Discussion on multidimensional fuzzy control

Zeev Zalevsky, David Mendlovic, and Eran Gur

Fuzzy-logic inference engines are in use in various disciplines such as control systems, medicine, and thelike. The use of optical tools to implement such engines may improve the performance and the flexibilityof inference procedures. The optical processor works in a two-dimensional environment, whereas theinference engine might have to handle more than two independent input channels. Here several ap-proaches to generating the first, to our knowledge, N-dimensional optical fuzzy processor are addressed.The first approach uses space multiplexing, the second approach uses polarization multiplexing, and thethird approach uses wavelength multiplexing to increase the dimension of the processor. © 2000 OpticalSociety of America

OCIS codes: 050.1970, 070.4560, 200.4660.

1. Two-Dimensional Fuzzy-Logic Inference Engine:Review

Fuzzy logic suggests an alternative for Boolean logic,especially in systems in which data are not full or areinaccurate. Fuzzy logic is also preferred to Booleanlogic in those cases in which the complexity of infer-encing is high. In fuzzy logic the inference engine isbased on rules rather than on mathematical calcula-tions, and small alterations in the model are not crit-ical ~whereas, in a standard Boolean engine, eachdeviation from the optimal model can result in a dev-astating outcome!. Moreover, fuzzy logic providesan analog-fashion control, and thus it yields smoothresults. The literature reveals many fields in whichthese advantages of fuzzy logic were put to use.1–3

A basic fuzzy-logic inference engine consists of fourstages4,5: data fuzzification, inferencing, defuzzifica-tion, and setting of a combination rule. A sketch ofthe connection between these stages is drawn in Fig. 1.A two-dimensional ~2-D! optical implementation of adual-input single-output processor is given in Fig. 2.6In the optical fuzzification stage the input data areconverted to a membership function ~MF! generated bya laser source and a beam shaper. An acousto-opticdeflector ~AOD! is responsible for shifting the laserbeam according to the input data. The inferencingstage requires classifying the input MF to a discrete

The authors are with the Faculty of Engineering, PhysicalElectronics, Tel Aviv University, 69978, Tel Aviv, Israel. D.Mendlovic’s e-mail address is mend@eng.tau.ac.il.

Received 7 June 1999; revised manuscript received 13 September1999.

0003-6935y00y020333-04$15.00y0© 2000 Optical Society of America

number of criteria. The match between a given cri-terion and a shifted MF is known as the membershipgrade ~MG! of the MF regarding the specific criterion.A set of rules is forced on the inputs. Each rule hasseveral premises and one or more conclusions. Theinput data are compared with the premises, and theoutput of each rule is the conclusion, accompanied by aMG formed from the MG’s that match its premises.In the 2-D optical setup given in Fig. 2 the rules areobtained with a 2-D microprism matrix. The pre-mises of each rule are proportional to the spatial loca-tion of its prism. The conclusion of the rule isobtained by the phase shift that the prism generates.The MG connected to this conclusion is proportional tothe amount of energy that passes through the prism.The third stage, called defuzzification, reassigns theoutput MG’s to the proper criteria. Therefore eachcriterion is attached to a single MG. In the opticalimplementation, a Fourier transform of the micro-prisms plane generates a set of peaks. Each peak isplaced in a location relative to the phase shift of itsgenerating rule prism, and the intensity of the peak isproportional to the MG of that rule’s conclusion. Inthe final stage a single value is obtained from combi-nation of the set of criteria and MG’s. The opticalequivalent of this combination rule is obtained by aposition sense detector ~PSD!. The output current ofthe PSD is proportional to the center of gravity of theincident beams.

As can be seen in Fig. 2, one input datum leads toshifts in the x axis, whereas the second datum leads toshifts in the y axis. Since the setup has only twoorthogonal coordinates, only a dual-input engine maybe implemented with the suggested setup. In Section2 we discuss alternative ways for obtaining an opto-electronic N-dimensional inference engine.

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2. N-Dimensional Optical Fuzzy Inference Engines

A. Space Multiplexing

The concept of space multiplexing is illustrated in Fig.3. On the right-hand side of Fig. 3 an example of a5 3 5, 2-D rule matrix is given for a specific case ofontrolling a pendulum on a cart.7 The x and the y

axes indicate the discrete criteria that also serve aspremises of rules. The five criteria in the case of aninverted pendulum on a cart, corresponding to twoinputs, are positive or negative large, positive or neg-ative small, and zero. The conclusion of each rule isgiven inside the box whose coordinates match the pre-mises of that rule. The larger 5 3 5 matrix containssubmatrices of the previous kind. Such an opticallyimplemented matrix is realized by a microprism array,similar to the one given in Fig. 4. Naturally, each ofthe submatrices is different, because it corresponds toa different set of inputs I1, I2. In this setup the MFwill experience a large displacement in x and y direc-tions, owing to the I1, I2 input data ~voltage inputs ofAOD’s, in the optical setup!. It will experience asmaller displacement in x and y directions, owing tothe I3, I4 input data ~voltage inputs of other AOD’s, inhe optical setup, leading to a total of four AOD’s in aour-input engine!. Such a setup can handle as many

