Interval-Valued Fuzzy Control in Space Exploration

This paper is a short overviewof our NASA-supported research into thepossibility of using interval-based intelligentcontrol techniques for space exploration. Interval-values fuzzy sets were introduced by L.Zadeh, J. A. Goguen, and especially by I. B.Türkşen; they were actively used in expert systemsby L. Kohout. Before we proceed to explain how touse them in fuzzy control, let us first explain whywe need to use them. I. Why intervals? Reasons Ia–c:Intervals naturally appear Ia. Traditional fuzzy control techniques start withthe expert’s degree of belief that are represented bynumbers from the interval [0, 1]. • This use of numbers may be natural when wedescribe physical quantities, for which there ex-ists a true value that can be, in principle, mea-sured with greater and greater accuracy.• However, for degrees of belief, numbers may notbe the most adequate representation. Indeed, how are the existing knowledge elicitationtechniques determine these numbers? • One of the possible techniques is to ask an ex-pert to estimate his or her degree of belief bya number on a scale, say, from 0 to 10. Then,when an expert estimates this degree of beliefby choosing, say, 6, we take 6/10 = 0.6 as thenumerical expression of the expert’s degree ofbelief.At first glance, this may sound like a reason-able assignment, but in reality, the fact that anexpert has chosen 6 does not necessarily meanthat the expert’s degree of belief is exactly equalto 0.6; it rather means that this degree of beliefis closer to 0.6 than to the other values betweenwhich we have asked the expert to choose (i.e., The authors are with the Computer Science Department ofthe University of Texas at El Paso, El Paso, TX 79968, USA,[email protected] and with the Department of MathematicalSciences, New Mexico State University, Las Cruces, NM, 88003,USA, [email protected]. This work was supported by NASAResearch Grant No. 9-757, and, in part, by NSF Grant No. EEC-9322370.to 0, 0.1, . . . , 0.5, 0.7, . . . , 0.9, 1.0). Mathemati-cally, values that are the closest to 0.6 form aninterval [0.55, 0.65]. In other words, the onlything that we can conclude based on this choiceis that the expert’s true degree of belief belongsto the interval [0.55, 0.65].In principle, we could try to get a more precisevalue of the degree of belief by asking an expertfor a value on, say, a scale from 0 to 100, buthardly anyone can distinguish between degreeof belief that correspond to, say, 63 and 64 onthis scale. Thus, the interval [0.55, 0.65] is thebest we can get.• Another way of determining the degree of beliefis to poll experts. If 6 experts out of 10 believethat, say, a given value of x is small, then wetake 6/10 = 0.6 as the degree of belief μsmall(x)that this value x is small.Polls have their own margins of uncertainty.Hence, from a poll, we cannot extract the exactratio of experts who believe that x is small; wecan, at best, find an interval of possible valuesof this ratio.In principle, to get a narrower interval, we canask more and more experts, but in reality, thenumber of experts is often limited, and askingall of them is not practically possible. As aresult, the interval of possible values is the bestwe can get. Similar conclusions can be obtained for all othermethods of eliciting the values. For all these meth-ods, an interval is a much more adequate descriptionof the expert’s degree of belief. Ib. Even if we manage to get narrow enough inter-vals for degrees of belief, so that these original de-grees of belief can be adequately described as num-bers, in the fuzzy control methodology, we need toprocess these numbers. The first processing consistsof applying and and or operations.These operations, in their turn, must also beelicited from an expert so that they would be mostadequate in describing what the experts mean whenthey use the corresponding connectives. We havealready seen that eliciting numbers leads, in reality, to intervals. The resulting uncertainty is evenworse if we try to elicit not a single number, butseveral different numbers that describe the desired functions f&(a, b) and f∨(a, b). As a result, insteadof a single pair of functions, we, most probably, willget an interval of possible functions. If we applythis interval function to numerical input values, weget an interval of possible results.Thus, even if we managed to avoid intervals onthe first stage, they will appear on the second stageof fuzzy control methodology: when we combine theoriginal degrees of belief into degrees of belief of dif-ferent rules. Ic. Even if we fix and and or operations, for thesame query, we can have different representationsin terms of “and”, “or”, and “not”. These differ-ent representations are equivalent in classical logic,but in fuzzy logic, they are not. As a result, de-pending on which representation we use, we may getdifferent numerical answers to the query. Hence, ifwe only know the query itself, and we are not surewhat “translation” into basic logical operations isthe best, it is natural to return not a single numer-ical value, but the entire interval of possible valuesof degrees of belief that correspond to different pos-sible translations.In [61], we show how to compute this interval fordifferent queries.