Membership-function-dependent Research Articles

Interval Type-2 Fuzzy-Model-Based Sampled-Data Control of an AUV Depth System with Input Saturation

This paper proposes a sampled-data fuzzy controller design technique for an autonomous underwater vehicle (AUV) depth system represented by an interval type-2 (IT-2) fuzzy model, considering input saturation. In the Takagi–Sugeno (T–S) fuzzy model of an AUV depth system, surge velocity is chosen as a premise variable. To address the associated uncertainty with this variable, we employ the IT-2 fuzzy modeling technique. Also, the controller proposed in this paper utilizes time-varying gains, ensuring superior exponential stability compared with traditional fixed gain approaches. Furthermore, a membership function-dependent (MFD) H∞ criterion is employed to enhance robustness for each subsystem individually. Taking into account the mentioned aspects, the controller design condition is derived in the form of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed method is validated through simulation examples.

Fuzzy membership-function-dependent design of aperiodic sample-data control scheme for nonlinear PMSG-based WECS with quantization measurements via refined looped Lyapunov functional

This manuscript investigates the fuzzy memory sampled-data control (MSDC) scheme for the nonlinear permanent magnet synchronous generator (PMSG)-based wind energy conversion system (WECS) with quantization measurements. First, due to the complex nonlinearity, a Takagi-Sugeno (T-S) fuzzy modeling approach is successfully applied to the dynamics of nonlinear PMSG-based WECS. Second, by using the information of both quantized aperiodic sampling pattern and signal transmission delay, an H∞-based fuzzy quantized controller is put forward to ensure the stable performance as well as disturbance suppression index of the considered PMSG model by utilizing the refined looped Lyapunov functional (RLLF) approach. This RLLF perfectly utilizes more realistic information about the time derivative of the fuzzy membership function and actual sampling pattern. Then, two new integral inequalities are introduced to approximate the quadratic integral terms arising from the derivative of RLLF. Next, we establish some less conservative stabilization results in terms of linear matrix inequalities (LMIs) using the constructed RLLF, along with the proposed integral inequalities and fuzzy membership function-dependent (MFD) switching ideas. Finally, the simulation studies of PMGS-based WECS are validated numerically, and then a comparative example is provided to emphasize the superiority and relevance of the suggested approach.

Security sliding mode control for T-S fuzzy systems under attack probability-dependent dynamic round-robin protocol

In this paper, the security control problem for T-S fuzzy systems is investigated through the sliding mode control (SMC) strategy. Consider deception attacks occurring randomly, a multi-modes round-robin protocol is proposed on the sensor-to-controller channel. Under this protocol, the number of transmitted nodes can be dynamically adjusted according to attack probability and system state so as to balance the relationship between system performance and communication transmission. The fuzzy sliding mode controller depending on the scheduling signal is constructed, whose membership functions (MFs) are different from the ones of the controlled system. In addition, by using the membership-function-dependent (MFD) stability analysis method, appropriate sufficient conditions are derived to ensure both the stochastic stability and the reachability. Finally, simulation results are presented to prove the proposed SMC strategy.

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.

Relaxed Stabilization of Event-Triggered Interval Type-2 T–S Fuzzy Positive Systems With Stochastic Actuator Failure

