# An Innovative Applied Control System of Helicopter Turboshaft Engines Based on Neuro-Fuzzy Networks
## 2 Materials and Methods  
### Development of Helicopter Turboshaft Engine Adaptive Control Law  
Based on [26; 27; 28], the helicopter TE regulation law mathematical model, grounded in the present text, can be outlined as follows:  
1. Engine (gas generator) operation is controlled by varying the fuel supply \(G_{T}\). It is the sole parameter influencing the gas generator's operating mode, as the compressor is fixed, and the first stage of the accessible turbine functions as a throttle with a constant cross-section. 2. Maintaining constant free turbine speed \(n_{FT}\) = const is key for safe helicopter piloting. To meet this condition, the automatic control system (ACS) adjusts \(G_{T}\) when the blade pitch angle of the main rotor \(\varphi_{m.r. }\) changes. 3. The control model is expressed as follows: * \(n_{FT}\) = const, if \(\varphi_{m.r. }\leq\varphi_{m.r.\max}\); * If \(\varphi_{m.r. }>\varphi_{m.r.\max}\), the ACS limits fuel supply, and \(n_{FT}\) starts to decrease to constrain one of the limiting parameters (e.g., maximum turbine speed \(n_{TC\max}\) or maximum power \(N_{e}\)). 4. The free turbine rotor speed regulator central equation when the fuel supply changes is as follows: \[n_{FT}= f(G_{T},\varphi_{m.r.},n_{TC},N_{e}).\] (1)
5. The power limitation condition is represented as follows: \[N_{e}\leq N_{e}^{\max}.\] (2)
6. Upon reaching the maximum power limit \(N_{e}=N_{e}^{\max}\), maintaining \(n_{FT}\) = const becomes impossible, and the \(n_{FT}\) frequency decreases. The central regulation equation for the free turbine while maintaining the rotational frequency \(n_{FT}\) can be expressed through the power balance:  
\[N_{FT}=N_{m.r.}+N_{power\ loss}. \tag{3}\]  
The free turbine power is expressed as follows:  
\[N_{FT}=\eta_{FT}.\frac{G_{T}\cdot\dot{Q}}{c_{p}.T_{G}^{*}}, \tag{4}\]  
where \(\dot{Q}=G_{T}\cdot H_{u}\). The variation in fuel supply based on the rotational frequency of the free turbine and required power is defined by the following expression:  
\[G_{T}=\frac{n_{FT}^{2}\cdot N_{m.r.}}{\eta_{FT}.H_{u}\cdot c_{p}.T_{G}^{*}}. \tag{5}\]Thrust control for the main rotor through the blade pitch angle is performed according to the following equation:  
\[T_{m.r.}=C_{T}(\varphi_{m.r.})\cdot\rho\cdot n_{FT}^{2}\cdot A_{m.r.}. \tag{6}\]  
The control relations for the blade pitch angle are represented as follows:  
\[\varphi_{m.r.}=f(T_{m.r.},H,P_{a},T_{a}). \tag{7}\]  
Considering changes in altitude and ambient temperature, correction factors for fuel flow are introduced:  
\[G_{T}(H,T_{a})=G_{T}\cdot\bigg{(}1+k_{H}\cdot\frac{H}{H_{max}}\bigg{)}\cdot \bigg{(}1+k_{T}\cdot\frac{T_{a}-T_{N}}{T_{N}}\bigg{)}. \tag{8}\]  
To ensure safe operation, restrictions are introduced on parameters such as maximum power \(N_{e}\), gas generator rotor r.p.m. \(n_{TC\text{max}}\), and the gas temperature in the front of the turbine \(T_{G}^{*\text{max}}\)[29; 30]:  
\[ \begin{array}{l}N_{e}=N_{e}^{\text{max}},\\ n_{TC}\leq n_{TC\text{max}},\\ T_{G}^{*}\leq T_{G}^{*\text{max}}.\end{array}  \tag{9}\]  
If any restriction is reached, the system adjusts fuel flow \(G_{T}\) to maintain engine operation within safe values:  
\[G_{T}=\min\Big{(}G_{T},G_{T}^{lim}\Big{)}. \tag{10}\]  
To ensure effective system operation across different flight modes, an adaptive control law is introduced which adjusts parameters based on the current power \(N_{e}\) and frequency \(n_{FT}\):  
\[G_{T}=G_{T}^{base}\cdot\bigg{(}1+\alpha\cdot\frac{N_{e}-N_{e}^{req}}{N_{e}^{ req}}\bigg{)}\cdot\bigg{(}1+\beta\cdot\frac{n_{FT}-n_{FT}^{req}}{n_{FT}^{req}} \bigg{)}. \tag{11}\]  
By combining all equations, a general control law is obtained:  
\[\varphi_{m.r.}=f(T_{m.r.},\varphi_{m.r.},H,P_{a},T_{a},T_{G}^{*}). \tag{12}\]  
To develop a mathematical model for the control law numerical solution in the \"pitch-throttle\" system for helicopter TEs, employing numerical optimization methods and discretization of the differential equation system describing engine dynamics proves effective [31; 32].