## 2 System Description
### State Space Model
The transfer function representing the UAV's steering inertia model (elevator model) is expressed as follows [30]:  
\[\frac{\delta_{e}(s)}{v(s)}=\frac{K_{s}}{\tau s+1} \tag{5}\]  
where \(v(s)\) is the control input voltage applied to the elevator's servomotor, \(K_{s}\) is the servomotor's gain constant, and \(\tau\) is the servomotor's time constant. As discussed in the previous studies, the values of \(K_{s}\) and \(\tau\) are set at \(-1\) and \(0.1\) s, respectively, for the Bluebird UAV [11; 30]. The resulting transfer function depicting the relationship between the elevator's servo control input \(v(s)\) and the UAV's pitch angle \(\theta(s)\) is derived in the following expression. \[\frac{p(s)}{v(s)}=\frac{171.1s+360.6}{s^{3}+13.98s^{2}+66.28s+264.7} \tag{6}\]The aforementioned transfer function is converted into the system's corresponding state space model. For this purpose, the transfer function is dissociated into two separate blocks, as shown below. \[\frac{m(s)}{v(s)}=\frac{1}{s^{3}+13.98s^{2}+66.28s+264.7},\ \frac{p(s)}{m(s)}=171.1s+360.6 \tag{7}\]  
The variable \(m(t)\) and its higher-order derivatives are chosen as the intermittent state variables of the system, relating the servomotor's rotation to the UAV's pitch angle. The expressions in Equation (8) are transformed into the time domain as shown below. \[\vec{m}(t)+13.98\vec{m}(t)+66.28\hat{m}(t)+264.7m(t)=v(t) \tag{8}\]  
\[p(t)=171.1\hat{m}(t)+360.6m(t) \tag{9}\]  
To improve the UAV's longitudinal reference tracking accuracy and robustness against bounded perturbations, a supplementary pitch-rate-error integral variable \(\varepsilon(t)\), expressed in (10), is also included in the system's mathematical model [33]. \[\varepsilon(t)=\int\varepsilon_{p}(t)dt \tag{10}\]  
suchthat, \(e_{p}(t)=\dot{\theta}_{r}-\dot{\theta}(t)\)  
where \(\dot{\theta}_{r}\) is the reference pitch rate of the UAV, and \(e_{p}(t)\) is the error between the reference and the actual pitch rates of the UAV. The error integral variable is rewritten as follows:  
\[\dot{\varepsilon}(t)=\dot{\theta}_{r}-\dot{\theta}(t) \tag{11}\]  
Since \(p(t)=\dot{\theta}(t)\); therefore, by substituting Equation (9) in the above equation, the following expression is derived. \[\dot{\varepsilon}(t)=\dot{\theta}_{r}-171.1\hat{m}(t)-360.6m(t) \tag{12}\]  
The state space model of a linear dynamical system is expressed as shown below. \[\dot{x}(t)=\mathbf{A}x(t)+\mathbf{B}u(t)+\mathbf{G}r(t),\ y(t)=\mathbf{C}x(t)+\mathbf{D}u(t) \tag{13}\]  
where \(x(t)\) is the state vector, \(y(t)\) is the output vector, \(u(t)\) is the voltage input, \(r(t)\) is the reference pitch rate, \(\mathbf{A}\) is the system matrix, \(\mathbf{B}\) is the control input matrix, \(\mathbf{G}\) is the reference input matrix, \(\mathbf{C}\) is the output matrix, and \(\mathbf{D}\) is the feed-forward matrix. The state, input, reference pitch, and output variables of the said UAV system are expressed as Equation (14). \[x(t)= \begin{bmatrix}m(t)&\hat{m}(t)&\hat{m}(t)&\dot{\varepsilon}(t)\end{bmatrix}  ^{T},\ u(t)=v(t),\ r(t)=\dot{\theta}_{r},\ y(t)=p(t) \tag{14}\]  
The state space matrices of Bluebird's UAV model are expressed in (15). \[\mathbf{A}= \begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ -264.7&-66.28&-13.98&0\\ -360.6&-171.7&0&0\end{bmatrix} ,\ \mathbf{B}= \begin{bmatrix}0\\ 0\\ 1\\ 0\end{bmatrix} ,\ \mathbf{G}= \begin{bmatrix}0\\ 0\\ 0\\ 1\end{bmatrix} , \tag{15}\]  
\[\mathbf{C}= \begin{bmatrix}360.6&171.1&0&0\end{bmatrix} ,\ \mathbf{D}=[0]\]