# An Innovative Applied Control System of Helicopter Turboshaft Engines Based on Neuro-Fuzzy Networks
## 2 Materials and Methods
### Development of Fuzzy Controller  
The research proposes the implementation of a fuzzy controller using a fuzzy neural network (Figure 5) for the developed control law (23) by structuring the model with fuzzification, inference, and defuzzification components, integrating the provided specific control laws. The fuzzy controller now takes the control error \(e(t)\) and the error change rate \(\frac{de(t)}{dt}\) as input parameters; the output is the control action \(u(t)\). This structure includes the fuzzification block, converting the control error \(e(t)\) and its change rate \(\frac{de(t)}{dt}\) into fuzzy values. An inference machine uses fuzzy rules to derive the appropriate control action based on the inputs. The defuzzification block converts the fuzzy output into a crisp control action. A fuzzy rule base contains rules linking \(e(t)\) and \(\frac{de(t)}{dt}\) to the control output \(u(t)\). The control law implements the control action according to the provided control law. The input parameters are as follows:  
1. The control error is calculated as \[e(t)=e_{n_{TC}}+e_{n_{FT}}+e_{T_{G}^{*}}+e_{T_{N}}+\ldots,\] (33) where \(e_{n_{TC}}=n_{TC}^{given}-n_{TC}(t)\), \(e_{n_{FT}}=n_{FT}^{given}-n_{FT}(t)\), \(e_{T_{G}^{*}}=T_{G}^{given}-T_{G}^{*}(t)\). It is noted that the components in (33) may vary depending on the input data amount. 2. The error change rate \(\frac{de(t)}{dt}\). The fuzzification process converts these inputs into fuzzy values using membership functions, i.e.,  
\[\mu_{e}(e(t))=\exp(-\frac{\left(e(t)-e_{e(t)}\right)^{2}}{2\cdot \sigma_{e(t)}^{2}}), \tag{34}\] \[\mu_{\frac{de(t)}{dt}}\left(\frac{de(t)}{dt}\right)=\exp(-\frac{ \left(\frac{de(t)}{dt}-\frac{de(t)}{dt}\right)^{2}}{2\cdot\sigma_{\frac{de(t)} {dt}}^{2}}).\]  
The inference machine applies fuzzy rules to the fuzzified inputs \(\overset{\sim}{e}(t)\) and \(\frac{d\overset{\sim}{e}(t)}{dt}\) to determine the control action. The fuzzy rule base can be written as follows:  
1. If \(e(t)\) is \"negative large\" and \(\frac{de(t)}{dt}\) is \"negative large\", then\(u(t)\) is \"large positive\". Rule 2 : If \(e(t)\) is \"small\" and \(\frac{de(t)}{dt}\) is \"zero\", then \(u(t)\) is \"zero\". Figure 5: The fuzzy controller scheme implemented as a fuzzy neural network. The inference process is performed using a fuzzy logic operator (e.g., minimum or product rule):  
\[\mu_{out}(u(t))=\min\biggl{(}\mu_{e}(e(t)),\mu_{\frac{de(t)}{dt}}\biggl{(}\frac{ de(t)}{dt}\biggr{)}\biggr{)}. \tag{36}\]  
Expression (36) gives the fuzzy output membership function for the control action \(u(t)\). Defuzzification converts the fuzzy output into a crisp control value using (27). The fuzzy rule base is constructed to link the control error \(e(t)\) and its change rate \(\frac{de(t)}{dt}\) to the control action \(u(t)\). The rules define how the control system should respond to different combinations of \(e(t)\) and \(\frac{de(t)}{dt}\). Example rules include the following:  
\[ \begin{array}{ll}\text{Rule 1 : If $e(t)$ is ``positive large'' and $\frac{de(t)}{dt}$ is ``positive small'', then $u(t)$ is ``decrease''.}\\ \text{Rule 2 : If $e(t)$ is ``negative small'' and $\frac{de(t)}{dt}$ is ``negative large'', then $u(t)$ is ``increase''.}\\ \end{array}  \tag{37}\]  
The fuzzy control action is used to regulate system parameters according to the given control law (23). Examples are given below:  
\[