- Email: [email protected]
Interval-Based Diagnosis of an Anaerobic Digester Pilot Plant
Copyright (j) IF AC Fault Detection, Supervision and Safety fo r Technical Processes, Budapest, Hungary, :WOO
INTERV AL-BASED DIAGNOSIS OF AN ANAEROBIC DIGESTER PILOT PLANT
J. P. Steyer,.f. Harmand, V. A1caraz-Gonzalez
Laboratoire de Biotechnologie de I'Environnement. LRE - INRA . Avenue des etangs. 11100 Narbonne. France . • : +33 (0) 468.425,151. Fax : +33 (0) 468.425.160.
Abstract: Anaerobic digestion processes are highly nonlinear time-varying processes. When used for WasteWater Treatment, they are subject to large input disturbances both of concentrations and flow rates of the polluted effluents to be treated. Furthermore, such processes still suffer from a lack of reliable and cheap sensors, As a consequence, in practice, these processes are usually not observable nor detectable and most state observation and control techniques cannot be applied. In such situations and under specific conditions that are highlighted in the paper, it is possible to design some "interval observers" which compute lower and upper bounds in-between which the real values of the unmeasured variables are guaranteed to lie, The use of such observers for diagnosis purposes (to detect unmeasured input changes and sensor faults) in an anaerobic digestion WWfP is investigated and discussed, Copyright C 2000 IFAC Keywords : Interval observers, Fault detection, Fault isolation, Biological process, Unknown inputs.
1. INTRODUCTION
concentrated wastes such as agricultural (e.g. plant residues, animal wastes) or food industry wastewaters. A number of studies on the supervision and the diagnosis of the anaerobic digestion processes have been reported: see for instance (Steyer et al., 1997) or (Genovesi et ai" 1999a) and related references. However, most model-based diagnosis techniques reported involve a strict characterization of the noises and the disturbances of the plants of interest. Thus, they are usually unable to guarantee any stability or performance properties if the actual disturbances differ significantly from their initial characterization. When adopting a systemic approach to design diagnostic systems for WWTPs, the input variations are considered as input disturbances which can have very different - possibly time-varying - characteristics depending on the class and origin - urban or industrial - of the wastewaters to be treated. Thus, these perturbations can have arbitrary properties : from non stationary stochastic signals to well defmed or noisy waveforms (sinusoidal, ramp or step-like signals). The problem is thus quite difficult and even more complex since a very limited number of sensors is available in the wastewater treatment industry. This lack of sensors has two crucial consequences. First, it makes the mode ling of the growth rate expressions and the identification of their parameters very difficult.
Due to the increasing complexity and necessity for safety of industrial processes, efficient monitoring and decision support systems are becoming more and more important. Indeed, even in normal operational conditions, several types of disturbances can be present and they can largely affect the operating conditions of the process. Hence, there is a clear need for advanced control in order to keep the system performance as close as possible to optimal. This is particularly true of biological processes for environmental purposes (e.g., biological WasteWater Treatment Plants or WWTP) where the state of the "living" part of the system must be closely monitored. Extensive surveys have been published and several international conferences have been held on this topic: see for examples [Andrews, 1994] and related references. Anaerobic digestion processes are commonly used as WWTP. These processes present a number of advantages. In particular, organic matter is degraded into a gas mixture of carbon dioxide and methane producing valuable fuel. In addition, this kind of process is able to operate under severe conditions : high strength effluents and short hydraulic retention time. It is therefore particularly well adapted to treat
177
oxygen demand (COD) of the influent and produces valuable energy (i.e., methane). The biological scheme involves several multi-substrate multiorganism reactions that are performed both in series and in parallel (see for example [Henze and Harremoes, 1983])
Second, it leads to non-detectable systems for which no observation nor control techniques can be used without additional usually nonrealistic assumptions on the model structure or parameter values. In such cases, however, if it is assumed that guaranteed lower and upper bounds on the unknown inputs are provided and if a basic observer structure can be found to verify a property called "cooperativity", then it is possible to design a system of two observers which has the objective of computing guaranteed lower and upper bounds for the unrneasured variables rather than estimating their exact values. Since these observers deliver bounds in-between which the real values of the unrneasured states are guaranteed to lie, they are called "interval observers".
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 [Bemard et aI., 2000] :
x, =lu, -aD)x,
x, =lu, -aD)x,
z= D(z' -z) s, = D(s: -S,}-k,JI,X, s] ~D(s: -Sz)+k ,u,X -k sJ.l:X: en =D(c~ -Cn)+k,(k,P"" +Z-Cn -S, )+k.JI,X, +k,JI , X, 2
This theory was originally developed for robust observation and control of a class of nonlinear systems: see for instance [Rapaport and Gouze, 1999] and related references or [Alcaraz et al., 1999b] for the application of robust nonlinear interval observers to biological processes. The use of such observers for estimating guaranteed intervals on unrneasured variables in WWTPs is particularly suitable since it allows the user to deal with both the nonlinearities of the systems, the unknown input disturbances and, in a more general sense, with any disturbance or uncertainty that can be considered as an unknown input.
(1)
I
where XI' Xl> SI' S2 and CT! are respectively the concentrations of acidogenic bacteria, methanogenic bacteria, COD, volatile fatty acids (VF A) and total inorganic carbon. The alkalinity (i.e., the buffer capacity) is represented by Z. PC(h is the constant CO2 partial pressure. The parameter a represents a proportionality parameter of experimental determination. D=D(t) is the dilution rate and is supposed to be a persisting input, i.e.
r
D{T ';iT> 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 S, given by and p, = P .... , K S
The problem to be solved in this paper can be expressed as follows : "Given i) the mass balance model of an anaerobic digestion process, ii) a set of input/output online measurements, iii) the total unknowledge of the reaction kinetics, iv) the knowledge of some bounds on the unmeasured inputs in-between the real unknown inputs are lying, how designing a diagnosis system capable of detecting and isolating a number of sensor faults?".
s,
+ ,
Parameters defmition and their values are listed in Table 1. In all cases, the upper index i indicates "influent concentration".
