## 2 System Description
### State Space Model  
The state-space model representing the UAV's longitudinal motion is expressed in (1) [11; 29]. \[ \begin{bmatrix}\dot{q}(t)\\ \dot{w}(t)\\ \dot{p}(t)\\ \dot{\theta}(t)\end{bmatrix} = \begin{bmatrix}X_{q}&X_{w}&X_{p}&-g(\cos\theta_{o })\\ Z_{q}&Z_{w}&Z_{p}&0\\ M_{q}&M_{w}&X_{p}&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}q(t)\\ w(t)\\ p(t)\\ \theta(t)\end{bmatrix} + \begin{bmatrix}X_{\delta}\\ Z_{\delta}\\ M_{\delta}\\ 0\end{bmatrix} \delta_{e}(t), \tag{1}\]  
\[\theta(t)= \begin{bmatrix}0&0&1&0\end{bmatrix}\begin{bmatrix}q(t)\\ w(t)\\ p(t)\\ \theta(t)\end{bmatrix} +[0]\delta_{e}(t)\]  
where \(q(t)\) is the forward velocity, \(w(t)\) is the vertical velocity, \(p(t)\) is the pitch rate, \(\theta(t)\) is the pitch angle, and \(\delta_{e}(t)\) represents the elevator's deflection angle. It is to be noted that \(p(t)=\dot{\theta}(t)\). The coefficients of the system matrix (\(X_{q}\), \(X_{w}\), \(X_{p}\), \(Z_{q}\), \(Z_{w}\), \(Z_{p}\), \(M_{q}\), \(M_{w}\), and \(M_{p}\)) are denoted as the stability derivatives, while the coefficients of the input matrix (\(X_{\delta}\), \(Z_{\delta}\), and \(M_{\delta}\)) are denoted as the control derivatives [29]. The parameter \(g=9.81\) m/s\({}^{2}\) represents the gravitational acceleration, while \(\theta_{o}\) is the elevator trim. The schematic representing the longitudinal flight dynamics of a UAV is shown in Figure 1. The coefficients of the system matrix and the input matrix are identified by considering the dynamics of the Bluebird UAV, described in [11; 30]. Figure 1: Schematic of the UAV’s longitudinal flight dynamics. Upon the substitution of the model parameter values, expressed in [11; 30], the nominal state space model of the UAV system is presented in Equation (2). \[ \begin{bmatrix}\dot{q}(t)\\ \dot{w}(t)\\ \dot{p}(t)\\ \dot{\theta}(t)\end{bmatrix} = \begin{bmatrix}-0.045&0.183&0&-0.241\\ -0.312&-1.945&1&0\\ 0.152&-22.511&-2.036&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}q(t)\\ w(t)\\ p(t)\\ \theta(t)\end{bmatrix} + \begin{bmatrix}-0.007\\ 0.124\\ -17.105\\ 0\end{bmatrix} \delta_{e}(t) \tag{2}\]  
The vertical movement of the UAV is categorized into the short-period model and the long-period model. For the purpose of this study, the short-period model is employed as it provides a more accurate representation than the phugoid model [30]. The change in the UAV's vertical velocity \(w(t)\) and its pitch rate \(p(t)\) are sufficient to depict the short-period motion. The forward flight speed is practically the same. The dynamics of the vertical motion are simplified if \(q(t)=0\), and consequently, the short-period model is represented as follows [11; 30]:  
\[ \begin{bmatrix}\dot{w}(t)\\ \dot{p}(t)\end{bmatrix} = \begin{bmatrix}-1.945&1\\ -22.511&-2.036\end{bmatrix}\begin{bmatrix}w(t)\\ p(t)\end{bmatrix} + \begin{bmatrix}0.124\\ -17.105\end{bmatrix} \delta_{e}(t) \tag{3}\]  
\[\dot{p}(t)= \begin{bmatrix}0&1\end{bmatrix}\begin{bmatrix}w(t)\\ p(t)\end{bmatrix} +[0]\delta_{e}(t)\]  
The short-period model occurs over a brief period, during which changes in pitch angle are significant. As a result, the pitch rate \(p(t)\) dominates the dynamics, while variations in vertical velocity \(w(t)\) remain relatively small and less significant in practical flight conditions [28]. Since short-period motion primarily consists of rapid pitch oscillations, the influence of \(w(t)\) is negligible compared to \(p(t)\). Therefore, deriving the transfer function solely in terms of \(p(t)\) simplifies the system without compromising essential dynamic behavior [11; 30]. Thus, the state-space model in Equation (3) is used to derive the system's transfer function from the elevator's deflection angle to the UAV's pitch rate. It is expressed in Equation (4). \[\frac{p(s)}{\delta_{e}(s)}=\frac{-17.11s-36.06}{s^{2}+3.98s+26.47} \tag{4}\]  
where \(s\) is the Laplace operator.