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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