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This paper is concerned with the online state estimation of a class of one-dimensional semilinear partial differential equation (PDE) systems considering piecewise measurements over the spatial domain. In the context of infinite-dimensional linear systems, it is well-known that the Kalman filter minimizes the mean square estimation error. For semilinear infinite-dimensional systems, the extended Kalman filter (EKF) is a widely used extension relying on successive linearizations of the estimation error dynamics. In this paper, we propose a computationally tractable implementation of the EKF using a sample-and-hold approach for which the optimal output injection operator associated to the proposed estimator is computed at each sampling time via the approximate solution of the infinite-dimensional Riccati equation. The performance of the observer is exemplified through numerical experiments which demonstrate the efficiency of the proposed approach.

The identification of accurate models for the anaerobic digestion process is essential for the characterization of the region that guarantees the conservation of the bacteria population. Traditional techniques involve the identification of nonlinear models based on data from the system. In this paper, we introduce data-driven techniques that allow the characterization of the system’s behavior via the approximation of the Koopman operator with the extended dynamic mode decomposition algorithm. We propose methods to reduce the order and dimension of the representation based on orthogonal polynomials.

Metabolic flux analysis is often (not to say almost always) faced with system underdeterminacy. Indeed, the linear algebraic system formed by the steady-state mass balance equations around the intracellular metabolites and the equality constraints related to the measurements of extracellular fluxes do not define a unique solution for the distribution of intracellular fluxes, but instead a set of solutions belonging to a convex polytope. Various methods have been proposed to tackle this underdeterminacy, including flux pathway analysis, flux balance analysis, flux variability analysis and sampling. These approaches are reviewed in this article and a toy example supports the discussion with illustrative numerical results., info:eu-repo/semantics/published

Geometric integrators allow preservation of specific geometric properties of the exact flow of differential equation systems, such as energy, phase-space volume, and time-reversal symmetry, and are particularly reliable for long-run integration. In this study, variable step size composition methods and Gauss methods are implemented in Matlab library integrators, and are tested with several representative problems, including the Kepler problem, the outer solar system and a conservative Lotka–Volterra system. Variable step size integrators often perform better than their fixed step size counterparts and the numerical results show excellent long time preservation of the Hamiltonian in these examples.

This paper investigates state and unknown input estimations in an Anaerobic Digestion Reactor (ADR) considering a simple two-stage reaction model describing acidogenesis and methanogenesis. From the sole methane outlet flow, this model allows to predict the inlet and outlet biodegradable suspended solid and volatile fatty acid concentrations, which are critical for process stability, as well as acidogenic and methanogenic biomasses. Continuous–discrete exogenous and unknown input formulations of the Extended Kalman Filter (EKF) are applied in this context and a numerical robustness analysis is achieved to characterize sensitivities to measurement noise and model uncertainties. The resulting observers are then validated experimentally with data from a lab-scale plant.

Underdetermined metabolic networks are usually investigated using Flux Variability Analysis (FVA) under the pseudo-steady state assumption of the internal metabolites. When a dynamic overview of the flux map is sought, the time variation of the cell uptake and excretion rates is deduced from extracellular dynamic mass balances and smoothing of the measurement data of the time evolution of the extracellular concentrations. Nevertheless, the resulting system of equations does not always admit a feasible solution under these constraints. Indeed, measurement data is affected by noise, whose processing (smoothing, etc) always entails some subjective user choices, and the network itself might not be perfectly suited to explain the several culture phases. To alleviate these adverse effects, the constraints can be relaxed by introducing coefficients of variation of the external fluxes, and by considering an interval representation of the fluxes. This work presents a systematic method to determine these coefficients of variation, along the time course of the culture, leading to an adaptive scheme where the coefficients are set to the tightest bounds. The methodology is applied to experimental data of cultures of hybridoma in batch and perfusion modes and compared to previously published results., SCOPUS: cp.phttps://www.elsevier.com/journals/ifac-papersonline/2405-8963/open-access-journal, info:eu-repo/semantics/published

The extended dynamic mode decomposition algorithm is a tool for accurately approximating the point spectrum of the Koopman operator. This algorithm provides an approximate linear expansion of non-linear discrete-time systems, which can be useful for system analysis and controller design. The accuracy of this algorithm depends heavily on the availability of a set of basis functions that provide the ability to capture the nonlinear dynamics of the underlying system. Recently, the use of orthogonal polynomials, along with reduction techniques for the dimension and maximum order of the polynomial basis, have been successfully used to approximate nonlinear systems with the additional benefit of using smaller datasets. This paper expands the current methods for selecting the set of observables for nonlinear systems with periodic behavior, which is prone to a representation in terms of trigonometric functions. The benefit of working with orthogonal polynomials is preserved by embedding the trigonometric functions into the orthogonal basis. The algorithm is illustrated with the data-driven modelling of an inverted pendulum in simulation and real-life experiments.

