Scaling of fuzzy controllers using the cross-correlation - IEEE Xplore

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Correspondence Scaling of Fuzzy Controllers Using the Cross-Correlation Rainer Palm U

Fig. 1. FC structure. Abstract-The paper deals with the optimal adjustment of input scaling factors for Fuzzy Controllers. The method bases on the assumption that in the stationary case an optimally adjusted input scaling factor meets a specific statistical input output dependence. A measure for the strength of statistical dependence is the correlation function and the correlation coefficient, respectively. Without loss of generality, the adjustment of input scaling factors using correlation functions is pointed out by means of a single input-single output (SIS0)-system. In the first section, the paper deals with the so-called equivalent gain which is closely connected to the cross-correlation of the controller input and the defuzzified controller output. The next section considers the computation of correlation functions and their representation inside the FC. The paper concludes with an example of a system of fuzzy rules controlling a redundant robot manipulator.

I. INTRODUCTION In most technical applications fuzzy controllers (FC) work in such a way that, on the basis of crisp desired inputs and crisp actual outputs, the system (plant) is controlled also by crisp manipulated variables. In this wide-spread case fuzziness is restricted only to the controller which is said to be more robust than conventional controllers [6], [3], [9]. Fig. 1 shows the general FC-structure. The operation of a FC of this type requires fuzzification of the inputs (e.g., error and change of error), each crisp input is attached to a subset of grades of membership depending on the a priori chosen subset of membership functions. The design of a FC requires information about the system to be controlled such as operating areas of the FC inputs and the manipulated variable of the FC output. For simplicity, in most cases the membership functions of the input and output variables are defined within normalized intervals (universes of discourse). In the case of normalized universes of discourse an appropriate choice of specific operating areas requires respective scaling or denormalization factors. An input scaling factor transformes a physical signal into the normalized universe of discourse of the controller input whereas the output denormalization factor provides a transformation of the defuzzified output signal from the normalized universe of discourse of the controller output into a physical manipulated variable. The importance of an optimal choice of input scaling factors is evidentely shown by the fact that ill scaling results in either shifting the operating area to the boundaries or utilizing only a small area of the normalized universe of discourse. On the other hand, the adjustment of the output denormalization factor affects the closed-loop gain which has direct influence on system stability. The behavior of the system controlled finally depends on the choice of the normalized transfer characteristics (control surface) of the controller. In the case of a predefined set of rules the control surface is mainly determined by shape and location of the input and output membership functions. Taking these influences into account for controller design one should pay attention to the following priority list: Manuscript received May 13, 1993; revised May 26, 1994. The author is with Siemens AG Corporate Research and Development, Dept. ZFE ST SN 4,Otto-Hahn-Ring 6, D-81730 Munich, Germany. IEEE Log Number 9406656.

L Fig. 2. Closed-loop system with a nonlinear FC.

The output denormalization factor has the most influence on stability and osziflation tendency. Because of its strongest impact on stability this factor is assigned to the first priority in the design process. Input scaling factors have the most influence on basic sensitivity of the controller with respect to optimal choice of the operating areas of the input signals. Therefore, input scaling factors are assigned to the second priority. Shape and location of input and output membership functions and, with this, the transfer function of the normalized controller influence positively the behavior of the system controlled in different areas of the state space provided that the operating areas of the signals are optimally chosen through a well adjusted input scaling factor. Therefore, this aspect is getting the third priority. Once, by means of some system analysis, the scaling factors and the parameters of the membership functions have been chosen the next task is to tune them to improve the systems’s behavior according to some optimization criteria. In this context tuning should consider the same priority list as for the design process. One, therefore, obtains the following tuning hierarchy: I) Tune the denormalization factor 2) Tune the input scaling factors 3) Tune the membership functions Most of the reports on tuning refer to membership functions to change the transfer characteristics of the controller [17], [20], 111. Examples for gain tuning can be found at [5], [7], [18], [2], 1191. Tuning of rules has been considered by [15], [14]. Many reports deal with integral criteria with respect to particular test signals such as step, pulse, and random functions, respectively [17], [20]. This paper deals with the second level of tuning hierarchy, namely with appropriate scaling of controller inputs which has the most influence on the sensitivity of the controller. It is assumed that both the rule set and the membership functions are predefined and kept constant during the tuning process. The input data are assumed to be statistically distributed satisfying a Gaussian distribution whose parameters are assumed to be unknown. A poor knowledge about the distribution parameters can be explained by slow varying plant parameters or drift in the sensory used for state observation. Here, we distinguish three relevant cases:

