Example 1. Classification by means of Total Fuzzy Similarities

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Example 2. Highway driving fluency

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Lecture IV

Example 3. Determining Athlete’s Aerobic and Anaerobic Thresholds

100 metres sprinter has to run a short distance very fact, therefore, he has to have muchtraining in the anaerobic zone (where his pulse close to maximal value), whilea long distance runner needs endurance, thus, he needs training in the aerobic zone.It is important for an athlete to test his aerobic and anaerobic thresholds regularly,these tests can be done e.g. on a running track ergometer [see Picture 1].

Aerobic and anaerobic thresholds are functions of blood lactate [mmol/l], ventilation;CO2 [l/min] and O2 uptake [%]. They, in turn, are functions of heartbeat [b/min]. A typical test starts with a 3 minutes worm up [pulse around 100 beat/min], then the load is increased every 2 - 3 minutes and blood lactate, ventilation, CO2, O2 uptakeand heartbeat are measured. A test lasts until volitional exhaustion [pulse near 200 b/min], this takes usually 20-30 minutes [A typical test protocol, see Picture 2].

Lactate Ventilation Pulse Order

Aerobic * lowest value ↑ *VE increasing * max - 40 +/- lactate, VE

threshold * ∆=0.2 mmol/l * O2% highest 10 +/- 5 [b/min] pulse

Anaerobic * rapid increasing * VE clearly ↑ * max - 20 +/- lactate, VE

threshold * 3 mmol/l * VCO2 ↑ 5 +/- 3 [b/min] VCO2,

    *O2% decreasing O2%

    * VE/VO2 ↑   VE/VO2

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To mimic a skilled sport medicine spacialist’s action when she/he is determing anaerobic threshold, we need only one rule (namely that one given above!)

Indeed, we count the degree of total fuzzy similarity between each measured valueand that given be the rule.

Clearly, all these consepts are fuzzy and can be expressed by fuzzy intervals. They are context dependent, too. For example ’Maximal pulse – 40’ depends on a respective measurement. We used the following membership function

FuzzyMaxMinus40.mw

Examples: open Matlab7>> cd C:\Matlab7\Mika\hemi>> aekSamples 33, 78

Example 4. Signalized isolated pedestrian crossing (See Picture 3.)

As long as there are no pedestrians, vehicles have green signal. If a pedestrian pushes a button and no cars are approaching the pedestrian will have immediately green signal.

In case there are vehicles approaching and pedestrians waiting, then vehicles’s greendepends on the following factors:* how long time have pedestrians been waiting for [a short/long/too long time]* how many vehicles are approaching [few/some/many]* what is the shortes cap between approaching vehicles [short/large]The situation is updated after every half second.

Experienced traffic engineers described the above fuzzy set as follows (See Picture 2).

There are 18 rules in our fuzzy IF-THEN rule base (green extension is prefered)This corresponds to all possible rules (3x3x2 = 18), therefore, the rule base is complete.

An example of a rule (given by traffic engineers): IF (pedestrians’ waiting time = short) [weight = 1] AND (approaching vehicles = few) [weight = 2] AND (gab between approaching vehicles = short) [weight = 1] THEN (extend vehicles green signal)

The output is always a crisp action (red or green). In a ’50-50 situation’ it is green.

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1

3 5 12 16 24 28

short

2 3 41 5

some

[sec] [] 1 2 3

1

[sec]

short

1

1

3 5 12 16 24 28

long

2 3 41 5some

[sec] [] 1 2 3

1

[sec]

large

1

1

3 5 12 16 24 28

too long

2 3 41 5

many

[sec] [] 1 2 3

1

[sec]

short

1

An example (assuming we would have only three rules)

3/8

Pedestrians’ waiting time = 14 sec Vehicles = 4 Gab between = 2 sec

0

½ ½

3/4½ ½ 9/16

0.1½ ½ 16/40

The actual traffic situation is most similar to the 2. rule, thus...

2

1

2

1

4

1 1201

2

1

2

1

4

3

4

1 121

2

1

2

1

10

1

4

1 121

Example 5. More complicated traffic signal control (See....)DVI Document

Traffic flow on the main street (phase A) is from two to ten times more intensive than traffic flow from the other direction. Normally the phase order is A-B-C-A…, however, if there are very few or no vehicles in thenext phase B or C, then these phases can be skipped. Thus, the order canbe e.g. A-C-A-B-C or A-B-A-B-C. Thus, the first task is to determinethe right phase order. Here the IF-THEN rules have a form

IF V(B) is medium AND V(C) is over saturated THEN phase is C Corresponding to Step 3 of the Algorithm, if the maximal total similarity is not unique, the phase with the longest waiting time will be fired.

