Delay-dependent Stability Research Articles

Delay-Dependent Stability Region for the Distributed Coordination of Delayed Fractional-Order Multi-Agent Systems

Delay and especially delay in the transmission of agents’ information, is one of the most important causes of disruption to achieving consensus in a multi-agent system. This paper deals with achieving consensus in delayed fractional-order multi-agent systems (FOMAS). The aim in the present note is to find the exact maximum allowable delay in a FOMAS with non-uniform delay, i.e., the case in which the interactions between agents are subject to non-identical communication time-delays. By proving a stability theorem, the results available for non-delayed networked fractional-order systems are extended for the case in which interaction links have nonequal communication time-delays. In this extension by considering a time-delay coordination algorithm, necessary and sufficient conditions on the time delays and interaction graph are presented to guarantee the coordination. In addition, the delay-dependent stability region is also obtained. Finally, the dependency of the maximum allowable delay on two parameters, the agent fractional-order and the largest eigenvalue of the graph Laplacian matrix, is exactly determined. Numerical simulation results are given to confirm the proposed methodologies.

Stability and Stabilization for Amplidyne Electrical Systems via a Novel Negative-Definiteness Lemma

It is arduous to determine the stability for amplidyne electrical systems (AESs) with time-varying delay. In this article, a novel approach with two independent tuning parameters is established to obtain the stability conditions for AESs with time-varying delay. The new method could optimize negative determination conditions by tuning two tuning parameters without producing extra decision variables. Some existing lemmas commonly employed in the literature are contained in the proposed method as exceptional cases. Then, an improved Lyapunov–Krasovskii (L–K) functional is proposed in which more delay and state information are employed. Based on the novel quadratic function negative-definiteness method and the proposed L–K functional, new delay-dependent stability and stabilization criteria are achieved for AESs with time-varying delay. Finally, numerical examples and simulation results are conducted to show the advantages of the proposed approach.

Input-to-State Stability for Time-Delay Systems With Large Delays

In this article, we consider the input-to-state stability (ISS) problem for a class of time-delay systems with intermittent large delays, which may cause the invalidation of traditional delay-dependent stability criteria. The topic of this article features that it proposes a novel kind of stability criterion for time-delay systems, which is delay dependent if the time delay is smaller than a prescribed allowable size. While if the time delay is larger than the allowable size, the ISS can be preserved as well provided that the large-delay periods satisfy the kind of duration condition. Different from existing results on similar topics, we present the main result based on a unified Lyapunov-Krasovskii function (LKF). In this way, the frequency restriction can be removed and the analysis complexity can be simplified. A numerical example is provided to verify the proposed results.

Non-fragile sampled-data control for synchronization of chaotic fractional-order delayed neural networks via LMI approach

This study addresses the master and slave synchronization problem of chaotic fractional-order delayed neural networks using a non-fragile sampled-data control (NFSDC) scheme which includes uncertainty information in the control gain matrix. An appropriate Lyapunov functional is constructed with information about sampling instants. A new, improved fractional-order inequality is developed to estimate the integral term. Then, based on the new integral inequality, the delay-dependent stability criteria are derived in the form of linear matrix inequality, which guarantees the asymptotic stability of the fractional-order error systems. This implies that the proposed NFSDC can synchronize the fractional-order master and slave systems. Moreover, numerical simulation results illustrate the synchronization nature and the influence of fractional derivatives in the system dynamics. Finally, it can be concluded that the proposed method and control scheme are effective.

The Stability Analysis for a 2 × 2 Conservation Law with PI Controller and PDP Controller

This article is concerned with the stability analysis of a 2 × 2 conservation law with PI and PDP boundary feedback control. In the PI controller case, the authors establish a sufficient and necessary stability condition for feedback parameters by Walton–Marshall criterion. Moreover, in PDP controller case, the characteristic equation is an exponential polynomial which includes two exponential arguments. Based on Ruan’s results on stability for quasi-polynomials and Schur–Cohn criterion on the roots inside the unit circle for a real coefficient polynomial, a complete characterization for system parameters and the feedback delay is presented. Also, the delay-independent and delay-dependent stability intervals are obtained in this article.

