Tsk model of fuzzy inference system in relation to different membership functions with examples

Introduction

Fuzzy logic begins with the concept of a fuzzy set, which differs from traditional sets by allowing elements to have degrees of membership rather than strict inclusion or exclusion. This flexibility enables fuzzy logic to handle uncertainties and ambiguities more effectively. A fuzzy inference system (FIS) is a method for making decisions based on fuzzy logic principles assembling input data according to a set of fuzzy rules in order to produce an output vector. There are two main types of fuzzy inference systems: Mamdani FIS and Sugeno FIS. The Mamdani FIS, pioneered by Ebrahim H. Mamdani (1974) [1] , was originally designed to control a steam engine and boiler using linguistic control rules provided by human operators. It defines the output of each rule as a fuzzy logic on the other hand, the Sugeno FIS, proposed by Takagi and Sugeno (1985) [2] and later revised by Sugeno and Kang (1988) [3] aims to systematically derive fuzzy rules from input-output datasets. In the Sugeno approach, the output is a crisp function of the input variables, denoted as f(X, Y). In the Mamdani approach, the inference is typically expressed as " | f×₁ is A₁ and X₂ is A₂ , then Y is B, where A₁ , A₂ , , and B are fuzzy sets defined over their respective universes of discourse. In contrast, the Sugeno model's output is computed using different defuzzification methods such as weighted arithmetic mean (WAM), weighted geometric mean (WGM), and weighted harmonic mean(WHM). These methods aggregate the outputs of each rule to produce a final crisp output. Overall, fuzzy inference systems provide a robust framework for decision-making in scenario's where traditional binary logic may fall short, allowing for flexible reasoning in uncertain environments. Mamdani fuzzy inference systems are especially well-suited for applications where rules are derived from human expert knowledge, such as in medical diagnostics. In these systems, rules are formulated as intuitive if-then statements that describe relationships between inputs and outputs. Each rule generates a fuzzy set as its output, determined by the membership functions and implication methods used in the fuzzy inference system (FIS). These fuzzy sets are then aggregated into a single fuzzy set that represents the combined output. To obtain a final crisp output value, the aggregated fuzzy set undergoes defuzzification using methods like centroid or weighted average. In contrast, Sugeno fuzzy inference systems, also known as Takagi-Sugeno-Kang fuzzy inference, simplify the output representation by using singleton output membership functions. These functions are either constant or linear functions of the input variables. This approach enhances computational efficiency compared to Mamdani systems because defuzzification involves calculating a weighted average or sum of a few data points rather than computing the centroid of a two-dimensional area. The essence of fuzzy logic lies in mapping an input space to an output space using a set of rules structured as if-then statements. These rules operate independently of each other, allowing flexibility in their evaluation order. The key terms and descriptors used in the rules, such as "hot" or "tall," are defined through membership functions that assign degrees of membership between 0 and 1 to each point in the input space. For instance, the concept of "tall" in a fuzzy set considers a range of heights rather than a strict boundary, accommodating nuances that classical sets cannot handle. Ultimately, fuzzy inference is a methodical approach to interpreting input values and determining corresponding output values based on a defined set of rules. It leverages the flexibility of fuzzy sets and membership functions to model complex systems where precise boundaries are impractical or insufficient.

Material and Methods

In Mamdani approach, if X₁ is A₁ and X₂ is A₂ then Y is B while in TSK, it is f(X,Y), where f(X,Y) is the crisp function of the FLC in the consequent, as proposed by Takagi,Sugeno and Kang. The overall aggregated value of the output will be determined by WAM, WGM and WHM defuzzification methods respectively. The i th rule of TSK model can be represented as: If X₁ is A₁ ⁱ and X₂ is A₂ ⁱ . . . . . .and Xₙ is Aₙ ⁱ , then Yⁱ=Σ ⱼ₌₀ ⁿaⁱj (Xⱼ) where a₀ , a₁ , a₂ , . . . . aₙ are constants being ×phantom0=1

The weight of the i th rule can be represented as Wⁱ=Π ⱼ₌₁ ⁿμ ⁱAⱼ (Xⱼ) The combined control action can be determined using three methods i.e. WAM, WGM and WHM respectively as

where k represents the total number of rules. Four rules of TSK model of FLC is given below: Rule-I: If x and y are small then z=-x+y+3 Rule-II: If x is small and y is large then z=-y+3 Rule-III: If x is large and y is small then z=-x+3 Rule-IV: If x and y are large then z=x+y+2

Membership Functions of Various Types (i) Triangular Membership function:

