Practical identifiability analysis of environmental models

Identifiability of a system model can be considered as the extent to which one can capture its parameter values from observational data and other prior knowledge of the system. Identifiability must be considered in context so that the objectives of the modelling must also be taken into account in its interpretation. A model may be identifiable for certain objective functions but not others; its identifiability may depend not just on the model structure but also on the level and type of noise, and may even not be identifiable when there is no noise on the observational data. Context also means that non-identifiability might not matter in some contexts, such as when representing pluralistic values among stakeholders, and may be very important in others, such as where it leads to intolerable uncertainties in model predictions. Uncertainty quantification of environmental systems is receiving increasing attention especially through the development of sophisticated methods, often statistically-based. This is partly driven by the desire of society and its decision makers to make more informed judgments as to how systems are better managed and associated resources efficiently allocated. Less attention seems to be given by modellers to understand the imperfections in their models and their implications. Practical methods of identifiability analysis can assist greatly here to assess if there is an identifiability problem so that one can proceed to decide if it matters, and if so how to go about modifying the model (transforming parameters, selecting specific data periods, changing model structure, using a more sophisticated objective function). A suite of relevant methods is available and the major useful ones are discussed here including sensitivity analysis, response surface methods, model emulation and the quantification of uncertainty. The paper also addresses various perspectives and concepts that warrant further development and use

Getting the most out of it: optimal experiments for parameter estimation of microalgae growth models

International audienceMathematical models are expected to play a pivotal role for driving microalgal production towards a profitable process of renewable energy generation. To render models of microalgae growth useful tools for prediction and process optimization, reliable parameters need to be provided. This reliability implies a careful design of experiments that can be exploited for parameter estimation. In this paper, we provide guidelines for the design of experiments with high informative content based on optimal experiment techniques to attain an accurate parameter estimation. We study a real experimental device devoted to evaluate the effect of temperature and light on microalgae growth. On the basis of a mathematical model of the experimental system, the optimal experiment design problem was formulated and solved with both static (constant light and temperature) and dynamic (time varying light and temperature) approaches. Simulation results indicated that the optimal experiment design allows for a more accurate parameter estimation than that provided by the existing experimental protocol. For its efficacy in terms of the maximum likelihood properties and its practical aspects of implementation, the dynamic approach is recommended over the static approach

On Feedback Identification of Unknown Biochemical Characteristics in an Artificial Lake

The problem of dynamical identification of unknown characteristics (states/parameters) in a biochemical model of an artificial lake with only inflow and given observations of some states is considered. An algorithm that solves this simultaneous state and parameter estimation problem and that is stable with respect to bounded informational noises and computational errors is presented. The algorithm is based on the principle of auxiliary models with adaptive controls. Convergence of the algorithm is proven and a convergence rate is derived. The performance of the algorithm is illustrated to a typical single-species environmental example