This article deals with the development of a reduced model describing the temporal evolution of the fructo-oligosaccharide production by Aureobasidium pullulans and the optimal process control. First, a reduced model is derived from a detailed model reproducing with good accuracy the dynamics of the fructo-oligosaccharide production. The reduced model is obtained using maximum likelihood principal component analysis and parameter identification based on a weighted least squares criterion. Next, the fructo-oligosaccharide concentration at an a priori undetermined time is maximized using Pontryagin maximum principle. The methodology is analyzed based on experimental data from batch and fed-batch cultures, and results are compared with those obtained with another simple model available in the literature.

The region of attraction of nonlinear dynamical systems is a critical factor in the analysis of equilibrium points. To have an approximation of the region of attraction, we propose the construction and analysis of eigenfunctions coming from the Koopman operator. The construction of eigenfunctions with associated unitary eigenvalue along with the location and local stability of equilibrium points provides a method to find the stable manifolds of saddle points, and in turn, the region of attraction of asymptotically stable points. The method relies on data collected from the trajectories of the underlying system and does not require the knowledge of the differential equation.

In this work, two metabolic networks representing the metabolism of CHO cells in fed-batch cultures are considered. The first metabolic network is relatively detailed and underdetermined with the available extracellular measurements, while the second is a reduced version of the former and is overdetermined. A dynamic metabolic flux analysis based on convex analysis (DMFCA) is applied to the detailed network, which allows the computation of the time evolution of bounded intervals. On the other hand, a linear optimization problem is solved for the reduced-size network, with either positivity constraints or box constraints inferred from DMFCA. In all cases, smoothing splines and mass balance differential equations are used to infer the time evolution of the uptake and excretion rates from experimental data. The analysis allows to get insight into CHO metabolism as well as to investigate the influence of the size of the metabolic network.

In this study, the steady states of an anaerobic digestion model are analytically computed and characterized. The influence of the dilution rate on the system states and methane outflow rate is illustrated by means of one-parameter bifurcation diagrams. A two-parameter bifurcation diagram is also computed. This diagram illustrates the combined effect of the dilution rate and inlet substrate concentration on the system steady states and allows to determine corrective actions (in terms of selecting appropriated inputs) for preserving system stability and efficiency. Expressions for all the boundaries and critical points appearing in the diagram are provided. Furthermore, the complex system behavior is illustrated by means of phase portraits. The paper emphasizes the methodology, which can also be applied to other bioprocesses taking place in continuously stirred tank reactors. The method allows practitioners to build two-parameter bifurcation diagrams by solving a set of algebraic equations. This implies low computational requirements.

We propose a robust state estimation approach for a linear reaction-convection-diffusion equation under bounded unknown disturbances. Inspired by sliding mode theory, an adequate discontinuous input function is designed to compensate for the effect of the unknown disturbances. Based on Filippov's solutions theorem, we report the existence of generalized solutions to the estimation error system subject to the discontinuous input. Based on a Lyapunov stability analysis, we show the asymptotic convergence of the estimation error. The observer is then designed under more relaxed and realistic assumptions by replacing the discontinuous input by a continuous approximation and by using adaptive techniques to compensate for the upper bound on the bounded disturbances which are rather assumed to be unknown. Numerical simulations are performed to illustrate the effectiveness of the proposed robust estimation approach.

This paper presents a framework for the multivariable robust control of perfusion animal cell cultures. It consists of a cascade control structure and an estimation algorithm, which provides the unmeasurable variables needed in the design of the control law, and ensures the regulation of the cell and glucose concentrations at imposed levels by manipulating the bleed and the dilution rates. The cascade control structure uses a feedback linearizing controller in the inner loop and linear (PI) controllers in the outer loops, and requires the measurement of the cell concentration and the glucose concentration in the bioreactor. Two approaches are provided: the first one assumes the availability of an approximate model of the process kinetics and uses an extended Kalman filter (EKF) to estimate the system states; the second approach does not require the prior knowledge of the process kinetics. These are estimated from the available measurements using sliding mode observers (SMO). A receding horizon optimization algorithm is employed to (periodically) tune the gains of the outer loop controllers. The proposed framework is easy to implement and tune, and may be applied to a general class of perfusion cell culture systems. Its effectiveness and robustness are illustrated by means of simulation results.

