## 3 Validation  
### Stability Analysis of the Proposed Path Tracking Method  
In this Section, a study of the stability characteristics of the proposed path-tracking method is conducted over a circular trajectory of radius R. Results may be generalized for any curve C by setting R equal to the local curvature of C. In order to focus on the performance of the path-tracking algorithm aircraft dynamics have been neglected; it is hereby assumed that the aircraft reproduces all commands instantaneously and without error. Figure 7 illustrates the system geometry for the examined case.  
For a path with fixed curvature:  
\[d_{11^{\prime}}=const \tag{18}\]  
Consequently,  
\[\delta=const=\sin^{-1}\biggl{(}\frac{d_{11^{\prime}}}{2R}\biggr{)} \tag{19}\]  
Figure 5: Schematic representation of the proposed path-tracking method.  
Figure 6: Path tracking in a supersonic climb example. The climb path is shown in red and the aircraft’s trajectory in black, empty circles corresponding to aircraft position at equal time intervals. The climb path is plotted over contours of Specific Excess Power, expressed in m/s.  
The rate of change of cross-track error \(d_{01}\) equals  
\[\dot{d_{01}}=-V\prime\sin(\beta+\delta) \tag{20}\]  
From the triangle \((x_{0}\ x_{0^{\prime}}\ x_{1^{\prime}})\):  
\[\beta=\tan^{-1}\biggl{(}\frac{d_{01}\cos\delta}{d_{01}\sin\delta+d_{11^{\prime}}} \biggr{)} \tag{21}\]  
From Equations (20) and (21), knowing that \(d_{11^{\prime}}\), \(\delta\) are constant:  
\[\dot{d_{01}}=-V^{\prime}\sin\biggl{(}\tan^{-1}\biggl{(}\frac{d_{01}\cos\delta }{d_{01}\sin\delta+d_{11^{\prime}}}\biggr{)}+\delta\biggr{)}=f(d_{01}), \tag{22}\]  
which is the system's state-space model. For equilibrium, the system's state vector must remain invariable. Consequently:  
\[\dot{d_{01}}=0\stackrel{{\eqref{eq:d01}}}{{\Rightarrow}}\beta=- \delta\Rightarrow\tan\beta=-\tan\delta \tag{23}\]  
Combining Equations (19), (21) and (23) yields  
\[d_{01_{trim}}=-2R\sin^{2}\delta, \tag{24}\]  
which corresponds to a point inside the circle where \(\dot{\vec{V}}_{0}\) becomes normal to \(\stackrel{{\rightarrow}}{{d_{01}}}\), as shown in Figure 7.  
Some steady-state cross-track error is thus unavoidable, given that a positive value of \(\delta\) is required for path-tracking; this, however, may become negligible if angle \(\delta\) is set at an adequately small value, i.e., \(d_{11^{\prime}}\ll R\).  
Angle \(\beta\) is bounded in the interval \((-\pi/2,+\pi/2)\), consequently, from Equations (19) and (21), \(d_{01}\) is also bounded:  
\[\beta\in\Bigl{(}-\frac{\pi}{2},\ +\frac{\pi}{2}\Bigr{)}\Rightarrow\mathrm{d}_{01 }\in(-2R,+\infty) \tag{25}\]  
From Equation (22), f is monotonous for \(d_{01}\in(-2R,+\infty)\), as a synthesis of monotonous functions:  
\[f(d_{01})=\left\{\begin{array}{l}>0\ d_{01}<d_{01_{trim}}\\ =0\ d_{01}=d_{01_{trim}}\\ <0\ d_{01}>d_{01_{trim}}\end{array}\right. \tag{26}\]  
Let \(=d_{01}-d_{01 trim}\), and a Lyapunov-candidate function \(V(x)=|x|\). Then:  
\[\dot{V}(x)=\frac{dV}{dx}\dot{f}(x)=sgn(x)\dot{d_{01}}(x) \tag{27}\]  
Figure 7: System geometry for a circular path.  
Using Equation (26)  
\[\dot{V}(x)<0\ \forall x\in\left(-2R-d_{01_{train}},+\infty\right)\backslash\{0\} \tag{28}\]  
Therefore, \(f(x)\) is asymptotically stable for all possible values of \(x\).