Non-singleton Type-2 Fuzzy Logic Systems Research Articles

The Enhanced Wagner–Hagras OLS–BP Hybrid Algorithm for Training IT3 NSFLS-1 for Temperature Prediction in HSM Processes

This paper presents (a) a novel hybrid learning method to train interval type-1 non-singleton type-3 fuzzy logic systems (IT3 NSFLS-1), (b) a novel method, named enhanced Wagner–Hagras (EWH) applied to IT3 NSFLS-1 fuzzy systems, which includes the level alpha 0 output to calculate the output y alpha using the average of the outputs y alpha k instead of their weighted average, and (c) the novel application of the proposed methodology to solve the problem of transfer bar surface temperature prediction in a hot strip mill. The development of the proposed methodology uses the orthogonal least square (OLS) method to train the consequent parameters and the backpropagation (BP) method to train the antecedent parameters. This methodology dynamically changes the parameters of only the level alpha 0, minimizing some criterion functions as new information becomes available to each level alpha k. The precursor sets are type-2 fuzzy sets, the consequent sets are fuzzy centroids, the inputs are type-1 non-singleton fuzzy numbers with uncertain standard deviations, and the secondary membership functions are modeled as two Gaussians with uncertain standard deviation and the same mean. Based on the firing set of the level alpha 0, the proposed methodology calculates each firing set of each level alpha k to dynamically construct and update the proposed EWH IT3 NSFLS-1 (OLS–BP) system. The proposed enhanced fuzzy system and the proposed hybrid learning algorithm were applied in a hot strip mill facility to predict the transfer bar surface temperature at the finishing mill entry zone using, as inputs, (1) the surface temperature measured by the pyrometer located at the roughing mill exit and (2) the time taken to translate the transfer bar from the exit of the roughing mill to the entry of the descale breaker of the finishing mill. Several fuzzy tools were used to make the benchmarking compositions: type-1 singleton fuzzy logic systems (T1 SFLS), type-1 adaptive network fuzzy inference systems (T1 ANFIS), type-1 radial basis function neural networks (T1 RBFNN), interval singleton type-2 fuzzy logic systems (IT2 SFLS), interval type-1 non-singleton type-2 fuzzy logic systems (IT2 NSFLS-1), type-2 ANFIS (IT2 ANFIS), IT2 RBFNN, general singleton type-2 fuzzy logic systems (GT2 SFLS), general type-1 non-singleton type-2 fuzzy logic systems (GT2 NSFLS-1), interval singleton type-3 fuzzy logic systems (IT3 SFLS), and interval type-1 non-singleton type-3 fuzzy systems (IT3 NSFLS-1). The experiments show that the proposed EWH IT3 NSFLS-1 (OLS–BP) system presented superior capability to learn the knowledge and to predict the surface temperature with the lower prediction error.

Dynamic adaptation of the PID’s gains via Interval type-1 non-singleton type-2 fuzzy logic systems whose parameters are adapted using the backpropagation learning algorithm

This work presents a new design to dynamically adapt the proportional, the integral and the derivative (PID) controller’s gains using three interval type-1 non-singleton type-2 fuzzy logic systems (IT2 NSFLS-1), one fuzzy system for each gain of the PID, being the first main contribution of this proposal. This assembly is named as hybrid IT2 NSFLS-1 PID. Each IT2 NSFLS-1 system requires two non-singleton input values each period of discrete time $$ \left( \varvec{k} \right) $$, (1) the error $$ \varvec{e}\left( \varvec{k} \right) $$ and its standard deviation $$ \varvec{\sigma e}\left( \varvec{k} \right) $$, and (2) the change of error $$ \Delta \varvec{e}\left( \varvec{k} \right) $$ and its standard deviation $$ \varvec{\sigma}\Delta \varvec{e}\left( \varvec{k} \right) $$, to calculate the corresponding adjustment $$ \Delta \varvec{KP}\left( \varvec{k} \right) $$, $$ \Delta \varvec{KI}\left( \varvec{k} \right) $$, and $$ \Delta \varvec{KD}\left( \varvec{k} \right) $$ for the PID controller’s gains $$ \varvec{K}_{\varvec{p}} \left( \varvec{k} \right) $$, $$ \varvec{K}_{\varvec{i}} \left( \varvec{k} \right) $$, and $$ \varvec{K}_{\varvec{d}} \left( \varvec{k} \right) $$. The second main contribution of this proposal is that the parameters of each IT2 NSFLS-1 system are tuned each period of discrete time $$ \left( \varvec{k} \right) $$ by the non-singleton backpropagation (BP) algorithm using the plant output error and its standard deviation, which are processed as non-singleton values together with its non-singleton partial derivatives with respect to each IT2 fuzzy system parameter. Then these updated gains are used by the PID controller to calculate the best control signal for the plant under control. The uncertainty and the mean value of the measurement are used to calculate the non-singleton error which is processed as (a) input and (b) as gradient vector by each of the three IT2 NSFLS-1 systems. Simulation results show that the proposed hybrid assembly presents the better performance than the next five benchmarking control systems (a) the classic Zeigler–Nichols PID controller, and (b) four hybrid assemblies using PID controller and fuzzy systems with fixed fuzzy rule bases (T1 SFLS, T1 NSFLS, IT2 SFLS, IT2 NSFLS-1). The proposed assembly produces the better performance in a shortest period of time and it maintains a stable behavior on the output of the second-order plant model subject to variations and noise.

