In 1960, Rudolf E. Kalman created what is known as the Kalman filter, which is a way to estimate unknown variables from noisy measurements. The algorithm follows the logic that if the previous state of the system is known, it could be used as the best guess for the current state. This information is first applied a priori to any measurement by using it in the underlying dynamics of the system. Second, measurements of the unknown variables are taken. These two pieces of information are taken into account to determine the current state of the system. Bayesian inference is specifically designed to accommodate the problem of updating what we think of the world based on partial or uncertain information. In this paper, we present a derivation of the general Bayesian filter, then adapt it for Markov systems. A simple example is shown for pedagogical purposes. We also show that by using the Kalman assumptions or “constraints”, we can arrive at the Kalman filter using the method of maximum (relative) entropy (MrE), which goes beyond Bayesian methods. Finally, we derive a generalized, nonlinear filter using MrE, where the original Kalman Filter is a special case. We further show that the variable relationship can be any function, and thus, approximations, such as the extended Kalman filter, the unscented Kalman filter and other Kalman variants are special cases as well.

Background: The focus of this study is online estimation of biomass concentration in fed-batch cultures. It describes a bioengineering software solution, which is explored for Escherichia coli and Saccharomyces cerevisiae fed-batch cultures. The experimental investigation of both cultures presents experimental validation results since the start of the bioprocess, i.e. since the injection of inoculant solution into bioreactor. In total, four strains were analyzed, and 21 experiments were performed under varying bioprocess conditions, out of which 7 experiments were carried out with dosed substrate feeding. Development of the microorganisms’ culture invariant generic estimator of biomass concentration was the main goal of this research. Results: The results show that stoichiometric parameters provide acceptable knowledge on the state of biomass concentrations during the whole cultivation process, including the exponential growth phase of both E. coli and S. cerevisiae cultures. The cell culture stoichiometric parameters are estimated by a procedure based on the Luedeking/Piret-model and maximization of entropy. The main input signal of the approach is cumulative oxygen uptake rate at fed-batch cultivation processes. The developed noninvasive biomass estimation procedure was intentionally made to not depend on the selection of corresponding bioprocess/bioreactor parameters. Conclusions: The precision errors, since the bioprocess start, when inoculant was injected to a bioreactor, confirmed that the approach is relevant for online biomass state estimation. This included the lag and exponential growth phases for both E. coli and S. cerevisiae. The suggested estimation procedure is identical for both cultures. This approach improves the precision achieved by other authors without compromising the simplicity of the implementation. Moreover, the suggested approach is a candidate method to be the microorganisms’ culture invariant approach. It does not depend on any numeric initial optimization conditions, it does not require any of bioreactor parameters. No numeric stability issues of convergence occurred during multiple performance tests. All this makes this approach a potential candidate for industrial tasks with adaptive feeding control or automatic inoculations when substrate feeding profile and bioreactor parameters are not provided.

This contribution discusses the main challenges related to successful application of automatic control systems used to control specific growth rate in industrial biotechnological processes. It is emphasized that, after the implementation of basic automatic control systems, primary attention shall be paid to the specific growth rate control systems because this process variable critically affects the physiological state of microbial cultures and the formation of the desired product. Therefore, control of the specific growth rate enables improvement of the quality and reproducibility of the biotechnological processes. The main requirements have been formulated that shall be met to successfully implement the specific growth rate control systems in industrial bioreactors. The relatively easy-to-implement schemes of specific growth rate control systems have been reviewed and discussed. The recommendations for selection of particular control systems for specific biotechnological processes have been provided.

The cell cultivation process in a bioreactor is a high-value manufacturing process that requires excessive monitoring and control compatibility. The specific cell growth rate is a crucial parameter that describes the online quality of the cultivation process. Most methods and algorithms developed for online estimations of the specific growth rate controls in batch and fed-batch microbial cultivation processes rely on biomass growth models. In this paper, we present a soft sensor – a specific growth rate estimator that does not require a particular bioprocess model. The approach for online estimation of the specific growth rate is based on an online measurement of the oxygen uptake rate. The feasibility of the estimator developed in this study was determined in two ways. First, we used numerical simulations on a virtual platform, where the cell culture processes were theoretically modeled. Next, we performed experimental validation based on laboratory-scale (7, 12, 15 L) bioreactor experiments with three different Escherichia coli BL21 cell strains.

For historic reasons, industrial knowledge of reproducibility and restrictions imposed by regulations, open-loop feeding control approaches dominate in industrial fed-batch cultivation processes. In this study, a generic gray box biomass modeling procedure uses relative entropy as a key to approach the posterior similarly to how prior distribution approaches the posterior distribution by the multivariate path of Lagrange multipliers, for which a description of a nuisance time is introduced. The ultimate purpose of this study was to develop a numerical semi-global convex optimization procedure that is dedicated to the calculation of feeding rate time profiles during the fed-batch cultivation processes. The proposed numerical semi-global convex optimization of relative entropy is neither restricted to the gray box model nor to the bioengineering application. From the bioengineering application perspective, the proposed bioprocess design technique has benefits for both the regular feed-forward control and the advanced adaptive control systems, in which the model for biomass growth prediction is compulsory. After identification of the gray box model parameters, the options and alternatives in controllable industrial biotechnological processes are described. The main aim of this work is to achieve high reproducibility, controllability, and desired process performance. Glucose concentration measurements, which were used for the development of the model, become unnecessary for the development of the desired microbial cultivation process.

In this paper, we continue our efforts to show how maximum relative entropy (MrE) can be used as a universal updating algorithm. Here, our purpose is to tackle a joint state and parameter estimation problem where our system is nonlinear and in a non-equilibrium state, i.e., perturbed by varying external forces. Traditional parameter estimation can be performed by using filters, such as the extended Kalman filter (EKF). However, as shown with a toy example of a system with first order non-homogeneous ordinary differential equations, assumptions made by the EKF algorithm (such as the Markov assumption) may not be valid. The problem can be solved with exponential smoothing, e.g., exponentially weighted moving average (EWMA). Although this has been shown to produce acceptable filtering results in real exponential systems, it still cannot simultaneously estimate both the state and its parameters and has its own assumptions that are not always valid, for example when jump discontinuities exist. We show that by applying MrE as a filter, we can not only develop the closed form solutions, but we can also infer the parameters of the differential equation simultaneously with the means. This is useful in real, physical systems, where we want to not only filter the noise from our measurements, but we also want to simultaneously infer the parameters of the dynamics of a nonlinear and non-equilibrium system. Although there were many assumptions made throughout the paper to illustrate that EKF and exponential smoothing are special cases ofMrE, we are not “constrained”, by these assumptions. In other words, MrE is completely general and can be used in broader ways.