This paper studies the membership-function-dependent (MFD) stability and positivity analysis of positive Interval Type-2 (IT2) Takagi-Sugeno (T-S) fuzzy fault-tolerant control systems under the event-triggered control framework. Considering the fact that parameter uncertainties may exist in nonlinear plants, the IT2 T-S fuzzy model as an effective mathematical tool is employed to approximately express a complex positive system with nonlinearities and uncertainties. In order to save communication resources and improve the security of system operation, an event-triggered fault-tolerant control strategy is then designed based on the Imperfectly Premise Matching (IPM) concept. In addition, for capturing positivity characteristic and matching with the linear co-positive Lyapunov function, a novel event-triggered condition in the 1-norm rather than in the 2-norm is developed to control the positive IT2 T-S fuzzy systems. Thereinto, the non-convex problem as the first obstacle is required to be cleared since feasible solutions usually cannot be obtained based on the non-convex conditions. For dealing with this problem, a valid approach is presented to conduct convexification with the help of a user-chosen constant vector. Besides, because of event-triggered action, the inherent mismatched issue of the premise variables of the fuzzy model and the ones of the fuzzy controller is encountered. Fortunately, the advanced MFD analysis, which is capable of bringing the information contained in the membership functions (MFs) into the stability and positivity conditions, provides an effective strategy to remove this barrier. Consequently, relaxed results which can accomplish the stability and positivity for wider positive systems are developed and displayed in sum-of-squares (SOS) form. Detailed simulation results are given to clarify the effectiveness of the designed event-triggered fault-tolerant control method.

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.

Interval type-2 polynomial fuzzy fault detection scheme with a multi-order homogeneous polynomial Lyapunov functions considering unmeasurable premise variables

A polynomial fuzzy fault detection scheme for sampled-output-measurements-based interval type-2 (IT2) polynomial-fuzzy-model-based (PFMB) systems is investigated in this paper, where the uncertainties in the premise variables (PVs) and membership functions (MFs) are described by IT2 fuzzy sets. Fully or partially unmeasurable PVs cause the parameter matrices of the polynomial fuzzy fault detection observer (PFFDO) to rely on the estimated states and the corresponding mismatching problems are further considered. Lyapunov stability theory is carried out with a novel multi-order homogenous polynomial Lyapunov functions (MHPLF) to introduce more information of the states when eliminating the partial derivatives, and the time-delays introduced by sampled-output measurements are handled by L-K functions. Unlike the membership-function-independent (MFI) approaches, the membership-function-dependent (MFD) approaches carry the information of the MFs for the relaxation of the stability constraints. Corresponding stable constraints in sum-of-squares (SOS) form are given to hold the asymptotic stability of the fault detection system with H∞ performance γ. A numerical example with many cases illustrates the effectiveness of the proposed techniques in uncertainty handling and conservativeness reduction, while an inverted pendulum example verifies the feasibility of the method on physical systems.

New Relaxed Static Output Feedback Stabilization of T–S Fuzzy Systems with Time-Varying Delays

This paper investigates the imperfect premise matching-based (IPMB) static output feedback (SOF) controller for T–S fuzzy systems with time-varying delays. Firstly, by employing integral inequality techniques, the membership-function-independent fuzzy SOF design methods in terms of LMIs are presented based on imperfect premise matching (IPM) strategy. The obtained stabilization conditions do not contain equality constraints, and the output matrices do not have rank constraints. Secondly, some suitable relaxation variables are employed, and more relaxed membership-function-dependent (MFD) stabilization conditions are obtained by considering the local boundary information of membership functions (MFs). Finally, two simulation examples are given to show the progressiveness of the proposed methods in this paper.

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.

Fault estimation and fault tolerant control for interval type-2 Takagi–Sugeno fuzzy systems via membership-function-dependent approach

This paper investigates the fault estimation (FE)-based fault tolerant control (FTC) technique to achieve the desired control performance for the nonlinear systems suffering from uncertainties, external disturbance and actuator faults using the interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy model. In this work, an IT2 fuzzy observer is built to simultaneously estimate both the system states and actuator faults, upon which a fault tolerant controller is proposed to guarantee the asymptotical stability of the closed-loop system with a prescribed H_{infty } performance level. Considering the bidirectional robustness interactions between the observer and FTC system, an integrated design technique is developed to address observer and FTC units together in one step to realize the required robustness within the whole closed-loop FTC system. By utilizing Lyapunov stability theory combined with the matrix inequality convexification techniques, a membership-function-dependent (MFD) FTC strategy is proposed where the information of membership functions is taken into account in the analysis for relaxation of the stability conditions. Additionally, to offer greater design flexibility and lower implementation cost to the fault tolerant controller, the imperfect premise matching (IPM) scheme is adopted, such that the premise membership functions of the fault tolerant controller can be chosen differently from those of IT2 fuzzy model. Finally, simulation results are provided to validate the effectiveness of the proposed FTC strategy.

