## 3 Methodology
### Plant Representation  
The RUAV model is configured in State Space representation. Due to the complex, highly coupled nature of the dynamic system, the differential equations are arranged in matrix form, making the controller design process simpler. The state space equations are shown in Equation (1) along with the corresponding block diagram for these equations in Figure 3, the matrices for deriving these differential equations can be found in Appendix A.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Author** & **Type** & **Category** & **Advantages** & **Disadvantages** \\ \hline
18 & [PERSON] [20] & Neural Network state estimation & Hybrid & Provides a high level of sensor noise attenuation for use with LQR & Larger level of computational complexity when compared with a Kalman filter \\ \hline
19 & [PERSON] et al. [21] & PIDNN & Hybrid & Optimal gains for PID controller improving the conventional Extremum seeking and Zigler Nichols algorithms & No consideration of the practical implementation of the controller, torques possible in simulation may not be feasible in real application \\ \hline
20 & [PERSON] et al. [22] & PID-GA & Hybrid & Genetic Algorithm (GA) used to tune PID gains out-performs Extremum seeking and Zigler Nichols in settling time and overshoot & Implemented on a simple dynamic system so may not transfer accurately to an RUAV. \\ \hline
21 & [PERSON] et al. [23] & LQG-GA & Hybrid & LQG tuned by GA out performs the same controller tuned manually in tracking performance. & Only the Feedback gain element of the LQG was tuned by GA, the Kalman filter was tuned manually. \\ \hline \end{tabular}
\end{table}
Table 1: Cont.  
\[\dot{x} = Ax+Bu \tag{1}\] \[y = Cx+Du\]  
The A matrix is the system matrix and contains the state information of the system. The RUAV model has eight states--longitudinal velocity (m), normal velocity (w), pitch rate (q), pitch displacement (\(\theta\)), lateral velocity (v), roll rate (p), roll displacement (\(\varphi\)) and yaw rate (r). The variable \(x\) is the state vector containing the current value of the states, in the case of the RUAV model, it is an eight-element vector with each element corresponding to a state in the A matrix.  
The B matrix is the Input matrix and describes the behavior of each input to the system. The RUAV has three inputs--pitch displacement (\(\theta\)), roll displacement (\(\varphi\)) and yaw rate (r). The \(u\) variable is the input vector (controller output), \(u\) contains the output of the controller, in this case, a three-element vector is required with each element corresponding to the size of the B matrix.  
The C matrix is the output matrix and details which of the state are measured as outputs in the system. The RUAV model has five measured outputs--pitch displacement (\(\theta\)), roll displacement (\(\varphi\)), yaw rate(r), pitch rate (q), roll rate (p).  
The D matrix is the feedthrough/feedforward matrix, this is used in the system when the system input directly effects the system output. In this case, the RUAV model does not have a D matrix and as such, has a 5 \(\times\) 3 matrix of zeros. The \(y\) variable details the output of the system.