Interval Type-2 Membership Functions Research Articles

Integrated Fault-Tolerant Control Design With Sampled-Output Measurements for Interval Type-2 Takagi-Sugeno Fuzzy Systems.

This article is concerned with the integrated design of fault estimation (FE) and fault-tolerant control (FTC) for uncertain nonlinear systems suffering from actuator faults and external disturbance. The uncertain nonlinear systems are characterized as the interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy model, and IT2 membership functions are employed to effectively handle uncertainties. A fuzzy observer, utilizing only sampled-output measurements, is applied to simultaneously estimate actuator faults and system states. Based on the estimation, the fault-tolerant controller is designed to ensure the system stability under a predefined H∞ performance. The sampling behavior complicates the system dynamics and makes the integrated FTC design more challenging. To confront this issue, the discontinuous Lyapunov functional technique is exploited to enhance stability results by considering the sampling characteristic, upon which FE and FTC units are co-designed in the linear matrix inequality (LMI) framework. To further relax stability criteria, the analysis process incorporates the bound information of membership functions through the membership-function-dependent (MFD) method. Additionally, the relationship of mismatched premise variables resulting from the sampling scheme is also taken into account. Moreover, considering the imperfect premise matching (IPM) framework, the proposed fault-tolerant controller provides greater flexibility in selecting the shapes of membership functions and number of fuzzy rules that can vary from the counterpart of the fuzzy system. Finally, the efficacy of the proposed FTC technique is validated through a detailed numerical example.

Stability analysis of interval type-2 sampled-data polynomial fuzzy-model-based control system with a switching control scheme

The stability of an interval type-2 (IT2) sampled-data (SD) polynomial fuzzy-model-based control system with a switching control scheme is studied in this paper. The uncertain nonlinear plant is depicted via an IT2 polynomial fuzzy model. To realize control, a switching IT2SD polynomial fuzzy controller is generated. This paper adopts a switching control scheme with a variable sampling period. The modeling domain consists of several sub-domains, and each sub-domain corresponds to a local IT2SD polynomial fuzzy controller. These local IT2SD polynomial fuzzy controllers form the switching IT2SD polynomial fuzzy controller. To aid in the stability analysis, this paper adopts a looped-functional-based technique. The imperfect premise matching concept is brought in to solve the mismatch dilemma caused by the SD control strategy and uncertainties. For decreasing the conservativeness, this paper takes into account the state information as well as the information of IT2 membership functions. The stability analysis is performed for each sub-domain, providing the potential for further relaxation. As polynomials exist in the stability conditions, this paper employs the sum-of-squares method for the stability investigation. The simulation outcomes confirm the efficacy of the proposed method.

Tracking Control for Nonlinear Systems With Actuator Saturation via Interval Type-2 T-S Fuzzy Framework.

In this work, the problem of tracking control for discrete-time nonlinear actuator-saturated systems via interval type-2 (IT2) T-S fuzzy framework is investigated. Improved on the (type-1) T-S fuzzy system, the IT2 T-S fuzzy system has a better capability for the expression of system uncertainty, and correspondingly, it will increase the difficulty of analysis, especially for the membership-functions-dependent (MFD) method. In addition, in this case, the control input nonlinearity caused by actuator saturation will complicate the stability analysis of the systems. We make an attempt to address the challenges that the information of membership functions (MFs) is underutilized or not utilized, by developing an MFD analysis approach, which allows the enhancement of design flexibility of IT2 fuzzy controller and effectiveness of lessening the conservativeness of the analysis result. The piecewise MFs which are formed by connecting the sample point on or close to the original IT2 MFs are utilized to approximate the original IT2 MFs, and the error between the piecewise MFs and the original upper and lower MFs is taken into account in the stability analysis. To acquire the linear matrix inequality-based (LMI-based) constraint, the actuator saturation is converted to a sector nonlinear issue. performance is considered to limit the difference between the reference system and the control saturated system. Examples are presented to illustrate the validity of the results.

