Chopped PID Control of Processes with Delay

For nearly 100 years PID controllers have been the standard for feedback control of a wide range of processes including autopilots, automotive cruise control, industrial heating, servo positioning, and motor speed control.   PID stands for Proportional-Integral-Derivative, also called Span-Reset-Rate in earlier times.  The PID controller reads the process variable, PV, such as temperature or position to be controlled.  The value is compared to a setpoint, SP, and use to calculate a control variable, CV, such as heater power or motor voltage to drive the process toward the setpoint.  PID controllers are “tuned” by adjusting three internal constants.  Ideally the tuning should allow close control of the process.  A properly tuned controller should bring the process up to the desired setpoint quickly and settle smoothly to the desired value without overshoot.  A controller that is too aggressively tuned will slew the process too hard and overshoot, then try to bring that process back, undershooting, and continue to oscillate around the setpoint.  A controller that is too conservatively tuned will react sluggishly and slowly approach the desired value, perhaps never reaching it.

PID works well for ordinary processes but has a problem controlling processes with a significant delay compared to the process response time constant.  An exotic example would be a remotely controlled lunar rover.  It may be able to turn in 1 second but nothing will be seen for 3 seconds and any error will take 3 seconds to change.  Imagine a simple earth based direction control where the rover hunts back and forth around the desired direction, tacking one way for 3 seconds and then the other for 3 seconds.  If a standard PID controller is tuned conservatively enough to eliminate the hunting it will be very sluggish compared to the potential speed of the overall controlled process.  A more familiar example is adjusting a bathroom shower.  If it’s too cold you turn down the cold faucet, but nothing happens for few seconds.  If you keep turning the cold down while waiting the water will end up too hot.  Then you start turning the cold back up and when the water at the shower head hits the right temperature it is already too cold coming up the pipe.  The problem here is that the temperature can be changed much faster than you can find out about it.  A human quickly hits upon the solution:  make a small adjustment, wait a bit, make another small adjustment, wait, and repeat until the temperature is correct.

Modern computerized process control uses strategies such as the Smith predictor which uses a mathematical model of the process to predict in advance what the process will do after the delay or System Inversion which uses a model of both the process and the controller for an improved future prediction.  These require a priori mathematical computer models of the process and controller.

Long before such sophistication existed there was a surprisingly robust technique called chopped or pulsed PID.  This basically emulates the human shower algorithm by chopping the PID controller on and off repeatedly.   This is still useful when a simpler solution is desired or when there is limited knowledge of the process model, as is usually the case.

The PID is first turned on briefly.  While ON the PID operates normally, driving the process toward the desired result.

The PID control is then turned off for a time approximating the process delay.  While OFF the proportional and derivative terms sample the error normally but have no other effect beyond keeping track of the rate of change.  If the integral gain is nonzero, the internal integral term is held unchanged.  If the process is integrating such as position control, the control variable is turned off (zeroed) while the PID is off.  If the process is a normal lag such as temperature or speed control, the control variable is held unchanged while the PID is off.

This adds two more constants to the existing three for PID.  The off time should approximate the delay, longer is more conservative, shorter is more aggressive.  An on time less than the process time constant exclusive of delay is more conservative and longer is less conservative.  Neither is particularly critical and the control scheme is quite robust.  When tuned properly, chopped PID responds and settles much more quickly and stably than standard PID for a process with delay.

Processes that involve pure delay include: fluid flow in pipes where the source is heated or cooled but the pipe outlet temperature is sensed, speed of light delay for control of satellites, pneumatic instrumentation and control systems where small pressure changes have to propagate through long thin tubes, and control over computer networks where there can be significant processing and/or communication delays.

Integrating Sampler

The standard point or impulse sampler that converts a continuous signal to a discrete data sequence has been well understood since 1924 thanks to Harry Nyquist.   It is often assumed that the impulse sampler is the only way to create a discrete sequence.  There are other sampling strategies that may have advantages in some applications.  Here is an example of one.

