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:

    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

  • 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


    _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


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


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


    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