# An Innovative Applied Control System of Helicopter Turboshaft Engines Based on Neuro-Fuzzy Networks
## 2 Materials and Methods
### Development of Fuzzy Controller
\begin{array}{ll}1.&\left(n^{given}_{TC}\right)_{lim}=f(T_{N}).\text{ This control law uses the error in gas generator rotor r.p.m. }\\ &e_{t_{TC}}=n^{given}_{TC}-n_{TC}(t)\text{ as one of the inputs.}\\ 2.&\text{G}_{\text{T}}=G_{\text{T}}^{base}+k_{G}\cdot\Delta n_{FT}(t).\text{ It can be controlled based on the fuzzy output $u(t)$, where u(t) affects $\Delta n_{\text{FT}}(t)$, the change in free turbine speed.}\\ 3.&\varphi_{m.r.}(t+\Delta t)=\varphi_{m.r.}(t)+k_{\varphi}\cdot\Delta T_{m.r. }(t),\text{ where the control action $u(t)$ determines the adjustment to the rotational speed $\varphi_{m.r.}(t).}\\ 4.&\text{The control actions must also satisfy the constraints given in the system $N_{e}=N_{e}^{\text{max}}$, $\frac{n_{TC}}{dt}\leq\frac{n_{TC}}{dt}$max, $T^{*}_{G}\leq T^{\prime}_{G}$.}\\ \end{array} \]  
The fuzzy neural network ensures that the control actions adhere to these limits by adjusting \(u(t)\) accordingly. The final control action \(u(t)\) is computed based on the fuzzified error \(e(t)\) and error change rate \(\frac{de(t)}{dt}\) passed through the inference machine and defuzzification block. The result is a crisp value that modifies the system parameters in real time:  
\[u(t)=\{\textit{Defuzzified output from fuzzy controller}\}. \tag{38}\]  
This fuzzy neural network design allows for the adaptive control of the helicopter TE parameters based on the error and error change rate, ensuring robust performance across different operating conditions. Fault tolerance is assessed by the system's ability to correctly compensate for the impact of failures on control and engine performance. A corrective control signal \(u_{c}(t)\) is generated based on fuzzy adaptation rules in the event that failures occur:  
\[u(t)=u_{0}(t)+u_{c}(t). \tag{39}\]  
It is assumed that \(d(t)\) is a failure vector, which is described through binary failure indicators (for example, sensor failure, drive failure):  
\[d(t)=[d_{1}(t),\,d_{2}(t),\,\ldots,\,d_{k}(t)]^{T} \tag{40}\]  
where \(d_{i}(t)\) takes the value 1 in the case of failure and 0 in the case of regular operation. A cost function / is introduced to evaluate system deviations from the target values:  
\[J=\int\limits_{0}^{T}\left\|x(t)-x_{ref}(t)\right\|^{2}+\gamma_{1}\cdot\|u(t) \|^{2}+\gamma_{2}\cdot\|d(t)\|^{2}dt. \tag{41}\]  
The system is considered fault-tolerant if \(J\leq J_{max}\), where \(J_{max}\) is the maximum permissible deviation level. After failure, the system applies an iterative recovery process,minimizing the cost function \(J\). Each iteration of correction takes the current failure vector \(d(t)\) into account and updates the corrective control signal \(u_{c}(t)\), i.e.,  
\[u_{c}^{(k+1)}=u_{c}^{(k+1)}-\gamma\cdot\frac{\partial J}{\partial u_{c}}. \tag{42}\]  
The proposed fault tolerance mathematical model in the helicopter TE fuzzy control system enables the simulation of failure impacts and adaptive control strategies to minimize their effects. Based on fuzzy logic, the system dynamically adjusts control actions, ensuring the stability and reliability of engine operation in real time.