆ = N −1 (wˆ − N x ) 1 2 2 1
(6)
is an exponential observer (i.e., xˆ1 (t ) converges exponentially towards x1 (t ) for any initial conditions). The small hat symbol "^" denotes "estimated" in all cases. Remark 1 : In some sense, on one hand, since there exists some hypothesis on the structure of the A matrix, this observer is less general than the one proposed in [BASTIN and DOCHAIN, 1990]. However, on the other hand, this structuration of the uncertainty allows the user to estimate unmeasured variables using less measurements than in the design proposed in [BASTIN and DOCHAIN, 1990]. Remark 2 : The general form of the proposed exponential observer is a generalization of a specific application that was proposed in [HADJ-SADOK et al., 1998].
IV - An interval observer Again, in this section, it is considered that the kinetics are unknown. In addition, it is assumed that H1cd are verified. In other words, some bounds on the initial conditions are available on the initial conditions and the vector b(t) is now unmeasured but some lower and upper bounds are known. In such a situation, notice that the model (1) can be no longer detectable. Thus, it is not possible to use the exponential observer developed in the previous section. However, its basic structure - and especially its property of being a kinetic independent transformation - will be used. The idea developed in the following is to design a set-valued observer in order to build guaranteed intervals for the unmeasured variables instead of estimating them precisely. Let us consider the following proposition :
Proposition 2 : Under hypotheses H1-H4, a robust interval observer for the system (1) is given by : For the upper bound: w& + = W (t )w + (t ) + X (t ) x (t ) + Mv + 2 + + w(0 ) = N x(0 ) + −1 + x$ 1 = N 1 w − N 2 x 2 For the lower bound: − − − w& = W (t )w (t ) + X (t ) x 2 (t ) + Mv w(0 ) − = N x(0 ) − − −1 − x$ 1 = N 1 w − N 2 x 2
(
(
with
[
]
~ M = N1 M N 2 M N 2 ,
~ N 2 = [ N 2 (i, j ) ] ,
)
(7)
)
1 + v = b1+ (t ) (b2 (t ) + b2− (t )) 1 (b2+ (t ) − b2− (t )) 2 2 +
T
and
T
1 + v − = b1− (t ) (b2 (t ) + b2− (t )) 1 (b2− (t ) − b2+ (t )) . 2 2 Now, let us prove that the set-valued observer specified in this proposition guarantees that − + x1− (t ) ≤ x1 (t ) ≤ x1+ (t ), ∀t ≥ 0 as soon as x(0) ≤ x(0 ) ≤ x(0) . First, let us recall the following result :
Lemma [SMITH, 1995] : Let w& = f(t,w) . This system is said to be a cooperative system if
∂f i (t,w) ≥ 0,∀i ≠ j . ∂w j
Furthermore, w(0) ≥ 0 implies w(t ) ≥ 0,∀t ≥ 0 . In addition, it is known that cooperative systems generate a monotone semiflow in the forward time direction. The properties of the nonlinear robust interval observer will be shown to be verified using the cooperative characteristics of the error dynamics defined hereafter in (8). With reference to the previous Lemma, the following hypothesis guarantees this property for the system under interest : Hypothesis H4 : A11ij (t ) ≥ 0, ∀i ≠ j . Now, it is to be shown that x1− (t ) ≤ x1 (t ) ≤ x1+ (t ), ∀t ≥ 0 as soon as x(0) ≤ x(0 ) ≤ x(0) . Let −
+
e + = x1+ − x1 and e − = x1 − x1− be the observation errors associated to (7) of the non measured state variables for the upper bound and for the lower bound respectively. For simplicity, e* is written for any of the errors e+ or e- since they have a dynamic with the same mathematical structure. It is then easy to verify that : e& * = We* + V * where V * =
1 k
( Mv
+
(8)
)
− Nb in the upper bound and V * =
1 k
( Nb − Mv ) −
in the lower bound case. It is
obvious, when regarding the definitions of e and e that e * (0) ≥ 0 . Then, from conditions H3b, H4 and the application of the previous Lemma, the system (8) is stable and cooperative with positive inputs generating a monotone semiflow in the forward time direction. Therefore, it is guaranteed that e * ≥ 0, ∀t thus, we deduce that x1− (t ) ≤ x1 (t ) ≤ x1+ (t ), ∀t ≥ 0 and the proposition 2 is proved. +
-
V - Application to a wastewater treatment process
V.1 - The anaerobic digester model Anaerobic digestion is among the oldest biological wastewater treatment processes, having first been studied more than a century ago. It 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 and produces valuable energy (i.e., methane). The biological scheme involves several multi-substrate 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 is considered [BERNARD et al., 1998] : X& 1 = (µ 1 − αD )X 1 X& 2 = (µ 2 − αD )X 2 & i Z = D(Z − Z ) & i S 1 = D(S − S1 ) − k 1 µ 1 X 1 S& = D(S i − S ) + k µ X − k µ X 2 2 1 1 3 2 2 2 C& = D (C i − C ) + k (k P + Z − C − S ) + k µ X + k µ X TI 7 8 CO TI 2 4 1 1 5 2 2 TI 1
2
TI
2
(9)
where X1, X2, S1, S2 and CTI are respectively the concentrations of acidogenic bacteria, methanogenic bacteria, COD, volatile fatty acids (VFA) and total inorganic carbon. The alkalinity (i.e., the buffer capacity) is represented by Z. PCO2 is the constant CO2 partial pressure. The parameter α represents a proportionality parameter of experimental determination. D = D(t) is the dilution rate and is supposed to be a persisting input, i.e.
∫
∞ 0
D(τ )dτ > 0 .
