Nonlinear Singular Systems Research Articles

Reachable Set Estimation for Singular Discrete‐Time Nonlinear Systems With Time‐Varying Delays

ABSTRACTIn this article, the reachable set estimation problems for a class of singular discrete‐time nonlinear systems (SDNS) with time‐varying delay (TVD) are presented. First, we employ a Lyapunov function scheme and utilize zero‐type free matrix equations. Based on this, a criterion is derived in the form of linear matrix inequalities (LMIs) to guarantee that the considered system to be regular, causal, and the state trajectory of the system is bounded. Second, by developing the singular systems decomposition technique into slow and fast subsystems and using the introduced auxiliary lemma, we establish that the state trajectory of the fast subsystem is also bounded, and the reachable set of singular nonlinear system is bounded within the ellipsoid. Finally, Numerical examples are presented to illustrate the effectiveness of the proposed method.

Designing a switching law for Mittag-Leffler stability in nonlinear singular fractional-order systems and its applications in synchronization

In this work, the stability and synchronization issue of switched singular continuous-time fractional-order systems with nonlinear perturbation are examined. Using the fixed-point principle and S-procedure lemma, a sufficient condition for the existence and uniqueness of the solution to the switched singular fractional-order system is first stated. Next, using the Lyapunov functional method in combination with some techniques related to singular systems and fractional calculus, a switching rule for regularity, impulse-free, and Mittag-Leffler stability is developed based on the formation of a partition of the stability state regions in convex cones. For synchronizing switched fractional singular dynamical systems, we propose a state feedback controller that ensures regularity, impulse-free, and Mittag-Leffler stable in the error closed-loop system. Finally, the ease of use and computational convenience of our proposed methods are illustrated by two numerical examples and a practical example about DC motor controlling an inverted pendulum accompanied by simulation results.

Observer-based consensus control for a class of nonlinear singular multi-agent systems

This paper studies the bounded consensus control problem for nonlinear singular multi-intelligence systems (NSMASs) based on observers, where the structure of nonlinear dynamics is completely unknown. Firstly, the unknown nonlinear terms are estimated by the linear parameterization method. However, since the system state is not directly available, the observers are used for state estimations, which leads to a double estimation problem in the parameterization process. Moreover, the distributed adaptive control protocols and the adaptive laws of unknown parameter estimations are developed in the leader-following and leaderless frameworks. Furthermore, the proposed control protocols achieve bounded consensus for the NSMASs. And the proposed nonlinear estimation methods guarantee bounded estimation performance of the unknown nonlinear terms in the double estimation problem. Finally, two examples are given to illustrate the effectiveness of the proposed methods.

Dissipativity-based robust filter design for singular fuzzy systems with dynamic quantization and event-triggered mechanism

This paper focus on the design issue of event-based dissipative filter for quantized nonlinear singular systems. To save the communication resource, we employ a dynamic quantizer to quantized the measurement output signal prior to transmitting it to the filter via digital communication. Furthermore, the paper also presents an event-triggered mechanism for determining the transmission of the quantized signal. The intention of this paper is to try to propose an event-triggered filter to ensure that the filtering error system (FES) is admissible and satisfies strictly dissipativity performance, in the presence of dynamic quantization and occurring uncertainties. The Lyapunov function is utilized to establish a sufficient condition for the FES to meet the strictly dissipative performance. Depending on the obtained conditions, the required filter parameters are proposed using the standard linear matrix inequality (LMI) technique. Ultimately, the effectiveness of the developed method is verified by an actual circuit simulation.

Finite‐time mixed ℋ∞$$ {\mathscr{H}}_{\infty } $$ and passive filtering for nonlinear singular systems under fading channels

AbstractThis article focuses on the finite‐time mixed and passive filtering problem for discrete‐time singular systems with randomly occurring nonlinear perturbations. The main purpose is to design a filter, which ensures the singular error system is stochastically finite‐time bounded and satisfies mixed and passive performance. By constructing an augmented matrix, the properties of matrix determinant and rank are used to prove that the singular system is regular and causal, Finsler's lemma and Projection lemma are used in the design process to acquire additional slack variable matrices, thereby enhancing the solution space with extra degrees of freedom. Then, the unknown parameters of the filter are obtained by solving the less conservative linear matrix inequalities. Finally, the feasibility of the proposed method is verified through a common numerical example and by controlling a DC motor for an inverted pendulum.

Reliable stabilization of discrete-time nonlinear singular systems with time-varying delays

The problem of delay-dependent reliable control for uncertain discrete-time nonlinear singular system with time-varying delays is investigated. A new set of delay-dependent sufficient conditions are established for the existence of a reliable controller which ensures the singular system to be regular, causal (absence of impulses) and stable for all admissible parameter uncertainties. First, the controller for singular system in the presence of known actuator fault matrix is designed. Further, the result is extended to study the robust stabilization of singular systems with unknown actuator fault matrix. The sensor failures of actuator are considered by a variable, which is varying in a given interval. The conditions for the existence of proposed controller design are established in the form of linear matrix inequality (LMI), which can be easily solved via standard numerical software. Finally, numerical examples with simulation results are provided to illustrate the usefulness of control design and results are compared with existing results to show the less conservatism of the proposed theory.

