# Modeling and Control of an Articulated Multibody Aircraft
## 4 Simulation and Results
### Control Design
#### 4.5.1 Longitudinal Control Simulation Study  
The eigenvalues of the \(\mathbf{A}_{long}\) matrix of the pre-augmented state-space model shown in Equation (68) indicate that the system is stable since all real parts of the eigenvalues are negative  
\[eig(\mathbf{A}_{long})= \begin{bmatrix}-17.73\pm 19.58\mathrm{i}\\ -0.15\pm 0.24\mathrm{i}\end{bmatrix}  \tag{71}\]  
For the longitudinal controller, the variable being controlled is the pitch angle; hence, \((\mathbf{y}_{long}=e_{\theta})\), furthermore, it is assumed that changes in \(\theta\) do not affect the speed of the aircraft. Therefore, the corresponding axial velocity \(u\) components are ignored. In addition, thrust is assumed to be constant, therefore, the effects of changes in thrust are ignored. The \(z_{B}\) axis velocity \(w\) was replaced with the angle of attack \(\alpha\) using the relationship \((\alpha=\frac{w}{u_{0}})\). The longitudinal augmented states considered are \(\mathbf{z}_{long}=[\alpha\;q\;\theta\;e_{\theta}]^{\prime}\), and the longitudinal control inputs: \(\mathbf{u}_{long}=[\delta_{e}\;\theta_{T}]^{\prime}\). The resulting augmented matrices are  
\[\hat{\mathbf{A}}_{long} = \begin{bmatrix}-2.66&0.68&0.025&0\\ -67.54&-8.74&0.64&0\\ 0&1&0&0\\ 0&0&-1&0\end{bmatrix} ,\quad\hat{\mathbf{B}}_{long}= \begin{bmatrix}0&0\\ -2.20&1.65\\ -6.43&-190.03\\ 0&0\end{bmatrix}  \tag{72}\] \[\hat{\mathbf{C}}_{long} =[0\;\;0\;\;0\;\;1],\qquad\hat{\mathbf{D}}_{long}=[0\;\;\;0]\]  
The obtained longitudinal linearized model was checked to be controllable and observable using MATLAB functions \(\mathrm{ctrb}(\hat{\mathbf{A}}_{long},\hat{\mathbf{B}}_{long})\) and \(\mathrm{obsv}(\hat{\mathbf{A}}_{long},\hat{\mathbf{C}}_{long})\), respectively. The observability and controllability matrices were fully ranked \((rank(\mathcal{O}_{long})=rank(\mathcal{C}_{long})=4)\). Therefore, the longitudinal model is controllable and observable. Three pitch control (PC) strategies were investigated based on the selection of weighting matrices \(\mathbf{R}_{long}\) to reflect the following:  
1. PC 1 --Prioritized use of elevator for pitch angle control, interpreted as _cheap_ elevator cost and _expensive_ abdominal pitch cost. 2. PC 2--Prioritized use of abdominal pitch for pitch angle control, interpreted as _expensive_ elevator cost and _cheap_ abdominal pitch cost. 3. PC 3--Relatively evenly prioritized use of elevator and abdominal pitch for pitch angle control, interpreted as _cheap_ elevator cost, _cheap_ abdominal pitch cost. The same state-weighting matrices \(\mathbf{Q}_{long}=diag[4\;6\;4\;500]\), which were selected using [PERSON]'s rule as a guide [140], were used for all cases considered. Then, the optimal gains were calculated using MATLAB's lqr(A,B,Q,R) function. A summary of the cases considered for pitch angle control are presented in Table 5. As the control objective was to track a reference pitch angle while keeping all states and control efforts within bounds, the performance characteristics for all cases are compared in terms of settling time, overshoot and steady-state error. The selected gains resulted in the closed-loop step response shown in Figure 4 for the states, while the control efforts are shown in Figure 5. The performance characteristics of the control strategies are summarized in Table 6.  
Figure 4: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline
**Pitch Control** & \(\mathbf{R}_{long}\) & \(\mathbf{K}_{long}\) \\
**(PC) Case** & \(\mathbf{R}_{long}\) & \(\mathbf{K}_{long}\) \\ \hline PC 1 & diag [\(2\times 10^{2}\)\(8\times 10^{7}\)] & \(\begin{bmatrix}-87.68&10.29&93.47&-158.11\\ 0&0&0&0\end{bmatrix}\) \\ PC 2 & diag [\(8\times 10^{7}\)\(2\times 10^{-2}\)] & \(\begin{bmatrix}0&0&0&0\\ 0.03&-17.30&-75.53&158.06\end{bmatrix}\) \\ PC 3 & diag [\(2\times 10^{-2}\)\(2\times 10^{-2}\)] & \(\begin{bmatrix}-12.78&-0.69&-1.98&4.22\\ 0.69&-17.29&-75.38&158.06\end{bmatrix}\) \\ \hline \end{tabular}
\end{table} 
Table 5: Simulation parameters for pitch control (PC) cases.