Interval Type-2 Takagi-Sugeno Fuzzy Systems Research Articles

Event-based predictive control for interval type-2 Takagi–Sugeno fuzzy systems under Kalman filter

In this article, the event-based predictive control problem for a class of interval type-2 Takagi–Sugeno fuzzy system is investigated. To reduce the influence of Gaussian noise, an improved Kalman filter is proposed, in which the system state is filtered or not depends on an event-triggered mechanism. Then, by designing the network predictive control scheme via interval type-2 fuzzy model, some sufficient conditions are derived, which guarantee the Schur stabilization of the concerned system subject to network delays. Finally, the correctness and effectiveness are verified by two simulation examples.

Event-triggered non-fragile state estimator design for interval type-2 Takagi–Sugeno fuzzy systems with bounded disturbances

In this paper, the state estimator design problem of interval type-2 Takagi–Sugeno fuzzy systems suffering from bounded disturbances is studied. To enhance the resilience of the estimator, a non-fragile design scheme is proposed to tackle the estimator gain variations. Meanwhile, an event-triggered communication mechanism is introduced for relieving the transmission burden over networks. To settle down the non-fragile estimator design issue subject to bounded disturbances and event-induced error, we propose a new definition of quadratic boundedness via the multiple Lyapunov functions. Based on this definition, a novel co-design method of estimator and event generator for fuzzy system models in the presence of both measurable and immeasurable premise variables is presented. In virtue of quadratic boundedness framework, less conservative conditions of the existence and quadratic stability of the fuzzy estimators are obtained, and the upper bound of estimation error is given explicitly. The desired estimator gains are determined by convex optimization technique using slack matrices. Two illustrative examples are exploited to validate the availability and superiority of the addressed design approach.

Distributed-Delay-Dependent Stabilization for Networked Interval Type-2 Fuzzy Systems With Stochastic Delay and Actuator Saturation

In this article, a distributed-delay-dependent method is proposed to deal with the stabilization for interval type-2 Takagi–Sugeno fuzzy systems subject to stochastic network delay and actuator saturation. To deal with the stochastic feature of network-induced delays, their probability density function is utilized to establish the distributed delay model, which is more practical than the traditional time-varying network delay model. Meanwhile, in order to enlarge the estimation of the domain of attraction, a polytopic strategy composed of a state vector and a distributed-delay-dependent vector is proposed to treat the saturation nonlinearity. By constructing a distributed-delay-dependent Lyapunov–Krasovskii functional, new and less conservative conditions are achieved to guarantee the stability of the established closed-loop system. Additionally, two simulation examples are carried out to illustrate the advantages of the proposed strategy.

Unknown Input Functional Observer Design for Discrete-Time Interval Type-2 Takagi–Sugeno Fuzzy Systems

This article proposes a novel unknown input functional observer design approach toward discrete-time interval type-2 Takagi–Sugeno fuzzy system models subject to measurable and unmeasurable premise variables. By constructing a new state vector that contains both the unknown inputs and the system states, functional observers are proposed for the cases with measurable and unmeasurable premise variables to estimate this new state vector for unknown input and/or state estimation. The observer design problem is converted into the solvability issue of a linear matrix equation involving observer gain matrices, and the existence conditions of the observers are explicitly obtained based on matrix rank analysis. Meanwhile, instead of solving the intricate Sylvester equation directly, the solution of the simplified matrix equation is employed to derive the observer gains. Moreover, the effectiveness and the superiority of the presented method are demonstrated via two illustrative examples.

Stability for IT2 T-S fuzzy systems under alternate event-triggered control

In this paper, an alternate event-triggered control is proposed to achieve stability of interval type-2 Takagi–Sugeno fuzzy systems. Comparing with the existing literature, this new control strategy displays an almost complete aperiodic feature which eliminates the conservativeness caused by time-triggered property of the traditional aperiodically intermittent control. Moreover, with two events being triggered alternately in this control strategy through examining two predetermined conditions, the efficiency of control can be further improved and the resources consumption can be greatly reduced. By employing the Lyapunov function and graph theory, several stability criteria are rigorously demonstrated. In addition, Zeno behavior is excluded in our system through obtaining a positive lower bound of the time interval between two triggering points. Subsequently, the validity of the presented strategy is evidenced by single-link robot arms systems. Finally, a numerical example is given to lend insight into the feasibility of our theoretical results.

