Deriving analytical structure of a type-2 fuzzy PD/PI controller

Cascade fuzzy control of a tubular chemical reactor

Fuzzy logic control based on the Takagi–Sugeno inference method has been applied for the yeast alcoholic fermentation running in a continuous-time biochemical reactor in this paper. A type-1 fuzzy PID controller was designed to temperature control in a biochemical reactor. The fuzzy PID controller was also designed using type-2 fuzzy sets. The advantage of the fuzzy control is that it can be used very successfully for control of strongly non-linear processes and processes that are difficult to model because of complicated reaction kinetics. Obtained simulation results confirm this fact. The disadvantage of the fuzzy control design lies in the time-consuming tuning of controllers. The subtractive clustering method was used to identify the rule base. This approach was chosen to minimize the number of rules of the designed fuzzy logic controllers and to simplify the fuzzy controller design. Simulation results confirm that fuzzy PID controllers can assure better performance than conventional PID controllers. Yeast alcoholic fermentation is an exothermic process and the cooling is necessary to maintain the optimal temperature in the reactor. Fuzzy PID control reduces coolant consumption in comparison with conventional PID control and so it assures energy-efficient control of continuous-time alcoholic fermentation. Using type-2 fuzzy sets in the fuzzy PID controller design is very promising as the type-2 fuzzy controller offers even better results than the type-1 fuzzy controller.

Compared with type-2 fuzzy sets, the secondary membership degree of interval type-3 fuzzy sets is an interval rather than crisp value, which makes interval type-3 fuzzy sets can obtain more degree of freedoms. This article studies an interval type-3 fuzzy PID controller based on interval type-3 fuzz sets. The framework of interval type-3 fuzzy PID controller is identical with type-2 fuzzy PID controller, but it contains more adjustment controller parameters and its type reduction procedure is more complex. In this paper, type reduction of interval type-3 fuzzy sets is derived from general type-2 fuzzy sets represented by α-plane and a direct NT type reduction algorithm is applied. The control effects of interval type-3 fuzzy PID controller are firstly tested by 2 nonlinear plants, the simulation results show that interval type-3 fuzzy PID controller has better control performance indexes than PID controller, type-1 fuzzy PID controller, interval type-2 fuzzy PID controller and general type-2 fuzzy PID controller. Furthermore, the interval type-3 fuzzy PID controller will be applied in rated voltage control of solid oxide fuel cells (SOFC) power plant. The output voltage control of SOFC is quite challenging because of the strong nonlinearity, limited fuel flow, and rapid variation of the load disturbance. The simulation results demonstrate the advantages and robustness of proposed interval type-3 fuzzy PID controller.

Fuzzy controllers and fuzzy sliding-mode controllers have found extensive use in a variety of applications. Generally, type-1 fuzzy sets are used for the membership functions of these controllers. However, in real-time applications, uncertainty associated with the available information always occurs. A type-2 fuzzy controller and a type-2 fuzzy sliding-mode controller are proposed that are able to solve the problem that the words used in the inference rules can mean different things to different people. Since the membership functions use type-2 fuzzy sets then, the proposed control schemes can handle the rule uncertainties when the operation is extremely uncertain and/or the membership grades cannot be exactly determined. The proposed control systems are applied to control a buck DC–DC converter. A comparison between a PI controller, a type-2 fuzzy controller and a type-2 fuzzy sliding-mode controller is made. The experimental results show that the type-2 fuzzy sliding-mode controller achieves the best control performance.

In this paper mathematical models of Mamdani type fuzzy PI or PD controllers via Center of Gravity (CoG) defuzzification are obtained. These controllers are called the simplest fuzzy PI or PD controllers as minimal number of fuzzy sets (two fuzzy sets on each of the two-input variables and three fuzzy sets on the output variable) are employed. The basic components of these models are L-type and Y-type membership functions on each input variable, trapezoidal or triangular membership functions on output variable, minimum / algebraic product t-norm, maximum OR operator, three linear fuzzy control rules, minimum / algebraic product inference and CoG defuzzification. Mathematical models of fuzzy controller via CoG defuzzification were not reported in the literature and are therefore presented for the first time in this paper.

