There are various structured ways to set PID gains. The mentioned use of optimal control is one, although if you go there you need to be careful about setting

There are various structured ways to set PID gains. The mentioned use of optimal control is one, although if you go there you need to be careful about setting your costs -- LQR and other optimal schemes assume a perfect model of the plant; the higher you set Q the higher the probability that your system will ring or go unstable. There's various robust control methods, all of which I haven't used in a Good Long While, because for the most part swept-sine measurements work nice. The one I've used most involves taking swept-sine measurements to get the plant response in Bode plot form, then using either Bode or Nyquist plots (or both) to tune my PID (or whatever controller I'm using). For a large class of industrial problems, swept-sine measurements won't do, either because measurements need to be undertaken on production lines that are in operation, and the operators get cranky about things like noticeable sinusoidal variations in the product (think aluminum foil or paper), or because even when operated within safe limits, large machines undergoing swept-sine measurements can be downright scary. In such cases one usually ends up using random excitation or steps (often called "bump testing" if you're in an oil refinery or a paper plant) and some sort of system identification step like ARMA. If you do end up doing testing followed by system ID, you'll most likely get an approximate plant transfer function -- so it's wise to either use a grain of salt when doing your optimal design, or to use some robust design method or other.