# Self-Evolving Fuzzy Controller Composed of Univariate Fuzzy Control Rules
## 1 Introduction
In [17], an evolving neuro-fuzzy controller based on the Taylor series neuro-fuzzy (TaSe-NF) model was proposed, in which the proposed method analyzes the error surface to obtain the fuzzy rule with the worst performance to be split, based on the presented criteria. A model-based evolving granular fuzzy control approach was proposed in [18], where the model's structure and parameters were adapted based on information extracted from uncertain data streams. In [19], an evolving probabilistic fuzzy neural controller using an asymmetric membership function was presented. In [20], an evolving fuzzy model-based controller for a generic hypersonic vehicle (HV) was proposed, where the control law was designed based on the fuzzy model, and the stability analysis was performed using the Lyapunov method. Self-evolving Type-2 control was also studied [21, 22, 23]. However, the above control applications focused on indirect control, in which the evolving strategy was proposed for the model. In the evolving fuzzy systems, when compared with identification and classification problems (and even with indirect control applications), just a few works have been proposed for direct control learning and evolving (and not for identification of a system or data model, even to be possibly used in indirect control learning). In order to design the entire fuzzy control structure in an online way, methodologies to design direct FLCs in an evolving way have been proposed. In [2,24], an evolving FLC design methodology was proposed, where the proposed method was initialized with no fuzzy control rules (the online process starts with an empty control structure). However, the proposed solution can lead to a complex controller with a huge number of control rules. Later, based on data clouds, the robust evolving cloud-based controller (RECCo) was proposed [25]. In RECCo, the antecedent part of the fuzzy control rules to be designed is defined by data clouds (instead of the typical membership functions). RECCo was used also in [26, 27, 28] and improved in [27, 28]. In [29], an evolving data cloud-based PID-like controller was proposed for uncertain nonlinear systems, in which, a stable recursive method (based on the Lyapunov function) was proposed to adapt the parameters aiming at fast convergence performance. However, the control rules' interpretability ability by a human operator was lost since the antecedent part of the control rules was defined by data clouds, which make the control rules very hard to be understood by human operators [26]. In [30], the problem of translating the control knowledge of a human expert operator into fuzzy control rules was addressed, where an approach was proposed to automatically design a Mamdani FLC iteratively. In [31], a methodology was proposed to online evolve a fuzzy control structure defined by univariate fuzzy control rules, which was based on the inverse function of the system under control. However, the works in [30; 31] only considered adding control rules during the evolving learning process. In order to reach an online self-organizing fuzzy control design, which can be easily interpretable by human operators, a new methodology is proposed in this paper, to online self-evolve direct FLCs. The fuzzy control rules' structure is defined by univariate rules. In this way, the interpretability ability of the controller is improved, as well as the complexity (the number of membership functions and fuzzy control rules) of the learned FLC controller is reduced. In an offline stage, the initial fuzzy control structure is designed using only the variables range (minimum and maximum admissible values). Then, the online stage, which occurs when the process is under control, corresponds to the evolving stage where fuzzy control rules can be added or removed from the current knowledge base (fuzzy control rules).