Like in any other mass balance model of biological processes, a strongly nonlinear kinetic behavior is present due to the reaction rates. These rates are given by : µ 1 = µ max 1
S1 K S + S1
and
S2
µ 2 = µ max 2
S KS + S2 + 2 KI Parameters definition and their values are listed in Table 1. In all cases, the upper “influent concentration”. 1
2
2
(10) 2 index i indicates
The nonlinear observers developed in sections III and IV are then applied to the dynamic process model (9) defining the state vector in the following way : x1 = X 1 x2 = X 2
x3 = CTI x4 = Z
x5 = S1 x6 = S 2
(11)
The model (9) thus takes the following matrix form : x&1 1 x& 0 2 x&3 k 4 = x& 4 0 x&5 − k1 x&6 k 2
0 0 0 0 0 x1 0 µ 1 x1 −αD 0 0 0 0 x2 −αD 1 µ 2 x2 0 − k 7 x3 DCTIi −( D + k 7 ) k 7 k5 + + 0 x4 −D 0 0 0 − D 0 x5 − k 3 − D x6
0
0 0 + k 7 k8 PCO DZ i DS1i DS2i
2
(12)
or simply x& = CR(t ) + A(t )x + b(t ) that is identical to model (1). It is then straightforward to apply the observers described in previous sections as follows. From eq. (12), it is clear that rank C = 2 and, from the conditions H2, only a minimum of two measurements is thus required to reconstruct the state space. However, the two biomass concentrations cannot be assumed to be on-line available since they are very difficult to measure. Hence, the model is to be rearranged in such a way that biomass concentrations are included in the estimated variables set. The two substrate concentrations are here used as on-line measurements in order to verify the hypothesis H2. In other words, S1and S2 were used to estimate X1, X2, CTI and Z. Once, the model is correctly set, it is easy to check that the matrix form (12) verifies the hypothesis H3b. Following the condition H2b, we obtain s = 4, which leads to the choice of N1 = I4 (i.e., for simplicity, the arbitrary choice k = 1 was made) and it verifies the hypothesis H3a. Therefore, N takes the form :
k k 1 3 1 N = [N 1 M N 2 ] = k1k3
0 k3 k2 k1 M M(k3 k 4 + k 2 k5 ) k1k5 0 0 k1k3 M
0M
k1k3 k1k3
0
(13)
that meets the requirement (4). Thus, A is split in the following form : 0 0 0 −αD 0 0 −αD −( D + k 7 ) k 7 A11 M A12 A = K . K = −D A21 M A22 K K K K
0 0
M M M M . M M
0 0 − k7 0 0 K K −D 0 0 − D 0 0 0
(14)
and the hypothesis H4 is then fulfilled. The observer is now exactly set like in equations (5), (6) and (7) and eq. (13) and (14) can be used. V.2 - Simulation results Simulations were carried out using the parameter values and operating conditions reported in Tables 1 and 2 respectively for the model (9). For the simulation runs, S1 and S2 measurements are supposed to be perfectly known. These measurements were taken directly from the model and they are presented in Figure 1. This simulation was carried out over a 50 days period at different dilution rates and at different input substrate concentrations (see Table 2). The CTIi concentrations reported in Table 2 have been calculated from the CO2 equilibrium. The two observation schemes were tested in two different cases. First, the asymptotic observer (Cf. eq. (6)) was used. The influent concentrations were here assumed to be known and an uncertainty about 25% of the true values was considered for the initial conditions, that is xˆ (0) = 0.75 x(0) . In the second case, (i.e., using the set-valued observer (7)), the influent concentrations were considered to be unknown but within a ± 10% boundary known region, that is xi,- = 0.9xi and xi,+ = 1.1xi In this case, an uncertainty of ± 50% was also considered around of true initial conditions, that is x(0)- = 0.5x(0) and x(0)+ = 1.5x(0). Both cases included totally unknown kinetics. Estimation results for the unmeasured states are presented in Figures 2 to 6. “Real state” values were directly obtained from the model.
Table 1 : Model parameters for simulation and experimental runs (from [BERNARD et al., 1998]) Parameter k1 k2 k3 k4 k5 k6 k7 k8 α µ max1
Meaning Yield coefficient for COD degradation Yield coefficient for fatty acid production Yield coefficient for fatty acid consumption Yield coefficient for CO2 production due to X1 Yield coefficient for CO2 production due to X2 Yield coefficient for CH4 production liquid/gas transfer rate Henry’s constant Proportion of dilution rate for bacteria Maximum acidogenic biomass growth rate
Value (simulations runs) 13.7 g COD/g X1 168.2 mmol VFA/g X1 1640 mmol VFA/g X2 230 mmol CO2/g X1 273 mmol CO2/g X2 1804 mmol CH4/g X2 500 day-1 22.3 mmol CO2/lt-atm 0.5 1.25 day-1
Value (experimental runs) 12.1 g COD/g X1 181.2 mmol VFA/g X1 1640 mmol VFA/g X2 169 mmol CO2/g X1 273 mmol CO2/g X2 1804 mmol CH4/g X2 200 day-1 22.3 mmol CO2/lt-atm 0.5 1.25 day-1
µ max2 KS1 KS2 KI2 Vl
0.5 day-1 7 g COD/l 6 mmol VFA/l 20 mmol VFA/l 1 m3
Maximum methanogenic biomass growth rate Saturation parameter associated with S1 Saturation parameter associated with S2 Inhibition constant associated with S2 Volume of the fermenter
0.85 day-1 7.65 g COD/l 18 mmol VFA/l 25 mmol VFA/l 0.948 m3
2.00
6
1.80 5 Acidogenic bacteria X1 (g/l)
1.60
S1, S2 (g/l)
4 COD, S1 VFA, S2
3
2
1
0
1.40 1.20 1.00 Real state
0.80 0.60
Estimated state (influent concentrations known)
0.40
Upper estimated state (influent concentrations unknown)
0.20
Lower estimated state (influent concentrations unknown)
0.00 0
10
20
30
40
0
50
10
20
30
40
50
Time (days)
Time (days)
Figure 2 : Estimation of acidogenic bacteria concentration
Figure 1 : Measurements of substrates S1 and S2 0.25
140 Real state
Methanodenic bacteria X2 (g/l)
Total inorganic carbon CTI (mmol/l)
Estimated state (influent concentrations known) Upper estimated state (influent concentrations unknown)
0.20
Lower estimated state (influent concentrations unknown)
0.15
0.10
0.05
120 100 80 60 Real state
40
Estimated state (influent concentrations known) Upper estimated state (influent concentrations unknown)
20
Lower estimated state (influent concentrations unknown)
0.00
0 0
10
20
30
40
50
0
10
20
Time (days)
Figure 3 : Estimation of methanogenic bacteria concentration
40
50
Figure 4 : Estimation of total inorganic carbon concentration
120
2.50
Sum of biomass (X1+X2) (g/l)
100
Alkalinity Z (meq/l)
30 Time (days)
80
60 Real state
40
Estimated state (influent concentrations known) Upper estimated state (influent concentrations unknown)
20
2.00
1.50
1.00
Real state Estimated state (influent concentrations known)
0.50
Upper estimated state (influent concentrations unknown) Lower estimated state (influent concentrations unknown)
Lower estimated state (influent concentrations unknown)
0
0.00 0
10
20
30
40
Time (days)
Figure 5 : Estimation of alkalinity concentration
50
0
10
20
30
40
Time (days)
Figure 6 : Estimation of sum of biomass
Table 2 : Operational conditions for the simulation runs (from [BERNARD et al., 1998])
50
t (days) D (day-1) S1i (g/l) S2i (mmol/l) Zi (mmol/l) CTIi (mmol/l) PT (atm)
Time Dilution rate Input COD Input volatile fatty acids Input alkalinity Input total inorganic carbon Total gas pressure
0 0.3333 9.55 9.55 66.36 236.25 1.30
15 0.5 9.55 9.55 66.36 236.25 1.30
25 0.8 9.55 9.55 66.36 236.25 1.30
30 0.8 14.325 14.325 99.54 354.37 1.30
50 0.8 14.325 14.325 99.54 354.37 1.30
V.3 - Experimental results The experimental runs were carried out in a 1 m3 upflow anaerobic fixed bed reactor for the treatment of industrial wine distillery vinasses obtained from local distilleries in the Narbonne area. The experimental model parameters are reported in the Table 1. These experimental runs are carried out over a 35 days period. The measurements of the dilution rate were performed on-line and they are represented in Figure 7. The measurements of Zi, S1i and S2i, were obtained off-line and they are represented in Figures 8 to 10. Like in the previous simulation runs, the CTIi concentration is in fact calculated from the CO2 equilibrium. The measurements of S1 and S2 were also taken off-line and they are represented in Figures 11 and 12.
