application of a robust interval observer to an anaerobic digestion process

Dev. Chem. Eng. Mineral Process. 13(3/4), pp. 267-278, 2005.

Application of a Robust Interval Observer to

an Anaerobic Digestion Process

V. Alcaraz-GonzBlez'*, J. Harmand', A. Rapaport2, J.P. Steyer', V. GonzBlez-Alvarez3, C. Pelayo-Ortiz3 INRA-LBE, Avenue des Etangs, 1 I1 00 Narbonne, France RVRA-Biomktrie, 2, Place Viala, 34060 Montpellier, France University of Guadalajara-CUCEI, Dept of Chemical Engineering.

Blvd. Gral. Marcelino Garcia Barragdn 1451, Guadalajara Jalisco, 44430. Mbxico

2

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A new robust state estimation scheme is proposed for chemical and biochemical processes. This scheme is a nonlinear interval observer which is robust in the face of uncertainties of the inputs, load disturbances and the nonlinearity of the system, while maintaining excellent stability conditions. We show that this observer yields satisfactory estimates of the unmeasured states within a guaranteed interval determined by the available measurements and some reasonable hypotheses about the observer inputs. For demonstrating the effectiveness and applicability of the proposed estimation scheme, we apply it in an actual anaerobic digestion pilot plant for wastewater treatment in the presence of uncertainties including measuring errors, sensor failures, time varying inputs and unmeasured disturbances. On-line implementation results reveal that the proposed robust nonlinear interval observer is a practical and encouraging approach to the robust state estimation in uncertain nonlinear processes.

Introduction The efficient monitoring, control and operation of a process demands a reliable knowledge of the essential variables of the process. However, in many cases, information about such variables is not always available on-line, or if it is available, it is usually plagued by time delays, or corrupted by noise. This measurement problem is a consequence of the inadequacy of available sensors or operational limitations. Then, in the absence of measurements of the essential variables, continuous estimates of them must be inferred from the available measurements. In order to solve this problem, many solutions have been proposed in the past. It is well known that the classical extended Kalman filter and the Luenberger observer allow the estimation of both the parameters and the states of the system. However, since these estimators are based on a linearized model of the process, the stability and convergence properties

* Author for correspondence (victor@ccip. udg. m).

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V. Alcaraz-Gonzalez, J. Hannand, A. Rapaport, et al.

are essentially local and valid only around an equilibrium point, which make their stability over wide ranges of operation difficult to guarantee. Several linearization methods have been also proposed (see for example [ l]), but they have the drawback that only local behavior can be guaranteed with limited performance and stability properties. Other estimation schemes based on geometric theory have been shown to guarantee convergence with linear output error dynamics but apply to an extremely restrictive class of plants [2]. The high gain [3], adaptive [4] and sliding mode [ 5 ] approaches have been also devised to solve the state reconstruction problem since it guarantees stability but their design involve conditions that must be assumed a priori or that are usually hard to verify [ 6 ] . Moreover, the application of such observers is not straightforward since they have very complex tuning procedures. Furthermore, where the inputs are unknown, the classical estimation schemes are not able to reconstruct the non-measured states from the available measurements due to the serious detectability problems imposed by the input conditions [7].

In this paper, a new robust observer for a class of lumped time varying non-linear models useful in chemical and biochemical processes is proposed. It is based on the knowledge of the intervals in which the unmeasured inputs of the process are likely to evolve [8, 91. In addition, a key structural property (called cooperativity) must hold to allow us to compute guaranteed intervals for the unmeasured states when bounds on the unknown initial conditions are given. We show that this Robust Interval Observer (NO) is capable of coping simultaneously with the problems posed by both the uncertainties in the inputs and the lack of knowledge of the nonlinearities, while maintaining excellent stability properties under the influence of the time varying parameters, system failures, load disturbances, and inputs. Conditions for the observer stability are also derived by analyzing the dynamics of the estimation error.

