Anaerobic Digestion Process Modeling Under Uncertainty: A Narrative Review

Growing concern about global climate change has led to considerable interest in investigating renewable energy sources such as the biological conversion of biomass to methane in an anaerobic environment. Through a series of complicated biochemical interactions, it uses various bacterial species to degrade biodegradable material in the feedstock. Due to the complex and interacting biochemical processes, anaerobic digestion has nonlinear dynamics. Anaerobic digestion is highly at risk of instabilities and uncertainties because of its dynamic and nonlinear behavior, uncertain feedstock quality, and sensitivity to the process’s environmental conditions. Therefore, effectively operating a biogas production unit necessitates a thorough understanding of the system’s uncertainties. The present study aims to identify and assess the sources and methods of coping with the uncertainties in anaerobic digestion processes through a narrative review. Moreover, the knowledge gap is also investigated to reveal the challenges and opportunities to have a robust model. The results indicate that the unpredictability of model parameters and input variables were the most significant source of uncertainty, and the Monte Carlo technique, confident interval, and interval observers, as well as sensitivity analysis were the most frequently used tools to cope with these uncertainties

Optimal control of fed-batch fermentation processes

Optimisation of a fed-batch fermentation process typically uses the calculus of variations or Pontryagin's maximum principle to determine an optimal feed rate profile. This often results in a singular control problem and an open loop control structure. The singular feed rate is the optimal feed rate during the singular control period and is used to control the substrate concentration in the fermenter at an optimal level. This approach is supported by biological knowledge that biochemical reaction rates are controlled by the environmental conditions in the fermenter; in this case, the substrate concentration. Since an accurate neural net-based on-line estimation of the substrate concentration has recently become available and is currently employed in industry, we are therefore able to propose a method which makes use of this estimation. The proposed method divides the optimisation problem into two parts. First, an optimal substrate concentration profile which governs the biochemical reactions in the fermentation process is determined. Then a controller is designed to track the obtained optimal profile. Since the proposed method determines the optimal substrate concentration profile, the singular control problem is therefore avoided because the substrate concentration appears nonlinearly in the system equations. Also, the process is then operated in closed loop control of the substrate concentration. The proposed method is then called "closed loop optimal control". The proposed closed loop optimal control method is then compared with the open loop optimal feed rate profile method. The comparison simulations from both primary and secondary metabolite production processes show that both methods give similar performance in a case of perfect model while the closed loop optimal control provides better performance than the open loop method in a case of plant/model mismatch. The better performance of the closed loop optimal control is due to an ability to compensate for the modelling errors using feedback

Polymer Reactor Modeling, Design and Monitoring

Polymers range from synthetic plastics, such as polyacrylates, to natural biopolymers, such as proteins and DNA. The large molecular mass of polymers and our ability to manipulate their compositions and molecular structures have allowed for producing synthetic polymers with attractive properties. new polymers with remarkable characteristics are synthesized. Because of the huge production volume of commodity polymers, a little improvement in the operation of commodity-polymer processes can lead to significant economic gains. On the other hand, a little improvement in the quality of specialty polymers can lead to substantial increase in economic profits

Multi-Rate Observers for Model-Based Process Monitoring

Very often, critical quantities related to safety, product quality and economic performance of a chemical process cannot be measured on line. In an attempt to overcome the challenges caused by inadequate on-line measurements, state estimation provides an alternative approach to reconstruct the unmeasured state variables by utilizing available on-line measurements and a process model. Chemical processes usually possess strong nonlinearities, and involve different types of measurements. It remains a challenging task to incorporate multiple measurements with different sampling rates and different measurement delays into a unified estimation algorithmic framework. This dissertation seeks to present developments in the field of state estimation by providing the theoretical advances in multi-rate multi-delay observer design. A delay-free multi-rate observer is first designed in linear systems under asynchronous sampling. Sufficient and explicit conditions in terms of maximum sampling period are derived to guarantee exponential stability of the observer, using Lyapunov’s second method. A dead time compensation approach is developed to compensate for the effect of measurement delay. Based on the multi-rate formulation, optimal multi-rate observer design is studied in two classes of linear systems where optimal gain selection is performed by formulating and solving an optimization problem. Then a multi-rate observer is developed in nonlinear systems with asynchronous sampling. The input-to-output stability is established for the estimation errors with respect to measurement errors using the Karafyllis-Jiang vector small-gain theorem. Measurement delay is also accounted for in the observer design using dead time compensation. Both the multi-rate designs in linear and nonlinear systems provide robustness with respect to perturbations in the sampling schedule. Multi-rate multi-delay observer is shown to be effective for process monitoring in polymerization reactors. A series of three polycondensation reactors and an industrial gas-phase polyethylene reactor are used to evaluate the observer performance. Reliable on-line estimates are obtained from the multi-rate multi-delay observer through simulation

