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APSC 150 Engineering Case Studies Case Study 3

APSC 150 Engineering Case Studies Case Study 3. Part 3: Automation Lecture 3.8 . Fuzzy Control John A. Meech Professor and Director of CERM3 Centre for Environmental Research in Minerals, Metals, and Materials. http://www.mining.ubc.ca/faculty/meech/apsc150.htm.

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APSC 150 Engineering Case Studies Case Study 3

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  1. APSC 150Engineering Case StudiesCase Study 3 Part 3: Automation Lecture 3.8. Fuzzy Control John A. Meech Professor and Director of CERM3 Centre for Environmental Research in Minerals, Metals, and Materials http://www.mining.ubc.ca/faculty/meech/apsc150.htm

  2. What is Fuzzy Logic? When the only tool you have is a hammer, all your problems look like a nail. - Lotfi Zadeh, University of California, Berkeley The Father of Fuzzy Logic • A method to develop approximate solutions that tolerate imprecision • Conventional mathematical models often demand a degree of precision that is difficult to achieve (adaptation may also be a problem) • Models may only work over a small region in time or space, particularly non-linear ones

  3. The Whole World is Fuzzy! • We all use FL everyday in a natural way without even realizing it • FL is a method that “computes with words” rather than with numbers • FL deals with how we think about control rather than modeling the process itself • A FL system is how we verbalize our understanding of the process

  4. 100 0 Degree of Belief (DoB) Fuzzy Set Terminology

  5. Fuzzy Day to Night

  6. Fuzzy Set Examples • An automobile changing lanes while passing • The position of the shoreline during tidal inflow or outflow • A door being closed or opened (it's ajar!) • A water valve being opened or closed • A glass of water (Is it half-full or half empty?) • The mixing together of two primary colours • The age of a young customer in a bar (is ID required?) • The time it takes to drive from home to work • The waiting time in a queue

  7. Fuzzy Logic versus Binary Logic • Binary Logic - things are either True or False • While that may be ultimately the case, as we grapple with trying to predict this state of affairs, the degree to which we believe something is true or false can change on a scale from 0 to 100 • Binary Logic deals with the set {0,100} • Fuzzy Logic deals with the set (0,100)

  8. Fuzzy Statements • I am 90% sure about this. • It is warm today.(same meaning in Yellowknife as in Miami?) • It may rain today. (where, when, how intense, for how long?) • A “recession” is a “decline in GDP” over 2 consecutive quarters. • A “depression” is a severe (10% GDP drop) or prolonged (3-4 year) recession. • “Read my lips: no new taxes” – G.H.W. Bush, 1988 • "It depends on what the meaning of the word 'is' is." – W.J. Clinton, 1998

  9. Paradoxes • A man says: Don't Trust Me. Should you trust him? If you do, then you don't! • A politician says: All politicians are liars. Is this true? If so, then he is not a liar. • A card states on one side: The sentence on the other side is false... • On the other side appears: The sentence on the other side is true... How do you interpret this card?

  10. Paradoxes Bertrand Russell's Famous Paradox: “All rules have exceptions.” Is this a rule? If so, then what is its exception? • The Liar's Paradox represented by "This sentence is false." can only be understood as a half truth. It can never be a true statement and it never can be a false statement.

  11. Paradoxes • Paradoxes all have the same form: “A statement S and its negation not-S both of which have the same truth-value t(S)” t(S) = t(not-S) • The two statements are both TRUE (1) and FALSE (0) at the same time. But bivalent logic states that negation produces the reverse truth value. Hence: t(not-S) = 1 - t(S)

  12. Paradoxes • Combining these two expressions, we get: t(S) = 1 - t(S) • This is contradictory since if S is true, then 1 = 0 and if S is false, then 0 = 1. But a fuzzy interpretation of truth values can solve for t(S) allowing it to assume a value other than the set {0, 1}. So: t(S) = 0.5 • With fuzzy logic, a Paradox reduces to a literal half-truth which can be considered the uncertainty inherent in every empirical statement and many mathematical expressions.

  13. Paradoxes Returning to the Liar’s Paradox, it must have a value of 0.5 on a truth scale from 0 to 1. But it is possible to generate similar sentences that can take on a value anywhere along the full range of the truth scale. For example: DoB (%) "This sentence is sometimes false." 50–100 "This sentence is rarely false." 0–50 Other sentences that use qualifiers, hedges, or modifiers such as "could be" or "might be“ provide a way to make all paradoxical sentences truly fuzzy. (and, perhaps, sensible!)