On a singular mathematical model for brain lactate kinetics

In this paper, we consider a singular nonlinear differential system that characterizes the intricate dynamics of brain lactate kinetics between cells and capillaries, as described by System (1.1) below. We begin by establishing the existence and uniqueness of nonnegative solutions for our system through the application of Schauder's fixed‐point theorem. Subsequently, we explore the behavior of these solutions as the viscosity term approaches zero, shedding light on the system's dynamic evolution in such scenarios. To provide empirical validation for our theoretical findings, we offer a series of numerical simulations. These simulations not only confirm the results we have obtained but also reinforce prior research, underscoring the model's efficiency in capturing the complexities of the brain's lactate kinetics. Our work contributes not only to the theoretical underpinning of this field but also to its practical implications, making it a valuable resource for both researchers and practitioners seeking to comprehend and manipulate these vital biological processes.

Event-Triggered Consensus Control for Nonlinear Singular Multi-Agent Systems under Directed Topology

Event-Triggered Consensus Control for Nonlinear Singular Multi-Agent Systems under Directed Topology

Multi-sensor information fusion filtering for delayed nonlinear stochastic singular systems with packet disorders

The fusion filtering problem is investigated for a class of time-varying nonlinear singular systems with packet disorders (PDs). The key idea is to transfer the original singular systems with multiple sensors to full-order augmented nonsingular systems by utilizing the matrix transformation method. The phenomenon of PDs accompanied with random transmission delays (RTDs) is considered, which exists in the channel from sensor to filter. A set of independent and identically distributed random variables obeying known probability distribution are introduced to model the RTDs. A centralized fusion filter is designed, which employs the rounding function of the mathematical expectation of the RTDs. Furthermore, the nonlinear function is linearized by utilizing the method of Taylor series expansion. Then, the desired upper bounds of the estimation error covariance are guaranteed and then locally minimized through appropriately designing the corresponding filter gains. Finally, a numerical example with comparisons is carried out to verify the feasibility of the proposed fusion filtering method.

Finite‐time robust simultaneous stabilization control for two nonlinear singular time‐delay systems

AbstractIn this paper, the finite‐time simultaneous stabilization problem of two nonlinear singular time‐delay Hamiltonian systems is investigated. Firstly, the two general nonlinear singular time‐delay Hamiltonian systems are transformed into their Hamiltonian differential‐algebraic forms by equivalent transformation, respectively, they are then converted to two strictly dissipative Hamiltonian forms via the state feedback method, and two strictly dissipative Hamiltonian systems are extended into a Hamiltonian one. Based on which, a suitable Hamiltonian function is constructed to analyze the finite‐time simultaneous stability and finite‐time robust simultaneous stabilization problem. Finally, an illustrative example is given to show the effectiveness of the results obtained. It should be pointed out that since there does not exist a finite‐time stability criterion on the nonlinear singular delay system, we adopt a new double compound asymptotic method.

Finite‐time adaptive stabilization and H∞$$ {H}_{\infty } $$ control for a class of nonlinear Hamiltonian singular systems with parametric perturbations

AbstractThis article addresses the finite‐time adaptive stabilization and finite‐time adaptive control problems of a class of nonlinear Hamiltonian singular systems with parametric perturbations. Under the suitable state feedback control scheme, a dissipative differential‐algebraic form of the nonlinear Hamiltonian singular system is obtained. According to this dissipative form, the finite‐time adaptive stabilization of the nonlinear Hamiltonian singular system with unknown parametric perturbations is investigated using the Hamiltonian function method. Then, considering the system with the external disturbances, a finite‐time adaptive controller is designed. Finally, the simulation results of the nonlinear singular circuit system demonstrate the efficacy of the controller designed in this article.

Mixed H∞‐based finite‐time passive filtering for a class of uncertain nonlinear singular systems

AbstractThis article discusses the finite‐time mixed and passive filtering design problem for uncertain nonlinear singular systems. The parameter uncertainties are constrained by well‐defined upper bounds on their magnitudes. The objective is to design a full‐order filter to ensure that the augmented singular system to be finite‐time boundedness(FTB) with and passivity performance. Firstly, a sufficient condition for achieving singular finite‐time boundedness with mixed and passivity performance for uncertain nonlinear singular system is derived by introducing a free matrix. Secondly, a novel criterion is established for analyzing finite‐time and passivity problem of filtering error singular system based on augmented matrix technique. Thirdly, the filtering design conditions are presented and the relevant parameters of the desired filter are determined using linear matrix inequality decoupling principle. Finally, a practical circuit system and a numerical example are provided to demonstrate the feasibility of the proposed scheme.