The paper is organized as follows . First, the model of the process under interest is recalled. Then, the structure of the nonlinear diagnostic system is presented and discussed. The use of interval observers is then highlighted with respect to the diagnosis objectives before simulations results are presented to illustrate the approach. Finally, some conclusions and perspectives are drawn.
The model (1) thus takes the following matrix form :
2. THE NONLINEAR MODEL OF THE ANAEROBIC DIGESTION PROCESS
x, x,
0
X,
k.
k,
x, x, x.
0 -k,
0 0
k,
-k,
0
-aD
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 (C02), It thus reduces the chemical
0
-aD +
0
178
0 0
P,x,
p,x,
+
DC~
+ k,k.Pco, DZ' DS; DS;
0 0
0
0 0
0 0
x, x, -(D+k,) k, 0 -k, x, -D 0 0 x, -D x, 0 -D x. 0
(2)
3. DIAGNOSING THE ANAEROBIC DIGESTION PROCESS
way, the bank of observers that are designed will be able to detect and isolate sensor faults while they are robust against the unknown inputs. In addition, since a set of interval observers only uses a subset of the available measurements, it is also possible to detect if a measurement does not belong anymore to its reconstructed interval. In such a case, it can be concluded that either the measurement is faulty or some hypotheses on the lower and upper bounds on the unknown inputs are violated. A discrepancy between the real process and the model could also be the reason of such results but, for simplicity, that cause is not considered here.
3.1 Objectives and structure ofthe diagnostic system
The objective of the work reported here is to design a diagnostic system able to detect and isolate a number of sensor faults in the presence of unmeasured timevarying input disturbances which are assumed to lie in a given interval and assuming that the reaction rates are totally unknown. More specifically, in this study, 6 measurements are available: SI' S2, Cti, Z, PC0 2 and QC02 which is a nonlinear function of the states [Bernard et al., 2000]. The objective is to detect and isolate a number of faults within a set of 14 faults that can occur : faults on the 6 available sensors plus the detection of a violation of the hypotheses on the 4 unmeasured inputs considered : sJn, s/n, Ctiin, zin. In other words in this last situation, the problem is to detect that the real input violates its pre-specified lower or upper bounds (this defines 8 faults).
3.2 The residual generation process
Based upon the available measurements, a bank of robust nonlinear interval observers, each of them using either only 2 or 3 measurements are designed. Then, a number of residuals can be computed as :
In the following we propose an multi-observer based approach (using a bank of observers) to solve this problem. Using the classical asymptotic observers first proposed in [Bastin and Dochain, 1990] (see also [Chen, 1992] and [Alcaraz et aI., 1999a]), all the observers designed are independent of the reaction rates. However, the most important inconvenience of these observers is their dependency on the inputs acting on the system. In the following, these observers are robustified with respect to these unknown inputs by the use of interval observers.
where j=l, ... ,nob is the index of the observer with nob the total number of observers, i=l, ... ,nj is the index of the residual with ni the number of residuals generated with the observer j, k= 1. .... ny is the index of the estimated bounds on the measured outputs denoted by y- and y+, y is the measurement vector, 1= 1, ... ,no/), m = 1, ... ,nobL ... and n (n=l, ... ,nus) is the index of the unmeasured states with nus their number, rr(j) (resp. ri+(j» is the ith residual generated by the direct difference between the kth measurement and the estimation of its lower (resp. upper) bound computed by the jlh observer. Finally, rZlm(xn-) (resp. rZlm(xn is the difference between the lower (resp. the upper) bounds estimated by the observers I and m.
In fact, an interval observer consists in a system of
two observers to reconstruct bounds (one for the lower bound and one for the upper bound) on the unmeasured variables instead of reconstructing the exact state. Basically, its synthesis is based on the structure of any existing observer (an observer that can be designed if the inputs are known, i.e. if the system is observable) under the additional condition that this classical observer is a cooperative system. This property ensures that, given a dynamical system, two tenn-by-tenn ordered initial conditions and some lower and upper bounds on the inputs inbetween the actual inputs are living, then the solutions of the two differential systems are also ordered. Further details about the interval observers and their use in the estimation and control of biological processes can be found in [Rapaport and Harrnand, 1998a, 1998b, 1999] and [Hadj-Sadok and Gouze, 1998a, 1998b, 1999).
+»
3.3 The residual evaluation process
Depending on the residual to be evaluated (3a or 3b), the evaluation process differs. Obviously, when considering (3a), the problem is to evaluate the sign of the residual while the residuals generated with (3b) would follow a more classical evaluation process as : r = 0 as f(t) = 0 { n~ 0 as f(t) 0
*
Since there exists uncertainty on the inputs, once generated by the bank of interval observers, the residuals are compared to larger thresholds (i.e. the upper bound minus the lower one) than usual. This
(4)
where r is the residual to be evaluated and f any fault under consideration.
179
Notice that evaluating the signs of the residuals generated with (3a) is equivalent to evaluate a residual with respect to a time-varying threshold. For fault isolation, since it is usually not possible to make one residual to be sensitive to only one fault, it is necessary to use a fault signature table.
"By definition, the r+(i) and r'{i) residuals are necessarily complementary with respect to the direct measurements used in their generation. This is why we note 1/0 or Oil in the corresponding raws 2 to 7 of the above table. In Practice, its means that if one is faulty, the other is not used for isolation.