In this study, the problem of developing simple dynamic models of an anaerobic digestion process is tackled using an identification procedure which proceeds in several consecutive steps. Starting from sets of experimental data describing the time evolution of several key component concentrations, i.e. biomass(es), substrates and products, the minimal number of macroscopic bioreactions required to represent the data at hand, as well as the parameters of the associated stoichiometry matrix, are determined using maximum likelihood principal component analysis. Then, the structure of the kinetic laws, together with their parameters, are identified using likelihood ratio tests to navigate through the branches of decision trees made of various kinetic structures. The effectiveness of the modelling procedure is illustrated with a simulated example of anaerobic digestion. As usual sensors only provide aggregate measurements of the component concentration, a deeper investigation of practical identifiability is ach...

This paper presents a brief evaluation of a start-up strategy for multi-species anaerobic digestion systems modelled as two-step reaction systems, where acidogenesis is described by Monod kinetics while the methanogenesis is described by Haldane kinetics. The start-up policy has been developed originally for single species systems with the aim of maximizing the biogas outflow rate. It consists of switching the dilution rate from minimum to maximum and then to the optimal value (bang-bang control) in order to bring the system from an arbitrary initial condition to the optimal set-point. This start-up strategy is applied to the multi-species system using an averaged model, which is usually the only model that can be identified for a multi-species system, as measuring individual biomasses is almost impossible in practice. Even the development of an accurate averaged model, fully characterizing the system dynamics based on the variation of the species proportions is difficult. The averaged models used in this...

Dynamic modeling of bioprocesses is of fundamental importance for optimization and control. The underlying macroscopic models, i.e., reaction scheme stoichiometry and kinetics, have to be derived and identified from experimental data. This study addresses the problem of determining the kinetics, and proposes a systematic procedure based on the use of decision graphs collecting well-established kinetic laws, and organizing them according to their complexity (i.e., number of unknown parameters). An algorithm for selecting the most likely kinetic structures, starting from the simplest models in the decision graphs, is developed based on the concepts of nested models and likelihood ratio tests. The potential of the procedure is illustrated with the determination of the synthesis rate of a biopolymer from the mutant strain Azotobacter vinelandii utilizing glucose.

This study presents an effective procedure for the determination of a biologically inspired, black-box model of cultures of microorganisms (including yeasts, bacteria, plant and animal cells) in bioreactors. This procedure is based on sets of experimental data measuring the time-evolution of a few extracellular species concentrations, and makes use of maximum likelihood principal component analysis to determine, independently of the kinetics, an appropriate number of macroscopic reactions and their stoichiometry. In addition, this paper provides a discussion of the geometric interpretation of a stoichiometric matrix and the potential equivalent reaction schemes. The procedure is carefully evaluated within the stoichiometric identification framework of the growth of the yeast Kluyveromyces marxianus on cheese whey. Using Monte Carlo studies, it is also compared with two other previously published approaches.

The concept of elementary flux vector is valuable in a number of applications of metabolic engineering. For instance, in metabolic flux analysis, each admissible flux vector can be expressed as a non-negative linear combination of a small number of elementary flux vectors. However a critical issue concerns the total number of elementary flux vectors which may be huge because it combinatorially increases with the size of the metabolic network. In this paper we present a fast algorithm that randomly computes a decomposition of admissible flux vectors in a minimal number of elementary flux vectors without explicitly enumerating all of them.

In this work a new approach to the control of simulated moving bed (SMB) chromatographic separation processes is presented. This approach is based on the combination of the wave theory and Multi-Model Predictive Control (MMPC). The wave theory provides the theoretical framework in which the control law is formulated whereas receding-horizon MPC is used for determining the appropriate controller parameters. As SMB plants are distributed parameter systems (DPS) with nonlinearity arising from the expression of the adsorption isotherms, classical numerical methods for the solution of DPS are computationally demanding for MPC purposes. Reduced-order models are therefore derived using the proper orthogonal decomposition (POD) technique. To ensure stability, the POD model is updated on-line, resulting in MMPC.

