A tunable multivariable nonlinear robust observer for biological systems

This paper presents a robust nonlinear asymptotic observer with adjustable convergence rate with a great potential of applicability for biological systems in which the main state variables are difficult and expensive to measure or such measurements do not exist. This observer scheme is based on the classical asymptotic observer, which is modified to allow the tuning of the convergence rate. It is shown that the proposed observer provides fast and satisfactory estimates when facing load disturbances, system failures and parameter uncertainty while maintaining the excellent robustness and stability properties of the classical asymptotic observer. The implementation of the tunable observer is carried out by numerical simulations of a mathematical model of an anaerobic digestion process used for wastewater treatment. The key results are examined and further developed.     

SOME THOUGHTS ON NUTRIENT LIMITATION IN ALGAE1

An empirical relation relating specific growth, rate in steady state systems to nutrient status with respect to more than one nutrient simultaneously is proposed, based on 3 experimentally verifiable postulates: (1) that uptake depends on the external substrate concentration; (2) that growth depends on the interval substrate concentration; and (3) in a steady state system specific rate of uptake (in the absence of significant, excretion) is necessarily the product of the specific growth rate and internal substrate concentration. The implications of this model are discussed in particular in respect to the concept of luxury consumption and Liebig's law of minimum. Some aspects of uptake in transient situations are also discussed.     

初始底物浓度对序批式培养光合细菌产氢动力学影响

实验研究了初始底物浓度对序批式培养光合细菌生长、降解及产氢过程的影响,根据最大比生长速率实验数据拟合得到其关于初始底物浓度影响的关联式,并在建立的修正Monod模型基础上建立了光合细菌比生长速率、基质比消耗速率和比产氢速率关于底物初始浓度影响的数学模型,模型预测值与实验结果在光合细菌生长期和稳定期内得到较好吻合,反映了光合细菌生长、降解和产氢过程中受底物初始浓度限制性和抑制性影响的基本规律。分析发现光合细菌生长、降解基质和产氢过程中最适底物浓度为50 mmol/L,初始底物浓度低于或高于该浓度时,光合细菌生长、降解及产氢过程都受到限制性或抑制性影响,且抑制性影响较限制性影响效果更明显;底物比消耗速率受初始底物浓度影响较小。     

Monitoring in a Lotka-Volterra model

The problem of monitoring arises when in an ecosystem, in particular in a system of several populations, observing some components, we want to recover the state of the whole system as a function of time. Due to the difficulty to construct exactly this state process, we look for an auxiliary system called an observer. This system reproduces this process with a certain approximation. This means that the solution of the observer tends to that of the original system. An important concept for this work is observability. This means that from the observation it is possible to recover uniquely the state process, however, without determining a constructive method to obtain it. If observability holds for the original system, it guarantees the existence of an auxiliary matrix that makes it possible to construct an observer of the system. The considered system of populations is described by the classical Lotka-Volterra model with one predator and two preys and the construction of its observer is illustrated with a numerical example. Finally, it is shown how the observer can be used for the estimation of the level of an abiotic effect on the population system.     

Observing the observer (I): meta-bayesian models of learning and decision-making

In this paper, we present a generic approach that can be used to infer how subjects make optimal decisions under uncertainty. This approach induces a distinction between a subject's perceptual model, which underlies the representation of a hidden "state of affairs" and a response model, which predicts the ensuing behavioural (or neurophysiological) responses to those inputs. We start with the premise that subjects continuously update a probabilistic representation of the causes of their sensory inputs to optimise their behaviour. In addition, subjects have preferences or goals that guide decisions about actions given the above uncertain representation of these hidden causes or state of affairs. From a Bayesian decision theoretic perspective, uncertain representations are so-called "posterior" beliefs, which are influenced by subjective "prior" beliefs. Preferences and goals are encoded through a "loss" (or "utility") function, which measures the cost incurred by making any admissible decision for any given (hidden) state of affair. By assuming that subjects make optimal decisions on the basis of updated (posterior) beliefs and utility (loss) functions, one can evaluate the likelihood of observed behaviour. Critically, this enables one to "observe the observer", i.e. identify (context- or subject-dependent) prior beliefs and utility-functions using psychophysical or neurophysiological measures. In this paper, we describe the main theoretical components of this meta-Bayesian approach (i.e. a Bayesian treatment of Bayesian decision theoretic predictions). In a companion paper ('Observing the observer (II): deciding when to decide'), we describe a concrete implementation of it and demonstrate its utility by applying it to simulated and real reaction time data from an associative learning task.     

