Fuzzy Approximation Research Articles

Event-triggered-based adaptive asymptotic tracking control for stochastic nonlinear time-delay systems with actuator and sensor faults

The issue of dynamic event-triggered adaptive asymptotic tracking control for stochastic nonlinear time-delay systems with actuator and sensor faults is investigated in this article. Utilizing fuzzy logic systems and projection operator, unknown time delays are calculated. Compared with the existing methods to overcome the effect of time delay, the estimation of the time delay information is considered to avoid the controller singularity problem. Moreover, to reckon the upper bound of fuzzy logic system approximation errors and other residuals, an error adaptive law is developed in the controller construction. With the assistance of stochastic Barbalat’s lemma and adaptive fuzzy control strategy, a novel asymptotic tracking fuzzy controller is proposed under multiple faults, which guarantees that the whole signals are bounded in probability. Besides, a dynamic event triggering control (DETC) is employed to save the communication resources. A practical example is supplied to demonstrate the effectiveness of the proposed control approach.

FLS-based AHOSM control for driverless vehicle steering system with parameter uncertainty

This paper aims to address the tracking control problem for the steer-by-wire (SbW) system with uncertain nonlinearity and parameter uncertainty. First, the adaptive fuzzy logic system (FLS) is constructed to realize the intelligent modeling of the SbW system. In addition, an adaptive higher-order sliding mode (AHOSM) control with dynamic gain is designed to overcome the lumped uncertainties including inaccurate model-parameter and fuzzy logic system approximation error, and has the advantage of eliminating the gain-overestimation phenomenon effectively without the prior knowledge about the bounds of approximation error and parameter uncertainty. Furthermore, theoretical analysis based on Lyapunov stability theory is provided to verify that the real sliding mode can be established. Finally, simulations and vehicle experiments are given to evaluate the effectiveness and superiority of the proposed methods.

Fuzzy-Based Adaptive Optimization of Unknown Discrete-Time Nonlinear Markov Jump Systems With Off-Policy Reinforcement Learning

This article explores a novel adaptive optimal control strategy for a class of sophisticated discrete-time nonlinear Markov jump systems (DTNMJSs) via Takagi–Sugeno fuzzy models and reinforcement learning (RL) techniques. First, the original nonlinear system model is represented by fuzzy approximation, while the relevant optimal control problem is equivalent to designing fuzzy controllers for linear fuzzy systems with Markov jumping parameters. Subsequently, we derive the fuzzy coupled algebraic Riccati equations for the fuzzy-based discrete-time linear Markov jump systems by using Hamiltonian–Bellman methods. Following this, an online fuzzy optimization algorithm for DTNMJSs as well as the associated equivalence proof is given. Then, a fully model-free off-policy fuzzy RL algorithm is derived with proved convergence for the DTNMJSs without using the information of system dynamics and transition probability. Finally, two simulation examples, respectively, related to the single-link robotic arm and the half-car active suspension are given to verify the effectiveness and good performance of the proposed approach.

Speed Tracking Control for Unmanned Driving Robot Vehicle Based on Fuzzy Adaptive Sliding Mode Control

To achieve accuracy and stable speed tracking under different conditions, a speed tracking method for the unmanned driving robot vehicle with sliding mode gains adjusted by a fuzzy adaptive system is proposed. Using the complex vector method and equivalent substitution method, kinematics and dynamics models of driving mechanical legs are established. Combined with models of driving and brake systems of manipulated vehicles, the longitudinal dynamics model is built. Then the longitudinal manipulation strategy is established. Based on the dynamics model of the mechanical leg, the sliding mode controller's control law is deduced. In addition, the fuzzy adaptive approximation is constructed and the fuzzy adaptive law is designed. Furthermore, the stability of the fuzzy sliding mode controller is proven using the Lyapunov function. Finally, through simulation and experiment, a comparison among the proposed method and several existing control methods demonstrates the effectiveness of the proposed method.

