Chebyshev model arithmetic for factorable functions

This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden

Guaranteed parameter estimation in nonlinear dynamic systems using improved bounding techniques

This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding - or approximating as closely as possible - the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. Exclusion tests rely on the ability to bound the solution set of the dynamic system for a given parameter subset and the tightness of these bounds is therefore paramount. Equally important is the time required to compute the bounds, thereby defining a trade-off. It is the objective of this paper to investigate this trade-off by comparing various bounding techniques based on interval arithmetic, Taylor model arithmetic and ellipsoidal calculus. When applied to a simple case study, ellipsoidal and Taylor model approaches are found to reduce the number of iterations significantly compared to interval analysis, yet the overall computational time is only reduced for tight approximation levels due to the computational overhead. © 2013 EUCA

Réduction du modèle ASM 1 pour la commande optimale des petites stations d'épuration à boues activées

L'adoption par l'Union Européenne de normes de rejets plus contraignantes implique une meilleure gestion des stations d'épuration. L'utilisation de modèles de simulation dynamique dans des schémas de commande en boucle fermée constitue une alternative intéressante pour répondre à ce problème.Sur la base du modèle ASM 1, un modèle réduit est ici élaboré pour le procédé à boues activées en aération séquentielle, en vue de la commande optimale du système d'aération. Les simplifications considérées sont de deux types : (i) les dynamiques lentes du système sont identifiées au moyen d'une méthode d'homotopie, puis éliminées du modèle ; (ii) des simplifications plus heuristiques consistant à prendre en compte un composé organique unique et à éliminer la concentration des composés organiques azotés sont ensuite appliquées. Elles conduisent à un modèle simplifié de 5 variables. L'application d'une procédure d'identification paramétrique permet alors de démontrer que le comportement dynamique du modèle simplifié est en bonne adéquation avec celui du modèle ASM 1 sur un horizon de prédiction de plusieurs heures, même lorsque les concentrations de l'influent ne sont pas connues. Il est également vérifié que le modèle proposé est observable et structurellement identifiable, sous des conditions d'aérobiose et d'anoxie, à partir des mesures en ligne des concentrations en oxygène dissous, ammoniaque et nitrate.Le modèle simplifié développé présente ainsi toutes les propriétés requises pour une future utilisation au sein de schémas de commande en boucle fermée, en vue de la commande optimale des petites stations d'épuration à boues activées.In order to meet the stricter wastewater effluent guidelines adopted by the European Union, wastewater treatment plants require better management strategies. Wastewater treatment process models have become a major tool to design closed-loop control schemes. However, the dynamic models that are currently used in the simulation of activated sludge treatment plants (ASM 1, ASM 2 and, more recently, ASM 3 models) are highly dimensional and are not appropriate for on-line implementation (e.g., for model predictive control or optimal control). It is therefore important to develop reduced models that could be used for this purpose.A reduced model was developed to describe the behaviour of alternating activated sludge treatment plants, with the aim of applying it to the optimal control of an aeration system. The reduction scheme was based on appropriate simplifications to the ASM 1 model (which is more appropriate for open-loop control). The objective was to verify if accurate predictions could be made time periods of several hours (about 8 h).The present results are related to an existing small-size wastewater treatment plant. This plant was designed for 15,000 population-equivalents (p.e.) and consists of a primary treatment stage (screening, grit removal, primary sedimentation), followed by a secondary treatment stage (biological treatment). The latter consists of a single aeration tank of about 2,050 m3 equipped with 3 turbines which are operated cyclically to create alternating aerobic and anoxic conditions. Ammonia is converted into nitrate during air-on periods (nitrification step) and nitrate is subsequently removed during air-off periods (denitrification step). It is important to note that a dynamic model, based on the ASM 1 model and calibrated from a set of input/output measurements over a one-day period (Chachuat, 2001), was used here as a reference to perform model reduction. The following two-level simplification procedure was applied :· A homotopy method was first used to establish relationships between the states and the dynamics of the system, via an eigenvalue decomposition. The components that are associated with the slowest dynamics are then assumed constant to reduce the state space dimension. Heterotrophic (XB,H) and autotrophic (XB,A) biomass and inert particulate organic compounds (XI) were detected as the slow state variables. It was found that the short-term predictions of the dynamic model were not affected by assuming that XI, XB,H and XB,A concentrations were constant. Eliminating these 3 state variables, along with the concentrations of soluble inert organic compounds (SI), resulted in a 7-dimensional dynamic model.· However, further simplifications were required to enable the on-line optimisation of the bioreactor aeration profiles with reasonable computational times. These simplifications consisted of taking into account the process specifications in order to reduce the state space dimension to 4 or 5, and were therefore based on more heuristic considerations. Both organic and nitrogenous compounds are under consideration: (i) a single organic compound (denoted as XDCO) is formed by adding soluble and particulate organic compound concentrations, and (ii) the mathematical expression that describes the organic nitrogen hydrolysis process is simplified so that the dynamics with respect to soluble and particulate organic nitrogen are independent.The two previous simplification steps produced a reduced 5-dimensional dynamic model with state variables XDCO, SNO, SNH, SND and SO. It should also be noted that the resulting model involved the parameters YH, iNBM, KS, KNO, KO,H, KNH,A, ηNO,g and ηNO,h that are identical to those defined in the original ASM 1 model by Henze et al. (1987). In addition, 7 specific parameters were defined defined (θ1, θ2, θ3, θ4, θ5, KDCO, KND). These new parameters exhibited rather slow temporal variation, thus agreeing with the general ASM 1 model for short time periods.Afterwards, a two-step procedure was applied to calibrate the model. This procedure first consisted of determining a reduced set of identifiable parameters by the use of both sensitivity and principal component analyses. Note that the inlet concentrations of organic compounds, ammonia nitrogen and soluble organic nitrogen may be considered as additional parameters since they are generally not measured on-line. The selected parameters (θ1, θ2, θ3) and inlet concentrations (XinDCO, SinNH) were then estimated by the application of a local gradient search method (successive quadratic programming, SQP). Comparisons between the dynamic behaviour of both reduced and ASM 1 models show that accurate predictions can be obtained over time periods of several hours (8 h). It was also shown that the reduced model was observable and structurally identifiable under aerobic and anoxic conditions from dissolved oxygen, ammonia and nitrate concentration measurements. These results therefore demonstrate the ability of the reduced model to be embedded into closed-loop control schemes.The conclusions from this work are twofold: (i) The reduced model can be used as a basis to construct an on-line observer to estimate the unmeasured state variables, the unknown (most sensitive) parameters and inlet concentrations; (ii) Non-linear model predictive control (NMPC) schemes can then be implemented to operate the aeration system so that the nitrogen discharge or the energy consumption are minimised (optimal control).The initial results demonstrate that the application of NMPC strategies is likely to give large reductions of nitrogen discharge with respect to usual operating strategies (e.g., oxygen or redox control). Such closed-loop control schemes are particularly efficient in dealing with large influent variations (inlet flow rate, concentration and composition) resulting from both human activities and climatic conditions, and inherent modelling uncertainties. However, an experimental validation of this control strategy on a pilot scale or an industrial scale is required to confirm these results