3.4 Application to the anaerobic process
From the above table, notice that all faults cannot be isolated. In particular, the following sets of faults can be defmed: a) SI' SI in', S2in' and Cin' corresponding to the signature [O,O,O,I,O,I,O]T or SI' Slin+' S2in+ and Cin+ correponding to the signature [0,0,1,0, I,O,Or, b) S2' CT! corresponding to the signature [O,O,I,O,I,O,I]T.
In our application, 5 observers were designed to compute a total of 7 residuals. 3 observers (numbered Obl to Ob3) were used to generate 6 residuals computed using different combinations of available measurements and the estimations of their upper and lower bounds (these residuals are "(3a)-like" residuals). The 2 other observers (numbered Ob4 and Ob5) were used to generate I "(3b)-like" residuals. In other words, these two last observers generate residuals computed in substracting the estimations of the lower or upper bounds on the same variable X 2• The definition of all observers is as follows (because of lack of space, only the residuals used and the evaluation rules are shown are shown) :
Obl :
However, it is particularly interesting to point out that faults on Z, PC02, QCD2 and Zin (respectively called c, e, f, and g) are isolable within all others. 4. SIMULATION RESULTS Sirnulations were carried out over a period of 280 days using input data showed in Figures I . The "fault occurrence protocol" we followed is defined in the following table (the faults, the instant of their occurrence and their type are specified (the "m" indice signifies "measured"» :
r+{I) = Z+(t)- Z{t) {r+{I) < 0 or {r-{I)=Z{t)-Z• -{t) , fault if r-{I)
+(2)-C+(t) C (t) {r+(2)<0 - TI - . TI , fault if or { r-(2)< 0 r-(2)= CTJ (t)-C; (t)
Table 1 : Simulation protocol Affect Fault d Time (d) variabl N° e I 10to 16 S2 2 30 to 35 SI
Ob2 : r
Ob3 :
r+(3)-S+(t) S(t) {r +(3)<0 - I - • I , fault if or { r-(3) < 0 r-(3) = SI (t)- SI-(t)
Ob4&5 : rm(Xn =IX;(4)- X;(5~, fault if r4/S(
X; »&
The signature table used in this study is given hereafter : Table J : Signature table/or the diagnosis system Z SI S2 CT! PC(h r+(1 ) 0 0 I/O 0 0 r'(1 ) 0/1 0 0 0 0 r+(2) I/O I/O I/O I/O 0 r'(2) 0/1 0/1 0/ 1 OIl 0 r+(3) I/O 1/0 I/O 0 0 r'(3) 0/1 OIl OIl 0 0 r4l s(
X;)
SI in
r+(I) r'(I) r+(2 ) r'(2 ) r+(3) r'(3) r41s(.)
0 0 I 0 I 0 0
0 Slin-
0 0 0 I 0 I 0
I S2in T 0 0
I 0 I 0 0
I S2in0 0 0
I 0 I 0
I Zin
I 0 I 0 0 0 0
T
QC(h 0 0 0 0 110 0/1
I
I
Zin0
T
I 0 I 0 0 0
Cin 0 0
I 0 I 0 0
Type of fault Constant Ramp
3
36 to 45
Z
False curve
4 5
61 to 63 90 to 95
CT! PCOz
Constant Ramp
6
115 to 120
QcOz
Ramp
7 8
138 to 168 169 to 196
SI in Slin
9
199 to 234
Ci•
Slin
10
240 to 250
Zin
11
258 to 268
Zin
Z ;n
Z;,·
Definition S2m=17 Slm- I,1498t-25.243
Zm--I/(t35.833)+97 CTI..=90 PCOzm-0.OO4t+O.24 QC02m=4.364t+548.7 See Figure See Figure
See Figure See Figure See Figure
Because of lack of space, only inputs and the most significant data are reported in the following. Inputs are presented in Figures la to le. Only one output is shown (see in Figure 2 the lower, upper bounds, the measured and the real values of Z). Two residuals are then pictured in Figures 3 and the fault signals corresponding to the evaluation of all residuals are fmally shown in Figure 4.
Cin0 0 0 I 0 I 0
180
09 011
J "1!
•
rrl
"
..
i I I
-~os
~/V
"
I~~
"
.
r
"
,
..
'.
-..
,.,
DJ
•
.. ,.,
-,.
..
~~.-----'~----"'m~---"'~;----'D;-----~3'-~
Figure la: Dilution rate
Figure le : Upper, lower bounds and real lin
3r------r------~----~------~----_r--_,
,- -.- -", :
I'-~--_r.:--'
..:~./--. -'-2~ -_\,
:
~~
4
1
!
:. / . -~-/1
ft,'!
\-
,
" "
"
,
•
..
"
..
""
-.
'"
DJ
"0
""
Figure 1b : Upper, lower and real S.in
"
• •. v
•
..
.
.. ,
t
..J '51
-«
""
""
Figure 2 : Lower, upper bounds, real (dashed line) and measured (dotted dashed line) l
"r--..-r-----~----~-----,-----~--__,
11O~ ,
,
I \ "
/;
, .::-..
10
1'--' , ,..,----"r~. ~
"
\!!
\~~
~'
.
!
...
-. , ...
..
..
..
,IO.~--~~~----~=-----~,~~--""~----~""~--~ -to
Figure le: Upper, lower and real Szin
Figure 3a : Residual "-(2)
"~-----r----_~-----r---_-----'_--,
"r--,--,-----~---~------~----~--,
f
.
;;I '
.. , ,
\ .~
r-- ··
"
.