Structured modeling and state estimation in a fermentation process: Lipase production by Candida rugosa

A simple structured mathematical model coupled with a methodology of state and parameter estimation is developed for lipase production by Candida rugosa in batch fermentation. The model describes the system according to the following qualitative observations and hypothesis: Lipase production is induced by extracellular oleic acid present in the medium. The acid is transported into the cell where it is consumed, transformed, and stored. Lipase is excreted to the medium where it is distributed between the available oil-water interphase and aqueous phase. Cell growth is modulated by the intracellular substrate concentration. Model parameters have been determined and the whole model validated against experiments not used in their determination. The estimation problem consists in the estimation of three state variables (biomass, intra- and extracellular substrate) and two kinetic parameters by using only the on-line measurement provided by exhaust gas analysis. The presented estimation strategy divides the complex problem into three subproblems that can be solved by stable algorithms. The estimation of biomass (X) and the specific growth rate (mu), is achieved by a recursive prediction error algorithm using the on-line measurement of the carbon dioxide evolution rate. mu is then used to perform an estimation of intracellular substrate and the other kinetic parameter related to substrate transport (A) by an adaptive observer. Extracellular substrate is then evaluated by means of the estimated values of intracellular substrate and biomass through the material balance of the reactor. Simulation and experimental tests showed good performance of the developed estimator, which appears suitable to be used for process control and monitoring. (c) 1995 John Wiley & Sons, Inc.     

Estimation and control techniques for continuous culture fermentation processes

This article presents results obtained when some modern estimation and control techniques are applied to a simulated fermentation process. The control structure uses a particular observer of the substrate concentration and assumes the biomass concentration is measurable. The overall structure has been tested for both external and parametric disturbances, with very good results.     

Reduced modeling and state observation of an activated sludge process

This article first proposes a reduction strategy of the activated sludge process model with alternated aeration. Initiated with the standard activated sludge model (ASM1), the reduction is based on some biochemical considerations followed by linear approximations of nonlinear terms. Two submodels are then obtained, one for the aerobic phase and one for the anoxic phase, using four state variables related to the organic substrate concentration, the ammonium and nitrate‐nitrite nitrogen, and the oxygen concentration. Then, a two‐step robust estimation strategy is used to estimate both the unmeasured state variables and the unknown inflow ammonium nitrogen concentration. Parameter uncertainty is considered in the dynamics and input matrices of the system. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009     

Adaptive fuzzy control with output feedback for H infinity tracking of SISO nonlinear systems

Observer-based adaptive fuzzy H(infinity) control is proposed to achieve H(infinity) tracking performance for a class of nonlinear systems, which are subject to model uncertainty and external disturbances and in which only a measurement of the output is available. The key ideas in the design of the proposed controller are (i) to transform the nonlinear control problem into a regulation problem through suitable output feedback, (ii) to design a state observer for the estimation of the non-measurable elements of the system's state vector, (iii) to design neuro-fuzzy approximators that receive as inputs the parameters of the reconstructed state vector and give as output an estimation of the system's unknown dynamics, (iv) to use an H(infinity) control term for the compensation of external disturbances and modelling errors, (v) to use Lyapunov stability analysis in order to find the learning law for the neuro-fuzzy approximators, and a supervisory control term for disturbance and modelling error rejection. The control scheme is tested in the cart-pole balancing problem and in a DC-motor model.     

Shoot: Root Allocation with Respect to C, N and P: an Investigation and Comparison of Resistance and Teleonomic Models

Two shoot: root allocation models are described: the transport-resistanceapproach, and a teleonomic (goal-seeking) method based on maximizingspecific growth rate when the system is growing exponentially.These are applied to two growth modes: exponential growth, andthe steady state where all variables are constant with no netgrowth. The dynamic behaviour after shoot defoliation is investigated:the damping/overshoot effects observed are highly dependenton the presence or absence of product inhibition of the inputprocess (e.g. plant substrate N may inhibit the uptake of mineralN by the plant). The teleonomic model is far more damped thanthe resistance model and may therefore be misleading if usedto interpret transient experiments. Ontogenetic effects on allocationare simulated by varying the scaling (with plant size) of thetransport resistances; this may give increasing allocation tothe shoot or the root with the passage of time. The two modelsresemble each other very closely as far as equilibrium responsesare concerned - this applies to exponential growth and to thesteady state. Increasing the nitrogen input may lead to loweror higher whole-plant carbohydrate levels. The response to increasingnitrogen input depends on the other inputs; for instance itcan be much curtailed by low phosphorus inputs. The responseto phosphorus input can be similarly limited.Copyright 1995,1999 Academic Press Partitioning, plant growth, simulation     

