Fuzzy Sets and Control

1. Fuzzy Sets and Control

2. Fuzzy Logic The definition of Fuzzy logic is a form of multi-valued logic derived from fuzzy set theory to deal with reasoning that is approximate rather than precise. Degree of Truths and probabilities range between 0 and 1. From Wikipedia

3. An example of Fuzzy Reasoning Fuzzy Set Theory defines Fuzzy Operators on Fuzzy Sets. The problem in applying this is that the appropriate Fuzzy Operator may not be known. For this reason, Fuzzy logic usually uses IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices. Rules are usually expressed in the form:IF variable IS property THEN action For example, an extremely simple temperature regulator that uses a fan might look like this:IF temperature IS very cold THEN stop fanIF temperature IS cold THEN turn down fanIF temperature IS normal THEN maintain levelIF temperature IS hot THEN speed up fan Notice there is no "ELSE". All of the rules are evaluated, because the temperature might be "cold" and "normal" at the same time to different degrees. The AND, OR, and NOT operators of boolean logic exist in fuzzy logic, usually defined as the minimum, maximum, and complement; when they are defined this way, they are called the Zadeh operators, because they were first defined as such in Zadeh's original papers. So for the fuzzy variables x and y: NOT x = (1 - truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y)) From Wikipedia

4. Fuzzy Set Definition The definition of a fuzzy set is given by the membership function elements of the universe of discourse U, can belong to the fuzzy set with any value between 0 and 1. The degree of membership of an element u when the universe of discourse U, is discrete and finite, it is given for a fuzzy set A by

5. Fuzzy Set Operations The union of two fuzzy sets is defined by The intersection of two fuzzy sets is defined by is defined by The complement of fuzzy set

6. Properties of Set Operations Most of the properties that hold for classical sets (e.g., commutativity, associativity and idempotence) hold also for fuzzy sets except for following two properties: Law of contradiction the intersection of a fuzzy set and its complement results in a fuzzy set with membership values of up to ½ and thus does not equal the empty set (as in the case of classical sets) Law of excluded middle

7. Intelligent Control • An intelligent control system is one in which a physical system or a mathematical model of it is being controlled by a combination of a knowledge-base, approximate (humanlike) reasoning, and/or a learning process structured in a hierarchical fashion. • Under this simple definition, any control system which involves fuzzy logic, neural networks, expert learning schemes, genetic algorithms, genetic programming or any combination of these would be designated as intelligent control.

8. Fuzzy Control • A fuzzy controller consists of three operations: (1) fuzzification, (2) inference engine, and (3) defuzzification. • A common definition of a fuzzy control system is that it is a system which emulates a human expert. In this situation, the knowledge of the human operator would be put in the form of a set of fuzzy linguistic rules. • The human operator observes quantities by observing the inputs, i.e., reading a meter or measuring a chart, and performs a definite action (e.g., pushes a knob, turns on a switch, closes a gate, or replaces a fuse) thus leading to a crisp action • The human operator can be replaced by a combination of a fuzzy rule-based system (FRBS) and a block called defuzzifier. The input sensory (crisp or numerical) data are fed into FRBS where physical quantities are represented or compressed into linguistic variables with appropriate membership functions. • These linguistic variables are then used in the antecedents (IF-Part) of a set of fuzzy rules within an inference engine to result in a new set of fuzzy linguistic variables or consequent (THEN-Part). Variables are combined and changed to a crisp (numerical) output.

9. Fuzzy Control Architecture System Defuzzifier Fuzzifier Rule Engine real numbers real numbers member ship values fuzzy sets Fuzzy Controller

10. Simple Rules – if antecedent then consequentEx: fuzzyControlForDec consequent- cuts tts fuzzy set and computes its area and moment fuzzifier – accepts real number inpuit and outputs its membership defuzzifier –accumulates the areas and moments and outputs the centroid common domain input, x

11. Simple Rule Controller fuzzyControlForDec provides the negative feedback to stabilize the integrator

12. Composite Rules – if antecedent1 and antecedent2 then consequentExample: fuzzControlFor2To1 AndFn outputs minimum of inputs x common domain input, x v common domain input, v

13. CompositeRule Controller fuzzyControlFor2To1 provides the control to settle the spring at zero

14. Linear Time Invariant Models

15. Inverted Pendulum Fuzzy Control Linearized inverted pendulum on a cart fuzzyControlFor2To1 provides the control to keep the stick stable

16. Modeling the Inverted Pendulum Source: http://www.engin.umich.edu/group/ctm/examples/pend/invpen.html Moment of Inertia: http://hyperphysics.phy-astr.gsu.edu/hbase/mi2.html#rlin

17. Inverted Pendulum – Swing Up Non-linear Model source: http://www.control.lth.se/publications/fulldocs/ast_fur96.pdf detect when angle reaches turn-off level detect when pendulum stops rising detect when pendulum stops rising reset theta to 0 when reach 2*pi starting from hanging configuration, rod can be made to reach inverted configuration with sufficient force acting until horizontal line is reached