Application to the Chemostat
This chapter is devoted to the stabilization of chemostats under measurement delays. Continuous stirred microbial bioreactors, often called chemostats, cover a wide range of applications; specialized “pure culture” biotechnological processes for the produ
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Application to the Chemostat
1 Introduction This chapter is devoted to the stabilization of chemostats under measurement delays. Continuous stirred microbial bioreactors, often called chemostats, cover a wide range of applications; specialized “pure culture” biotechnological processes for the production of specialty chemicals (proteins, antibiotics, etc.) as well as large-scale environmental technology processes of mixed cultures such as wastewater treatment. The dynamics of the chemostat is often adequately represented by a simple dynamic model involving two state variables, the microbial biomass X and the limiting organic substrate s [55]. For control purposes, one input is usually considered: the dilution rate D. A general model for microbial growth on a limiting substrate in a chemostat is of the form: X_ ¼ ðμðsÞ DÞX s_ ¼ DðS0 sÞ γ 1 μðsÞX
ð1:1Þ
where S0 is the feed substrate concentration, μ(s) is the specific growth rate, and γ > 0 is a biomass yield factor (constant). As seen in what follows, one important example is anaerobic digestion, which finds many applications, e.g., in wastewater treatment, sludge management, and energy from biomass. In equation (1.1), the specific growth rate μ(s) is given by an empirical correlas tion, e.g., the Monod equation μðsÞ ¼ Kμmax , or the Haldane equation μðsÞ ¼ μmax s s2 , S þs K S þsþK
I
with all parameters μmax, KS, KI, . . . being positive. In all types of empirical kinetic equations, the function μ(s) is a C1 function that satisfies
© Springer International Publishing AG, CH 2017 I. Karafyllis, M. Krstic, Predictor Feedback for Delay Systems: Implementations and Approximations, Systems & Control: Foundations & Applications, DOI 10.1007/978-3-319-42378-4_6
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6 Application to the Chemostat
μðsÞ > 0 for all s > 0 : μð0Þ ¼ 0
However, the specific properties of the function μ(s) (its monotonicity) affect the number and stability properties of the chemostat’s steady states. For example, in the case of Monod kinetics (which is an increasing function) under constant D and S0, there can only be one nontrivial steady state and it is always globally asymptotically stable, but the situation is entirely different in the case of Haldane kinetics (due to the change in monotonicity). The problem of the stabilization of a nontrivial steady state of the chemostat was considered and solved in [56]. However, here we consider the stabilization problem for the case where the measurements are sampled and delayed: Y ðtÞ ¼ ½ Xðτi r Þ
sðτ i r Þ 0
ð1:2Þ
where fτi g1 i¼0 is a partition of ℜþ and r > 0 is the measurement delay. This situation is typical for all (bio)chemical processes, for which the measurement of concentration of (bio)chemical species is involved. Indeed, (bio)chemical analyzers need some time to analyze the sample, which is taken periodically from the bioreactor. Therefore, there is need to compensate for the measurement delay and take into account sampling. Although there is (usually) no uncertainty in the sampling times, usually there i
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