The paper is organized as follows. First, a generalized nonlinear model considered in this study is described. Second, a dynamic nonlinear asymptotic observer for unknown lunetics is introduced. Then, the equations of this observer are used to synthesize a new nonlinear robust interval observer, assuming that input disturbances are unknown but that they belong to a prescribed bounded set. Finally, the proposed nonlinear N O is tested on-line using actual data from an anaerobic digestion pilot plant before some conclusions and perspectives are drawn.

The Generalized Process Model Consider the following general nonlinear time varying lumped model, which describes a large number of chemical and biochemical processes:

.(t) = Cf(x( t ) , t )+ A(t)x(t)+b(t) ...( 1)

where x(t) E 'W is the state vector ( i e . concentrations of the chemical and or biochemical species involved in the process); f(x(t),t) E 9Ir denotes the vector of nonlinearities (including reaction rates); and C E 'iRflxr represents a matrix of constant coefficients (e.g. stoichiometric or yield coefficients). The time varying matrix A(t) E W"'" is the state matrix while b(t) E %" gathers the inputs (e.g. mass and/or energy feeding rate vector) and/or other possibly time varying functions (e.g. the gaseous outflow rate vector, if any).

268

Application of a Robust Interval Observer to an Anaerobic Digestion Process

The following hypotheses formally describe the frame of uncertainties and the minimum knowledge on the system that is necessary to design the interval observer proposed here.

Hypotheses H1: Ax(?),?) is l l l y unknown. A(?) is known for each t 1 0 . m state variables are measured on-line. C is constant and known. A(t) is bounded; that is, there exist two constant matrices A' and A' such as A - I A ( ~ ) I A + . Initial conditions of the state vector are unknown but guaranteed bounds are given as x-(o) I X(O) I x+(o) . The input vector b(t) is unknown but within guaranteed, possibly time varying, bounds and it is given by K ( t ) I b(t) I b'(t) .

From hypothesis HI c, the state space may be split in such a way that Equation (1) can be rewritten as:

where the m measured state variables have been grouped in the xz(t) vector (dim x2(t) = m) and the variables that have to be estimated are represented by x l ( t ) (dim xl( t ) =s= n - m). Matrices All(?) E 'iJl"', A12(t) E 'dxm, &I(?) E 'iJlmxs, &(t) E 91mxm, CI E dXr, C2 E 91mxr9 b1 E %' and b2 E 'iJlm are the corresponding partitions of A(?), C and b(t), respectively.

An Asymptotic Observer A necessary condition for designing interval observers is that a known-input observer exists (i.e. any observer that can be derived if b(t) is known) [8]. If such an observer exists and if b(t) is unknown (but with known lower and upper bounds), the structure of this observer can be used to build an interval observer. In thls section, in order to cover this first requirement, an asymptotic observer is introduced. Indeed, in addition to being a known-input observer, this estimation scheme maintains the stability property of its asymptotic basis and shows excellent robustness properties against the uncertainties on the nonlinearities (i.e. it permits the exact cancellation of the nonlinear terms). Thus, in a first step, only the hypotheses H f a - H f e will be necessary. For this purpose, it is assumed, in principle, that b(t) is known and that there are no restrictions on the initial conditions. Let us introduce the following hypothesis related to the selection of x2 and, consequently, on the construction of Equation (2):

Hypothesis H2: rank C2 = rank C

Then, under hypotheses Hla-e and H2, when b(t) is known, the following system:

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V. Alcaraz-Gonzcilez, J. Harmand, A . Rapaport, et al.

k(?) = W(t)G(t) + X(f)X, ( t ) + Nb(?) G(0) = Ni(0) ;,(+ N$(?)- N2x2( t ) )