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1) Known mean (e.g., mean = 0) and unknown deviation. 2) Unknown mean and known deviation. 3) Mean and deviation are unknown. Considering a controller with multi input and single output cases 1-3 can be processed on-line measuring the linear dependence between each input and output signal of the controller. A measure for linear input-output dependence of a transfer element is the crosscorrelation function and the cross-correlation coefficient, respectively. First, the shift of the signal’s mean value along the universe of discourse ensures the signal to meet the relevant operating area of the control surface. Second, once the relevant region is reached the tuning procedure keeps on changing the particular input scaling factor until the goal, the cross-correlation coefficient to be a certain value near its maximum, is reached. It is shown that for Gaussian input signals a given FC can be imaginarily replaced by an equivalent gain which strongly depends on the nonlinear transfer characteristic of the FC [4]. This method allows the utilization of linear system theory even in the case of nonlinear elements within the control loop. Therefore, an appropriate choice of the equivalent gain has a great influence on the behavior of the closed-loop system. The equivalent gain can be expressed in terms of the standard deviation of the input and the input-output cross-correlation function. The claim is that for a stationary input a certain amount of signal amplitudes around the operating point of the FC should be linearily transmitted by the FC. As already mentioned, a measure for linear input-output dependence of a transfer element is the cross-correlation function. Hence, if a specific linearity between input and output is required one has to adjust the standard deviation in such a way that a corresponding cross-correlation coefficient is met. For a given SISO FC structure the only parameter to influence the equivalent gain is the scaling factor for the input signal. This result can easily be extended to the multi inpudmulti output case (MIMO) if the individual states of the plant to be controlled are noncorrelated with each other. The method presented deals with the optimal adjustment of scaling factors for FCs with the help of the input-output cross-correlation [lo]. If the distribution of the signal is a priori known the method can be characterized as a design approach only by consideration of the nonlinear FC without closing the control loop. A subsequent section deals with the representation of correlation functions inside the FC. The last section gives an example of how to tune scaling factors in the process of kinematical control of a redundant robot manipulator.

11. INPUT OUTPUT CORRELATION FOR A FC

+A

./

05

2

Fig. 3. Symmetrical nonlinear transfer characteristic of a FC with limits.

more, let the desired value w include Gaussian noise. We then obtain non-Gaussian noise at the output of the FC because of its nonlinear transfer characteristic. On the other hand, we suppose the system to filter out all frequencies causing a non-Gaussian distribution. In this way, we expect to have Gaussian noise at the output of the system and, with this, at the adder where the desired value w is compared with the actual value 2. With these assumptions the scaled signal e, = (w - 2) . sc is also of Gaussian type. From nonlinear system theory we know the describing function for sinusoidal signals and the equivalent gain for signals with noise [16]. The basic idea of this method is to substitute imaginarily the nonlinear element in a closedloop system by a linear one whose gain depends on the amplitude eo (for sinusoidal inputs) or variance a: (for noise) of the controller input. The main purpose of this approach was the stability test of the nonlinear system to be controlled with the means of linear control theory. This method is here adopted for the adjustment of scalings of a given FC. Let e be a stationary and ergodic Gaussian process. Furthermore, let

where li (oz)-equivalent gain corresponding to a specific FC transfer characteristic v-noise which is noncorrelated with e. Let, finally

A . Equivalent GairtSISO Case

Fuzzy controllers with crisp inputs and outputs can be considered as multidimensional nonlinear transfer elements with upper and lower limits. Let, furthermore, the transfer characteristics at the origin be described by

be the linear cross-correlation function for

T

= 0, and

the expected value. For simplicity the mean values of e, E, and v are equal to zero This can be justified by the following reasons: 1) Most important applications deal with the error e or error vector e = (e, 6 , . . .)T at the controller’s input so that the respective input signal is centralized. 2) A stationary signal can, with the knowledge of its mean value, always be centered so that a centralized signal at the controller input can be assumed. Let the system to be controlled be linear or, within the operating area, linearizable with lowpass characteristic (see Fig. 2). Further-

E[e]= 0;

E[F]= 0;

E [ v ] = 0.

