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NASA NTRS1993

Nonlinear rescaling of control values simplifies fuzzy control

Traditional control theory is well-developed mainly for linear control situations. In non-linear cases there is no general method of generating a good control, so we have to rely on the ability of the experts (operators) to control them. If we want to automate their control, we must acquire their knowledge and translate it into a precise control strategy. The experts' knowledge is usually represented in non-numeric terms, namely, in terms of uncertain statements of the type 'if the obstacle is straight ahead, the distance to it is small, and the velocity of the car is medium, press the brakes hard'. Fuzzy control is a methodology that translates such statements into precise formulas for control. The necessary first step of this strategy consists of assigning membership functions to all the terms that the expert uses in his rules (in our sample phrase these words are 'small', 'medium', and 'hard'). The appropriate choice of a membership function can drastically improve the quality of a fuzzy control. In the simplest cases, we can take the functions whose domains have equally spaced endpoints. Because of that, many software packages for fuzzy control are based on this choice of membership functions. This choice is not very efficient in more complicated cases. Therefore, methods have been developed that use neural networks or generic algorithms to 'tune' membership functions. But this tuning takes lots of time (for example, several thousands iterations are typical for neural networks). In some cases there are evident physical reasons why equally space domains do not work: e.g., if the control variable u is always positive (i.e., if we control temperature in a reactor), then negative values (that are generated by equal spacing) simply make no sense. In this case it sounds reasonable to choose another scale u' = f(u) to represent u, so that equal spacing will work fine for u'. In the present paper we formulate the problem of finding the best rescaling function, solve this problem, and show (on a real-life example) that after an optimal rescaling, the un-tuned fuzzy control can be as good as the best state-of-art traditional non-linear controls.

Vanlangingham, H.↗

NASA NTRS1992

Gaussian membership functions are most adequate in representing uncertainty in measurements

In rare situations, like fundamental physics, we perform experiments without knowing what their results will be. In the majority of real-life measurement situations, we more or less know beforehand what kind of results we will get. Of course, this is not the precise knowledge of the type 'the result will be between alpha - beta and alpha + beta,' because in this case, we would not need any measurements at all. This is usually a knowledge that is best represented in uncertain terms, like 'perhaps (or 'most likely', etc.) the measured value x is between alpha - beta and alpha + beta.' Traditional statistical methods neglect this additional knowledge and process only the measurement results. So it is desirable to be able to process this uncertain knowledge as well. A natural way to process it is by using fuzzy logic. But, there is a problem; we can use different membership functions to represent the same uncertain statements, and different functions lead to different results. What membership function do we choose? In the present paper, we show that under some reasonable assumptions, Gaussian functions mu(x) = exp(-beta(x(exp 2))) are the most adequate choice of the membership functions for representing uncertainty in measurements. This representation was efficiently used in testing jet engines to airplanes and spaceships.

Kreinovich, V.↗