The objective of this paper is to report on the development of a method of lines (MOL) toolbox within MATLAB, and especially, on the implementation and test of a moving grid algorithm based on the equidistribution principle. This new implementation includes various spatial approximation schemes based on finite differences and slope limiters, the choice between several monitor functions, automatic grid adaptation to the initial condition, and provides a relatively easy tuning for the non-expert user. Several issues, including the sensitivity of the numerical results to the tuning parameters, are discussed. A few test problems characterized by solutions with steep moving fronts, including the Buckley-Leverett equation and an extended Fisher-Kolmogorov equation, are investigated so as to demonstrate the algorithm and software performance.

Mathematical modeling of sedimentation has attracted considerable attention in the past decades, and nowadays, one of the most popular models of secondary settler in activated sludge processes is the one proposed by Takacs et al. [I. Takacs, G.G. Patry, D. Nolasco, A dynamic model of the clarification-thickening process, Water Res. 25 (10) (1991) 1263–1271]. This model is based on a discretization in finite volumes (or layers) of the spatial domain, and in a rather inconsistent way, the number of layers is usually considered as a model parameter chosen so as to fit experimental data. In this study, a simple convection–diffusion partial differential equation (PDE) model is first formulated and solved using a Method of Lines strategy allowing the use of various spatial discretization methods with largely improved accuracy and efficiency. Model parameters are estimated using experimental data collected in batch settling experiments by De Clercq [J. De Clercq, Batch and continuous settling of activated sludge: in-depth monitoring and 1D compression modeling, Ph.D. Thesis, Universiteit Gent, Faculteit Ingenieurswetenschappen, Belgium, 2006], showing the good model predictive capability. Finally, the PDE settler model is coupled with a standard ASM1 representation of the activated sludge process, and implemented in a MATLAB dynamic simulator.

The development and acceptance of model-based monitoring tools in the bioprocess industry is made difficult by the usually large uncertainty associated with the process model. A natural approach to handle this issue is the design of adaptive state observers for the joint estimation of the process state and some of the uncertain model parameters. However, the state extension is often restricted to a few parameters only, for which observability conditions are satisfied with the available measurement information. In this study, this latter issue is circumvented by the combination of two observers: (a) a receding-horizon observer is designed for the joint estimation of the state and uncertain model parameters, and (b) an asymptotic observer, which provides state estimates independently of the kinetic model, is used to provide the missing additional information to the receding-observer, thus avoiding observability loss. This paper derives the properties of this combined state estimation scheme and demonstrates its performance with a realistic simulation case study of animal cell cultures.

The optimization, monitoring and control of simulated moving bed (SMB) units require the use of a process model and the estimation of the model parameters. A systematic numerical procedure for determining parameters of SMB models from batch experiments is presented and evaluated. The unknown parameters are estimated by minimizing a cost function measuring the difference between experimental and simulated concentration profiles. In contrast with previous studies, parameter identifiability is studied and errors on the estimated parameters are calculated. A sensitivity analysis is used to design the experiments and to compare the identifiability of different chromatographic models. Then, the influence of local minima is evaluated by applying the numerical procedure on fictitious measurements generated from a model with known parameters.

Simultaneous perturbation stochastic approximation (SPSA) is a gradient-based optimization method which has become popular since the 1990s. In contrast with standard numerical procedures, this method requires only a few cost function evaluations to obtain gradient information, and can therefore be advantageously applied when identifying a large number of unknown model parameters, as for instance in neural network models or first-principles models. In this paper, a first-order SPSA algorithm is introduced, which makes use of adaptive gain sequences, gradient smoothing and a step rejection procedure to enhance convergence and stability. The algorithm performance is illustrated with the estimation of the most-likely kinetic parameters and initial conditions of a bioprocess model describing the evolution of batch animal cell cultures.

This paper is concerned with a pilot-scale fixed-bed biofilter used for nitrogen removal from municipal wastewater. Process modelling yields a set of mass balance partial differential equations describing the evolution of the component concentrations along the biofilter. Based on sets of experimental data collected over several months, unknown model parameters are estimated by minimizing an output-error criterion. The resulting distributed parameter model and a few pointwise measurements of nitrate, nitrite, and ethanol concentrations are then used to design observers allowing the unmeasured biomass concentrations to be reconstructed on-line. First, it is demonstrated that asymptotic observers are not applicable to the given model structure. Then, a receding-horizon observer is designed and tested, showing a very satisfactory performance.