. . .(3)

where W(?)=(N,A , , (? )+N,A , , ( t ) )N; ' , X(?)= N,A, , ( t )+N,A, , (?)-W(?)N, N , E 'WX' is an arbitrary invertible matrix, N, = -N,C,C: with N , E %"' , C: is the generalized pseudo-inverse of C, and N = [ N , i N,]; is an asymptotic nonlinear observer independent of the nonlinearities f(x(f),t) for the nonlinear time-varying lumped model (1) [lo]. Now, let e(t)= ; , ( t ) -x,(?) be the observation errors associated to Equation (3). It is easy to verify that e follows the dynamics: e(?)= W,(t)e(t) with W,(t)= N; 'W(?)N, . Note that under hypothesis Hle, it is possible to find two constant matrices We- and We+ such that We- I We (f) .S W: Vf 2 0. Thus, in order to guarantee the stability of Equation (3), the following hypotheses are also introduced:

Hypotheses H3: a ) WC:, 2 0,Vi f j , b) We- and We+ are Hurwitz stable

Lemma 1: Under hypotheses Hle-fand H3 the asymptotic observer Equation (3) is stable. That is, i , ( t ) converges asymptotically towards xl(?) for any set of initial conditions. The proof of this lemma can be found in [ 111.

Remark 1: Up to now, with the exception of the existence of N;' , no other restrictions on N I have been introduced. However, without loss of generality, it will be assumed for simplicity that N , is proportional to the identity matrix (i. e . , N , = kIs where k is an arbitrary, real and positive constant parameter). Other reasons for this choice will be discussed below.

The Robust Interval Observer (NO) Let us recall the following result:

= f ( ( , t ) + g(t) . This system is said to be a cooperative system if af,(t,t)/a(, 20,Vi # j . It implies that if g(t) 2 0 \It 2 0 and f ( O ) 2 0, then

The proof of this lemma can be found in [12]. Now, let hypotheses H1f-S be verified. In other words, some bounds are now available for the initial conditions and b(t) is assumed to be unmeasured, but within known lower and upper (and possibly time varying) bounds. In such a situation, notice that the model of Equation( 1) may be no longer observable. Consequently, it is not possible to design an asymptotic observer such as Equation (3). Nevertheless, we can use its basic stable structure and its property of being independent of the nonlinearities, to design a robust set-valued observer in order to build guaranteed intervals for the unmeasured variables instead of estimating them precisely.

Lemma 2: Let

t ( t ) 2 0, vt 2 0 .

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'

'For the up er bound:

y(& = NxP"

For the lower bound:

y(& = N x P ) -

6' f ) = W()w+ (f)+ X(f)X, (f )+ Mv + (f)

x; (f) = N,' w + (t)- N 2 X 2 (f))

w - f ) = W(f)w - (f)+ x(t)X, (f)+ Mv - (f)

x; ( t ) = N ; , w - (1)- N 2 X 2 (f))

* . .(4)

is a stable robust interval observer for the system of Equation (l) , and guarantees that x; (f) I x, ( t ) I x,* (t), Vf 2 0 given x'(0) I x(0) I x+(O) if hypotheses H1-H3 hold.

Proof. Let ec = x,' -x, and e- = x, -x; be the observation errors associated to Equation (4) which are related to the unmeasured state variables for the upper bounds and for the lower bounds, respectively. For simplicity, we refer to e* as any of the errors e' or e- since their dynamics have the same mathematical structure; then, it is straightforward to show that:

i' = W,(t)e* + V * ...( 5 )

with V * = N;'(Mv+ - Nb) in the upper bound case and V * = N;'(Nb - M v - ) in the lower bound case. Since the RIO (Equation 4) is built on the structural basis of the asymptotic observer (Equation 3), its existence is ensured for hypotheses HI-H2. It is obvious (when considering the definitions of e+ and e - ) that if hypothesis Hlfholds, then e*(O) 2 0 . Thereafter, the positivity of V* is ensured by hypothesis Hlg and the choice of N,=kIs as established in the Remark 1. Thus, from hypotheses H3 and the application of Lemma 2, the system (Equation 5 ) is cooperative and stable. Therefore, it is guaranteed that e* 2 0, Vt 2 0, and thus, x; (f) I x, (f) I x: (t), Vf 2 0 .

