## 3 Modeling of the Rotational Speed System of the Propeller  
The analysis of the basic principle model is crucial for the development of a proper controller to improve the control quality of the rotational speed system. With regard to the aim of this study, the schematic diagram of the motor that conveys detailed information about the electrical components of the motor is shown in Figure 2, where the no-load current is known and is compensated by the ESC. Figure 1: The prototype electric Micro Air Vehicle (MAV). The equations that describe the motor electrical components are as follows [24]:  
\[u = R_{a}i_{a}+L_{a}\frac{\mathrm{d}i_{a}}{\mathrm{d}t}+e_{a} \tag{1a}\] \[e_{a} = \frac{1}{K_{V}}\omega\] (1b) \[T = K_{Q}i_{a} \tag{1c}\]  
where \(u\) (V) is the armature voltage, \(i_{a}\) (A) the armature current, \(R_{a}\) (\(\Omega\)) the armature coil resistance, \(L_{a}\) (H) the armature coil inductance, \(e_{a}\) (V) the back electromotive force, \(K_{V}\) (rad/s/V) the motor voltage constant, \(K_{Q}\) (Nm/A) the torque constant, \(T\) (Nm) the torque generated by the motor, \(\omega\) (rad/s) the rotational speed of the motor. Considering that the MAV performs low-speed flight missions, the propeller approximately produces a torque according to [25]  
\[T_{p}=k_{p}\omega_{p}^{2} \tag{2}\]  
where \(k_{p}\) (Nm s\({}^{2}\)/rad\({}^{2}\)) is the propeller moment constant, \(T_{p}\) (Nm) the propeller torque, \(\omega_{p}\) (rad/s) the rotational speed of the propeller. The following conditions hold  
\[\omega_{p}=\omega,\,T_{p}=T \tag{3}\]  
under the assumption that the propeller is mounted on a propeller shaft which is driven through a rigid connection with the main drive shaft of the motor. Substituting Equations (2) and (3) into Equation (1) yields  
\[a_{1}\omega\dot{\omega}+a_{0}\omega^{2}+b_{0}\omega=u \tag{4}\]  
where \(a_{1}=\dfrac{2L_{a}k_{p}}{K_{Q}}\), \(a_{0}=\dfrac{R_{a}k_{p}}{K_{Q}}\), and \(b_{0}=\dfrac{1}{K_{V}}\). The characteristics of the rotational speed system in Equation (4) are a nonlinear differential equation, and the coefficients of the equation are difficult and expensive to accurately obtain in practice, which results in the fact that most existing autopilots for MAVs with electric propellers, such as the widely used Pixhawk(r) (Zurich, Switzerland) autopilot [26], only employ a proportional-integral-differential (PID) controller to simply and directly control the altitude without a closed-loop rotational speed control structure under fairly stable conditions. The behavior of the rotational speed directly and greatly affects the flight dynamics, and then indirectly and greatly affects the flight performance, such as the positioning accuracy and stability. The control of the rotational speed is therefore important to the improvement of the flight performance of the MAVs. In addition, some reasonable assumptions must be made preliminarily to simplify the adaptive control design as follows:  
\[a_{1}>0,a_{0}>0,b_{0}>0 \tag{5a}\] \[u>0,\,\omega>0 \tag{5b}\]  
It is finally noted that such assumptions are based on the fact that the rotational speed of the propeller remains high enough to lift the MAV. Figure 2: The schematic diagram of the motor. The derivative of the Lyapunov function candidate can then be rewritten as follows:  
\[u = R_{a}i_{a}+L_{a}\frac{\mathrm{d}i_{a}}{\mathrm{d}t}+e_{a} \tag{1a}\]0  
Thus, we can conclude that \(\dot{V}\) is a negative definite. From Equations (8), (12) and (15), we can obtain the following adaptive laws:  
\[\left\{ \begin{array}{l}\dot{k}_{a_{1}}=\lambda_{a_{1}}\left(e+\mu\right) \dot{\omega},\\ \dot{k}_{a_{0}}=\lambda_{a_{0}}\left(e+\mu\right)\omega,\\ \dot{k}_{b_{0}}=\lambda_{b_{0}}\left(e+\mu\right),\\ \dot{k}_{b}=\lambda_{b}\left(e+\mu\right)r.\end{array} \right. \tag{18}\]  
It is worth noting that, in contrast with most existing adaptive laws, the term \(\mu\) introduced here can contribute to the improvement of the convergence rates of adjustable parameters, as discussed below.