Fuzzy Rule-Based System
6.6 Fuzzy Inference System
6.6.4 Takagi–Sugeno Fuzzy Method (TS Method)
Output Distribution
Z*
Fig. 6.3.Defuzzification using the center of mass Output
Distribution
Z*
Fig. 6.4.Defuzzification using the mean of maximum
Fuzzy Inputs
In summary, Fig. 6.5 shows a two input Mamdani FIS with two rules. It fuzzi- fies the two inputs by finding the intersection of the crisp input value with the input membership function. It uses the minimum operator to compute the fuzzy AND for combining the two fuzzified inputs to obtain a rule strength.
It clips the output membership function at the rule strength. Finally, it uses the maximum operator to compute the fuzzy OR for combining the outputs of the two rules.
124 6 Fuzzy Rule-Based System
IF StrengthRule THEN
x
x
y
y
X0 y0
z
z
z Input
Distribution
Output Distribution and
and
Fig. 6.5.A two input, two rule Mamdani FIS with a fuzzy input
but it can be any other functions that can appropriately describe the output of the output of the system within the fuzzy region specified by the antecedent of the rule. When f(x, y) is a first-order polynomial, we have thefirst-order Sugeno fuzzy model. Whenf is a constant, we then have thezero-order Sugeno fuzzy model, which can be viewed either as a special case of the Mamdani FIS where each rule’s consequent is specified by a fuzzy singleton, or a special case of Tsukamoto’s fuzzy model where each rule’s consequent is specified by a membership function of a step function centered at the constant. Moreover, a zero-order Sugeno fuzzy model is functionally equivalent to a radial basis function network under certain minor constraints.
The first two parts of the fuzzy inference process, fuzzifying the inputs and applying the fuzzy operator, are exactly the same. The main difference between Mamdani and Sugeno is that the Sugeno output membership func- tions are either linear or constant. A typical rule in a Sugeno fuzzy model has the form
IF Input 1 =xAND Input 2 =y, THEN Output isz=ax+by+c.
For a zero-order Sugeno model, the output levelzis a constant (a=b = 0).The output levelziof each rule is weighted by the firing strengthwi of the rule. For example, for an AND rule with Input 1 = xand Input 2 = y, the firing strength is
wi= AndMethod(F1(x), F2(y)),
where F1,2(·) are the membership functions for Inputs 1 and 2. The final output of the system is the weighted average of all rule outputs, computed as
Final output =
!N i=1wizi
!N
i=1wi
.
Input 1
Input MF
Output MF Rule Weight (firing strength)
Output Level w
Z
z = ax+by+c Input MF
F1(x)
F2(y) Input 2
x
y
AND
Fig. 6.6.Sugeno rule
1. Fuzzify inputs
1. poor
2.
3.
good
excellent
rule 2 has no dependency on input
rancid
2. Apply fuzzy operation (OR - mex)
3. Apply implication method (prod)
tip-cheap
tip-average then
then
then lip = generous z3 = (generous) food is rancid
delicious
food is delicious service is poor
If
service is good
input 2 output
input 2 If
service is excellent If
or
cr
z1(cheap) z1
z2 w1
w2
w3
4. Aggregation
5. Defuzzify (weighted average)
service = 3 food = 8
tip - 16.3%
z3 z2(average)
Fig. 6.7.Fuzzy tipping model
A Sugeno rule operates as shown in Fig. 6.6.
Figure 6.7 shows the fuzzy tipping model developed in previous sections of this manual adapted for use as a Sugeno system. Fortunately, it is frequently the case that singleton output functions are completely sufficient for the needs of a given problem. As an example, the system tippersg.fis is the Sugeno type representation of the now-familiar tipping model (Fig. 6.7).
• a = readfis(‘tippersg’);
gensurf(a)
The above command gives the surface view of fuzzy tipping model as shown in Fig. 6.8. The easiest way to visualize first-order Sugeno systems is to think of each rule as defining the location of a “moving singleton.” That is,
126 6 Fuzzy Rule-Based System
0 0
2 2
4 4
6 6
service food
8 10
8 10 10
fp15
20
Fig. 6.8.Surface view
the singleton output spikes can move around in a linear fashion in the output space, depending on what the input is. This also tends to make the system notation very compact and efficient. Higher-order Sugeno fuzzy models are possible, but they introduce significant complexity with little obvious merit.
Sugeno fuzzy models whose output membership functions are greater than first-order are not supported by the Fuzzy Logic Toolbox.
Because of the linear dependence of each rule on the input variables of a system, the Sugeno method is ideal for acting as an interpolating supervisor of multiple linear controllers that are to be applied, respectively, to differ- ent operating conditions of a dynamic nonlinear system. For example, the performance of an aircraft may change dramatically with altitude and Mach number. Linear controllers, though easy to compute and well suited to any given flight condition, must be updated regularly and smoothly to keep up with the changing state of the flight vehicle. A Sugeno FIS is extremely well suited to the task of smoothly interpolating the linear gains that would be applied across the input space; it is a natural and efficient gain scheduler.
Similarly, a Sugeno system is suited for modeling nonlinear systems by inter- polating between multiple linear models.
Because it is a more compact and computationally efficient representation than a Mamdani system, the Sugeno system lends itself to the use of adaptive techniques for constructing fuzzy models. These adaptive techniques can be used to customize the membership functions so that the fuzzy system best models the data.