Uniform step-by-step observer for aerobic bioreactor based on super-twisting algorithm

This paper describes a fixed-time convergent step-by-step high order sliding mode observer for a certain type of aerobic bioreactor system. The observer was developed using a hierarchical structure based on a modified super-twisting algorithm. The modification included nonlinear gains of the output error that were used to prove uniform convergence of the estimation error. An energetic function similar to a Lyapunov one was used for proving the convergence between the observer and the bioreactor variables. A nonsmooth analysis was proposed to prove the fixed-time convergence of the observer states to the bioreactor variables. The observer was tested to solve the state estimation problem of an aerobic bioreactor described by the time evolution of biomass, substrate and dissolved oxygen. This last variable was used as the output information because it is feasible to measure it online by regular sensors. Numerical simulations showed the superior behavior of this observer compared to the one having linear output error injection terms (high-gain type) and one having an output injection obtaining first-order sliding mode structure. A set of numerical simulations was developed to demonstrate how the proposed observer served to estimate real information obtained from a real aerobic process with substrate inhibition.     

Nonlinear predictive control for maximization of CO2 bio-fixation by microalgae in a photobioreactor

In the framework of environment preservation, microalgae biotechnology appears as a promising alternative for CO₂ mitigation. Advanced control strategies can be further developed to maximize biomass productivity, by maintaining these microorganisms in bioreactors at optimal operating conditions. This article proposes the implementation of Nonlinear Predictive Control combined with an on-line estimation of the biomass concentration, using dissolved carbon dioxide concentration measurements. First, optimal culture conditions are determined so that biomass productivity is maximized. To cope with the lack of on-line biomass concentration measurements, an interval observer for biomass concentration estimation is built and described. This estimator provides a stable accurate interval for the state trajectory and is further included in a nonlinear model predictive control framework that regulates the biomass concentration at its optimal value. The proposed methodology is applied to cultures of the microalgae Chlorella vulgaris in a laboratory-scale continuous photobioreactor. Performance and robustness of the proposed control strategy are assessed through experimental results.     

Estimation of neutral lipid and carbohydrate quotas in microalgae using adaptive interval observers

Under stress conditions, microalgae are known to accumulate large amounts of neutral lipids and carbohydrates, which can be used for biofuel production. However, on-line measurement of microalgal biochemical composition is a difficult task which makes the microalgal process rather difficult to manage. In this paper, we propose a so called adaptive interval observer for the on-line estimation of neutral lipid and carbohydrate quotas in microalgae. The observer is based on a change of coordinates that involves a time-varying gain. We introduce dynamics for the gain, whose trajectory converges toward a predefined optimal value (which maximizes the convergence rate of the observer). The observer performance is illustrated with experimental data of Isochrysis sp. cultures under nitrogen limitations and day–night cycle. The proposed observer design appears to be a suitable robust estimation technique.     

Estimating biofilm reaction kinetics using hybrid mechanistic-neural network rate function model

This work describes an alternative method for estimation of reaction rate of a biofilm process without using a model equation. A first principles model of the biofilm process is integrated with artificial neural networks to derive a hybrid mechanistic-neural network rate function model (HMNNRFM), and this combined model structure is used to estimate the complex kinetics of the biofilm process as a consequence of the validation of its steady state solution. The performance of the proposed methodology is studied with the aid of the experimental data of an anaerobic fixed bed biofilm reactor. The statistical significance of the method is also analyzed by means of the coefficient of determination (R²) and model efficiency (ME). The results demonstrate the effectiveness of HMNNRFM for estimating the complex kinetics of the biofilm process involved in the treatment of industry wastewater.     

