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Lab 2: Fuzzy Systems

Fuzzy Systems

Objective

  • to construct a Mamdani fuzzy system using the scikit-fuzzy Python library
  • to evaluate the result of the constructed fuzzy system

Note

Install the scikit-fuzzy Python library in your environment before proceeding with the lab.

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conda install -c conda-forge scikit-fuzzy

Fuzzy control system for a train

  1. Consider a fuzzy control system to control the brake and throttle of a train based on the speed of the train and the distance of the train to the next stop.

  2. Import the skfuzzy, skfuzzy.control, and numpy.

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    import numpy as np
from skfuzzy import control as ctrl
from skfuzzy import membership as mf

Initialise inputs and outputs

  1. Speed and distance are the inputs of the system whereas brake and throttle are the outputs.

  2. The ranges for the variables are:

    Variable Range
    Speed 0 - 85 km/h
    Distance 0 - 3000 m
    Brake 0 - 100%
    Throttle 0 - 100%

  3. As the inputs will be the antecedents of the rules, construct the variables speed and distance as skfuzzy.control.Antecedent objects. 

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    speed = ctrl.Antecedent(np.arange(0, 85, 0.1), 'speed')

  4. The initialisation function for skfuzzy.control.Antecedent object takes 2 arguments, the first is the universe of the variable, i.e. the values the variables can take, the second is the label of the variable. The initialisation function for skfuzzy.control.Consequent is similar.

  5. The label and the range of the variable can be accessed using .label and .universe respectively.

Task: Initialise the variables distance as Antecedent object, and brake and throttle as Consequent objects. (Outputs of the system will be consequents of the rules)

Define membership functions for fuzzy sets of variables

  1. The fit vectors of the fuzzy sets for the linguistic variables are given as follows:

    • speed (0 to 85 km/h)

      Linguistic value Fit vector
      Stopped (1/0, 0/2)
      Very slow (0/1, 1/2.5, 0/4)
      Slow (0/2.5, 1/6.5, 0/10.5)
      Medium fast (0/6.5, 1/26.5, 0/46.5)
      Fast (0/26.5, 1/70, 1/85)

    • distance (0 to 3000 m)

      Linguistic value Fit vector
      At (1/0, 0/2)
      Very near (0/1, 1/3, 0/5)
      Near (0/3, 1/101.5, 0/200)
      Medium far (0/100, 1/1550, 0/3000)
      Far (0/1500, 1/2250, 1/3000)

    • brake (0 to 100%)

      Linguistic value Fit vector
      No (1/0, 0/40)
      Very slight (0/20, 1/50, 0/80)
      Slight (0/70, 1/83.5, 0/97)
      Medium (0/95, 1/97, 0/99)
      Full (0/98, 1/100)

    • throttle (0 to 100%)

      Linguistic value Fit vector
      No (1/0, 0/2)
      Very slight (0/1, 1/3, 0/5)
      Slight (0/3, 1/16.5, 0/30)
      Medium (0/20, 1/50, 0/80)
      Full (0/60, 1/80, 1/100)

  2. The skfuzzy.membership module provides the following membership functions:

    Membership function Description
    skfuzzy.membership.dsigmf(x, b1, c1, b2, c2) Difference of two fuzzy sigmoid membership functions
    skfuzzy.membership.gauss2mf(x, mean1, ...) Gaussian fuzzy membership function of two combined Gaussians
    skfuzzy.membership.gaussmf(x, mean, sigma) Gaussian fuzzy membership function
    skfuzzy.membership.gbellmf(x, a, b, c) Generalized Bell function fuzzy membership generator
    skfuzzy.membership.piecemf(x, abc) Piecewise linear membership function (particularly used in FIRE filters)
    skfuzzy.membership.pimf(x, a, b, c, d) Pi-function fuzzy membership generator
    skfuzzy.membership.psigmf(x, b1, c1, b2, c2) Product of two sigmoid membership functions
    skfuzzy.membership.sigmf(x, b, c) The basic sigmoid membership function generator
    skfuzzy.membership.smf(x, a, b) S-function fuzzy membership generator
    skfuzzy.membership.trapmf(x, abcd) Trapezoidal membership function generator
    skfuzzy.membership.trimf(x, abc) Triangular membership function generator
    skfuzzy.membership.zmf(x, a, b) Z-function fuzzy membership generator

  3. The fit vector of a linguitic value can be assigned to a linguistic variable using

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    speed['stopped'] = mf.trimf(speed.universe, [0, 0, 2])
speed['very slow'] = mf.trimf(speed.universe, [1, 2.5, 4])

    Task: Assign all fuzzy sets to the linguistic variables.

  4. The fuzzy set diagram of a linguistic variable can be viewed using .view()

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    speed.view()

    Task: Check if the fuzzy set diagrams match the fit vectors.

