# 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
The \"pitch-throttle\" system [33] regulates fuel flow and the angle of attack of the main rotor blades to maintain the free turbine, which requires rotational frequency \(n_{FT}\), power \(N_{e}\), and rotor thrust \(T_{m.r.}\). The free turbine rotational frequency \(n_{FT}\) depends on fuel flow and the main rotor resistance torque. In differential form, the equation can be expressed as follows:  
\[J_{FT}\cdot\frac{dn_{FT}(t)}{dt}=M_{FT}(t)-M_{m.r.}(t). \tag{13}\]  
The free turbine moment \(M_{FT}(t)\) depends on fuel flow \(G_{T}(t)\) and gas temperature \(T_{G}^{*}\):  
\[M_{FT}(t)=\eta_{FT}\cdot\frac{G_{T}(t)\cdot H_{u}}{c_{p}\cdot T_{G}^{*}(t)}. \tag{14}\]  
The main rotor thrust \(T_{m.r. }(t)\) depends on the blade pitch angle \(\varphi_{m.r. }(t)\) and the free turbine speed \(n_{FT}(t)\):  
\[T_{m.r.}(t)=C_{T}(\varphi_{m.r.}(t))\cdot\rho(t)\cdot n_{FT}^{2}(t)\cdot A_{m.r.}. \tag{15}\]For numerical solutions, the Euler method is applied for time integration with a time discretization step \(\Delta t\)[34]. The discretized equation for free turbine speed is defined as follows:  
\[n_{FT}(t+\Delta t)=n_{FT}(t)+\frac{\Delta t}{f_{FT}}.(M_{FT}(t)-T_{m.r.}(t)). \tag{16}\]  
To ensure numerical stability, the time step \(\Delta t\) must satisfy a stability condition. Specifically, \(\Delta t\) should be less than a certain threshold determined by the system's dynamic properties, such as the inertial moment \(J_{FT}\) and the maximum expected rate of the torque difference (\(M_{FT}-T_{m.r. }\)) change. For stability, a common guideline is \(\Delta t<\frac{I_{FT}}{\max\left|\frac{d(M_{FT}(t)-T_{m.r.}(t))}{dt}\right|}\). The discretized equation for the main rotor thrust takes the following form:  
\[T_{m.r.}(t+\Delta t)=C_{T}(\varphi_{m.r.}(t))\cdot\rho(t)\cdot n_{FT}^{2}(t+ \Delta t)\cdot A_{m.r.}. \tag{17}\]  
The Development of the Algorithm for the Discretized Equation for the Main Rotor Thrust Numerical Solution  
An algorithm for the numerical solution is proposed, consisting of the following steps:  
1. Initialization of the initial conditions as follows: \[n_{FT}(0)=n_{FT0},\,\varphi_{m.r.}(0)=\varphi_{m.r.0},\,G_{T}(0)=G_{T0},\,H(0) =H_{0},\,T_{a}(0)=T_{a0}.\] (18)
2. Computation is performed at each time step: * Fuel flow \(G_{T}(t)\) and the blade pitch angle \(\varphi_{m.r. }(t)\) are updated according to the \"pitch-throttle\" system control law based on the specified target parameters \(n_{FT}^{req}\) and \(T_{m.r.}^{req}\). * The free turbine rotor speed \(n_{FT}(t+\Delta t)\) is calculated. * The thrust \(T_{m.r. }(t+\Delta t)\) is calculated. * External parameters \(H(t+\Delta t)\), \(T_{a}(t+\Delta t)\), and \(P_{a}(t+\Delta t)\) are updated. 3. Deviations are assessed. Deviations in turbine speed and thrust from target values are calculated according to the following expression: \[\Delta n_{FT}(t)=n_{FT}(t)-n_{FT^{*}}^{req}\,\Delta T_{m.r.}(t)=T_{m.r.}(t)-T_{ m.r.}^{req}\] (19)
4. Control signals are adjusted as follows: * If \(|\Delta n_{FT}(t)|>\varepsilon_{n}\), then then fuel flow \(G_{T}(t)\) is adjusted. * If \(|\Delta n_{FT}(t)|>\varepsilon_{T}\), then the blade pitch angle \(\varphi_{m.r. }(t)\) is adjusted. 5. Transition to the next time step is carried out according to the following expression: \[G_{T}=G_{T}^{base}\cdot\bigg{(}1+\alpha\cdot\frac{N_{e}-N_{e}^{req}}{N_{e}^{ req}}\bigg{)}\cdot\bigg{(}1+\beta\cdot\frac{n_{FT}-n_{FT}^{req}}{n_{FT}^{req}} \bigg{)}. \tag{11}\]0 (20)  
Based on the above, the proposed adaptive control law for fuel flow and the blade pitch angle involves adjusting fuel flow \(G_{T}(t)\) and the pitch angle \(\varphi_{m.r. }(t)\) according to deviations in rotational speed and thrust:  
\[ \begin{array}{c}G_{T}(t+\Delta t)=G_{T}(t)+k_{G}\cdot\Delta n_{FT}(t),\\ \varphi_{m.r.}(t+\Delta t)=\varphi_{m.r.}(t)+k_{\varphi}\cdot\Delta T_{m.r.}(t ).\end{array}  \tag{21}\]  
The system continues integration until the deviations in rotational speed \(n_{FT}\) and thrust \(T_{m.r.