  14. Fuzzy Linguistic Hedges • Linguistic expressions "flavour" our certainty in a concept or fact • A hedge is a qualifier used to avoid total commitment or to make a statement more vague or more definite • The Random House Word Menu by Stephen Glazier lists 5 categories of qualifiers that include: - Limitations and Conditions 325 entries - Approximations and Generalizations 150 entries - Emphasizers 85 entries - Maximizers and Superlatives 105 entries - Absolutes and Guarantees 185 entries

  15. Fuzzy Linguistic Hedges • English is full of rich linguistic terms to provide "shades of grey" to a concept. Consider the following set of words: beautiful, pretty, gorgeous, voluptuous, sexy, attractive, stunning, handsome, fabulous, marvellous, outstanding, cute, remarkable, lovely, magnificent, extraordinary, etc. • Each may describe a person’s appearance, but the meaning is different depending on the word or context in which it is used. • Notice how your mind instantly switches context as you move from one word to another. • The term "handsome", for example, is often reserved for males while "pretty" and "gorgeous" typically describe a female.

  16. Degree of Fuzziness

  17. FL is Part of AI • AI = Artificial Intelligence • Components of AI • Expert Systems (rule base structure) • Fuzzy Logic (uncertainty management) • Artificial Neural Networks (learning method) • Genetic Algorithms (optimization search) • SWARM Intelligence (agent-based approach)

  18. How a Fuzzy Controller Works • Adaptation and Extension are done with ease in one or more of four ways: • Redefining fuzzy sets dynamically • Redefining the rule base dynamically • Changing the Inferencing method on the fly • Changing the Defuzzification method on the fly

  19. Example Rule • Rules are constructed as spoken by an experienced operator (an Expert): If CURRENT DRAW is LOW Then INCREASE FEEDRATE A LOT Provided SCREEN BIN LEVEL is not TOO-HIGH • A set of rules like this provides a way to implement control rapidly and effectively

  20. Rule Operations } • ANDing takes the Minimum DoB • ORing takes the Maximum DoB • The Net Degree of Truth of the premise is assigned to the conclusion statement using the following equation: DoB(conc) = NdT * CF / 100 • where CF = Certainty Factor of the rule conclusion statement (If no value is given, it is taken as 100%) NdT

  21. Example } } IF “A” DoB(A) = 95 AND “B” DoB(B) = 85 OR “C” DoB(C) = 88 AND “D” DoB(D) = 75 THEN “F” CF = 90 85 85 } 75 So the value of the NdT = 85 And “F” is assigned a DoB = 85*90/100 = 76.5%

  22. Example (inclusive OR) } IF “A” DoB(A) = 95 AND (“B” DoB(B) = 85 OR “C”) DoB(C) = 88 AND “D” DoB(D) = 75 THEN “F” CF = 90 } 75 88 So the value of the NdT = 75 And “F” is assigned a DoB = 75*90/100 = 67.5%

  23. Secondary Crushing Plant

  24. YES YES YES NO NO NO Feed Rate Fuzzy Control Rule-Set Current Draw HIGH Current Draw MEDIUM-HIGH Current Draw OK Current Draw MEDIUM-LOW Current Draw LOW OR AND AND AND Screen Bin Level HIGH Screen Bin Level OK Screen Bin Level OK Screen Bin Level OK OR Chamber Level HIGH Feed Rate Change NEGATIVE-BIG Feed Rate Change NEGATIVE-SMALL Feed Rate Change NO-CHANGE Feed Rate Change POSITIVE-SMALL Feed Rate Change POSITIVE-BIG Feed Rate Change (discrete) = Weighted Average of Fuzzy Set Supremums based on Respective Degrees of Belief

  25. Rule-Base Matrix NB = Negative-Big PB = Positive-Big NS = Negative-Small PS = Positive Small NC = No-Change

  26. Fuzzy Sets (membership functions) 100 0 Low Medium Low Medium Medium High High Degree of Belief 40 42 44 46 48 50 Current Draw (amps) 100 0 Not-OK Low OK High Degree of Belief 0 20 40 60 80 100 Bin Level (%)

  27. Output Fuzzy Singletons Negative Big Negative Small No Change Positive Small Positive Big 100 0 Degree of Belief -50 -10 0 +10 +50 Feed Rate Change (tph)

  28. Controller Performance – crisp sets PB PS NC NS NB SCREEN BIN LEVEL = LOW CRISP SETS Current Draw (amps) Feed Rate Change (tph) LOW MEDIUM MEDIUM MEDIUM HIGH LOW HIGH

  29. Controller Performance – fuzzy sets PB PS NC NS NB SCREEN BIN LEVEL = LOW FUZZY SETS Current Draw (amps) Feed Rate Change (tph) LOW MEDIUM MEDIUM MEDIUM HIGH LOW HIGH

  30. Using a Fuzzy Confidence Level Normally C = 0, but it may be reasonable for C to be chosen to exclude low belief information

  31. Results – Best Production

  32. Results - Stability

  33. How to Build a Fuzzy Controller • Begin by selecting a specific output • Fan Speed (for example) • Select at least three terms to characterize the output variable (five is better) • Off • Low • Medium • High • Very High • Determine the minimum and maximum level of the output variable (discrete value) Min = 0 Max = 100