Delay partitioning approach to the delay-dependent stability of discrete-time systems with anti-windup

ABSTRACT In this digital era, the basis of every smart instrument is discrete signal models e.g. in Networked control systems, Cyber physical systems etc. It has been shown that time-delays are unavoidable during the digital implementation of an engineering system. Therefore, the stabilization of discrete time delayed systems is gaining the high importance [1–10]. Although a lot of literature is found on the stabilization of time delayed systems for a long time using the construction of proper non-negative Lyapunov functional. Recalling some existing results on this issue, the LMI-based stability conditions are obtained by its forward difference negative-definite in direction to claim the less conservative results [15–25]. In order to seek less conservative stability criteria, this paper introduces an anti-windup scheme appended with Wirtinger inequality, reciprocal convex approach and delay partitioning of a discrete-time delayed systems by using Lyapunov Krasovskii functional. To accomplish this task, delay partitioning technique may be utilized to develop improved stability conditions for the considered system. The Wirtinger-based inequality and reciprocal convex approach has been employed to derive less conservative results. On employing the delay partitioning, a novel linear matrix inequality-based criterion is proposed to stabilize such systems. The considered Lyapunov-Krasovskii functional includes the information of intermediate delay to acknowledge the delay information implicitly that ensures the considered system to be regular, impulse free and stable in terms of linear matrix inequalities. The estimation of the attraction basin is to ensure that the state remains inside the level set of a certain Lyapunov function. Numerical simulation verifies that the presented method reduces conservatism than the existing results.

Stabilization of nonlinear fractional order faulty systems with time-varying delay

This paper focuses on the issue of stabilization of nonlinear fractional order time-delay systems in the presence of fault. A general class of nonlinear fractional order systems is considered in which fault causes variations in the system dynamic and actuators. Besides, time-varying delay is noticed in the system equations that is of high importance in control of real-life systems. In order to facilitate the problem of controller design for such systems, a precise method for modeling complex nonlinear systems, namely, Takagi–Sugeno fuzzy model, is adopted. Regarding a Caputo derivative–based Lyapunov–Krasovskii functional, new delay-dependent stabilization conditions are established in the form of linear matrix inequalities. Eventually, two examples including a truck–trailer and Lorenz system are simulated in order to evaluate the results of this research work.

Delay-Dependent Stability Condition in Finite Frequency Domain: Input-Output Approach

Over finite frequency ranges, this study analyzes delay-dependent stability criteria for time-varying delay systems. To develop new delay-dependent stability conditions in finite frequency domains for the continuous time-varying delay systems, we construct an input-output technique for delay-dependent systems utilizing the two-term transformation model paired with Scaled-Small Gain (SSG) approach. Sufficient conditions are obtained and described in terms of linear matrix inequalities (LMIs) constraints based on a novel delay-dependent finite frequency inequality. As an illustration, a numerical example has been supplied to test the correctness of the current approach.

Delay-dependent stability switches in a delay differential system

In this study, stability properties of a linear delay differential system $ x'(t) = -ax(t-\tau)-by(t) $, $ y'(t) = -cx(t)-dy(t-\tau) $ are considered, where $ a $, $ b $, $ c $, and $ d $ are real numbers and $ \tau>0 $. Some explicit necessary and sufficient conditions are presented for the zero solution of the system to be asymptotically stable. The results demonstrate that delay-dependent stability switches in the system can occur not only when $ bc<0 $ but also when $ a>0 $, $ b>0 $, $ c>0 $, and $ d>0 $. Some examples are provided to illustrate the delay-dependent stability switches. The proof technique is based on careful analysis of the existence and the transversality of characteristic roots on the imaginary axis.

