SummaryThis study addresses the issue of developing an adaptive combined integral fuzzy sliding mode (ACIFSM) control for a class of time‐delay nonlinear descriptor systems represented by Takagi–Sugeno (TS) fuzzy models through the linear matrix inequality (LMI) framework. The systems being investigated are susceptible to both exogenous disturbances and uncertainties. The systems under study are subject to uncertainties as well as exogenous disturbances. The matched uncertainty is represented in a uniform structure that can be adjusted using the adaption laws. By adopting a novel fuzzy sliding manifold with a time‐delay that is based on a LMI, it is proved that the resulting closed‐loop system can ensure the reaching condition, hence ensuring its stability. According to the constructed fuzzy sliding manifold, a novel ACIFSM controller is proposed, and its structure is determined by the adaptive laws and the control input matrices. Finally, two examples are provided to demonstrate the validity of the methods employed in this research.

For nonlinear descriptor systems, this article presents a novel neural network interval observer. The framework of this interval observer is based on the combination of monotone systems theory and radial basis function neural networks, as the expansion of the definition of the general interval observer. Additionally, the K -means algorithm is used in the design process of the neural network interval observer to achieve the optimal distribution of hidden nodes, thus avoiding the drawbacks of traditional neural network methods. The number of hidden nodes is significantly decreased, and the persistence of excitation conditions is secured. Also, the interval estimation and convergence performance of the observer can be obtained. Eventually, the validity of the proposed method is illustrated by a numerical example.

Sampled output data driven convergent fault estimation for continuous nonlinear descriptor systems

This work proposes a discrete-time output regulator for nonlinear descriptor systems. The approach consists in two parts: 1) the design of a convex nonlinear stabilizer computed by means of the so-called non-quadratic Lyapunov functions, 2) the design of a linear regulator computed through the so-called Francis equations for descriptor models; the designing conditions of both parts are in terms of linear matrix inequalities. The effectiveness of the proposal is illustrated via numerical examples.

ABSTRACTThis article studies the state and disturbance simultaneous estimation problem for a class of nonlinear descriptor systems with using a dynamic event‐triggered mechanism. refers to incremental quadratic constraints, which can provide a unified description of many types of common nonlinear functions. To reduce the negative impact of measurement outliers on the identification estimation, a stubborn state and disturbance observer co‐design scheme is proposed for the first time by embedding dynamic saturation output estimation errors. Meanwhile, a dynamic event‐triggered mechanism is introduced to avoid the need for continuously available output information, which can reduce the pressure on communication resources. By constructing a Lyapunov function, existence conditions of the stubborn state and disturbance observer are obtained in the form of a convex optimization problem so that the error estimation maintains an acceptable estimation performance. Finally, simulation examples illustrate the universality and stubbornness of the proposed observer.

This paper proposes an observer-based proportional Derivative (O-BPD) fuzzy controller for uncertain discrete-time nonlinear descriptor systems (NDSs). Representing NDSs with the Takagi–Sugeno fuzzy model (T-SFM), the proportional derivative (PD) feedback method can be utilized in the fuzzy controller design via the Parallel Distributed Compensation (PDC) concept, such that the noncausal problem and impulse behavior are avoided. A fuzzy observer is proposed to obtain unmeasured states to fulfill the PD fuzzy controller. Moreover, uncertainties and transient response performances are taken into account for the NDSs. Then, a stability analysis process and corresponding stability conditions are derived from the Lyapunov theory with the robust control method and the pole constraint. Different from existing research, the Singular Value Decomposition (SVD) and the projection lemma are utilized to transfer the stability conditions into the Linear Matrix Inequation (LMI) form. Because of this reason, the conservatism of the analysis process can be reduced by eliminating the restriction on the positive definite matrix in the Lyapunov function. By giving the proper center and radius parameters of the pole constraint in the O-BPD fuzzy controller design process, the expected transient responses can be obtained for different designers and different practical applications. Finally, the effectiveness and applicability of the proposed O-BPD fuzzy controller are demonstrated by two examples of the simulation.

This research focuses on the development of state-difference feedback controllers for discrete-time (DT) nonlinear descriptor systems. Discrete-time nonlinear DA systems consist of difference and algebraic equations and play a crucial role in describing dynamic behavior and capturing the constraints or relationships within the system. However, analytical stability may pose additional challenges due to the unique characteristics of the system. Utilizing fuzzy model-based techniques, the DT nonlinear DA system discussed in this study can be effectively represented using the Takagi–Sugeno (T-S) fuzzy model. After linearizing the nonlinear system through the T-S fuzzy model, traditional linear control techniques become applicable. These techniques are then applied to T-S fuzzy systems to establish stability criteria. This article chooses the Lyapunov function as the method used to analyze system stability. Additionally, we use a free-weighting matrix to introduce additional degrees of freedom. In summary, this paper presents simulation results and discussions to verify the effectiveness of the proposed design approach.

