Polynomial Fuzzy Systems Research Articles

A Novel Convexification Method for Control Synthesis Analysis of Continuous-Time Saturated Positive Polynomial Fuzzy Systems Under Imperfect Premise Matching

A Novel Convexification Method for Control Synthesis Analysis of Continuous-Time Saturated Positive Polynomial Fuzzy Systems Under Imperfect Premise Matching

Estimation of the domain of attraction for continuous-time saturated positive polynomial fuzzy systems based on novel analysis and convexification strategies

In this paper, the domain of attraction (DOA) of the continuous-time positive polynomial fuzzy systems subject to input saturation is estimated by using the level set of the linear copositive Lyapunov function. To relax the estimation of DOA, the restriction on the level set is removed by embedding the expression of the level set into the stability conditions and positivity conditions. Referring to the nonconvex terms caused by above novel analysis strategy, some polynomial inequality lemmas are proposed to handle them; the nonconvex terms caused by imperfect premise matching (IPM) nonlinear membership functions are dealt with by sector nonlinear methods and advanced Chebyshev membership-function-dependent (MFD) methods. In this advanced MFD method, the state space segmentation and polynomial order selection of the Chebyshev approximation method are improved based on breakpoints of the first derivative and curvature, respectively, which is helpful to reduce the conservatism and computational burden of the result. Thus, this advanced Chebyshev MFD method not only optimizes the convexification strategy, but also can further be extended to estimate the DOA when it is used to introduce the membership functions information for convex stability and positivity conditions. Finally, a numerical example and the lipoprotein metabolism and potassium ion transfer nonlinear model are presented to validate the effectiveness and feasibility of the aforementioned analysis and convexification strategies in the expansion of DOA estimation.

Observer‐based control for nonlinear Hadamard fractional‐order systems via SOS approach

AbstractPractical stability refers to the notion that the origin is not an equilibrium point (EP) and that the system states tend to converge toward a sphere centered at the origin. The first goal of this paper is to analyze the concept of “practical stability” in Caputo–Hadamard fractional‐order derivative (CHFOD) systems. Then, using the Lyapunov approach, a polynomial fuzzy (PF) observer‐based controller for stabilizing CHFOD PF systems is created. The observer‐based control is innovative since it was created and proven using the sum‐of‐squares (SOS) method. In conclusion, a numerical illustration is provided to corroborate the theoretical findings.

Predictive Controller for Large-Scale Fuzzy Polynomial Systems

This paper presents an innovative method to tackle the predictive-control challenges associated with large-scale fuzzy polynomial systems comprising interconnected polynomial fuzzy systems. This study models large-scale nonlinear fuzzy systems in a polynomial framework, which can reduce the number of fuzzy rules. We derive the conditions for controller synthesis in the main theorem using the Lyapunov theory and sum-of-squares technique. Simulation results confirm the validity and efficiency of this approach.

Relaxed Stability and Non-weighted $$L_2$$-Gain Analysis for Asynchronously Switched Polynomial Fuzzy Systems

AbstractThis paper concerns the stability and non-weighted $$L_2$$ L 2 -gain analysis issues for the switched systems subject to asynchronous switching and external disturbance. By taking full advantage of the admissible edge-dependent average dwell time (AED-ADT) switching signal, a stability criterion of asynchronously switched systems is proposed to achieve a lower non-weighted $$L_2$$ L 2 -gain index than the existing results. Furthermore, the dynamics of switched systems are described by using the polynomial fuzzy model instead of the T-S fuzzy model, which can represent a wider range of nonlinear plants. For such asynchronously switched polynomial fuzzy systems, the relaxed conditions are derived through the membership-function-dependent (MFD) technique and novel high-order multiple Lyapunov functions (MLFs) construction method to guarantee less conservative stability and non-weighted $$L_2$$ L 2 -gain results. Finally, the superiority of the proposed approaches are verified through two simulation examples.

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Disturbance rejections of polynomial fuzzy systems under equivalent-input-disturbance estimator approach

This paper proposes an integrated robust stabilization and anti-disturbance control scheme for nonlinear systems using a polynomial fuzzy model approach. To estimate unknown disturbances, an equivalent-input-disturbance (EID) estimator is employed. The proposed approach incorporates a novel fuzzy model assisted by EID estimator into the sum-of-squares-based approach, utilizing a polynomial fuzzy model-based observer to estimate the disturbance effect. A suitable fuzzy rule-based control law is developed by utilizing the parallel distributed compensation approach and the output of the EID estimator. To ensure stability, fuzzy membership functions are converted into sum-of-square polynomials using a polynomial curve fitting approach, allowing their exact shape information to be used in the stability condition. The addressed system is transformed into an augmented system by incorporating the system, observer, and filter states, simplifying the analysis. Gain matrices for the controller and observer are obtained using Lyapunov stability theory and sum-of-squares methods to confirm asymptotic stabilization of the fuzzy system. Two numerical examples are presented to demonstrate the effectiveness of the proposed control design method.