Application to a Wastewater Treatment Process An Anaerobic Digestion Model Anaerobic digestion is a very complex biological wastewater treatment process in which organic matter, represented by the Chemical Oxygen Demand (COD) is degraded into a gas mixture of methane (CHd and carbon dioxide (COz). The biological scheme involves several multi-substrate multi-organism reactions that are performed both in series and in parallel [ 131. This biotechnological process is a highly nonlinear time-varying system in which the kinetic parameters are badly or poorly

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V. Alcaraz-Gonzalez, J. Harmand, A . Rapaport, et al.

known. Furthermore, the lack of sensors for the key variables makes it very difficult to monitor and control. Here, we consider a model of an anaerobic digestion process carried out in a continuous fixed bed bioreactor for the treatment of agricultural wastewater [ 141, where the input concentrations are unknown and only the interval on which they are likely to change is known. This model can be represented in the following matrix form:

t

-aD 0 0 0 0 0 0 -aD 0 0 0 0 0 0 - ( D + k , ) k , 0 - k , 0 0 0 - D 0 0 0 0 0 0 - D 0 0 0 0 0 0 - D

...( 6 )

or simply i = C’({(t),t)+ A(t){(t)+ b(t) which matches exactly model (Equation 1)

with x(t) = { ( t ) . In this model, 4, = X,, 6, = X2, 4, = S, , 4, = S, and 4, = C , are, respectively, the concentrations of acidogenic bacteria, methanogenic bacteria, COD, Volatile Fatty Acids (VFA) and total inorganic carbon. The variable 5, = Z is linked to the total alkalinity and represents the sum of strong ions in the medium. At chemical equilibrium at pH=7 it can be reasonably assumed that Z = S, + [HCO;] where [HCO;]is the bicarbonate ion concentration. It is supposed that no other ion significantly influences 2. D = D(t) corresponds to the dilution rate and it is supposed to be a persisting input, i.e. [ D(.r)d.r > 0. a (0 I a I 1) reflects the

hydrodynamic heterogeneity of the process: a = 1 corresponds to an ideal fixed bed reactor, whereas a= 0 corresponds to an ideal continuous reactor. Pco, is the C02 partial pressure. In all cases, the upper index i indicates “influent concentration”. The highly nonlinear nature of the anaerobic digestion process comes mainly from the reaction rates, which are given by:

... (7) s2 and P2 =Po Sl 4, +s, PI = P l n r n l ~

K,, +S2 + 2 1’ 2 72


Design of the Interval Observer It is straightforward to apply the observer scheme (Equation 4). By inspecting the system of Equation (6), it is evident that rank C = 2. Hence, from hypothesis H2, a minimum of two measured states is required to reconstruct the state space.

In this application they are the two substrate concentrations, S, and S2, which are then used as measured states in order to estimate guaranteed intervals on XI, X , CTI and Z. Thus, A m a y be split to yield:

A(t) = All All

A,, i A,,

. . . . . . .

-aD 0 0 o i o 0 -aD 0 0 . 0 0

0 - 0 ; 0 0

- ( D + k , ) k, i 0 -k,

. . . . . . ... . . . . . . . . . . 0 i - D 0

i 0 - D

...( 8)

which clearly shows that the only time sensitive variable is the dilution rate D which varies between 0.05 5 D(t) 11.15. Then, in agreement with hypothesis Hle, we set A - = AID*, I, and A' = AID=ao5 .

Without loss of generality, and in agreement with the Remark 1 , NI can be arbitrarily chosen as N , = I4 and hence, N takes the form:

1 N = [ N , ; N , ] = - klk,

.. .(9) 0 0 k,k, 0 i(k,k4 +k,k5) k,k,

0

rk,k, 0 0 0 i k, 0 k,k, 0 0 i k,

0 0 0 k,k,i 0

while W(t) and W,(t) are given by:

On the other hand, X(t) and M are represented, respectively, by:

...( 10)

.. . ( 1 1 )

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V. Alcaraz-Gonzdlez, J. Harmand, A. Rapaport, et al.