1.20
120
1.00
100
0.80
80
Alkalinity Z (meq/l)
Dilution rate (1/day)
Again, the two observer schemes were tested in two different cases. In the first case (i.e., using the asymptotic observer (6)), influent concentrations were known and an uncertainty of 25% of the true initial conditions values was taken into account, that is xˆ (0 ) = 0.75 x(0 ) . The measurements of the influent concentrations were directly taken from experimental values showed in Figures 8 to 10. The second case (i.e., using the set-valued observer (7)) considers that the influent concentrations are unknown but a known boundary region was used as it can be seen in Figures 8 to 10. Here, an uncertainty of ± 50% around the true initial conditions was also used, that is x(0)- = 0.5x(0) and x(0)+ = 1.5x(0). In the experimental runs, estimation results for the unmeasured states are presented in Figures 13 to 16 and they demonstrate the large interest of this approach.
0.60
0.40
60 Real state
40
Estimated state (influent concentrations known)
0.20
20
0.00
0
Upper estimated state (influent concentrations unknown) Lower estimated state (influent concentrations unknown)
0
5
10
15 20 Time (days)
25
30
Figure 7 : Experimental on-line dilution rate
35
0
10
20
30
40
50
Time (days)
Figure 8 : Experimental measures of influent alkalinity and boundary region
140
20 18
Experimental points Upper bound
100
Influent VFA (mmol/l)
Influent COD (g/l)
120
Lower bound
16 14 12 10 8 6
80 60 Experimental points Lower bound Upper bound
40
4
20
2 0
0 0
5
10
15 20 Time (days)
25
30
35
Figure 9 : Experimental influent COD concentration and boundary region
0
5
10
15 20 Time (days)
25
30
35
Figure 10 : Experimental influent VFA concentration and boundary region
6
35 30
5
25 VFA S2 (mmo/l)
COD S1 (g/l)
4
3
20 15
2 10 1
5
0
0 0
5
10
15 20 Time (days)
25
30
35
Figure 11 : Effluent measurements on chemical oxygen demand (COD).
0
5
10
15 20 Time (days)
25
30
35
Figure 12 : Effluent measurements on volatile fatty acids (VFA). 0.40
2.50
0.35 Methanogenic bacteria X2 (g/l)
Acidogenic bacteria X1 (g/l)
2.00
1.50
1.00 Predictions of the model Estimated state (influent concentrations known)
0.50
Upper estimated state (influent concentrations unknown)
0.30 0.25 0.20 0.15
Predictions of the model Estimated state (influent concentrations known)
0.10
Upper estimated state (influent concentrations unknown)
0.05
Lower estimated state (influent concentrations unknown)
Lower estimated state (influent concentrations unknown)
0.00
0.00 0
5
10
15 20 Time (days)
25
30
Figure 13 : Estimation of acidogenic bacteria concentration
35
0
5
10
15 20 Time (days)
25
30
Figure 14 : Estimation of methanogenic bacteria concentration.
35
100
90
90
80
80
70
70
Alkalinity Z (meq/l)
Total inorganic carbon CTI (mmol/l)
100
60 50 40 Predictions of the model
30
40 Predictions of the model Estimated state (influent concentrations known)
20
Upper estimated state (influent concentrations unknown)
10
50
30
Estimated state (influent concentrations known)
20
60
Upper estimated state (influent concentrations unknown) Lower estimated state (influent concentrations unknown)
10
Lower estimated state (influent concentrations unknown)
Measured data
0
0 0
5
10
15 20 Time (days)
25
30
Figure 15 : Estimation of total inorganic carbon concentration.
35
0
5
10
15 20 Time (days)
25
30
Figure 16 : Estimation of alkalinity concentration.
VI - Conclusions In this paper, an exponential observer and an interval observer for a general class of lumped models useful in chemical and biochemical engineering processes are proposed. Conditions on the general structure of the model are established for designing such observers. These observers were satisfactorily tested in simulations and validated experimentally on a 1 m3 fixed bed reactor for the treatment of industrial distillery wine vinasses. The exponential observer scheme presents a good accuracy and fast convergence properties in the presence of noise due to operating disturbances. The interval observer presents also very good convergence and predicts correctly the dynamical bounds respected by the true values of the non measured variables. Because of the large interest of these observers at the experimental scale, their use in robust nonlinear control schemes with application to continuous bioreactors is actually under study.