Relaxed static output stabilization of polynomial fuzzy control systems by Lagrange membership functions

AbstractThis article is concerned with the stability analysis of the static output‐feedback polynomial fuzzy‐model‐based (SOF PFMB) control systems through designing a novel membership grade integration (MGI) approach. The nonconvex problems of the SOF PFMB control systems are convexificated into the convex conditions by introducing block diagonal positive‐definite Lyapunov matrix and nonsingular transformation matrix. We proposed new approximate membership functions, that is, Lagrange membership functions (LMFs), which can be introduced into the stabilization process to relieve the conservativeness of stability results. The LMFs are general representations of piecewise‐linear membership functions, which makes the number of stability conditions not limited by the number of sample points. In a fixed subdomain, arbitrary sample points can be employed by the LMFs method and achieve higher approximation capability by increasing more sample points so that membership grades can be incorporated into the system analysis. Furthermore, a novel MGI approach, including the information of premise variables and LMFs, is proposed making the stability conditions more relaxed. Finally, two simulation examples show the merits of the developed techniques.

Observer‐Based Exponential Stabilization for Conformable Polynomial Fuzzy Systems With Time Delay

Observer‐Based Exponential Stabilization for Conformable Polynomial Fuzzy Systems With Time Delay

Event-triggered output feedback control design for polynomial fuzzy systems

This paper presents a novel approach to event-triggered output feedback control of polynomial fuzzy systems that exhibit unmeasurable system states. To stabilize the system state, a fuzzy model-based event-triggered control scheme is proposed based on output measurement information. The transmission frequency is modulated using a periodic event-trigger protocol, which is scheduled by resorting to a set of historically released packets to ensure better dynamic performance. The event-triggering condition uses the current samples to determine the next trigger and also takes into account the triggering time-dependent gain parameter, thereby significantly reducing network burden. The proposed approach ensures the existence of both polynomial controller gain and event-triggered parameters by satisfying sufficient conditions based on the sum-of-squares approach and the polynomial fitting approximation algorithm. The addressed system’s stability conditions are solved using MATLAB SOSTOOLS. Three numerical examples, including an invented pendulum model, illustrate the superiority and applicability of the methodology. The main contributions of this work are the introduction of a triggering instant dependent periodic event-triggered output feedback control design, which reduces network traffic and enhances dynamic performance. The approach is validated through numerical examples, which demonstrate its superiority and applicability.

Event-triggered control design with varying gains for polynomial fuzzy systems against DoS attacks

This paper presents an innovative event-triggered control scheme for addressing the stabilization problem of polynomial fuzzy systems under the influence of Denial-of-Service (DoS) attacks. The proposed controller utilizes a sampling-based event-triggered mechanism to reduce communication resources and avoid Zeno behavior. Furthermore, a novel polynomial fuzzy model-based control system is developed to investigate the impact of periodic DoS attacks and the addressed event-triggered mechanism on system stability. To improve system performance, control gains are updated at each triggering instant. The exponential stability criteria are formulated in the form of sum-of-square constraints, supported by a triggering instant dependent piecewise Lyapunov-Krasovskii functional and an online asynchronous premise reconstruction approach. Finally, the efficiency and usefulness of the theoretical findings are validated through simulation examples.

Event-Triggered Positive Filter Design for Positive Polynomial Fuzzy Systems via Premise Integration and Membership-Function-Dependent Methods

This paper focuses on the design of the positive fuzzy filter with event-triggered mechanism for positive polynomial fuzzy-model-based (PPFMB) system. The main objective is to design a positive and stable fuzzy filter which makes the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> performance index of the estimated output signal as good as possible with limited networked bandwidth resources. Since the positive constraints exist in the positive systems, while the disturbance signal of the output signal and the transmission delay in the event-triggered mechanism are taken into account in this research, the augmented system is rebuilt by introducing the weight coefficient of disturbance signal, and a novel linear copositive Lyapunov function (LCLF) is proposed. In order to adapt to the analysis framework based on linear Lyapunov function, a novel linear event-triggered condition is proposed. Furthermore, to handle the mismatched premise variables caused by the event-triggered mechanism, the premise variable associated with event-triggered instants are integrated into the interval of continuous premise variable with the help of the novel event-triggered condition, which means that the original multi-dimensional type-1 membership functions (MFs) are transformed into single-dimensional interval type-2 MFs. Then, an interval type-2 membership-function-dependent (IT2MFD) method is employed to introduce MFs information into the resultant conditions, so that more optimized <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> performance index is obtained. Finally, an example with simulation results is given to verify the effectiveness of all the fuzzy filter design strategies proposed in this paper for optimizing performance index.

