# Dynamic Analysis of Fuzzy Systems
## 1 Introduction  
The most common models in fuzzy control, the Mamdani model [1] and Takagi-Sugeno (T-S) model [2], are traditionally used to model non-linear systems. The main difference lies in the consequent part of the fuzzy rules. Mamdani fuzzy systems use fuzzy sets as the consequents of the rules, while T-S fuzzy systems use affine functions of the input variables as the consequents of the rules. From a linguistic perspective, Mamdani-type fuzzy systems are understandable since fuzzy linguistic variables are used in both premises and consequents [3]. On the other hand, in the T-S model, the number of free parameters is restricted by a suitable structure of the consequent functions, and the antecedent of each rule defines a validity region for the corresponding affine consequent model. It can be considered an optimization problem [4; 5]. This work deals with the dynamic analysis of non-linear systems rather than control design. Therefore, the Mamdani model would be the appropriate choice. The structure of the fuzzy model using the product-sum-gravity method was used [6; 7] to obtain a flat surface in the inference map and then modify the membership functions with linguistic operators. The min-max-gravity method [8] does not obtain a flat surface in the inference map due to the non-linearity of these operators. Knowledge of a system is essential for applying fuzzy logic. Combining it with linguistic information (by experts) and numerical data improves the ability to represent complex non-linear functions with simple local linear models [9]. Although these linguistic descriptions are not precise, they provide essential information about the system. A human operator can determine a set of successful control rules based solely on the linguistic descriptions of the process. This is based on Mamdani Fuzzy Inference Systems in linguisticsynthesis [10]. In recent work, in [11], the authors develop the \"Linguistic Compositional Variable\" to represent mole fractions of components of each phase as a fuzzy variable. As a result, the number of rules is significantly reduced. In [12], the authors use the cloud model to represent the linguistic variables. Instead, [13] uses a Mamdani Fuzzy Inference Network (MFIN) with an advanced optimization technique. The study of evaluative linguistic expressions has crossed the field of theoretical linguistics and has attracted interest in very different research areas, such as artificial intelligence, psychology, or cognitive linguistics [14]. Applying a simple arithmetic operation on the fuzzy sets will modify them. It tends to consider the adverb \"very\" interpretation as the result of squaring a membership function, which causes the membership function to take smaller values, concentrating on the larger ones. [PERSON] [15] called this concentration. The opposite effect is dilation, which would correspond to the interpretation of the adverb \"more or less\", which would be implemented as the square root of the membership function. These are linguistic variables used to describe overly complex or ill-defined phenomena typical of non-linear systems. These modifiers have been used for interpretable context adaptation [16; 17]. In [18], an extended fuzzy hedge set is used to create a more detailed classification of client sentiments, achieving better performance than without them. In some approaches, they are used to create more compact rules, such as [19]. The basic first-order system is described with a single real root, while the underdamped second-order system has a single pair of complex roots. Many devices and processes are practically well-modelled by either of these two classes, and complex systems can be considered a combination of several first-order and/or second-order forms [20]. In previous work, the authors have studied the transient response of feedback systems [21], using the Takagi-Sugeno fuzzy model for system identification [22; 23; 24] or applying fuzzy logic to improve Kalman filtering [25; 26].