Many of the attacks leveled against fuzzy logic seem to support the time-honored axiom that ignorance is the firmest basis from which to criticize. Some, however, are well-considered. One observation in particular-I have read it in various places-is worth looking into. Simply stated, it is:
Successful applications of fuzzy logic are successful because of factors other than the use of fuzzy logic.
Although the following response is subjective, I hope my observations are honest. I base them on the approximately 20 fuzzy design projects with which I have been directly involved, ranging from simple control systems to rather complex analysis models. In each case, the goal was not to solve the problem using fuzzy logic, but rather to solve the problem successfully and in a cost-effective manner. In each case, the solution included fuzzy logic.
Two widely claimed benefits of fuzzy systems are system representation in intuitive terms, and ease of development and modification. Two less-often-stated benefits are flexibility in input selection and controlled input quantization. I have addressed the first two in previous columns; the other two require some discussion.
Flexibility in input selection: For a given system design, the solution model often defines what inputs we will use. For example, in physical control systems, we work with proportional error, its integral, its derivative, and, sometimes, its second derivative. The characteristics of these inputs, their behavior in the presence of various types of noise, and how they interact with each other are all well-understood. If a system parameter or characteristic is not part of the model, we would not normally consider its possible use, and we would not know how to use it.
Working with a rule-based system makes choice of inputs an open issue. Inputs combine logically in rule conditions. Adding an input means adding it to the conditions of existing rules and creating additional rules. The choice of system parameters to use as inputs is no longer a part of the model being implemented, but rather a critical part of the design process.
Controlled quantization: In a rule-based system, the input space is the universe of all possible input combinations. Part of the design process is to identify those regions within the overall input space that will be active and to create rules for them. If the entire input space is valid, that is, if any combination of inputs is possible during system operation, rules must cover the entire input space.
The system designer divides the active input space into regions. A single rule governs each region. The designer typically defines each region by identifying sections of individual inputs and logically combining them. The size of each region in the overall input space is what I call its "quantization." The phrase "controlled quantization" refers to the ability to define each region size independent of the size of other regions. Its meaning also includes the ability to tighten or loosen quantization (to decrease or increase region size) easily with minimum impact to the rest of the system.
These four capabilities-system representation in intuitive terms, ease of development and modification, flexibility in input selection, and controlled input quantization-exist not because a system is fuzzy, but rather because rules represent its system operation. The operational model developed in this manner generates a response function: an output defined for collections of inputs. You can do this using if...then...rules and by other semantic techniques as well.
This conclusion is worth repeating: The four capabilities listed exist not because a system is fuzzy, but rather because it is rule-based. Concerning these four traits, the observation that opened this column is correct-sort of.
As powerful as rule-based representation is, the use of fuzzy logic strengthens it in a number of ways.
Output interpolation: A problem associated with crisp, rule-based systems relates to the dividing of the input space by rule conditions. As changing inputs drive the system from one input region to another, the output undergoes a discontinuous transition. This situation occurs as the rule associated with the first region stops firing and the rule associated with the second region starts firing.
Decreasing the size of the input regions reduces the size and impact of these discontinuous steps; the number of rules increases as well. Alternatively, if system performance allows, filtering the output removes the higher frequency step components.
A fuzzy handling of the rule base eliminates these output discontinuities by interpolating between adjacent commanded outputs. Smooth transitions between output levels is accomplished by overlapping input regions, defining these regions with fuzzy membership functions, allowing multiple rules to simultaneously fire when inputs are in these overlapping regions, and combining the multiple actions into a single output. This is done without increasing the number of rules. In fact, through judicious use of input-function overlap, the number of rules, when compared with a crisp implementation, can actually be reduced.
Robustness: A second problem associated with a crisp rule base is its brittleness. The transition of inputs from a region governed by a rule to a region for which there is no rule, thereby causing system failure, is abrupt; there is no warning. The same transition in a fuzzy system is gradual, allowing for a graceful transition from being fully operational to being nonoperational. In feedback systems, this transition is especially valuable, allowing for operation to be controlled back into a controlled region.
Linguistic variables: Linguistic variables, implemented with fuzzy membership functions, are at the heart of Zadeh's original view of fuzzy complex systems models. Although formulated to allow modeling of extremely complex systems, linguistic variables have been the basis for designing simple and moderately complex fuzzy systems. In such systems, the use of linguistic variables is starting to give way to automatically generated variables-for example, using neural nets or genetic algorithms. However, in sparsely populated, nonoptimized, extremely complex systems, where human expertise remains a necessary part of the design process, linguistic variables will continue to be of great value.
Fuzzy logic allows not only for system operation to be expressed in fuzzy-based linguistic variables, but also for the handling of imprecise and uncertain inputs and intermediate values in a straightforward manner. Other, nonfuzzy techniques for dealing with imprecision exist, but treating such inputs as being fuzzy is easily and effectively handled.
These four traits-output interpolation, robustness, effective linguistic variables, and straightforward handling of uncertain or imprecise input-add significant power to a rule-based architecture. All are a result of using fuzzy logic.
In addition, the four traits are easily observable. There are others, but they are less tangible. By strengthening rule-based representations, fuzzy logic has allowed designers to break from previous system-modeling techniques into models that are intuitively based.
The key phrase is to "break from previous system-modeling techniques." Because using fuzzy logic for system design is new, the number of design procedures is small, and there are few cookbook approaches. The fuzzy-system designer must think through the best way of doing things rather than adapting how someone else did something similar-a situation that can only enhance the opportunity for powerful results.
One example, mentioned previously, is that the designer is free to look at nontraditional system inputs, incorporating them into rule conditions. Critics have said that Japanese consumer goods, touted as being superior because they are fuzzy, are actually superior because they use additional sensors. In general, the use of a fuzzy rule base greatly facilitates adding more sensors and taking advantage of the information they provide. Both claims-that a fuzzy system with additional sensors is more powerful only because of the fuzzy logic and that it is more powerful only because of the additional sensors-are incorrect. The power comes from a fuzzy rule base's ability to effectively use the additional sensors.
So on the surface, I agree with the opening observation: Fuzzy logic tends not to be the driving force in fuzzy-logic systems. However, we can't stop there. Well-designed fuzzy-logic systems work and work well. I see fuzzy logic not as a driving force, but as an enabling force, increasing capability across many facets of existing techniques. Its strength is subtle, yet powerful.