1. 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

2. OUTLINE I.Fuzzy Logic Overview II.Constructing a Fuzzy Logic Controller III.Practical example on Matlab/simulink environment IV.Conclusions V.Reference

3. I. Fuzzy Logic Overview  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

4.  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

5.  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

6.  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 I. Fuzzy Logic Overview

7. II. Constructing a Fuzzy Logic Controller  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:

8.  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). Figure 1: Fuzzy sets will always be triangular functions defined on equally spaced points Where: and k = i+j-1 II. Constructing a Fuzzy Logic Controller

9. 1. Without scaling factor Let From We have II. Constructing a Fuzzy Logic Controller

10. 2. Scaling factor Let II. Constructing a Fuzzy Logic Controller From and We have

11. Table 1: Fuzzy control rules II. Constructing a Fuzzy Logic Controller

12. 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?

13. Command and Output (Scope) Error (Scope2): Derivative of Error (Scope3): Controller Output (Scope4): III. Practical example on Matlab/simulink environment

14. Respond of PD controller Respond of FLC with scaling factor Respond of FLC without scaling factor III. Practical example on Matlab/simulink environment

15. 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.

16. 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

18. Without scaling factor With scaling factor III. Practical example on Matlab/simulink environment