"O~----~"~---~"'~---~"'~----~D;----~"';---' -to
..
-.
.~.---i-=-----~~----~~~----~""~----~,.,,-~
Figure Id : Upper, lower and real Gin
Figure 3b : Residual r+(2)
181
as well as CONACyT (Mexican National Council of Science and Technology) for the financial support provided to this study.
6. REFERENCES A1caraz V., A. Genovesi, J. Harmand, A. V. Gonzalez, A. Rapaport and J. P. Steyer (1999a) : "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, pp. 225-235, Girona, Spain, February 24-26. A1caraz V., J. Harmand, D. Dochain and J. P. Steyer (I 999b) : "A robust asymptotic observer for a class of nonlinear systems", submitted to Chemical Engineering Science. Alcaraz V., J. Harmand, A. Rapaport, J. P. Steyer, V. Gonzalcz and C. Pelayo-Ortiz (l999c) : "Robust interval observers for a class of uncertain non linear systems : application to a wastewater treatment process", submitted to Journal of
Figure 4 : From the lower to the upper curve : fault signals corresponding to the isolation of the set offaults a to/
Figure 4 shows very good results when confronted with the "fault protocol" followed. However, a number of false-alarms exist (for instance, see the I't and the 3rd detections on the first fault signal in Figure 4). These false alann are due to a slow convergence of some residuals. Notice, however, that these false alarm disappear once the residuals have converged. As expected, set of faults b are perfectly detected and isolated as such (see the second curve in figure 4 which corresponds to faults N°l and 4). The same observations can be done concerning the third curve which corresponds to fault N°3. The 4th curve in Figure 4 indicates two faults on PC02' In fact, the first one is a false alann (again due to the Iow convergence of one of the residuals). The second fault detected corresponds to fault N°5. The same observations can be done concerning the fifth curve which corresponds to fault N°6. Finally, the two last curves allows a correct detection and isolation of faults N°W and 11. Notice that faults N°9 is not detected. This non-detection is due to the fact that the fault has not enough effect on the corresponding residuals to be detected. This highlights the limitations of the interval-based approach which can be quite conservative depending on the magnitude of the intervals chosen for the inputs of the process.
Process Control. Andrews J. F. (1994) : "Dynamic control of wastewater treatment plants", Environmental Science Technology, vo\. 28, n09, pp. 434A-440A. Bastin G. and D. Dochain (1990) : "On-line estimation and adaptive control of bioreactors", Elsevier, Amsterdam, Oxford, New York, Tokyo, 379 pages. Bemard 0 ., Z. Hadj-Sadok, D. Dochain, A. Genovesi and J. P. Steyer (2000) : "Dynamical model development and parameter identification for an anaerobic wastewater treatment process", submitted to Water Research. Chen L (1992) : "Modelling, identifiability and control of complex biotechnological systems", PhD Thesis, Universite Catholique de Louvain, Belgium. Henze M. and P. Harremoes (1983) : "Anaerobic treatment of wastewater in fixed film reactors - a literature review", Water Science Technology., vo\. IS, nOl, pp. 1-101. Steyer J. P.., D. Rolland, J. C . Bouvier and R. Moletta (1997) : "Hybrid fuzzy neural network for diagnosis - application to the anaerobic treatment of wine distil1ery wastewater in a fluidized bed reactor", Water Science and Technology, vo\. 36, n06-7, pp. 209-217. Genovesi A., J. Harmand. and J. P. Steyer ( 1999), "Integrated fault detection and isolation : application to a winery's wastewater treatment plant", Applied Intelligence (APIN). Special Issue in Environment and AI, to appear. Hadj-Sadok M. Z. and J. L. Gouze (1998a) : "Bounds estimation for uncertain models of wastewater treatment processes", in IFAC-EurAgEng International Workshop on "Decision and Control in Waste Bio-Processing". IEEE Iternational Coni. on Control and Applications. Trieste, Italy. Hadj-Sadok M. Z. and J. L. Gouze (1998b) : "Interval observers for uncertain models of wastewater treatment processes".
5. CONCLUSIONS AND PERSPECTIVES
CONTROLO'98 3rd Portuguese Conference on Automatic Control. Coimbra, Portugal. Hadj-Sadok M Z., and J. L. Gouze (1999) : "Comparison between two interval for wastewatcr treatment processes". European Control Conference ECC'99, Karlsruhe, Germany, 1999.
This paper presented a new interval-based approach for the diagnosis of biological systems. Based on the comparison between the measured outputs and their estimated bounds within which they should live, the technique detects either a sensor fault or the violation of the basic hypotheses made during the design step. Such observers wiII be implemented soon on a real WWTP. An other research axe which is explored now is to extend the use of this technique to systems which presents a spatial gradient of concentrations, such as fixed bed reactors modeled by distributed parameter nonlinear models.
Rapaport A. and J. Harmand (1998a) : "Robust Regulation of a Bioreactor in a Highly Uncertain Environment n , in IFAC-
EurAgEng International Workshop on -Decision and Control in Waste Bio-Processingn. Waste-Decision'98, 8 pages (CDROM), Narbonne, France, 25-27 February 1998. Rapaport A. and J. Harmand (1998b) : "Robust Nonlinear Control of a Class of Partially Observed Processes : Application to Conference on Control Continuous Bioreactors", Applications. CCA'98, Trieste, Italy, 1998. Rapaport A. and J. Harmand (1999) : "Robust regulation of a class of partia1\y observed nonlinear continuous bioreactors", submitted to Journal of Process Control. Rapaport, A. and J. L. Gouze (1999) : "Practical observers for uncertain affine output injection systems", IEEE Tansactions on Automatic Control, submitted.