Delay-dependent stability conditions for fundamental characteristic functions

This paper is devoted to the investigation on the stability for two characteristic functions $f_1(z) = z^2+pe^{-z\tau }+q$ and $f_2(z) = z^2+pz e^{-z\tau }+q$, where $p$ and $q$ are real numbers and $\tau >0$. The obtained theorems describe the explicit stability dependence on the changing delay $\tau $. Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.

An Extended-Matrix-Separation-Based Summation Inequality on Stability of Discrete-Time Delayed Lur'e System*

This paper investigates the stability of discrete-time Lur'e systems with a time-varying delay. Firstly, an extended-matrix-separated-based summation inequality is established to estimate the summation term containing the system state and the forward difference of the state. Then, an improved delay-product-type Lyapunov-Krasovskii functional (LKF) is proposed, which involves the characteristics of matrices refined and the information of delay squared. By using a quadratic function negative transformation lemma based on matrix-injection, a delay-dependent stability criterion is obtained. Finally, a numerical example demonstrates the advantages of the proposed criterion.

Noise-to-state and delay-dependent stability of highly nonlinear hybrid stochastic differential equation with multiple delays

This study is intended to discuss the noise-to-state stability (NSS) and the delay-dependent stability (DDS) of the highly nonlinear hybrid stochastic differential equation (HNHSDE) with multiple delays. Distinct from the white noise considered in the existing literature, we introduce the coloured noise characterised by a finite second-order process into the HNHSDE. Under some mild assumption conditions, we demonstrate that the HNHSDE exists a unique global solution (GS). Moreover, some criteria regarding the NSS in the qth moment of the GS are established. Furthermore, several theorems are provided for guaranteeing the pth moment asymptotic stability and the almost surely asymptotic stability of the GS with the aid of the delay-dependent Lyapunov functional and stochastic analysis theory. At last, two illustrative examples are given to show the feasibility of our developed theory findings.

Delay-dependent and order-dependent conditions for stability and stabilization of fractional-order memristive neural networks with time-varying delays

The stability and stabilization problems of fractional-order memristive neural networks with time-varying delays are addressed. Based on Jensen-based integral inequalities, novel delay-dependent and order-dependent stability conditions for fractional-order memristive neural networks with time-varying delays are established. The proposed stability criterion is in terms of linear matrix inequalities and is easy to be verified and applied. Then, a linear feedback control law that stabilizes fractional-order memristive neural networks with time-varying delays is designed by utilizing the obtained stability conditions. Two numerical examples are used to illustrate the effectiveness and less conservativeness of the proposed results.

Delay-Dependent Stability Analysis of Load Frequency Control Systems With Electric Vehicles.

This article investigates the problem of delay-dependent stability for the one-area load frequency control (LFC) system with electric vehicles (EVs). Two closed-loop models of the LFC system with EVs are proposed, including the model based on the model reconstructed technique and the model with uncertain parameters that considers state of charge. By employing the Lyapunov-Krasovskii functional method, two delay-dependent stability criteria are presented for the systems under study such that a more accurate admissible delay upper bound (ADUB) can be obtained. Case studies are finally carried out to disclose the interrelationship between the ADUB, PI controller gains, and other parameters of the EVs.

Robust Stability Analysis and Feedback Control for Uncertain Systems With Time-Delay and External Disturbance

This article addresses the delay-dependent Takagi–Sugeno (T–S) fuzzy state feedback control and exponential admissibility analysis for a class of T–S fuzzy singular uncertain systems. First, the T–S fuzzy model is employed to approximate the singular uncertain system with time-varying delay, saturation input, and unmatched disturbance. Second, the delay-dependent T–S fuzzy state feedback controller is designed by employing the T–S fuzzy model. Third, the free-weighting matrices and delay-dependent Lyapunov–Krasovskii functional with multiple integral terms are employed to derive the delay-dependent exponential admissibility conditions and prescribed H-infinity performance is guaranteed. Compared with previous works, the delay-dependent T–S fuzzy state feedback controller is designed for the T–S fuzzy singular uncertain system to relax system design conditions. The convex hull lemma is employed to convert the closed-loop system with saturation input into the closed-loop system without saturation input to enhance controller design flexibility. The Schur complement lemma and Gronwall Bellman lemma are employed to derive the less conservative delay-dependent stability conditions for determining controller gain matrices. The exact invariant set with less conservativeness is employed to convert the controller design problem into linear matrix inequalities (LMIs) optimization constraints to reduce computation complexity of solving LMIs. Finally, simulation examples are presented to show the effectiveness of the proposed methods.

