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Fuzzy Logic. Priyaranga Koswatta. Mundhenk and Itti, 2007. Advantages of Fuzzy Controllers Minimal mathematical formulation Can easily design with human intuition Smoother controlling Faster response. Agenda. General Definition Applications Formal Definitions Operations Rules
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Priyaranga Koswatta • Mundhenk and Itti, 2007
Advantages of Fuzzy Controllers • Minimal mathematical formulation • Can easily design with human intuition • Smoother controlling • Faster response
Agenda • General Definition • Applications • Formal Definitions • Operations • Rules • Fuzzy Air Conditioner • Controller Structure
General Definition Fuzzy Logic - 1965 Lotfi Zadeh, U.C. Berkeley • superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth • central notion of fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value on the range [0.0, 1.0], with 0.0 representing absolute Falseness and 1.0 representing absolute Truth. • deals with real world vagueness
Applications • ABS Brakes • Expert Systems • Control Units • Bullet train between Tokyo and Osaka • Video Cameras • Automatic Transmissions
Formal Definitions • Definition 1: Let X be some set of objects, with elements noted as x. • X = {x}. • Definition 2: A fuzzy set A in X is characterized by a membership function mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of membership" of x in A increases. • Definition 3: A is EMPTY iff for all x, mA(x) = 0.0. • Definition 4: A = B iff for all x: mA(x) = mB(x) [or, mA = mB]. • Definition 5: mA' = 1 - mA. • Definition 6: A is CONTAINED in B iff mA mB. • Definition 7: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)). • Definition 8: C = A INTERSECTION B where: mC(x) = MIN(mA(x), mB(x)).
http://www.seattlerobotics.org/Encoder/mar98/fuz/flindex.htmlhttp://www.seattlerobotics.org/Encoder/mar98/fuz/flindex.html • http://www.cs.cmu.edu/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html • http://plato.stanford.edu/entries/logic-fuzzy/
Operations A B A B A B A
Mechanical Design of a very inexpensive Line-Following Robot
Very Basic Control Theory Your speed towards a goal or away from an object should be proportional to the distance from it. If you want to get to a goal in an optimal amount of time you want to move quickly. However, you need to decelerate as you grow near the target so you can have more control. Speed ∝ distance-to-target
Very Basic Control Theory In systems with momentum (i.e. the real world) we have to worry about more complex acceleration and deceleration. We can overshoot our target or stop short! You increase your rate of deceleration based on how close you are to a goal or obstacle. You can also integrate over the distance to a goal to create a steady state. This is the basic idea behind a PID controller. Proportional Integral Derivative The physical derivation of PID can be tricky, we will avoid it for now. However this part of an extremely interesting topic!
IDEA! Lets just hack a fuzzy controller together and avoid some math. The gods will curse us…. But if it works, that may be all that matters! Derive rule of thumb ideas for speed and direction If I’m 6 meters from the obstacle, am I far from it?
Try some fuzzy rules… Lets look at adjusting trajectory first then we will look at speed… If an obstacle is near and center, turn sharp right or left. If an obstacle is far and center, turn soft left or right. If an obstacle is near, turn slightly right or left, just in case. Etc…
The robot works in continuous time The fuzzy rules should change slightly at each time step. We don’t want the robot to jerk to a new trajectory too quickly. Smooth movements tend to be better. How often we need to update the controller is dependant on how fast we are moving. For instance: If we update the controller 30 times a second and we are moving < 1 kph then movement will be smooth. Ideally, the commands issued from the fuzzy controller should create an equilibrium with the observations.
Our robot has momentum We have somewhat implicitly integrated the notion of momentum. This is why we would slow down as we get closer to an obstacle What about more refined control of speed and direction? Use the derivative of speed and trajectory to increase or decrease the rate of change. Thus, if it seems like we are not turning fast enough, then turn faster and visa versa.
Controller Structure • Fuzzification • Scales and maps input variables to fuzzy sets • Inference Mechanism • Approximate reasoning • Deduces the control action • Defuzzification • Convert fuzzy output values to control signals
Rule Base • Fan Speed • Set stop {0, 0, 0} • Set slow {50, 30, 10} • Set medium {60, 50, 40} • Set fast {90, 70, 50} • Set blast {, 100, 80} Air Temperature • Set cold {50, 0, 0} • Set cool {65, 55, 45} • Set just right {70, 65, 60} • Set warm {85, 75, 65} • Set hot {, 90, 80}
Membership function is a curve of the degree of truth of a given input value default: The truth of any statement is a matter of degree Rules Air Conditioning Controller Example: • IF Cold then Stop • If Cool then Slow • If OK then Medium • If Warm then Fast • IF Hot then Blast