as four inputs simultaneously. Clearly, if the large5 3 5 ~actually 25 3 25! matrix is used as a submatrixf a larger 5 3 5 matrix, we can increase the dimension

of the controller more and more.This setup has two main drawbacks. First, more

input data means larger optical rule plates. An extrainput enlarges the required spatial environment by a

Fig. 2. 2-D optical fuzzy inference engine. I1, I2, inputs; O1, out-ut.

Fig. 1. Fuzzy inference engine. The inputs pass through fourtages ~left to right: fuzzification, inference engine, defuzzification,

and application of a combination rule! before output is established.

34 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000

factor equal to the number of criteria. Second, severeerrors in the output may occur, owing to small errors inthe input. Assume that two 5 3 5 rule plates, similarto the one given in Fig. 3, are used for consecutivevalues of I1 ~say, zero and positive small!. Then, forexample, if the input fits a positive large value of I3, forthe smaller value of I1, even a slight increase in theinput might shift us to the larger I1 with I3 equal to thenegative large value. Therefore the conclusion will bethe opposite of what the rules should have produced.Such an error is critical regarding the quality of theinference procedure.

B. Wavelength Multiplexing

In light of the problems associated with the space-multiplexing setup, an alternative approach is nowpresented. In this approach the light source is poly-chromatic, and a rotating grating determines whichwavelength will propagate along the processors opticalaxis, since the diffraction angle changes with the wave-length8 ~Fig. 5!. The grating is controlled by addi-tional data input. If all lenses are refractive, then allwavelengths require the same length of a 4-f setup.The equation describing the transfer function of thespherical lens is given by

T(x, y) 5 expF 2 ik2f

(x2 1 y2)Gexp~iknD0!, (1)

where k is the wave number, n the index of refractionnside the lens, D0 the width of the lens, and f the focal

length of the lens.The Fresnel approximation for the Kirchhoff–

Fig. 3. 5 3 5 criteria rule matrix for a four-input inference engine.he 2-D matrix on the left-hand side contains 2-D 5 3 5 subma-

trices. Each block indicated by R ~rule table! contains a 5 3 5 ruletable as given on the right-hand side. NL, negative large; NS,negative small; Z, zero; PS, positive small; PL, positive large.

Fig. 4. 2-D optical fuzzy inference engine for four input channels.I1, I2, I3, I4, inputs; O1 output. Note that the first two AOD’s areimaged onto the plane of the final two.

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Helmholz integral, describing free-space propagation,is given by

U(x0, y0) 5exp~ikz!

ilz *̀`

*̀`

exp{ik2z@~x 2 x0)

2

1 ~y 2 y0)2#}U(x, y)dxd, (2)

where l is the wavelength, z the distance along the zaxis, ~x, y! the input coordinates, and ~x0, y0! the out-put coordinates.

It is a known fact that propagating a distance f,assing through a lens with a focal length f, and then

again propagating a distance f yields a scaled Fouriertransform of the input field distribution @this can beseen when we combine Eqs. ~1! and ~2!#. The scalingdepends on the wavelength, but the location of trans-formation does not. Thus the 4-f setup ~containingtwo sequential Fourier transforms! is identical for allwavelengths.

In this way, regardless of the wavelength, the ruleplate should be placed at one precalculated plane. Ifall lenses are refractive, all the information is obtainedin the PSD plane, and one can use a color plate toweight the different wavelengths ~e.g., the plate willblock red but transfer blue with only a small intensitydecrease!. We can also use the responsivity of thePSD as given by Fig. 6 ~the detector reacts differentlyto different wavelengths, and usually there is a wave-length that yields a maximum response, at least local-ly!. If we choose wavelengths beneath the one withmaximum responsivity, the PSD output will increaseas a function of the wavelength. If, however, wechoose wavelengths above the peak, the PSD outputwill decrease as a function of the wavelength. Now,assume the Fourier lens, before the PSD, is diffractive~i.e., a Fresnel zone plate!.

The principle behind a diffractive lens is given now.

Fig. 5. Rotating plate with grating for choosing active wavelengthfrom a polychromatic light source. The entire setup is given in ~a!and the wavelength discriminator in ~b!.