1 0 0 0 klk3 0 0 0

Note that hypothesis H3a is automatically fulfilled. Furthermore, due to hypothesis Hle, we may choose W; and We+ as W; = WeID=l Is and We. = WelD=ao5, respectively, such that hypothesis H3b is also fulfilled. Note also that the Lemma 2 holds.

Estimation Results The RIO was tested on-line in a series of experimental runs conducted over a 35-day period in a 1000-litre upflow anaerobic fixed bed reactor for the treatment of industrial wine vinasses obtained from local distilleries in Narbonne, France (see Figure 1). The matrices of Equations (8) to (12) involved in the design of the RIO were evaluated by using the parameters listed in Table 1. For the experimental implementation of the observer, the influent concentrations were considered unknown and only a known boundary region was supplied for the estimation calculations. Figure 2 depicts the corresponding bounded intervals for SI i , S,' and 2.

R-1 V-1 V-2 E-1 K1 F-1 s- 1 FixabBsd Dilution NaOH Hial Wdar Uhrdilmion G8r-Uquid Bioneclor Syrlam Storago Tank ExchangH Haalar Syatam Phase Sapardor

P-1 P-1A P-2 P-3 P-4 P-5 Feeding Mixing Recycling NaOH Fsiding NaOH Mixing Filland Pump Pump Pump Pump lo Dilulinn Pump lo Returning

Systsm Rscycling Pump

Figure 1. Schematic representation of the anaerobic fuced bed bioreactor.

2 74


ir, f il I*’

50

-

2 75

V. Alcaraz-Gonzalez, J. Hannand, A. Rapaport, et al.

For comparison purposes and to show that hypothesis Hlg holds, we also added the off-line readings (taken at irregular time periods) of these inputs. It is worth mentioning that ths data set was not used in the on-line implementation of the RIO. The variables CT/, XI’ and X,’ were assumed negligible. We also introduced in the RIO calculations actual experimental data for the dilution rate, D(t), as well as the partial C02 pressure which were measured on-line in the anaerobic fixed bed reactor. These on-line readings are shown in Figure 3. Substrate concentration measurements, SI and S2 (shown in the Figure 4) were available from infrequent on-line assays. For the RIO implementation, these measurements were assumed to remain constant between the assays.

Table 1. Model parameters @om [14]).

Parameter Meaning kl Yield coefficient for COD degradation

Yield coefficient for fatty acid production Yield coefficient for fatty acid consumption Yield coefficient for C02 production due to XI Yield coefficient for COZ production due to Xz Yield coefficient for CH4 production Liquidgas transfer rate Proportion of dilution rate for bacteria Maximum acidogenic biomass growth rate Maximum methanogenic biomass growth rate Saturation parameter associated with SI

Value 12.1 kg COD/kg XI

18 1.2 mol VFAkg XI 1640 mol VFA/kg Xz

169 mol COz/kg XI 273 mol C02/kg X2

1804 mol CH4/kg Xz 200 d-‘ 0.5 (dimensionless)

1.25 d-‘ 0.85 d-‘ 7.65 kg COD/m3

KS2 Saturation parameter associated with Sz 18 (mol V F N ~ ~ ) ’ ” K,, Inhibition constant associated with S, 25 mol VFA/m3

Figure 5 shows the estimation results for the intervals of the unmeasured states CTI and Z. Note how the interval bounds estimated by the interval observer envelop correctly these unmeasured states. For all the other unmeasured states, the RIO showed excellent robustness and stability properties and provided satisfactory estimation results in the event of highly corrupted measurements and operational failures. Observe, in particular, the robustness of the FUO around day 25 when the inlet concentrations drastically increased and when a major disturbance occurred at day 3 1, due to an operational failure, resulting in a rapid fall of both the dilution rate (which actually fell to zero) and the substrate concentration readings. Off-line readings of cT1 and Z (not used in the state estimation calculations) were also added to validate the proposed RIO design (see Figure 5).

However, in this application, the convergence rate cannot be tuned by the user since it is determined by the operational conditions. Therefore, the compromise between the convergence rate and robustness is not fully achieved until the estimation error dynamics reach steady state. Further studies are now being conducted to provide results of this interval observer with tunable gains.