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.

An application of interval type-2 fuzzy model based control system for generic aircraft

This paper presents a design application of Interval Type-2 (IT2) Takagi–Sugeno (T–S)​ Fuzzy Model Based (FMB) control system for generic aircraft. The IT2 T–S FMB flight control system consists of an IT2 T–S fuzzy model and a fuzzy controller connected in a closed-loop. The IT2 T–S​ fuzzy model is obtained by linearizing the nonlinear aircraft dynamics about various representative points (equilibrium points) of the flight envelope with some fuzzy rules. The aircraft flight envelope parameters, i.e., operating altitude and aircraft speed are characterized as premise parameters and elements of stability and control derivative matrix are identified as consequent parameters of fuzzy model. Longitudinal dynamics of X-29A research aircraft is selected for design of IT2 T–S FMB control system. To achieve the optimum design flexibility, an imperfectly matched premise and membership function dependent (MFD) stability analysis is considered. In MFD stability conditions, the information of flight envelope parameters is included to capture the nonlinearity pertaining to variation in flight conditions. The closed-loop response of IT2 T–S FMB controller is presented at trim equilibrium points by taking four initial angle of attack flight conditions. The performance analysis of designed controller is also presented and discussed in comparison with Fuzzy LQR and LQR based optimal controllers. The simulation results reveal that proposed IT2 T–S FMB controller not only stabilizes the aircraft dynamics but also provides improved transient performance as compared with Fuzzy LQR and LQR based optimal controllers. This demonstrates the utility of IT2 T–S FMB control system for aircraft/ UAV’s related application.

Robust Stability and Stabilization Conditions for Nonlinear Networked Control Systems With Network-Induced Delay via T–S Fuzzy Model

Robust stability and stabilization problems for a class of discrete-time nonlinear systems are studied in this article. The control signal is transmitted to the nonlinear systems through a lossy communication channel in which time-varying network-induced delay occurs. A Takagi–Sugeno (T–S) fuzzy model is used to represent the nonlinear systems. Using membership function boundary information, a novel membership-function-dependent (MFD) analysis approach is presented. Unlike most previous results, the proposed MFD analysis approach does not need to increase the number of slack variables to reduce conservatism. Combining auxiliary matrices and this approach, we propose some new fuzzy Lyapunov–Krasovskii functionals (FLKFs) that do not require the restrictions of positive definiteness imposed on Lyapunov matrices. Using the FLKF and MFD analysis approach, a sufficient condition in the form of linear matrix inequalities (LMIs) is developed to guarantee the stability of T–S fuzzy time-varying delay systems. Based on the results for stability, less conservative stabilization conditions are also derived. Finally, numerical examples show that less conservative results can be obtained by the proposed approach. The inverted pendulum system with state delay or network-induced delay is also studied to verify the effectiveness of the suggested approach.

Sliding-Mode Control of T-S Fuzzy Systems Under Weighted Try-Once-Discard Protocol.

This article investigates the sliding-mode control (SMC) problem for a class of T-S fuzzy systems with communication constraints. The weighted try-once-discard (WTOD) protocol is utilized to arrange the transmission order of actuator nodes, under which only one node is permitted to obtain access to the communication network at any time. A key issue is how to integrate the WTOD protocol into the design of the SMC system. To this end, an update rule is proposed for the actuators to obtain the actual actuator signals. And then, a membership-function-dependent (MFD) sliding surface and a token-dependent sliding-mode controller are designed. By taking the information of membership functions into stability and reachability analysis, MFD sufficient conditions are derived such that both the asymptotic stability of the closed-loop fuzzy systems and the reachability of the specified sliding surface can be guaranteed. Moreover, the cone complementary linearization (CCL) algorithm is employed to convert nonlinear inequalities into a minimization problem with linear constraints. Finally, a numerical example is utilized to illustrate the proposed SMC design approach.

