Tutorial 3 - Fuzzy Set PDF

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Fuzzy Set...

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Chair of Automatic Control Engineering Univ. Prof. Dr.-Ing./Univ. Tokio habil. Martin Buss 1. Task: Fuzzy set Let

1. Compute

0.7

 0 A(x) = x + 3  −x − 1   0 x−1 B(x) =  −x + 3

Computational intelligence Exercise 3

if x < −3 or x > −1 if − 3 ≤ x ≤ −2 if − 2 < x ≤ −1 if x < 1 or x > 3 if 1 ≤ x ≤ 2 if 2 < x ≤ 3

A.

2. Using the standard definitions, sketch a picture of B c . 3. Now, let

Compute B ∩ C

 if x < 2 or x > 4 0 C(x) = x−2 if 2 ≤ x ≤ 3  −x + 4 if 3 < x ≤ 4

TUM WS 21/22

2.Task: Fuzzy controller For a fuzzy controller as follow

e✲

u✲

FuzzyController

The controller is specified as follows: Linguistic control rules: R1:

IF

(e = SMALL),

THEN (u = SMALL)

R2:

IF

(e = LARGE),

THEN

(u = LARGE)

The following basic quantities apply to the variables e and u E = { 0, 20, . . . . . . 100 } U = { 0, 20, . . . . . . 100 } Membership functions for e or u in case of SMALL (S) and LARGE (L) {e/µS (e)} = {0/1.0 ; 20/0.8 ; 40/0.6 ; 60/0.4 ; 80/0.2 ; 100/0.0} {e/µL (e)} = {0/0.0 ; 20/0.2 ; 40/0.4 ; 60/0.6 ; 80/0.8 ; 100/1.0} {u/µS (u)} = {0/1.0 ; 20/0.8 ; 40/0.6 ; 60/0.4 ; 80/0.2 ; 100/0.0} {u/µL (u)} = {0/0.0 ; 20/0.2 ; 40/0.4 ; 60/0.6 ; 80/0.8 ; 100/1.0} Answer the following questions: 1. Determine the membership values γi of IF-Part. 2. Determine the local membership functions to the rules R1 and R2 with Correlation min implication, which are µR1 und µR2 u 0

e

0 20 40 60 80 100

20

40

60

80

100

µR1

u 0

20

40

60

80

100

0 20 40 60 80 100

e

µR2

3. Aggregate the local membership functions of the rule base by fuzzy OR operator (MAX-combination: µΣ ) or by the fuzzy arithmetic sum (µΣ ) 1 2 u 0

20

40

u 60

80

0 20 40 e 60 80 100

100

0

µΣ 1

20

40

60

0 20 40 e 60 80 100

a) Center of gravity (COG) (µΣ ) 1 b) Mean of Maxima Method (MeOM) µΣ 1 c) Center of gravity (COG) (µΣ ) 2 where u ∈ R. 0

20

40

60

100

µΣ 2

4. Defuzzification the outputs of the fuzzy controller according to

e

80

80

100

ua ub uc 5. Draw a graph of the outputs of the 3 kinds of defuzzification and make a discussion. 6. Repeat steps 2. to 4.a) and 5. once more for {u/µS (u)} = {0/1.0} and {u/µL (u)} = {100/1.0}

3. Task: Application of fuzzy controller The movement of a point-shaped, controllable vehicle is described by the following kinematic relationship: x˙ = v0 · cos[ϕ(x, y)] y˙ = v0 · sin[ϕ(x, y )] . x and y are in the Cartesian coordinate system, v0 is a constant and the magnitude of the velocity vector. ϕ(x, y) is the angle relative to the x-axis. A driver has done the following parking operation with the above vehicle:

y / mm

1500 1000 500

Parkposition 0 -4000

-3000

-2000

-1000

0 x / mm

Figure 1: Trajectory of the vehicle while parking Use fuzzy techniques to design a control law that imitates the driver. In the fuzzy system, the inputs are x: horizontal distance from the vehicle to the park position and y: vertical distance from the vehicle to the park position. The output of the system is ϕ(x, y). a) Draw the graph of the membership functions for the fuzzification of input signals. For the input space, draw 3 membership functions (Description words: close, far,very far) for x and 2 membership functions (Description words: Small, Large) for y (Use the bell curves as the membership functions). 1500

y / mm

y / mm

1500 1000 500

0 0

0 -4000

1

-3000

-2000

m (y)

-1000

0

x / mm 1

m (x)

0.8 0.6 0.4 0.2 0 -4000

-3000

-2000

-1000

x / mm

0

b) Based on an analysis of the trajectories, draw the graph of 3 membership functions (Description words: Zero, Middle, Large) for the angle ϕ(x, y) as the output (Use the bell curves as the membership functions). µ(Angle) ✻

✲

0

-45

-90

c) Give a control rule set with 4 rules to solve the parking task.

Angel Angle / Degree...