B. Control Design

Choose appropriate values for $PO$ and $t_p$ in both pitch and yaw axes satisfying the design specifications.

For these selected values of $PO$ and $t_p$ within the design specs, compute the desired values of damping ratio $\zeta$ and natural frequency $\omega_n$ using the following equations: $PO = \left(e^{ \frac{-\zeta \pi}{\sqrt{1 - \zeta^2}}}\right)*100\%$ $t_p = \displaystyle \frac{\pi}{\omega_d}$ $\omega_d = \omega_n \sqrt{1 - \zeta^2}$

Using the aforementioned desired values of $\zeta$ and $\omega_n$ and the system parameters calculated in part A, compute the proportional gain $K_p$ and the derivative gain $K_d$ individually for pitch and yaw controllers as follows: Note: Let subscript $\text{op}$ represent open-loop (actual system parameters), and subscript $\text{cp}$ represent closed-loop (desired parameters). Pitch: $K_p = \displaystyle \frac{\omega_{n_\text{cl}}^2 - \omega_{n_\text{op}}^2}{\overline{K}_{\theta \theta}}$ $K_d = \displaystyle \frac{2 \zeta_{\text{cl}} \omega_{n_\text{cl}} - 2 \zeta_{\text{op}} \omega_{n_\text{op}}}{\overline{K}_{\theta \theta}}$ Yaw: $K_p = \displaystyle \frac{\omega_{n_\text{cl}}^2}{\overline{K}_{\psi \psi}}$ $K_d = \displaystyle \frac{2 \zeta_{\text{cl}} \omega_{n_\text{cl}} - \displaystyle \frac{1}{\tau_{\text{op}}}}{\overline{K}_{\psi \psi}}$ Note: In the above calculations, use the normalized thrust gains obtained using the steady-state method.

Add an integral term, $K_i$, to the pitch controller. Choose $K_i$ to be a value in the range $1\%-10\%$ of $K_p$. Caution: Integral feedback, while improving steady-state response, introduces lag. Large values of integral gain, while reducing damping, can lead to loss of stability of the closed loop system.

Initially, set the feedforward gains ($K_{\theta-to-\psi}$ and $K_{\psi-to-\theta}$) to 0, i.e., ff_py = 0 and ff_yp = 0 in the MATLAB code. Eventually, you will choose appropriate values for the feedforward gains such that the coupling effect is minimized. For example, choose $K_{\theta-to-\psi}$ in Figure 1, to cancel the torque from the pitch rotor, i.e. , $K_{\theta-to-\psi} = \displaystyle - \frac{\overline{K}_{\psi\theta}}{\overline{K}_{\psi\psi}} \qquad \qquad (1)$ Likewise, choose $K_{\psi-to-\theta}$ to cancel the torque from the yaw rotor, i.e. , $K_{\psi-to-\theta} = \displaystyle - \frac{\overline{K}_{\theta\psi}}{\overline{K}_{\theta\theta}} \qquad \qquad (2)$ Feedforward controller minimizes the coupling effects. Use Figure 1 as a reference for how to add the two feedforward paths.

First consider simulation validation of your controller using decoupled dynamics and decoupled controller (i.e., no feedforward between pitch and yaw channels). 1. Set the values of $\overline{K}_{\theta\psi} = 0$ in the pitch plant Simulink block and $\overline{K}_{\psi\theta} = 0$ in the yaw plant Simulink block. This will remove the coupling effects between pitch and yaw dynamics. 2. Set the integral gain of the pitch controller $K_i = 0$. 3. Simulate the closed loop system response with yaw-only command: $\theta_d = 0 \degree$and $\psi_d = 30 \degree$. Save the data. 4. Check whether your controller meets the desired specifications for the yaw command. Since decoupled system dynamics are used in this step, you will likely see a close match between your simulation results and the desired specifications. 5. Set the integral gain of the pitch controller $K_i \approx 1\% - 10\%$ of $K_p$ (refer to Step 4 of Controller Design). 6. Simulate the closed loop system response with pitch-only command: $\theta_d = 10 \degree$and $\psi_d = 0 \degree$. Save the data. 7. Check if your pitch controller meets all the desired specifications. If it fails to meet the $PO$ and $t_p$ specifications, relax the specifications and/or tune the pitch PID controller gains to try to meet the specifications. Save the tuned results.

Next, consider the simulation evaluation of the feedback controller for the system with coupled dynamics. 1. Set the values of $\overline{K}_{\theta\psi}$ in the pitch plant Simulink block and $\overline{K}_{\psi\theta}$ in the yaw plant Simulink block to the estimated values from Part A: System Identification. This accounts for the coupled dynamics. 2. Simulate the closed-loop response of the system with pitch-only command: $\theta_d = 10 \degree$and $\psi_d = 0 \degree$. Save the data. 3. Simulate the closed-loop response of the system with yaw-only command: $\theta_d = 0 \degree$and $\psi_d = 30 \degree$. Save the data. 4. Simulate the closed-loop response of the system with simultaneous pitch and yaw commands: $\theta_d = 10 \degree$and $\psi_d = 30 \degree$. Save the data.

Set the feedforward gains $K_{\theta-to-\psi}$ and $K_{\psi-to-\theta}$ based on Eqs. 1 and 2 (marked as ff_py and ff_yp in the MATLAB code). These gains are different from the normalized cross-thrust torque gains that you set to make the coupled dynamics.

Simulate the closed-loop response of the system with pitch-only command: $\theta_d = 10 \degree$and $\psi_d = 0 \degree$. Save the data.

Simulate the closed-loop response of the system with yaw-only command: $\theta_d = 0 \degree$and $\psi_d = 30 \degree$. Save the data.

Simulate the closed-loop response of the system with simultaneous pitch and yaw commands: $\theta_d = 10 \degree$and $\psi_d = 30 \degree$. Save the data.

Yaw-only command case: State if the design specifications were met, either including figures with design specs marked (Step 3) OR including a table with the $PO_{\psi}$, $t_{p_\psi}$, $e_{ss_\psi}$ and $\max |V_\psi|$values recorded from the response (Step 4).

Pitch-only command case: State if the design specifications were met, either including figures with design specs marked (Step 7) OR including a table with the $PO_{\theta}$, $t_{p_\theta}$, $e_{ss_\theta}$ and $\max |V_\theta|$values recorded from the response (Step 7).

Pitch and yaw combined case: State if the design specifications were met, either including figures with design specs marked (Step 4) OR including a table with the $PO_{\theta}$, $t_{p_\theta}$, $e_{ss_\theta}$, $\max |V_\theta|$, $PO_{\psi}$, $t_{p_\psi}$, $e_{ss_\psi}$ and $\max |V_\psi|$values recorded from the responses (Step 4).

Pitch and yaw combined case: State if the design specifications were met, either including figures with design specs marked (Step 4) OR including a table with the $PO_{\theta}$, $t_{p_\theta}$, $e_{ss_\theta}$, $\max |V_\theta|$, $PO_{\psi}$, $t_{p_\psi}$, $e_{ss_\psi}$ and $\max |V_\psi|$values recorded from the responses (Step 4).