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plc fuzzy - 26.1 26. FUZZY LOGIC <TODO - Find an implementation platform and add section> Topics: • Fuzzy logic theory; sets, rules and solving • Objectives: • To understand fuzzy logic control. • Be able to implement a fuzzy logic controller . 26.1 INTRODUCTION Fuzzy logic is well suited to implementing control rules that can only be expressed verbally, or systems that cannot be modelled with linear differential equations. Rules and membership sets are used to make a decision. A simple verbal rule set is shown in A Fuzzy Logic Rule Set. These rules concern how fast to fill a bucket, based upon how full it is. 1. If (bucket is full) then (stop filling) 2. If (bucket is half full) then (fill slowly) 3. If (bucket is empty) then (fill quickly) Figure 26.1 A Fuzzy Logic Rule Set The outstanding question is "What does it mean when the bucket is empty, half full, or full?" And, what is meant by filling the bucket slowly or quickly. We can define sets that indicate when something is true (1), false (0), or a bit of both (0-1), as shown in plc fuzzy - 26.2 Fuzzy Sets. Consider the bucket is full set. When the height is 0, the set membership is 0, so nobody would think the bucket is full. As the height increases more people think the bucket is full until they all think it is full. There is no definite line stating that the bucket is full. The other bucket states have similar functions. Notice that the angle function relates the valve angle to the fill rate. The sets are shifted to the right. In reality this would probably mean that the valve would have to be turned a large angle before flow begins, but after that it increases quickly. 1 1 bucket is full stop filling 0 0 1 1 bucket is half full fill slowly 0 0 1 1 bucket is empty fill quickly 0 0 height angle Figure 26.2 Fuzzy Sets Now, if we are given a height we can examine the rules, and find output values, as shown in Fuzzy Rule Solving. This begins be comparing the bucket height to find the membership for bucket is full at 0.75, bucket is half full at 1.0 and bucket is empty at 0. Rule 3 is ignored because the membership was 0. The result for rule 1 is 0.75, so the 0.75 membership value is found on the stop filling and a value of a1 is found for the valve angle. For rule 2 the result was 1.0, so the fill slowly set is examined to find a value. In this case there is a range where fill slowly is 1.0, so the center point is chosen to get angle a2. These two results can then be combined with a weighted average to get 0.75 a1 + 1.0 a2 - angle = --------------------------------------------- 0.75 + 1.0 . plc fuzzy - 26.3 1. If (bucket is full) then (stop filling) 1 1 bucket is full stop filling 0 0 height angle a1 2. If (bucket is half full) then (fill slowly) 1 1 bucket is half full fill slowly 0 0 height a2 angle 3. If (bucket is empty) then (fill quickly) 1 1 bucket is empty fill quickly 0 0 height angle Figure 26.3 Fuzzy Rule Solving An example of a fuzzy logic controller for controlling a servomotor is shown in A Fuzzy Logic Servo Motor Controller [Lee and Lau, 1988]. This controller rules examines the system error, and the rate of error change to select a motor voltage. In this example the set memberships are defined with straight lines, but this will have a minimal effect on the controller performance. plc fuzzy - 26.4 vdesired verror Fuzzy Vmotor Motor vactual Imotor Logic Power Servo + Motor - Controller Amplifier a The rules for the fuzzy logic controller re; 1. If verror is LP and d /dt verror is any then Vmotor is LP. 2. If verror is SP and d/dt verror is SP or ZE then V . motor is SP 3. If verror is ZE and d/dt verror is SP then V motor is ZE. 4. If verror is ZE and d/dt verror is SN then V motor is SN. 5. If verror is SN and d/dt verror is SN then V motor is SN. 6. If verror is LN and d/dt verror is any then V motor is LN. The sets for v , d/dt verror , and Vmotor are; error d verror /dt verror Vmotor 1 1 1 0 0 0 LN rps rps/s V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 1 1 1 SN 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 1 1 1 ZE 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 1 1 1 SP 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 1 1 1 LP 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 Figure 26.4 A Fuzzy Logic Servo Motor Controller Consider the case where verror = 30 rps and