## 1 Introduction
### Existing Solutions and Literature Review
A control system that would ensure optimal performance, even in the case of an unknown mathematical model of the fermentation process, would be very useful [21]. Following a careful review of published results, by studying and simulating the assessment of various modern control theories, and considering the known dynamic properties of bioreactors, a particular theory of adaptive control stands out. It represents conclusively one of the formidable theories for the development of control systems of bioreactors. Adaptive control systems are able to adapt themselves with a view to assuring optimal behaviour of the control system under real circumstances. The development of adaptive systems was especially intense in the years 1960-1990, when many different adaptive control concepts appeared, some ofwhich survived and are still in use with certain additions. Today, we divide most of the adaptive control systems into three large groups:  
* Gain Scheduling Control (GSC), which is actually a link between conventional control and adaptive control. Because of its transparency and simplicity, gain scheduling is probably the most widely used non-linear control technique. To use this approach, it is necessary to find measurable quantities, called scheduling quantities, which correlate well with the process dynamics. The controller's parameters are changed through measurement of the scheduling quantities. * Self-Tuning Control (STC), also known as indirect or explicit adaptive control, consists of three separate modules: controlled plant parameter identification, tuning of the controller parameters, and control law realisation. An STC system utilises two information channels: a conventional control feedback loop, which enables information about controlled plant output to affect controller output, and an additional \"parameters\" channel, which facilitates the adaptation of the controller's parameters based on the controlled plant parameters' identification. * Model Reference Adaptive Control (MRAC), also known as direct or implicit adaptive control, applies a reference model to specify the performance of a controlled plant. A comparison of the reference performance index and the real performance index is made directly by comparing the outputs (or the states) of the reference model and the controlled plant. An MRAC system generates an additional control signal, so that the response of the controlled plant to the control is close to that given by the reference model. There are two basic structures of the model reference system, namely, parallel and series, and two basic implementations of the adaptive approach, namely, signal synthesis adaptation and parameter adaptation. The most popular MRAC structure is a parallel configuration with signal synthesis adaptation. Major progress in the field of MRAC systems was made by [PERSON] (1958), where [PERSON]'s stability theory and [PERSON]'s hyper stability theory were used for designing MRAC systems with guaranteed stability. MRAC is important, because of its easy implementation and high-speed adaptation. It can be used in a variety of situations. Due to the listed advantages, we used the MRAC theory to develop a control system for the control of the fermentation process in a batch bioreactor.