Unmeasurable Premise Variables Research Articles

Fuzzy Reset-Based H ∞ Unknown Input Observer Design for Uncertain Nonlinear Systems With Unmeasurable Premise Variables

This paper proposes an $H_{\infty }$ reset unknown input observer (R-UIO) based on the Takagi-Sugeno (T-S) fuzzy model for the state estimation of nonlinear uncertain systems. Firstly, $H_{\infty }$ unknown input observer (UIO) is designed for TSFM-based nonlinear systems with measurable and unmeasurable premise variables. Then, according to the importance of observers based on unmeasurable premise variables, the results on UIO is modified to propose R-UIO. The sufficient conditions for the stabilization of the estimation error are derived in terms of linear matrix inequalities (LMIs). The proposed R-UIO benefits from less computation complexity to find the feasible parameters, improvement of the estimation process in viewpoints of convergence speed and overshoot. To verify the effectiveness of the recommended approaches, the methods are applied to a practical system.

Takagi-Sugeno Fuzzy Fault Detector Design with Finite-Frequency Specifications for Autonomous Ground Vehicles

This paper investigates the steering fault detection problem for autonomous ground vehicles (AGVs). Using an observer-based approach, a new fuzzy fault detector for steering actuator is designed for safety concern. To this end, a two degrees-of-freedom (2-DOF) nonlinear vehicle model is adopted to represent the nonlinear dynamics of AGVs. Since it is not easy to measure the lateral velocity in practice, this model is then represented in a specific Takagi-Sugeno (TS) fuzzy form with nonlinear consequents. In contrast to the conventional TS fuzzy modeling, it allows separating the unmeasured premise variables in the local nonlinear consequent, which enables a more effective way to deal with the challenging issue of unmeasured premise variables. Moreover, to minimize the effect of disturbances on system performance and maximize that of actuator faults on the generated residual, both H∞ disturbance attenuation index and H_fault sensitivity index are taken into account in a finite-frequency domain. The conditions to design fault detection TS fuzzy observer are derived using Lyapunov stability method. The design procedure can be reformulated as an optimization problem under linear matrix inequalities, efficiently solved by standard numerical solvers. Simulation results are given to verify the fault detection performance of the proposed method.

Robust Zonotopic Observer Design: Avoiding Unmeasured Premise Variables for Takagi-Sugeno Fuzzy Systems

Abstract This paper addresses the design of a robust set-based interval observer for a nonlinear discrete-time system affected by system uncertainty (state disturbance and measurement noise) using a Takagi-Sugeno (TS) fuzzy model including an unmeasurable premise variable. The effect of unmeasurable premise variable and uncertainties is considered as of unknown but bounded nature, i.e., in the set-membership framework. A zonotopic representation of a set towards reducing set operations to simple matrix calculations is used to bound the state estimation provided by the interval observer-based approach. Furthermore, the criterion-based approach and H∞ performance technique are considered in order to compute the observer gain to achieve robustness. Finally, an example is employed to both illustrate and discuss the effectiveness of the proposed approach.

Switching TS fuzzy model-based dynamic sliding mode observer design for non-differentiable nonlinear systems

This paper presents four sliding mode observers (SMOs) based on a novel approach in Takagi–Sugeno (TS) fuzzy modeling of multi-input multi-output (MIMO) non-linear systems that have non-differentiable operating points. A comprehensive approach is proposed to using the TS fuzzy model (TSFM) in the field of non-differentiable nonlinear systems, where the TSFM is an approximation with high accuracy and a negligible error (2ε) of the nonlinear model. Furthermore, the considered system can be with measurable or unmeasurable premise variables. The observers are synthesized for the above two cases and dynamic observers for state estimations of MIMO non-linear Lipschitz systems. The dynamic gain of the observer is established from inspiring state-space representation of an LTI system with error as input, internal states, and the gain of the observer as output. The dynamics used in the gain of the observer will increase the degrees of freedom in the design procedure and a generalization to the one used in previous works. The proposed method is applicable for continuous-time, but not necessarily differentiable, nonlinear systems. Considering the inherent strongly nonlinear and coupling performance of the plant, the switching method driven by states is presented. This paper presents a comparison of four SMOs and multiple-model adaptive estimation (MMAE) for benchmark hydraulic wind power transfer (HWPT). Simulation results demonstrate improvement in the state observation convergence rate and simplicity and universality of the proposed approach.