In this paper, we report on the development of a MATLAB library for the solution of partial differential equation systems following the method of lines. In particular, we focus attention on upwind finite difference schemes and grid adaptivity, i.e., grid movement or grid refinement. Several algorithms are presented and their performance is demonstrated with illustrative examples including a fixed-bed reactor with periodic flow reversal, a model of flame propagation, and the Korteweg–de Vries equation.

In this paper, we investigate the dynamics and interaction properties of recently discovered “embedded solitons” in an extended fifth-order KdV model inspired by water waves in the presence of surface tension. The dynamical behaviour of the solitons can be efficiently followed by using a moving or an adaptive finite difference mesh in combination with a suitable time-integrator. We will demonstrate this numerically for different types of wave solutions, such as solitary waves, multihumped waves, and interacting waves.

In this study, a class of dynamic models based on metabolic reaction pathways is analyzed, showing that systems with complex intracellular reaction networks can be represented by macroscopic reactions relating extracellular components only. Based on rigorous assumptions, the model reduction procedure is systematic and allows equivalent `input–output' representations of the system to be derived. The procedure is illustrated with a few examples, and a comparison is made with another recently published method for generating and evaluating macroscopic reaction schemes.

In this study, we consider partial differential equation problems describing nonlinear wave phenomena, e.g., a fully nonlinear third order Korteweg-de Vries (KdV) equation, the fourth order Boussinesq equation, the fifth order Kaup–Kupershmidt equation and an extended KdV5 equation. First, we develop a method of lines solution strategy, using an adaptive mesh refinement algorithm based on the equidistribution principle and spatial regularization techniques. On the resulting highly nonuniform spatial grids, the computation of high-order derivative terms appears particularly delicate and we focus attention on the selection of appropriate approximation techniques. Finally, we solve several illustrative problems and compare our computational approach to conventional solution techniques.

Together with some on-line measurements, a reliable process model is the key ingredient of a successful state observer design. In common practice, the model parameters are inferred from experimental data so as to minimize a model prediction error, e.g. so as to minimize an output least-squares criterion. In this procedure, no care is actually exercised to ensure that the unmeasured model states are sensitive to the measured states. In turn, if sensitivity is too low, the resulting state observer will probably generate poor estimates of the unmeasured states. To alleviate these problems, a new parameter identification procedure is proposed in this study, which is based on a cost function combining a conventional prediction error criterion with a state estimation sensitivity measure. Minimization of this combined cost function produces a model dedicated to state estimation purposes. A thorough analysis of the procedure is presented in the context of bioreactor modeling, including parameter identification, model validation and design of extended Kalman filters and full horizon observers.

Anaerobic fermentation is an important process used for recycling solid organic waste, which leads to a significant reduction of the waste volume with the production of biogas as a positive side effect. For state observation and control purposes, a mathematical representation of the process is required. However, anaerobic fermentation is far too complex to be described in full metabolic details, due to the variety of responsible microorganisms and the unknown and time-varying waste composition. The level of complexity of the description is limited by the amount and quality of available experimental data, which can be used for model identification. In practice, the derivation of a dynamic process model involves the following steps: (i) the selection of suitable macroscopic reaction schemes and kinetic structures, (ii) the estimation of the unknown model parameters from experimental data by minimizing a maximum-likelihood criterion, (iii) the estimation of the unknown measurement variances, (iv) the estimation of the covariance matrix of the parameter estimates and (v) the validation of the obtained model. In this study, attention is focused on these several steps, and a dynamic model of a complex anaerobic process is inferred from infrequent measurements of global variables. The experimental data are obtained from six experiments carried out in a small-scale continuous bioreactor under different feed and (controlled) acidity conditions.

State estimation is a significant problem in biotechnological processes, due to the general lack of hardware sensor measurements of the variables describing the process dynamics. The objective of this paper is to review a number of software sensor design methods, including extended Kalman filters, receding-horizon observers, asymptotic observers, and hybrid observers, which can be efficiently applied to bioprocesses. These several methods are illustrated with simulation and real-life case studies.