Fuzzy transform-based approximation method for solving fractional semi-explicit differential-algebraic equations

We present an efficient numerical method to approximate the solution of a system of fractional-order linear semi-explicit differential-algebraic equations with variable coefficients. The method is based on the use of the direct and inverse fuzzy transforms ( $${\mathcal {F}}$$ -transforms). By employing this method, we obtain an analytical approximate solution to the main problem in terms of flexible basic functions. The non-local property of fuzzy transforms helps us to have an efficient method for problems involving non-singular kernels. The error analysis and convergence evaluation of the method are demonstrated in detail. We give some examples to illustrate the significant features of the method.

On two new types of fuzzy rough sets via overlap functions and corresponding applications to three-way approximations

Fuzzy rough sets (FRSs) can be served as a useful technique to cope with inconsistent data by providing a pair of lower and upper fuzzy rough approximations of a fuzzy concept. Diversified FRS models have been proposed over the years. Meanwhile, t-norms and t-conorms, as two binary aggregation operators, play a critical part in the construction of fuzzy rough approximation operators. Nevertheless, one of common properties of t-norms and t-conorms is that they all fulfill associativity. Additionally, the associativity plays an important role in discussing properties of FRSs. Therefore, if the binary aggregation function does not satisfy associativity, how to define FRS models and discuss the relevant properties is a topic worthy of further investigation. Overlap functions, as a new kind of not necessary associative binary aggregation operator, have been continuously investigated by many scholars in theory and practical applications. Therefore, based on the advantages of overlap functions, and as the extension models for FRSs, we propose two new types of FRSs via overlap functions and their residual implications (or called IO-implications) in this paper. More specifically, we first give the definition of the 1st type of FRSs and study some essential properties. Subsequently, the definition of the 2nd type of FRSs is presented and the relevant properties are also investigated. Furthermore, the relationship between the 1st type of FRSs and the 2nd type of FRSs is described. That is, in a fuzzy approximation space, the 2nd type of lower approximation is contained in the 1st type of lower approximation, while the 1st type of upper approximation is included in the 2nd type of upper approximation when the overlap function has 1 as the identity element. At the end, a novel three-way decision (3WD) method is presented on an information table by using the new type of FRSs, and its practicality and superiority are verified via a numerical example and a series of experimental analyses.

Bionic adaptive fault-tolerant control of non-Gaussian stochastic attitude hypersonic vehicle

This study investigates an adaptive fault-tolerant control (FTC) for hypersonic flight vehicles (HFVs) with incipient faults and non-Gaussian stochastic output attitudes. In the nonlinear HFV dynamics, a hybrid fuzzy approximation method achieves the linearization, then the stochastic outputs are transformed into probability density functions (PDFs) via rational square root B-spline. The disturbance and faults are estimated simultaneously by an adaptive augmented observer. Then actuator faults are compensated by an bionic adaptive fault-tolerant controller to ensure that the output PDFs accurately track the expected PDFs, thereby matching actual attitude angles with the desired ones, the bionic prey adaptive law can make FTC accurately repair the incipient fault deviations. Lyapunov theory proves the robust stability of the scheme, and simulation illustrates the effectiveness.

ADAPTIVE TRACKING CONTROL FOR FRACTIONAL-ORDER NONLINEAR UNCERTAIN SYSTEMS WITH STATE CONSTRAINTS VIA COMMAND-FILTERING AND DISTURBANCE OBSERVER

In this paper, the state-constrained adaptive control problem for a class of uncertain fractional-order nonlinear systems with unknown external disturbances is studied. Using the fuzzy approximation and backstepping control layout, an adaptive disturbance-observer-based control method is proposed, which can effectively estimate the unknown external disturbances through the designed disturbance observers. As we know, the standard backstepping control has inherent computational complexity. Therefore, a design approach of fractional-order command filter is introduced, which takes advantage of fractional-order command filter to pre-estimate the virtual input signal and its fractional derivative. In addition, barrier Lyapunov functions are exploited to cope with the fractional states-constrained control problem. The combination of barrier Lyapunov function technique and backstepping design not only ensures excellent tracking performance, but also enhances the robustness of the system. Furthermore, based on the fractional Lyapunov approach, the relevant stability analysis is established, which demonstrates that all states of the system remain within their constrained scope, and all signals of the closed-loop system are bounded. Finally, the effectiveness of the proposed scheme is verified by numerical simulation examples of RLC circuit and horizontal platform system.

