Stability Of Time-delay Systems Research Articles

Delay margin computation for single area load frequency control system with time delay

In this paper, the delay margin is computed for a load frequency control system with time delay of a single area using delay-independent stability analysis method. The delay-independent stability analysis method is used to consider some characteristics. The delay-independent stability of time-delayed systems may lose stability if their eigenvalue spectrum lies on the imaginary axis for delays. The stability is guaranteed for the delay margin, which must be known. A method for computing the delay margin for load frequency control system with delay is presented. In this method, the system's characteristic equation is converted into frequency-dependent equation. Then the spectral radius of the matrix pair is used to compute the crossing frequency at which eigenvalues cross the imaginary axis. The crossing frequencies and angles are computed using sweeping tests and binary recursive algorithm. At the end, the effectiveness of the proposed method is demonstrated using a case study of load frequency control system with delay of a single area.

Complex Dynamics and PID Control Strategies for a Fractional Three-Population Model

In recent decades, there have been many studies on Hopf bifurcation and population stability with time delay. However, the stability and Hopf bifurcation of fractional-order population systems with time delay are lower. In this paper, we discuss the dynamic behavior of a fractional-order three-population model with pregnancy delay using Laplace transform of fractional differential equations, stability and bifurcation theory, and MATLAB software. The specific conditions of local asymptotic stability and Hopf bifurcation for fractional-order time-delay systems are determined. A fractional-order proportional–integral–derivative (PID) controller is applied to the three-population food chain system for the first time. The convergent speed and vibration amplitude of the system can be changed by PID control. For example, after fixing the values of the integral control gain ki and the differential control gain kd, the amplitude of the system decreases and the convergence speed changes as the proportional control gain kp decreases. The effectiveness of the PID control strategy in complex ecosystem is proved. The numerical simulation results are in good agreement with the theoretical analysis. The research in this paper has potential application values concerning the management of complex population systems. The bifurcation theory of fractional-order time-delay systems is also enriched.

Global exponential stability of nonlinear time-delay system under impulsive control with distributed delay

This paper investigates the global exponential stability of nonlinear time-delay system (NTDS) under impulsive control with distributed delay. By establishing a connection between the distributed delay in impulsive control and the number of impulsive times, a novel Razumikhin-type impulsive inequality is formulated. Using this inequality and the concept of the average impulsive interval, some delay-independent sufficient conditions of the global exponential stability of NTDS are derived. This indicates that the upper bounds of the impulsive interval and state time delay are no longer constrained, and the mutual constraint between the distributed delay in impulse and state time delay is removed, which greatly improves the results of recent studies. Finally, two numerical examples are presented to showcase the validity and practicality of the results.

Almost sure stability and stabilization of variable dual switching time-delay systems

Almost sure stability and stabilization of variable dual switching time-delay systems

New negative-determination conditions for cubic polynomials with applications to time-varying delay systems

This paper studies the stability of time delay systems. The Lyapunov-Krasovskii functional (LKF) method is used for our study, in which a novel negative-determination criterion for cubic polynomials is proposed. An improved stability criterion of time delay system is obtained by the new method. The effectiveness of the proposed method is verified by some numerical examples and reduced conservativeness can be obtained.

ASYMMETRIC FUNCTIONAL-BASED APPROACHTO EXPONENTIAL STABILITY OF LINEAR DISTRIBUTEDTIME-DELAY SYSTEMS

In this paper, the problem of exponential stability of linear time-delay systems with mixed discrete and distributed delays is studied. Based on an asymmetric Lyapunov-Krasovskii functional approach, sufficient conditions are derived in terms of linear matrix inequalities to guarantee the exponential convergence of the system state trajectories with a prescribed decay rate. The efficacy of the obtained results is demonstrated by a given numerical example and simulations.

Improved Stability Analysis of Neural Networks with Time Delay Based on Variable Augmented Free Weight Matrix

This article investigates the stability of time-delay neural network system by the free-weighting matrices based on variable-augmentation.In the process of accurately evaluating the stability of delayed neural network systems, the conservative simplification of stability criteria has received widespread attention. In this paper, a variable-augmented-based free-weighting matrix method is applied to time-varying delayed neural network systems, and unnecessary free weighting matrices are removed. Some results with fewer free-weighting matrices are obtained, while maintaining their stability and conservatism. Finally, a specific neural network numerical example was used to demonstrate the effectiveness of this method.

