# A Takagi-Sugeno Fuzzy Inference System for Developing a Sustainability Index of Biomass

## Abstract

**:**

## 1. Introduction

## 2. Fuzzy Inference System: A Theoretical Review

_{1}is A

_{1}and ...and X

_{n}is A

_{n}then Y is B”, where A, A

_{n}, B are fuzzy sets [15]. The knowledge base, which comprehends general knowledge concerning a problem domain, joins antecedents with consequences or premises with conclusions [16] (see Figure 1). The most commonly used fuzzy inference technique was proposed by Mamdani. However, in Mamdani-type FIS the number of rules grows with the number of premise-part variables. As the number of rules grows the activity of assembling rules can become very burdensome and sometimes it becomes difficult to comprehend the relationships between the premises and consequences [17]. A Sugeno-type method (or Takagi-Sugeno-Kang) has fuzzy inputs and a crisp output (linear combination of the inputs). It is computationally efficient and suitable to work with optimization and adaptive techniques, so it is very adequate for control problems, mainly for dynamic nonlinear systems [18]. Sugeno method develops a systematic approach to generate fuzzy rules from a given input-output data set. It changes the consequent (then part) of Mamdani rule with a function (Equation) of the input variables. The T-S style fuzzy rule is: IF x is A AND y is B THEN z is f (x, y) where x, y and z are linguistic variables, A and B are fuzzy sets on universe of discourses X and Y and f (x, y) is a mathematical function [18]. Sugeno-type FIS uses weighted average to compute the crisp output while Mamdani-type FIS uses the technique of defuzzification of a fuzzy output. The first two parts of the fuzzy inference process, fuzzifying the inputs and applying the fuzzy operator, are the same [9]. The main difference is that the Sugeno output membership functions are either linear or constant [9]. In Figure 2 different types of fuzzy systems are shown. Type two is Mamdani FIS with output function based on overall fuzzy output, while type three is the Takagi-Sugeno fuzzy inference.

**Figure 2.**Different types of Fuzzy inference system [30].

## 3. Fuzzy Inference for Measuring the Sustainability of Biomass

^{j}rules, where p is the number of linguistic terms per input variable. As the dimension and complexity of a system increases, the size of the rule base increases exponentially [13]. In this system 81 if-then rules have been defined; in the following Figure 4 some of these are reported as examples.

^{–1}. Data from the literature [31] attributes an energy output value of between 260 GJ ha

^{–1}and 530 GJ ha

^{–1}for miscanthus, between 240 GJ ha

^{–1}and 600 GJ ha

^{–1}for the giant reed, between 155 GJ ha

^{–1}and 252 GJ ha

^{–1}for the cardoon and, finally, between 334 GJ ha

^{–1}and 507 GJ ha

^{–1}for sorghum. This input variable has three fuzzy sets: “Low”, “Medium”, and “High”. MFs of “Low” and “High” are trapezoidal, while MF of “Middle” is triangular. Their Equations (1)–(3) are shown below:

^{-1}) refers to the total quantity of various fertilizers (N + K

_{2}O + P

_{2}O

_{5}) used on the crops [32,33].

## 4. Discussion and Testing

Input Data | ||||
---|---|---|---|---|

Crops | Energy Output (GJ ha^{–1}) | Energy O/I (ratio) | Fertilizer (kg/ha^{–1}) | Pesticide (score) |

Sweet Sorghum | 334–507 | 10–32 | 245–3900 | 3–10 |

Hemp | 128–270 | 5–20 | 156–295 | 0 |

Miscanthus | 260–530 | 12–66 | 152–252 | 0–15 |

Giant Reed | 240–600 | 11–75 | 167–227 | 0 |

Cardoon | 155–252 | 7–31 | 149–459 | 0–4 |

Switchgrass | 174–435 | 8–54 | 96–146 | 42 |

Crops | FSBI |
---|---|

Sweet Sorghum | 0.482–0.701 |

Hemp | 0.337–0.561 |

Miscanthus | 0.481–0.659 |

Giant Reed | 0.487–0.9 |

Cardoon | 0.303–0.482 |

Switchgrass | 0.288–0.482 |

## 5. Conclusions

## Conflicts of Interest

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**MDPI and ACS Style**

Cavallaro, F.
A Takagi-Sugeno Fuzzy Inference System for Developing a Sustainability Index of Biomass. *Sustainability* **2015**, *7*, 12359-12371.
https://doi.org/10.3390/su70912359

**AMA Style**

Cavallaro F.
A Takagi-Sugeno Fuzzy Inference System for Developing a Sustainability Index of Biomass. *Sustainability*. 2015; 7(9):12359-12371.
https://doi.org/10.3390/su70912359

**Chicago/Turabian Style**

Cavallaro, Fausto.
2015. "A Takagi-Sugeno Fuzzy Inference System for Developing a Sustainability Index of Biomass" *Sustainability* 7, no. 9: 12359-12371.
https://doi.org/10.3390/su70912359