On Covering-Based Rough Intuitionistic Fuzzy Sets

Intuitionistic Fuzzy Sets (IFSs) and rough sets depending on covering are important theories for dealing with uncertainty and inexact problems. We think the neighborhood of an element is more realistic than any cluster in the processes of classification and approximation. So, we introduce intuitionistic fuzzy sets on the space of rough sets based on covering by using the concept of the neighborhood. Three models of intuitionistic fuzzy set approximation space based on covering are defined by using the concept of neighborhood. In the first and second model, we approximate IFS by rough set based on one covering (C) by defining membership and non-membership degree depending on the neighborhood. In the third mode, we approximate IFS by rough set based on family of covering (Ci) by defining membership and non-membership degree depending on the neighborhood. We employ the notion of the neighborhood to prove the definitions and the features of these models. Finlay, we give an illustrative example for the new covering rough IF approximation structure.

Robust, Accurate, and Fast Decentralized Power Sharing Mechanism for Isolated DC Microgrid Using Droop-Based Sliding-Mode Control

In this paper, a droop-based sliding mode controller (DBSMC) is designed for power sharing in an isolated DC microgrid (DC MG), with improved nonlinear droop model, in the presence of bounded structured uncertainties and external disturbances. The proposed DBSMC strategy does not require any communication link. The design process of the proposed controller is such that the trade-off between the two factors proportional power sharing and precise voltage regulation reaches the minimum possible. The creative design of the controller greatly reduced the control action stress, in such a way that, the occurrence of various uncertainties in the nonlinear droop model is compensated by applying small intelligent changes in the control parameters, using a fuzzy approximator. To evaluate the efficiency of the designed DBSMC controller, simulations are performed on an isolated DC MG with DC-DC buck configuration in MATLAB software. Finally, the real-time digital simulation results utilizing real-time digital power system simulator (RTDS) corroborate the performance of the proposed DBSMC controller. The simulation results obtained from MATLAB and the RTDS verify the robust, accurate and fast function of the proposed droop-based controller in power sharing of the isolated DC MG in the presence of variable operating conditions, parametric uncertainties and external disturbances.

Fuzzy Adaptive Output-Feedback Constrained Trajectory Tracking Control for HFVs With Fixed-Time Convergence

This article proposes an output-feedback fixed-time trajectory tracking control methodology for hypersonic flight vehicles subject to asymmetric output constraints. In contrast to the state of the art, the most distinguishing feature of our control design lies in avoiding using conventional recursive design methods (e.g., backstepping technique, dynamic surface control, etc.) and in not relying on full-state availability. In the velocity control loop, an asymmetric integral barrier Lyapunov function is adopted to confine velocity variable within a well-defined compact set all the time. In the altitude control loop, after utilizing its cascaded property and proposing a novel scaling function, the original constrained system is transformed to an unconstrained one, which facilitates the control design and stability analysis. Moreover, the proposed control algorithm only involves one fuzzy logic approximator as well as one fixed-time differentiator in the transformed system and guarantees that the tracking errors of velocity and altitude converge into the user-defined residual sets within fixed time. Several comparative simulations have been conducted to highlight the superiorities of the developed method.

Adaptive Fuzzy Backstepping Asymptotic Disturbance Rejection of Multiagent Systems With Unknown Model Dynamics

This article presentsan adaptive fuzzy asymptotic disturbance rejection scheme for multiagent mechanical systems with unknown model dynamics and unknown frequency disturbances. The mechanical motion equations are expressed as the parameterized equations and the disturbances are described by unknown parametric exogenous system. The unknown parametric exogenous system is transformed into the canonical model with unknown disturbances as the inputs. The disturbance filter provides estimations of the state vector in the canonical model such that disturbances are represented as parameterized forms with exponentially decaying errors. The disturbance rejection can be converted into the adaptive control. Based on the adaptive vectorial fuzzy backstepping, the disturbance rejection formation controller is designed with the fuzzy logic systems approximating the model unknown dynamics. The adaptive robust control term attenuates fuzzy approximation errors. In contrast to the existing results, the adaptive fuzzy disturbance rejection formation control realizes the estimation and rejection for unknown frequency disturbances without requiring the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge of model dynamics. Based on the Barbalat’s lemma, it is shown that the adaptive disturbance rejection controller achieves the asymptotic formation tracking. Application on the surface vessel formation confirms the availability.

