Why use fuzzy logic for control ?

Controlling a system means that some characteristics of this system are monitored, and, depending on the values of these characteristics, different controls are applied. An algorithm that transforms sensor inputs into corresponding control values is called a control strategy. The previous chapters deal with the traditional approach of control systems design that consists of the following:

• First, one tries to to describe the behaviour of the system in precise mathematical terms, i.e., one comes up with the exact model of the system.

• Second, one tries to describe in precise terms what one wants to achieve. One wants the control that is the best in the sense of some criterion.

• Now that the controlled system is described in precise mathematical terms, and the objective function is described in the same manner, it can be determined for each control strategy and for each initial state how exactly the system will change and what the resulting value of the control will be. The main goal is then to find the control strategy for which the resulting value of the objective function is the largest possible one. This is a well-defined mathematical optimisation problem, and traditional control theory has developed many methods for solving this problem and designing the corresponding control strategies.

Traditional control theory has many important applications. There are, however, practical cases when this theory is not applicable. Indeed, to apply the traditional control theory, one must

• know the model of the controlled system,
• know the objective function formulated in precise terms, and
• be able to solve the corresponding mathematical design problem.

If one of these conditions is not satisfied, then traditional control methodology is not applicable, as in the following cases:

• Sometimes, the model and the objective function is known, but the design problem cannot be solved. This is when the design problem is very complicated, time consuming or when the problem is new and algorithms for solving it have not yet been developed. For example, parking a car is an example of a problem that traditional control theory has not considered until recently.

• Sometimes, the model is known, but the objective function is unknown. For example, if a control system for a vehicle is designed, the intended goal is to make the ride most comfortable, but there is no well-accepted formalism of what comfortable means.

• Sometimes, one does not even know the model of the controlled system. In many practical applications one can in principle measure all the possible variables and determine the model exactly, but this will increase the cost drastically. In other practical situations, the main goal of the controlled system is to explore the unknown, e.g., to control a rover over a terrain of unknown type, or to control surgery instruments. In such situations, the entire objective of the control is to learn as much about the system, and one cannot have a precise model of this system before the control is over.

If traditional control methodology cannot be applied, how can one control? Often, there is an additional expert knowledge available, for example, expert operators who successfully control the desired system. Expert operators know how to operate a plant. Therefore it is desirable to extract the control rules from the expert and use this knowledge in an automatic control system. At first glance, the problem seems very simple. Since the person is a real expert, one simply ask her multiple questions like ``suppose that is equal to 1.2, is equal to -2.7, ..., what is ?'' After asking all these questions, one will get many pattern, from which one will be able to extrapolate the function using one of the known methods. Alas, there are two problems with this idea:

• There is a computational problem. Since one needs to ask a question for each combination of sensor readings, one may end up having to ask too many questions that takes years.

• There is a more serious problem that makes it in most cases impossible to implement. If one asks a car driver a question like ``you are driving at 80km/h when a car which is 20m in front of you slows down to 50km/h, for how many seconds do you hit the brakes?'', nobody will give a precise number.

An expert cannot usually express his knowledge in precise numerical terms, like ``hit the brakes for 1.27s'', but he can formulate his knowledge by using words from natural language. The knowledge, which one can extract from an expert consists of statements like ``if the velocity is a little bit smaller than maximum, hit the breaks for a while''.

For the fuzzy control methodology one has to

• know the expert's control rules formulated by words from natural language and
• one wants to produce a precise control strategy.

The methodology that transform the informal expert control rules into a precise control strategy is called fuzzy control. The idea was first proposed by Zadeh, and the methodology itself was first proposed and applied by Mamdani. In this chapter it is described exactly how this transformation is done.