Multiplying both sides of (2) with e and computing the expected values one obtains

with

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E[e . ~]--cross-correlation Re, mE[e2]-variance U : . Because of E[e . U] = 0 one obtains for the equivalent gain

values leads, with conditions [8], [9], for the ith row to

With regard to scaled Gaussian distributed input signals e, we then obtain the equivalent gain

Kt(&1))

= Re(n-l),Aue(n-l))

(1 1)

U : ( - - 1)

From (11) we obtain the advantageous result that the individual equivalent gains can be derived independently from each other. Therefore, the next steps regarding input scaling can be done by considering only the SISO case without loss of generality.

B. Equivalent G a i 4 l M O Case For the MIMO case let

C. Input Scaling The input e is scaled by means of

with

where sc-scaling factor, e, -scaled input.

and Let the scalar output signal E be computed by the center of gravity

where Moreover, we assume zero mean for all elements of e, ii, and v pu E ( 0 , l ) d e g r e e of membership,

V i ,k

E[e(”)I = 0;

E[nz]= 0;

E[v,] = 0.

(8)

Finally, we assume the following cross correlations to be zero

vi # L vi = k

E[e(*). e(k)]= 0; E [ e ( i ). vk] = o

~ [ v . ,v k ] = o

(9)

U

E (A, B)-universe

of discourse.

(15)

For simplicity we assume the denormalization factor of E to be one. Furthermore, let e ( t ) and E ( t ) be stationary and ergodic processes. The standard deviations of the scaled signal e,(t) and nonscaled signal e ( t ) are connected in the same way as the signals e, and e are

with From (6) and (16) we get

Both gain K and cross-correlation R,,s reach their maximum values in the case of maximum linear input-output connection. The normalized correlation coefficient

reaches its maximum at

= 1. R e , , reaches its minimum at

Re,,lmin = Multiplication of (10) e, 2 , .

..,

and computing the expected

/E

if the transfer characteristic is symmetrical with respect to the mean

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distribution

-

.TI

optimum

KI\/

Fig. 4. Linear transfer characteristic with limits. value Es = E[e,(t)](see Fig. 3). This is related to a relay transfer characteristic because for a very large standard deviation compared with the width 2 . A of the interval considered every symmetrical transfer characteristic acts as a relay function. Moreover, it is evident that a shift of the mean E, of the distribution to the limits of the transfer characteristic of the FC leads to decreasing input-output correlation of the controller. The extreme point is reached when the largest part of the distribution is covered by one of the branches at which the control output is always a constant value. For this case we obtain Re,c 0. Optimal input scaling actually means searching for an optimal ues with respect to the interval [-A, +A] of the FC between its limits. We assume the optimal scaling for the following case. We start the searching procedure with a large sc (which means a large g e e = ueslmax). Then, keeping u e sconstant, we change Es stepwise by adding AF, to Fs which corresponds to a shifting of the distribution of the input signal along the V,-axis. The result is a curve B(Es,u e s )with a single maximum R(&,ar,)Jmaxwith @E8 + A E S , a e s # ) R(Es,ae,)

Vk(Es,aea) (19)

and u e s= const.

After that we decrease a e s by Aue3 and change the result R(Vs,ues ; Aues) in the same way. Because of the monotonic function R ( E , , ~ e , ) l ~ , = c o n s t we obtain a higher maximum R(~,,u,~) aslbefore ~~~ R ( E s 3 uecl- A u t . ) m a x

> &(Ea> ae,)max.

(20)

We stop the searching procedure at R(E,,aes)2 1 - ff

with condition (19) where a E (0, I) is a free parameter. If condition (19) is not fulfilled we obtain a plateau. In this case the domain of the FC is assumed to be insufficient with respect to the given standard deviation a e s .Hence, one has to increase the scaling factor sc until (19) is met. Fig. 5 shows a typical R@,, a,,)-plot and Fig. 6 shows the corresponding block scheme. We choose the free parameter cy such that for a linear FC characteristic between the upper and lower limit (see Fig. 4) the standard deviation ae, of the scaled signal e, is set equal to the interval A of the controller ff

= 1 - fi(Fs,ue,)a,,=A.

