Intelligent Systems

General Flashcards on the information required

Definition of Fuzzy Logic

Fuzzy Logic uses words and sentences to describe complex environments. It exploits vagueness and the inherent ill definition of most real world problems to characterise uncertainties, using fuzzy sets to express how certain we are that a property is true, false, or somewhere in between.

Principle of Incompatibility

States that the precision and transparency of a model are incompatible properties. White box models are transparent (interpretable) but less precise, black box models are more precise but less transparent. Fuzzy logic finds a compromise point to both properties for an optimal outcome.

Compositional Rule of Inference

Find the output fuzzy set B that corresponds to the input fuzzy A, taking into account the knowledge (relational matrix) R

B = A ∘ R

𝜇_B(v) = max[ min[ 𝜇_A(u), 𝜇_R(u,v)]]

• Given the knowledge ‘R’ and an input set ‘A’, we are always guaranteed an answer fuzzy set ‘B’

• The quality of the output set ‘B’ will depend on how rich the knowledge represented by ‘R’ is

• This is equivalent to an interpolation problem

Extension Principle

Given a fuzzy set A defined in the Universe of Discourse U relating to the variable u, and another fuzzy set B defined in the Universe of Discourse V for the variable v, mapped via a function g such that: v = g(u)

B = g(A) = { (v,𝜇_B(v)) ; v = g(u) 𝜇_B(v) = 𝜇_A(u) }

Mamdani Type Systems

• Antecedents (inputs) and Consequents (outputs) are represented by fuzzy sets

• Fuzzy rules are fully transparent

• Fuzzy rules can easily be maintained by expert knowledge

• Most commonly used option when fuzzy logic is the logic of choice

Takagi-Sugeno-Kang Type Systems

• Only Antecedents are fuzzy sets, Consequents are represented by deterministic functions

• Fuzzy rules are not fully transparent

• Fuzzy rules are not easy to maintain

• Computationally more efficient

ANFIS (Adaptive Network Fuzzy Inference System) Architecture

A hybrid architecture that uses the interpolating power of ANN (Artificial Neural Networks) to optimise the TSK-based fuzzy rules of a fuzzy system using input and output data

ANFIS Advantages

• Fuzzy rules obtained automatically

• Fuzzy MFs optimised automatically

• Raw input/output data needed (without normalisation)

ANFIS Disadvantages

• Input/output data needs to be provided

• Type and number of MFs for the inputs needs to be defined

• Type of TSK functions needs to be defined

PD Fuzzy Control

Uses Error (Kp) and Error Rate (Kd)

Output is control signal

PI Fuzzy Control

Uses Error (Ki) and Error Rate (Kp)

Output is an incremental control signal, which when passed through an integrator gives a control signal

Input and Output Scaling

Scaling all domains of the inputs and outputs which relate to the definitions of all fuzzy sets in an interval (e.g. [-1 +1]).

Done to avoid the problems with under-exploitation or violation of the domain which the Universe of Discourse exists within

Properties for a fuzzy rule base

1. Completeness

2. Continuity

3. Consistency

Completeness

The rule base should always fire at least one fuzzy rule

Continuity

There is no gap between neighbouring rules (always some overlap)

Consistency

There are no contradictory rules - no different decisions for the same input.

ANFIS Network Diagram

Rule 1: IF x is A₁ AND y is B₁ THEN f₁ = a₁x + a₂y + c₁

Rule 2: IF x is A₂ AND y is B₂ THEN f₂= b₁x + b₂y + c₂

ANFIS Layer 1

Fuzzification Layer:

• Inputs = Inputs to the system

• Outputs = Membership Values

ANFIS Layer 2

Rule Layer:

• Inputs = Membership Values

• Outputs = Weighting

E.g. W = 𝜇_A1(x) . 𝜇_B1(y)

ANFIS Layer 3

Normalisation Layer:

• Inputs = Weights

• Outputs = Normalised Weights

E.g. W̅₁ = W₁/(W₁ + W₂)

ANFIS Layer 4

Defuzzification Layer:

• Inputs = Inputs to the system and Normalised Weights

• Outputs = Weighted deterministic functions

E.g. W̅₁f₁ = W̅₁(a₁x + a₂y + c₁)

ANFIS Layer 5

Summation Layer:

• Inputs = Weighted deterministic functions

• Outputs = Sum of weighted deterministic functions

E.g. f = W̅₁f₁ + W̅₂f₂