Okay, I showed you a Paper Study in the last column (Electronic Design, Nov. 6, p. 146). It was a schematic diagram of a servo controller that I hadn't yet built. I asked you to save that column because you'll need it this month. I told you it was pretty likely to work, and work well. Did it? Yes, it worked very well. Almost perfectly.
First, I built the "model" of the truck (Fig. 2 of the previous column), and by turning the pots, I could make the "truck's speed" increase (by "stepping on the gas") and decrease, as expected. Hey, a truck trying to pick up speed is JUST like an integrator. (Well, like a leaky integrator, because the faster it goes, the less it's able to accelerate fast. And if you take your foot off the gas, it tends to slow down.)
Then I built the main PID servo (Fig. 1 of the previous column). With the integrator disconnected, I connected it up to the "truck model" and started it up. The P and D terms alone made a surprisingly crisp servo. All I had to do to get fast settling and no overshoot was to set those two pots at 100%. Swoop, the output "accelerated" right up to "53.4 mph," ±0.5 mph, and held there with no overshoot (Scope photo 1). The vertical scale factor is 2 mph per division, at *0.835 seconds per horizontal division. Looking at the three lower waveforms, the one on the left shows the "speed" of a lightly loaded ("8-ton") truck with fast acceleration, and the middle one reveals a heavily loaded ("40-ton") truck, accelerating at a slower rate. The right waveform shows the "8-ton" truck on an "upgrade." You can see that the limiter is stable for all kinds of loads and biases. The upper waveforms are just the outputs of the differentiator. In a short while we'll discuss why it worked so well, and why I picked the R and C values that I did.
Now, with just the P and D amplifiers connected, the gain was not infinite. The proportional path had a gain of about 5 × 16 = 80. Therefore, if one was "driving uphill" as fast as it would go on a steep upgrade, there would be a little error, perhaps -0.3 mph (i.e., 53.1 mph), which was necessary to let the throttle go nearly wide open (-7 V). Conversely, if the "truck" was on a "downgrade," the speed error might be +0.3 mph, because the main servo path had to pull the gas pedal "up" until it was nearly off (-0.2 V, e.g.)—even if the driver kept his foot FLOORED. Some people might say that ±0.3 mph with no overshoot is pretty good. It's acceptable to many truck drivers. But I wanted to make it much better. (Have you ever seen any Fuzzy Logic (FL) controllers that had accuracy better than that on upgrades and downgrades?)
Next, I connected up the integrator stage. Sure enough, the integrator would "wind up" and cause a 1.8-mph overshoot as the speed limiter was starting to cut in—up to 55.2 mph—exactly as I had expected. See Scope photo 2 with the same two "trucks." But after it settled, the integrator would trim the error to exactly 53.4 mph, as desired. Not too bad!
Then I added the anti-wind-up circuit. Son of a gun, that worked, too! It kept the integrator's output at 0 V until the speed limiter began to cut in. Now, the speed would rise right up to 53 mph ±0.2 mph, and promptly settle out to 53.4 mph, exactly, within better than 0.1 mph. That's what I expected (Scope photo 3).
I fooled around with higher gain and lower gain, and things worked as expected. I fooled around with heavier versus lighter trucks. I even got data (and a scope photo) on an "88-ton truck," but we decided not to print it because it was very well-behaved and boring. The control was always well-behaved. I tried cutting back on the gain for the D term and got some slow overshoot. It wasn't really BAD, maybe 0.5 mph. That's what I had anticipated. Then I set that gain back to 100%.