(21)

This means, we have an input signal probability P = 0.68 for the linear region of the FC. If the characteristic of the controller between its limits is nonlinear, however, (see Fig. 3) one obtains automatically

L___1

I

Fig. 6. Block scheme for tuning of scaling factors.

an sc such that uea< A. Namely, it is clear that the cross-correlation

R as a linear operation meets its maximum value when, for a given standard deviation u e s the , function between the limits of the transfer characteristic of the FC is linear. If the function between the limits is nonlinear one obtains for the same u e sa lower value for R. This, however, corresponds to a smaller standard deviation me, and, with this, a smaller scaling factor sc if a linear function between the limits is assumed.

D.Numerical Example Suppose a standard deviation a, = 1 of the input signal e ( t ) . For a FC with a line? characteristic between the limits as shown in Fig. 7(a) we obtain R(E,,a,,),,s=lo= 0.95 and cy = 0.05. This corresponds to a scaling factor s, = 10 and the scaled standard deviation ae, = 10. For a FC with a sinusoidal characteristic as shown in Fig. 7(b) we obtain for the same a = 0.05 a scaling factor s, = 7 and the scaled standard deviation aes= 7. 111. INTERNAL REPRESENTATION OF CORRELATION FUNCTIONS WITHIN AN FC An interesting aspect is the way of representing the crosscorrelation between input x ( t ) and output E ( t ) of a FC in terms of membership functions. This is motivated by the fact that insight into correlation mechanisms within a fuzzy controller can be helpful for: 1) Initial choice of membership functions. 2) Tuning of shape and location of membership functions. 3) Tuning of the rules on a numerical level. In this section the internal representation of expected values and correlation functions is shown for the continuous and the discrete case.

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B . The Discrete. Case Let S be a relation between the membership functions pz and pu

which has been obtained from the following set of rules IF X = X 1 THEN U = U1 IF X = X N THEN U = UN.

In the next step the signal x s is classified into one of the m intervals for which holds

A . The Continuous Case

The calculation of the correlation function R[x,%] first of all requires the comprehension of the expected value E[E] that can be formulated in the following way

E [ z ]= $@I = lim

l, 1 I+’ [IB 1

+T

Edt

-T

A

(22)

zsE I, with

I, = [ x , , x 2 + 1 ) and i E (1,m).

After classifying the scaled value zs we obtain the corresponding membership vector pz,EI,

= (p.1,p.z,...,pL.1,...,pzm) = ( c J ~ c J ~ . . . > ~ ~ . . . ~ o ) .

With the input vector p z , c ~ ,and the relational matrix S we obtain by means of the o-inference (e.g., max min inference)

Jfpu(t)du

Due to the fact that only pu depends on t the integrals can be exchanged

( p u l i p u ~ ,’.. > p u n ) = s 1 1

s12

5’21

SZZ

... . . ’ Szn

Sm1 Sm2 = (Szl, Szz,. . ‘ 7

... Smn Stn).

We define pUN(u,t ) =

PU(t)

J,”

Pu ( t ) d u

(23)

where p , is ~ the normalized grade of membership at time t so that we finallv obtain

(27)

The scalar output for x s E I , yields

A

s,,, U , ’

rB

For a scaled stochastic input process x s ( t ) the index i changes

Then, the discrete analogy of (24) is In consideration of (23) we obtain and of (26) and consequently

E [ z .z]=

kB

U .

E [ X ( t ). f L N ( U , t ) ] d U .

(26)

The correlation between a crisp input x ( t ) and the corresponding defuzzified output E ( t ) is equal to the weighted mean value of the correlation functions between the input x ( t ) and the normalized grades of membership p ~ ( ~ , tThis ) . result can be extended to the MIMO-case, easily.

This result can also be extended to the MIMO case. It shows the relationship between the input signal x ( t ) and the “firing” of the intemal elements S,, of the relational matrix S . This can, though not being the topic of this paper, be utilized for tuning the elements S,, to obtain a desired correlation level between x ( t ) and particular elements of S . This, finally, means changing the rules of a FC on a numerical level in a different way than described by [15].