Output Feedback Stabilization of High-Order Nonlinear Time-Delay Systems With Low-Order and High-Order Nonlinearities

In this paper, the output feedback stabilization of a class of high-order nonlinear time-delay systems with more general low-order and high-order nonlinearities is investigated. By constructing the new Lyapunov-Krasovskii functional and reduced-order observer, based on homogeneous domination theory together with the adding a power integrator method, an output feedback controller is developed to guarantee the equilibrium of the closed system globally uniformly asymptotically stable.

New exponentially weighted inequality and its application to finite-time stability of descriptor time-delay systems

The paper considers the finite-time stability of descriptor time-delay systems. Firstly, we establish a set of exponentially weighted orthogonal polynomials, and based on this, we obtain a new exponentially weighted inequality(EWI). Secondly, an augmented Lyapunov-like functional (LLF) is established driven by the presented EWI. Finally, by applying the EWI and the LLF, some sufficient conditions for ensuring the finite-time stability of the system are obtained. A small partial element equivalent circuit model and two numerical examples are used to verify the effectiveness of the method.

Event-Triggered Impulsive Control for Input-to-State Stabilization of Nonlinear Time-Delay Systems.

This article investigates the event-triggered impulsive control (ETIC) problem for a class of nonlinear time-delay systems subject to exogenous disturbances. An original event-triggered mechanism (ETM) which utilizes the information of system state and external input is constructed based on Lyapunov function approach. To achieve the input-to-state stability (ISS) of the considered system, some sufficient conditions are presented, in which the underlying relationship among ETM, exogenous input, and impulse action is established. Furthermore, the possible Zeno behavior induced by the proposed ETM is excluded simultaneously. As an application, the design criterion of ETM and impulse gain is put forward for a class of impulsive control systems with delay according to the feasibility of some linear matrix inequalities (LMIs). Finally, two examples with numerical simulations are provided to confirm the effectiveness of the developed theoretical results, where the synchronization issue of delayed Chua's circuit is considered.

A novel stability criterion and precise control for linear time delay systems based on regional configuration of the poles

This paper presents the concept of region stability and provides criteria for region stability of linear time delay systems, which can reveal the dynamic and steady-state performance of the systems more precisely. Corresponding design schemes for stabilization and tracking control that can accurately control various performance of time delay systems have also been explored. First, in the light of the connection between the poles and the dynamic properties of the system, the concept of region stability is given to describe the finer dynamic behavior of time delay systems. The criteria for the region stability are also presented. Second, the region stabilization methods are investigated, which can ensure that the system satisfies a certain dynamic performance by setting the eigenvalues in a certain convex region. Third, a precise tracking control of the linear time delay systems is addressed as an application of region stabilization. It can control the steady state performance and transient response of the tracking signal more precisely. Finally, three instances are provided to display the superiority of the new method for the performance indexes of the linear time delay systems.

Safe Stabilization for Stochastic Time-Delay Systems

Safe Stabilization for Stochastic Time-Delay Systems

General single/multiple integral inequalities and their applications to stability of time-delay systems

This paper is concerned with the stability of time delay systems. A new type of multiple integral inner product and a new class of multiple integral inner product space are firstly introduced. Based on them, general single/multiple integral inequalities are developed, and some previous integral inequalities including single and multiple integral inequalities can be regarded as their special cases. Secondly, suitable Lyapunov–Krasovskii functionals (LKF) are constructed. On the basis of the proposed integral inequalities, the derivative of the LKF can be gauged more accurate than those in the literature. As a result, some improved stability criteria of time-delay systems are formulated in terms of linear matrix inequalities. Through two numerical examples, it is shown that these criteria can provide larger delay upper bounds than some existing ones, showing the effectiveness and superiority of the proposed method.

