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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
Priyaranga Koswatta
  • Mundhenk and Itti, 2007

Advantages of Fuzzy Controllers

  • Minimal mathematical formulation
  • Can easily design with human intuition
  • Smoother controlling
  • Faster response
  • General Definition
  • Applications
  • Formal Definitions
  • Operations
  • Rules
  • Fuzzy Air Conditioner
  • Controller Structure
general definition
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
  • ABS Brakes
  • Expert Systems
  • Control Units
  • Bullet train between Tokyo and Osaka
  • Video Cameras
  • Automatic Transmissions
formal definitions
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.cs.cmu.edu/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html
  • http://plato.stanford.edu/entries/logic-fuzzy/


A  B A  B A

very basic control theory
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


􀂀 Speed ∝ distance-to-target

very basic control theory1
Very Basic Control Theory

􀂀 In systems with momentum (i.e. the real world) we

have to worry about more complex acceleration and


􀁹 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!


􀂀 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


􀂀 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



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
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
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


The truth of any statement is a matter of degree


Air Conditioning Controller Example:

  • IF Cold then Stop
  • If Cool then Slow
  • If OK then Medium
  • If Warm then Fast
  • IF Hot then Blast