Robust $ {H}_{\infty} $ output feedback finite-time control for interval type-2 fuzzy systems with actuator saturation

<abstract><p>The finite-time $ {H_\infty } $ performance of the interval type-2 Takagi-Sugeno fuzzy system (IT2 T-S) in presence of immeasurable states and input saturation is investigated. At first, an observer associated with IT2 T-S states is considered to address the problem of immeasurable states. After that, the input saturation is described based on the polyhedron model, and accordingly, a robust $ {H_\infty } $ observer-based finite-time controller is proposed via non-PDC algorithm. Then, on the basis of the Lyapunov function method and LMIs theory, the sufficient conditions for the finite time stability of fuzzy systems are derived. At last, the feasibility of the designed algorithm is verified by two examples of the nonlinear mass-spring-damping system and tunnel diode circuit system, respectively.</p></abstract>

Stability Analysis for Interval Type-2 Fuzzy Systems by Applying Homogenous Polynomially Membership Functions Dependent Matrices and Switching Technique

In this article, the homogenous polynomially membership functions dependent (HPMFD) matrices are used to study the interval type-2 Takagi–Sugeno fuzzy systems. First, some necessary notations of the HPMFD matrices are introduced. Next, based on these notations, the time derivative of the HPMFD matrices is discussed and a switching method is proposed to ensure that the time derivative of the HPMFD matrices is negative. Then, a HPMFD controller is designed and new stabilization conditions are obtained by using the HPMFD Lyapunov function. In the end, the simulations show that the method in this article is less conservative than the existing ones in the literatures.

Faulty actuator-based control synthesis for interval type-2 fuzzy systems via memory state feedback approach

This study focuses on the memory state feedback control problem for a family of time delayed interval type-2 Takagi-Sugeno fuzzy systems with external disturbance and actuator faulty input. Notably, an actuator fault model is implemented in the control design. In this study, the membership functions, the premise variables of the controller and the considered plant model are chosen freely. By employing the Lyapunov–Krasovskii functional method, the linear matrix inequality approach and the Abel lemma-based finite sum inequality, a set of sufficient conditions that guarantees the stability of the considered system is obtained via the extended passivity performance index. The developed theoretical results are validated with an artificial example and an application-oriented example.

Admissibilisation of singular interval type‐2 Takagi–Sugeno fuzzy systems with time delay

This study investigates the admissibility analysis and stabilisation problems for singular interval type-2 Takagi–Sugeno fuzzy systems with time delay. A generalised integral inequality method is used to obtain the delay-dependent condition. The criteria for admissibility analysis and controller synthesis are given in terms of linear matrix inequalities. In order to reduce the conservatism of the system, some free weighting matrices and advanced integral inequalities are introduced. Finally, two illustrative examples are exhibited to demonstrate the effectiveness of the proposed method.

Adaptive Second-Order Strictly Negative Imaginary Controllers Based on the Interval Type-2 Fuzzy Self-Tuning Systems for a Hovering Quadrotor With Uncertainties

We introduce a new adaptive control technique based on the second-order strictly negative imaginary (SNI) controller, coupled with the interval type-2 fuzzy self-tuning mechanism. We demonstrate the performance of our controllers in the three control loops of the AR.Drone quadrotor to achieve stable and balanced hover performance in flights with a variable Center-of-Gravity (CoG). To facilitate an automatic tuning mechanicsm in our SNI controller, we employ the knowledge-based interval Type-2 Takagi-Sugeno fuzzy system, known for its ability to accommodate the footprint-of-uncertainties (FoU). The proposed adaptive control law is implemented to compensate the dynamic variations in the quadrotor's CoG when the mass of the payload is constantly varied. The robustness and the effectiveness of the proposed control technique are highlighted not only using extensive computer simulations, but also through numerous real-time flight tests. Besides, we also conduct the stability analysis based on the SNI theorem.

