# Critically Leveraging Theory for Optimal Control of Quadrotor Unmanned Aircraft Systems
## 2 The Structure and Mathematical Model of the Quadrotor
### E-SSPC for Orbital Tracking  
In this section, the control law is designed to solve the problem of tracing the set trajectory. In the context of a linear Model Predictive Control (MPC) state space framework incorporating model error (E-SSPC), two predictive controllers are synthesized based on the model error. The first controls the pitch through \(\mathrm{U}_{1}\), and the second uses this signal to generate a time parameter in linear motions in the x- and y-axes to compute the two virtual inputs \(\bar{\mathrm{u}}_{x},\bar{\mathrm{u}}_{y}\). To design the controller, Equation (8) can be rewritten in the state space form as follows:  
\[\dot{\bar{\mathrm{z}}}(\mathrm{t})=\mathrm{f}\big{(}\overline{\bar{\mathrm{z}} }(\mathrm{t}),\mathrm{u}_{\bar{\mathrm{z}}}(\mathrm{t})\big{)} \tag{15}\]  
where \(\overline{\bar{\mathrm{z}}}(\mathrm{t})=\big{[}\mathrm{x}(\mathrm{t})\quad \mathrm{u}_{0}(\mathrm{t})\quad\mathrm{y}(\mathrm{t})\quad\mathrm{v}_{0}( \mathrm{t})\quad\mathrm{z}(\mathrm{t})\quad\mathrm{w}_{0}(\mathrm{t})\big{]}^{\prime}\) represents the state space vector of the system. The components, \(\mathrm{u}_{0}(\mathrm{t}),\mathrm{v}_{0}(\mathrm{t}),\mathrm{w}_{0}(\mathrm{t})\) correspond to the linear velocity components of the Quadrotor center in the internal coordinate system representation. Based on Equation (8) and the updated state space vector, the kinematic equation of the system can be formulated, which is utilized for the design of the control algorithm as follows:  
\[\dot{\bar{\mathrm{z}}}(\mathrm{t})= \begin{bmatrix}\mathrm{u}_{0}(\mathrm{t})\\ \mathrm{u}_{\mathrm{x}}(\mathrm{t})\frac{\mathrm{U}_{1}(\mathrm{t})}{\mathrm{ m}}+\frac{\mathrm{A}_{\mathrm{x}}(\mathrm{t})}{\mathrm{m}}\\ \mathrm{u}_{y}(\mathrm{t})\frac{\mathrm{U}_{1}(\mathrm{t})}{\mathrm{m}}+\frac{ \mathrm{A}_{\mathrm{x}}(\mathrm{t})}{\mathrm{m}}\\ -\mathrm{g}+(\mathrm{cos}\theta(\mathrm{t})\mathrm{cos}\[PERSON](\mathrm{t}))\frac{ \mathrm{U}_{1}(\mathrm{t})}{\mathrm{m}}+\frac{\mathrm{A}_{\mathrm{x}}(\mathrm{ t})}{\mathrm{m}}\end{bmatrix}  \tag{16}\]  
where:  
\(\mathrm{u}_{\mathrm{x}}(\mathrm{t})=\mathrm{cos}\,\[PERSON](\mathrm{t})\mathrm{sin} \theta(\mathrm{t})\,\mathrm{cos}\Psi(\mathrm{t})+\mathrm{sin}\Phi(\mathrm{t}) \mathrm{sin}\Psi(\mathrm{t})\);  
\(\mathrm{u}_{y}(\mathrm{t})=\mathrm{cos}\,\Phi(\mathrm{t})\mathrm{sin}\theta( \mathrm{t})\,\mathrm{sin}\Psi(\mathrm{t})-\mathrm{sin}\Phi(\mathrm{t})\mathrm{ cos}\Psi(\mathrm{t})\).  
Equation (8) demonstrates that the motion along the x- and y-axes is dependent on the control input \(\mathrm{U}_{1}\). In fact, \(\mathrm{U}_{1}\) is the magnitude of the total thrust designed to achieve the desired linear motion, while \(\mathrm{u}_{\mathrm{x}}\), \(\mathrm{u}_{y}\) can be viewed as the directions of \(\mathrm{U}_{1}\) causing motions in the x- and y-axes.  
The goal of this approach is to ensure that the UAV follows a predetermined trajectory and to minimize deviation. However, due to the fact that the set coordinates change with time, we need a virtual device with the same model as the Quadrotor:  
\[\dot{\overline{\dot{\mathrm{z}}}}_{\mathrm{r}}=\mathrm{f}\big{(}\overline{ \dot{\mathrm{z}}}_{\mathrm{r}}(\mathrm{t}),\mathrm{u}_{\bar{\mathrm{z}}_{ \mathrm{r}}}(\mathrm{t})\big{)} \tag{17}\]  
where:  
\(\mathrm{u}_{\bar{\mathrm{z}}_{\mathrm{r}}}(\mathrm{t})=\big{[}\mathrm{u}_{ \mathrm{x}}\mathrm{r}\mathrm{t}\mathrm{t}\mathrm{t}\mathrm{t}\mathrm{t}\mathrm{ t}\mathrm{t  
where \(\dot{\widetilde{\xi}}(t)=\widetilde{\xi}(t)-\widetilde{\xi}_{x}(t)\) is the error vector and \(\widetilde{\mathsf{u}}_{\xi_{x}}(t)=\mathsf{u}_{\xi_{x}}(t)-\mathsf{u}_{\xi x}\) is the control input error. The matrices A(t) and B(t) are the Jacobi matrices of system (15). Moreover, to facilitate tracking performance in the presence of continuous perturbations, the error vector now incorporates an integral component of the position error. Considering the error vector:  
\[\mathsf{x}_{\xi_{x}}(t)=