Acknowledgment The authors gratefully acknowledge the ECOS-Nord program for French-Mexican scientific cooperation as well as CONACyT (Mexican National Council of Science and Technology) for the financial support provided to this study. References • Bastin G. and Dochain D. : “On-line estimation and adaptive control of bioreactors”, Elsevier, 1990, 379 pages. • Ben Y. C., Roux G. and Dahhou B. : "Multivariable Adaptive Predictive Control of Nonlinear Systems : Application to a Multistage Wastewater Treatment Process", European Control Conference ECC'95, Roma, Italy, 1995. • Bernard O., Dochain D., Genovesi A., Punal A., Perez Alvarino D., Steyer J.P. and Lema, J. : “Software sensor design for an anaerobic wastewater treatment plant”, in IFAC-EurAgEng International Workshop on "Decision and Control in Waste Bio-Processing", WASTE-DECISION'98, 8 pages (CD-ROM), Narbonne, France, 25-27 February 1998. • Busawon K. : “Sur les observateurs des systèmes non-linéaires et le principe de séparation”, PhD Thesis, Université Claude Bernard, Lyon, France, 1996. • Dochain D. and Pauss A. : “On-line estimation of microbiological specific growth rates : an illustrative case study”, Can. J. Chem. Eng., vol. 47, pp. 327-336, 1988. • Farza K., Busawon K. and Hammouri H. : “Simple Nonlinear Observers for On-line Estimation of
35
•
•
•
•
•
•
Kinetic Rates in Bioreactors”, Automatica, vol. 34, n°3, pp. 301-318, 1998 Gauthier J.P. and Kupka I. : “Observability and Observers for Nonlinear Systems”, SIAM J. Control and Optim., vol. 34, n°4, pp. 975-994, 1994 Hadj-Sadok M Z., Gouzé J.L. and Rapaport A. : “State observers for uncertain models of activated sludge processes”, in IFAC-EurAgEng International Workshop on "Decision and Control in Waste Bio-Processing", WASTE-DECISION'98, 8 pages (CD-ROM), Narbonne, France, 25-27 February 1998. Henze M. and Harremoes P. : “Anaerobic Treatment of Wastewater in Fixed Film Reactors- A Literature Review”, Water Science and Technology., vol. 15, n°1, pp. 1-101, 1983. Rapaport A. : “Information state and guaranteed value for a class of min-max nonlinear optimal control problems”, 6th IEEE Mediterranean Conference on Control and Systems, Alghero, Italy, 911 June 1998. a Rapaport A., Harmand J. : "Robust Regulation of a Bioreactor in a Highly Uncertain Environment", in IFAC-EurAgEng International Workshop on "Decision and Control in Waste Bio-Processing", WASTE-DECISION'98, 8 pages (CD-ROM), Narbonne, France, 25-27 February 1998. b Rapaport A., Harmand J. : "Robust Nonlinear Control of a Class of Partially Observed Processes : Application to Continuous Bioreactors", Conference on Control Applications, CCA'98, Trieste, Italy, 1998. • Smith H.L. : “Monotone Dynamical Systems. An introduction to the Theory of Competitive and Cooperative Systems”, AMS Mathematical Surveys and Monographs, vol. 41, pp. 31-53, 1995.
Return to report
ROBUST EXPONENTIAL NONLINEAR INTERVAL OBSERVERS FOR A CLASS OF LUMPED MODELS USEFUL IN CHEMICAL AND BIOCHEMICAL ENGINEERING. APPLICATION TO A WASTEWATER TREATMENT PROCESS. Alcaraz-González V.1, Genovesi A.1, Harmand J.1, González A.V.3, Rapaport A.2 and Steyer J.P. 1 1
INRA-LBE, Avenue des Etangs, 11100 Narbonne - France. Fax : 33-468-425-160 [email protected], [email protected], [email protected], [email protected] 2 INRA-Biométrie, 2, Place Viala, 34060 Montpellier - France. Fax : 33-499-521-427. [email protected] 3 Departamento de Ingeniería Química de la Universidad de Guadalajara. Blvd. Marcelino Garcia Barragán y Calzada Olímpica, 44860 Guadalajara Jalisco - México. Fax : 523-619-4028. [email protected]
Abstract :
In this paper, two new robust observers are detailed for a class of lumped models useful in chemical and biochemical engineering. The first observer is an exponential nonlinear observer which is robust against uncertainties on the kinetics of the process. To cope with additional disturbances on the inputs, thus leading to a non detectable system, a second observer is designed, that is a nonlinear interval observer. With an a priori knowledge of the bounds concerning the unknown initial conditions, this interval observer also provides guaranteed intervals for the unmeasured variables. Simulation results are presented using a model of an anaerobic digestion process for the treatment of an industrial wine distillery wastewater. Finally, the observers are experimentally tested using data obtained from a 1 m3 continuous fixed bed bioreactor.
Keywords : Robust nonlinear state observers, nonlinear mass balance models, interval, wastewater treatment processes, anaerobic digestion.
I - Introduction Working on the control of biological processes – and especially of biological wastewater treatment processes - raises a number of very challenging problems. Indeed, if a number of on-line sensors providing state information are today available at the industrial scale, they are still very expensive and their maintenance is usually time consuming. Furthermore, biological processes are highly nonlinear time varying systems in which kinetic parameters are badly or poorly known. To overcome these difficulties, the notion of software sensors has been introduced in the early eighties. In fact, these software sensors simply consist in observers designed to estimate unmeasured states from the available on-line measurements and they have been demonstrated to be of a great help for biological processes (see for example [DOCHAIN and PAUSS, 1988], [BASTIN and DOCHAIN, 1990] [GAUTHIER and KUPKA, 1994], [BEN et al., 1995], [BUSAWON, 1996] or [FARZA et al., 1998]). However, in all previously cited schemes, the knowledge of all the inputs of the process, including for example the substrate input concentration, is needed. Unfortunately, if this knowledge can be available for biological processes used in the food or in the pharmaceutical industries, this is normally not the case in the wastewater treatment field. As a consequence, part of the process input vector is considered as unmeasured input disturbances and classical observer schemes cannot be used since the system can
be not observable nor detectable depending on the available on-line measurements. In this paper, conditions for designing robust nonlinear observers for a class of nonlinear models are derived. Based upon the work of [RAPAPORT, 1998], [RAPAPORT and HARMAND, 1998a-b], existence conditions of these observers are derived assuming that part of the input vector is not measured. As a consequence, it is possible that the considered system is not detectable. In other terms, it is then not possible to estimate the unmeasured states from the available on-line measurements. The new observers derived in this paper are called set-observers since they allow the user to reconstruct a guaranteed interval on the unmeasured states instead of reconstructing their precise numerical values. The only requirement is to know an interval in which the unmeasured inputs of the process evolve. The paper is organized as follows. First, the classical nonlinear asymptotic observer for unknown kinetics proposed in [BASTIN and DOCHAIN, 1990] is extended in some sense without any restriction on the inputs. Then, the equations of this observer are used to build a new nonlinear robust interval observer assuming that input disturbances are unknown but that they belong to a prescribed bounded set. This methodology allows one to compute guaranteed intervals for the unmeasured states when bounds on the unknown initial conditions are given. Simulation and experimental results are then provided and finally, conclusions and perspectives are drawn.