Optimal feeding strategies by adaptive mesh selection for fed-batch bioprocesses

The objective of this contribution is the design of optimal feeding strategies for fed-batch bioprocesses, where complex dynamic models with input and state constraints are present. For the solution of this dynamic optimization problem a transformation to a finite dimensional optimization problem is made using piecewise linear control profiles. The optimization of these profiles is performed by a sequential approach, that includes an ODE solver for the solution of the model ODE's. Further an adaptive mesh selection algorithm was investigated for an appropriate discretization of the control profiles. The implementation of the resulting optimal feeding profiles is shown for a process example, namely the production of nikkomycin by Streptomyces tendae. This implementation uses a hierarchical process control framework, that consists of components for process monitoring, state estimation, and trajectory control.     

Modeling and observer design for recombinant Escherichia coli strain

A mathematical model for recombinant bacteria which includes foreign protein production is developed. The experimental system consists of an Escherichia Coli strain and plasmid pIT34 containing genes for bioluminescence and production of a protein, β-galactosidase. This recombinant strain is constructed to facilitate on-line estimation and control in a complex bioprocess. Several batch experiments are designed and performed to validate the developed model. The design of a model structure, the identification of the model parameters and the estimation problem are three parts of a joint design problem. A nonlinear observer is designed and an experimental evaluation is performed on a batch fermentation process to estimate the substrate consumption.     

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.     

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.     

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.     

Monitoring in a predator–prey systems via a class of high order observer design

The goal of this work is the monitoring of the corresponding species in a class of predator–prey systems, this issue is important from the ecology point of view to analyze the population dynamics. The above is done via a nonlinear observer design which contains on its structure a high order polynomial form of the estimation error. A theoretical frame is provided in order to show the convergence characteristics of the proposed observer, where it can be concluded that the performance of the observer is improved as the order of the polynomial is high. The proposed methodology is applied to a class of Lotka–Volterra systems with two and three species. Finally, numerical simulations present the performance of the proposed observer.     

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.     

Endogenous model state and parameter estimation from an extensive batch experiment

In this paper an extensive batch experiment of endogenous process behavior in an aerobic biodegradation process is presented. From these experimental data, comprising measurements of MLVSS (mixed liquor volatile suspended solids) and respiration rate, in a first step the states and unknown parameters in a four-compartmental model are reconstructed analytically. Subsequently, for a selected set of states and parameters, using the results of the previous step, a recursive state estimation procedure, in particular an Extended Kalman filter-based observer, is applied to deal with the noise properties of the data appropriately. From this it appears that the initially proposed model structure, and especially the hydrolysis term, has to be modified.     

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.     

Robust and reliable estimation via recursive nonlinear dynamic data reconciliation based on cubature Kalman filter

Since measurements of process variables are subject to measurements errors as well as process variability, data reconciliation is the procedure of optimally adjusting measured date so that the adjusted values obey the conservation laws and constraints. Thus, data reconciliation for dynamic systems is fundamental and important for control, fault detection, and system optimization. Attempts to successfully implement estimators are often hindered by serve process nonlinearities, complicated state constraints, and un-measurable perturbations. As a constrained minimization problem, the dynamic data reconciliation is dynamically carried out to product smoothed estimates with variances from the original data. Many algorithms are proposed to solve such state estimation such as the extended Kalman filter (EKF), the unscented Kalman filter, and the cubature Kalman filter (CKF). In this paper, we investigate the use of CKF algorithm in comparative with the EKF to solve the nonlinear dynamic data reconciliation problem. First we give a broad overview of the recursive nonlinear data dynamic reconciliation (RNDDR) scheme, then present an extension to the CKF algorithm, and finally address the issue of how to solve the constraints in the CKF approach. The CCRNDDR method is proposed by applying the RNDDR in the CKF algorithm to handle nonlinearity and algebraic constraints and bounds. As the sampling idea is incorporated into the RNDDR framework, more accurate estimates can obtained via the recursive nature of the estimation procedure. The performance of the CKF approach is compared with EKF and RNDDR on nonlinear process systems with constraints. The conclusion is that with an error optimization solution of the correction step, the reformulated CKF shows high performance on the selection of nonlinear constrained process systems. Simulation results show the CCRNDDR is an efficient, accurate and stable method for real-time state estimation for nonlinear dynamic processes.     

