## 5 Control Laws--Nominal Case
### Backstepping Design  
A disadvantage of the controller design in the previous section is that the scalar proportional gain \(k_{p}\) is restrictive. In this section, we present a backstepping controller that allows for a matrix proportional gain, which gives the flexibility for the control to be more aggressive along certain body-fixed axes, which is important due to geometric and aerodynamic asymmetries of aircraft. In the previous section, the proportional action defines a _torque_ that is aligned with the axis of shortest rotation. The backstepping controller, on the other hand, defines a _desired angular velocity_ that generates a geodesic curve on the sphere. To this end, define the virtual control signal  
\[\varphi(\eta,\eta_{d},\omega_{d}^{\perp})\triangleq-\kappa e_{\eta}+\Pi_{\eta} ^{\perp}\omega_{d}^{\perp}\in\mathrm{T}_{\eta}S^{2}, \tag{32}\]  
where \(\kappa\in\mathbb{R}_{+}\) is a user specified parameter. We will show that \(\omega^{\perp}=\varphi(\eta,\eta_{d},\omega_{d}^{\perp})\) solves the kinematic reduced-attitude tracking problem (see the proof of Proposition 2). Now, introduce the tracking-error signal  
\[z\triangleq\underbrace{\left(\omega^{\perp}-\varphi(\eta,\eta_{d},\omega_{d}^ {\perp})\right)}_{\in\mathrm{T}_{\eta}S^{2}}+\underbrace{\left(\omega^{\parallel }-\omega_{d}^{\parallel}\right)}_{\in\mathrm{N}_{\eta}S^{2}}=\omega-\bar{ \omega}_{d}, \tag{33}\]  
Figure 4: Block diagram of a guidance, navigation and control (GNC) architecture for a fixed-wing UAV. where \(\bar{\omega}_{d}=\varphi(\eta,\eta_{d},\omega_{d}^{\perp})+\omega_{d}^{\parallel}\). Note that \(\bar{\omega}_{d}=\omega_{d}-\kappa\epsilon_{\eta}\) and \(z\) can be written as \(z=e_{\omega}+\kappa\epsilon_{\eta}\). Due to orthogonality properties, \(z\) defined as in (33) has the nice property that as \(z\) converges to zero, \(\omega^{\perp}\to\varphi(\eta,\eta_{d},\omega_{d}^{\perp})\), which stabilizes the desired reduced-attitude, and at the same time, \(\omega^{\parallel}\) converges to \(\omega_{d}^{\parallel}\). Consider the tracking error dynamics (18), and for \(k_{1}>0,K_{2}=K_{2}^{\top}>0\), define the control input as  
\[u=u_{\text{trim}}+\frac{1}{V_{a}^{2}}B^{-1}\big{[}u_{pd}+u_{\text{ff}}-\Delta( v_{r},t)\big{]} \tag{34}\]  
where  
\[u_{pd} =-k_{1}e_{\eta}-K_{2}z \tag{35}\] \[u_{\text{ff}} =J\dot{\omega}_{d}-S(J\bar{\omega}_{d})\bar{\omega}_{d}-V_{a}D \bar{\omega}_{d}, \tag{36}\]  
and the matrix \(K_{2}\) is chosen such that  
\[\lambda_{\min}^{K_{2}}-\lambda_{\max}^{I}(c_{\omega_{d}}+\kappa)\geq\gamma, \tag{37}\]  
for some \(\gamma>0\). Then the following holds:  
1. There are two closed-loop equilibria, given by \((\eta,z)=(\pm\eta_{d},0)\). 2. The equilibrium \((\eta,z)=(-\eta_{d},0)\) is unstable. 3. The desired equilibrium \((\eta,z)=(\eta_{d},0)\) is almost globally asymptotically stable. 4. The desired equilibrium \((\eta,z)=(\eta_{d},0)\) is locally exponentially stable. In addition, if the initial conditions \(\eta(0),\eta_{d}(0),z(0)\) satisfy \[\Psi(\eta(0),\eta_{d}(0)) <2\] (38) \[k_{1}\Psi(\eta(0),\eta_{d}(0))+\frac{1}{2}z^{\top}(0)Jz(0) <2k_{1},\] (39) then the energy function \(V_{2}(t)\triangleq k_{1}\Psi(\eta,\eta_{d})+(1/2)z^{\top}Jz\) converges exponentially to zero. 5. \(\omega^{\perp}\to\omega_{d}^{\perp}\) and \(\omega^{\parallel}\to\omega_{d}^{\parallel}\) as \(t\to\infty\). Proof. : See Appendix C.  
As for the previous design, a statement similar to Remark 8 holds true also here.