Delay-dependent Stability Criteria Research Articles

Synchronous and non synchronous delay-dependent robust wide-area controllers for power system

Wide-area control is highly efficacious for maintaining sufficient damping of the low-frequency oscillations in the power system. The Wide-Area Measurement System (WAMS) provides wide-area signals to the control site through a communication channel. Even though the wide-area signal improved the dynamic behavior of the power system, a time delay usually exists during the transmission, which is unavoidable. The damping performance of the power system is degraded due to the time delay. In this paper, synchronized and non-synchronized feedback configurations are considered, and the time delay margin is calculated. The Lyapunov theory is adopted for investigating the synchronous and non-synchronous delay-dependent stability criteria for the power system. A lead–lag and a robust H∞ wide-area damping controller (WADC) are proposed to measure the delay margin for synchronized and non-synchronized delay feedback. The relationship between the damping performance and the delay margin is established. It is observed that a non-synchronous feedback-based controller can endure a larger time delay variation than a synchronous feedback-based controller. The nonlinear simulation is carried out to verify the effectiveness of the proposed controllers in Kundur’s two-area system and the IEEE-39 bus test system.

Stability and stabilization of systems with a cyclical time-varying delay via delay-product-type looped-functionals

In this paper, the stability analysis and synthesis of systems with a cyclical time-varying delay are investigated. At first, the entire delay interval is divided into intervals of monotonically increasing and monotonically decreasing alternating cycles. Then, a Lyapunov-Krasovskii functional (LKF) with two delay-product-type looped-functionals is constructed based on different delay monotone intervals. For the generated quadratic function with respect to delay in the derivative of LKF, a matrix-injection-based method is utilized to obtain the negative-definite conditions of LKF’s derivative. A delay-dependent stability criterion with less conservatism is derived and the corresponding state-feedback control strategy is given consequently. Finally, a numerical example is utilized to show the superiority and the effectiveness of the proposed method, and a single-area power system under load frequency control is applied to verify the feasibility of the proposed control strategy.

Improved stability and stabilisation conditions of uncertain switched time-delay systems

This article is concerned with the stability and stabilisation of switched time-delay systems (STDSs) with exponential uncertainty. Based on the Hurwitz convex combination and the energy attenuation principle, an improved state-dependent switching strategy is proposed, which switches to the next modes to obey the minimum energy. This approach fully considers the system dynamic of subsystems, which is more general. Considering the complex switching and delay dynamics, a mode-dependent Lyapunov–Krasovskii functional (LKF) that contains a triple integral term is constructed. The generalised free-matrix-based integral inequality (GFMBII) is used to estimate the integral terms in the derivative of the LKF, and an improved delay-dependent stability criterion is established in the form of linear matrix inequalities (LMIs). Further, to guarantee the stability of the STDSs with a large time-varying delay, a controller that considers the time delay and the exponential uncertainty is designed. Under this controller, a less conservative delay-dependent robust stabilisation criterion for STDSs with exponential uncertainty is established. The validity of the proposed methods is validated by two numerical examples and an application in river pollution control.

Stability Analysis of Discrete-Time Neural Networks With a Time-Varying Delay: Extended Free-Weighting Matrices Zero Equation Approach.

This research investigates the stability of discrete-time neural networks (DNNs) with a time-varying delay by using the Lyapunov-Krasovskii functional (LKF) method. Recent researches acquired some less conservatism stability criteria for time-varying delayed systems via some augmented LKFs. However, the forward difference of such LKFs resulted in high-degree time-varying delay-dependent polynomials. This research aims to develop some augmented state-related vectors and the corresponding extended free-weighting matrices zero equations to avoid the appearance of such high-degree polynomials and help to provide more freedom for the estimation results. Besides, an augmented delay-product-type LKF is also established for ameliorating the stability conditions of the time-varying delayed DNNs. Then, based on the above methods and Jensen's summation inequality, the auxiliary-function-based summation inequality, and the reciprocally convex matrix inequality, some less conservatism stability criteria for time-varying delayed DNNs are formulated. The validity of the proposed time-varying delay-dependent stability criteria is illustrated by two numerical examples.