Online nonlinear sequential Bayesian estimation of a biological wastewater treatment process

Online estimation of unknown state variables is a key component in the accurate modelling of biological wastewater treatment processes due to a lack of reliable online measurement systems. The extended Kalman filter (EKF) algorithm has been widely applied for wastewater treatment processes. However, the series approximations in the EKF algorithm are not valid, because biological wastewater treatment processes are highly nonlinear with a time-varying characteristic. This work proposes an alternative online estimation approach using the sequential Monte Carlo (SMC) methods for recursive online state estimation of a biological sequencing batch reactor for wastewater treatment. SMC is an algorithm that makes it possible to recursively construct the posterior probability density of the state variables, with respect to all available measurements, through a random exploration of the states by entities called ‘particle’. In this work, the simplified and modified Activated Sludge Model No. 3 with nonlinear biological kinetic models is used as a process model and formulated in a dynamic state-space model applied to the SMC method. The performance of the SMC method for online state estimation applied to a biological sequencing batch reactor with online and offline measured data is encouraging. The results indicate that the SMC method could emerge as a powerful tool for solving online state and parameter estimation problems without any model linearization or restrictive assumptions pertaining to the type of nonlinear models for biological wastewater treatment processes.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

State estimation of a Uricase production process with Candida Utilis

The present work discusses the implementation of a Kalman filtering procedure in a state estimation of a batch Uricase production process with Candida Utilis. An unstructured model of the process is used for the estimation procedure. The observability is thoroughly investigated and a Kalman filter is applied afterwards as a powerful and precise state estimation tool. The estimates in all cases of observability are presented, compared and discussed.     

On-line estimation of biodegradation in an unsaturated soil

The objective of this study was to develop a model-based estimator of biodegradation in unsaturated soil. This would allow real-time assessment of the efficiency of treatment bioprocesses, such as bioventilation and biopile, and eventually permit optimization through the implementation of control strategies. Based on a reduced-order model, an asymptotic observer was designed to estimate on-line the contaminant concentration, using carbon dioxide measurement. Two observer-based estimators were built to approximate: (1) the specific microbial growth rate; and (2) the biocontact kinetics representing the soil resistance to contaminant biodegradation. State observers and parameter estimators were confronted with the experimental results of biodegradation in microcosms. Hexadecane was used as the model compound, representing petroleum hydrocarbons. Three water contents, corresponding to 20%, 50% and 80% of the water-holding capacity, were tested. The asymptotic observer is able to predict hexadecane depletion with an error on the overall time trajectories of 13%, 8% and 4% for the dry, intermediate and wet soils, respectively, which is acceptable given that all the biokinetic parameters were identified from a biodegradation experiment in liquid phase. The observer-based estimator of the specific microbial growth rate, based on the CO₂ measurement, was successfully calibrated using the off-line measurements of hexadecane as validation data, and allowed estimation of the time when biodegradation switched from a microbial to a biocontact limitation. The biocontact kinetics was also identified on-line, using an estimator based on the hexadecane not in biocontact. These results are very encouraging with respect to the potential for on-line assessment of the performance of treatment bioprocesses in unsaturated soils.     

Modeling the microbial growth and temperature profile in a fixed-bed bioreactor

Aiming to scale up and apply control and optimization strategies, currently is required the development of accurate plant models to forecast the process nonlinear dynamics. In this work, a mathematical model to predict the growth of the Kluyveromyces marxianus and temperature profile in a fixed-bed bioreactor for solid-state fermentation using sugarcane bagasse as substrate was built up. A parameter estimation technique was performed to fit the mathematical model to the experimental data. The estimated parameters and the model fitness were evaluated with statistical analyses. The results have shown the estimated parameters significance, with 95 % confidence intervals, and the good quality of process model to reproduce the experimental data.