On the well posedness of a mathematical model for a singular nonlinear fractional pseudo-hyperbolic system with nonlocal boundary conditions and frictional damping terms

<abstract><p>This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system with frictional damping terms. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedness of the nonlinear fractional system. This mathematical model of pseudo-hyperbolic systems arises mainly in the theory of longitudinal and lateral vibrations of elastic bars (beams), and in some special case it is propounded in unsteady helical flows between two infinite coaxial circular cylinders for some specific boundary conditions.</p></abstract>

Distributed optimization of nonlinear singularly perturbed multi-agent systems via a small-gain approach and sliding mode control

This paper addressed the challenging problem of distributed optimization for nonlinear singular perturbation multi-agent systems. The main focus lies in steering the system outputs toward the optimal points of a globally objective function, which was formed by the combination of several local functions. To achieve this objective, the singular perturbation multi-agent system was initially decomposed into fast and slow subsystems. Compared to traditional methods, robustness in reference-tracking signals was ensured through the design of fast-slow sliding mode controllers. Additionally, our method ensured robustness against errors between reference signals and optimal values by employing a distributed optimizer to generate precise reference signals. Furthermore, the stability of the entire closed-loop system was rigorously guaranteed through the application of the small-gain theorem. To demonstrate the efficacy of the proposed approach, a numerical example was presented, providing empirical validation of its effectiveness in practical scenarios.

Dynamic PDC and HMM-based asynchronous output feedback strict dissipative control for nonlinear singular incomplete jumping systems with actuator saturation

This paper investigates the asynchronous output feedback strict dissipative control for nonlinear singular incomplete jumping systems with actuator saturation via dynamic parallel distributed compensation technique and hidden Markov model mechanism. Dynamic parallel distributed compensation technique is used to design the asynchronous fuzzy dynamic output feedback controller and hidden Markov model is employed to describe the asynchronous phenomenon between the singular Markov jump system modes and the dynamic output feedback controller modes. Novel conditions of stochastic admissibility and strict dissipativity for the fuzzy closed-loop singular Markov jump systems are derived based on singular value decomposition technique and improved L-K functional. The asynchronous dynamic output feedback controller gains are developed in terms of linear matrix inequalities and the minimization optimization scheme is proposed by established ellipsoid. Single-link robot arm simulation results illustrate the practicability and effectiveness of the proposed methods.

Biorthogonal flatlet multiwavelet collocation method for solving the singular nonlinear system with initial and boundary conditions

PurposeThe numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.Design/methodology/approachThe approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.FindingsAccording to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.Originality/valueThe key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

Dissipative control for switched nonlinear singular systems with dynamic quantization

This paper focuses on the state feedback control for discrete-time switched Takagi–Sugeno fuzzy singular systems with input quantization. The considered singular systems combine the characteristics of switched hybrid systems and Takagi–Sugeno (T–S) fuzzy systems. The main goal is to co-design the controller and the quantizer based on the interval adjustment rule and satisfy the prescribed dissipative performance such that the quantized closed-loop singular (QCLS) system is stochastically admissible. In addition, the design conditions of the minimum quantization range are obtained by optimizing related parameters. Finally, a tunnel diode circuit example is presented to demonstrate the effectiveness of the proposed new design method.

Finite-time H∞ control for a class of Takagi–Sugeno fuzzy singular systems

This article studies the finite-time stabilization and finite-time H∞ control problems for a class of Takagi–Sugeno fuzzy singular systems and proposes a finite-time H∞ fuzzy controller to enable the system to cope with external disturbances, and the closed-loop system state converges to the equilibrium point. Before designing the controller, impulse controllability of the Takagi–Sugeno fuzzy singular system is discussed. Then, a finite-time H∞ fuzzy controller is designed, and the finite-time stabilization conditions of the Takagi–Sugeno fuzzy singular system are expressed by linear matrix inequality. Finally, the simulation outcomes of a nonlinear singular system demonstrate that the designed controller has a good suppression effect on external disturbances, and the closed-loop system state converges to the equilibrium point in finite-time T and remains at the equilibrium point after T.

Some stability results on non-linear singular differential systems with random impulsive moments

This paper studies the exponential stability for random impulsive non-linear singular differential systems. We established some new sufficient conditions for the proposed singular differential system by using the Lyapunov function method with random impulsive time points. Further, to validate the theoretical results' effectiveness, we finally gave two numerical examples that study with graphical illustration and an additional example involving matrices with complex entries, proving the results to be true in that case as well.

Robust stabilization and adaptive H∞ control of nonlinear singular systems via static output feedback

This article studies robust stabilization and adaptive [Formula: see text] control problems for a class of continuous-time nonlinear singular systems by static output feedback. First, under sufficient conditions which the system is impulse controllable, the static output feedback controller is designed, and the stabilization problem of the system is discussed. Second, the [Formula: see text] and adaptive [Formula: see text] controllers are designed for the continuous-time nonlinear singular systems with external disturbances. Compared with the existing conclusions, the controller can be designed for more nonlinear singular systems. Finally, an example of a nonlinear singular circuit is given, and the simulation results verify the validity of the proposed controllers.