Optimization for Interval Type-2 Polynomial Fuzzy Systems: A Deep Reinforcement Learning Approach

It is known that the interval type-2 (IT2) fuzzy controllers are superior compared to their type-1 counterparts in terms of robustness, flexibility, etc. However, how to conduct the type reduction optimally with the consideration of system stability under the fuzzy-model-based (FMB) control framework is still an open problem. To address this issue, we present a new approach through the membership-function-dependent (MFD) and deep reinforcement learning (DRL) approaches. In the proposed approach, the reduction of IT2 membership functions of the fuzzy controller is completing during optimizing the control performance. Another fundamental issue is that the stability conditions must hold subject to different type-reduction methods. It is tedious and impractical to resolve the stability conditions according to different type-reduction methods, which could lead to infinite possibility. It is more practical to guarantee the holding of stability conditions during type-reduction rather than resolving the stability conditions, the MFD approach is proposed with the imperfect premise matching (IPM) concept. Thanks to the unique merit of the MFD approach, the stability conditions according to all the different embedded type-1 membership functions within the footprint of uncertainty (FOU) are guaranteed to be valid. During the control processes, the state transitions associated with properly engineered cost/reward function can be used to approximately calculate the deterministic policy gradient to optimize the acting policy and then to improve the control performance through determining the grade of IT2 membership functions of the fuzzy controller. The detailed simulation example is provided to verify the merits of the proposed approach.

Toward Effective Uncertainty Management in Decision-Making Models Based on Type-2 Fuzzy TOPSIS

Over the past century, there has been a dramatic increasing interest in the multi-criteria group decision-making (MCGDM) technique, with a considerable amount of studies published regarding it. One of the well-known approaches in the MCGDM paradigm is Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). The integration of the TOPSIS method with fuzzy set theory has proven to be successful in various applications. Recently, a wide array of publications has proposed implementing a type-2 fuzzy set with TOPSIS. However, the additional degree of uncertainty represented by type 2 has largely been ignored, especially in a few specific mathematical operations in the model. We propose constructing interval type-2 fuzzy membership functions (IT2 MFs) using interval-based data gathered from a survey, where this is used to generate a new scale to represent ratings for each alternative. This procedure utilized all information gathered from decision makers. In addition, we present a complete algorithm for TOPSIS based on IT2 fuzzy sets (IT2 FSs) which preserve the interval-based form output. The output in the form of intervals offers decision makers (DMs) with more detailed information, enabling them to make more nuanced decisions. This can include cautious decisions when intervals are wider and overlapping. Although understanding the exact meaning of these intervals and their widths in a decision-making context is challenging, this paper introduces a systematic method for connecting input uncertainty to output uncertainty in the TOPSIS technique. This approach establishes a solid foundation for future research. Thus far, no other researchers have suggested a data-driven method that combines TOPSIS with fuzzification and provides intervals as the final output.

Improved interval type-2 fuzzy K-means clustering based on adaptive iterative center with new defuzzification method

Interval Type-2 Fuzzy K-means (IT2FKM) is an efficient clustering algorithm, which mainly focuses on further describes the uncertainty of the data sets by introducing interval type-2 fuzzy membership function. Since the estimation center obtained by two membership functions is an interval type-1 fuzzy set, it is necessary to use the defuzzification method to obtain a crisp cluster center. The existing IT2FKM algorithms adopt the generalized centroid method (GC) in the process of type-reduction and defuzzification for iterative clustering center. But unfortunately, this method simply uses a single numerical value 0.5 as the type-reduction coefficient to balance and synthesize for the left value and the right value of an interval cluster center, while ignoring the distribution of the clusters. This defect makes IT2FKM fail to update the optimal cluster center when facing with imbalanced clusters, which affect the accuracy in some cases. To address this failing, in this paper, some adverse effects caused by generalized centroid method when facing with imbalanced clusters are analyzed in detail. Thereafter, an improved type-reduction method of iterative cluster centers for IT2FKM clustering is proposed considering the imbalanced clustering distributions. On the basis of which, an enhanced IT2FKM algorithm is further developed. The enhanced IT2FKM algorithm has an average classification accuracy of 97.5% when faced with four datasets with different degrees imbalance, which is about 3.25% higher than other clustering algorithms. The effectiveness and superiority of the proposed algorithm are further demonstrated by some experimental analysis of UCI standard datasets.

