Homework 8 Min Max “Temperature is low” AND “Temperature is middle” Fuzzy Logic Membership Function Homework 8 Min “Temperature is low” AND “Temperature is middle” Max “Temperature is low” OR “Temperature is middle”

Homework 8 Algebraic product Algebraic sum Fuzzy Logic Membership Function Homework 8 Algebraic product “Temperature is low” AND “Temperature is middle” Algebraic sum “Temperature is low” OR “Temperature is middle”

Homework 8 Bounded product Bounded sum Fuzzy Logic Membership Function Homework 8 Bounded product “Temperature is low” AND “Temperature is middle” Bounded sum “Temperature is low” OR “Temperature is middle”

Further Fuzzy Set Operations Fuzzy Logic Fuzzy Control Further Fuzzy Set Operations Dilation Concentration

Fuzzy Logic Fuzzy Control Fuzzy Control Loop

Fuzzy Logic Fuzzy Control Fuzzy Inference Prior to fuzzy control, the followings must be defined: Fuzzy membership functions Fuzzy logic operators Fuzzy rules, including fuzzy linguistic value and linguistic variable The processing steps in a fuzzy control include: Fuzzification Implication / Inference Core Accumulation Defuzzification

Fuzzy Rules Example of a fuzzy rule while “Driving a Car”: Fuzzy Logic Fuzzy Control Fuzzy Rules Example of a fuzzy rule while “Driving a Car”: “IF the distance to the car in front is small, AND the distance is decreasing slowly, THEN decelerate quite big” The question that arises: Given a certain distance and a certain change of distance, what (crisp) value of acceleration should we select?

Definition of Fuzzy Membership Functions Fuzzy Logic Fuzzy Control Definition of Fuzzy Membership Functions v. small Distance small perfect big v. big moderate Distance decrease slow fast very fast v. slow Acceleration –small zero +small +big –big

Observation/ measurement Fuzzy Logic Fuzzy Control Fuzzification Observation/ measurement Observation/ measurement v. small Distance small perfect big v. big moderate Distance decrease slow fast very fast v. slow Acceleration –small zero +small +big –big Distance between small and perfect Distance decrease can be moderate or fast What acceleration should be applied?

Observation/ measurement Fuzzy Logic Fuzzy Control Implication of Rules Observation/ measurement v. small Distance small perfect big v. big Acceleration –small zero +small +big –big 0.55 RULE 1: IF distance is small THEN decelerate small Inference core: Clipping Clip the fuzzy membership function of “–small” at the height given by the premises (0.55). Later, the clipped area will be considered in the final decision

Observation/ measurement Fuzzy Logic Fuzzy Control Implication of Rules Observation/ measurement moderate Distance decrease slow fast very fast v. slow Acceleration –small zero +small +big –big 0.7 RULE 2: IF distance decrease is moderate THEN keep the speed Inference core: Clipping Clip the fuzzy membership function of “zero” at the height given by the premises (0.7). Later, the clipped area will be considered in the final decision

Fuzzy Logic Fuzzy Control Accumulation From each rule, a clipped area is obtained. But, in the end only one single output is wanted. How do we make a final decision? –small zero +small +big –big Acceleration Rule 1 Rule 2 In the accumulation (aggregation) step, all clipped areas are merged into one merged area (taking the union). Rules with high premises will contribute large clipped area to the merged area. These rules will “pull” that merged area towards their own central value.

Fuzzy Logic Fuzzy Control Defuzzification –small zero +small +big –big Acceleration Center of gravity Crisp value In this last step, the returned value is the wanted acceleration. Out of many possible ways, the center of gravity is the commonly used method in defuzzification.

Fuzzy Logic Fuzzy Control Inference Core There are two approaches that can be used for inference core: acceleration 1. Clipping approach: 0.55 Min-Operator Fuzzification value Membership function acceleration 2. Scaling approach: 0.55 Algebraic Product

Review on Center of Gravity Fuzzy Logic Fuzzy Control Review on Center of Gravity Rectangle Triangle

Review on Center of Gravity Fuzzy Logic Fuzzy Control Review on Center of Gravity Isosceles Trapezoid Trapezoid

Summary of Fuzzy Control Fuzzy Logic Fuzzy Control Summary of Fuzzy Control Fuzzify inputs, determine the degree of membership for all terms in the premise. Apply fuzzy logic operators, if there are multiple terms in the premise (min-max, algebraic, bounded). Apply inference core (clipping, scaling, etc.) Accumulate all outputs (union operation i.e. max, sum, etc.) Defuzzify (center of gravity of the merged outputs, max-method, modified center of gravity, height method, etc)

Limitations of Fuzzy Control Fuzzy Logic Fuzzy Control Limitations of Fuzzy Control Definition and fine-tuning of membership functions need experience (covered range, number of MFs, shape). Defuzzification may produce undesired results (needs redefinition of membership functions).

Acceleration adj. [m/s2] Fuzzy Logic Fuzzy Control Homework 9 A fuzzy controller is to be used in driving a car. The fuzzy membership functions for the two inputs and one output are defined as below. v. small Distance to next car [m] small perfect big v. big 0 5 10 15 20 25 1 Speed change [m/s2] constant growing declining –small zero +small +big –big Acceleration adj. [m/s2] –2 –1 0 1 2 –10 –5 0 5 10

Fuzzy Logic Fuzzy Control Homework 9 (Cont.) A fuzzy controller is to be used in driving a car. The fuzzy rules are given as follows. Rule 1: IF distance is small AND speed is declining, THEN maintain acceleration. Rule 2: IF distance is small AND speed is constant, THEN acceleration adjustment negative small. Rule 3: IF distance is perfect AND speed is declining, THEN acceleration adjustment positive small. Rule 4: IF distance is perfect AND speed is constant, THEN maintain acceleration.

Fuzzy Logic Fuzzy Control Homework 9 (Cont.) Using Min-Max as fuzzy operators, clipping as inference core, union operator as accumulator, and center of gravity method as defuzzifier, find the output of the controller if the measurements confirms that distance to next car is 13 m and the speed is increasing by 2.5 m/s2.

Fuzzy Logic Fuzzy Control Homework 9A A driver of an open-air car determine how fast he drives based on the air temperature and the sky conditions. The corresponding fuzzy membership functions can be seen here.

Fuzzy Logic Fuzzy Control Homework 9A (Cont.) After years of experience, he summarizes his personal driving rules as follows: Rule 1: IF it is sunny AND warm, THEN drive fast. Rule 2: IF it is partly cloudy AND hot, THEN drive slow. Rule 3: IF it is partly cloudy, THEN drive fast. You are now assigned to design a fuzzy control with the following requirements: Fuzzy logic operators: algebraic sum / product Inference core: scaling Accumulator: union operator Defuzzification: center of gravity method The speed limit is 120 km/h. How fast will the driver go if in one day the temperature is 65 °F and the cloud cover is 25 %?