Define rules

  1. The rules for this system are displayed in the following fuzzy association memory (FAM) representaion table.

    Distance
    At Very near Near Medium far Far
    Speed Stopped Full brake
    No throttle    		 Full brake
    Very slight throttle
    Very slow Full brake
    No throttle    		 Medium brake
    Very slight throttle    		 Slight brake
    Very slight throttle
    Slow Full brake
    No throttle    		 Medium brake
    Very slight throttle    		 Very slight brake
    Slight throttle
    Medium fast Very slight brake
    Medium throttle    		 No brake
    Full throttle
    Fast Very slight brake
    Medium throttle    		 No brake
    Full throttle

  2. Rule can be defined using skfuzzy.control.Rule(antecedent, consequent, label). To define the first rule, i.e. if distance is 'at' and speed is 'stopped', then full brake and no throttle, 

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    rule1 = ctrl.Rule(distance['at'] & speed['stopped'], (brake['full'], throttle['no']))

    If the antecedent consists of multiple parts, they can be combined using operators | (OR), & (AND), and ~ (NOT).

    If the consequent consists of multiple parts, they can be combined as a list/tuple.

    Task: Define all the rules. Then combine all the rules in a list, i.e. rules = [rule1, rule2, ...].

Construct the fuzzy control system

  1. The train control system can be constructed with

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    train_ctrl = ctrl.ControlSystem(rules=rules)

  2. A skfuzzy.control.ControlSystemSimulation object is needed to simulate the control system to obtain the outputs given certain inputs.

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    train = ctrl.ControlSystemSimulation(control_system=train_ctrl)

  3. To obtain the values for brake and throttle given that speed is 30 km/h and distance is 6 m,

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    # define the values for the inputs
train.input['speed'] = 30
train.input['distance'] = 2000

# compute the outputs
train.compute()

# print the output values
print(train.output)

# to extract one of the outputs
print(train.output['brake'])

  4. To view the results in the graph,

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    brake.view(sim=train)
throttle.view(sim=train)

View the control/output space

  1. The control/output space allows us to identify if the outputs fit our expectation.

  2. Construct an empty 3D space with 100-by-100 x-y grid.

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    x, y = np.meshgrid(np.linspace(speed.universe.min(), speed.universe.max(), 100),
                   np.linspace(distance.universe.min(), distance.universe.max(), 100))
z_brake = np.zeros_like(x, dtype=float)
z_throttle = np.zeros_like(x, dtype=float)

  3. Loop through every point and identify the value of brake and throttle of each point. As the specified rules are not exhaustive, i.e. some input combinations do not activate any rule, we will set the output of such input combinations to be float('inf'). 

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    for i,r in enumerate(x):
  for j,c in enumerate(r):
    train.input['speed'] = x[i,j]
    train.input['distance'] = y[i,j]
    try:
      train.compute()
    except:
      z_brake[i,j] = float('inf')
      z_throttle[i,j] = float('inf')
    z_brake[i,j] = train.output['brake']
    z_throttle[i,j] = train.output['throttle']

  4. Plot the result in a 3D graph using the matplotlib.pyplot library.

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    import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

def plot3d(x,y,z):
  fig = plt.figure()
  ax = fig.add_subplot(111, projection='3d')

  ax.plot_surface(x, y, z, rstride=1, cstride=1, cmap='viridis', linewidth=0.4, antialiased=True)

  ax.contourf(x, y, z, zdir='z', offset=-2.5, cmap='viridis', alpha=0.5)
  ax.contourf(x, y, z, zdir='x', offset=x.max()*1.5, cmap='viridis', alpha=0.5)
  ax.contourf(x, y, z, zdir='y', offset=y.max()*1.5, cmap='viridis', alpha=0.5)

  ax.view_init(30, 200)

plot3d(x, y, z_brake)
plot3d(x, y, z_throttle)

Fuzzy tipping recommendation system

  1. A fuzzy expert system is designed to identify the percentage of tips a customer will give based on the service and the food the customer received.

  2. The system has service and food as inputs, and tips as output.

  3. The fit vectors of the fuzzy sets for the linguistic variables are given as follows:

    • service (0 to 10)

      Linguistic value Fit vector
      Poor (1/0, 0/5)
      Average (0/0, 1/5, 0/10)
      Good (0/5, 1/10)

    • food (0 to 10)

      Linguistic value Fit vector
      Poor (1/0, 0/5)
      Average (0/0, 1/5, 0/10)
      Good (0/5, 1/10)

    • tips (0 to 30%)

      Linguistic value Fit vector
      Low (1/0, 0/15)
      Medium (0/0, 1/15, 0/30)
      High (0/15, 1/30)

  4. The rules are displayed in the following fuzzy association memory (FAM) representaion table.

    Food
    Poor Average Good
    Service Poor low tips low tips medium tips
    Average low tips medium tips high tips
    Good medium tips high tips high tips

Task: Construct the fuzzy inference system.

Task: Modify the membership functions of the input 'service' to

Linguistic value Fit vector
Poor (1/0, 0/3)
Average (0/2, 1/5, 0/8)
Good (0/6, 1/10)