Fuzzy logic (FL) has been regarded as a useful methodology in modelling. The modeling performance of FL heavily relies on the modeling capability of a fuzzy set (FS). However, the MF of a FS cannot be arbitrarily accurate in practice and thus the centroid as a measure of a FS cannot be accurate to any degree. In face of inaccurate MF, the sensitivity of the centroid of a FS to the deviation of its MF from the ideal MF determines the accuracy of the centroid. To evaluate the modeling capability of a type-1 (T1) FS and an interval type-2 (IT2) FS, this paper will present a comparative study of the sensitivity of the centroid of a T1 FS and an IT2 FS to the deviation of their MFs from the ideal MFs. In this paper, equations relating the amount that the centroid of a T1 FS and an IT2 FS vary with the deviation of their MFs from the ideal MFs were established. Based on the derived equations, it was found that an IT2 FS is able to provide an additional design freedom to vary the sensitivity of its centroid compared to a T1 FS. This unique characteristic may enable an IT2 FS to minimize the sensitivity of the centroid and thus to achieve higher modeling accuracy. Furthermore, guidelines for designing an IT2 FS were provided.

This paper presents a comparative study between the firefly algorithm (FA) and the galactic swarm optimization (GSO) method, where the performance of both methods is observed and tested in the optimization of a fuzzy controller for path tracking of an autonomous mobile robot. The main contribution of this work is finding the best method that generates an optimal vector of values for the membership function optimization of the fuzzy controller. This with the goal of improving the performance of the controller and thus the trajectory generated by the autonomous robot is closer to the desired trajectory. It should be noted that the fuzzy controller that is optimized is an interval type-2 fuzzy controller, which has a greater capability for managing uncertainty than a type-1 fuzzy controller. In this case, the limiting membership functions in the interval type-2 fuzzy sets are themselves type-1 fuzzy sets that define the footprint of uncertainty. Type-2 fuzzy controllers have been shown in previous works to handle better the control of robotic systems under noisy and dynamic conditions and this is why their optimal design is very important. Simulation results show that GSO outperforms FA in the optimal design of interval type-2 fuzzy controllers.

The Wang-Mendel Approach (WMA) focuses on combining the numerical as well as linguistic information for achieving greater explainability for inference models. The standard WMA models the linguistic information using type-1 (T1) fuzzy sets (FSs), which have a reduced capability to model the semantics of linguistic information. Therefore, we propose a novel Enhanced WMA, which models the linguistic information using the type-2 (T2) FSs. Further, our Enhanced T2 FS-based WMA can be modified to reflect the use of interval type-2 (IT2) FSs, for modelling linguistic uncertainty. IT2 FSs are suitable when better uncertainty handling capabilities are required compared to T1 FSs, however, at a computational cost lesser than the T2 FSs. Performance of Enhanced WMA is demonstrated through a real-world crop-yield prediction problem in smart agriculture and an additional exemplar application on users’ satisfaction ratings. Further, we have compared our approach with the performance obtained from the T1 FS-based WMA and the original estimations given in the original data. We found that our Enhanced WMA provides more precise estimates than the other two with 95% confidence level. To the best of our knowledge, this is the first proposal of a T2 FSs method for enhancing the modeling of linguistic uncertainty in the WMA.

Modeling, design and control of a 4-arm delta parallel manipulator employing type-1 and interval type-2 fuzzy logic-based techniques for precision applications

In this paper, we develop the parametric graded mean integration representation (PGMIR) for the interval type 2 fuzzy sets (IT2FS). The PGMIR value can be served as the type reduced or defuzzification of an IT2FS. A set of IT2FSs can then be ordered by the corresponding PGMIR values. Firstly, the parametric representation of the embedded fuzzy sets is defined for the general IT2FS. The IT2FS is characterized by the set of embedded fuzzy sets induced from the parametric function. Then, the graded mean integration representation (GMIR) of a type 1 fuzzy set (T1FS) is extended to the IT2FS through the parametric form of an IT2FS. The GMIR of the parametric embedded T1FS is defined and is called the PGMIR. In this manner, the IT2FSs are defuzzified and can be ranked by sorting the PGMIR values. The average value of the PGMIR is used to determine IT2FS ranking. A numerical example is presented to show that the proposed approach is superior to the ranking method from IT2FS centroid.

An adaptive controller is proposed using a novel interval type-2 fuzzy cerebellar model articulation controller (T2FCMAC). The T2FCMAC realizes an interval type-2 fuzzy logic system based on the structure of the cerebellar model articulation controller (CMAC). Due to the better ability of handling uncertainties provided by type-2 fuzzy sets, it can solve some complicated problems with outstanding effectiveness than type-1 fuzzy sets. In addition, an intelligent control system is proposed; this control system is comprised of a T2FCMAC and an auxiliary compensation controller. The T2FCMAC is utilized to approximate a perfect controller and the parameters of T2FCMAC are on-line tuned by the derived adaptive laws based on a Lyapunov function. The auxiliary compensation controller is designed to suppress the influence of residual approximation error between the perfect controller and the T2FCMAC. Finally, a Chua's chaotic circuit is performed to verify the effectiveness of the proposed control scheme. The simulation results confirm that the proposed intelligent adaptive control system can achieve favorable tracking performance.

