# Backstepping Adaptive Neural Network Control for Electric Braking Systems of Aircrafts
## 2 Mathematical Model of the EMA
### Ideal EMA Mathematical Model  
The brushless DC motor (BLDCM) is usually controlled through a variable armature voltage. To simplify the mathematical model of the EMA, the following assumptions are made: the current fluctuation caused by the commutations is ignored, the magnetic condition of the motor is unsaturated, and the hysteresis losses and the eddy-current losses are ignored. The voltage equation and torque equation of the BLDCM are expressed as:  
\[\left\{ \begin{array}{l}\dot{L}=-Ri-E_{a}+U_{a}\\ \dot{J}\omega_{m}=-B_{v}\omega_{m}+T_{e}-T_{L}\\ T_{e}=k_{T}\dot{t},E_{a}=k_{m\
eq}\omega_{m}\end{array} \right., \tag{1}\]  
where \(U_{a}\) is the armature voltage, \(R\) is the stator resistance, \(i\) is the armature current, \(L\) is the stator inductance, \(E_{a}\) is the back electromotive force, \(k_{cmf}\) is the back electromotive force constant, \(J\) is the inertia moment, \(\omega_{m}\) is the mechanical speed of the motor, \(B_{v}\) is the viscous damping coefficient, \(T_{e}\) is the electromagnetic torque, \(k_{T}\) is the torque constant, and \(T_{L}\) is the load torque. The relationship between the vertical motion of the ball screw and the rotary motion of the motor is shown as:  
\[\dot{x}_{ema}=L_{0}\omega_{m}/2\pi\eta, \tag{2}\]  
where \(x_{ema}\) is the axis displacement of the EMA, \(L_{0}\) is the lead of the ball screw, and \(\eta\) is the transmission ratio of the reduction gear. Figure 1: Structure of the electric braking system in aircraft. By considering the essential characteristic of the electric braking system, the output braking pressure of the EMA can be calculated as:  
\[p=c_{b}(x_{cma}-x_{b}), \tag{3}\]  
where \(p\) is the braking pressure of the EMA, \(c_{b}\) is the stiffness coefficient of the brake disc and \(x_{b}\) is the lateral displacement of the brake disc. If the lateral displacement of the brake disc is assumed to be zero, Equation (3) can be simplified as:  
\[p=c_{b}x_{cma}. \tag{4}\]  
The load torque of the motor (\(T_{L}\)) can be divided into two parts: part of it is used to overcome the resistance torque of the ball screw, and the remainder is used to overcome the drive torque of the ball screw. Then, the load torque can be calculated using the following equation:  
\[T_{L}=T_{rt}+T_{dt}, \tag{5}\]  
where \(T_{rt}\) is the resistance torque of the ball screw and \(T_{dt}\) is the drive torque of the ball screw. In the EMA of the electric braking system, the resistance torque \(T_{rt}\) is assumed to be a constant value, and the drive torque \(T_{dt}\) is related to the output braking pressure \(p\). In most studies, \(T_{dt}\) is equivalent to a linear function of the braking pressure as:  
\[T_{dt}=pL_{0}/2\pi\eta. \tag{6}\]  
However, this simplification has a significant effect on the control performance of the electric braking system. By analyzing the electric braking system, it can be found that when the changing rate of the braking pressure \(\dot{p}\) is positive, the drive torque is a resistance to the motor. When the changing rate of the braking pressure \(\dot{p}\) is negative, the drive torque is a motivation to the motor, as shown in Equation (7):  
\[T_{dt}=p\mathrm{sgn}(\dot{p})L_{0}/2\pi\eta \tag{7}\]  
To decrease the chattering of the braking pressure control effect, a saturation function is introduced instead of the sign function. According to Equations (2) and (4), the load of the motor can be expressed as:  
\[T_{L}=T_{rt}+p\mathrm{sat}(\omega_{m},\tau)L_{0}/2\pi\eta, \tag{8}\]  
where the saturation function is defined as:  
\[\mathrm{sat}(\omega_{m},\tau)=\left\{ \begin{array}{ll}1,&\omega_{m}>\tau\\ \omega_{m}/\tau,&|\omega_{m}|\leq\tau\\ -1,&\omega_{m}<-\tau\end{array} \right., \tag{9}\]  
where \(\tau\) is the boundary coefficient of the saturation function. By taking consideration of Equations (1)--(9), selecting the state variables as \(x_{i}=[p,\omega_{m},i]\), the ideal mathematical model of the EMA can be expressed as:  
\[\left\{