Attitude PD / PID Controller (Simulink)

Goal: Create angle command controller with zero overshoot + robustness and disturbance resistance

Plant: Rigid axis with damping (viscous, against applied torque)

$\dot{\theta}=\omega,\quad I\dot{\omega}=\tau-b,\omega$

Parameters: I=0.05 kg·m², b=0.002 N·m·s. Actuator max: max_tau

Controller

• PD tune for near-critical damping, next add a small integral gain to reject bias
• Preset gains used in figures: Kp=2.4, Kd=1.35, Ki=0 (PD controller)
• Disturbance demo presets: Ki=0.03 with anti-windup (back-calculation)

Tuning method I used: construct closed loop using dynamics + control eqns, match to second-order target, solve for gains.

Dynamics: $\dot{\theta}=\omega,\quad I\dot{\omega}=\tau-b,\omega$

Control: $\tau = K_p(\theta_c-\theta) - K_d,\dot{\theta}$

Closed loop (PD)

$$ \ddot{\theta} + \frac{b+K_d}{I},\dot{\theta} + \frac{K_p}{I},\theta = \frac{K_p}{I},\theta_c $$

Match to second-order target (approx. critical damping, $\zeta\ = 1$)

$$ \ddot{\theta} + 2\zeta,\omega_n,\dot{\theta} + \omega_n^2,\theta = \omega_n^2,\theta_c $$

So gains are:

$$ K_p = I,\omega_n^2,\qquad K_d = 2,\zeta,I,\omega_n - b $$

I picked $\omega_n$ for desired speed (using torque limit), then did minor tweaks by hand

Note d/dt($\theta_c$) = 0 as goal point is held constant.

How to run

1. Open attitude_pd.slx
2. Edit Model Workspace parameters:
   My values: I=0.05, b=0.002, Kp=2.4, Kd=1.35, Ki=0, theta_step=deg2rad(30), max_tau=0.02
3. Run
4. Robustness check: Set I to desired values (I chose 0.045, 0.055) and rerun
5. Disturbance check: connect tau_dist (disturbance torque) step at desired time and value (e.g. 0.018 N·m, t = 4 s)

Note: use either the internal output limiter or external saturation block (not both). Limits: ±max_tau

Observations

• Without disturbance torque, a PD controller works well (0 overshoot, Ki = 0)
• With disturbance torque, select nonzero Ki (PID), and enable back calculation anti windup for no overshoot

Results

• Block setup:
• Theta vs. commanded theta:
• Commanded torque: .
• Robustness (±10% moment of inertia): .
• Disturbance Torque Applied: PD (notice offset) and PID+AW (no more offset):
  

Tested with Simulink R2025A

License: MIT