Modeling and Estimation of Biological Plants

Estimating the state of a dynamic system is an essential task for achieving important objectives such as process monitoring, identification, and control. Unlike linear systems, no systematic method exists for the design of observers for nonlinear systems. Although many researchers have devoted their attention to these issues for more than 30 years, there are still many open questions. We envisage that estimation plays a crucial role in biology because of the possibility of creating new avenues for biological studies and for the development of diagnostic, management, and treatment tools. To this end, this thesis aims to address two types of nonlinear estimation techniques, namely, the high-gain observer and the moving-horizon estimator with application to three different biological plants. After recalling basic definitions of stability and observability of dynamical systems and giving a bird's-eye survey of the available state estimation techniques, we are interested in the high-gain observers. These observers may be used when the system dynamics can be expressed in specific a coordinate under the so-called observability canonical form with the possibility to assign the rate of convergence arbitrarily by acting on a single parameter called the high-gain parameter. Despite the evident benefits of this class of observers, their use in real applications is questionable due to some drawbacks: numerical problems, the peaking phenomenon, and high sensitivity to measurement noise. The first part of the thesis aims to enrich the theory of high-gain observers with novel techniques to overcome or attenuate these challenging performance issues that arise when implementing such observers. The validity and applicability of our proposed techniques have been shown firstly on a simple one-gene regulatory network, and secondly on an SI epidemic model. The second part of the thesis studies the problem of state estimation using the moving horizon approach. The main advantage of MHE is that information about the system can be explicitly considered in the form of constraints and hence improve the estimates. In this work, we focus on estimation for nonlinear plants that can be rewritten in the form of quasi-linear parameter-varying systems with bounded unknown parameters. Moving-horizon estimators are proposed to estimate the state of such systems according to two different formulations, i.e., "optimistic" and "pessimistic". In the former case, we perform estimation by minimizing the least-squares moving-horizon cost with respect to both state variables and parameters simultaneously. In the latter, we minimize such a cost with respect to the state variables after picking up the maximum of the parameters. Under suitable assumptions, the stability of the estimation error given by the exponential boundedness is proved in both scenarios. Finally, the validity of our obtained results has been demonstrated through three different examples from biological and biomedical fields, namely, an example of one gene regulatory network, a two-stage SI epidemic model, and Amnioserosa cell's mechanical behavior during Dorsal closure

Applications of Mathematical Models in Engineering

The most influential research topic in the twenty-first century seems to be mathematics, as it generates innovation in a wide range of research fields. It supports all engineering fields, but also areas such as medicine, healthcare, business, etc. Therefore, the intention of this Special Issue is to deal with mathematical works related to engineering and multidisciplinary problems. Modern developments in theoretical and applied science have widely depended our knowledge of the derivatives and integrals of the fractional order appearing in engineering practices. Therefore, one goal of this Special Issue is to focus on recent achievements and future challenges in the theory and applications of fractional calculus in engineering sciences. The special issue included some original research articles that address significant issues and contribute towards the development of new concepts, methodologies, applications, trends and knowledge in mathematics. Potential topics include, but are not limited to, the following: Fractional mathematical models; Computational methods for the fractional PDEs in engineering; New mathematical approaches, innovations and challenges in biotechnologies and biomedicine; Applied mathematics; Engineering research based on advanced mathematical tools

Liquid Transport Pipeline Monitoring Architecture Based on State Estimators for Leak Detection and Location

This research presents the implementation of optimization algorithms to build auxiliary signals that can be injected as inputs into a pipeline in order to estimate —by using state observers—physical parameters such as the friction or the velocity of sound in the fluid. For the state estimator design, the parameters to be estimated are incorporated into the state vector of a Liénard-type model of a pipeline such that the observer is constructed from the augmented model. A prescribed observability degree of the augmented model is guaranteed by optimization algorithms by building an optimal input for the identification. The minimization of the input energy is used to define the optimality of the input, whereas the observability Gramian is used to verify the observability. Besides optimization algorithms, a novel method, based on a Liénard-type model, to diagnose single and sequential leaks in pipelines is proposed. In this case, the Liénard-type model that describes the fluid behavior in a pipeline is given only in terms of the flow rate. This method was conceived to be applied in pipelines solely instrumented with flowmeters or in conjunction with pressure sensors that are temporarily out of service. The design approach starts with the discretization of the Liénard-type model spatial domain into a prescribed number of sections. Such discretization is performed to obtain a lumped model capable of providing a solution (an internal flow rate) for every section. From this lumped model, a set of algebraic equations (known as residuals) are deduced as the difference between the internal discrete flows and the nominal flow (the mean of the flow rate calculated prior to the leak). The residual closest to zero will indicate the section where a leak is occurring. The main contribution of our method is that it only requires flow measurements at the pipeline ends, which leads to cost reductions. Some simulation-based tes

