## 3 Hybrid Compensation Scheme
### Adaptive NDI Controller for Inertia Uncertainty  
Theoretically, the designed nominal NDI feedback controller (15) can make the aircraft track a given flight command, but in actual flight, the attitude angle and attitude angular velocity channels are easily affected by the external flight environment, which makes the aircraft have parameter uncertainty. The uncertainty of the moment of inertia is considered here, and it needs to be suppressed to improve the robustness of the flight control system. Model the drift characteristics of the aircraft's moment of inertia parameters, and express the uncertainty of the moment of inertia in the following random form:  
\[I_{y}=I_{y0}(1+\Delta I_{y}) \tag{19}\]  
where \(I_{y}\) represents the rotational inertia around the y-axis, \(I_{y0}\) represents the nominal value of the parameter \(I_{y}\), \(\Delta I_{y}\) represents the increment of the parameter \(I_{y}\), and its range is \(|\Delta I_{y}|\leq I_{y1}\), and \(I_{y1}\) is positive. In order to enhance the robustness of the system, an adaptive NDI controller is designed to make the aircraft immune to the uncertainty of the rotational inertia parameters, and the system output accurately tracks the given reference signal. The block diagram of the adaptive NDI controller is shown in Figure 2.  
The adaptive NDI controller design process is as follows:  
Consider the system reference model, that is, the model without uncertainty as follows:  
\[\dot{x}_{m}=A_{m}x_{m}+B_{m}u \tag{20}\]  
Figure 2: Adaptive NDI control structure diagram of longitudinal HFV model. The actual system model with rotational inertia uncertainty is as follows:  
\[\dot{x}_{p}=A_{p}(t)x_{p}+B_{p}(t)u \tag{21}\]  
Based on the reference model, the designed nominal NDI control law (see Section 2.2 for the derivation process) is as follows:  
\[u=g^{-1}(x)(\dot{x}_{c}-Ke) \tag{22}\]  
where \(e=x-x_{c}\), \(K\) is the control bandwidth matrix. The generalized error vector of the system is  
\[e_{1}=x_{m}-x_{p} \tag{23}\]  
and the equation with the generalized error as the state vector is  
\[\dot{e}_{1}=A_{m}e_{1}+[A_{m}-A_{p}(t)-B_{p}(t)\Theta_{1}(t)]x_{p}+[B_{m}-B_{p} (t)\Theta_{2}(t)]u \tag{24}\]  
In order to make the dynamic response of the uncertain system (21) to the input \(u\) exactly the same as the dynamic response of the reference model (20) to the input \(u\), the adaptive law adjusts \(\Theta_{1}(t)\) and \(\Theta_{2}(t)\) so that the uncertain system model matches the reference model, i.e.,  
\[A_{m}=A_{p}(t)+B_{p}(t){\Theta_{1}}^{*} \tag{25}\]  
\[B_{m}=B_{p}(t){\Theta_{2}}^{*} \tag{26}\]  
The function with \"*\" represents the optimal state when the model is completely matched, and Equation (20) can be written as  
Figure 2.0  
where the errors of the two adjustable functions are a 1 \(\times\) 3-order matrix and a 1 \(\times\) 1-order matrix, respectively, and satisfy  
Figure 2.1  
Figure 2.2  
**Theorem 1. **_For the generalized error system (24), there is a Lyapunov function that contains the state error \(e_{1}\) and the adjustable function errors (28) and (29). If there are dimensionally adjustable parameters P, \(R_{1}\), and \(R_{2}\) such that the derivative of the Lyapunov function satisfying the condition is less than zero, then the system (24) is stable._  
**Proof of Theorem 1. ** The Lyapunov theory is used to design the adaptive laws, as shown below:  
Figure 2.3  
where \(tr(\cdot)\) represents the trace of the matrix, that is, the sum of the diagonal elements, and the Lyapunov function can be derived from time to obtain:  
\[ \begin{array}{l}\dot{V}=\frac{1}{2}{e_{1}}^{T}(PA_{m}+A_{m}^{T}P)e_{1}+tr( \dot{\widetilde{\Theta}}_{1}^{T}R_{1}^{-1}\widetilde{\Theta}_{1}^{T}+x_{p}{e_ {1}}^{T}PB_{m}{\Theta_{2}}^{*-1}\widetilde{\Theta}_{1})\\ +tr(\dot{\widetilde{\Theta}}_{2}^{T}R_{2}^{-1}\widetilde{\Theta}_{2}^{T}+{u_{ 1}}^{T}PB_{m}{\Theta_{2}}^{*-1}\widetilde{\Theta}_{2})\\ \leq 0\end{array}