Interval type-2 fuzzy neural networks with asymmetric MFs based on the twice optimization algorithm for nonlinear system identification

This paper proposes a novel algorithm twice optimization for interval type-2 fuzzy neural networks with asymmetric membership functions (TOIT2FNN-AMF), for nonlinear system identification problems. The proposed TOIT2FNN-AMF uses an asymmetric Gaussian interval type-2 membership function to enhance the network’s ability to describe and solve nonlinear and uncertain problems. The twice optimization algorithm consists of structure learning and parameter learning. Firstly, this paper proposes a multi-strategy adaptive differential evolution (MSADE) algorithm as the first optimization algorithm, which is used to determine the structure and the initial values of the parameters of the TOIT2FNN-AMF. It applies the root mean square error (RMSE) of the TOIT2FNN-AMF as the fitness function to determine the structure (number of rules) and initial parameters of the IT2FNN by searching for the RMSE values under different structures. When the fitness value reaches the minimum, that is, the RMSE value of the TOIT2FNN-AMF, the corresponding number will become the optimal one of fuzzy rules of the TOIT2FNN-AMF. Then, the second optimization algorithm of the TOIT2FNN-AMF turns into a hybrid optimization algorithm composed of an adaptive moment estimation (Adam) algorithm and recursive least squares (RLS) algorithm. Adam is used to optimize the antecedent parameters of TOIT2FNN-AMF rules, so as to maintain rapid convergence without generating oscillation during the training process; RLS is used to optimize the consequent parameters of TOIT2FNN-AMF rules, so that the network parameters can be optimized rapidly. In this way, the problems of excessive parameters to be adjusted and excessive slow convergence of the network can be solved. Finally, this paper evaluates the proposed TOIT2FNN-AMF by testing on problems of nonlinear system identification and chaotic time-series prediction. The simulation results are compared with those of similar methods in the existing literatures, which demonstrates that the proposed TOIT2FNN-AMF model yields a lower RMSE value and a simpler network structure than the other type-2 fuzzy neural networks (T2FNNs).

Reduced-Order Extended Dissipative Filtering for Nonlinear Systems With Sensor Saturation via Interval Type-2 Fuzzy Model

The system nonlinearity, sensor saturation, and the uncertainty will hamper the analysis and affect the control performance. Filtering is a signal processing method which facilitates the system analysis and synthesis by signal estimation or noise suppression. To achieve generalized filtering problem for nonlinear systems with sensor saturations with lower computational burden, this article addresses the reduced-order extended dissipative filter design for nonlinear sensor-saturated system which is modeled by interval type-2 (IT2) T–S fuzzy system. For IT2 T–S fuzzy systems, the main challenge exists in the acquisition of the information in IT2 membership functions (MFs) for analysis and design. A membership-function-dependent (MFD) method is applied to capture the information of the MFs for reducing the conservativeness introduced by MFs not involved in the analysis. An extended dissipative filtering method, under imperfect premise matching (IPM) concept that the membership functions of the filter are different from those of the model, is proposed for sensor-saturated IT2 fuzzy systems. The proposed method has a high flexibility in parameters adjustment of both the filter and the design condition, including extended dissipative matrices, approximation MFs, and sensor saturation degree, one can freely choose the parameters according to the required performance of the fuzzy filter. A numerical example is given to demonstrate the effectiveness of the results.

Stabilization Analysis and Impulsive Controller Design for Positive Interval Type-2 Polynomial Fuzzy Systems

This article investigates the polynomial fuzzy impulsive control problem for positive nonlinear systems subject to parameter uncertainties. The positive nonlinear systems are represented as positive interval type-2 (IT2) polynomial fuzzy-model-based systems, while the parameter uncertainties are captured by the IT2 membership functions (MFs). Considering that the controller being designed needs to be implementable and practical, in addition to ensuring the positivity and stability of the system, there are still some potential requirements for the controller, such as lower implementation cost and control cost. To reduce the implementation cost, the IT2 polynomial fuzzy impulsive controller is designed under the imperfect premise matching concept that the premise MFs and the number of rules of fuzzy controller are different from those of fuzzy model. To reduce the control cost, the impulsive controller that can tolerate larger impulse control interval (ICI) is designed by the following two novel methods. The first one is the proposed impulse-time-dependent discretized polynomial copositive Lyapunov function whose Lyapunov variable is designed as the interpolation polynomial function of time. The second one is the advanced IT2 membership-function-dependent (IT2MFD) analysis method which is improved by a novel footprint of uncertainty partitioning method proposed by this paper, so that this advanced IT2MFD method can alleviate the negative impact of parameter uncertainties on the ICI. Finally, a simulation example using lipoprotein metabolism and potassium ion transfer model as the nonlinear system to be controlled is used to verify the effectiveness of the proposed impulsive control methods.