d/dt verror = 1 rps/s. Rule 1to 6 are calculated in Rule Calculation. plc fuzzy - 26.5 d 1. If verror is LP and /dt verror is any then V . motor is LP 1 1 1 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 A ANY V LUE 30rps (so ignore) 17V (could also This has about 0.6 (out of 1) membership have chosen some value above 17V) d 2. If v motor is SP error is SP and /dt verror is SP or ZE then V . the AND means take the the OR means take the lowest of the two highest of the two memberships memberships 1 1 1 1 rps rps/s V 0 0 0 rps/s 0 -100 -50 0 50 100 -6 -3 0 3 6 -6 -3 0 3 6 0 6 12 18 24 1rps/s 30rps 1rps/s 14V This has about 0.4 (out of 1) membership d 3. If verror is ZE and /dt verror is SP then Vmotor is ZE. 1 1 1 rps/s 0 rps 0 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 the lowest results in 0 set 30rps 1rps/s membership This has about 0.0 (out of 1) membership plc fuzzy - 26.6 d 4. If verror is ZE and /dt verror is SN then Vmotor is SN. 1 1 1 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 1rps/s the lowest results in 0 set 30rps membership This has about 0.0 (out of 1) membership d 5. If verror is SN and /dt verror is SN then Vmotor is SN. 1 1 1 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 1rps 30rps This has about 0.0 (out of 1) membership d 6. If verror is LN and /dt verror is any then Vmotor is LN. 1 1 1 0 0 0 V rps rps/s -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 ANY VA LUE 30rps This has about 0 (out of 1) membership Figure 26.5 Rule Calculation The results from the individual rules can be combined using the calculation in plc fuzzy - 26.7 Rule Results Calculation. In this case only two of the rules matched, so only two terms are used, to give a final motor control voltage of 15.8V. n V motor membershipi i V motor = i=1 - --------------------------------------------------------------------- n membershipi i=1 V motor = 0.6 17V + 0.4 14V = 15.8V --------------------------------------------------- 0.6 + 0.4 Figure 26.6 Rule Results Calculation 26.2 COMMERCIAL CONTROLLERS At the time of writing Allen Bradley did not offer any Fuzzy Logic systems for their PLCs. But, other vendors such as Omron offer commercial controllers. Their controller has 8 inputs and 2 outputs. It will accept up to 128 rules that operate on sets defined with polygons with up to 7 points. It is also possible to implement a fuzzy logic controller manually, possible in structured text. 26.3 REFERENCES Li, Y.F., and Lau, C.C., “Application of Fuzzy Control for Servo Systems”, IEEE International Conference on Robotics and Automation, Philadelphia, 1988, pp. 1511-1519. plc fuzzy - 26.8 26.4 SUMMARY • Fuzzy rules can be developed verbally to describe a controller. • Fuzzy sets can be developed statistically or by opinion. • Solving fuzzy logic involves finding fuzzy set values and then calculating a value for each rule. These values for each rule are combined with a weighted average. 26.5 PRACTICE PROBLEMS 26.6 PRACTICE PROBLEM SOLUTIONS 26.7 ASSIGNMENT PROBLEMS 1. Find products that include fuzzy logic controllers in their designs. 2. Suggest 5 control problems that might be suitable for fuzzy logic control. 3. Two fuzzy rules, and the sets they use are given below. If verror = 30rps, and d/dtverror = 3rps/s, find Vmotor. plc fuzzy - 26.9 d error is ZE) and ( /dt verror is ZE) then (V 1. If (v motor is ZE). d error is SP) or ( /dt verror is SP) then (V 2. If (v motor is SP). d verror /dt verror Vmotor 1 1 1 SN 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 1 1 1 ZE 0 rps 0 rps/s 0 V -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 1 1 1 SP rps rps/s V 0 0 0 -100 -50 0 50 100 -6 -3 0 3 6 0 6 12 18 24 4. Develop a set of fuzzy control rules adjusting the water temperature in a sink. 5. Develop a fuzzy logic control algorithm and implement it in structured text. The fuzzy rule set below is to be used to control the speed of a motor. When the error (difference between desired and actual speeds) is large the system will respond faster. When the difference is smaller the response will be smaller. Calculate the outputs for the system given errors of 5, 20 and 40. plc fuzzy - 26.10 100% Big Error error 0% 10 30 100% Small Error error 0% 10 30 20 Big Output error 50 Small Output 5 error 50 if (big error) then (big output) if (small error) then (small output)

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