Dissipativity-Based Integrated Fault Estimation and Fault Tolerant Control for IT2 Polynomial Fuzzy Systems With Sensor and Actuator Faults

The integrated fault estimation and fault-tolerant control scheme is developed in this paper for a series of interval type-2 polynomial fuzzy systems with both sensor faults and actuator faults, where the bi-directional influence between fault estimation unit and fault-tolerant control unit is investigated. Considering the existence of sensor faults, unmeasurable premise variables are investigated for more general situations and Class III state/fault estimation observers are established for the final fault estimation and fault-tolerant control purposes. To increase design flexibility and reduce physical implementation complexity, the proposed method allows the observer and original system to share asynchronous membership functions and a different number of fuzzy rules. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\mathcal{Q,S,R}) \,-\, \alpha$</tex-math></inline-formula> dissipative performance index is also introduced to fulfill a wider vary of perfor-mance requirements. Membership-function-dependent stability constraints are given in the format of bi-linear polynomial matrix inequalities to obtain less conservative results, which are computed by a two-step path-following method. Superiority and validity are demonstrated by an inverted pendulum example in terms of estimation errors, fault-tolerant control performance and control inputs.

Observer-based control of polynomial fuzzy fractional-order systems

In this study, the Lyapunov technique is used to analyze the observer-based control problem for polynomial fuzzy fractional order (PFFO) models. The case of polynomial matrices with unmeasurable states is considered to increase the applicability of the PFFO models in the design problem. In this regard, we offer two design procedures. First, the design conditions are presented in a one-step procedure. In this design, the non-convex conditions are transformed in a set of sum of squares (SOS) by introducing a new symbolic variable except the state vector [Formula: see text] and its estimated [Formula: see text]. The obtained SOS conditions are presented involve three independent symbolic variables which increase both computing complexity and conservatism. To get around this shortcoming, a second design method is developed. By using the suggested method, SOS conditions requiring just two separate symbolic variables may be obtained. The architecture is shown in two stages; however, the observer and controller gains are computed in a single stage in order to further minimize conservatism. In order to demonstrate the utility of the suggested theoretical analysis, a simulated example is then provided.

Event-Triggered Sliding-Mode Control for Polynomial Fuzzy Singular Systems

This article focuses on the problem of the observer-based sliding-mode control (SMC) for polynomial fuzzy singular systems via a novel dynamic event-triggered (DET) mechanism. First, the DET mechanism is introduced to reduce the transmission burden in the communication network. Second, the DET-based observer is constructed to estimate the unavailable state variables. On this basis, the integral sliding variable is provided, and the observer dynamics can be obtained. A sufficient condition is then proposed to make sure that the whole closed-loop system composed of the observer dynamics and the estimation error system is admissible and passive by utilizing the sum-of-squares approach. Meanwhile, the sliding surface parameter matrix, the observer gain, and the trigger weighting matrix are co-designed. It should be emphasized that the common equality constraints, nonstrict and nonlinear inequality constraints are removed through equivalent sets and several inequality techniques in this design process. Moreover, the DET observer-based SMC law is derived to assure the reachability of the specified sliding surface. Finally, the theoretical results are verified by simulation results.

Robust [formula omitted] sliding mode control for uncertain polynomial fuzzy stochastic singular systems

This paper considers the robust H∞ sliding mode control (SMC) problem for uncertain polynomial fuzzy stochastic singular systems. The interest is focused on designing a polynomial fuzzy sliding mode controller, and two assumptions in most integral SMC methods for fuzzy stochastic systems are removed. By constructing an integral sliding surface, the equivalent control law is obtained, and the sliding mode dynamics is derived. With the help of the sum-of-squares method, for sliding mode dynamics, the mean-square admissibility and the H∞ performance are guaranteed. Furthermore, the sliding mode controller is established. The effectiveness and advantages of the proposed design method are finally revealed via a practical example of the modified bio-economic model and a numerical example.

