Design of a Cascade Observer for a Model of Bacterial Batch Culture with Nutrient Recycling

Miled El Hajji and Alain Rapaport

Additional information is available at the end of the chapter http://dx. doi. org/10.5772/52997

1. Introduction

Microbial growths and their use for environmental purposes, such as bio-degradations, are widely studied in the industry and research centres. Several models of microbial growth and bio-degradation kinetic have been proposed and analysed in the literature. The Monod’s model is one of the most popular ones that describes the dynamics of the growth of a biomass of concentration X on a single substrate of concentration S in batch culture [15,18]:

S = -*£>X, X=},(S)X. (1)

image041 Подпись: (2)

where the specific growth rate ц(- ) is:

with цтах, Ks and Yare repsectively the maximum specific growth rate, the affinity constant and the yield coefficient. Other models take explicitly into account a lag-phase before the growth, such as the Baranyi’s [1-3] or the Buchanam’s [6] ones. These models are well suited for the growth phase (i. e. as long as a substantial amount of substrate remains to be converted) but not after [18], because the accumulation of dead or non-viable cells is not taken into account. Part of the non-viable cells release substrate molecules, in quantities that are no longer negligible when most of the initial supply has been consumed. The on-line observation of the optical density of the biomass provides measurements of the total biomass, but not of the proportion among dead and viable cells. Some tools allow the distinction between viable and dead cells but do not detect non-viable non-dead ones [22].

In this work, we consider an extension of the model (1) considering both the accumulation of dead cells and the recycling of part of it into substrate, and tackle the question of parameters

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and state reconstruction. To our knowledge, this kind of question has not been thoroughly studied in the literature. Models of continuous culture with nutrient recycling have already been studied [4, 5, 9, 12-14, 16, 20, 21, 24, 25] but surprisingly few works considers batch cultures. A possible explanation comes from the fact that only the first stage of the growth, for which cell mortality and nutrient recycling can be neglected, is interested for industrial applications. Nevertheless, in natural environment such as in soils, modelling the growth end is also important, especially for biological decontamination and soil bioremediation.

Moreover, we face a model for which the parameters are not identifiable at steady state. Then, one cannot apply straightforwardly the classical estimation techniques, that usually requires the global observability of the system. Estimation of parameters in growth models, such as the Baranyi’s one, are already known to be difficult to tackle in their differential form [11]. In addition, we aim here at reconstructing on-line unmeasured state variables (amounts of viable and non-viable cells), as well as parameters. For this purpose, we propose the coupling of two non-linear observers in cascade with different time scales, providing a practical convergence of the estimation error. Design of cascade observers in biotechnology can be found for instance in [17, 23], but with the same time scale.