Fuzzy approximation-based adaptive tracking control for high-order nonlinear systems with asymmetric full-state constraints

This paper studies the adaptive tracking control for a class of constrained high-order nonlinear systems with uncertainties. First, a type of improved asymmetric log-type barrier Lyapunov functions (LBLFs) is constructed to handle the full-state constraints. Second, fuzzy systems are built to approximate the unknown functions in the control system. An adaptive controller is designed on the basis of adding a power integrator technique. Consequently, the designed controller can ensure that the constraints are not breached, while the tracking error approaches an arbitrarily small neighborhood of the origin. The designed controller’s performance can be guaranteed with the disappearance of the constraints. The computational burden of fuzzy approximation is mitigated by the adaptive gain-update law. A numerical example and a practical example are presented to verify the effectiveness of the proposed method.

Fuzzy Adaptive Protective Control for High-Speed Trains: An Outstretched Error Feedback Approach

This paper presents a fuzzy adaptive protective control method for autonomous high-speed trains (HSTs) automatic operation using a new outstretched error feedback design approach. In order to stabilizing the error dynamics with respect to target position and speed profiles of controlled HSTs, nonlinear transformation in prescribed performance control methodology is used to convert running states subject to protective (constrained) information, imposed by automatic train protection (ATP) subsystem, to new coordinates in unconstrained form. By blending a new outstretched error feedback and fuzzy approximation, it is guaranteed above-mentioned errors are kept within regions characterized by error boundary (or prescribed performance) functions and control parameters simultaneously, which can be adjusted to arbitrarily small even without error decreasing boundary functions. Fuzzy approximation is used in compensating unknown running resistances. It is rigorously proved that the resulting closed-loop system is stable in sense of Lyapunov stability in the presence of unknown resistance, containing basis and aerodynamic resistances with uncertain parameters and piecewise continuously slope resistance over varying gradient profile. The proposed control is demonstrated to be effective by comparative simulations of train G1 running on Beijing-Shanghai railway line.

Adaptive depth tracking of underwater vehicles considering actuator saturation: Theory, simulation and experiment

This paper investigates underwater vehicles' saturated depth tracking problem via an adaptive fuzzy sliding mode controller (AFSMC) with an anti-saturation auxiliary compensator (ASAC). To suppress the chattering effect due to the discontinuity of the sliding mode control, a saturation function boundary layer is first applied in the AFSMC. Subsequently, a model-free fuzzy approximation system with a sliding mode surface as its input is designed for complex uncertainty compensation. Thereafter, a two-layer cascade ASAC is then attached to the controller to reject inherent actuator saturation efficiently. Based on the Lyapunov theorem and Barbalat's lemma, the stability of the entire closed-loop system is proved. Finally, the outperformance of the proposed depth tracking law is verified by numerical simulations and water tank experiments. Namely, there are improvements in tracking accuracy and energy consumption compared to the proportional integral differential algorithm.

Rotary Flexible Joint Control Using Adaptive Fuzzy Sliding Mode Scheme.

An adaptive fuzzy sliding control (AFSMC) approach is adopted in this paper to address the problem of angular position control and vibration suppression of rotary flexible joint systems. AFSMC consists of fuzzy-based singleton control action and switching control law. By adjusting fuzzy parameters with the self-learning ability to discard system nonlinearities and uncertainties, singleton control based on fuzzy approximation theory can estimate the perfect control law of feedback linearization. To enhance robustness, an additional switching control law is incorporated to reduce the approximation error between the derived singleton control action and the perfect control law of feedback linearization. AFSMC's closed-loop stability will be demonstrated via sliding surface and Lyapunov function analysis of error function. In order to demonstrate the effectiveness of the AFSMC in tracking performance as well as its ability to respond to model uncertainties and external perturbations, simulations are carried out using Simulink and Matlab in order to demonstrate how well it adapts to these situations. Based on these results, it can be concluded that the AFSMC performs satisfactorily in terms of tracking.