Optimization-based domain reduction in guaranteed parameter estimation of nonlinear dynamic systems

This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding-or approximating as closely as possible-the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. In order to enhance the convergence of this procedure, we investigate the use of optimization-based domain reduction techniques for tightening the parameter boxes before partitioning. We construct such bound-reduction problems as linear programs from the polyhedral relaxation of Taylor models of the predicted outputs. When applied to a simple case study, the proposed approach is found to reduce the computational burden significantly, both in terms of CPU time and number of iterations. © IFAC

Dynamic coupling of photoacclimation and photoinhibition in a model of microalgae growth.

International audienceThe development of mathematical models that can predict photosynthetic productivity of microalgae under transient conditions is crucial for enhancing large-scale industrial culturing systems. Particularly important in outdoor culture systems, where the light irradiance varies greatly, are the processes of photoinhibition and photoacclimation, which can affect photoproduction significantly. The former is caused by an excess of light and occurs on a fast time scale of minutes, whereas the latter results from the adjustment of the light harvesting capacity to the incoming irradiance and takes place on a slow time scale of days. In this paper, we develop a dynamic model of microalgae growth that simultaneously accounts for the processes of photoinhibition and photoacclimation, thereby spanning multiple time scales. The properties of the model are analyzed in connection to PI-response curves, under a quasi steady-state assumption for the slow processes and by neglecting the fast dynamics. For validation purposes, the model is calibrated and compared against multiple experimental data sets from the literature for several species. The results show that the model can describe the difference in photosynthetic unit acclimation strategies between Dunaliella tertiolecta (n-strategy) and Skeletonema costatum (s-strategy)