Evaluation of Sampling-Based Methods for Sensitivity Analysis: Case Study for the E. coli Food Safety Process Risk Model

This article evaluates selected sensitivity analysis methods applicable to risk assessment models with two-dimensional probabilistic frameworks, using a microbial food safety process risk model as a test-bed. Six sampling-based sensitivity analysis methods were evaluated including Pearson and Spearman correlation, sample and rank linear regression, and sample and rank stepwise regression. In a two-dimensional risk model, the identification of key controllable inputs that can be priorities for risk management can be confounded by uncertainty. However, despite uncertainty, results show that key inputs can be distinguished from those that are unimportant, and inputs can be grouped into categories of similar levels of importance. All selected methods are capable of identifying unimportant inputs, which is helpful in that efforts to collect data to improve the assessment or to focus risk management strategies can be prioritized elsewhere. Rank-based methods provided more robust insights with respect to the key sources of variability in that they produced narrower ranges of uncertainty for sensitivity results and more clear distinctions when comparing the importance of inputs or groups of inputs. Regression-based methods have advantages over correlation approaches because they can be configured to provide insight regarding interactions and nonlinearities in the model.     

Effects of the observation method (direct v. from video) and of the presence of an observer on behavioural results in veal calves

This study aimed at assessing the effect of the observation method (direct or from video) and the effect of the presence of an observer on the behavioural results in veal calves kept on a commercial farm. To evaluate the effect of the observation method, 20 pens (four to five calves per pen) were observed by an observer for 60 min (two observation sessions of 30 min) and video-recorded at the same time. To evaluate the effect of the presence of the observer in front of the pen, 24 pens were video-recorded on 4 consecutive days and an observer was present in front of each pen for 60 min (two observation sessions of 30 min) on the third day. Behaviour was recorded using instantaneous scan sampling. For the study of the observer's effect, the analysis was limited to the posture, abnormal oral behaviour and manipulation of substrates. The two observation methods gave similar results for the time spent standing, but different results for all other behaviours. The presence of an observer did not affect the behaviour of calves at day level; however, their behaviour was affected when the observer was actually present in front of the pens. A higher percentage of calves were standing and were manipulating substrate in the presence of the observer, but there was no effect on abnormal oral behaviour. In conclusion, direct observations are a more suitable observation method than observations from video recordings for detailed behaviours in veal calves. The presence of an observer has a short-term effect on certain behaviours of calves that will have to be taken into consideration when monitoring these behaviours.     

Hybrid bounded error observers for uncertain bioreactor models

In this paper, we build bounded error observers for a common class of partially known bioreactor models. The main idea is to construct hybrid bounded observers “between” high gain observer, which has an adjustable convergence rate but requires perfect knowledge of the model, and asymptotic observer which is very robust towards uncertainty but has a fixed convergence rate. An hybrid bounded error observer which reconstructs the two state variables is constructed considering two steps: first step is similar to a high gain observer meaning that fast convergence rate but error depending on the knowledge of the model are obtained; second step is a switch to an observer similar to the asymptotic one meaning that fixed convergence rate towards an error as small as desired is obtained. Thus, a better convergence rate of estimated variables than the classical asymptotic observer is obtained.     

On-line state estimation of biomass based on acid production in Zymomonas mobilis cultures

The concentrations of biomass, substrate and product are very important state variables of almost every bioprocess and generally unable to be measured directly in?situ due to the lack of reliable sensors. In this paper, an adaptive observer of the biomass concentration is proposed for an anaerobic fermentation process where only the measurement of the acid product is available on-line. The observer was tested to be effective by several experiments under various operating conditions. In this experimental system, an auto-sampling device was connected between the bioreactor for the fermentation of Zymomonas mobilis and a HPLC so that the concentrations of glucose and ethanol could be directly measured through such implementation.     

