Pole Placement

With full state feedback, a controller can put the closed-loop poles almost anywhere. You choose how fast and how damped you want the response, and the gains follow directly. The catch is the effort: faster poles need much more force.

Desired speed ωn Desired damping ζ

Choosing the gains

A cart pushed by force $u$ is a double integrator, $\ddot{p} = u$. With feedback $u = -k_1 p - k_2 \dot p$, the closed loop is $\ddot p + k_2 \dot p + k_1 p = 0$. Matching it to the desired $s^2 + 2\zeta\omega_n s + \omega_n^2$ gives

$$k_1 = \omega_n^2 \qquad k_2 = 2\zeta\omega_n$$

For larger systems, Ackermann's formula or a numerical solver does the same matching.

What to notice

Double the speed $\omega_n$ and the cart returns twice as fast, but the peak force grows about four times, since $k_1 = \omega_n^2$. Real motors have limits, so you cannot place poles as far left as you like. Low damping gives overshoot. This is the trade-off that LQR, on the next page, handles more gracefully by weighing error against effort.