## 3 The Adaptive Attitude Control Strategy of the MAV with Actuator Dynamics
### The Online Parameter Estimation Method for the MAV  
Referring to Equation (5), the linearized model of the pitch axis is given by:  
\[\dot{q}=M_{u}^{\hat{q}}u+M_{v}^{\hat{q}}v+M_{u}^{\hat{q}}w+M_{\theta}^{\hat{q} }\phi+M_{\phi}^{\hat{q}}\phi+M_{\psi}^{\hat{q}}\psi+M_{q}^{\hat{q}}\psi+M_{q}^{ \hat{q}}\phi+M_{p}^{\hat{q}}r+M_{\phi_{p}}^{\hat{q}}\delta_{p\mu}+M_{\delta_{val} }^{\hat{q}}\delta_{val}+M_{\delta_{p\mu}}^{\hat{q}}\delta_{p\mu}+M_{\delta_{p\mu }}^{\hat{q}}\delta_{p\mu}+\tau_{e}^{\hat{q}} \tag{6}\]  
where \(M_{u}^{\hat{q}},M_{v}^{\hat{q}},M_{w}^{\hat{q}},M_{\theta}^{\hat{q}},M_{\varphi }^{\hat{q}},M_{q}^{\hat{q}},M_{\varphi}^{\hat{q}},M_{\theta}^{\hat{q}},M_{ \theta_{p}}^{\hat{q}},M_{\delta_{val}}^{\hat{q}},M_{\delta_{p\mu}}^{\hat{q}},M _{\delta_{p\mu}}^{\hat{q}}\in\Theta\); \(\tau_{e}^{\hat{q}}\) is the corresponding element of \(E\), which is added to represent the unknown trim error. Given the corresponding wind tunnel test data, there exists \(M_{v}^{\hat{q}},M_{\theta}^{\hat{q}},M_{\varphi}^{\hat{q}},M_{r}^{\hat{q}},M_{ \delta_{p\mu}}^{\hat{q}}=0\). Thus, Equation (6) can be simplified as:  
\[\dot{q}=M_{u}^{\hat{q}}u+M_{w}^{\hat{q}}w+M_{q}^{\hat{q}}q+M_{p}^{\hat{q}}p+M_ {\delta_{p\mu}}^{\hat{q}}\delta_{p\mu}+M_{\delta_{val}}^{\hat{q}}\delta_{rol}+M _{\delta_{p\mu}}^{\hat{q}}\delta_{rot}+\tau_{e}^{\hat{q}} \tag{7}\]  
For simplicity in the design, the transfer function of the actuator is specified by the input-output relation:  
\[G_{a}\left(s\right)=\frac{\delta_{p\mu}\left(s\right)}{\delta_{p\mu}\left(s \right)}=\frac{1}{T_{a}s+1} \tag{8}\]  
It must be pointed out, however, that the proper control system for the electric servo actuator can easily be developed by means of cascade compensation networks or an actuator compensator, and many related algorithms are available for a high-performance servo control system in the literature. Substituting Equation (8) into Equation (7) yields:  
\[T_{a}\ddot{q}+\left(1-T_{a}M_{q}^{\hat{q}}\right)\dot{q}-M_{q}^{\hat{q}}q=M_{ \delta_{p\mu}}^{\hat{q}}\delta_{p\mu i}+M_{u}^{\hat{q}}u_{d}+M_{w}^{\hat{q}}w_{ d}+M_{p}^{\hat{q}}p_{d}+M_{\delta_{p\mu}}^{\hat{q}}\delta_{roll}+M_{\delta_{ p\mu}}^{\hat{q}}\delta_{roud}+\tau_{e}^{\hat{q}} \tag{9}\]  
where \(T_{a}\) and \(M_{q}^{\hat{q}}\) should satisfy the following constraints:  
\[1-T_{a}M_{q}^{\hat{q}}>0 \tag{10a}\] \[M_{q}^{\hat{q}}<0 \tag{10b}\]  
which explicitly indicate that actuator dynamics can potentially destabilize the overall system of the MAV. We now define the identification model of the system in Equation (9) as follows:  
\[T_{a}\ddot{q}_{m}+a_{q}\dot{q}_{m}+a_{q}q_{m}=b_{g\mu i}\delta_{p\mu i}+b_{a}u _{d}+b_{w}w_{d}+b_{p}\,p_{d}+b_{g_{val}}\delta_{roud}+b_{g_{ret}}\delta_{roud}+ b_{e} \tag{11}\]  
where the identification parameters should satisfy the following constraints:  
\[a_{q}>0,\,a_{q}>0 \tag{12}\]  
The error between \(q_{m}\) and \(q\) is expressed as:  
\[e=q_{m}-q \tag{13}\]  
Substituting Equations (9) and (11) into Equation (13) yields:\[T_{a}\ddot{e}+a_{q}\dot{e}+a_{q}e=-\Delta a_{q}\dot{q}-\Delta a_{q}q+\Delta b_{g_{ \mu\alpha}}\delta_{p\mu i}+\Delta b_{u}u_{d}+\Delta b_{w}u_{d}+\Delta b_{p}\,p_{ d}+\Delta b_{g_{\mu\alpha}}\delta_{rold}+\Delta b_{g_{\mu\alpha}}\delta_{rold}+ \Delta b_{e} \tag{14}\]  
where:  
\[ \begin{cases}\Delta a_{q}=a_{q}+M_{q}^{\dot{q}}\\ \Delta a_{q}=a_{q}-1+T_{a}M_{q}^{\dot{q}}\\ \Delta b_{u}=b_{u}-M_{u}^{\dot{q}}\\ \Delta b_{w}=b_{w}-M_{u}^{\dot{q}}\\ \Delta b_{p}=b_{p}-M_{p}^{\dot{q}}\\ \Delta b_{g_{\mu\alpha}}=b_{g_{\mu\alpha}}-M_{g_{\mu\alpha}}^{\dot{q}}\\ \Delta b_{g_{\mu\alpha}}=b_{g_{\mu\alpha}}-M_{\delta_{r\alpha}}^{\dot{q}}\\ \Delta b_{e}=b_{e}-\tau_{e}^{\dot{q}}\end{cases}  \tag{15}\]  
However, \(u_{d},w_{d},p_{d},\delta_{rold}\) and \(\delta_{rold}\) are also difficult to measure and are, therefore, not available.