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IV. APPLICATION TO A REDUNDANT MANIPULATOR ARM The application sample of this section deals with the control of the kinematic of a redundant robot arm (the inverse task) whose static transfer characteristic is highly nonlinear but, in contrast to this, whose intemal dynamics consists of a simple dead time element or a delay. Contents of this section is, however, not to describe the whole problem of solving the inverse task in the case of kinematical redundancy. This has already been discussed at [8] in detail. In the following, only the aspect of appropriate tuning of the input scaling factors required for kinematical control of the robot arm is considered. In Section I1 it has been shown that a shift of the mean value of an input signal leads to a reduction of the correlation coefficent. The following example deals with input signals whose signs do not change during the control process. Moreover, the input scaling factor does not only affect the standard deviation but the mean value, too. The task is the optimal choice of scaling factors so that the distribution of the corresponding input signal is located within the corresponding operating area of the FC. The basic assumption is that an optimal scaling factor is obtained when the statistical input-output dependence meets its maximum

Fig. 8. Motion of a redundant robot arm.

For the “Scaled output” z , = z according to the ith link yields

z z z i

lic’,=ol+

m a .

z

To be independent of any change of sign in the control loop the absolute value of R has been chosen. The problem of kinematical control of a redundant manipulator arm can be simply described as follows: The effector (gripper, tool) of the planar robot arm is supposed to follow a predefined path (see Fig. 8). The robot kinematic is constructed in such a way that the individual links of the manipulator are able to avoid both extemal obstacles and intemal restrictions (e.g., boundaries of the iinks). In the special case the motion of each link is, in addition to the given effector task, determined by 1) distance h between link and middle position, 2) distance s between link and wall, 3) distance d between link and obstacle. The distances h, s, and d are evaluated by fuzzy attributes (e.g., s = Positive Small) and their membership functions. For each link a fuzzy rule base provides the corresponding correction of the joint angle. The actions z (angle corrections) of each link are evaluated also by fuzzy attributes (e.g., z = Positive Big angle correction). Although the process to be controlled is highly nonlinear in the large it can be considered as linear for small angle corrections. The distances h, s and d are scaled so that they fit the predefined normalized universes of discourse. For the intemal restriction “Scaled distance between the ith link and its middle position” h , . ~= h yields h negative big: HNB = ( p ~ ~ ~ ( h ) / h ) , h negative small: HNS = ( p ~ ~ ~ ( h ) / h ) , h positive big: HPB = ( p ~ p ~ ( h ) / h ) , h positive small: HPS = ( p ~ ~ ~ ( h ) / h ) , V h E H. For the extemal restriction “Scaled distance between the ith link and some wall” S ~ N= s yields s s

big: SIB = ( ~ s I B ( s ) / s ) . small: SIS = ( ~ s I s ( s ) / s ) , vs E

s.

For the extemal restriction “Scaled distance between the ith link and some obstacle” d,N = d yields d d

big: DIB = ( ~ D I B ( s ) / s ) , small: DIS = ( ~ D I s ( s ) / s ) , Vd E D.

z

negative big: ZNB = ( p z ~ ~ ( h ) / h ) , negative small: ZNS = ( p z ~ ~ ( h ) / h ) , negative zero: ZNZ = ( p z ~ ~ ( h ) / h ) , positive big: ZPB = ( p z ~ ~ ( h ) / h ) , positive small: ZPS = (pz~s(h)/h), positive zero: ZPZ = (pzpz(h)/h), v z E 2.

All membership functions p vary only within a predefined standard interval h, s, d, z E [MAX, MINI. Outside this interval the value of p is either 0 or 1. Furthermore, all fuzzy sets are normal, i.e. there is an h, s, d or z with a corresponding p = 1. To obtain an appropriate motion for each link the following set of rules has been applied IF OR

(SIS AND DIS AND (HNS OR HPS)) (SIS AND HNB AND DIB) THEN ZNZ

IF OR

(SIS AND H P B AND DIS) (SIB AND HPS AND DIB) THEN ZNS

IF OR

(SIS AND DIB AND (HNS OR HPS)) ((SIS OR SIB) AND H P B AND DIB)

IF

(SIS AND HNB AND DIS) THEN ZPZ

IF OR

(SIB AND DIS AND ( H P S O R H P B ) ) (SIB AND HNS AND DIB) THEN ZPS

IF OR

(SIB AND DIS AND (HNSORHNB)) (SIB AND HNB AND DIB) THEN ZPB

THEN ZNB

The scalar output value has been computed by the center of gravity

Within the rules for the operations AND and OR the MAX and MIN operator, respectively, have b_een chosen. The correlation coefficient R for discrete points of time t3 has been applied conceming the distance s between each link and the wall

fils, 21 -

c;=lsJiJ-~(c;==,

%E;==, 23)

Jc;==l- :(c;=l . Jc;=, SJ

z32

-

t (e;=, zJ)2 (31)

Figs. 9 and 10 show the change of the correlation coefficient R where ss is the scaling factor for distance s.