Functional observer–controller method for unmeasured premise variables Takagi-Sugeno systems with external disturbance

A functional observer–controller design problem of Takagi-Sugeno fuzzy systems with unmeasured premise variables and external disturbance is investigated in this paper. First, robust state feedback controller is designed based on a relaxed fuzzy Lyapunov approach. With an estimation membership functions of transformation technique, a fuzzy functional observer is introduced to estimate the unmeasured premise variables and system control input. By using a robust separation principle, the observer and controller errors system is proved to be stable and robust to external disturbance. Finally, the estimation and control effects of the proposed method are verified through several engineering examples.

Fault detection in finite frequency domain for T-S fuzzy systems with partly unmeasurable premise variables

This paper is concerned with the problem of finite frequency fault detection (FD) for a class of T-S fuzzy systems with partly unmeasurable premise variables. To fully use the available information of the considered T-S fuzzy system, an FD observer whose premise variables consist of the measurable premise variables and the estimations of unmeasurable ones of the fuzzy model is designed. However, the membership functions of the fuzzy model and the observer to be designed are unsynchronized, such that the existing FD methods based on parallel distribute compensation (PDC) strategy may be infeasible. To overcome this difficulty, a novel non-PDC H∞/H− unknown input observer is designed to achieve the purpose of fault detection and provide less conservative results. Finally, two simulation examples are given to illustrate the validity and merits of the presented FD scheme.

An observer-based robust H∞ controller design for uncertain Takagi–Sugeno fuzzy systems with unknown premise variables using particle swarm optimisation

This paper deals with the observer-based robust H∞ controller design for the Takagi–Sugeno (T-S) continuous fuzzy systems with uncertainties and external disturbances in the case where the premise variables are not measurable. The main objective of this work is to provide robust stability conditions in terms of relaxed LMIs (Linear Matrix Inequalities) for uncertain and disturbed T-S fuzzy systems with unmeasurable premise variables. Using some matrix transformations and lemmas, sufficient conditions of robust stabilisation are derived in the form of a set of relaxed LMIs guaranteeing an optimal value of the H∞ performance level. The proposed LMI constraints are solved using particle swarm optimisation (PSO) algorithm to obtain the robust controller and observer gains. The proposed method has the advantage of having less predefined parameters against the available methods in the literature. The obtained H∞ control scheme ensures asymptotic stability of the closed-loop system as well as strong disturbance attenuation based on the quadratic Lyapunov function, a fuzzy observer and a fuzzy controller. Two simulation examples are provided to show the effectiveness of the proposed approach.

Event-triggered observer-based sliding mode control for T-S fuzzy systems via improved relaxed-based integral inequality

This paper performs the observer-based exponential stabilization criterion for event-triggered fuzzy systems using observer-based sliding mode fuzzy control subject to limited communication resources and unmeasurable premise variables. The nonlinear control system is modeled through a fuzzy model by considering the observer-based sliding mode control with event triggered mechanism. A discrete-time event-triggered scheme has been proposed to determine the unmeasurable states of the proposed system. To deal with the integral factors occurring in the derivate of the Lyapunov–Krasovskii functional (LKF), a new relaxed-based integral inequality has been introduced as a combination of Wirtinger-based inequality and reciprocally convex lemma together with the proposed Lemma. By employing this new integral inequality and by the Lyapunov stability theory, the exponential stability conditions with the prescribed observer and controller gain matrices are attained for ensuring the exponential stability of the T-S fuzzy system. Finally, the proposed theoretical results are applied to the truck-trailer model to validate the effectiveness of the proposed approach.

A Unified Framework for Asymptotic Observer Design of Fuzzy Systems With Unmeasurable Premise Variables

This article develops a unified framework to design fuzzy-model-based observers of general nonlinear systems for both discrete-time and continuous-time cases. This observer problem is known as a challenging task due to the mismatch caused by the unmeasurable premise variables. To deal with this major challenge, we propose to rewrite the nonlinear system as a specific fuzzy model with two types of local nonlinearities: measurable and unmeasurable. Then, a differential mean value theorem for vector-valued functions is applied to local unmeasurable nonlinearities. This allows to represent the estimation error dynamics in a special polytopic form involving measurable membership functions and unknown but bounded time-varying parameters. Using Lyapunov-based arguments, design conditions in terms of linear matrix inequalities are derived to guarantee the asymptotic convergence of the estimation error. Three illustrative examples are given to demonstrate the interests of the new fuzzy observer framework in reducing: 1) the design conservatism and 2) the numerical complexity of the fuzzy observer structure for real-world applications.