In this chapter, finite difference (FD) approximations and the method of lines, which combine FD with available time integrators, are discussed. First, a convection diffusion-reaction PDE is used to introduce a few basic FD schemes and addresses the concept of stability of the numerical scheme. As a rule of thumb, centered FD schemes appear as good choices for approximating second-order (diffusion) operators, whereas upwind FD schemes are preferable to first-order (convection) operators. Once stability is ensured, accuracy can be adjusted by selecting appropriately the fineness of the spatial grid and the order of the FD approximation (i.e. the stencil of points on which the FD approximation is built). The calculation of FD approximation can be conveniently organized using differentiation matrices, which are easy to manipulate in MATLAB, SCILAB, or OCTAVE. FD can also be efficiently computed on arbitrarily spaced grids using an algorithm due to Fornberg. This chapter continues with a presentation of different methods to take the boundary conditions into account. Boundary conditions are an essential part of the IBVP definition, and several methods for translating the BCs in the code implementation are presented. Finally, attention is paid to the computation of the Jacobian matrix of the ODE system, which is used by various solvers.

In this chapter, the weighted residual methods are introduced. These methods represent the solution as a series of basis functions whose coefficients are determined to make the PDE (and BC) residuals as small as possible (in an average sense). In particular, attention is focused on the Galerkin and collocation methods, and the use of global and local basis functions, the latter leading to the famous Finite Element Method (FEM). Together with finite difference methods, FEM is one of the most popular methods for solving IBVP. Various basis functions can be used in conjunction with the FEM, especially Lagrange or Hermite polynomials. The construction of the FEM matrices is presented in detail for these two types of polynomials, when using a Galerkin method, and for Hermite polynomials, when using an orthogonal collocation approach. It is also possible to compute, using experimental or simulation data, optimal basis functions, i.e., the ones that are able to capture most of the solution features with a limited number of functions. This is the essence of the proper orthogonal decomposition (POD). Several examples are studied including the heat equation and the Brusselator.

Whereas the previous chapters are exclusively dedicated to lumped systems (systems of dimension 0 described by ODEs) and distributed parameter systems in one spatial dimension, this chapter touches upon the important class of problems in more space dimensions, as well as problems with time-varying spatial domains. Both are difficult topics and the ambition of this chapter is just to give a foretaste of possible numerical approaches. Finite difference schemes on simple 2D domains, such as squares, rectangles or more generally convex quadrilaterals, are first introduced, including several examples such as the heat equation, Graetz problem, a tubular chemical reactor, and Burgers equation. Finite element methods, which have more potential than finite difference schemes when considering problems in 2D, are then discussed based on a particular example, namely FitzHugh-Nagumo model. This example also gives the opportunity to apply the proper orthogonal decomposition method to derive reduced-order models. Finally, the problematic of time-varying domains is introduced via another particular application example related to freeze drying. The main idea here is to use a transformation so as to convert the original problem into a conventional one with a time-invariant domain.Whereas the previous chapters are exclusively dedicated to lumped systems (systems of dimension 0 described by ODEs) and distributed parameter systems in one spatial dimension, this chapter touches upon the important class of problems in more space dimensions, as well as problems with time-varying spatial domains. Both are difficult topics and the ambition of this chapter is just to give a foretaste of possible numerical approaches. Finite difference schemes on simple 2D domains, such as squares, rectangles or more generally convex quadrilaterals, are first introduced, including several examples such as the heat equation, Graetz problem, a tubular chemical reactor, and Burgers equation. Finite element methods, which have more potential than finite difference schemes when considering problems in 2D, are then discussed based on a particular example, namely FitzHugh-Nagumo model. This example also gives the opportunity to apply the proper orthogonal decomposition method to derive reduced-order models. Finally, the problematic of time-varying domains is introduced via another particular application example related to freeze drying. The main idea here is to use a transformation so as to convert the original problem into a conventional one with a time-invariant domain.

This paper focuses on modeling of a simulated moving bed process (SMB) dedicated to the separation of racemic mixtures. In the first approach, a true moving bed model is derived, which assumes an equivalent counter-current movement of the solid phase. The good agreement between the model and the real system is demonstrated with experimental results. Then, a more rigorous approach is developed, which considers the system as an arrangement of static chromatographic columns and takes into account periodic switching. Attention is focused on model formulation and numerical solution techniques in order to develop efficient dynamic simulation programs.