Analysis and Design of Interval Type-2 Polynomial-Fuzzy-Model-Based Networked Tracking Control Systems

Highly nonlinear systems exist in many real-world applications. The control design of such systems is challenging even for existing advanced control theory. Besides, when there is uncertainty in the nonlinear system and networked control strategy is considered, the problem will become even more complicated. To confront the problem, for the first time, this article provides a solution for the tracking control design of the nonlinear networked control systems (NCSs) subject to uncertainty under the polynomial event-triggered mechanism (PETM). In the proposed tracking control scheme, the nonlinearity in the NCSs is effectively represented by a polynomial fuzzy model while the uncertainty is handled by the interval type-2 (IT2) fuzzy sets. The tracking control objective is to properly design the event-triggered IT2 polynomial controller, which drives the states of the plant to track those of the stable reference system under the PETM. Different from the existing works in the literature, the event-triggering condition can be varied according to the state to improve the flexibility and capacity of event-triggered mechanism. Furthermore, in some works reported in the literature, the perfect match of premise variables is assumed. However, this assumption is extremely difficult to be fulfilled in the event-triggered control applications. To address the long-standing challenge of an intrinsic mismatch issue of the premise variables due to event-triggering mechanism, in the proposed analysis, the imperfect premise matching (IPM) concept is adopted along with the membership-function-dependent (MFD) approach to facilitate the stability analysis and control synthesis. The stability conditions are summarized as sum-of-squares (SOS). The H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> index is utilized to evaluate the tracking control performance, and the performance can be improved through minimizing the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> index. A detailed simulation example is provided to verify the effectiveness of the proposed method.

Output Feedback and Stability Analysis of Positive Polynomial Fuzzy Systems

This article investigates the polynomial fuzzy output feedback (PFOF) control synthesis problem for positive polynomial fuzzy systems. When the states of a positive system cannot be fully obtained, the output feedback control strategy is a good method to stabilize the control system. From fuzzy control point of view, we employ the polynomial fuzzy model rather than the Takagi–Sugeno fuzzy model, and this kind of models can express a wider range of nonlinear positive systems. However, the polynomials in the system matrices and the feedback control gain matrices will make the nonconvex stability conditions more difficult to deal with. Thereby, in order to crack the hard nut, a nonzero transformation vector is introduced in this article to deal with the nonconvex problem skillfully. Furthermore, the imperfect premise matching technique is taken into account so that the implement of the controller is more simple and money-saving. In addition, the augmented dynamic of the positive polynomial fuzzy-model-based (PPFMB) control system is investigated to facilitate the stability and positivity analysis, and the basic conditions in the light of sum of squares (SOSs) are derived. Besides, the advanced membership function dependent (MFD) technique is used so that a great of useful information of MFs is extracted to improve the relaxation of the results. Finally, the effectiveness of the theoretical findings is illustrated by a simulation example.

Synchronization Analysis for Nonlinear Complex Networks With Reaction-Diffusion Terms Using Fuzzy-Model-Based Approach

The synchronization of a class variable-coefficient complex networks with reaction-diffusion terms is investigated. The objective of this article is to synthesize a state-feedback controller, which is designed based on a Takagi-Sugeno (T-S) fuzzy model of the variable-coefficient complex network with reaction-diffusion terms such that the closed-loop system is synchronized. With the support of Green formula and some matrix inequality techniques, and utilizing the upper and lower bounds of the membership functions, membership-function-dependent (MFD) synchronization conditions guaranteeing the system's synchronization are obtained in the form of linear matrix inequalities (LMIs). A numerical example is presented to verify the analysis results and illustrate the effectiveness of the proposed synchronization conditions.