Avoiding Unmeasured Premise Variables in Designing Unknown Input Observers for Takagi–Sugeno Fuzzy Systems

This letter investigates the design of unknown input (UI) observers for a large class of nonlinear systems using Takagi-Sugeno (TS) fuzzy modeling. To avoid the well-known issue on the unmeasured premise variables in fuzzy observer design, we reformulate the nonlinear systems in a TS fuzzy form with local nonlinear models. A particular feature of these so-called N-TS fuzzy models is that all the unmeasured nonlinearities are isolated in a nonlinear consequent. Together with a judicious use of the differential mean value theorem, the N-TS fuzzy reformulation enables an effective framework to design fuzzy UI observers. Based on an UI decoupling technique, no specific information on the UI is required for fuzzy observer design. The asymptotic estimations of both the state and the UI are guaranteed with fuzzy Lyapunov arguments. The observer gains can be effectively computed following an LMI-based design procedure. Numerical illustrations are given to demonstrate the interests of the proposed method over related existing results.

Unknown Dynamics Decoupling to Overcome Unmeasurable Premise Variables in Takagi–Sugeno Observer Design

This letter discusses a new approach to overcome unmeasurable premise variables in observer synthesis for Takagi-Sugeno models. The main idea is based on the decoupling of the nonlinear dynamics in order to manage unmeasured state existing into the membership functions. The obtained structure is a system with membership functions subjected only to measured variables. The stability analysis of the observer is carried out using Lyapunov theory. The observer gains were computed from the resolution of the Linear Matrix Inequality constraints. The present result alleviates the strong conditions assumed in the design of observers for TS systems with unmeasurable premise variables. Simulation results are provided to demonstrate the effectiveness of the proposed approach.

Optimal Interval Observer for Switched Takagi–Sugeno Systems: An Application to Interval Fault Estimation

The main goal of this article is to design interval observers for continuous nonlinear switched systems. The nonlinear modes are described by the multimodel approach of Takagi-Sugeno (T-S) fuzzy systems where premise variables depending on the state vector that is unmeasurable. In this article, we propose T-S interval observers that consider the unmeasurable premise variables as bounded uncertainties under common assumptions that additive disturbances as well as measurement noises are unknown but bounded. The stability and the nonnegativity conditions are given in terms of linear matrix inequality to ensure simultaneously the convergence and the nonnegativity of error dynamics. Furthermore, in the absence of measurement noises, optimal gains attenuating the effect of additive disturbances are computed using H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> approach to improve the accuracy of the present interval observers. Theoretical results are finally applied to a numerical example to highlight the performance of the proposed method.

Designing Fuzzy Descriptor Observer with Unmeasured Premise Variables for Head-Two-Arms-Trunk System

Abstract Using the technique of unknown input observer, this paper aims at estimating internal variables of people living with a complete spinal cord injury (SCI). The goal is to provide a better understanding on the sitting control strategy of SCI people. The observer design is based on a Head-Two-Arms-Trunk (H2AT) model, belonging to a class of nonlinear descriptor systems. For observer design, this model is represented in a specific Takagi-Sugeno (TS) fuzzy form with nonlinear consequents. In contrast to previous fuzzy estimation results based on conventional TS fuzzy modeling, the new TS formulation allows separating all unmeasured premise variables in the nonlinear consequent parts. This contributes to reduce the computational burden of the observer design and the structural complexity of the designed fuzzy observer. In particular, the new formulation enables a more effective way to deal with unmeasured premise variables. Using Lyapunov stability theorem, sufficient conditions to design the unknown input observer are derived in the form of linear matrix inequalities, conveniently solved by convex optimization techniques. Simulation results demonstrate the effectiveness of the proposed observer design.

Robust Fault Detection for switched Takagi-Sugeno systems with unmeasurable premise variables: Interval-Observer-based approach

Abstract This paper deals with the problem of robust fault detection for continuous-time switched Takagi-Sugeno (T-S) fuzzy models. A procedure based on interval observers is proposed. First, an interval observer is designed under the assumption that the disturbances as well as the uncertainties are unknown but bounded. Stability and nonnegativity properties are given in terms of Linear Matrix Inequalities (LMIs) taking into account disturbances attenuation. Next, residual intervals generated by the interval observer are used for fault detection decision. Finally, a numerical example is provided to show the usefulness of this approach.

Takagi‐Sugeno tracking control of a fermentation process with respect to asymmetric constraints

SummaryThis article deals with the tracking problem for nonlinear biological process with asymmetric constraints on the controller. Using the fuzzy logic Takagi‐Sugeno (TS), the nonlinear process is modeled by a fuzzy model and a parallel distributed compensation (PDC) controller with respect asymmetric constraints is designed. The main contribution of this work is to extend the available results, in the context of a TS formulation, for symmetrical constraints to asymmetrical ones. Hence, sufficient conditions are obtained for the first time, in terms of linear matrix inequalities (LMIs), for the existence of an asymmetrical constrained PDC controller. Moreover, fuzzy observer based controller with unmeasured premise variables is designed. The performance and effectiveness of the method are illustrated in simulation by considering a fermentation process.

