# ADM1-Based Robust Interval Observer for Anaerobic Digestion Processes

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Jose Luis Montiel-Escobar

Vctor Alcaraz-Gonzalez

Hugo Oscar Mendez-Acosta

Victor Gonzalez-Alvarez

Universidad de Guadalajara, CUCEI,

Guadalajara, Jalisco, Mexico

Research Article

ADM1-Based Robust Interval Observer forAnaerobic Digestion Processes

A robust state estimation scheme is proposed for anaerobic digestion (AD) processes to

estimate key variables under the most uncertain scenarios (namely, uncertainties on

the process inputs and unknown reaction and specific growth rates). This scheme

combines the use of the IWA Anaerobic Digestion Model No. 1 (ADM1), the interval

observer theory and a minimum number of measurements to reconstruct the unmea-

sured process variables within guaranteed lower and upper bounds in which they

evolve. The performance of this robust estimation scheme is evaluated via numerical

simulations that are carried out under actual operating conditions. It is shown that

under some structural and operational conditions, the proposed robust interval

observer (RIO) has the property of remaining stable in the face of uncertain process

inputs, badly known kinetics and load disturbances. It is also shown that the RIO is

indeed a powerful tool for the estimation of biomass (composed of seven different

species) from a minimum number of measurements in a system with a total of 32

variables from which 24 correspond to state variables.

Keywords: Biochemical process; Biomass estimation; Nonlinear systems; State estimation

Received: December 22, 2011; revised: April 12, 2012; accepted: May 4, 2012

DOI: 10.1002/clen.201100718

1 Introduction

Major problems exist in the anaerobic digestion (AD) processes

(paralleled in the chemical and food-processing industries) con-

cerned with the on-line estimation of parameters and variables that

determine the process behaviour. The fundamental problem is that

the key variables cannot be measured at a rate that enables their

efficient regulation. Often a variable of interest must be determined

indirectly from other measurable properties and even if a variable is

easily measured, its valuemay be corrupted by the presence of noise.

Furthermore, time delays which may accompany certain measure-

ments also pose serious control problems which can lead to insta-

bility of the controlled process. Besides the measurement problems,

the process itself may be subject to random, un-modelled upsets

which must be considered and dealt with in order to achieve satis-

factory control of AD processes [1]. A method particularly suited for

this purpose is the proposal of on-line estimation schemes also

known as software sensors.

The idea behind state estimation is to optimally determine (in

some sense) the values of the process states based upon themeasured

variables and a dynamic model of the process which is used to infer

un-measurable states and to predict the process states between

measurements. Thus, the success of state estimators depends

strongly on the accuracy of the model. For biochemical processes,

manymathematicalmodels at the cell level have been developed and

used to predict substrate consumption, cell growth and cell compo-

sition, product formation, etc. [25]. The progress in understanding

of cellular metabolic processes and the regulation system structure

for specific pathways have made it possible to establish mechanistic,

structured models including many of the fundamental processes

involved in cellular metabolism of complex biochemical processes.

In the particular case of AD, the International Water Association

(IWA) has created a task group for mathematical modelling of

anaerobic process [6], with the goal to construct a common platform

for AD processes modelling and benchmarking and to increase model

application in research, development and optimization of such a

process. The resulting model was the IWA Anaerobic Digestion

Model No. 1 (ADM1), which comprises the several stages in bio-

chemical and physico-chemical processes occurring in AD [7].

Biochemical processes include substrate disintegration, hydrolysis,

acidogenesis, acetogenesis and methanogenesis carried out by seven

bacterial groups, whereas physico-chemical processes take into

account ion associationdissociation and gasliquid transfer aspects.

On the other hand, there has been an increasing interest in recent

years to develop new state and parameter estimation schemes to

reduce the deficiencies of classical state estimators (namely, Kalman

filters and Luenberger observers) encountered in areas like process

control. New state estimators, called asymptotic observers (AO) and

the adjustable asymptotic observers (AAO) have been designed and

implemented in several wastewater treatment processes in the state

estimation of key variables that are further applied in efficient

control schemes. However, their success have been limited since

these observers require the full knowledge of all the input variables

of the plant that otherwise, may lead to a non-observable and even

undetectable system. In order to overcome this problem, a class of

Correspondence: V. Alcaraz-Gonzalez, Departamento de Ing. Qumica,Universidad de Guadalajara CUCEI, Blvd. Marcelino Garca Barragan1451, 44430 Guadalajara, Jalisco, MexicoE-mail: victor.alcaraz@cucei.udg.mx

Abbreviations: AD, anaerobic digestion; AAO, adjustable asymptoticobservers; AO, asymptotic observers; IO, interval observers; ODE,ordinary differential equations; RIO, robust interval observer; WWTP,wastewater treatment process

Clean Soil, Air, Water 2012, 40 (9), 933940 933

2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.clean-journal.com

observers, named interval observers (IO), have been recently pro-

posed to deal with the estimation problem in lumped AD processes

described by ordinary differential equations (ODE) [8, 9] and

extended to distributed parameter systems [10]. The main charac-

teristics of the IOs is that they are able to give guaranteed interval

estimations of the state variables rather than the exact estimation of

them, if an upper and a lower bound (i.e., an interval) for each one of

the unmeasured process inputs are given.