Design and application of Nagar-Bardini structure-based interval type-2 fuzzy logic systems optimized with the combination of backpropagation algorithms and recursive least square algorithms

This paper designs a kind of Nagar-Bardini structure based on interval type-2 (IT2) fuzzy logic systems for permanent magnetic drive uncertain parameter forecasting. For each fuzzy rule, the consequent, antecedent and input measurement of IT2 membership functions (MFs) are selected as the Gaussian type-2 MFs with uncertain standard deviations. Backpropagation algorithms and recursive least square algorithms are used to optimize the antecedent, input measurement and consequent parameters of fuzzy logic system (FLS) forecasters, respectively. Compared with the corresponding singleton and non-singleton type-1 (T1) FLSs, two Monte Carlo simulation instances on the basis of data of PMD process show the effectiveness and superiority of non-singleton IT2 FLSs according to the convergence analysis.

Altitude control of a quadcopter using interval type-2 fuzzy controller with dynamic footprint of uncertainty

Footprint of uncertainty (FOU) characteristics of interval type-2 membership functions (IT2-MFs) play an important role in the performance and robustness of interval type-2 fuzzy controllers (IT2-FCs). In literature, fixed FOU structures are used in almost all IT2-FC designs. In this study, an IT2-FC with dynamic FOU is proposed to provide high performance and robustness in the altitude control of a quadcopter. A proportional–derivative (PD) type FC with Takagi–Sugeno rule structure is used in the design procedure of the proposed controller. Additionally, input variables (error and derivative of error) are defined by using triangular IT2-MFs. To provide a dynamic FOU structure, the height of the lower MF (LMF) of each interval type-2 fuzzy set is defined as a function of the system error. In this way, FOU levels of IT2-MFs are adjusted dynamically since the height value of the LMF directly determines the FOU level of the IT2-MF. To evaluate the effectiveness of the proposed controller, comparison studies are performed under different conditions, including unmodeled measurement noises, an external disturbance, and a system parameter uncertainty. A classical PD, a type-1 fuzzy PD, and an interval type-2 fuzzy PD with fixed FOU controllers are used in the comparisons. The comparison results demonstrate that the proposed interval type-2 fuzzy PD controller with dynamic FOU exhibits better performance and more robustness than the other controllers.

ℓ₂-ℓ∞ Control of Discrete-Time State-Delay Interval Type-2 Fuzzy Systems via Dynamic Output Feedback.

This article investigates the design of the l2-l∞ dynamic output-feedback (DOF) controller for interval type-2 (IT2) T-S fuzzy systems with state delay. For nonlinear systems, the IT2 fuzzy model is an efficient modeling method which can better express uncertainties than the (type-1) fuzzy model. In addition, state delay is also a general factor that affects system performance. After analyzing the stability of the system, based on convex linearization and the projection theorem, this article proposes a delay-dependent output-feedback controller design method. The IT2 membership functions (MFs) of the fuzzy controller are chosen to be different from those of the model so as to increase the freedom of controller selection. A membership-function-dependent (MFD) method based on the staircase MFs is applied to relax the stability analysis results. Finally, a numerical simulation example is given to illustrate the effectiveness of the results.

Observer-based interval type-2 fuzzy PID controller for PEMFC air feeding system using novel hybrid neural network algorithm-differential evolution optimizer

In this paper, a novel hybrid Neural Network Algorithm-Differential Evolution (NNA-DE) optimizer which integrates both NNA and DE is proposed. The proposed hybrid NNA-DE optimizer demonstrates better performance compared to the standard NNA and the other state-of-the-art optimization algorithms. The proposed hybrid NNA-DE algorithm is then employed for optimizing an observer-based interval type-2 fuzzy PID (OB-IT2FPID) controller applied to the proton exchange membrane fuel cell (PEMFC) air feeding system. The whole design parameters of the OB-IT2FPID controller including the scaling factors, interval type-2 membership function parameters and the footprint-of-uncertainty (FOU) are optimized using the proposed hybrid NNA-DE algorithm. The results show that the proposed NNA-DE optimized OB-IT2FPID controller achieves better performance in terms of set-point tracking, disturbance rejection and the time-domain performance indices. The robustness of the proposed NNA-DE optimized OB-IT2FPID controller against parametric uncertainty in the PEMFC air-feeding system is tested. The proposed controller demonstrated better robustness against parameter uncertainty in the system. Processor-in-the-Loop (PIL) approach is adopted to validate the performance of the proposed controller on embedded control hardware.