AknowJedgement :
The authors gratefully acknowledge the ECOS-Nord program for French-Mexican scientific cooperation
182
INTERV AL-BASED DIAGNOSIS OF AN ANAEROBIC DIGESTER PILOT PLANT
J. P. Steyer,.f. Harmand, V. A1caraz-Gonzalez
Laboratoire de Biotechnologie de I'Environnement. LRE - INRA . Avenue des etangs. 11100 Narbonne. France . • : +33 (0) 468.425,151. Fax : +33 (0) 468.425.160.
Abstract: Anaerobic digestion processes are highly nonlinear time-varying processes. When used for WasteWater Treatment, they are subject to large input disturbances both of concentrations and flow rates of the polluted effluents to be treated. Furthermore, such processes still suffer from a lack of reliable and cheap sensors, As a consequence, in practice, these processes are usually not observable nor detectable and most state observation and control techniques cannot be applied. In such situations and under specific conditions that are highlighted in the paper, it is possible to design some "interval observers" which compute lower and upper bounds in-between which the real values of the unmeasured variables are guaranteed to lie, The use of such observers for diagnosis purposes (to detect unmeasured input changes and sensor faults) in an anaerobic digestion WWfP is investigated and discussed, Copyright C 2000 IFAC Keywords : Interval observers, Fault detection, Fault isolation, Biological process, Unknown inputs.
1. INTRODUCTION
concentrated wastes such as agricultural (e.g. plant residues, animal wastes) or food industry wastewaters. A number of studies on the supervision and the diagnosis of the anaerobic digestion processes have been reported: see for instance (Steyer et al., 1997) or (Genovesi et ai" 1999a) and related references. However, most model-based diagnosis techniques reported involve a strict characterization of the noises and the disturbances of the plants of interest. Thus, they are usually unable to guarantee any stability or performance properties if the actual disturbances differ significantly from their initial characterization. When adopting a systemic approach to design diagnostic systems for WWTPs, the input variations are considered as input disturbances which can have very different - possibly time-varying - characteristics depending on the class and origin - urban or industrial - of the wastewaters to be treated. Thus, these perturbations can have arbitrary properties : from non stationary stochastic signals to well defmed or noisy waveforms (sinusoidal, ramp or step-like signals). The problem is thus quite difficult and even more complex since a very limited number of sensors is available in the wastewater treatment industry. This lack of sensors has two crucial consequences. First, it makes the mode ling of the growth rate expressions and the identification of their parameters very difficult.
Due to the increasing complexity and necessity for safety of industrial processes, efficient monitoring and decision support systems are becoming more and more important. Indeed, even in normal operational conditions, several types of disturbances can be present and they can largely affect the operating conditions of the process. Hence, there is a clear need for advanced control in order to keep the system performance as close as possible to optimal. This is particularly true of biological processes for environmental purposes (e.g., biological WasteWater Treatment Plants or WWTP) where the state of the "living" part of the system must be closely monitored. Extensive surveys have been published and several international conferences have been held on this topic: see for examples [Andrews, 1994] and related references. Anaerobic digestion processes are commonly used as WWTP. These processes present a number of advantages. In particular, organic matter is degraded into a gas mixture of carbon dioxide and methane producing valuable fuel. In addition, this kind of process is able to operate under severe conditions : high strength effluents and short hydraulic retention time. It is therefore particularly well adapted to treat
177
oxygen demand (COD) of the influent and produces valuable energy (i.e., methane). The biological scheme involves several multi-substrate multiorganism reactions that are performed both in series and in parallel (see for example [Henze and Harremoes, 1983])
Second, it leads to non-detectable systems for which no observation nor control techniques can be used without additional usually nonrealistic assumptions on the model structure or parameter values. In such cases, however, if it is assumed that guaranteed lower and upper bounds on the unknown inputs are provided and if a basic observer structure can be found to verify a property called "cooperativity", then it is possible to design a system of two observers which has the objective of computing guaranteed lower and upper bounds for the unrneasured variables rather than estimating their exact values. Since these observers deliver bounds in-between which the real values of the unrneasured states are guaranteed to lie, they are called "interval observers".
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 [Bemard et aI., 2000] :
x, =lu, -aD)x,
x, =lu, -aD)x,
z= D(z' -z) s, = D(s: -S,}-k,JI,X, s] ~D(s: -Sz)+k ,u,X -k sJ.l:X: en =D(c~ -Cn)+k,(k,P"" +Z-Cn -S, )+k.JI,X, +k,JI , X, 2
This theory was originally developed for robust observation and control of a class of nonlinear systems: see for instance [Rapaport and Gouze, 1999] and related references or [Alcaraz et al., 1999b] for the application of robust nonlinear interval observers to biological processes. The use of such observers for estimating guaranteed intervals on unrneasured variables in WWTPs is particularly suitable since it allows the user to deal with both the nonlinearities of the systems, the unknown input disturbances and, in a more general sense, with any disturbance or uncertainty that can be considered as an unknown input.
(1)
I
where XI' Xl> SI' S2 and CT! are respectively the concentrations of acidogenic bacteria, methanogenic bacteria, COD, volatile fatty acids (VF A) and total inorganic carbon. The alkalinity (i.e., the buffer capacity) is represented by Z. PC(h is the constant CO2 partial pressure. The parameter a represents a proportionality parameter of experimental determination. D=D(t) is the dilution rate and is supposed to be a persisting input, i.e.
r
D{T ';iT> 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 S, given by and p, = P .... , K S
The problem to be solved in this paper can be expressed as follows : "Given i) the mass balance model of an anaerobic digestion process, ii) a set of input/output online measurements, iii) the total unknowledge of the reaction kinetics, iv) the knowledge of some bounds on the unmeasured inputs in-between the real unknown inputs are lying, how designing a diagnosis system capable of detecting and isolating a number of sensor faults?".
s,
+ ,
Parameters defmition and their values are listed in Table 1. In all cases, the upper index i indicates "influent concentration".