  34. How to Build a Fuzzy Controller • Select two variables that would affect the level of the output variable • Room Temperature • Relative Humidity • Select at least three terms to characterize these two input variables (here we have 5) • Temperature • Cold • Cool • OK • Hot • Very Hot Min = 0 Max = 30 Min = 0 Max = 100 • Humidity • Low • Medium • Medium-High • High • Very High

  35. How to Build a Fuzzy Controller • Formulate a set of rules to link each input state combination to an output state Off Off Low Medium High Off Off Low Medium High Off Low High High Medium Off Low Medium Very-High High Low High Very-High Very-High Medium

  36. How to Build a Fuzzy Controller • Create Fuzzy Sets for Input Variables 1 & 2 100 0 Very Hot Cold Cool OK Hot Degree of Belief 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Temperature °C 100 0 Very High Degree of Belief Low Medium Medium High High 0 10 20 30 40 50 60 70 80 90 100 Relative Humidity (%)

  37. How to Build a Fuzzy Controller • Create Fuzzy Singletons for Output Variable Very High Low Medium High Off 100 0 Degree of Belief 0 10 20 30 40 50 60 70 80 90 100 Fan Speed (%)

  38. 100 0 DoB Ai Bi Ci Di Universe of Discourse Temperature Sub-sets • Consider each fuzzy set as a trapezoid Fuzzy Subset i Sub-set A B C D Cold 0 0 0 14 Cool 0 14 14 20 OK 14 20 20 25 Hot 20 26 26 30 Very Hot 25 30 30 30

  39. 100 0 DoB Aj Bj Cj Dj Universe of Discourse Humidity Sub-sets • Consider each fuzzy set as a trapezoid Fuzzy Subset j Sub-set A B C D Low 0 0 0 50 Medium 0 45 50 65 Medium-High 50 65 70 80 High 70 82 82 95 Very-High 80 95 100 100

  40. Membership (DoB) Equations • Generic equation for each set is as follows: • Let “x” = measured temperature (°C) • DoB(Xi)left = Min(100, Max(0,100*(x - Ai)/(Bi - Ai))) • DoB(Xi)right = Min(100, Max(0,100*(Di - x)/(Di - Ci))) • DoB(Xi)final = Min(DoBleft, DoBright) • Let “y” = measured humidity (%) • DoB(Yj)left = Min(100, Max(0,100*(y - Aj)/(Bj - Aj))) • DoB(Yj)right = Min(100, Max(0,100*(Dj - y)/(Dj - Cj))) • DoB(Yj)final = Min(DoBleft, DoBright)

  41. Example x = 22 and y = 81 Temperature DoB(Cold) = 0 DoB(Cool) = 0 DoB(OK) = Min(Min(100, Max(0,100*(22 - 14)/(20- 14))), (Min(100, Max(0,100*(25 - 22)/(25 - 20)))) = Min(Min(100, 133), Min(100, 60)) = 60 DoB(Hot) = Min(Min(100, Max(0,100*(22 - 20)/(26- 20))), (Min(100, Max(0,100*(26 - 22)/(26 - 20)))) = Min(Min(100, 33), Min(100, 66)) = 33 DoB(Very-Hot) = 0

  42. Example x = 22 and y = 81 Humidity DoB(Low) = 0 DoB(Medium) = 0 DoB(Medium-High) = 0 DoB(High) = Min(Min(100, Max(0,100*(81 - 70)/(82- 70))), (Min(100, Max(0,100*(95 - 81)/(95 - 82)))) = Min(Min(100, 92), Min(100, 108)) = 92 DoB(Very-High) = Min(Min(100, Max(0,100*(81 - 80)/(95- 80))), (Min(100, Max(0,100*(100 - 81)/(100 - 100)))) = Min(Min(100, 7). Min(100,  )) = 7

  43. Pass the Input DoBs through the Rules • Take the Minimum DoB and apply it to the respective Output subset 60 33 7 7

  44. Defuzzification Supremum DoB(medium) = 60 40 DoB(high) = 7 or 33 70 DoB(very-high) = 7 100 Accumulation Method Fan Speed = (60*40 + 7*70 + 33*70 + 7*100)/(60+7+33+7) = (2400 + 490 + 2310 + 700)/107 = 5200/107 = 48.6 Maximum Method Fan Speed = (60*40 + 33*70 + 7*100)/(60+33+7) = (2400 + 2310 + 700)/100 = 5410/100 = 54.1

  45. Adaptation • The system can be easily adapted to another variable by one or more of the following: • Allow fuzzy sets to move along the Universe of Discourse • Change rule base • Change method of inferencing • Change method of defuzzification • In the previous example, if we feel cautious then the accumulation method might be preferred whereas, if we are prepared to take risks then the maximum method could be used.

  46. Questions ?

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