## 5 Control Laws--Nominal Case
### Control Design Based on an Energy-Like Lyapunov Function  
**Proposition 1**.: _Consider the tracking error dynamics (18), and for \(k_{p}>0,K_{d}=K_{d}^{\top}>0\), define the control input as_  
\[u=u_{trim}+\frac{1}{V_{a}^{2}}B^{-1}\big{[}u_{pd}+u_{ff}-\Delta(v_{r},t)\big{]}, \tag{26}\]  
_where_  
\[u_{pd} =-k_{p}e_{\eta}-K_{d}e_{\omega} \tag{27}\] \[u_{ff} =J\dot{\omega}_{d}-S(J\omega_{d})\omega_{d}-V_{a}D\omega_{d}, \tag{28}\]  
_and the matrix \(K_{d}\) is chosen such that_  
\[\lambda_{\min}^{K_{d}}-\lambda_{\max}^{I}c_{\omega_{d}}\geq\gamma, \tag{29}\]  
_for some \(\gamma>0\). Then the following holds:_  
1. _There are two closed-loop equilibria, given by_ \((\eta,e_{\omega})=(\pm\eta_{d},0)\)_._
2. _The equilibrium_ \((\eta,e_{\omega})=(-\eta_{d},0)\) _is unstable._
3. _The desired equilibrium_ \((\eta,e_{\omega})=(\eta_{d},0)\) _is almost globally asymptotically stable._
4. _The desired equilibrium_ \((\eta,e_{\omega})=(\eta_{d},0)\) _is locally exponentially stable. In addition, if the initial conditions_ \(\eta(0),\eta_{d}(0),e_{\omega}(0)\) _satisfy_ \[\Psi(\eta(0),\eta_{d}(0)) <2\] (30) \[k_{p}\Psi(\eta(0),\eta_{d}(0))+\frac{1}{2}e_{\omega}^{\top}(0)Je _{\omega}(0) <2k_{p},\] (31) _then the energy-like function_ \(V(t)\triangleq k_{p}\Psi(\eta,\eta_{d})+(1/2)e_{\omega}^{\top}Je_{\omega}\) _converges exponentially to zero._
5. \(\omega^{\perp}\rightarrow\omega_{d}^{\perp}\) _and_ \(\omega^{\parallel}\rightarrow\omega_{d}^{\parallel}\) _as_ \(t\rightarrow\infty\)_._  
Proof.: See Appendix B.  
**Remark 8**.: _If Assumption 4 is not satisfied, it is not difficult to show that the result still holds if \(K_{d}\) is chosen such that \(\lambda_{\min}^{K_{d}}>\lambda_{\max}^{I}c_{\omega_{d}}+V_{\max}\sigma_{\max} ^{\mathrm{sym}(D)}\)._  
**Remark 9**.: _The region of exponential convergence to the desired equilibrium point can be made (almost) arbitrarily large by increasing \(k_{p}\) (\"semi-global\" property). However, the region of convergence can never include the unstable equilibrium point and its corresponding unstable manifold [53]._  
Figure 4 shows a block diagram that illustrates how this controller integrates into a typical guidance, navigation and control (GNC) architecture for a fixed-wing UAV. The references for reduced-attitude, angular velocity, and angular acceleration are generated by some outer-loop guidance controller, and the reduced-attitude control law is combined with a control law for airspeed control, e.g., a PI-controller [1]. The controller uses estimates of the rotation matrix \(R\), the angular velocity \(\omega\), as well as the relative velocity \(v_{r}\), which are all made available through a state estimation module. The use of \(v_{r}\) is relaxed in Section 6.  
The control law (26) is based on proportional action that is proportional to the error term (14), which defines an axis of revolution for the direct, shortest rotation connecting \(\eta\) and \(\eta_{d}\) (forming a _geodesic_ curve on the sphere). This is convenient when dealing with large rotation errors and is a property that is not shared with controllers based on Euler angles. A comparison between a geodesic controller like (26) and one based on Euler angles is presented in [72], which indicates that the geodesic controller spends less control energy than the controller based on Euler angles.