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4.5

-1.0

1

Fig. 9. Simulation results for ss with

Sh

= 20 and Sd = 120.

_j_

SS 4.5

The basis of the method is a well-known approach of the nonlinear control theory where under certain conditions for Gauss-distributed input signals a nonlinear control element can be imaginarily replaced by the so-called equivalent gain which is closely related to the inputoutput cross-correlation coefficient of the controller. The claim of the method presented is that for a stationary input signal a certain amount of its amplitudes should be linearily transmitted through the FC. Under the condition that the system to be controlled is approximately a lowpass filter one is able to adjust the input scaling factors at runtime measuring the cross-correlation coefficient. It has been pointed out that the method, widely explained with the example of a SISO system, can easily be extended for MIMO systems if the states to be controlled are noncorrelated with each other. Another case comes up if the distribution of the regarding input signal is known. In this case a tuning procedure becomes pointless and one is therefore able to design the input scaling factors from the beginning so that the controller meets its input sensitivity required. Once, however, either mean or deviation of the input signal is unknown tuning with respect to the procedure described is justified. In an example it is shown that the kinematical configuration of a redundant robot arm is controlled by means of a set of fuzzy rules. This example shows the applicability of the correlation method for the adjustment of input scaling factors in the MISO-case even if the system to be controlled is highly nonlinear. Further research with respect to tuning of scaling factors is needed with respect to the appearance of non-Gaussian signals. This could be done by transforming a non-Gaussian distributed signal into a series of Gaussian distributed signals (Gram-Charlier series) and further processing according to the approach mentioned above. This, however, tends to be too expensive as to the amount of computing time. Therefore, it seems to be more efficient to apply distribution free methods being suboptimal but more pragmatical. Another topic is the treatment of inputs that are not crisp but fuzzy sets [13], [12]. Just as in the case of crisp inputs even for fuzzy inputs one can adopt the method of equivalent gain for optimal utilization of input universes of discourse of a given FC [ 1I].

REFERENCES

-1.0

Fig. 10. Simulation results for ss with

Sh

= 100 and s d = 60.

The other scaling factors are S h = 20 and S d = 120 (see Fig. 9). The peak of R is lying at about s, = 80. Fig. IO-shows a similar situation for S h = 100 and Sd = 60. The peak of R lies at ss = 80 again. The result is that although there is a certain change in the curvature of R(s,), depending on a different choice of the other ssaling factors sd and S h , the abscissa of the maximum value of R does not change. This finally illustrates the independence of the location of the maxima of different correlation coefficient curves. V. CONCLUSION

Many control applications show that most fuzzy controllers are designed in such a way that the universes of discourse conceming the membership functions used are normalized according to a standard interval. This leads to the task of an appropriate choice of scaling factors for inputs and outputs. Together with an appropriate adjustment of membership functions Input/output scaling forms a tuning hierarchy in which input scaling gets the second priority after tuning the output scalings and before tuning the membership functions. Optimal adjustment of input scaling factors serves as a mean to influence the basic sensitivity of the controller and is the basis for the relevant utilization of the operating areas of the input signals.

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[12] R. Palm and D. Driankov, “Fuzzy inputs,” FUZZ IEEE ’94, Orlando, June 2629, 1994. [13] D. Driankov, R. Palm, and H. Hellendoom, “Fuzzy control with fuzzy inputs: The need for new rule semantics,” FUZZ IEEE ’94, Orlando, June 26-29, 1994. [ 141 P. Xian-Tu, “Generating rules for fuzzy logic controllersby functions,” Fuzzy Sets and Syst., vol. 36, pp. 83-89, 1990. [15] T. J. Procyk and E. H. Mamdani, “A linguistic self-organizing process controller,’’ Automatics vol. 15. New York: Pergamon, 1979, pp. 15-30. [ 161 H. Schlitt, “Stochastische Vorgaenge in linearen und nichtlinearen Regelkreisen.” Braunschweig: Vieweg, 1968.

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