II - The considered model In this paper, chemical and biochemical processes that can be described by the following general nonlinear model are considered : x& = C R(x(t ),t ) + A(t )x (t ) + b(t )
(1)
where x ∈ ℜn is the state vector, R ∈ ℜm denotes the reaction rate vector and C ∈ ℜn×m represents the constant stoichiometric coefficients matrix. The matrix A ∈ ℜn×n explicits the linear dependence between the state variables while b ∈ ℜn is a vector gathering all the terms of the model that are not a function of the state. Let us now assume that the state space can be split in such a way that (1) can be rewritten as : x&1 = C1 R(x (t ),t ) + A11 (t )x1 (t ) + A12 (t )x2 (t ) + b1 (t ) x&2 = C2 R( x(t ),t ) + A21 (t )x1 (t ) + A22 (t )x2 (t ) + b2 (t )
(2)
where the measured variables have been grouped in the x2 vector and the variables that has to be estimated are represented by x1. We note dim x1 = s and we define r = dim x2 = n - s. Matrices C1 ∈ ℜs×m, C2 ∈ ℜr×m are the corresponding partitions of C and A11 ∈ ℜs×s, A12 ∈ ℜs×r, A21 ∈ ℜr×s and A22 ∈ ℜr×r are the corresponding partitions of A. The following hypotheses are furthermore introduced : Hypotheses H1 : a) R is unknown. b) A and C are known. c) Measurements on x2 are continuously available. d) Guaranteed bounds on the initial conditions of the state vector are known with x(0)≤ x(0) ≤ x(0)+. e) There may also exist uncertainties on some parameters involved in b(t) (but not contained in A) and only some bounds of these are known such that b-(t) ≤ b(t) ≤ b+(t).
Note : The operator ≤ applied between vectors should be understood as a collection of inequalities between components.
III - An exponential observer In this section, only the hypotheses H1a-H1c will be necessary. Thus, it is assumed that the vector b(t) is perfectly known. In addition, m < n and C has a rank p with p ≤ m < n. Then, the following additional hypotheses are introduced : Hypotheses H2 : a) rank C2 = rank C b) r = dim x2 ≥ rank C Under hypotheses H2, it is possible to make a change of variables to design a reduced order observer regarding the following procedure. Let N ∈ ℜs×n be a matrix representing a linear combination of the state variables, so that an auxiliary variable w can be defined as : w(t)
=
Nx(t)
(3) Introducing this change of variable into (1), it is straightforward to establish that the new system will be kinetic independent if and only if N satisfies : NC = 0
(4)
Under hypotheses H2, the solution of the algebraic system (4) allows the choice of at least s rows of NT. In other words, if N1 ∈ ℜ s×s , N 2 ∈ ℜ s×r are the corresponding partitions of N, then we can arbitrary
choose N 1T to compute N 2T = −(C2T ) C1T N1T . Assumptions H2 ensure that (C 2T ) , the right pseudo−1
−1
inverse of C2T (in a general sense) exists. The kinetic independence property of the observer is guaranteed by (4). Thus, the last condition to be satisfied is that N 1−1 exists in order to recover the original state. The following additional hypotheses are then introduced : Hypotheses H3 : a) N1 is invertible. b) The dynamic of the system z& = A11 (t )z is exponentially stable. The first condition is satisfied if N1 = kIs is chosen where k is a real and positive arbitrary parameter. The second condition is satisfied if the real part of any eigenvalues of A11(t) are strictly negative for each t ≥ 0 . Under hypothesis H3a the derivative of the linear transformation (3) can be done and the original model (1) rewritten as a function of w and x2 as follows : w& = W (t )w + X (t )x 2 + Nb(t )
(5)
where W (t ) = N1 A11 ( t ) N1−1 , X (t ) = N1 A12 ( t ) + N 2 A22 (t ) − W (t ) N 2 . System (5) is then a linear and kinetics independent model which is only a function of w and x2.
Proposition 1 : Under hypotheses H1a-H1c, H2 and H3, when the vector b(t) is perfectly known, the following system:
wˆ& = W (t )wˆ (t ) + X (t )x 2 (t ) + Nb(t ) wˆ (0) = Nxˆ (0) xˆ = N −1 (wˆ − N x ) 1 2 2 1
(6)
is an exponential observer (i.e., xˆ1 (t ) converges exponentially towards x1 (t ) for any initial conditions). The small hat symbol "^" denotes "estimated" in all cases. Remark 1 : In some sense, on one hand, since there exists some hypothesis on the structure of the A matrix, this observer is less general than the one proposed in [BASTIN and DOCHAIN, 1990]. However, on the other hand, this structuration of the uncertainty allows the user to estimate unmeasured variables using less measurements than in the design proposed in [BASTIN and DOCHAIN, 1990]. Remark 2 : The general form of the proposed exponential observer is a generalization of a specific application that was proposed in [HADJ-SADOK et al., 1998].
