# Optimized Dissolved Oxygen Fuzzy Control for Recombinant _Escherichia coli_ Cultivations
## 1 Introduction  
Cultivation of _Escherichia coli_ has been extensively studied, focusing on high productivity, oxygen availability, medium composition, acetate production, feeding control strategy, bioreactor type, and operation policy [1, 2, 3, 4]. A major challenge is to ensure an adequate and continuous dissolved oxygen supply in the cultivation medium during the process. It is well documented that oxygen deprivation can cause harmful effects on _E. coli_ growth and biomass productivity due to the accumulation of inhibitory by-products (such as acetate) [5, 6, 7, 8]. In a series of works, [PERSON] and [PERSON] (1999) studied the correlation between oxygen availability and _E. coli_ cell growth. At first, high oxygen availability can lead to a high biomass yield, with low acetate production. However, at a certain point, oxygen surplus becomes detrimental for the cultivation (decreasing both biomass yield and specific growth rate and increasing acetate production) [9, 10, 11]. Given the low solubility of oxygen in water and mechanical stirring limitations of the bioreactor (limited agitation speed to avoid shearing and limited oxygen delivered by the gas sparge), the dissolved oxygen control is highly nonlinear and represents a challenging control problem [12, 13]. Regarding the current control strategies, the majority struggles with dissolved oxygen concentration (DOC) peaks and non-smooth behavior of air and oxygen flow rates. In this context, efforts were made in order to improve the DOC control in bioreactors. To overcome difficulties with varying process dynamics, adaptive controllers have been suggested. [PERSON] et al. (2018) proposed a one step ahead DOC controller. Process modeling was based on material balance equations, and it was linearized via first order Euler approximation. Its parameters were re-evaluated by a modified recursive least squares method, and, as for the one-step-ahead predictor, the extended Kalman filter was applied in order to reduce noise effects. Manipulated variables were air flow rate and agitation speed. Results showed an improvement in performance and stability compared to the classic PID controller [PERSON] et al. (2018). Aiming to create a control strategy independent from complex modeling equations, [PERSON] et al. (2018) employed a modified AutoRegressive-Moving-Average model. Its parameters were re-evaluated by a recursive linear regression method. Authors presented an indirect pole placement controller with air flow rate and agitation speed as manipulated variables. Although positive results were obtained for wild _E. coli_ cultivation, the controller had mild problems trying to maintain DOC near 30% during the induction phase [PERSON] et al. (2018). [PERSON] and [PERSON] (2018) and [PERSON] et al. (2018) designed adaptive controllers based on gain-scheduling. Both process models were based on material balance equations. In the first work, five operation points were stablished based on manipulated variable stirrer speed. In the second work, 16 operation points were stablished based on agitation speed and air flow rate. In Reference [PERSON] et al. (2018), gain scheduling was combined with a PID controller, whilst, in Reference [PERSON] et al. (2018), the technique was combined with a PI controller. Both adaptive controllers were tested in simulations and validated in cultivations. Results, in general, were satisfactory, although the first controller had tunning issues, and the second presented difficulties with moderate disturbances. [PERSON] et al. (2018) designed a Model Reference Adaptive Control scheme, based on MIT rule [PERSON] et al. (2018), applied to DOC control. In this strategy, the controller parameters were adjusted so that the output of the process tracked the output reference model (with the same reference input). For the simulations, the process model was designed based on experimental data. As in Reference [PERSON] et al. (2018), the single manipulated variable was stirrer speed. Simulations showed satisfactory results comparing with a PI controller.