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, run it on the real thing, and see where the poles really went. This is the s-plane, live: an instrument more than a game, and where the series keeps its picture of what a pole is.
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. Left of σ = 1 the design asks for kd < 0, positive velocity feedback; the loop stays stable, since 2 + kd = 2σ > 0, but it is a strange thing to ask of a servo. The left half-plane is not free real estate.
On the model, two gains put two poles anywhere you like, and the further left you drag them the faster the servo answers. Then you run the experiment. The real actuator has a time constant the model left out, which adds a third pole; as your gains grow, that pole pulls the pair you placed toward the axis and, past some point, across it. The blue dots are where the poles really went. Every model omits something, and gain is what the omission punishes. The skill is knowing how far to push before it comes back.
Any linear system answers a nudge with a sum of modes, one per pole. The real part sets how quickly a mode dies, 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. Left of the axis modes decay, on it they ring forever, to the right they grow until something breaks. So the left half-plane is where stable things live, and feedback is the art of moving poles into it, cheaply, and keeping them there when the world differs from the model.
Poles describe the linearisation. They say nothing about how large a disturbance a nonlinear system survives; that is a region of attraction, which Kick is about. And a system can have every pole in the left half-plane and still be a poor design, if a zero sits in the right one.
Two gains, two poles: write the characteristic polynomial you want and match coefficients. Along a ray from the origin the damping ratio ζ is constant, so the overshoot is; on a circle the natural frequency is constant, so the speed is. ζ = 0.7 is the compromise everyone learns first. With the whole state fed back, as in Stable is Still with four gains for four poles, any pattern is reachable, and LQR is one principled way to choose it.
The price of a fast pole is gain, and gain is what unmodelled dynamics punish: the actuator's pole, sensor noise, the force limit. On this servo the design turns odd before it turns unstable, asking for kd < 0 left of σ = 1. Placement tells you what the gains must be; it does not tell you whether you should want them.
The actuator cannot deliver its force instantly; it lags with time constant τ, a pole at −1/τ the model does not have. For slow designs it is far away and harmless. Demand speed and the gains grow, the ignored lag eats the loop's phase margin, and one branch of the root locus, the picture Walter Evans drew in 1948 with a spirule, crosses into the right half-plane. Run the experiment and the blue dots show the crossing.
Every plant in this series hides something its manual leaves out: a delay, a lag, friction, an unmodelled pole. The textbook answer for the model is not the best answer for the world, which is the honest state of the field and the reason par can be beaten. Robustness is the study of how much omission a design survives; margins are its currency, and every increment of speed is bought with some of it.