Type-1 fuzzy PID controllers are widely used in industrial control systems due to their well-established theoretical foundation, simplicity of structure, and ease of operation. However, as control systems become increasingly complex and demands for higher control performance intensify, the limitations of Type-1 fuzzy controllers become more apparent. Additionally, the difficulty in tuning PID parameters and the inability to adjust these parameters online as the controlled system changes further constrain the effectiveness of traditional PID controllers. To address these challenges, this paper introduces a generalized Type-2 fuzzy logic control approach. The generalized Type-2 fuzzy controller extends the capabilities of interval Type-2 fuzzy controllers, offering enhanced ability to handle higher-order uncertainties. Moreover, the integration of memristor devices enables online adaptive tuning of controller parameters. By combining the generalized Type-2 fuzzy logic with memristor-based PID control, a novel memristor-based controller was designed, which allows for real-time adaptive adjustment of the controller parameters. Theoretical analysis is followed by simulations and experimental validation. The results demonstrate that the proposed memristor-based controller with generalized Type-2 fuzzy logic exhibits superior robustness. Specifically, when evaluated using the Integral of Time-weighted Absolute Error (ITAE) performance index, the controller achieves improvements of 65.9% and 40.2% compared to traditional PID and Type-1 fuzzy PID controllers, respectively. These findings confirm the excellent tracking performance and robustness of the proposed control system.

In this study, we will present the novel application of Type-2 (T2) fuzzy control into the popular video game called flappy bird. To the best of our knowledge, our work is the first deployment of the T2 fuzzy control into the computer games research area. We will propose a novel T2 fuzzified flappy bird control system that transforms the obstacle avoidance problem of the game logic into the reference tracking control problem. The presented T2 fuzzy control structure is composed of two important blocks which are the reference generator and Single Input Interval T2 Fuzzy Logic Controller (SIT2-FLC). The reference generator is the mechanism which uses the bird's position and the pipes' positions to generate an appropriate reference signal to be tracked. Thus, a conventional fuzzy feedback control system can be defined. The generated reference signal is tracked via the presented SIT2-FLC that can be easily tuned while also provides a certain degree of robustness to system. We will investigate the performance of the proposed T2 fuzzified flappy bird control system by providing comparative simulation results and also experimental results performed in the game environment. It will be shown that the proposed T2 fuzzified flappy bird control system results with a satisfactory performance both in the framework of fuzzy control and computer games. We believe that this first attempt of the employment of T2-FLCs in games will be an important step for a wider deployment of T2-FLCs in the research area of computer games.

This paper presents a practical implementation of an interval type-2 fuzzy logic controller for two stages photovoltaic system consisting of DC- DC boost converter and three-phase inverter. The power converter circuits are nonlinear and have uncertainties. These uncertainties cause noise and delay in the response of power systems. A photovoltaic system has also uncertainties caused by itself. The change in temperature and irradiation cause the change in parameters of photovoltaic systems. Conventional controllers and type-1 fuzzy controllers can’t overcome uncertainties. The main objective of this study is to show the superiority of interval type-2 fuzzy logic controller to overcome the uncertainties. Interval type-2 fuzzy logic controller algorithm was developed for DC- DC boost converter and three-phase space vector modulated inverter. To make comparison, the system was controlled by a type-1 and an interval type-2 fuzzy controller respectively. Simulation studies and real time application were performed. Settling time, overshoot and ripple voltage were investigated for the boost converter. Current and voltage harmonic analysis, frequency and phase angle data were analyzed for three-phase inverter. According to the simulation results and the practical results, interval type-2 fuzzy logic controller offered better solution for the whole system.

This article provides a high-level retrospective of type-2 fuzzy sets and fuzzy logic systems. It explains how type-2 fuzzy sets can be used to model membership function uncertainties, and how by doing this smoother performance can be obtained than by using type-1 fuzzy sets. It also summarizes the notation that should be used for type-2 fuzzy sets, describes four important mathematical representations for these fuzzy sets, explains the differences between type-1 and type-2 fuzzy logic systems and which of the four representations is most useful when designing an optimal type-2 fuzzy logic system, provides a very useful strategy for optimal designs of fuzzy logic systems – one that guarantees performance improvement as one goes from a type-1 fuzzy logic system to a type-2 fuzzy logic system design – , and describes four methods for simplifying the designs of type-2 fuzzy logic systems. Finally, it explains why type-2 fuzzy sets can capture two kinds of linguistic uncertainties simultaneously (the uncertainty of an individual and the uncertainties of a group about a word), whereas type-1 fuzzy sets cannot, and that such type-2 fuzzy set word models are what should be used to implement Zadeh’s Computing With Words paradigm.