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A more interesting example, and one dear to the heart of robot builders everywhere, is the ability to control the position of a motor-driven mechanism. In this example, both X and Y are in terms of a physical position and the control output Z is the power applied to the motor. Since Z is not just an on/off switch we have more subtlety available in this controller. Ignoring a few complexities for a moment, we can expand on equation (17-1): Z KP Y X 17-3 This, and equation (17-1), are proportional controls, since their outputs are in proportion to the error (Y X). The new factor KP is a gain control. It lets us
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Fig. 17-3. Proportional control.
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adjust how quickly the mechanism moves toward the goal. Figure 17-3 shows this formula in action. The square pulse is the target value, while the wiggly trace is the actual value of the system under control. The top trace has a proportional gain of 0.75 and shows a fair amount of overshoot. The oscillation that follows is called ringing. After this, it settles down into a steady line a little bit under the set point. The bottom trace has a smaller gain of 0.25 and avoids the overshoot, but it s settling point is a bit lower than the set point. Larger proportional gains settle closer to the set point but have more overshoot and more ringing. Smaller gains don t overshoot, but then they never really reach the goal either. If there were only a way to get rid of that offset. . . . Instead of using just the current error in position, we could accumulate, or add up, the error over time. This way, even a small offset would add up over time and could move our mechanism closer to its goal. This is shown in equation (17-4): e Y X eI eI e Z KP e KI eI This describes proportional-integral control. We broke the (Y X ) error term out into its own symbol e. Then we keep adding our instantaneous error e to the accumulated, or integral, error eI. Adding the new integral gain KI lets us control how important this integral error is to the control. KI needs to be kept small to avoid massive ringing 17-4
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Fig. 17-4.
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Proportional-integral control.
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(Fig. 17-4). In both traces KP is set to 0.25. Note the extra ringing in the top trace with KI set to 0.03. The subtly smaller KI of 0.02 reaches the goal with a minimum of fuss. It is, however, slow in reaching the goal. Is there a way to make the response a bit snappier Actually, having to react to a massive change in goal is unusual. Goal states are more likely to change slowly. In Fig. 17-5 we see the PI controller following a smoothly changing signal. There is one more term, a derivative, that is used when there is a time delay between the action output and the sensed input. This term tries to predict the future based on the error history: et 1 et et Yt Xt eI eI e eD et et 1 Z KP e KI eI KD eD We introduce time into the equation, where t is the current time and t 1 is the previous time slice. This completes the control, creating a proportionalintegral-derivative control or, more simply, a PID control. This is just one, simple, form of the PID math there are others that give better results at the expense of more calculation complexity. The PID needs 17-5
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Fig. 17-5.
PI control with gradual change.
to be tuned to match a specific problem. While the formulas are reasonably generic, the gain factors need to be adjusted to get the best control. Now for one of those complexities that we ignored earlier. What are the units we are generating for Z Right now, Z is in terms of the input. If the inputs X and Y were scaled to the range [0. . .1] by dividing them by their maximum values, the output is then in terms of percent power. This can then be converted into a power level for a motor or other mechanism.
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