# Modeling and Control of an Articulated Multibody Aircraft
## 4 Simulation and Results
### Control Design
#### 4.5.2 Lateral-Directional Simulation Study  
Estimating the eigenvalues of the \(\mathbf{A}_{lat}\) matrix of the pre-augmented state-space model in Equation (69) reveals that the system is unstable due to the existence of a positive real part in the characteristic roots.  
\[eig(\mathbf{A}_{lat})=\begin{bmatrix}1.02\pm 3.67\text{i}\\ -86.05\\ -2.68\\ 0\end{bmatrix} \tag{73}\]  
For the yaw angle controller, the variable being controlled is the yaw angle; hence, (\(\mathbf{y}_{lat}=e_{\psi}\)). The \(y_{B}\) axis velocity \(v\) was replaced with the sideslip angle, \(\beta\), using the relationship \(\beta=\frac{v}{u_{0}}\). The lateral augmented states considered were \(\mathbf{z}_{lat}=[\beta\;p\;r\;\phi\;\psi\;e_{\psi}]^{\prime}\), and the control inputs: \(\mathbf{u}_{lat}=[\delta_{a}\;\psi_{T}]^{\prime}\). The resulting augmented matrices are  
\[\hat{\mathbf{A}}_{lat}=\begin{bmatrix}-0.025&-0.032&-0.99&1.22&0&0\\ -14.09&-85.94&12.47&0&0\\ 0.45&-0.014&-0.49&-23.33&0&0\\ 0&1&-0.02&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&-1&0\end{bmatrix},\quad\hat{\mathbf{B}}_{lat}=\begin{bmatrix}-0.032& -7.11\\ -18.71&0\\ 0.32&-137.27\\ 0&0\\ 0&0\\ 0&0\end{bmatrix} \tag{74}\]  
Checking the lateral-directional linearized model controllability and observability using MATLAB functions \(\text{ctrb}(\hat{\mathbf{A}}_{lat},\hat{\mathbf{B}}_{lat})\) and \(\text{obsv}(\hat{\mathbf{A}}_{lat},\hat{\mathbf{C}}_{lat})\), respectively resulted in fully ranked matrices \((rank(\mathcal{O}_{lat})=rank(\mathcal{C}_{lat})=6)\). Therefore, the lateral-directional model is controllable and observable.  
Again, three yaw control (YC) strategies were investigated based on the selection of weighting matrices \(\mathbf{R}_{lat}\) to reflect the following:  
1. YC 1--Prioritized use of aileron for yaw angle control, interpreted as _cheap_ aileron cost, _expensive_ abdominal yaw cost.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Characteristic** & **PC 1** & **PC 2** & **PC 3** \\ \hline Settling time (s) & 3.08 & 2.97 & 2.96 \\ Overshoot (\%) & 3.84 & 3.82 & 3.81 \\ Steady-state error (\%) & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of performance characteristics for the different pitch control (PC) cases.  
Figure 5: Time histories of the longitudinal control inputs: (**a**) elevator deflection (\(\delta_{e}\)). (**b**) abdominal pitch angle (\(\theta_{T}\)).  
2. YC 2--Prioritized use of abdominal yaw for yaw angle control, interpreted as _expensive_ aileron cost, _cheap_ abdominal yaw cost.
3. YC 3--Relatively evenly prioritized use of elevator and abdominal pitch for yaw angle control, interpreted as _cheap_ aileron cost, _cheap_ abdominal yaw cost.  
The same state-weighing matrix \(\mathbf{Q}_{lat}=diag[6\ 6\ 6\ 3\ 3\ 500]\) was used for all cases considered. The choice of the state-weighting matrix was also influenced by the need to minimize the sideslip when the aircraft is turning. The estimated optimal gains, as well as other parameters for the yaw controllers, are presented in Table 7. The state trajectory and control effort responses are shown in Figures 6