Membership-Function-Dependent Stabilization of Event-Triggered Interval Type-2 Polynomial Fuzzy-Model-Based Networked Control Systems

In this article, the stability analysis and control synthesis of interval type-2 (IT2) polynomial-fuzzy-model-based networked control systems are investigated under the event-triggered control framework. The nonlinear dynamics in the plant is efficiently represented by an IT2 polynomial fuzzy model that the IT2 membership functions are utilized to capture the uncertainties in the plant. An event-triggered IT2 polynomial fuzzy controller is then designed to stabilize the nonlinear model subject to uncertainties. The stability conditions of the closed-loop control system are summarized in the form of sum-of-squares. Under the imperfectly premise matching (IPM) concept, the membership-function-dependent (MFD) approach is applied to endow the polynomial fuzzy controllers with more flexibility in terms of number of rules and premise membership functions. In the MFD approach under the IPM concept, both the number of rules and the shape of membership functions in the fuzzy models and controllers can be different. Also, the information of IT2 membership functions of the polynomial fuzzy model and controller is considered and adopted to further relax the stability conditions. Furthermore, the intrinsic mismatched issue of the premise variables of the fuzzy model and controllers due to the event-triggering mechanism is handled by the MFD approach. A detailed simulation example is provided to verify the effectiveness of the proposed event-based control strategy.

Tracking control design of interval type-2 polynomial-fuzzy-model-based systems with time-varying delay

In this paper, the tracking control design for the interval type-2 (IT2) polynomial-fuzzy-model-based (PFMB) control system subject to time-varying delay situation is investigated. The tracking control system is formed of the IT2 polynomial fuzzy model representing a nonlinear system with time-varying delay, the stable reference model and the IT2 polynomial fuzzy controller. The control objective is to design a proper IT2 polynomial fuzzy controller which is capable of driving the states of the polynomial fuzzy model to track those in the reference model and the tracking performance is evaluated and improved by the H∞ performance index. Also, to handle the uncertainty in the membership functions, the property of IT2 fuzzy sets is utilized to enhance the fuzzy controller’s robustness against uncertainty. In addition, considering the effect of time-varying delay, the Lyapunov–Krasovskii functional based approach is adopted to facilitate the delay-dependent stability analysis. Stability conditions depending on the time-varying delay characteristic with the consideration of H∞ performance are obtained in terms of sum-of-squares (SOS). Furthermore, the information of the IT2 membership functions is employed in the stability analysis to relax the stability conditions, both membership-function-independent (MFI) and membership-function-dependent (MFD) approaches are presented to develop the stability conditions. Simulation examples are presented to verify the effectiveness of the proposed tracking control approach.

A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis

This paper reviews the stability analysis of continuous-time fuzzy-model-based (FMB) control systems, with emphasis on state-feedback control techniques, which is an essential issue received a great deal of attention in the fuzzy control community. It gives an overview of the milestones and the trend of developments and achievements for the past decades. Focusing on the stability analysis of FMB control systems, it summarizes the issues in the four fundamental and essential aspects, namely, the types of membership-function matching, types of Lyapunov functions, types of stability analysis and the techniques of stability analysis are discussed. To start with systematic discussion, the FMB control systems are categorized into three types of membership-function matching, namely, perfectly, partially and imperfectly matched premises, regarding the premise membership functions and the number of rules used in the fuzzy model and fuzzy controller which forms the FMB control system. The features of each category are thoroughly discussed from theoretical to practical point of view. Various types of Lyapunov functions available in the literature for conducting stability analysis and their characteristics are then reviewed. Especially, the focus of this paper is to promote the concept of membership-function-dependent (MFD) stability analysis, which makes use of the information of membership functions aiming to relax the stability conditions compared with the dominant membership-function-independent (MFI) stability analysis in the literature. The techniques for MFI and MFD stability analysis are then discussed in details, which provide some solid ideas to analyze the stability of FMB control systems. More importantly, it sheds light on the fact that MFD stability analysis demonstrates a greater potential than the MFI one for relaxing the conservativeness of stability analysis results, which points a promising research direction for this topic. The purposes of this paper are to provide a comprehensive update for the stability analysis of FMB control systems to the researchers in the field and serve as a quick guide for the potential researchers who want to enter the field.