Closing the gap from nominal to robust asymptotic stabilization of time delay systems

SummaryThe control problem of a general class of perturbed linear invariant time‐delay systems within the Sliding Mode Control framework is addressed. We introduce a novel class of sliding manifolds with a structure not previously reported in the literature, which constructively emerges from complete type functionals of the corresponding nominal time‐delay systems and depends on the whole state of the system. A fundamental conclusion derived from this proposal is that every time‐delay system with matched perturbations, whose associated nominal system is exponentially stabilizable, is robustly asymptotically stabilizable. In fact, the introduced sliding manifold enables to construct discontinuous and continuous sliding mode control‐based algorithms that mitigate different classes of perturbations.

Stabilization of time-delay nonlinear systems using Takagi-Sugeno fuzzy models

We propose a controller design method for time-delay nonlinear systems with delays affecting both the states and the inputs, represented by Takagi-Sugeno fuzzy models with nonlinear consequents. To handle the nonlinearities in the consequents we assume that they are slope-bounded. Linear matrix inequality conditions are formulated to design the controller. The obtained results are compared to state-of-the-art approaches and illustrated on two examples.

Absolute stability of time-delayed Lurie direct control systems with unbounded coefficients

Absolute stability of time-delayed Lurie direct control systems with unbounded coefficients

On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation

This work is dedicated to the stability analysis of time-delay systems with a single constant delay using the Lyapunov–Krasovskii theorem. This approach has been widely used in the literature and numerous sufficient conditions of stability have been proposed and expressed as linear matrix inequalities (LMI). The main criticism of the method that is often pointed out is that these LMI conditions are only sufficient, and there is a lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel–Legendre inequality or orthogonal polynomial-based inequalities. The interest of these methods relies on their hierarchical structure with a guarantee of reduction of the level of conservatism. However, the convergence is still an open question that will be answered for the first time in this paper. The objective is to prove that the stability of a time-delay system implies the feasibility of these scalable LMI, at a sufficiently large order of the Legendre polynomials. Moreover, the proposed contribution is even able to provide an analytic estimation of this order, giving rise to a necessary and sufficient LMI for the stability of time-delay systems.

Sampled-data output tracking control based on T–S fuzzy model for cancer-tumor-immune systems

This study investigates the tracking control issue for a cancer-tumor immune nonlinear model, which explains the interaction between cancer cells and the immune system in the body. To achieve this, the nonlinear model is first converted into linear sub-models using the Takagi–Sugeno (T–S) fuzzy methodology. The tracking and control of cancer cell proliferation while preserving immune cells is proposed using a sample-data output tracking control technique. Time-dependent discontinuous Lyapunov Krasovskii functional is recommended based on the improved Wirtinger inequality, this may provide additional information about the sawtooth structure of the sampling interval. Less conservative stability criteria are extracted having the form of linear matrix inequalities (LMIs) in consequence of extended reciprocally convex matrix inequality and Wirtinger-based integral inequality. The proposed approach achieves the stabilization of an augmented time delay system that connects with a given fuzzy nonlinear model and tracking error system. Furthermore, the proposed approach reduces the impact of external disturbances under the H∞ norm bound.

Finite-time robust adaptive simultaneous stabilisation of nonlinear time-delay systems with actuator saturation

This paper focuses on finite-time robust adaptive simultaneous stabilisation for nonlinear time-delay systems, taking into account actuator saturation and unknown parameters. Firstly, by employing adaptive control strategies and finite-time stability theory, a finite-time adaptive simultaneous stabilisation scheme for the nonlinear time-delay systems is developed. Secondly, utilising the orthogonal decomposition method, a finite-time robust adaptive simultaneous stabilisation scheme for nonlinear time-delay systems with external disturbances is proposed. Finally, simulations demonstrate the effectiveness of the provided control scheme.

Observer based stabilization of linear time delay systems using new augmented LKF

This paper addresses the stabilization problem of linear time-varying delay systems with unmeasurable states. A novel augmented Lyapunov–Krasovskii functional (LKF) is proposed that effectively accounts for the impact of time delays, and an observer based stabilization controller is developed employing linear matrix inequality (LMI) based optimization technique. The utilization of extended reciprocally convex matrix inequality (ERCMI) is employed in this work to establish less conservative stabilization conditions within the framework of linear matrix inequalities (LMIs). By formulating a convex optimization problem, the observer gain and controller gains are determined. Simulation results are used to validate the design, and two numerical examples are considered to prove the usefulness of the proposed method over existing methods.