The paper is organized as follows . First, the model of the process under interest is recalled. Then, the structure of the nonlinear diagnostic system is presented and discussed. The use of interval observers is then highlighted with respect to the diagnosis objectives before simulations results are presented to illustrate the approach. Finally, some conclusions and perspectives are drawn.
The model (1) thus takes the following matrix form :
2. THE NONLINEAR MODEL OF THE ANAEROBIC DIGESTION PROCESS
x, x,
0
X,
k.
k,
x, x, x.
0 -k,
0 0
k,
-k,
0
-aD
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 (C02), It thus reduces the chemical
0
-aD +
0
178
0 0
P,x,
p,x,
+
DC~
+ k,k.Pco, DZ' DS; DS;
0 0
0
0 0
0 0
x, x, -(D+k,) k, 0 -k, x, -D 0 0 x, -D x, 0 -D x. 0
(2)
3. DIAGNOSING THE ANAEROBIC DIGESTION PROCESS
way, the bank of observers that are designed will be able to detect and isolate sensor faults while they are robust against the unknown inputs. In addition, since a set of interval observers only uses a subset of the available measurements, it is also possible to detect if a measurement does not belong anymore to its reconstructed interval. In such a case, it can be concluded that either the measurement is faulty or some hypotheses on the lower and upper bounds on the unknown inputs are violated. A discrepancy between the real process and the model could also be the reason of such results but, for simplicity, that cause is not considered here.
3.1 Objectives and structure ofthe diagnostic system
The objective of the work reported here is to design a diagnostic system able to detect and isolate a number of sensor faults in the presence of unmeasured timevarying input disturbances which are assumed to lie in a given interval and assuming that the reaction rates are totally unknown. More specifically, in this study, 6 measurements are available: SI' S2, Cti, Z, PC0 2 and QC02 which is a nonlinear function of the states [Bernard et al., 2000]. The objective is to detect and isolate a number of faults within a set of 14 faults that can occur : faults on the 6 available sensors plus the detection of a violation of the hypotheses on the 4 unmeasured inputs considered : sJn, s/n, Ctiin, zin. In other words in this last situation, the problem is to detect that the real input violates its pre-specified lower or upper bounds (this defines 8 faults).
3.2 The residual generation process
Based upon the available measurements, a bank of robust nonlinear interval observers, each of them using either only 2 or 3 measurements are designed. Then, a number of residuals can be computed as :
In the following we propose an multi-observer based approach (using a bank of observers) to solve this problem. Using the classical asymptotic observers first proposed in [Bastin and Dochain, 1990] (see also [Chen, 1992] and [Alcaraz et aI., 1999a]), all the observers designed are independent of the reaction rates. However, the most important inconvenience of these observers is their dependency on the inputs acting on the system. In the following, these observers are robustified with respect to these unknown inputs by the use of interval observers.
where j=l, ... ,nob is the index of the observer with nob the total number of observers, i=l, ... ,nj is the index of the residual with ni the number of residuals generated with the observer j, k= 1. .... ny is the index of the estimated bounds on the measured outputs denoted by y- and y+, y is the measurement vector, 1= 1, ... ,no/), m = 1, ... ,nobL ... and n (n=l, ... ,nus) is the index of the unmeasured states with nus their number, rr(j) (resp. ri+(j» is the ith residual generated by the direct difference between the kth measurement and the estimation of its lower (resp. upper) bound computed by the jlh observer. Finally, rZlm(xn-) (resp. rZlm(xn is the difference between the lower (resp. the upper) bounds estimated by the observers I and m.
In fact, an interval observer consists in a system of
two observers to reconstruct bounds (one for the lower bound and one for the upper bound) on the unmeasured variables instead of reconstructing the exact state. Basically, its synthesis is based on the structure of any existing observer (an observer that can be designed if the inputs are known, i.e. if the system is observable) under the additional condition that this classical observer is a cooperative system. This property ensures that, given a dynamical system, two tenn-by-tenn ordered initial conditions and some lower and upper bounds on the inputs inbetween the actual inputs are living, then the solutions of the two differential systems are also ordered. Further details about the interval observers and their use in the estimation and control of biological processes can be found in [Rapaport and Harrnand, 1998a, 1998b, 1999] and [Hadj-Sadok and Gouze, 1998a, 1998b, 1999).
+»
3.3 The residual evaluation process
Depending on the residual to be evaluated (3a or 3b), the evaluation process differs. Obviously, when considering (3a), the problem is to evaluate the sign of the residual while the residuals generated with (3b) would follow a more classical evaluation process as : r = 0 as f(t) = 0 { n~ 0 as f(t) 0
*
Since there exists uncertainty on the inputs, once generated by the bank of interval observers, the residuals are compared to larger thresholds (i.e. the upper bound minus the lower one) than usual. This
(4)
where r is the residual to be evaluated and f any fault under consideration.
179
Notice that evaluating the signs of the residuals generated with (3a) is equivalent to evaluate a residual with respect to a time-varying threshold. For fault isolation, since it is usually not possible to make one residual to be sensitive to only one fault, it is necessary to use a fault signature table.
"By definition, the r+(i) and r'{i) residuals are necessarily complementary with respect to the direct measurements used in their generation. This is why we note 1/0 or Oil in the corresponding raws 2 to 7 of the above table. In Practice, its means that if one is faulty, the other is not used for isolation.