The method of lines (MOL) is a procedure for the numerical integration of partial differential equations (PDEs). Briefly, the spatial (boundary value) derivatives of the PDEs are approximated algebraically using, for example, finite differences (FDs). If the PDEs have only one initial value variable, typically time, then a system of initial value ordinary differential equations (ODEs) results through the algebraic approximation of the spatial derivatives. If the PDEs are strongly convective (strongly hyperbolic), they can propagate sharp fronts and even discontinuities, which are difficult to resolve in space. Experience has demonstrated that for these systems, some form of upwinding is generally required when replacing the spatial derivatives with algebraic approximations. Here we investigate the performance of various forms of upwinding to provide some guidance in the selection of upwind methods in the MOL solution of strongly convective PDEs.

Although the use of adaptive grid methods in one spatial dimension is now basically well understood and research is mainly devoted to problems in higher spatial dimensions, only a few methods are now available in the form of software packages and it may still be difficult for the user to find a path through the maze of methods published in the literature. In this paper, we examine five adaptive grid codes based on the method of lines, i.e., two local refinement algorithms and three moving grid algorithms. We illustrate and compare the use of the several methods with the numerical solution of a number of test problems from science and engineering, e.g., a model of a single-step reaction with diffusion, Burgers' equation, a model of flame propagation, and soliton solutions of the cubic Schrodinger equation. From the user point of view, we address important questions concerning the accuracy, temporal performance, ease of use (tuning) and simplicity of implementation.

This paper proposes a method of lines solution procedure with time and space adaptation for one-dimensional systems of partial differential equations whose solutions display steep moving fronts. The spatial remeshing algorithm, which is a variation of the method published by Sanz-Serna and Christie and an extension suggested by Revilla, is a static remeshing method based on equidistribution principles. The selection of the several algorithm components, i.e., grid placement criterion, spatial discretization scheme, time integrator, adaptation frequency, and parameter tuning, are investigated and illustrated with several test examples, i.e., the cubic Schrodinger equation, a model of a single-step reaction with diffusion, and a model of flame propagation.

Parameter estimation in distributed parameter systems is usually accomplished by minimizing an output least square criterion, which is defined implicitly through the solution of the model equations. This paper addresses itself to the numerical procedure used to compute the criterion gradient with respect to the unknown parameters. Several methods ranging from the straigthforward finite difference approximations to the more involved adjoint variable method are described and their relative merits are highligthed. An experiment design procedure based on the sensitivity matrix is presented. The methods for gradient computation and experiment design have been succesfuily applied to several test examples and are illustrated in this paper by a convective-diffusion problem.

This paper describes the interfacing of the simulation software DSS/2 and the optimization program OPTDES in order to build a PC environment for simulation and parameter estimation of dynamical systems with distributed parameters. The efficiency of these FORTRAN software tools is illustrated by a furnace modelling study. The nonlinear partial differential equations of the model are discretized by the finite-difference method and the orthogonal collocation on finite elements, and the simulation results are compared. It is shown that the use of differential-algebraic solvers to handle the boundary conditions and an effective grid point selection lead to a significant reduction of the result equations. The unknown model parameters are estimated from experimental data by minimizing an output least-square error criterion with a hybrid sequential quadratic programming-generalized reduced gradient algorithm.

The dynanlic behavior of an experimental multizone electrical fumace is described by nonlinear partial differential equations. On this basis, a nonlinear distributed observer of the load temperature profiles is designed. The resulting observer equations are discretized by orthogonal collocation on finite elements and implemented on-line on a PC. The observer shows very good properties: easy tuning, robustness and small computer resources. Some experimental results and computer simulations are presented and discussed. Moreover, a comparison is petfonned between the observer and a distributed extended Kalman filter, including on-line perfonnances and computational implementation aspects. It is shown that the observer requires less computational effort for sinlilar perfomances.

International audience; In this paper, two modelling approaches are proposed for wine-making fermentations. The first one is largely based on the first principle modelling approach and considers the main yeast physiological mechanisms. The model accurately predicts the fermentation kinetics of more than 80% of a large number of experiments performed with 20 wine yeast strains, 69 musts and different fermentation conditions. Thanks to the wide domain of validity of the model, a simulator based on this model coupled to a thermal model was developed to help winemakers to optimize tank management. It predicts the end of the fermentation and changes in the rate of fermentation but furthermore includes an optimization module based on fuzzy logic which allows, via temperature profiles and nitrogen addition strategies, to decrease the duration of fermentation and the energy requirements at winery scale according to user specifications. The objective of the second modelling approach is the development of a mathematical model of the fermentation process including some minority by-products known as characteristic flavour compounds. It refers to metabolic engineering and accounts for the intracellular behaviour of the yeast Saccharomyces cerevisiae by using approaches like the Metabolic Flux Analysis (MFA) and the Elementary Flux Modes (EFMs). A state of the art describes the application of these methods in the restrained field of wine-making/fermentation conditions and underlines the potential of such approaches.