In this contribution, we devise a robust interval observer (RIO)

based on the ADM1 model to reconstruct 14 state variables of the

ADM1 model using only 10 measured state variables. This robust

observer is capable of coping simultaneously with the problems

posed by both the uncertainties in the process inputs and the lack

of knowledge of the nonlinearities. We show that, under some

structural and operational conditions, the RIO has the property of

remaining stable under the influence of time varying parameters,

system failures, load disturbances, unknown kinetics and inputs.

Based upon the work of [8, 11, 12], existence conditions of this

observer are derived by assuming that only guaranteed lower and

upper limits on both process inputs and initial conditions are avail-

able. In Section 2, a generalized model is firstly presented in order to

be used as a basis of construction of the RIO under a mathematical

point of view. Thus, some hypotheses as well as some structural and

operational conditions are stated and then the general form of

the AO-based RIO is shown. In Section 3, the RIO is adapted and

implemented by numerical simulations to the ADM1 model, whose

results are also depicted and discussed in this section. Finally, some

conclusions and perspectives are made.

2 Materials and methods

2.1 A generalized model

The following general nonlinear time-varying lumped model is

introduced

_xt C fxt; t Atxt bt (1)

where x(t)2Rn is the state vector, C2Rnr represents a matrix ofconstant coefficients (e.g., stoichiometric or yield coefficients) while

f(x(t),t)2Rr denotes the vector of nonlinearities corresponding toprocess kinetics. The state matrix is represented by the time varying

matrix A(t)2Rnn and finally, b(t)2Rn groups process inputs (e.g.,mass feeding rate) and/or other possibly time varying functions (e.g.,

gaseous outflow rate).

The partial knowledge and the uncertainties of the system are

expressed in the following hypotheses:

Hypotheses H1 [9, 13, 14]:

(a) A(t) is known and bounded 8t 0; i.e., there exist two constantmatrices A and A such as A At A

(b) C is known and constant. Additionally, it is considered that rank

C2 rank C in Eq. (2)(c) Initial conditions of the state vector are unknown but guaran-

teed bounds are given as: x0 x0 x0(d) The input vector b(t) is unknown but guaranteed bounds,

possibly time varying, are given as: bt bt bt(e) m states that variables are measured on-line.

Note: Inequalities in hypotheses H1 ad should be understood as

element-by-element.

From hypothesis H1e, Eq. (1) can be split in the following

form:

_x1t C1 fxt; t A11tx1t A12tx2t b1t_x2t C2 fxt; t A21tx1t A22tx2t b2t

(2)

where x12Rs, (with s nm) represents the vector of variables tobe estimated while x22Rm represents the m state variables thatare measured. Matrices A11(t)2Rss, A12(t)2Rsm, A21(t)2Rms,A22(t)2Rmm, C12Rsr, C22Rmr, b12Rs and b22Rm are thecorresponding partitions of A(t), C and b(t), respectively.

2.2 The robust interval observer

The main requirements for the application of the proposed

observer scheme are [9, 13, 14]: (i) the existence of a known-input

observer (in the present case, an AO was chosen because of

its robustness against the badly known process kinetics [11]);

(ii) an interval in which initial conditions as well as the

non-measured inputs of the process evolve (these requirements

are fulfilled with the accomplishment of hypotheses H1ad

[9, 13, 14]); (iii) a system property called cooperativity must hold

[15]. This last property consists basically in that all the elements

of the state matrix of a system are all negative/zero or all positive/

zero. The direct consequence of this property in the proposed

estimation approach is that the RIO estimates guaranteed

intervals of the unmeasured states instead of their exact values.

Needless to say, the actual values are inside the estimated

intervals.

Now, rather than providing the lengthy derivation of the RIO, the

key assumptions and requirements for its application are discussed

in this section. For this purpose an IO is devised by assuming that

nonlinearities f(x(t),t) are fully unknown and that both, the input

disturbances and the initial conditions, are also unknown, but

bounded. Thus, under hypotheses H1, the following set-valued RIO

[9, 13, 14]:

For the upper bound:

_wt Wtwt Xtx2t Mvtw0 N x0

x^1 t N11 wt N2x2t

For the lower bound:

_wt Wtwt Xtx2t Mvtw0 N x0

x^1 t N11 wt N2x2t

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

(3)

with

M N1...N2

..

.~N2; ~N2 jN2;ijj;

vt b1 t1

2b2 t b2 t

1

2b2 t b2 t

T

vt b1 t1

2b2 t b2 t

1

2b2 t b2 t

T

Xt N1A12t N2A22t WtN2

934 J. L. Montiel-Escobar et al. Clean Soil, Air, Water 2012, 40 (9), 933940

2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.clean-journal.com

where N12Rss is an arbitrary invertible matrix, N2 N1C1C12with N22Rsr, C12 is the generalized pseudo-inverse of C2,

N N1...N2 and Wt N1A11t N2A21t N11 is cooperative

[15]; guarantees that x1 t x1 t x1 t ; 8t 0. Further detailsabout the design and construction of this observer can be found

in [9, 13, 14].