IV - An interval observer Again, in this section, it is considered that the kinetics are unknown. In addition, it is assumed that H1cd are verified. In other words, some bounds on the initial conditions are available on the initial conditions and the vector b(t) is now unmeasured but some lower and upper bounds are known. In such a situation, notice that the model (1) can be no longer detectable. Thus, it is not possible to use the exponential observer developed in the previous section. However, its basic structure - and especially its property of being a kinetic independent transformation - will be used. The idea developed in the following is to design a set-valued observer in order to build guaranteed intervals for the unmeasured variables instead of estimating them precisely. Let us consider the following proposition :
Proposition 2 : Under hypotheses H1-H4, a robust interval observer for the system (1) is given by : For the upper bound: w& + = W (t )w + (t ) + X (t ) x (t ) + Mv + 2 + + w(0 ) = N x(0 ) + −1 + x$ 1 = N 1 w − N 2 x 2 For the lower bound: − − − w& = W (t )w (t ) + X (t ) x 2 (t ) + Mv w(0 ) − = N x(0 ) − − −1 − x$ 1 = N 1 w − N 2 x 2
(
(
with
[
]
~ M = N1 M N 2 M N 2 ,
~ N 2 = [ N 2 (i, j ) ] ,
)
(7)
)
1 + v = b1+ (t ) (b2 (t ) + b2− (t )) 1 (b2+ (t ) − b2− (t )) 2 2 +
T
and
T
1 + v − = b1− (t ) (b2 (t ) + b2− (t )) 1 (b2− (t ) − b2+ (t )) . 2 2 Now, let us prove that the set-valued observer specified in this proposition guarantees that − + x1− (t ) ≤ x1 (t ) ≤ x1+ (t ), ∀t ≥ 0 as soon as x(0) ≤ x(0 ) ≤ x(0) . First, let us recall the following result :
Lemma [SMITH, 1995] : Let w& = f(t,w) . This system is said to be a cooperative system if
∂f i (t,w) ≥ 0,∀i ≠ j . ∂w j
Furthermore, w(0) ≥ 0 implies w(t ) ≥ 0,∀t ≥ 0 . In addition, it is known that cooperative systems generate a monotone semiflow in the forward time direction. The properties of the nonlinear robust interval observer will be shown to be verified using the cooperative characteristics of the error dynamics defined hereafter in (8). With reference to the previous Lemma, the following hypothesis guarantees this property for the system under interest : Hypothesis H4 : A11ij (t ) ≥ 0, ∀i ≠ j . Now, it is to be shown that x1− (t ) ≤ x1 (t ) ≤ x1+ (t ), ∀t ≥ 0 as soon as x(0) ≤ x(0 ) ≤ x(0) . Let −
+
e + = x1+ − x1 and e − = x1 − x1− be the observation errors associated to (7) of the non measured state variables for the upper bound and for the lower bound respectively. For simplicity, e* is written for any of the errors e+ or e- since they have a dynamic with the same mathematical structure. It is then easy to verify that : e& * = We* + V * where V * =
1 k
( Mv
+
(8)
)
− Nb in the upper bound and V * =
1 k
( Nb − Mv ) −
in the lower bound case. It is
obvious, when regarding the definitions of e and e that e * (0) ≥ 0 . Then, from conditions H3b, H4 and the application of the previous Lemma, the system (8) is stable and cooperative with positive inputs generating a monotone semiflow in the forward time direction. Therefore, it is guaranteed that e * ≥ 0, ∀t thus, we deduce that x1− (t ) ≤ x1 (t ) ≤ x1+ (t ), ∀t ≥ 0 and the proposition 2 is proved. +
-
V - Application to a wastewater treatment process
V.1 - The anaerobic digester model Anaerobic digestion is among the oldest biological wastewater treatment processes, having first been studied more than a century ago. It 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 and produces valuable energy (i.e., methane). The biological scheme involves several multi-substrate 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 is considered [BERNARD et al., 1998] : X& 1 = (µ 1 − αD )X 1 X& 2 = (µ 2 − αD )X 2 & i Z = D(Z − Z ) & i S 1 = D(S − S1 ) − k 1 µ 1 X 1 S& = D(S i − S ) + k µ X − k µ X 2 2 1 1 3 2 2 2 C& = D (C i − C ) + k (k P + Z − C − S ) + k µ X + k µ X TI 7 8 CO TI 2 4 1 1 5 2 2 TI 1
2
TI
2
(9)
where X1, X2, S1, S2 and CTI are respectively the concentrations of acidogenic bacteria, methanogenic bacteria, COD, volatile fatty acids (VFA) and total inorganic carbon. The alkalinity (i.e., the buffer capacity) is represented by Z. PCO2 is the constant CO2 partial pressure. The parameter α represents a proportionality parameter of experimental determination. D = D(t) is the dilution rate and is supposed to be a persisting input, i.e.
∫
∞ 0
D(τ )dτ > 0 .
Like in any other mass balance model of biological processes, a strongly nonlinear kinetic behavior is present due to the reaction rates. These rates are given by : µ 1 = µ max 1
S1 K S + S1
and
S2
µ 2 = µ max 2
S KS + S2 + 2 KI Parameters definition and their values are listed in Table 1. In all cases, the upper “influent concentration”. 1
2
2
(10) 2 index i indicates
The nonlinear observers developed in sections III and IV are then applied to the dynamic process model (9) defining the state vector in the following way : x1 = X 1 x2 = X 2
x3 = CTI x4 = Z
x5 = S1 x6 = S 2
(11)
The model (9) thus takes the following matrix form : x&1 1 x& 0 2 x&3 k 4 = x& 4 0 x&5 − k1 x&6 k 2
0 0 0 0 0 x1 0 µ 1 x1 −αD 0 0 0 0 x2 −αD 1 µ 2 x2 0 − k 7 x3 DCTIi −( D + k 7 ) k 7 k5 + + 0 x4 −D 0 0 0 − D 0 x5 − k 3 − D x6
0
0 0 + k 7 k8 PCO DZ i DS1i DS2i
2
(12)
or simply x& = CR(t ) + A(t )x + b(t ) that is identical to model (1). It is then straightforward to apply the observers described in previous sections as follows. From eq. (12), it is clear that rank C = 2 and, from the conditions H2, only a minimum of two measurements is thus required to reconstruct the state space. However, the two biomass concentrations cannot be assumed to be on-line available since they are very difficult to measure. Hence, the model is to be rearranged in such a way that biomass concentrations are included in the estimated variables set. The two substrate concentrations are here used as on-line measurements in order to verify the hypothesis H2. In other words, S1and S2 were used to estimate X1, X2, CTI and Z. Once, the model is correctly set, it is easy to check that the matrix form (12) verifies the hypothesis H3b. Following the condition H2b, we obtain s = 4, which leads to the choice of N1 = I4 (i.e., for simplicity, the arbitrary choice k = 1 was made) and it verifies the hypothesis H3a. Therefore, N takes the form :
k k 1 3 1 N = [N 1 M N 2 ] = k1k3
0 k3 k2 k1 M M(k3 k 4 + k 2 k5 ) k1k5 0 0 k1k3 M
0M
k1k3 k1k3
0
(13)
that meets the requirement (4). Thus, A is split in the following form : 0 0 0 −αD 0 0 −αD −( D + k 7 ) k 7 A11 M A12 A = K . K = −D A21 M A22 K K K K
0 0
M M M M . M M
0 0 − k7 0 0 K K −D 0 0 − D 0 0 0
(14)
and the hypothesis H4 is then fulfilled. The observer is now exactly set like in equations (5), (6) and (7) and eq. (13) and (14) can be used. V.2 - Simulation results Simulations were carried out using the parameter values and operating conditions reported in Tables 1 and 2 respectively for the model (9). For the simulation runs, S1 and S2 measurements are supposed to be perfectly known. These measurements were taken directly from the model and they are presented in Figure 1. This simulation was carried out over a 50 days period at different dilution rates and at different input substrate concentrations (see Table 2). The CTIi concentrations reported in Table 2 have been calculated from the CO2 equilibrium. The two observation schemes were tested in two different cases. First, the asymptotic observer (Cf. eq. (6)) was used. The influent concentrations were here assumed to be known and an uncertainty about 25% of the true values was considered for the initial conditions, that is xˆ (0) = 0.75 x(0) . In the second case, (i.e., using the set-valued observer (7)), the influent concentrations were considered to be unknown but within a ± 10% boundary known region, that is xi,- = 0.9xi and xi,+ = 1.1xi In this case, an uncertainty of ± 50% was also considered around of true initial conditions, that is x(0)- = 0.5x(0) and x(0)+ = 1.5x(0). Both cases included totally unknown kinetics. Estimation results for the unmeasured states are presented in Figures 2 to 6. “Real state” values were directly obtained from the model.