미세조류 배양 광생물반응기 시스템의 생산성 향상을 위한 근 실시간 추정 및 최적화

학위논문 (박사)-- 서울대학교 대학원 : 화학생물공학부, 2015. 8. 이종민.This thesis has presented the near real-time optimization procedures for productivity improvement of microalgal photobioreactor system under mixotrophic cultivation. Microalgae have been suggested as a promising feedstock for producing biofuel because of their potential for lipid production. However, the development of large-scale algal biodiesel production has been limited by the high production cost of algal biomass. Therefore it is necessary to improve the economic feasibility by reducing costs or increasing productivity. In order to have an economically sound algal bioprocess, this thesis tries to optimize the operating conditions by manipulating nutrient (carbon and nitrogen sources) flow rates and light intensity. For this purposes, it is need to develop a dynamic model that describes algal growth and lipid accumulation in order to support the development of algal bioprocesses, their scale up, optimization and control. However, there are some difficulties in applying model-based control strategies to microalgal cultivation systems. Microalgae cultivation systems are network of complex biochemical reactions manipuated by enzyme kinetics. Modelling of these complex biological systems accurately is difficult task since metabolism inside the cells makes systems have uncertainties. In addition to model uncertainties arising from complex biosystem dynamics, on-line measurement of important variables, especially in lipid is limited and difficult to realize in practice, which makes optimal bioreactor operation a challenging task. To cope with such problems, this thesis focused on the modelling, estimation of lipid concentration, and optimization of photobioreactor systems. At first, the model was developed based on the Droop model, and the optimal input design using D-optimality criterion was performed to compute the system input profile, to estimate parameters more accurately. From the experimental observations, the newly defined yield coefficient was suggested to represent the consumption of lipid and nitrogen within the cell, which reduces the number of parameters with more accurate prediction. Furthermore, the lipid consumption rate was introduced to reflect the experimental results that lipid consumption is related to carbon source concentration. The model was validated with experiments designed with different initial conditions of nutrients and input changes, and showed good agreement with experimental observations. After that, estimation of lipid concentration from other measurable sources such as biomass or glucose sensor was studied. Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF) were compared in various cases for their applicability to photobioreactor systems. Furthermore, simulation studies to identify appropriate types of sensors for estimating lipid were also performed. Finally, to maximize the biomass and lipid concentration, various optimization methods were investigated in microalgal photobioreactor system under mixotrophic conditions. Lipid concentration was estimated using UKF with other measurable sources and used as lipid data for performing model predictive control (MPC). In addition, maximized biomass and lipid trajectory obtained by open-loop optimization was used as a reference trajectory for traking by MPC. Simulation studies with experimental validation were performed in all cases and significant improvement in productivities of biomass and lipid was obtained when MPC applied. However, it was observed that lag phase occurs while manipulating feed flow rate, which considered to come from large amount of inputs introduced suddenly. This is important phenomena can make model-plant mismatches and needs to be researched more for the optimization of microalgal photobioreactor in reality.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 1 2. Experiment and data anlysis . . . . . . . . . . . . . . 5 2.1 Microalgae and media composition . . . . . . . . . . 5 2.2 Photobioreactor system and conditions . . . . . . . . 7 2.3 Method for data analysis . . . . . . . . . . . . . . . . 8 2.3.1 Biomass measurement . . . . . . . . . . . . . 8 2.3.2 Glucose measurement . . . . . . . . . . . . . 8 2.3.3 Glycine measurement . . . . . . . . . . . . . 9 2.3.4 Lipid measurement . . . . . . . . . . . . . . . 9 3. Modelling of photobioreactor system . . . . . . . . . . 11 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Classic growth models . . . . . . . . . . . . . . . . . 13 3.2.1 Monod model . . . . . . . . . . . . . . . . . 13 3.2.2 Cell quota model . . . . . . . . . . . . . . . . 14 3.3 Development of photobioreactor model . . . . . . . . 15 3.4 Optimal experimental design . . . . . . . . . . . . . 19 3.5 Parameter estimation . . . . . . . . . . . . . . . . . . 21 3.6 Results and Discussion . . . . . . . . . . . . . . . . . 23 3.6.1 Simulation and experimental results . . . . . . 23 3.6.2 Modification of the photobioreactor model . . 25 3.6.3 Validation of the model . . . . . . . . . . . . 30 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 32 4. Estimation of lipid concentration . . . . . . . . . . . . 34 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 34 4.2 Photobioreactor model . . . . . . . . . . . . . . . . . 36 4.3 Estimator algorithms : EKF, UKF, PF . . . . . . . . . 38 4.3.1 Extended Kalman Filter (EKF) . . . . . . . . 38 4.3.2 Unscented Kalman Filter (UKF) . . . . . . . . 40 4.3.3 Particle Filter (PF) . . . . . . . . . . . . . . . 42 4.4 Simulation studies . . . . . . . . . . . . . . . . . . . 44 4.4.1 Case study 1 : effect of system noise covariance (Q) 46 4.4.2 Case study 2 : effect of disturbances . . . . . . 48 4.4.3 Case study 3: effect of parametric mismatches 51 4.4.4 Case study 4 : types of equipments . . . . . . 52 4.5 Experimental results . . . . . . . . . . . . . . . . . . 54 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . 56 5. Optimization . . . . . . . . . . . . . . . . . . . . . . . 57 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Microalgal photobioreactor model . . . . . . . . . . . 58 5.3 State estimation . . . . . . . . . . . . . . . . . . . . 60 5.4 Optimization . . . . . . . . . . . . . . . . . . . . . . 64 5.4.1 Manual operation based on algal growth characteristic 64 5.4.2 Open-loop optimization . . . . . . . . . . . . 64 5.4.3 Model predictive control . . . . . . . . . . . . 66 5.5 Results and Discussion . . . . . . . . . . . . . . . . . 70 5.5.1 Manual operation based on algal growth characteristic 70 5.5.2 Open-loop optimization . . . . . . . . . . . . 72 5.5.3 Model predictive control . . . . . . . . . . . . 74 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . 78 6. Concluding Remarks . . . . . . . . . . . . . . . . . . . 79Docto

Optimal parametric sensitivity control of a fed-batch reactor

The paper presents an optimal parametric sensitivity controller for estimation of a set of parameters in an experiment. The method is demonstrated for a fed-batch bioreactor case study for optimal estimation of the half-saturation constant KS and the parameter combination µmaxX/Y in which µmax is the maximum specific growth rate, X is the biomass concentration, and Y the yield coefficient. The resulting parametric sensitivity controller for the parameter KS is utilized in two sequential experiments using a ‘bang–bang-singular’ control strategy. Comparison with an optimal solution for the weighted sum of squared sensitivities for both parameters are compared with the individual cases where only one specific parametric output sensitivity is controlled. The parametric uncertainty is handled in a completely deterministic way as to arrive at a control law that maximizes the parametric output sensitivity

Optimização de estratégias de alimentação para identificação de parâmetros de um modelo de E. coli. utilização do modelo em monitorização e controlo