3 Results and discussion

3.1 Application to an anaerobic digestion process

The AD system considered in this paper consists of a liquid-phase

continuous stirred tank reactor (CSTR) type bioreactor, with a single

input and a single output stream. Thus, according with ADM1, this

_j1

..

.

_j13

..

.

_j22

_j23

_j24

2666666666666666664

3777777777777777775

C1

C2

264

375

r5

r6

r7

..

.

r11

r12

26666666666664

37777777777775

A11

..

.A12

. . . : . . .

A21...

A22

266664

377775

j1

..

.

j13

..

.

j22

j23

j24

266666666666666664

377777777777777775

Dji1

..

.

Dji13 kLaSHCO3 KHCO2Pgas;CO2 ...

Dji22 16kLa KH;H2Pgas;H2Dji23 64kLa KH;CH4Pgas;CH4

Dji24

2666666666666666664

3777777777777777775

(4)

or simply _jt Crjt; t Atjt bt which matches exactly Eq. (1) with xt jt.

jt x1...x2T

with x1t XC XCH Xpr Xli Xsu Xaa Xfa XC4 Xpro Xac XH2 XI SIC SIN T

x2t Ssu Saa Sfa Sva Sbu Spro Sac SH2 SCH4 SI T

The partitions of matrices A(t) and C are, respectively, given by:

A11

D kdis 0 0 0 kdec;Xsu kdec;Xaa kdec;Xfa kdec;XC4 kdec;Xpro kdec;Xac kdec;XH2 0 0 0fch;xckdis D khyd;CH 0 0 0 0 0 0 0 0 0 0 0 0fpr;xckdis 0 D khyd;pr 0 0 0 0 0 0 0 0 0 0 0fli;xckdis 0 0 D khyd;li 0 0 0 0 0 0 0 0 0 0

0 0 0 0 D kdec;Xsu 0 0 0 0 0 0 0 0 00 0 0 0 0 D kdec;Xaa 0 0 0 0 0 0 0 00 0 0 0 0 0 D kdec;Xfa 0 0 0 0 0 0 00 0 0 0 0 0 0 D kdec;XC4 0 0 0 0 0 00 0 0 0 0 0 0 0 D kdec;Xpro 0 0 0 0 00 0 0 0 0 0 0 0 0 D kdec;Xac 0 0 0 00 0 0 0 0 0 0 0 0 0 D kdec;XH2 0 0 0

fxI;xckdis 0 0 0 0 0 0 0 0 0 0 D 0 0s1kdis s2khyd;CH s3khyd;pr s4khyd;li s13kdec;Xsu s13kdec;Xaa s13kdec;Xfa s13kdec;XC4

s13kdec;Xpro s13kdec;Xac s13kdec;XH2 0 D kLa 0ankdis 0 0 0 bnkdec;Xsu bnkdec;Xaa bnkdec;Xfa bnkdec;XC4 bnkdec;Xpro bnkdec;Xac bnkdec;XH2 0 0 D

26666666666666666666666666666666666664

37777777777777777777777777777777777775

A12 0

A21

0 khyd;CH 0 1 ffa;likhyd;li 0 0 0 0 0 0 0 0 00 0 khyd;pr 0 0 0 0 0 0 0 0 0 0

0 0 0 ffa;likhyd;li 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

fsI;xckdis 0 0 0 0 0 0 0 0 0 0 0 0 0

26666666666666666666664

37777777777777777777775

Clean Soil, Air, Water 2012, 40 (9), 933940 ADM1-Based Robust Interval Observer 935

2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.clean-journal.com

system can be represented in the following matrix form:

with

an Nx fxI;xcNI fsI;xcNI fpr;xcNaabn Nbac NxcA122R1410

In this model, Xj and Sj denote the concentrations of the differ-

ent bacterial populations and all the other chemical and biological

species that are present in the system, respectively. The units

for all state variables are given in kg COD/m3 except those of SINand SIC whose units are, respectively, kmol N/m

3 and kmol C/m3.

In all cases, the upper index i or in, indicates influent concen-

tration. Pgas,k denotes the partial pressure of the kth gas while

DD(t) qin/Vliq 0 is the dilution rate (d1). Vector rjt; t 2R8includes all the highly nonlinear functions that describe the bio-

chemical reactions in the system, including specific biomass

growth rates. We have to point out that the elements of r are

r5 to r12 as they appear in the original ADM1model description (we

prefer to keep this description to facilitate the easier reading and

interpretation of this contribution among readers familiarized

with ADM1 model). Thus, the matrix product Cr of Eq. (4) is

consistent. Detailed definitions of the different functions and their

values, as well as equations for the gas phase and ionic balance can

be found in the technical report of the ADM1 [6] and in [16]. The

complete list of ADM1 state variables considered in this contri-

bution is shown in Tab. 1.

Table 1. State variables in ADM1

Description Statesymbol

Stoichiometricunit

Soluble inerts Si kg COD/...

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