Generating Type 2 Trapezoidal Fuzzy Membership Function Using Genetic Tuning

Fuzzy inference system (FIS) is a process of fuzzy logic reasoning to produce the output based on fuzzified inputs. The system starts with identifying input from data, applying the fuzziness to input using membership functions (MF), generating fuzzy rules for the fuzzy sets and obtaining the output. There are several types of input MFs which can be introduced in FIS, commonly chosen based on the type of real data, sensitivity of certain rule implied and computational limits. This paper focuses on the construction of interval type 2 (IT2) trapezoidal shape MF from fuzzy C Means (FCM) that is used for fuzzification process of mamdani FIS. In the process, upper MF (UMF) and lower MF (LMF) of the MF need to be identified to get the range of the footprint of uncertainty (FOU). This paper proposes Genetic tuning process, which is a part of genetic algorithm (GA), to adjust parameters in order to improve the behavior of existing system, especially to enhance the accuracy of the system model. This novel process is a hybrid approach which produces Genetic Fuzzy System (GFS) that helps to enhance fuzzy classification problems and performance. The approach provides a new method for the construction and tuning process of the IT2 MF, based on the FCM outcomes. The result is compared to Gaussian shape IT2 MF and trapezoid IT2 MF generated by the classic GA method. It is shown that the proposed approach is able to outperform the mentioned benchmarked approaches. The work implies a wider range of IT2 MF types, constructed based on FCM outcomes, and an optimum generation of the FOU so that it can be implemented in practical applications such as prediction, analytics and rule-based solutions.

Finite-time extended dissipativity control for interval type-2 fuzzy systems with resilient memory sampled-data controller

In this work, the finite-time extended dissipativity of the interval type-2 (IT2) fuzzy systems with probabilistic time-varying delay is discussed via resilient memory sampled-data control. To enable the stability analysis and control combination, an IT2 fuzzy model is employed to represent the dynamics of nonlinear systems of which the parameter uncertainties are taken by IT2 membership functions distinguish by the lower and upper membership functions. The main objective of this paper is to design a resilient memory sampled-data controller such that the resulting closed-loop system is finite-time bounded and satisfies extended dissipative performance. Moreover, the solvability of the derived conditions not only depends on the size of the delay but also on the probabilistic distribution of the delay taking values in some interval, thus probabilistic delay protocol is encountered in the IT2 fuzzy model. By employing suitable Lyapunov-Krasovskii functional (LKF) along with Wirtinger-based inequality, a set of sufficient conditions ensuring the finite-time extended dissipative performance for IT2 fuzzy systems are derived in terms of linear matrix inequalities (LMIs). Finally, two numerical simulations are presented to reveal the effectiveness of the developed technique.

Interval type-2 fractional fuzzy inference systems: Towards an evolution in fuzzy inference systems

Following type-1 fractional fuzzy inference systems presented recently as the new generation of fuzzy inference systems, interval type-2 fractional fuzzy inference systems (IT2FFISs) as a leap further ahead in the evolution of fuzzy inference systems (FISs) are introduced in this article. The IT2FFISs, which are outlined in this article, add to the armamentarium of FISs some particular concepts such as interval type-2 fractional membership functions, type-2 fractional translation rule, type-2 fracture index, the concept of switching, the entanglement, the degeneracy concept, and so forth. An IT2FFIS exploits not only the tolerance for the uncertainty in the interpretation of the meaning of a word, but also the relevance between the quality and quantity levels of the given information to infer an answer to an inference query. The IT2FFISs make an increase in machine intelligence quotient possible by an increase in the range of FISs order rather than their type. Moreover, the synergy of the concepts coming with various modes of IT2FFISs such as the aggressive mode opens a gate to a space of fuzzy systems outputs which used to be indiscoverable. Furthermore, it is demonstrated that as the type-2 fracture index approaches zero, the space of IT2FFISs outputs contracts and eventually it coincides the space of IT2FISs output when the fracture index is equal to zero. It is also proved that, provided a particular order of the IT2FFIS is taken into account, independent of the problem in question, a typical IT2FIS never leads to results which are more satisfactory than those obtained by the IT2FFIS corresponding to the typical IT2FIS.