Let m0, m₁ , m₂ , m₃ . . . . , mₙ be the coded values of amount of attribute corresponding to the linguistic input variable as X₀ , X₁ , X₂ , … , Xₙr∈ spectively using triangular membership function in relation to the fuzzy set A. Also, let m₀ ³ , m₁ ³ , m₂ ³ m₃ ′ . . . , m _ n-1 ; be any points lying in between (m₀ , m₁) , (m₁ , m₂) , (m₂ , m₃) , (m₃ , m₄) (mₙ₋₁ , mₙ) respectively, then using the principle of similarity of triangle, the degree of membership lying in between (m₀ , m₁) , (m₁ , m₂) , (m₂ , m₃) , (m₃ , m₄) , . . . . , (mₙ₋₁ , mₙ) for the triangular membership function defined below is

Type and level of attribute

, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑥𝑥 𝑛𝑛-1 𝑏𝑏𝑡𝑡𝑥𝑥 𝑛𝑛 (ii) A mixture of triangle and trapezoidal membership function:

Similarly, let p₀ , p₁ , p₂ , p₃ . . . , pₙ be the coded values of amount of another attribute corresponding to the linguistic input variable as y₀ , y₁ , y₂ , … , yₙ respectively using triangular membership function in relation to the fuzzy set B. Also, let p₁ ; p₂ ′ p₃ ³ … , pₙ₋₂ be any points lying in between (mathsfp₁ , mathsfp₂) , (mathsfp₂ , mathsfp₃) , (mathsfp₃ , mathsfp₄) , (mathsfp₄ , mathsfp₅) , … , (mathsfp mathsfn-2)′ pₙ₋₁ ′) respectively, then using the principle of similarity of triangle, the degree of membership lying in between (mathsfp₁ , mathsfp₂) , (mathsfp₂ , mathsfp₃) , (mathsfp₃ , 4 ₄) , . . . . , (mathsfp mathsfn-2 ′ , mathsfp mathsfn-1 ′) for the following triangular membership function is defined as

Type and level of another attribute:

Based on similar lines, the degree of membership function will be determined as and when required for the triangle or trapezoidal shape of the membership function.

(iii) A right and left S-shaped membership function:

On the other hand, if the shape of the curve lies from the left side,then we have where m represents fuzzification factor.

Setting of Rules for Controller Action

(i) Rule of one attribute (M) in relation to another type of attribute (P) if we have the levels of both attributes, equal. In the same way, if rows and columns are being given weights as per the natural numbers, that is first row weighs 1, second row 2 , … , m rows as m and n rows as n, then the value of y lying in between m th rows and n th column in a mx n matrix is equal to mx value of Input 1+nx value of Input 2. Similarly, other values of y can be determined.

Examples Example 3.1

Solve for Input 1 as 13.0 and Input 2 as 2.

Input:I₁ of 13.0 may be called very low or low. Input:I₂ of 2.5 may be called very NR or NR Using principle of similar triangle μ w=(25-13)/(25-10)=( 12)/(15)=0.8

Similarly, μ L=(13-10)/(25-10)= [2] /(15)=0.2

In the same way, μ _ VNR=(3-2.5)/ [2,1] =(0.5)/ [3] =0.25

That is I₁ belongs to the set of very low or low and. I₂ belongs to the set of very NR or NR.Then, the fired set of rules can be constructed as I₁ is very low and I₂ is very NR I₁ is very low and I₂ is NR I₁ is low and I₂ is very NR I₁ is low and I₂ is NR Then their corresponding weights can be computed as The crisp value of the output is then determined by calculating the weighted mean of the weights and values of y, as shown below:

Conclusions

Despite the various membership functions discussed and applied earlier, such as Generalized Bell-shaped, Sigmoidal, Pi-shaped, and Open Right, these functions can effectively describe the inherent variability in control problems, especially within the dynamic and non-linear systems of the TSK fuzzy model. The Mamdani approach remains widely favored for capturing expert knowledge, despite imposing significant computational demands. In contrast, the TSK model is noted for its computational efficiency. This model integrates seamlessly with optimization and adaptive techniques, making it exceptionally suitable for control problems in fuzzy inference systems. Moreover, it circumvents the need for time consuming defuzzification methods required by the Mamdani approach. Furthermore, a combination of different membership functions can be employed to address and solve diverse control problems across various fuzzy models. This versatility extends to applications in fields like machine learning and artificial intelligence, where fuzzy logic plays a crucial role in handling uncertainty and imprecision effectively.

References

1. Mamdani EH. Application of Fuzzy Algorithms for Control of Simple Dynamic Plant. Proceedings of the IEEE. 1974;121(12):1585-1588.
2. Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE transactions on systems, man, and cybernetics. 1985;15(1):116-132.
3. Sugeno M, Kang G. Structure identification of fuzzy model. Fuzzy Sets and Systems. 1988; 28(1):15-33.