3.4 Application to the anaerobic process
From the above table, notice that all faults cannot be isolated. In particular, the following sets of faults can be defmed: a) SI' SI in', S2in' and Cin' corresponding to the signature [O,O,O,I,O,I,O]T or SI' Slin+' S2in+ and Cin+ correponding to the signature [0,0,1,0, I,O,Or, b) S2' CT! corresponding to the signature [O,O,I,O,I,O,I]T.
In our application, 5 observers were designed to compute a total of 7 residuals. 3 observers (numbered Obl to Ob3) were used to generate 6 residuals computed using different combinations of available measurements and the estimations of their upper and lower bounds (these residuals are "(3a)-like" residuals). The 2 other observers (numbered Ob4 and Ob5) were used to generate I "(3b)-like" residuals. In other words, these two last observers generate residuals computed in substracting the estimations of the lower or upper bounds on the same variable X 2• The definition of all observers is as follows (because of lack of space, only the residuals used and the evaluation rules are shown are shown) :
Obl :
However, it is particularly interesting to point out that faults on Z, PC02, QCD2 and Zin (respectively called c, e, f, and g) are isolable within all others. 4. SIMULATION RESULTS Sirnulations were carried out over a period of 280 days using input data showed in Figures I . The "fault occurrence protocol" we followed is defined in the following table (the faults, the instant of their occurrence and their type are specified (the "m" indice signifies "measured"» :
r+{I) = Z+(t)- Z{t) {r+{I) < 0 or {r-{I)=Z{t)-Z• -{t) , fault if r-{I)
+(2)-C+(t) C (t) {r+(2)<0 - TI - . TI , fault if or { r-(2)< 0 r-(2)= CTJ (t)-C; (t)
Table 1 : Simulation protocol Affect Fault d Time (d) variabl N° e I 10to 16 S2 2 30 to 35 SI
Ob2 : r
Ob3 :
r+(3)-S+(t) S(t) {r +(3)<0 - I - • I , fault if or { r-(3) < 0 r-(3) = SI (t)- SI-(t)
Ob4&5 : rm(Xn =IX;(4)- X;(5~, fault if r4/S(
X; »&
The signature table used in this study is given hereafter : Table J : Signature table/or the diagnosis system Z SI S2 CT! PC(h r+(1 ) 0 0 I/O 0 0 r'(1 ) 0/1 0 0 0 0 r+(2) I/O I/O I/O I/O 0 r'(2) 0/1 0/1 0/ 1 OIl 0 r+(3) I/O 1/0 I/O 0 0 r'(3) 0/1 OIl OIl 0 0 r4l s(
X;)
SI in
r+(I) r'(I) r+(2 ) r'(2 ) r+(3) r'(3) r41s(.)
0 0 I 0 I 0 0
0 Slin-
0 0 0 I 0 I 0
I S2in T 0 0
I 0 I 0 0
I S2in0 0 0
I 0 I 0
I Zin
I 0 I 0 0 0 0
T
QC(h 0 0 0 0 110 0/1
I
I
Zin0
T
I 0 I 0 0 0
Cin 0 0
I 0 I 0 0
Type of fault Constant Ramp
3
36 to 45
Z
False curve
4 5
61 to 63 90 to 95
CT! PCOz
Constant Ramp
6
115 to 120
QcOz
Ramp
7 8
138 to 168 169 to 196
SI in Slin
9
199 to 234
Ci•
Slin
10
240 to 250
Zin
11
258 to 268
Zin
Z ;n
Z;,·
Definition S2m=17 Slm- I,1498t-25.243
Zm--I/(t35.833)+97 CTI..=90 PCOzm-0.OO4t+O.24 QC02m=4.364t+548.7 See Figure See Figure
See Figure See Figure See Figure
Because of lack of space, only inputs and the most significant data are reported in the following. Inputs are presented in Figures la to le. Only one output is shown (see in Figure 2 the lower, upper bounds, the measured and the real values of Z). Two residuals are then pictured in Figures 3 and the fault signals corresponding to the evaluation of all residuals are fmally shown in Figure 4.
Cin0 0 0 I 0 I 0
180
09 011
J "1!
•
rrl
"
..
i I I
-~os
~/V
"
I~~
"
.
r
"
,
..
'.
-..
,.,
DJ
•
.. ,.,
-,.
..
~~.-----'~----"'m~---"'~;----'D;-----~3'-~
Figure la: Dilution rate
Figure le : Upper, lower bounds and real lin
3r------r------~----~------~----_r--_,
,- -.- -", :
I'-~--_r.:--'
..:~./--. -'-2~ -_\,
:
~~
4
1
!
:. / . -~-/1
ft,'!
\-
,
" "
"
,
•
..
"
..
""
-.
'"
DJ
"0
""
Figure 1b : Upper, lower and real S.in
"
• •. v
•
..
.
.. ,
t
..J '51
-«
""
""
Figure 2 : Lower, upper bounds, real (dashed line) and measured (dotted dashed line) l
"r--..-r-----~----~-----,-----~--__,
11O~ ,
,
I \ "
/;
, .::-..
10
1'--' , ,..,----"r~. ~
"
\!!
\~~
~'
.
!
...
-. , ...
..
..
..
,IO.~--~~~----~=-----~,~~--""~----~""~--~ -to
Figure le: Upper, lower and real Szin
Figure 3a : Residual "-(2)
"~-----r----_~-----r---_-----'_--,
"r--,--,-----~---~------~----~--,
f
.
;;I '
.. , ,
\ .~
r-- ··
"
.
"O~----~"~---~"'~---~"'~----~D;----~"';---' -to
..
-.