A Modular Implementation Scheme for Nonsingleton Type-2 Fuzzy Logic Systems With Input Uncertainties

Using nonsingleton (NS) input fuzzifiers along with type-2 fuzzy systems to handle uncertainties in inputs has received some attention in recent years. However, the NS fuzzification schemes proposed so far have a limited impact because they are restricted to a particular kind of fuzzy sets only. This paper proposes a modular implementation scheme for NS type-2 fuzzy logic systems with input uncertainties. The proposed implementation scheme constitutes a generalized fuzzification which is independent of the forms/shapes of the fuzzy sets, i.e., both the NS fuzzifiers and the membership functions. To investigate the effectiveness of the proposed scheme, type-2 fuzzy logic controllers for three different applications, airplane altitude control, obstacle avoidance for a mobile robot, and a wall following robot, are developed. Additionally, for mapping NS fuzzifiers for real sensors, a sensory noise pattern recognition stage is also developed. Despite encountering various kinds of membership functions and input uncertainties along these three applications, the proposed scheme is able to successfully deal with all three applications. Moreover, the performance results in all three application setups show that NS fuzzification can improve the robustness of type-2 fuzzy logic systems against input uncertainties.

Finishing mill strip gage setup and control by interval type-1 non-singleton type-2 fuzzy logic systems

The setup and control of the finishing mill roll gap positions required to achieve the desired strip head thickness as measured by the finish mill exit X-ray gauge sensor is made by an intelligent controller based on an interval type-2 fuzzy logic system. The controller calculates the finishing mill stand screw positions required to achieve the strip finishing mill exit target thickness. The interval type-2 fuzzy head gage controller uses as inputs the transfer bar thickness, the width and the temperature at finishing mill entry, the strip target thickness, the width and the temperature at finishing mill exit, the stand work roll diameter, the stand work roll speed, the stand entry thickness, the stand exit thickness, the stand rolling force, and the %C of the strip. Taking into account that the measurements and inputs to the proposed system are modeled as type-1 non-singleton fuzzy numbers, we present the so called interval type-1 non-singleton type-2 fuzzy logic roll gap controller. As reported in the literature, interval type-2 fuzzy logic systems have greater non-linear approximation capacity than that of its type-1 counterpart and it has the advantage to develop more robust and reliable solutions than the latter. The experiments of these applications were carried out for three different types of coils, from a real hot strip mill. The results proved the feasibility of the developed system for roll gap control. Comparison against the mathematical based model shows that the proposed interval type-2 fuzzy logic system equalizes the performance in finishing mill stand screw positions setup and enhances the achieved strip thickness under the tested conditions characterized by high uncertainty levels.

Adaptive Non-singleton Type-2 Fuzzy Logic Systems: A Way Forward for Handling Numerical Uncertainties in Real World Applications

Real world environments are characterized by high levels of linguistic and numerical uncertainties. A Fuzzy Logic System (FLS) is recognized as an adequate methodology to handle the uncertainties and imprecision available in real world environments and applications. Since the invention of fuzzy logic, it has been applied with great success to numerous real world applications such as washing machines, food processors, battery chargers, electrical vehicles, and several other domestic and industrial appliances. The first generation of FLSs were type-1 FLSs in which type-1 fuzzy sets were employed. Later, it was found that using type-2 FLSs can enable the handling of higher levels of uncertainties. Recent works have shown that interval type-2 FLSs can outperform type-1 FLSs in the applications which encompass high uncertainty levels. However, the majority of interval type-2 FLSs handle the linguistic and input numerical uncertainties using singleton interval type-2 FLSs that mix the numerical and linguistic uncertainties to be handled only by the linguistic labels type-2 fuzzy sets. This ignores the fact that if input numerical uncertainties were present, they should affect the incoming inputs to the FLS. Even in the papers that employed non-singleton type-2 FLSs, the input signals were assumed to have a predefined shape (mostly Gaussian or triangular) which might not reflect the real uncertainty distribution which can vary with the associated measurement. In this paper, we will present a new approach which is based on an adaptive non-singleton interval type-2 FLS where the numerical uncertainties will be modeled and handled by non-singleton type-2 fuzzy inputs and the linguistic uncertainties will be handled by interval type-2 fuzzy sets to represent the antecedents’ linguistic labels. The non-singleton type-2 fuzzy inputs are dynamic and they are automatically generated from data and they do not assume a specific shape about the distribution associated with the given sensor. We will present several real world experiments using a real world robot which will show how the proposed type-2 non-singleton type-2 FLS will produce a superior performance to its singleton type-1 and type-2 counterparts when encountering high levels of uncertainties.

HYBRID LEARNING ALGORITHM FOR INTERVAL TYPE-2 FUZZY LOGIC SYSTEMS

A new learning methodology based on a hybrid algorithm for interval type-2 fuzzy logic systems (FLS) parameter estimation is presented. The new proposal consists of recursive least-squares (RLS) with back-propagation (BP) and square-root filter (REFIL) with BP combinations. A system with the proposed methodology was simulated to test its capability for output surface temperature prediction of a transfer bar at a hot strip mill (HSM) scale breaker (SB) entry zone for three different types of coils. The inputs of the system are the surface temperature of the transfer bar and the traveling time required to reach the SB entry zone. The inputs are modelled as interval singleton, type-1, or type-2 fuzzy sets depending on whether the noise was taken into account or not. The following combinations were simulated: (a) interval singleton type-2 FLS (type-2 SFLS), (b) interval type-1 non-singleton type-2 FLS (type-2 NSFLS-1) and (c) interval type-2 non-singleton type-2 FLS (type-2 NSFLS-2), where the bold face terms indicate the kind of input. Modelling results show the feasibility of the proposed methodology to be implemented into an intelligent process controller for the HSM industry.