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The Equivalence between fuzzy logic
controllers and PD controllers for single
input systems
Professor: Chi-Jo Wang
Student: Nguyen Thi Hoai Nam
Student ID: MA02B202
OUTLINE
I.
Fuzzy Logic Overview
II. Constructing a Fuzzy Logic Controller
III. Practical example on Matlab/simulink environment
IV. Conclusions
V. Reference
I. Fuzzy Logic Overview
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
Fuzzy logic is a logic that describes fuzziness. As fuzzy logic
attempts to model humans’ sense of words, decision making and
common sense, it is leading to more human intelligent machines.
Fuzzy logic was introduced by Jan Lukasiewicz in the 1920s,
scrutinised by Max Black in the 1930s, and rediscovered,
extended into a formal system of mathematical logic and
promoted by Lotfi Zadeh in the 1960s
Fuzzy logic is a set of mathematical principles for knowledge
representation based on degrees of membership rather than on
the crisp membership of classical binary logic. Unlike two-valued
Boolean logic, fuzzy logic is multivalued
I. Fuzzy Logic Overview
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A fuzzy set is a set with fuzzy boundaries, such as short, average
or tall for men’s height. To represent a fuzzy set in a computer, we
express it as a function and then map the elements of the set to
their degree of membership. Typical membership functions used in
fuzzy expert systems are triangles and trapezoids.
A linguistic variable is used to describe a term or concept with
vague or fuzzy values. These values are represented in fuzzy sets
Fuzzy rules are used to capture human knowledge. A fuzzy rule is
a conditional statement in the form:
IF x is A THEN y is B
where x and y are linguistic variables and A and B are
linguistic values determined by fuzzy sets.
I. Fuzzy Logic Overview


Fuzzy inference is a process of mapping from a given input to an
output by using the theory of fuzzy sets. The fuzzy inference
process includes four steps: fuzzification of the input variables,
rule evaluation, aggregation of the rule outputs and
defuzzification
The two fuzzy inference techniques are the Mamdani and Sugeno
methods. The Mamdani method is widely accepted in fuzzy expert
systems for its ability to capture expert knowledge in fuzzy rules.
However, Mamdani-type fuzzy inference entails a substantial
computational burden
I. Fuzzy Logic Overview
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
To improve the computational efficiency of fuzzy inference, Sugeno
used a single spike, a singleton, as the membership function of the
rule consequent. The Sugeno method works well with optimisation
and adaptive techniques, which makes it very attractive in control,
particularly for dynamic nonlinear systems
Why Should We Use Fuzzy Controllers?
• Very robust
• Can be easily modified
• Can use multiple inputs and outputs sources
• Much simpler than its predecessors
• Very quick and cheaper to implement
II. Constructing a Fuzzy Logic Controller
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For a single input and single output system, we prove that when a
proportional derivative (PD) controller is given, one can design a
fuzzy logic controller whose output is identical to that of the PD
controller. We also prove that if a fuzzy logic controller using
specified fuzzy logic operations is given, there is a PD controller
whose output is identical to that of the fuzzy logic controller
How do we design the fuzzy logic controllers has the respond
similar to PD controllers?
The relationship between Kd, Kp, and the inputs, output of fuzzy
controller through Differential equation:
II. Constructing a Fuzzy Logic Controller

Create the membership values
We will only be considering single input and single output systems
whether they are linear or nonlinear. Our fuzzy sets will always be
triangular (spike) functions defined on equally spaced points
(Fig. 1).
Where:
and k = i+j-1
Figure 1: Fuzzy sets will always be triangular functions defined on equally spaced points
II. Constructing a Fuzzy Logic Controller
1. Without scaling factor
Let
From
We have
Where:
e(n) must be bounded in an interval [−𝑎𝑒 ,𝑎𝑒 ]
∆e(n) must be bounded in an interval [−𝑎𝑑 ,𝑎𝑑 ]
u(n) must be bounded in an interval [−𝑎𝑢 ,𝑎𝑢 ]
II. Constructing a Fuzzy Logic Controller
2. Scaling factor
Let
From
and
We have 𝑢 𝑛 = 0.5𝑆𝑢 𝑆𝑒 𝑒 𝑛 + 0.5𝑆𝑢 𝑆𝑑 ∆𝑒 𝑛
II. Constructing a Fuzzy Logic Controller

Specify the rule table
Our fuzzy control rules will be of the form:
𝐼𝐹 𝑒 𝑛 𝑖𝑠 𝐸𝑖 𝑎𝑛𝑑 ∆𝑒 𝑛 𝑖𝑠 𝐷𝑗 𝑇𝐻𝐸𝑁 𝑢 𝑛 𝑖𝑠 𝑈𝑘
Where k = i+j-1
Table 1: Fuzzy control rules
III. Practical example on Matlab/simulink environment
Traditional PD Control Design with Kp=298.6, Ki=0, Kd=5.34
How do we design the fuzzy logic controller has the similar respond
to the PD controller given above?
III. Practical example on Matlab/simulink environment
Command and Output (Scope)
Derivative of Error (Scope3):
1.4
2
reference
respond of PD controller
Derivative of Error
1.2
0
1
-2
0.8
-4
0.6
-6
0.4
-8
0.2
0
0
0.5
1
1.5
Error (Scope2):
2
2.5
3
3.5
4
4.5
5
1.2
-10
Controller Output (Scope4):
0
0.5
1
1.5
2
2.5
3
3.5
4
5
300
error
Controller output
1
250
0.8
200
0.6
150
0.4
100
0.2
50
0
0
-0.2
4.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-50
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
III. Practical example on Matlab/simulink environment
Respond of FLC without scaling factor
Respond of PD controller
1.4
1.4
reference
respond of PD controller
reference
respond of FLC without scaling factor
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0
0.5
1
1.5
2
Respond of FLC with scaling factor
1.4
reference
respond of FLC with scaling factor
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
2.5
3
3.5
4
4.5
5
IV. Conclusions
For a single input system, whether it is a linear or a nonlinear
system, we have shown that one can construct a fuzzy logic
controller equivalent to a given PD controller, and that a fuzzy logic
controller designed with prescribed fuzzy logic operations is
essentially a PD controller.
In practice, however, one often finds that a fuzzy logic controller
performs better than a PD controller even for a single input system.
V. Reference
[1] Byung Soo Moon, "Equivalence between fuzzy logic controllers and PI
controllers for single input systems“, Fuzzy Sets and Systems, 69 (1995)
105 113
[2] Michael Negnevitsky,” Artificial Intelligence, A guide to Intelligent
Systems“, Addison Wesley
[3] http://faculty.stut.edu.tw/~tang/project/paper/09_Equivalence_FLC.pdf