.~.---i-=-----~~----~~~----~""~----~,.,,-~
Figure Id : Upper, lower and real Gin
Figure 3b : Residual r+(2)
181
as well as CONACyT (Mexican National Council of Science and Technology) for the financial support provided to this study.
6. REFERENCES A1caraz V., A. Genovesi, J. Harmand, A. V. Gonzalez, A. Rapaport and J. P. Steyer (1999a) : "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, pp. 225-235, Girona, Spain, February 24-26. A1caraz V., J. Harmand, D. Dochain and J. P. Steyer (I 999b) : "A robust asymptotic observer for a class of nonlinear systems", submitted to Chemical Engineering Science. Alcaraz V., J. Harmand, A. Rapaport, J. P. Steyer, V. Gonzalcz and C. Pelayo-Ortiz (l999c) : "Robust interval observers for a class of uncertain non linear systems : application to a wastewater treatment process", submitted to Journal of
Figure 4 : From the lower to the upper curve : fault signals corresponding to the isolation of the set offaults a to/
Figure 4 shows very good results when confronted with the "fault protocol" followed. However, a number of false-alarms exist (for instance, see the I't and the 3rd detections on the first fault signal in Figure 4). These false alann are due to a slow convergence of some residuals. Notice, however, that these false alarm disappear once the residuals have converged. As expected, set of faults b are perfectly detected and isolated as such (see the second curve in figure 4 which corresponds to faults N°l and 4). The same observations can be done concerning the third curve which corresponds to fault N°3. The 4th curve in Figure 4 indicates two faults on PC02' In fact, the first one is a false alann (again due to the Iow convergence of one of the residuals). The second fault detected corresponds to fault N°5. The same observations can be done concerning the fifth curve which corresponds to fault N°6. Finally, the two last curves allows a correct detection and isolation of faults N°W and 11. Notice that faults N°9 is not detected. This non-detection is due to the fact that the fault has not enough effect on the corresponding residuals to be detected. This highlights the limitations of the interval-based approach which can be quite conservative depending on the magnitude of the intervals chosen for the inputs of the process.
Process Control. Andrews J. F. (1994) : "Dynamic control of wastewater treatment plants", Environmental Science Technology, vo\. 28, n09, pp. 434A-440A. Bastin G. and D. Dochain (1990) : "On-line estimation and adaptive control of bioreactors", Elsevier, Amsterdam, Oxford, New York, Tokyo, 379 pages. Bemard 0 ., Z. Hadj-Sadok, D. Dochain, A. Genovesi and J. P. Steyer (2000) : "Dynamical model development and parameter identification for an anaerobic wastewater treatment process", submitted to Water Research. Chen L (1992) : "Modelling, identifiability and control of complex biotechnological systems", PhD Thesis, Universite Catholique de Louvain, Belgium. Henze M. and P. Harremoes (1983) : "Anaerobic treatment of wastewater in fixed film reactors - a literature review", Water Science Technology., vo\. IS, nOl, pp. 1-101. Steyer J. P.., D. Rolland, J. C . Bouvier and R. Moletta (1997) : "Hybrid fuzzy neural network for diagnosis - application to the anaerobic treatment of wine distil1ery wastewater in a fluidized bed reactor", Water Science and Technology, vo\. 36, n06-7, pp. 209-217. Genovesi A., J. Harmand. and J. P. Steyer ( 1999), "Integrated fault detection and isolation : application to a winery's wastewater treatment plant", Applied Intelligence (APIN). Special Issue in Environment and AI, to appear. Hadj-Sadok M. Z. and J. L. Gouze (1998a) : "Bounds estimation for uncertain models of wastewater treatment processes", in IFAC-EurAgEng International Workshop on "Decision and Control in Waste Bio-Processing". IEEE Iternational Coni. on Control and Applications. Trieste, Italy. Hadj-Sadok M. Z. and J. L. Gouze (1998b) : "Interval observers for uncertain models of wastewater treatment processes".
5. CONCLUSIONS AND PERSPECTIVES
CONTROLO'98 3rd Portuguese Conference on Automatic Control. Coimbra, Portugal. Hadj-Sadok M Z., and J. L. Gouze (1999) : "Comparison between two interval for wastewatcr treatment processes". European Control Conference ECC'99, Karlsruhe, Germany, 1999.
This paper presented a new interval-based approach for the diagnosis of biological systems. Based on the comparison between the measured outputs and their estimated bounds within which they should live, the technique detects either a sensor fault or the violation of the basic hypotheses made during the design step. Such observers wiII be implemented soon on a real WWTP. An other research axe which is explored now is to extend the use of this technique to systems which presents a spatial gradient of concentrations, such as fixed bed reactors modeled by distributed parameter nonlinear models.
Rapaport A. and J. Harmand (1998a) : "Robust Regulation of a Bioreactor in a Highly Uncertain Environment n , in IFAC-
EurAgEng International Workshop on -Decision and Control in Waste Bio-Processingn. Waste-Decision'98, 8 pages (CDROM), Narbonne, France, 25-27 February 1998. Rapaport A. and J. Harmand (1998b) : "Robust Nonlinear Control of a Class of Partially Observed Processes : Application to Conference on Control Continuous Bioreactors", Applications. CCA'98, Trieste, Italy, 1998. Rapaport A. and J. Harmand (1999) : "Robust regulation of a class of partia1\y observed nonlinear continuous bioreactors", submitted to Journal of Process Control. Rapaport, A. and J. L. Gouze (1999) : "Practical observers for uncertain affine output injection systems", IEEE Tansactions on Automatic Control, submitted.
AknowJedgement :
The authors gratefully acknowledge the ECOS-Nord program for French-Mexican scientific cooperation
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