Table 1 : Model parameters for simulation and experimental runs (from [BERNARD et al., 1998]) Parameter k1 k2 k3 k4 k5 k6 k7 k8 α µ max1
Meaning Yield coefficient for COD degradation Yield coefficient for fatty acid production Yield coefficient for fatty acid consumption Yield coefficient for CO2 production due to X1 Yield coefficient for CO2 production due to X2 Yield coefficient for CH4 production liquid/gas transfer rate Henry’s constant Proportion of dilution rate for bacteria Maximum acidogenic biomass growth rate
Value (simulations runs) 13.7 g COD/g X1 168.2 mmol VFA/g X1 1640 mmol VFA/g X2 230 mmol CO2/g X1 273 mmol CO2/g X2 1804 mmol CH4/g X2 500 day-1 22.3 mmol CO2/lt-atm 0.5 1.25 day-1
Value (experimental runs) 12.1 g COD/g X1 181.2 mmol VFA/g X1 1640 mmol VFA/g X2 169 mmol CO2/g X1 273 mmol CO2/g X2 1804 mmol CH4/g X2 200 day-1 22.3 mmol CO2/lt-atm 0.5 1.25 day-1
µ max2 KS1 KS2 KI2 Vl
0.5 day-1 7 g COD/l 6 mmol VFA/l 20 mmol VFA/l 1 m3
Maximum methanogenic biomass growth rate Saturation parameter associated with S1 Saturation parameter associated with S2 Inhibition constant associated with S2 Volume of the fermenter
0.85 day-1 7.65 g COD/l 18 mmol VFA/l 25 mmol VFA/l 0.948 m3
2.00
6
1.80 5 Acidogenic bacteria X1 (g/l)
1.60
S1, S2 (g/l)
4 COD, S1 VFA, S2
3
2
1
0
1.40 1.20 1.00 Real state
0.80 0.60
Estimated state (influent concentrations known)
0.40
Upper estimated state (influent concentrations unknown)
0.20
Lower estimated state (influent concentrations unknown)
0.00 0
10
20
30
40
0
50
10
20
30
40
50
Time (days)
Time (days)
Figure 2 : Estimation of acidogenic bacteria concentration
Figure 1 : Measurements of substrates S1 and S2 0.25
140 Real state
Methanodenic bacteria X2 (g/l)
Total inorganic carbon CTI (mmol/l)
Estimated state (influent concentrations known) Upper estimated state (influent concentrations unknown)
0.20
Lower estimated state (influent concentrations unknown)
0.15
0.10
0.05
120 100 80 60 Real state
40
Estimated state (influent concentrations known) Upper estimated state (influent concentrations unknown)
20
Lower estimated state (influent concentrations unknown)
0.00
0 0
10
20
30
40
50
0
10
20
Time (days)
Figure 3 : Estimation of methanogenic bacteria concentration
40
50
Figure 4 : Estimation of total inorganic carbon concentration
120
2.50
Sum of biomass (X1+X2) (g/l)
100
Alkalinity Z (meq/l)
30 Time (days)
80
60 Real state
40
Estimated state (influent concentrations known) Upper estimated state (influent concentrations unknown)
20
2.00
1.50
1.00
Real state Estimated state (influent concentrations known)
0.50
Upper estimated state (influent concentrations unknown) Lower estimated state (influent concentrations unknown)
Lower estimated state (influent concentrations unknown)
0
0.00 0
10
20
30
40
Time (days)
Figure 5 : Estimation of alkalinity concentration
50
0
10
20
30
40
Time (days)
Figure 6 : Estimation of sum of biomass
Table 2 : Operational conditions for the simulation runs (from [BERNARD et al., 1998])
50
t (days) D (day-1) S1i (g/l) S2i (mmol/l) Zi (mmol/l) CTIi (mmol/l) PT (atm)
Time Dilution rate Input COD Input volatile fatty acids Input alkalinity Input total inorganic carbon Total gas pressure
0 0.3333 9.55 9.55 66.36 236.25 1.30
15 0.5 9.55 9.55 66.36 236.25 1.30
25 0.8 9.55 9.55 66.36 236.25 1.30
30 0.8 14.325 14.325 99.54 354.37 1.30
50 0.8 14.325 14.325 99.54 354.37 1.30
V.3 - Experimental results The experimental runs were carried out in a 1 m3 upflow anaerobic fixed bed reactor for the treatment of industrial wine distillery vinasses obtained from local distilleries in the Narbonne area. The experimental model parameters are reported in the Table 1. These experimental runs are carried out over a 35 days period. The measurements of the dilution rate were performed on-line and they are represented in Figure 7. The measurements of Zi, S1i and S2i, were obtained off-line and they are represented in Figures 8 to 10. Like in the previous simulation runs, the CTIi concentration is in fact calculated from the CO2 equilibrium. The measurements of S1 and S2 were also taken off-line and they are represented in Figures 11 and 12.
1.20
120
1.00
100
0.80
80
Alkalinity Z (meq/l)
Dilution rate (1/day)
Again, the two observer schemes were tested in two different cases. In the first case (i.e., using the asymptotic observer (6)), influent concentrations were known and an uncertainty of 25% of the true initial conditions values was taken into account, that is xˆ (0 ) = 0.75 x(0 ) . The measurements of the influent concentrations were directly taken from experimental values showed in Figures 8 to 10. The second case (i.e., using the set-valued observer (7)) considers that the influent concentrations are unknown but a known boundary region was used as it can be seen in Figures 8 to 10. Here, an uncertainty of ± 50% around the true initial conditions was also used, that is x(0)- = 0.5x(0) and x(0)+ = 1.5x(0). In the experimental runs, estimation results for the unmeasured states are presented in Figures 13 to 16 and they demonstrate the large interest of this approach.