Parameter identification for biological models

This thesis concerns the identification of dynamic models in systems biology. and is structured into two parts. Both parts concern building dynamic models from observed data, but are quite different in perspective, rationale and mathematics. The first part considers the development of novel identification techniques that are particularly tailored to (molecular) biology and considers two approaches. The first approach reformulates the parameter estimation problem as a feasibility problem. This reformulation allows the invalidation of models by analysing entire parameter regions. The second approach utilises nonlinear observers and a transformation of the model equations into parameter free coordinates. The parameter free coordinates allow the design of a globally convergent observer, which in turn estimates the parameter values, and further, allows to identify modelling errors or unknown inputs/influences. Both approaches are bottom up approaches that require a mechanistic understanding of the underlying processes (in terms of a biochemical reaction network) leading to complex nonlinear models. The second part is an example of what can be done with classical, well developed tools from systems identification when applied to hitherto unattended problems.In particular, part two of my thesis develops a modelling framework for rat movements in an experimental setup that it widely used to study learning and memory.The approach is a top down approach that is data driven resulting in simple linear models

Risk-aware and Robust Approaches for Machine Learning-supported Model Predictive Control for Iterative Processes

The recent advances in machine learning have catalyzed a renewed interest in machine-learning-supported model predictive control. Machine learning promises to facilitate modeling and improve the process' performance. Nevertheless, it brings some challenges: For instance, as the connection with physics law is (partially) lost, machine learning models can provide wildly inaccurate results. It is therefore necessary to provide control methods that take the model uncertainty of these models into account. Uncertainties are even more important for iterative processes - processes that do not operate at a steady state - due to the large changes in the process conditions during operation. In this work, two methods for data-driven uncertainty modelling are proposed. The first method uses Gaussian processes to learn the model uncertainty and neural networks to learn the nominal model. It provides an simple way to summarize the uncertainty of the model into a single parameter, which can be used by a model predictive controller to make risk-aware decisions. This method, while being simple, does not guarantee constraint satisfaction. The second method is based on tube-based model predictive control and can guarantee constraint satisfaction. It is based on the concept of the "safe set": a set where a tube-based MPC has a feasible solution. We show that, under some assumptions, the safe set enlarges at every iteration of the process, potentially allowing increased performance. Finally, a novel Python library for machine-learning-based model predictive control, called HILO-MPC, is presented. This library interfaces with TensorFlow and PyTorch and provides easily-accesible tools for defining control and estimation problem using machine learning model

Model-Based State Estimation for Fault Detection under Disturbance

The measurement of process states is critical for process monitoring, advanced process control, and process optimization. For chemical processes where state information cannot be measured directly, techniques such as state estimation need to be developed. Model-based state estimation is one of the most widely applied methods for estimation of unmeasured states basing on a high-fidelity process model. However, certain disturbances or unknown inputs not considered by process models will generate model-plant mismatch. In this dissertation, different model-based process monitoring techniques are developed and applied for state estimation under uncertainty and disturbance. Case studies are performed to demonstrate the proposed methods. The first case study estimates leak location from a natural gas pipeline. Non-isothermal state equations are derived for natural gas pipeline flow processes. A dual unscented Kalman filter is used for parameter estimation and flow rate estimation. To deal with sudden process disturbance in the natural gas pipeline, an unknown input observer is designed. The proposed design implements a linear unknown input observer with time-delays that considers changes of temperature and pressure as unknown inputs and includes measurement noise in the process. Simulation of a natural gas pipeline with time-variant consumer usage is performed. New optimization method for detection of simultaneous multiple leaks from a natural gas pipeline is demonstrated. Leak locations are estimated by solving a global optimization problem. The global optimization problem contains constraints of linear and partial differential equations, integer variable, and continuous variable. An adaptive discretization approach is designed to search for the leak locations. In a following case study, a new design of a nonlinear unknown input observer is proposed and applied to estimate states in a bioreactor. The design of such an observer is provided, and sufficient and necessary conditions of the observer are discussed. Experimental studies of batch and fed-batch operation of a bioreactor are performed using Saccharomyces cerevisiae strain mutant SM14 to produce β-carotene. The state estimation of the process from the designed observer is demonstrated to alleviate the model-plant mismatch and is compared to the experimental measurements