The second task is to control A’s green ending, that is, the decision of the moment when the green of the first signal group of phase A can be terminated, so that the first signal group of phase B or C can be started. These rules are of the following form

IF A is a Few AND Queue is Medium THEN Extension is Short

A Similarity Model and a Mamdani style fuzzy model was constructedat H.U.T, Lab of Traffic and Transportation. The results were compared…

These fuzzy models ware tested by a traffic simulator ‘HUTSIM’ and the results were as follows:

veh/hour 200 400 600 800 1000 1200 1400 1600

Veh. Density ratio10/1

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Tot.sim 13.4 1.921 14 2.061 13 1.562 13 1.661 14.4 1.405 14.7 1.653 15.2 1.766 18.3 2.694

Mamdani 12.1 2.406 12.2 1.941 12.5 1.182 13.2 2.197 13.6 1.216 14.9 1.135 16.1 2.051 17.4 2.255

Tot.sim PS 12 2.447 11.8 1.893 12.3 1.291 11.7 2.046 13.9 1.952 13.2 1.825 13.3 2.398 13.8 2.952

Mamdani PS 12 2.447 11.8 1.893 12.3 1.291 12.4 1.937 14.3 1.741 14.8 1.544 15.9 2.536 18.3 3.123

veh/hour 200 400 600 800 1000 1200 1400 1600

Veh. Density ratio10/2

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Tot.sim 12.7 3.351 12.5 1.363 12.3 1.957 13.4 1.651 13.9 1.307 15.3 2.029 16.4 1.299 22.5 4.960

Mamdani 10.3 2.559 13.1 1.893 12.3 1.874 13.8 1.612 13.9 1.364 15.7 1.590 17.3 2.160 20.6 3.296

Tot.sim PS 10.8 2.826 11.4 1.123 12.1 1.579 13.5 2.259 14.4 1.536 14.5 2.137 14.6 1.516 16.1 3.049

Mamdani PS 10.8 2.826 11.4 1.123 12.1 1.579 13.5 2.259 14.4 1.536 15 1.914 17 1.564 19.8 3.124

veh/hour 200 400 600 800 1000 1200 1400 1600

Veh. Density ratio10/5

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Averagedelay

StdevAverage

delayStdev

Tot.sim 12.8 2.239 12.7 1.366 12.5 1.488 13.5 1.328 14.2 1.172 16.2 1.911 16.5 1.552 24.4 6.223

Mamdani 11.6 2.079 12.2 1.473 12.4 1.970 14.2 1.790 15 2.035 17 1.905 16.9 2.015 30.2 10.845

Tot.sim PS 11.9 2.345 12.6 1.390 12.6 1.556 13.9 1.461 14.2 1.364 16.4 1.800 17.4 2.014 22.1 3.567

Mamdani PS 11.9 2.345 12.6 1.390 12.6 1.556 13.9 1.461 14.8 1.483 17.1 1.323 16.8 2.035 23.7 4.024

Tables 3,4 and 5. T-junction; Average vehicles delays and Standard deviations

Example 6. Real-Time Reservoir Operation

* Lake Päijänne is located in the Southern part of Finland. * It’s water runs to the Golf of Finland via River Kymijoki.* Each year Päijänne is frozen at least 5 months and lots of snow is accumulated.* In spring floods caused by melting snow would be typical if Päijänne was not regulated.* The water reference level is a function of date given by law.* Based e.g. on ‘snow water equivalent’, human experts are able to regulate several dams such that water level. can be kept close to the reference level* The task was to create a formal control system to do the same job.

A control system was created at Helsinki University of Technology,Laboratory of Water Resource Management by Tanja Dudrovin.

The model consists of two real-time submodels; the first submodel sets up a reference water level (WREF) for each time step. Given this reference level, the observed water level (W), and the observed inflow (I), the second submodel makes the decision on how much water should be released from the reservoir during the next time step.

For the snowmelt season, WREF value is dependent on the snow water equivalent (SWE) and can be inferred for each time step with the rules:

IF SWE is smaller than average/average/larger than average/much larger than average THEN WREF is high/middle/low/very low.

In the second sub model, the rules have a form

IF W is very low/low/objective/high/very high AND I is very small/small/large/very largeTHEN release is exceptionally small/very small/small/quite small/quite large/large/very large/exceptionally large. [Membership functions see Picture 4].

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To calibrate the corresponding fuzzy set, a data of real control actions collected during 1975-1985 was used.

RESULTS

The model was tested using data from the years 1985-1996. The Sugeno method was chosen for comparison against the Total FuzzySimilarity. With both methods the system was kept the same as much as possible.

To apply the Sugeno method the, defuzzification was performed using a weighted average.

The performances of the two methods were almost indistinguishable. With the Total Fuzzy Similarity the water level targets during the summer were sometimes better fulfilled, but the release tended to fluctuate more, and the limitation on change in release was more relevant.

The model performance was generally good, but the model did not capture expert thinking in the most exceptional circumstances. (- later the modelwas completed by an extra subsystem to do the job.)

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