# Self-Evolving Fuzzy Controller Composed of Univariate Fuzzy Control Rules
## 1 Introduction  
The globalization of markets, environmental legislation restrictions, the necessity of having an efficient management of energy and sustainable resources, zero defect trends, customer pressure to reduce costs, personalized products, low product lifetime, and several other facts have resulted in a significant increase in industrial process complexity. Industrial processes have increasingly exhibited nonlinear behaviors and other complex characteristics, such as unknown and time-varying dynamics, constraints, and disturbances. Considering the above facts and trends of the \"Industry 4.0\" strategy, more advanced industrial control solutions with high levels of efficiency, flexibility, reliability, and performance are currently required. Motivated by these problems, as an advanced industrial control solution, fuzzy logic controllers (FLCs), which are rule-based systems that allow control of complex ill-defined processes using the experience of expert operators, have been applied in a wide variety of industrial processes [1]. However, there still exist many difficulties in designing FLCs for complex nonlinear industrial processes, using only human knowledge to control complex processes, and there is not a standard approach to translate this knowledge into fuzzy control rules [2]. The design of fuzzy logic systems has attracted many researchers, many of them working on identification, classification, control, and decision making problems. Regarding the fuzzy control systems, the design methods can be generally distinguished as indirect or direct methods, where indirect methods, the FLC is designed through operator control knowledge, and in indirect methods, the FLC is designed through operator knowledge about the process [3,4]. However, the number of proposed direct fuzzy control design methodologies are significantly less when comparing to identification, classification, and indirect control applications. Evolutionary algorithms have been used to design fuzzy controllers [5, 6, 7, 8, 9, 10, 11]. However, such methods (e.g., GA, PSO, and ABC) are computationally heavy, mainly offline, and do not consider changes in the dynamics of the process, when the process is under control. In order to deal with systems characterized by changing the characteristics, evolving systems have recently emerged as promising methods that adapt their structure (and parameters, as happens with adaptive systems) to new operating conditions, changing system dynamics, drifting situations, and non-stationary environments [12]. It is important to note that evolving systems self-change or organize their structure (and parameters) according to novelties such as, for example, new regions of operation, anomalies, unknown environment, drifts, and shifts, which differs from adaptive systems, which only adapt their parameters, having the model/control structure fixed. Several relevant evolving methodologies have been proposed to design fuzzy systems. In [13], a review of evolving fuzzy and neuro-fuzzy methodologies for clustering, regression, identification, and classification applications was presented, in which the addition, merging, splitting, and removing evolving mechanisms were presented and discussed. Additionally, in [12], an overview of the following evolving systems was performed: fuzzy rule-based, neuro-fuzzy, Cauchy possibilistic clustering, granular neural network, and fuzzy linear regression tree. However, in these reviews [12,13], control applications were not considered. Focusing on control applications, the work in [14] presented an approach for on-line designing indirect T-S (Takagi-Sugeno)fuzzy controllers with evolving learning in a recursive way. A fuzzy model reference adaptive control was proposed in [15], in which the evolving fuzzy model (eFuMo) method was used to learn the controller's fuzzy model. In addition to [15], an adaptive law was proposed in [16] in order to guarantee global stability.