A position servo under a PD controller: two closed-loop poles, exactly where you put them. Except the real actuator has a lag the model omits. Design on the model, test on the real thing, and see where the poles really went.
The plant is G(s) = 1 / (s(s+2)). The controller is u = kp(r − y) − kdẏ, and the two gains follow from where you drag the pole. The cost is settling time in seconds, plus four times the overshoot, plus a hundredth of the control energy; an unstable loop scores 99. The textbook design puts the poles at damping 0.7 and natural frequency 4. The left half-plane is not free real estate.
Any linear system answers a nudge with a sum of modes, each of the form est. The numbers s are its poles, and they are complex: the real part sets how quickly a mode dies away, the imaginary part how fast it oscillates while doing so. A pole at −3 ± 3j is a wobble at three radians per second that shrinks by a factor of e every third of a second. Two poles, one picture, the whole response.
With feedback you can move the poles. For this servo two gains give two poles, and you can put them anywhere: match the polynomial above and the gains fall out. That is pole placement, and it is why the dashed rays are drawn. Along a ray the damping ratio ζ is constant, so the overshoot is constant; at a fixed distance from the origin the natural frequency is constant, so the speed is. The textbook answer, ζ = 0.7, is the compromise everyone learns first: fast, with a small overshoot.
The real servo's actuator cannot deliver a force instantly; it lags with its own time constant, which adds a third pole the model does not have. For slow designs it does not matter. But as you drag the pole left, demanding a faster response, the gains grow, and the ignored lag eats the loop's phase margin until the real poles drift toward the right half-plane. Run the experiment and the blue dots show where the poles actually went. Every model omits something. The skill is knowing how far you can push before the omission comes back to bite.
The picture of where poles travel as a gain increases is called the root locus, drawn by Walter Evans in 1948 with a protractor and a spirule. On this servo, one branch of it eventually crosses the axis. That crossing is the game.