0.60
0.40
60 Real state
40
Estimated state (influent concentrations known)
0.20
20
0.00
0
Upper estimated state (influent concentrations unknown) Lower estimated state (influent concentrations unknown)
0
5
10
15 20 Time (days)
25
30
Figure 7 : Experimental on-line dilution rate
35
0
10
20
30
40
50
Time (days)
Figure 8 : Experimental measures of influent alkalinity and boundary region
140
20 18
Experimental points Upper bound
100
Influent VFA (mmol/l)
Influent COD (g/l)
120
Lower bound
16 14 12 10 8 6
80 60 Experimental points Lower bound Upper bound
40
4
20
2 0
0 0
5
10
15 20 Time (days)
25
30
35
Figure 9 : Experimental influent COD concentration and boundary region
0
5
10
15 20 Time (days)
25
30
35
Figure 10 : Experimental influent VFA concentration and boundary region
6
35 30
5
25 VFA S2 (mmo/l)
COD S1 (g/l)
4
3
20 15
2 10 1
5
0
0 0
5
10
15 20 Time (days)
25
30
35
Figure 11 : Effluent measurements on chemical oxygen demand (COD).
0
5
10
15 20 Time (days)
25
30
35
Figure 12 : Effluent measurements on volatile fatty acids (VFA). 0.40
2.50
0.35 Methanogenic bacteria X2 (g/l)
Acidogenic bacteria X1 (g/l)
2.00
1.50
1.00 Predictions of the model Estimated state (influent concentrations known)
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Upper estimated state (influent concentrations unknown)
0.30 0.25 0.20 0.15
Predictions of the model Estimated state (influent concentrations known)
0.10
Upper estimated state (influent concentrations unknown)
0.05
Lower estimated state (influent concentrations unknown)
Lower estimated state (influent concentrations unknown)
0.00
0.00 0
5
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15 20 Time (days)
25
30
Figure 13 : Estimation of acidogenic bacteria concentration
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0
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Figure 14 : Estimation of methanogenic bacteria concentration.
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100
90
90
80
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Alkalinity Z (meq/l)
Total inorganic carbon CTI (mmol/l)
100
60 50 40 Predictions of the model
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40 Predictions of the model Estimated state (influent concentrations known)
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Upper estimated state (influent concentrations unknown)
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Estimated state (influent concentrations known)
20
60
Upper estimated state (influent concentrations unknown) Lower estimated state (influent concentrations unknown)
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Lower estimated state (influent concentrations unknown)
Measured data
0
0 0
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15 20 Time (days)
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Figure 15 : Estimation of total inorganic carbon concentration.
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Figure 16 : Estimation of alkalinity concentration.
VI - Conclusions In this paper, an exponential observer and an interval observer for a general class of lumped models useful in chemical and biochemical engineering processes are proposed. Conditions on the general structure of the model are established for designing such observers. These observers were satisfactorily tested in simulations and validated experimentally on a 1 m3 fixed bed reactor for the treatment of industrial distillery wine vinasses. The exponential observer scheme presents a good accuracy and fast convergence properties in the presence of noise due to operating disturbances. The interval observer presents also very good convergence and predicts correctly the dynamical bounds respected by the true values of the non measured variables. Because of the large interest of these observers at the experimental scale, their use in robust nonlinear control schemes with application to continuous bioreactors is actually under study.
Acknowledgment The authors gratefully acknowledge the ECOS-Nord program for French-Mexican scientific cooperation as well as CONACyT (Mexican National Council of Science and Technology) for the financial support provided to this study. References • Bastin G. and Dochain D. : “On-line estimation and adaptive control of bioreactors”, Elsevier, 1990, 379 pages. • Ben Y. C., Roux G. and Dahhou B. : "Multivariable Adaptive Predictive Control of Nonlinear Systems : Application to a Multistage Wastewater Treatment Process", European Control Conference ECC'95, Roma, Italy, 1995. • Bernard O., Dochain D., Genovesi A., Punal A., Perez Alvarino D., Steyer J.P. and Lema, J. : “Software sensor design for an anaerobic wastewater treatment plant”, in IFAC-EurAgEng International Workshop on "Decision and Control in Waste Bio-Processing", WASTE-DECISION'98, 8 pages (CD-ROM), Narbonne, France, 25-27 February 1998. • Busawon K. : “Sur les observateurs des systèmes non-linéaires et le principe de séparation”, PhD Thesis, Université Claude Bernard, Lyon, France, 1996. • Dochain D. and Pauss A. : “On-line estimation of microbiological specific growth rates : an illustrative case study”, Can. J. Chem. Eng., vol. 47, pp. 327-336, 1988. • Farza K., Busawon K. and Hammouri H. : “Simple Nonlinear Observers for On-line Estimation of
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Kinetic Rates in Bioreactors”, Automatica, vol. 34, n°3, pp. 301-318, 1998 Gauthier J.P. and Kupka I. : “Observability and Observers for Nonlinear Systems”, SIAM J. Control and Optim., vol. 34, n°4, pp. 975-994, 1994 Hadj-Sadok M Z., Gouzé J.L. and Rapaport A. : “State observers for uncertain models of activated sludge processes”, in IFAC-EurAgEng International Workshop on "Decision and Control in Waste Bio-Processing", WASTE-DECISION'98, 8 pages (CD-ROM), Narbonne, France, 25-27 February 1998. Henze M. and Harremoes P. : “Anaerobic Treatment of Wastewater in Fixed Film Reactors- A Literature Review”, Water Science and Technology., vol. 15, n°1, pp. 1-101, 1983. Rapaport A. : “Information state and guaranteed value for a class of min-max nonlinear optimal control problems”, 6th IEEE Mediterranean Conference on Control and Systems, Alghero, Italy, 911 June 1998. a Rapaport A., Harmand J. : "Robust Regulation of a Bioreactor in a Highly Uncertain Environment", in IFAC-EurAgEng International Workshop on "Decision and Control in Waste Bio-Processing", WASTE-DECISION'98, 8 pages (CD-ROM), Narbonne, France, 25-27 February 1998. b Rapaport A., Harmand J. : "Robust Nonlinear Control of a Class of Partially Observed Processes : Application to Continuous Bioreactors", Conference on Control Applications, CCA'98, Trieste, Italy, 1998. • Smith H.L. : “Monotone Dynamical Systems. An introduction to the Theory of Competitive and Cooperative Systems”, AMS Mathematical Surveys and Monographs, vol. 41, pp. 31-53, 1995.
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