FORC · Lecture 4

PD, PID, and state feedback

Simeon Nedelchev

PD response

I\ddot e+k_d\dot e+k_pe=0

\omega_n=\sqrt{k_p/I},\qquad \zeta=\frac{k_d}{2\sqrt{Ik_p}}

$I=0.5$ , $\omega_n=4$ , $\zeta=0.8$

$k_p=8$ , $k_d=3.2$

A 1 rad error requests 8 N m before gravity compensation.

The actuator may deliver less than the requested torque.

The ideal linear response then no longer describes the motion.

Persistent error accumulates while the actuator clips.

Conditional integration can stop accumulation that worsens saturation.

Continuous-time reasoning needs a discrete-time check.

High gains, delay, and a large timestep can destabilize the implementation.

Compare PD and bias compensation with identical torque limits.

Explain the tracking error and saturation traces.