New results on sliding mode control for polynomial fuzzy singular systems with time‐varying delay

AbstractThis article focuses on the problem of sliding mode control (SMC) for polynomial fuzzy singular systems with the matched uncertainty, time‐varying delay and different control input matrices. An integral sliding surface with different control input matrices and polynomial matrices is established. The existence of the equivalent controller can be determined according to the given sum‐of‐squares conditions. By choosing an appropriate augmented Lyapunov–Krasovskii functional and utilizing a generalized free‐matrix‐based integral inequality, a less conservative admissibility condition of sliding mode dynamics is proposed in the form of sum‐of‐squares. It is noteworthy that due to the choice of augmented Lyapunov–Krasovskii functional, the traditional methods cannot transform the nonlinear terms in the admissibility condition of sliding mode dynamics into linear terms. Here, several inequalities are introduced to overcome this difficulty. Furthermore, the SMC law is proposed to ensure the reachability of the sliding surface. The simulation results show that the proposed method is less conservative and effective.

Robust fault-tolerant control design for polynomial fuzzy systems

This paper focuses on the robust H∞ fault-tolerant control issue for a family of polynomial fuzzy systems subject to actuator faults and external disturbances. Especially, asynchronous fuzzy membership function technique is considered to minimize the expenses of controller implementation. To derive the robust stability conditions, actuator fault factors are taken into the account in the closed-loop system formulation. Moreover, to utilize the exact shape information of fuzzy membership functions in the stability criterion, they are approximated as sum-of-square polynomials with the aid of polynomial curve fitting method. In addition, to reduce the conservatism, a generalized polynomial Lyapunov function is considered. However, there are some non-convex terms in the stability conditions due to the construction of the generalized Lyapunov function that consists of all state-dependent polynomial matrix. Thus, some existing convex methods for general polynomial fuzzy systems are rendered inapplicable. To deal with this issue, the actual non-convex stability conditions converts into a convex optimization problem by introducing some additional convex constraints. Moreover, the membership-function-dependent robust stability conditions of the closed-loop system are provided as sum-of-square convex polynomial constraints. Lastly, the usefulness and superiority of the designed controller are illustrated by the simulation examples.

Descriptor Representation-Based Guaranteed Cost Control Design Methodology for Polynomial Fuzzy Systems

This paper presents a descriptor representation-based guaranteed cost design methodology for polynomial fuzzy systems. This methodology applies the descriptor representation for presenting the closed-loop system of the polynomial fuzzy model with a parallel distributed compensation (PDC) based fuzzy controller. By the utility of descriptor representation, the guaranteed cost control (GCC) design analysis can utilize polynomial fuzzy slack matrices for obtaining less conservative results. The proposed GCC design is presented as the sum-of-squares (SOS) conditions. The application of polynomial fuzzy slack matrices leads to the double fuzzy summation issue in the control design. Accordingly, the copositive relaxation works out the problem well and is adopted in the control design analysis. The GCC design minimizes the upper limit of a predesignated cost function. According to the performance function, two simulation examples are provided to demonstrate the validity of the proposed GCC design. In these two examples, the proposed design obtains superior results.

Static Output Feedback Fault Tolerant Control and Relaxed Stability Analysis of Positive Polynomial Fuzzy Systems

The increased demand for safety in control systems has motivated the research for fault-tolerant controllers (FTCs). This article presents the static output feedback FTC design and relaxed stability analysis for positive nonlinear systems on account of polynomial fuzzy models. A polynomial copositive Lyapunov function candidate is proposed for the stability analysis of positive polynomial fuzzy-model-based (PPFMB) control systems for the first time. This method develops the decision variable on the polynomial copositive Lyapunov function candidate to be a polynomial vector and also makes it independent of the form of the positive polynomial fuzzy model. Thus, it can be applied to a more extensive range of PPFMB control systems and potentially generates more relaxed stability analysis results. To achieve the stability analysis and control synthesis, the following challenges are overcome: 1) approximate the nonconvex terms in stability and positivity conditions into convex ones through the idea of matrix decomposition; 2) eliminate the nonconvex conditions resulting from the derivative of the polynomial Lyapunov function by employing an sum-of-squares (SOS) optimization method; 3) relax the results of stability analysis and control design by introducing the advanced membership-function-dependent technique to develop both the stability conditions and the positivity conditions. The relaxed conditions are obtained on the basis of SOS form so that the MATLAB third-party toolbox, SOSTOOLS, can be used to find feasible solutions. Finally, two simulation examples are presented to illustrate the effectiveness of the proposed method.