Signal Processing for Control - Signal Processing Systems - page 272

larger force can be applied without locking the wheels when they are rolling on dry
pavement. Thus, the key practical problem for ABS brakes is how to determine how
much force can be applied without locking the wheels. In fact, there is as yet no
known way to measure this directly. It must be estimated. The control, based on this
estimate, must perform at least as well as an unassisted human driver under every
possible circumstance.
The key practical idea of ABS brakes is to frequently change the amount of
braking force to probe for the best possible value. Probing works roughly as follows.
Apply a braking force and determine whether the wheels have locked. If they have,
reduce the braking force to a level low enough that the wheels start to rotate again.
If they have not, increase the braking force. This is repeated at a reasonably high
frequency.
Probing cannot stop and must be repeated frequently because the slipperiness
of the road surface can change rapidly. It is important to keep the braking force
close to its optimal value all the time. Balancing the dual objectives of quickly
detecting changes in the slipperiness of the surface and keeping the braking force
close to its optimal value is one of the keys to successful ABS brakes. Note that this
tradeoff is typical of adaptive control problems where the controller has to serve the
dual objectives of providing good control and providing the information needed to
improve the controller.
The details of the algorithm for combining probing and control are compli-
cated [ 2 ] . The point here is that the complicated algorithm is a life saving application
of digital signal processing for control. Two other points are important. In order for
ABS brakes to be possible, it was first necessary to develop a hydraulic braking
system that could be electronically modulated. That is, the improved controller
depends on a new actuator. Second, there is very little control theory available to
assist in the design of the controller upon which ABS braking depends.
There are three reasons why control theory is of little help in designing ABS
brakes. First, the interactions between tires and the road surface are very complex.
The dynamic distortion of the tire under loading plays a fundamental role in traction
(see [ 8 ] for a dramatic example). While there has been research on appropriate
mathematical models for the tire/road interaction, realistic models are too complex
for control design and simpler models that might be useful for controller design are
not realistic enough.
Second, the control theory for systems that have large changes in dynamics as
a result of small changes in the control—exactly what happens when a tire loses
traction—is only now being developed. Such systems are one form of hybrid system.
The control of hybrid systems is a very active current area of research in control
theory.
Third, a crucial component of ABS brakes is the electronically controllable
hydraulic brake cylinder. This device is obviously needed in order to implement the
controller. Control theory has very little to say about the physical devices needed to
implement a controller.