Introduction

One feature that you will find in almost every control system is hysteresis and proportional feedback, and if you're like many people, you don't even know that you're using it every day when you drive a car. If you don't drive a car, then you've been a passenger in a car and I'll tell the story from that viewpoint.

Hysteresis and Proportional Feedback

Have you even been in a car or bus where the driver alternately floors the accelerator and jams on the brakes? At the end of the ride, you either had a sore neck or an upset stomach or both because the driver was not using physical cues to regulate or smooth out the riding experience for his passengers.

That's what hysteresis and proportional feedback are for, and it's what makes cruise control such an effective way to smooth out the ride and to save on gas too. Let's get some basic definitions sorted out...

Any non-trivial control system has the following minimal characteristics, and I'll continue with the cruise control example to make things clear.

Target Value
The desired operating point of the overall system, which is speed.
Measured Value
The actual operating point of the system. The measured value is affected by external factors such as hills, and internal factors such as the amount of fuel delivered to the engine.
Difference Value
This is the difference between the target value and the measured value.
Control Input
This is the main adjusting point of the control system. The ammount of fuel delivered to the engine is the primary control input to the cruise control.
Control Algorithm
The control algorithm determines how to best regulate the control input to make the difference value as close to zero as possible. It does this by periodically looking at the difference value and adjusting the control input.

There, I've just explained in very simple terms how a closed loop control system works, and you really need to understand this part or the rest of your robot building carreer will be filled with clumsy and jerky creations. So keep reading.

As you might have guessed, all of the magic is really in the last item, the control algorithm. By making effective use of hysteresis and proportional feedback, the control algorithm allows the cruise control to smoothly accelerate and decelerate as the measured speed changes from the set speed. The sample period also has an effect which some examples will make clear.

Bang Bang Control

Imagine a control algorithm that does the following: If the speed is less than the set point press the gas pedal all the way down. If the speed is more than the set point release the gas pedal completely.

OK. Some of you actually drive like this. And it's dumb. It's dumb because it wastes gas and makes you and your passengers feel sick. And it's hard on the engine and transmission. As soon as the actual speed is less than the set speed, the accelerator gets jammed down. The engine revs higher, the transmission may kick down a gear and the car will accellerate.

Eventually, the speed is just above the set point, and the gas pedal is released. Now the car is actually pushing the motor, and eventually the car slows down and the cycle starts all over again.

And that's why it's called bang bang control. The gas is either fully pressed or fully released and the ride is pretty jerky. This type of control is used in only the simplest systems, such as older thermostats for furnaces and air conditioners.

To smooth things out a bit, and to make the on/off cycle take a bit longer, the designers generally add hysteresis, which we'll discuss next.

Adding Hysteresis

You can think of hysteresis as a sort of "guard band" around the desired set point of the system. In other words, the algorithm is changed so that the accelerator is pressed when the actual speed is less than the set point minus a small constant. Similarly, the accelerator is released when the actual speed is greater than the setpoint plus a small (possibly different) constant.

If you think about it, the ride will still be jerky in this case, and in fact it's going to feel even worse because the car will change speeds less often, but with a greater range of speed at each change.

How is hysteresis helping here? Well, it's not. For systems with a large lag between application of a control signal and the resulting output (like a furnace or air conditioner) the hysteresis reduces the number of control cycles by making each cycle longer. For short lag systems like cars, the result is an even jerkier ride.

We can add another refinement to the system that will smooth things out for everyone, called proportional control.

Adding Proportional Control

As you might guess, the next step is to apply a more sophisticated control signal to the gas pedal, one that allows the pedal to be set to a specific value depending on the error signal. More precisely, after every sample of the error or difference signal, we'll apply a little more or a little less gas.

Depending on how big the error signal is, we'll change the gas pedal setting by bigger or smaller ammounts. That's how this control method gets the proportional part of its name.

It's pretty clear then how the cruise control will work. If the car is standing still and we set the speed to 100 km/hr, the control algorithm will add more gas initially, and then less and less until the error signal approaches zero, at which point the gas setting will not change.

That's fine as long as the car is on a level roadway. If there is a hill, the speed will go down a little, at which point the algorithm adds a little more gas. Eventually, the speed goes back up to 100 km/hr and the gas pedal is pressed down a bit further.

Sooner or later, the car will get to the crest of the hill, and then it will pick up speed as gravity helps the engine. Now the control algorithm will back off the gas pedal, and eventually the speed will equalize.

Finally, the road levels out again, and the car slows down, so the control system adds gas. The end result is that the speed transitions should be so smooth you don't even realize they are happening. A happy by-product of letting the cruise control do the speed regulation is that you will save gas by avoing excessive acceleration and braking as you try to keep the car going at a constant speed.

Conclusion

We've gone over the basics of control systems, but how does this apply to building LEGO Mindstorms robots? Simple. By understanding how the robot can use the basic ideas in this article, you too can build robots that will operate smoothly do their jobs with a minimum of wasted energy.

I'll show many of these principles in action in the article on Using the Light Sensor to do basic navigation.