Output Feedback Model Predictive Control of Interval Type-2 T–S Fuzzy System With Bounded Disturbance

In this paper, the problem of output feedback model predictive control (MPC) for interval Type-2 Takagi–Sugeno fuzzy systems with bounded disturbance is investigated. The output feedback MPC approach includes an offline design of the state observer to estimate true states and predict bounds of future estimation error sets, and an online problem that optimizes the controller gains to stabilize the closed-loop observer system. The dynamics of the estimation error system is determined by the offline designed observer gain, and bounds of which are online refreshed by scaling a minimal robust positively invariant (RPI) set via a scalar. The optimized controller gains steer the current estimated state from an RPI set into another one such that future estimated states are invariant in the subsequent RPI set. Convergence of the estimation error system and stability condition on the closed-loop observer system in terms of linear matrix inequalities are derived using the technique of S-procedure. The estimation error and estimated state converge within the corresponding time-varying RPI sets, and therefore, recursive feasibility of the optimization problem and input-to-state stability of the closed-loop observer system with respect to the estimation error and bounded disturbance are ensured. For reducing the online computational burden, a lookup table that stores the offline calculated controller gains with corresponding regions of attraction is offline constructed for online searching real-time controller gains. A simulation example is given to show the effectiveness of the approach.

Interval-Boundary-Dependent Control for Interval Type-2 T–S Fuzzy Systems

This paper introduces a state-feedback controller for interval type-2 Takagi–Sugeno fuzzy systems that are described as a convex combination of linear subsystems via normalized time-varying fuzzy weighting parameters, which are unknown but belong to intervals. The introduced structure employs lower and upper interval boundaries directly, rather than alternative parameters generated by the interval boundaries in the studies, which does lead to a stabilization condition of parameterized linear matrix inequalities (LMIs), associated only with the parameters and their interval boundaries. The exact relationship among the parameters and their interval boundaries can, therefore, be fully used to convert parameterized LMIs into LMIs, which provides a great feature distinguished from those in the studies. Two illustrative examples demonstrate the effectiveness and robustness of the derived controller.

A sufficient condition on a general class of interval type-2 Takagi-Sugeno fuzzy systems with linear rule consequent as universal approximators

Whether a rule-based interval type-2 fuzzy system has the ability to approximate any continuous multivariate function arbitrarily well is a fundamentally important question for fuzzy control and modeling. The only approximation results available are the preliminary ones that we previously obtained. They state that two general classes of the interval T2 fuzzy systems, one for the Mamdani type and the other for the TS type, are universal approximators. We now further our investigation on the same T2 TS fuzzy systems to establish a quantitative sufficient approximation condition. These TS fuzzy systems use the linear rule consequent. Their input fuzzy sets are interval T2 and can be in any shape (e.g., Gaussian or trapezoidal) and the fuzzy AND operators in the rules can be any one type or mixed types. We first proved that the fuzzy systems could uniformly approximate any multivariate polynomials arbitrarily accurately and then utilized the Weierstrass approximation theorem to produce the sufficient approximation condition. The condition is a formula calculating the number of the input fuzzy sets needed to achieve any given approximation accuracy (the number of fuzzy rules needed can be easily computed afterward). A numerical example is provided for illustration.

Model reduction for interval type-2 Takagi–Sugeno fuzzy systems

This paper investigates the problem of model reduction for interval type-2 (IT2) fuzzy systems subject to D stability constraints. The membership functions and the number of rules can be freely chosen and they are different between the original system and the reduced-order system. By introducing some slack matrices and utilizing Lyapunov stability theory, the existence condition of model reduction is obtained to guarantee that the reduced-order model can approximate the original system with an H∞ performance. The parameters of the reduced-order system in the condition can be obtained by standard software. Finally, some simulation results are provided to demonstrate the effectiveness of the proposed results.

State feedback control of interval type-2 Takagi–Sugeno fuzzy systems via interval type-2 regional switching fuzzy controllers

The interval type-2 Takagi–Sugeno fuzzy systems have been proposed to handle nonlinear systems subject to parameter uncertainties. In this paper, a new type of state feedback controller, namely, interval type-2 regional switching fuzzy controller, is proposed to conceive less-conservative stabilisation conditions, which is switched by basing on the values of system states. To further reduce the conservativeness in the stability analysis, the information of lower and upper membership functions is also considered. Stability conditions for the interval type-2 fuzzy closed-loop systems are presented in the form of linear matrix inequalities (LMIs). Simulation examples are provided to illustrate the effectiveness of the proposed method.