Interval type-2 fuzzy neural networks ( IT2FNNs ) can be seen as the hybridization of interval type-2 fuzzy systems ( IT2FSs ) and neural networks ( NNs ). Thus, they naturally inherit the merits of both IT2FSs and NNs. Although IT2FNNs have more advantages in processing uncertain, incomplete, or imprecise information compared to their type-1 counterparts, a large number of parameters need to be tuned in the IT2FNNs, which increases the difficulties of their design. In this paper, big bang-big crunch ( BBBC ) optimization and particle swarm optimization ( PSO ) are applied in the parameter optimization for Takagi-Sugeno-Kang ( TSK ) type IT2FNNs. The employment of the BBBC and PSO strategies can eliminate the need of backpropagation computation. The computing problem is converted to a simple feed-forward IT2FNNs learning. The adoption of the BBBC or the PSO will not only simplify the design of the IT2FNNs, but will also increase identification accuracy when compared with present methods. The proposed optimization based strategies are tested with three types of interval type-2 fuzzy membership functions ( IT2FMFs ) and deployed on three typical identification models. Simulation results certify the effectiveness of the proposed parameter optimization methods for the IT2FNNs.

This paper aims to provide a clear explanation of the role of the footprint of uncertainty (FOU) parameters on the control signal generation and, thus, to increase the interpretability of specially structured interval type-2 (IT2) fuzzy logic controllers, namely single input IT2 fuzzy PID (SI-IT2-FPID) controller. In this context, we extend the analysis performed for SI-IT2-FPID controllers by providing the effect of the FOU parameters on control surface (CS) generation. We show that, by only adjusting a single parameter that shapes the FOU, it is possible to generate commonly employed CSs without a requirement of an optimization assistance. In order to validate our theoretical analysis, we present comparative real world quadcopter flight tests. The real-time experimental results show that the SI-IT2-FPID controller can achieve better control performance in the presence of uncertainties and strong wind conditions when compared to its type-1 and conventional counterparts. We believe that the results of this study will open the doors to a wider use of SI-IT2-FPID controllers in real world control applications as the proposed structure is easy to design and feasible to deploy, especially in real-time control systems.

Recent results on fuzzy control have shown that interval type-2 (IT2) fuzzy logic controllers (FLCs) might achieve better control performance due to the additional degree of freedom provided by the footprint of uncertainty (FOU) in their IT2 fuzzy sets. However, the design and robust stability analysis of the IT2-FLCs are still challenging problems due to their relatively more complex internal structure. In this paper, we will derive the explicitly fuzzy mapping (FM) of a single-input IT2-FLC (SIT2-FLC) to present design methods and investigate its robustness. The analytical information of the IT2-FM will give the opportunity to provide explanations on the roles of the FOU parameters by taking advantage of the well-developed framework of nonlinear control theory. Comparative theoretical explorations will be presented on the differences between the type-1 (T1) FM and IT2-FM to clearly show the role of the FOU on the robust control system performance. It will be proven that the robust stability of the IT2 fuzzy system is guaranteed with the aids of the well-known Popov-Lyapunov method. Moreover, analytical design methods are presented for SIT2-FLCs to generate commonly employed control curves by only tuning the size of the FOUs without a need of an optimization procedure. It will be theoretically shown that the FOU gives the opportunity to the SIT2-FLC to generate commonly employed nonlinear control curves while also providing a certain degree of robustness which cannot be accomplished by its T1 counterpart. The presented results provide theoretical explanations on the role of the FOU on the performance and robustness of the SIT2-FLC.

The interval type-2 fuzzy Proportional-Integral (PI) controller (IT2-FPI) might be able to handle high levels of uncertainties to produce a satisfactory control performance, which could be potentially due to the robust performance as a result of the smoother control surface around the steady state. However, the transient state and disturbance rejection performance of the IT2-FPI may degrade in comparison with the type-1 fuzzy PI (T1-FPI) counterpart. This drawback can be resolved via general type-2 fuzzy PI controllers which can provide a tradeoff between the robust control performance of the IT2-FPI and the acceptable transient and disturbance rejection performance of the type-1 PI controllers. In this paper, we will present a zSlices-based general type-2 fuzzy PI controller (zT2-FPI), where the secondary membership functions (SMFs) of the antecedent general type-2 fuzzy sets are adjusted in an online manner. We will examine the effect of the SMF on the closed-system control performance to investigate their induced performance improvements. This paper will focus on the case followed in conventional or self-tuning fuzzy controller design strategies, where the aim is to decrease the integral action sufficiently around the steady state to have robust system performance against noises and parameter variations. The zSlices approach will give the opportunity to construct the zT2-FPI controller as a collection of IT2-FPI and T1-FPI controllers. We will present a new way to design a zT2-FPI controller based on a single tuning parameter where the features of T1-FPI (speed) and IT2-FPI (robustness) are combined without increasing the computational complexity much when compared with the IT2-FPI structure. This will allow the proposed zT2-FPI controller to achieve the desired transient state response and provide an efficient disturbance rejection and robust control performance. We will present several simulation studies on benchmark systems, in addition to real-world experiments that were performed using the PIONEER 3-DX mobile robot that will act as a platform to evaluate the proposed systems. The results will show that the control performance of the self-tuning zT2-FPI control structure enhances both the transient state and disturbance rejection performances when compared with the type-1 and IT2-FPI counterparts. In addition, the self-tuning zT2-FPI is more robust to disturbances, noise, and uncertainties when compared with the type-1 and interval type-2 fuzzy counterparts.