Doutoramento em Engenharia Química e BiológicaOs principais objectivos desta tese são: o desenho óptimo de experiências para a identificação de coeficientes de rendimento de um modelo não estruturado de um processo de fermentação semicontínua de Escherichia coli; a verificação experimental das trajectórias de alimentação obtidas por simulação; o desenvolvimento de estratégias de monitorização avançada para a estimação em linha de variáveis de estado e parâmetros cinéticos; e por fim o desenvolvimento de uma lei de controlo adaptativo para controlar a taxa específica de crescimento, com base em estratégias de alimentação de substrato com vista à maximização do crescimento e/ou produção. São apresentadas metodologias para o desenho óptimo de experiências, que visam a optimização da riqueza informativa das mesmas, quantificada por índices relativos à Matriz de Informação de Fisher. Embora, o modelo utilizado para descrever a fermentação semi-contínua de E. coli não esteja ainda optimizado em termos cinéticos e de algumas dificuldades encontradas na implementação prática dos resultados obtidos por simulação para o desenho óptimo de experiências, a qualidade da estimativa dos parâmetros, especialmente os do regime respirativo, é promissora. A incerteza das estimativas foi avaliada através de índices relacionados com o modelo de regressão linear múltipla, índices relativos à matriz de Fisher e pelo desenho das correspondentes elipses dos desvios. Os desvios associados a cada coeficiente mostram que ainda não foram encontrados os melhores valores. Procedeu-se também à investigação do papel do modelo dinâmico geral no desenho de sensores por programação. Foram aplicados três observadores – observador estendido de Kalman, observador assimptótico e observador por intervalo – para estimar a concentração de biomassa, tendo sido avaliado e comparado o seu desempenho bem como a sua flexibilidade. Os observadores estudados mostraram-se robustos, apresentando comportamentos complementares. Os observadores assimptóticos apresentam, em geral, um melhor desempenho que os observadores estendidos de Kalman. Os observadores por intervalo apresentam vantagens em termos de implementação prática, apresentando-se bastante promissores embora a sua validação experimental seja necessária. É apresentada uma lei de controlo adaptativo com modelo de referência que se traduz num controlo por antecipação/retroacção cuja acção de retroacção é do tipo PI, para controlar a taxa específica de crescimento. A robustez do algoritmo de controlo foi estudada por simulação numérica gerando dados “pseudo reais”, por aplicação de um ruído branco às variáveis medidas em linha, por alteração do valor de referência, por alteração do valor da concentração da glucose na alimentação e variando os valores nominais dos parâmetros do modelo. O estudo realizado permite concluir que a resposta do controlador é em geral satisfatória, sendo capaz de manter o valor da taxa específica de crescimento na vizinhança do valor de referência pretendido e inferior a um valor que conduz à formação de acetato, revestindo-se este facto de grande importância numa situação real, em especial, numa fermentação cujo objectivo seja a produção, nomeadamente de proteínas recombinadas. Foram ainda, analisados diferentes métodos de sintonização dos parâmetros do controlador, podendo concluir-se que, em geral, o método de sintonização automática com recurso à regra de adaptação dos parâmetros em função do erro relativo do controlador foi o que apresentou um melhor desempenho global. Este mecanismo de sintonização automática demonstrou capacidade para melhorar o desempenho do controlador ajustando continuamente os seus parâmetros.The main objectives of this thesis are: the optimal experiment design for yield coefficients estimation in an unstructured growth model for Escherichia coli fed-batch fermentation; the experimental validation of the simulated feed trajectories; the development of advanced monitoring strategies for the on-line estimation of state variables and kinetic parameters; and at last the development of an adaptive control law, based on optimal substrate feed strategies in order to increase the growth and/or the production. Methodologies for the optimal experimental design are presented, in order to optimise the richness of data coming out from experiments, quantified by indexes based on the Fisher Information Matrix. Although the model used to describe the E. coli fed-batch fermentation is not optimised from the kinetic properties point of view and the fact that some difficulties were encountered in practical implementation of the simulated results obtained with the optimal experimental design, the estimated parameter quality, especially for the oxidative regimen, is promising. The estimation uncertainty was evaluated by means of indexes related with multiple linear regression model, indexes related to the Fisher matrix as well as by the construction of the related deviation ellipses. The deviations associated to each coefficient show that the best values were not yet found. The role of the general dynamical model was also investigated in which concerns the design of state observers, also called software sensors. The performance of three observer classes was compared: Kalman extended observer, assimptotic observer and interval observer. The studied observers showed good performance and robustness, being complementary of each other. Assimptotic observers showed, in general, a better performance than the Kalman extended observer. Interval observers presented advantages concerning practical implementation, showing a promising behaviour although experimental validation is needed. A model reference adaptive control law is presented and can be interpreted as a PI like feedforward/feedback controller, for specific growth rate control. Algorithm robustness was studied using “pseudo real” data obtained by numerical simulation, by applying a white noise to the on-line measured variables, by modifying the set-point value, by changing the glucose concentration value of the feed rate and varying the nominal model parameter value. The study made allowed to conclude that the controller response is, generally, satisfactory being able to keep the specific growth rate value in the proximity of the desired set-point and lower than the value that permits acetate formation, which is of major importance namely for real cases, specially, in a fermentation which objective was the production of recombinant proteins. Different tuning devices for controller parameters were analysed being the better performance achieved by the automatic tuning method with an adaptation rate as a function of the controller relative error. This automatic tuning mechanism was able to improve the controller performance adjusting continuously its parameters