Dissipativity-Based Fault Detection for Uncertain Switched Fuzzy Systems With Unmeasurable Premise Variables

This paper investigates the problem of fault detection for uncertain switched fuzzy systems with unmeasurable premise variables. A dissipativity-based fault detection observer is designed. The observer order can be adjusted within a certain range. By solving a set of linear matrix inequalities, the sufficient condition can be obtained to ensure that under arbitrary switching signal, the residual dynamic is strictly $(Q,S,R)$ - $\kappa$ -dissipative. Furthermore, the elaborate $H_\infty$ performance and the elaborate $H_-$ performance are obtained, which can be treated as expansions of the traditional $H_\infty$ performance and $H_-$ performance. The parameter uncertainties are decoupled from the residual. Simulation results are presented to verify the proposed techniques.

Fractional Order Unknown Inputs Fuzzy Observer for Takagi–Sugeno Systems with Unmeasurable Premise Variables

This paper presents a new procedure for designing a fractional order unknown input observer (FOUIO) for nonlinear systems represented by a fractional-order Takagi–Sugeno (FOTS) model with unmeasurable premise variables (UPV). Most of the current research on fractional order systems considers models using measurable premise variables (MPV) and therefore cannot be utilized when premise variables are not measurable. The concept of the proposed is to model the FOTS with UPV into an uncertain FOTS model by presenting the estimated state in the model. First, the fractional-order extension of Lyapunov theory is used to investigate the convergence conditions of the FOUIO, and the linear matrix inequalities (LMIs) provide the stability condition. Secondly, performances of the proposed FOUIO are improved by the reduction of bounded external disturbances. Finally, an example is provided to clarify the proposed method. The obtained results show that a good convergence of the outputs and the state estimation errors were observed using the new proposed FOUIO.

Finite-Frequency H-/H∞ Fault Detection for Discrete-Time T-S Fuzzy Systems With Unmeasurable Premise Variables.

This paper investigates a finite-frequency H-/H∞ fault detection method for discrete-time T-S fuzzy systems with unmeasurable premise variables. To minimize the effect of uncertainties on system performance and maximize that of actuator faults on the generated residual, both the H∞ disturbance attenuation index and finite-frequency H- fault sensitivity index are utilized. Since the premised variables are unmeasurable, the existing generalized Kalman-Yakubovich-Popov lemma cannot be directly extended to these nonlinear systems. In this paper, the conditions of allowing one to design the proposed H-/H∞ fault detection observer are established and transformed into linear matrix inequalities. Some scalars and slack matrices are introduced to bring extra degrees of freedom in observer design. Finally, a single-link robotic manipulator model is utilized to illustrate that the proposed technique can detect faults with smaller amplitude than that required by a normal H∞ observer technique.

Further Study on Observer Design for Continuous-Time Takagi-Sugeno Fuzzy Model With Unknown Premise Variables via Average Dwell Time.

This article further studies the problem of observer design for the continuous-time Takagi-Sugeno (T-S) fuzzy system with unmeasurable premise variables. A membership function-dependent Lyapunov function is designed to obtain the observer-based controller. Different from the existing results, a switching method is proposed to deal with the time derivative of membership functions. Several problems such as, too many parameters, and the small local stabilization region in the existing papers are solved by applying the switching method. In addition, two algorithms are designed to obtain the controller gains and observer gains. In the end, two examples are provided to demonstrate the effectiveness of the proposed approach.

Positive observer design for switched positive T–S fuzzy delayed systems with dwell time constraints

In this paper, the positive observer design problem is investigated for switched positive Takagi–Sugeno (T–S) fuzzy delayed systems (SPFDSs) with dwell time constraints. First, by using a clock-dependent co-positive Lyapunov–Krasovskii function (CDCLKF), sufficient conditions are derived for stability of SPFDSs with dwell time. Then, the observer design problem of SPFDSs is separately considered for the measurable and unmeasurable premise variable cases. On the basis of the obtained stability results, sufficient conditions are presented to check the existence of observers for SPFDSs with measurable and unmeasurable premise variables. Through minimizing the system matrix norm of the error dynamics, the optimized observer gains for SPFDSs with unmeasurable premise variables are achieved, and the errors between the system states and their estimations are reduced. All the clock-dependent conditions are relaxed into sum of squares (SOS) program via the SOS approximation approach. At last, two examples are given to show the validity of our results.