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200

160

120 f i

80 E

40 u

z 3 w

F:

0

Time ( d )

Figure 5. Alkulinity and total inorganic carbon concentrations. Estimated upper and lower bounds ( j and off-line readings (0).

Conclusions In this paper a robust interval observer for a general class of lumped time varying nonlinear models useful in chemical and biochemical engineering processes is proposed. Conditions on the general structure of the model as well as on the stability were established for designing such an observer. This observer was satisfactorily tested on-line in a pilot plant used for the treatment of industrial distillery wine vinasses. The interval observer exhibited excellent convergence and stability properties, and predicted correctly the dynamical bounds where the unmeasured states actually evolve. The performance of the robust interval observer during large disturbances is particularly encouraging in that the change of operating conditions is corrected well in spite of process failures, input uncertainties and sensor malfunctioning.

Acknowledgments For the support that made this study possible, the authors grateNly acknowledge the ECOS-ANUIES Program (Project: M97-B01), the French ADEME/AGRICE Program (No DVNAC 9901028 W A D E M E / AGRICE Contract), the PROMEP Program, and the TELEMAC project (IST-2000-28156).

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Linearization. IEEE Trans. Aufomatic Confrol., Vol. AC-3 1, No. 1. 2. Alvarez, J. 2000. Nonlinear State Estimation with Robust Convergence. J. Process Confrol., 10,SO-71.

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3. Gauthier, J.P., Hammouri, H., and Othman S. 1992. A Simple Observer for Nonlinear Systems. Aplication to Bioreactors. IEEE Transactions on Automatic Control., 37(6), 875-880.

4. Bastin, G., and Dochain, D. 1990. On-line estimation and adaptive control of bioreactors. Elsevier Science Publisher B.V., The Netherlands.

5 . Walcot, B., and Zak, S. H. 1986. Observation of Dynamical Systems in the Presence of Bounded NonlinearitieslLIncertainties. 25th Conference on Decision and Control.. Athens, Greece, pp.961-966.

6. Misawa, E.A., and Hedrick, J.K. 1989. Nonlinear Observers. A State-of-the-Art. Survey, Trans.,

7. Skelton, R.E. 1988. Dynamic Systems Control: Linear systems analysis and synthesis. John Wiley, New York.

8. Rapaport, A., and Harmand, J. 1998. Robust Regulation of a Bioreactor in a Highly Uncertain Environment. IFAC-EurAgEng International Workshop on Decision and Control in Waste Bio- Processing, WASTE-DEClSlON'98.. Narbonne, France, 8 pages (CD-ROM).

9. Gouzt, J.L., Rapaport A,, and Hadj-Sadok, M.Z. 2000. Interval observers for uncertain biological systems. Ecological Modelling., Vol. 133, pp.45-56.

10. Chen, L. 1992. Modelling, Identifiability and Control of Complex Biotechnological Systems., PhD Thesis. Universitt Catholique de Louvain, Louvain, Belgium.

11. Alcaraz-Gonzhlez, V., Harmand, J., Dochain, D., Rapaport, A,, Stller, J.P., Pelayo Ortiz, C., and Gonztilez-Alvarez, V. 2003. A Robust Asymptotic Observer for Chemical and Biochemical Reactors. ROCOND 2003, Milan, Italy, 6 pages (CD-ROM).

12. Smith, H.L. 1995. Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, AMSMathernatical Surveys and Monographs, Vol. 41, pp.3 1-53.

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14. Bernard, O., Dochain, D., Genovesi, A., Perez-Alvarino, A.D., Steyer, J.P., and Lema, J. 1998. Software sensor design for an anaerobic wastewater treatment plant. IFAC-EurAgEng Internotional Workshop on Decision and Control in Waste Bio-Processing. WASTE-DECISIONPB., Narbonne, France, 8 pages (CD-ROM).

ASME., Vol. 11 1, pp.344-352.

Received 17 March 2004; Accepted after revision: 15 June 2004.

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