## 3 Incremental Nonlinear Dynamic Inversion Control Law Design
### Angular Acceleration Estimation
where \(e_{1}=z_{10}-x\) and \(e_{2}=z_{20}-f_{m}\) are the estimation error. The error equation can be rewritten as  
\[\dot{e}=A_{e}e+Ew \tag{17}\]  
where  
\[e= \begin{bmatrix}e_{1}\\ e_{2}\end{bmatrix} ,A_{e}= \begin{bmatrix}-l_{1}&1\\ -l_{2}&0\end{bmatrix} ,E= \begin{bmatrix}0\\ -1\end{bmatrix}  \tag{18}\]  
The observer gains are determined using pole placement [19], where the choices of \(l_{1}=2\omega_{o}\) and \(l_{2}=\omega_{o}{}^{2}\) are made. Here, \(\omega_{o}>0\) represents the observer bandwidth. The characteristic polynomial of \(A_{e}\) can be expressed as  
\[\lambda(s)=s^{2}+l_{1}s+l_{2} \tag{19}\]  
Under the assumption that the parameter \(w\) is bounded, it can be concluded that the ESO achieves bounded-input-bounded-output (BIBO) stability. When the system reaches a steady state, referring to Equation (16), we have  
\[-l_{1}e_{1}+e_{2}=-l_{2}e_{1}-w=0 \tag{20}\]  
The absolute value of the steady-state error is  
\[|e_{1}|=\frac{|w|}{l_{2}},|e_{2}|=l_{1}\frac{|w|}{l_{2}} \tag{21}\]  
Finally, the estimated angular acceleration can be expressed as  
\[\dot{a}=z_{20}+a_{m} \tag{22}\]  
Rearranging Equation (22) using Laplace transform, we obtain  
\[\dot{a}=S^{\prime}(s)x_{s}+T^{\prime}(s)a_{m} \tag{23}\]  
where  
\[S^{\prime}(s)=\frac{l_{2}}{s^{2}+l_{1}s+l_{2}}s;\ T^{\prime}(s)=\frac{s^{2}+l_ {1}s}{s^{2}+l_{1}s+l_{2}} \tag{24}\]  
A comparison between Equations (13) and (24) reveals that they share many similarities in their forms, with the main difference being the location of the part \(l_{1}\)s. In the case of the CF, the part \(K_{p}s\) is present in the numerator of the angular rate transfer function. However, for ESO, the part \(l_{1}s\) is in the numerator of the model information transfer function. By setting the observer bandwidth to \(\omega_{0}\)= 30 rad/s and choosing \(l_{1}=K_{p}\) and \(l_{2}=K_{I}\), the Bode diagrams of the CF and ESO are compared in Figure 4.  
Figure 4a clearly illustrates that the magnitude of the ESO transfer function \(S^{\prime}(s)\) decreases beyond the observer bandwidth, while the magnitude of the CF transfer function \(S(s)\) remains constant. Furthermore, it is observed that \(S^{\prime}(s)\) exhibits a high degree of high-frequency attenuation with a loop roll-off slope of approximately 20 dB per decade. Consequently, the ESO exhibits superior noise suppression capabilities when compared with the CF. In Figure 4b, both the CF and ESO demonstrate high-pass characteristics for model information, but it is noteworthy that ESO passes more model information in the low-frequency range. Furthermore, in order to compare the time-domain properties of the two angular acceleration estimators, simplified roll mode dynamics that do not consider the coupling effect are used. The roll mode dynamics can be expressed as  
\[\dot{p}=\overline{L}_{\delta_{a}}\delta_{a}+\overline{L}_{p}p \tag{25}\]where \(p\) is the roll angular rate; \(\delta_{a}\) is the aileron deflection angle; and \(\overline{L}_{\delta_{a}}=-133\) s\({}^{-2}\) and \(\overline{L}_{p}=-3.4\) s\({}^{-1}\) are the roll control parameter and damping parameter, respectively. The roll angular rate is measured by a gyroscope with its noise \(\sigma^{2}=4.0\times 10^{-7}\) (rad/s)\({}^{2}\) and bias \(\varepsilon=3.0\times 10^{-5}\) rad/s [32]. The aileron deflection angle is modeled as a sine function and can be expressed as  
\[\delta_{a}=A_{n}\sin(2\pi f_{n}) \tag{26}\]  
where \(A_{n}\) is the amplitude and \(f_{n}\) is the input frequency. The on-board model input \(\dot{p}_{m}\) can be calculated as  
\[\dot{p}_{m}=\overline{L}_{\delta_{a}}\delta_{a}+\overline{L}_{p}p_{s} \tag{27}\]  
where \(p_{s}\) is the gyroscope output with measurement noise. The selected aileron input has an amplitude of 2 deg and a frequency of 2 Hz, and the estimation results are presented in \[-l_{1}e_{1}+e_{2}=-l_{2}e_{1}-w=0 \tag{20}\]0 The estimated angular rates of ESO and CF are nearly identical, and both estimators exhibit excellent filtering performance, as shown in Figure 5a.