We write Eq. ~1! and take a different lens width D1 5D0 1 d. If d satisfies the following connection

knd 5 2pm, (3)

here m is an integer, then for the given wave num-er k ~or wavelength l! the addition ~or subtraction!

in width does not affect the transfer function of thelens. Hence the lens may be cut down in size toseveral wavelengths. The condition of Eq. ~3! maybe expressed in terms of wavelength

d 5 ml, (4)

hus suggesting that the lens should be cut by annteger number of wavelengths. Thus for differentavelengths a different portion of the lens is to be

ut. A polychromatic beam passing through a dif-ractive lens designed for a given wavelength willield a Fourier transform of the input in a distance for that wavelength only. Other wavelengths willenerate the Fourier image in a different locationlong the z axis, representing the focal planes forhose wavelengths.

Equation ~5! describes a one-dimensional object u~x!multiplied by a linear phase

up~x! 5 u~x!exp~iax!. (5)

Performing a Fourier transform on Eq. ~5! will yieldthe following result:

Up~h! 5 *̀`

up~x!exp~ 2 i2pxv!dx 5 USv 2a

2pD, (6)

hich indicates that the new Fourier transform is ahifted version of the original one. In diffraction op-ics the Fourier transform depends on the wave-

Fig. 6. Spectral response of a typical PSD. If the controller useswavelengths beneath the peak value ~900 nm!, then increasing thewavelength will increase the response, whereas use of the higherwavelengths results in the opposite characteristics.

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length, and, since the microprisms create a linearphase change, for each wavelength the informationwill experience a different shift in the Fourier plane.If, however, a set of detectors is placed in differentlongitudinal distances from the Fourier lens ~distanc-es corresponding to different focal lengths!, the shiftswill be the same, and we will need to calculate thecenter of gravity of all the detectors.

C. Polarization Multiplexing

Now we propose to use the polarization degree of free-dom for increasing the amount of controlled data.Here we use a polarized beam splitter ~PBS! or a Wol-aston prism. The PBS will create two separate con-rol channels, each of them similar to the controlhannel given in Fig. 2. The new setup is given inig. 7. Both channels can work simultaneously, eachandling two inputs. Since the channels have no mu-ual impact, it is important to divide the four inputsisely between the two channels. At the far end of

he setup the two channels are united, by use of an-ther PBS. The final PBS may be replaced with aolarizer. Rotating this polarizer will change theeights between TE and TM polarizations according toq. ~7!, and, if the rotation is controlled by yet another

input data, the dimension of the setup increases:

PTE 5 Ptotal cox~u!, PTM 5 Ptotal sin~u!,

P 5 power, u 5 rotation angle. (7)

olarization multiplexing does not allow for actual rulemplementation, since the channels are nondependent.owever, we may increase the number of inputs to

our by enlarging the spatial environment by the num-er of criteria only. As we recall, in the space-ultiplexing setup, an increase from two to four inputs

equired enlarging the spatial environment by the cri-eria number in each direction, leading to a total en-argement of the square of the number of criteria.dding extra beam splitters and generating a larger

Fig. 7. Four-input optical inference engine, with two 2-D chan-nels. Each channel propagates with a different polarization, andthus interaction between the channels is prevented.

36 APPLIED OPTICS y Vol. 39, No. 2 y 10 January 2000

umber of control channels will enable handling ofore input data. However, if the original rule com-

lexity is high, the implementation will draw furtherrom the truth, and control quality will decrease.

3. Discussion and Conclusions

Three optical implementations for N-dimensional con-trol schemes were given in Sections 1 and 2. Theoptions are space consuming ~space multiplexing!, lim-ited to one extra dimension ~wavelength multiplexing!or not a full true implementation of the rule plates~polarization multiplexing!. Yet each of these meth-ods has some advantages, mainly since these are theonly approaches for implementing an N-input infer-ence engine by use of optics. Comparing the threeimplementations shows that the space-multiplexingapproach requires the largest setup, spacewise, but isthe only setup that fully implements four-dimensionalcontrol. The probability for error is rather large inthe space-multiplexing scheme, and the other twomethods appear to lack this flaw. However, one can-not overlook the fact that the wavelength multiplexingand the polarization multiplexing lack the flexibility ofthe space multiplexing and thus do not fully imple-ment the fuzzy inference engine. As far as complexityis concerned, the wavelength-multiplexing setup re-quires the lowest number of components and thereforeis the easiest to construct. However, the wavelength-multiplexing setup requires a rotating plate, a me-chanical apparatus that restricts the control speed.

4. Summary

In this paper we have demonstrated a 2-D optical fuzzyinference engine. The basic dual-input engine ~2-Dontrol scheme! was shown and its limitations stated.hree optical implementations for N-dimensional con-rol schemes were illustrated ~space multiplexing,avelength multiplexing, polarization multiplexing!,nd a discussion regarding their pros and cons wasiven.

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