Participation of Electric Vehicles in a Delay-Dependent Stability Analysis of LFC Considering Demand Response Control

Today, time-varying delays may result from a communication network’s engagement in frequency control services. These delays have an impact on the effectiveness of the load frequency control (LFC) system, which can occasionally lead to power system instability. The electric vehicle (EV) can be used as a beneficial source for the LFC system with the development of demand-side response due to its vehicle-to-grid capacity. Although demand response control has certain advantages for the power system, communication networks used in LFC systems result in time delays that reduce the stability of the LFC schemes. A stability study of an LFC system, comprising an EV aggregator with two additive time-varying delays, is demonstrated in this work. An enhanced Lyapunov–Krasovskii functional (LKF), which incorporates time-varying delays using the linear matrix inequality approach, is used to perform a delay-dependent stability analysis of the LFC to determine the stability zone and criterion. In conclusion, the efficiency of the proposed stability criterion is validated by making use of the thorough simulation findings.

Stability Analysis of Haptic Systems With Time-Varying Delay via a Delay-Product-Type Lyapunov–Krasovskii Functional

In this brief, the stability of the haptic system with a time-varying delay is studied by developing a new stability criterion. Firstly, the haptic system model with a time-varying delay is established. Next, based on the Lyapunov-Krasovskii stability theory, a delay-product-type Lyapunov-Krasovskii functional is constructed. Then, a new delay-dependent stability criterion with less conservatism due to more delay information is derived by applying an auxiliary function-based integral inequality. Finally, a case study is carried out to show the stable boundaries of different delays. The results show that, for the same delay boundary, the stable upper bound of the haptic system increases first and then decreases with the rise of the virtual damping. Moreover, increasing the time delay has negative effects and makes the system unstable.

Influence evaluation of signal transmission delay on supplementary damping controller for SVC in power system

Supplementary damping controller (SDC) for static var compensator (SVC) can be used to damp interarea low-frequency oscillations, but the influence of signal transmission delay on damping performance is generally neglected in the previous research. Because the delay may deteriorate damping performance or even cause instability for power system, this work intensively investigates the influence of signal transmission delay on input signal selection and parameter setting in SDC design. A novel evaluation method based on delay-dependent stability criterion and model order reduction is proposed to obtain the tradeoff between damping performance and delay margin for the SDC to be designed. Delay margin, determined by the proposed method, is taken as a critical index to evaluate the delay influence on SDC design. Case studies are carried out on two-area four-generator power system. Results show that for input signals with higher mode observability by residue method, such as current magnitude and active power of transmission line, delay margin decreases with the increase of SDC gain or delay variation ratio, which indicates that appropriately setting the gain can ensure SDC with better damping performance and larger delay margin.

Delay‐dependent stability of highly nonlinear neutral stochastic functional differential equations

AbstractThis article focuses on the delay‐dependent stability of highly nonlinear hybrid neutral stochastic functional differential equations (NSFDEs). The delay dependent stability criteria for a class of highly nonlinear hybrid NSFDEs are derived via the Lyapunov functional. The stabilities discussed in this article include stability, asymptotically stability and exponential stability. A numerical example is given to illustrate the criteria established.

A novel approach to exponential stability in mean square of stochastic difference systems with delays

By a novel approach, we present in this paper for the first time, some explicit criteria for the exponential stability in mean square of solutions of general stochastic difference systems with time-varying delays. Both delay-independent and delay-dependent stability criteria are provided. A discussion and illustrative examples are given.