On the Stock Estimation for a Harvested Fish Population

We consider a stage-structured model of a harvested fish population and we are interested in the problem of estimating the unknown stock state for each class. The model used in this work to describe the dynamical evolution of the population is a discrete time system including a nonlinear recruitment relationship. To estimate the stock state, we build an observer for the considered fish model. This observer is an auxiliary dynamical system that uses the catch data over each time interval and gives a dynamical estimate of the stock state for each stage class. The observer works well even if the recruitment function in the considered model is not well known. The same problem for an age-structured model has been addressed in a previous work (Ngom et al., Math. Biosci. Eng. 5(2):337–354, 2008).     

Control in bioreactors showing gradients

In large-scale bioreactors gradients often occur as a result of non-ideal mixing. This phenomenon complicates design and control of large-scale bioreactors. Gradients in the oxygen concentration can be modeled with a two-compartment model of the liquid phase. Application of this model had been suggested for the control of the dissolved oxygen concentration with a batch gluconic acid fermentation process as the model system. The control system consists of a controller, an observer and a parameter estimator. In this work, the controller design is reconsidered and, in simulation experiments, the performance of the control system has been investigated. When the parameter values are known, the controller in combination with the observer works adequately. The parameter estimator, however, yields incorrect parameters, which are caused by a coupling between two parameters. This causes a deviation of the estimated states from the process states. The simulation results suggest that a priori knowledge of the parameters is required for application of the model for control and state estimation.     

Cell Fate Forecasting: A Data-Assimilation Approach to Predict Epithelial-Mesenchymal Transition

Epithelial-mesenchymal transition (EMT) is a fundamental biological process that plays a central role in embryonic development, tissue regeneration, and cancer metastasis. Transforming growth factor-β (TGFβ) is a potent inducer of this cellular transition, which is composed of transitions from an epithelial state to intermediate or partial EMT state(s) to a mesenchymal state. Using computational models to predict cell state transitions in a specific experiment is inherently difficult for reasons including model parameter uncertainty and error associated with experimental observations. In this study, we demonstrate that a data-assimilation approach using an ensemble Kalman filter, which combines limited noisy observations with predictions from a computational model of TGFβ-induced EMT, can reconstruct the cell state and predict the timing of state transitions. We used our approach in proof-of-concept “synthetic” in silico experiments, in which experimental observations were produced from a known computational model with the addition of noise. We mimic parameter uncertainty in in vitro experiments by incorporating model error that shifts the TGFβ doses associated with the state transitions and reproduces experimentally observed variability in cell state by either shifting a single parameter or generating “populations” of model parameters. We performed synthetic experiments for a wide range of TGFβ doses, investigating different cell steady-state conditions, and conducted parameter studies varying properties of the data-assimilation approach including the time interval between observations and incorporating multiplicative inflation, a technique to compensate for underestimation of the model uncertainty and mitigate the influence of model error. We find that cell state can be successfully reconstructed and the future cell state predicted in synthetic experiments, even in the setting of model error, when experimental observations are performed at a sufficiently short time interval and incorporate multiplicative inflation. Our study demonstrates the feasibility and utility of a data-assimilation approach to forecasting the fate of cells undergoing EMT.     

Model for the feedback control system of bacterial growth. II. Growth in continuous culture

A mathematical model is developed that describes substrate limited bacterial growth in a continuous culture and that is based upon the conceptual framework elaborated in a previous paper for describing the feedback control system of cell growth [S. Bleecken, (1988). J. theor. Biol. 133, 37.] Central to the theory are the ideas that the limiting substrate is converted into low molecular weight building blocks of macromolecular synthesis which again are converted into biomass (RNA and protein) and that the rates of RNA and protein synthesis are controlled by the intracellular concentration of building blocks. It is shown that a continuous culture can be simulated by two interconnected feedback control systems the actuating signals of which are limiting substrate concentration and the intracellular concentration of building blocks, respectively. Three types of steady-states are found to appear in a continuous culture, besides the well-known stable steady-state of the whole culture there exist two batchlike steady-states of the biotic part of the culture which are metastable. The model is used to analyse the steady-states and their stability properties as well as the dynamic responses of biomass, RNA, protein, building block and substrate concentrations to changes in environmental conditions. Especially the inoculation of a continuous culture and the effects of step changes in dilution rate, inlet substrate concentration and growth temperature are studied in detail. Relations between the growth behaviour of a single cell and that of a continuous culture are derived. The RNA to protein ratio is introduced as a rough measure of the physiological state of cells and it is shown that a cell reacts to environmental changes with a simple pattern of basic responses in growth rate and physiological state. There are reasons to assume that the model presented is the minimal version of a structured model of bacterial growth and represents an optimum compromise between biological relevance and mathematical practicability.