## Materials and Methods
### Adaptive Control Algorithm
#### 2.2.1 Fuzzy-Based Algorithm Design  
To create a fuzzy-based adaptive PI controller, input and output variables for the fuzzy system need to be defined. Since the function of the fuzzy model is to adapt the PI controller parameters, controller gain \(K_{c}\) and integration time constant \(T_{i}\) parameters were selected as the output variables. Both \(T_{i}\) and \(K_{c}\) depend on the process variables as described in [19]:  
\[K_{c}\propto k_{0}w \tag{10}\]  
\[T_{i}\propto\frac{k_{1}}{OUR/w+k_{2}} \tag{11}\]  
where \(k_{0}\), \(k_{1}\) and \(k_{2}\) are fixed parameters, the values of which are subject to optimization and were calculated and discussed in detail in [19]. The aim of the optimization was to find a set of the parameters that minimizes the tracking error (e.g., integral time absolute error (ITAE)) of the designed control system when using the Equations (10) and (11) for controller parameter adaptation. Since the PI controller parameters mainly depend on \(OUR\) and \(w\), both process variables need to be used in the fuzzy system. To decrease the complexity of the fuzzy system, the ratio of these two process variables was used as the input for the fuzzy system. According to the heuristic knowledge available, the following rules were created for the fuzzy system:  
IF OUR/w is poor, THEN T\({}_{i}\) is big  
IF OUR/w is excellent, THEN T\({}_{i}\) is small  
IF OUR/w is poor, THEN K\({}_{c}\) is small  
IF OUR/w is excellent, THEN K\({}_{c}\) is big  
Both for the input and output of the fuzzy model, two Gaussian membership functions were selected. To avoid the extensive overlapping of the membership functions, the number of used membership functions per parameter was decreased to two (see Figure 2). Whether a fuzzy application is implemented successfully depends on a number of factors. One of these are the membership functions of the fuzzy model. However, these are usually decided upon subjectively. One way to improve the performance of the fuzzy model is the use of genetic algorithm (GA). Various studies have shown that GAs and other heuristic algorithms are able to improve the systems performance by optimizing the membership functions [27; 28; 29]. Due to its simplicity and straightforwardness, a genetic algorithm [30] was selected to find the optimal membership function parameters. GAs adapt a direct analogy of natural evolution to perform global optimization in order to solve highly complex problems. It assumes that the possible problem solution is individual and can be described by a set of parameters. These parameters are coded as genes of a chromosome and can be structured by a string of concatenated values. Variable representation is defined by the encoding scheme that can be represented by various forms like binary or real numbers, depending on the used data. The search space of the data is usually defined by the problem. In this research, the fuzzy model membership function parameters were coded in the chromosomes. At the beginning, an initial  
Figure 1: General structure of the fuzzy-based adaptive PI control system for specific growth rate control. chromosome is randomly generated. Then, the fitness values of all chromosomes are evaluated by calculating the objective function in a decoded form. The integral time absolute error (ITAE) value of the SGR setpoint tracking was selected as the fitness function since this criterion was also used for the systems performance evaluation. Determined by the fitness of each individual, the best chromosomes are grouped, based on the lowest ITAE value, and the group with the lowest value is then selected in the selection process. The genetic operators like crossover and mutation are applied to this selected population in order to improve the next generation solution. The process is repeated until the population converges to the global minimum or another termination criterion is reached. In the reproduction phase, the fitness value of each chromosome is evaluated, and it is used in the selection process to provide bias towards fitness individuals.