The bad news is that the introduction of the integrator term degrades stability, and for stable systems (a reasonable expectation, after all) the integrator τ or system time constant is five to ten times that of the proportional τ. Unlike the proportional τp of 3.2 milliseconds, integrator τi is on the order of 25 msec.

The ten-micron move in our frictionless air bearing example, which took a mere 18 milliseconds, is about 150 msec with a conventional stage. While there are a series of tricks (gain scheduling, friction bias, backlash compensation, and so on) that can be used to try to correct the problem, the issue remains.

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Traditional system: Absolute force value

Select figure to enlarge.

Friction is the main drawback of conventional stages, largely because it makes proportional servocontrol terms useless for small moves. To better understand the problem, consult the graph below. Here, the vertical axis is absolute value of force (which would be torque in a rotary-driven system) and the horizontal axis is position error, both positive and negative.

If we look at the proportional term of a servo loop, it produces a force that is linearly proportional to the error — hence its name. For example, if an error of 1,000 counts results in 100 Newtons of force, then an error of 500 counts would produce 50 Newtons, and so on.

The response of the proportional term is shown by the red line; its slope is the servo stiffness, in Newtons per meter; as it happens, this can be readily calculated - it is equal to m·ω02/4, where m is the mass in kilograms, and ω0 is the servo bandwidth in radians per second. The more familiar servo bandwidth in Hz. is simply ω0/2π.

Returning to the graph, note the horizontal line just above the X axis. This corresponds to the friction in the system, in units of Newtons. The force developed by the proportional term acts to drive the moving element of the stage towards zero position error. A problem arises, however, when the force due to the proportional term is less than or equal to the frictional force. At this point, we're stuck: the stage is, say, 50 µm from the target position. The proportional term responds with 5 N of force, but with 6 N of frictional force, nothing happens. The stage motion has encountered the friction boundary, at which the proportional term fails.