In general, the construction of observers for nonlinear descriptor systems depends on the solvability of a linear matrix inequality involving system matrices, and it is based on the system’s nonlinearity. Therefore, the type of nonlinearity present in the system heavily affects the observer design process. There are significant developments in the literature for observer design for descriptor systems with various types of nonlinearity. Motivated by this, the current work reviews the literature on observer design for nonlinear descriptor systems with an extensive discussion on the type of nonlinearities that are considered. Here, an analysis and the comparison on the most common nonlinearities is presented, providing a roadmap to all researchers in the field. Furthermore, less common nonlinearities have been identified, presenting under-explored areas within the literature, and can open new domains for future research.

Machine Learning based Fault Estimation of NonLinear Descriptor Systems

Machine learning-based fault estimation of nonlinear descriptor systems

A robust observer based on the nonlinear descriptor systems application to estimate the state of charge of lithium-ion batteries

One way in which nonlinear descriptor systems of (index-k) naturally arise is through semiexplicit differential-algebraic equations. The study considers the nonbilinear dynamical systems which are described by the class of higher-index differential-algebraic equations (DAEs). Their nature is analysed both quantitatively and qualitatively, and stability characteristics are presented for their solution. Higher-index differential-algebraic systems seem to show inherent shaky around their solution manifolds. The often use of logarithmic norms is for the estimation of stability and perturbation bounds in linear ordinary differential equations (ODEs). The question of how to apply the notation of logarithmic norms to nonlinear DAEs has long been an open question. Other problem extensions including nonlinear dynamics and nonbilinear DAEs need subtle modification of the logarithmic norms. The logarithmic norm is combined by conceptual focus with the finite-time stability criterion in order to treat nonbilinear DAEs with the aim of covering some unbounded operators. This means we obtain the perturbation bounds from differential inequalities for a norm by the use of the relationship between Dini derivatives and semi-inner products. A numerical result obtained when tested on the nonbilinear mechanical system with a larger scale showed that the method was highly efficient and accurate and particularly suitable for nonbilinear DAEs.

Abstract The problem of designing state and adaptive disturbance observers for a class of nonlinear descriptor systems with the disturbance generated by an unknown exogenous system is addressed in this paper. Especially, the nonlinear terms satisfy incremental quadratic constraints parameterized by a bunch of multiplier matrices, which can represent various classes of nonlinearities. For three different cases in the output equation, the state and adaptive disturbance observers are proposed. Some sufficient conditions formulated in terms of linear matrix inequalities and algebraic constraints are derived to guarantee that state and disturbance error systems, simultaneously, are asymptotically stable with performance level. The availability of proposed observers is validated by two practical examples.

Abstract In this article, the design and analysis of fault tolerant control (FTC) are investigated for stochastic nonlinear descriptor systems by using the sliding mode approach. First, an integral sliding mode‐based descriptor observer (ISM‐DO) is constructed to estimate the state and fault. The proposed ISM‐DO retains the basic characteristics of descriptor systems, meanwhile, it is not only suitable for dual normalized descriptor systems, but also for non‐dual normalized descriptor systems. It is more universal than the existing sliding mode fault tolerant control (SMFTC) methods. In the sequel, a novel restricted equivalent form based algorithm is developed to solve the sliding parameters. The proposed method provides a general algorithm, and it does not impose additional rank constraints. Further, a new adaptive sliding mode fault tolerant controller is constructed to stabilize the fault system, the assumption related to the diffusion matrix is removed. Based on these ingredients, the asymptotically mean square admissibility of the closed‐loop fault system and the observer gain matrix are got by a sufficient linear matrix inequality condition. The effectiveness of the proposed method is verified by simulations.

Adaptive observer design for a class of nonlinear descriptor systems

This paper concerns the design of a generalized functional observer for Takagi–Sugeno descriptor systems. Furthermore, a generalized structure is herein introduced for purposes of estimating linear functions of the states of descriptor nonlinear systems represented into a Takagi–Sugeno descriptor form. The originality of the functional generalized observer structure is that it provides additional degrees of freedom in the observer design, which allows for improvements in the estimation against parametric uncertainties. The effectiveness of the developed design is illustrated by a nonlinear model of a single link robotic arm with a flexible link. A comparison between the functional generalized observer and the functional proportional observer is given to demonstrate the observer performances.

We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling since it is invariant under power-conserving interconnection, congruence transformations and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity and numerical methods to check these properties.After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analysed, and time discretization and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed.The properties of port-Hamiltonian descriptor systems and their use in modelling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.

Improved estimation of algebraic states for nonlinear descriptor systems by integration of evolutionary algorithms within EKF framework

This paper presents an alternative to design a static output feedback controllers for nonlinear descriptor systems. The methodology is based on the direct Lyapunov method, from which, after a convex rewriting of the original nonlinear systems, conditions in the form of linear matrix inequalities are obtained. The proposal is shown to be more relaxed than previous ones in two ways; first, unmesaurable nonlinearities can be directly considered and, second, more flexibility can be obtained with the selection of different slack variables; such advantages are illustrated via numerical examples.

A fuzzy integral sliding-mode parallel control approach for nonlinear descriptor systems