This work applies the quasi-LPV technique to the design of robust observers for a class of bioreactors. The system nonlinearities are modeled in terms of two time varying parameter vectors, θ(t) and δ(t). The vector θ(t) contains all nonlinear terms that are only function of the measurements, whereas the remaining terms are lumped into the vector δ(t). Then, a θ(t) parameter-dependent Luenberger-like observer is proposed, where the design conditions are given in terms of linear matrix inequality constraints. These conditions ensure regional stability w.r.t. to a set of admissible initial conditions and also minimizes an upper-bound on the L 2-gain of the error system. These results are applied to a high cell density bioreator.

In this paper, attention is focused on a parabolic partial differential equation (PDE) modeling sedimentation in a secondary settler and the proper formulation of the problem boundary conditions (i.e., the conditions prevailing at the feed, clear water and sludge outlets). The presence of a diffusion term in the equation not only allows the reproduction of experimental observations, as reported in a number of works, but also makes the numerical solution of the initial-boundary value problem significantly easier than the original conservation law (which is a nonlinear hyperbolic PDE problem requiring advanced numerical techniques). A Method of Lines (MOL) solution strategy is then proposed, based on the use of finite differences or spectral methods, and on readily available time integrators. The efficiency and flexibility of the general procedure are demonstrated with various numerical simulation results.

Optimal steady-state reactor temperature profiles in classic jacketed tubular reactors often exhibit a (nearly) trapezoidal shape along the reactor, i.e., in a first part the temperature is raised, then a constant temperature interval follows, and (possibly) the temperature is lowered towards the outlet. The practical realisation of these optimal reactor temperature profiles is, however, complex because of the spatially varying jacket fluid temperature profile that is required for the constant reactor temperature part. Since similar trapezoidal temperature profiles are encountered in reverse flow configurations, this reactor type may be an alternative to implement the optimal profiles in practice. This article conceptually proves the feasibility of this concept and provides an optimisation procedure for its design and operation. (C) 2007 Elsevier Ltd. All rights reserved. [**] ispartof: Chemical Engineering Science vol:62 issue:17 pages:4675-4688 status: published

Secondary settlers, which ensure the separation of the activated sludge from the treated effluent, are described by distributed parameter models taking temporal and spatial variations into account. Over the years, a number of mathematical models have been proposed, including Kynch, Takacs and Hamilton models. The objective of this study is threefold: ( a ) to highlight the influence of the model formulation on the numerical solution procedure, ( b ) to propose a modified expression of the settling velocity, which avoids sharp spatial variations reported in the literature, and ( c ) to develop a numerical solution procedure following a method of lines strategy rather than the classical “tanks-in-series” approach.

In this study, a class of dynamic models based on metabolic reaction pathways is analyzed, showing that systems with complex intracellular reaction networks can be represented by macroscopic reactions relating extracellular components only. Based on rigorous assumptions, the model reduction procedure is systematic and allows an equivalent 'input-output' representation of the system to be derived. The procedure is illustrated with a few examples.

State observers generate estimates of non-measured variables based on a mathematical model of the process and some available hardware sensor signals. On the one hand, exponential observers, such as Luenberger observers or Kalman filters, have an adjustable rate of convergence, but strongly rely on the accuracy of the process model. On the other hand, asymptotic observers use a state transformation in order to avoid using the (usually uncertain) kinetic model, but have a rate of convergence imposed by the process dilution rate. In an attempt to combine the advantages of both techniques, a hybrid observer is developed, which evaluates a level of confidence in the process model and, accordingly, evolves between the two above-mentioned limit cases (exponential or asymptotic observer). In particular, attention is focused on a hybrid “Luenberger-asymptotic” observer, for which a rigorous stability/convergence analysis is provided. The efficiency and usefulness of the proposed observer is demonstrated with a bioprocess application example.