Hierarchical Fuzzy Rough Approximations With Three-Way Multigranularity Learning

The approximation learning of fuzzy concepts associated with fuzzy rough sets and three-way decisions is a useful technology for the representation, learning, and transformation of uncertain knowledge. However, how to integrate the advantages of fuzzy rough approximations and three-way approximations has rarely been carefully investigated so far. In this article, we focus on exploring the connection and interplay of these two methods, and further propose the hierarchical fuzzy rough approximations in the dynamic fuzzy open-world environment. We utilize the temporal-spatial perspectives of three-way decisions to construct multigranularity structures and implement multigranularity learning. The temporality of data and the spatiality of model parameters are both considered in such frameworks. Subsequently, we discuss the interpretation and representation of fuzzy three-way regions in fuzzy rough sets with some definitions and properties. The max, double, and aggregated evaluation-based models of three-way approximations are proposed to gradually transform a fuzzy concept to a crisp concept by the time-evolving attributes. Finally, the comparative experimental results between static and dynamic approaches demonstrate the effectiveness of our proposed hierarchical approximation learning models.

Some fuzzy Korovkin type approximation theorems via power series summability method

This article provides a power series summability-based Korovkin type approximation theorem for any sequence of fuzzy positive linear operators. Using the notion of fuzzy modulus of smoothness, we also derive an associated approximation theorem concerning the fuzzy rate of convergence of these operators. Furthermore, through an example, we illustrate that our summability- based Korovin type theorem has an advantage over the fuzzy Korovkin type theorem proved in the seminal paper by Anastassiou (Stud. Univ. Babeş-Bolyai Math L(4):3–10, 2005)

A Fuzzy Approximation for FCS-MPC in Power Converters

Standard model predictive control is an optimization-based control strategy that can handle multiple control objectives and system nonlinear constraints. However, it typically suffers from the limitation of the uncertainties in practical systems, such as external unknown disturbances and parametric uncertainties. Motivated by aforementioned limitation, in this article, a novel robust model predictive control framework, endowed with the merits of fuzzy logic system and finite control-set model predictive control solution, is proposed. The main objective of this article is to enhance the system robustness while guaranteeing adaptability to different conditions. More specifically, a fuzzy approximation point of view, which has a good potential to approximate the unknown nonlinear functions, is deployed and incorporated into the proposed design, which allows one to explicitly take the system nonlinear dynamics and uncertainties into account. The novelty of the proposed methodology relies on the fact that any prior knowledge and explicit information of system model parameters are not required, thereby resulting in considerable enhancement of robustness. Furthermore, the input-to-state stability of the approximation error system is proven through Lyapunov analysis, and it demonstrates that the estimated errors are uniformly ultimately bounded. Finally, the interest of the proposal is experimentally confirmed for modular multilevel converter.

Adaptive Fuzzy Output-Feedback Control for Switched Uncertain Nonlinear Systems With Full-State Constraints.

This article investigates an adaptive fuzzy tracking control approach via output feedback for a class of switched uncertain nonlinear systems with full-state constraints under arbitrary switchings. The adaptive observer and controller are designed based on fuzzy approximation. The main characteristic of discussed systems is that the state variables are not available for measurement and need to be kept within the constraint set. In order to estimate the unmeasured states, the adaptive fuzzy state observer is constructed. To guarantee that all the states do not violate the time-varying bounds, the tangent barrier Lyapunov functions (BLF-Tans) are selected in the design procedure. Based on the common Lyapunov function method, the stability of considered systems is analyzed. It is demonstrated that all the signals in the resulting system are bounded, and all the states are limited in their constrained sets. Furthermore, the simulation example is used to validate the effectiveness of the presented control strategy.