Dual Clutch Transmission

Elements of irrelevant tech are much like animals in a zoo, interesting to look at but, for the most part, not very good pets if you have to live with them.  Every once in a while one of the animals escapes into the general population.  A case in point is the dual clutch transmission.  These started appearing in production cars about 15 years ago although they have a longer history in various prototypes, specialty sports cars, and Porsche Le Mans cars.  Prior to that, the two standard choices for consumer car transmissions were manual (stick shift) or conventional automatic.  Manual transmissions are simple, rugged, and efficient but not for everyone.  Modern conventional automatic transmissions are the result of 90 years of refinement.  They routinely provide reliable, quiet performance far in excess of 100,000 miles.

The dual clutch “automatic” transmission is very complex with a large number of moving parts, many more than a conventional transmission.  It is effectively two manual transmissions, one (as an example) for 1st, 3rd, 5th, and 7th gear and the other for reverse, 2nd, 4th, and 6th gear.  It has two clutches to switch the engine torque between the two gear trains, two counter shafts, and two sets of synchronizers.  It contains multiple electric motors, reduction gears, and actuators to activate the two clutches and the shift forks for the various gears on each side and a computer to run the whole thing.  This tends to make them fairly expensive.

The two sides take turns as the transmission is shifted between gears.  As an illustration of normal operation assume the transmission has just up shifted to 3rd with the odd side now driving.  The disengaged even side is in 2nd and will be moved to 4th to get ready for the next shift.  To make that shift, the odd side clutch is disengaged and the even side clutch is carefully engaged to supply power in 4th gear.  Note that there is no fluid coupling or torque converter as in a conventional automatic transmission so the clutch engagement must be handled as carefully and smoothly as with the clutch on a manual transmission to avoid jerks.  One problem here is that the transmission has to guess which gear you will need next.  For a straight acceleration up through the gears, or for a normal downshift pattern coming to a stop, this is not a problem.  However, if you slowed down for a turn and want to speed up again, the off side transmission will find itself in the wrong gear and will have to shift back internally before it can transfer between the two clutches to apply power in the proper gear.  On paddle shift versions this means that sometimes the dual clutch transmission shifts when you hit the paddle and sometimes it shifts later.

When a new tuned and adjusted dual clutch transmission leaves the factory it shifts as smoothly and as quietly as a conventional automatic.  As it is driven, all the electric motors, reduction gears, actuators, bearings, and clutches start to wear.  After a few thousand miles the transmission develops a slight whir with sliding noises when it shifts, especially during turns in traffic.  When you take it to the dealer they will tell you there is nothing wrong with it as they are all like that.  The reason the dealer tells you that is that they are all like that.  After a few more thousand miles the whir turns into a whine of electric motor gears and the sliding noise turns into the shift forks clanking.  Once again the dealer assures you that they are all like that.  They offer to upgrade the software in the transmission, which doesn’t appear to do anything.  After a few more thousand miles the actuators are worn enough, and there is so much slop in the linkages, that the clutches can no longer be operated smoothly, so the car jerks and shudders when pulling away from a stop.  This was not a problem for the Porsche 956 Le Mans car, but it only had to run for 24 hours.

The dual clutch transmission is irrelevant tech.

 

The Cost of Speed

Traveling over 15 miles per hour takes a pair of cross trainers for $50.

Traveling over 150 miles per hour takes a Corvette for $50,000.

Traveling over 1500 miles per hour takes an F15 Eagle for $50,000,000.

Traveling over 15,000 miles per hour takes a space shuttle program at $150 billion for three operating shuttles or $50,000,000,000 each.

To go 10 times faster you have to spend 1000 times as much money.

Irrelevant Tech

When my great-grandfather needed to warm his farmhouse, he would cut down a tree, chop it into firewood, and carry a portion back to put in the fireplace. If he ever wondered about future generations automating that chore, he might have imagined a mechanical man performing a similar task, much like the Tin Woodsman from The Wizard of Oz. He could not have imagined that his grandchildren would warm and cool their homes with the turn of a dial, taking advantage of technology and infrastructure that are inevitable only in hindsight.

The Tin Woodsman is irrelevant tech.