A dynamic fuzzification approach for interval type-2 membership function development: case study for QoS planning

A dynamic fuzzification approach for interval type-2 membership function development: case study for QoS planning

Membership-Function-Dependent Control Design and Stability Analysis of Interval Type-2 Sampled-Data Fuzzy-Model-Based Control System

This article investigates the design and stability analysis of the interval type-2 (IT2) sampled-data (SD) fuzzy-model-based (FMB) control system with the optimal guaranteed cost performance. An IT2 Takagi–Sugeno (T-S) fuzzy model is applied to describe the dynamics of the nonlinear systems where the parameter uncertainties are captured by the lower and upper membership functions. To conduct the stability analysis for the SD FMB control system, a looped-functional approach taking the advantage of the information about the sampling periods is employed. Because of the SD control strategy, the state will be sampled at each sampling instant and the control signal generated by the IT2SD fuzzy controller will be kept by the zero-order holder during the sampling period, which will result in mismatched membership grades between IT2 T-S fuzzy model and IT2SD fuzzy controller that leads to the complexity in carrying out stability analysis. Thanks to the imperfect premise matching concept, which allows the difference on the number of rules and the premise membership functions between model and controller, the design of the IT2SD fuzzy controller can be more flexible. To further relax the stability conditions and minimize the upper bound of the guaranteed cost index, the membership-function-dependent stability analysis approach which can make use of the features of the IT2 membership functions is adopted. The performance of the control system can also be adjusted through the choice of the weighting matrices in the cost function. The stability conditions building on the Lyapunov stability theory and the performance conditions building on the concept of the guaranteed cost control in the shape of linear matrix inequalities are established to assure the system stability and acquire the optimal guaranteed cost performance. The proposed IT2SDFMB control design is tested on the inverted pendulum system and the simulation results verify the effectiveness of the proposed approach.

A Fast Inference and Type-Reduction Process for Constrained Interval Type-2 Fuzzy Systems

Constrained interval type-2 (CIT2) fuzzy sets have been introduced to preserve interpretability when moving from type-1 to interval type-2 (IT2) membership functions. Although they can be used to produce type-2 fuzzy systems with enhanced explainability, so far, the latter comes at the expense of high computational cost. Specifically, the exhaustive type-reduction method for CIT2 Mamdani systems has been shown to be too slow to be used in practical applications and even the current approximation procedure is much slower than modern type-reduction algorithms used for IT2 fuzzy sets. In this article, a novel type-reduction procedure for CIT2 sets is presented, based on the concept of switch indices. The algorithm is applied on a real-world classification problem and compared to other type-reduction approaches used in IT2 and CIT2 systems. In the case studies presented, the new algorithm is significantly faster than the exhaustive and sampling CIT2 approaches while keeping the high level of interpretability of the type-reduction operation that characterizes CIT2 fuzzy sets.

Event-Triggered Prescribed-Time Fuzzy Control for Space Teleoperation Systems Subject to Multiple Constraints and Uncertainties

Limited by the operation time window and working space, space teleoperation tasks need to be completed within an expected time while ensuring that the end effector meets the physical constraints. Meanwhile, the interaction with unknown environments would cause uncertainty in the closed-loop system, which brings great challenges to the control design. To solve the above problems, the control performance issue for a class of space teleoperation systems subject to multiple constraints and interaction uncertainties is investigated in this article. The force interaction with the human operator/space environment is represented by interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy systems, where the uncertain equivalent mass and damping parameters can be effectively described and captured by IT2 membership functions. In order to reduce the communication burden and satisfy the constraints of settling time, transient-state performance and operating space, a time-varying threshold event-triggered control scheme together with exponential-type Lyapunov function is developed for the first time. We show that, with the proposed controller, the synchronization tracking errors are guaranteed to converge to a user-defined residual set within preassigned settling time, and never exceed the prescribed range despite unknown control direction and actuator faults, which solves the long-standing constraint issue with more flexibility due to the fact that the related constraints can be arbitrarily specific within the physically available range. Moreover, the convergence set is only dependent on fewer user-defined parameters rather than approximation errors, which provides an effective analysis technique to deal with the difficulty that the convergence accuracy is difficult to calculate quantitatively in the presence of unknown disturbance. Detailed simulation results are provided to show the effectiveness and merit of the proposed control strategy.