Exponential stability for discrete-time impulsive positive singular system with time delays

This paper investigates the impulsive stability analysis issues of discrete-time positive singular systems with time delay. First, the paper addresses the positivity problem of the system by providing sufficient conditions. Next, a new method based on state transformations is presented to derive a new delay-dependent criterion for the exponential stability of impulsive positive singular systems. Finally, the effectiveness of the proposed conditions is validated through three numerical examples.

Enhanced cubic function negative-determination Lemma on stability analysis for delayed neural networks via new analytical techniques

This brief studies the stability issue of neural networks with respect to time-varying delay. First, an enhanced cubic functions negative-definiteness determination Lemma is proposed by using partitioning technique and Taylor’s formula. Compared with the previous results, this Lemma can eliminate the conservatism of constraint conditions by interval-decomposition method. Second, for the sake of the relaxed positive-definiteness determination, an asymmetric Lyapunov–Krasovskii functional is presented. Third, a resulting time-varying delay-dependent stability criterion is derived by making use of some inequality techniques. Finally, the advantage of the new approach are verified theoretically and numerically.

Stability of port-Hamiltonian systems with mixed time delays subject to input saturation

In this paper, we investigate the stability of port-Hamiltonian systems with mixed time-varying delays as well as input saturation. Three types of time delays, including state delay, input delay, and output delay, are all assumed to be bounded. By introducing the output feedback control law and utilizing serval Lyapunov–Krasovskii functionals, we present three delay-dependent stability criteria in terms of the linear matrix inequality. Meanwhile, we use Wirtinger’s inequality, constraint conditions, and Lyapunov–Krasovskii functionals of triple and quadruple integral form to obtain less conservative results. Some numerical examples demonstrate and support our results.

Synchronization of time-delayed complex networks with delay-partitioning: A Chua’s circuit networks application

This article addresses the almost surely global exponential synchronization (ASGES) for complex networks in the delay-partitioning framework. Under impulsive control, by dividing impulse intervals and time delay, the information of the impulse intervals can be more fully utilized. Firstly, a novel type of Lyapunov functional is introduced by applying a practical switching law, which combines mode-dependent average dwell-time (MDADT) and transition probability (TP). The use of the partition method makes the relationship between the undelayed state and the delayed state can be better handled, delay-dependent stability criteria are established with less conservatism. Secondly, under the action of impulsive control and fault-tolerant control, a controller synthesis algorithm is provided. Moreover, a corollary extends the results to intermittent control. Finally, an example of Chua’s circuit networks is provided to illustrate the effectiveness of proposed method, which can obtain larger allowable time-delay.

An improved delay-dependent stabilization criterion of linear time-varying delay systems: An iterative method

An improved delay-dependent stabilization criterion of linear time-varying delay systems: An iterative method

Intelligent Power Control of Inverter Air Conditioners in Power Systems: A Brain Emotional Learning-Based Approach

Inverter air-conditioning (IAC) units have been proved to be effective in frequency regulation by providing flexible capacities. This paper proposes a brain emotional learning (BEL)-based controller to provide the IACs with control signals to be efficiently involved in the frequency regulation in power systems. The BEL-based controller can learn quick-auto, making it appropriate in systems facing uncertainty. To assess the BEL controller performance in realistic conditions, the uncertainties as a consequence of variations in system parameters and load level are considered. The goal is to use the BEL controller to increase the IAC units' ability to track regulation signals accurately in uncertain circumstances. The controller is compared to a fuzzy-PI control, a proportional control scheme, a model predictive control and a linear quadratic regulator control. A delay-dependent stability criterion is used to calculate the highest time delay in the IACs response under which the system maintains stability. In addition, this paper presents an BEL-based coordinator to coordinate the IACs and traditional generation units for compensating considerable frequency variations caused by the time delays. Case studies are accomplished on a multi-area power system in MATLAB/Simulink environment. Eventually, real-time verifications by OPAL-RT real-time digital simulator on the simulated power system are executed to assess the control method.

Improved delay-dependent stability of superlinear hybrid stochastic systems with general time-varying delays

In the recent paper (Fei et al., 2019), the study of delay-dependent stability of hybrid stochastic differential delay equations (SDDEs) was generalized to superlinear ones (namely, do not satisfy the usual linear growth condition). However, the theory developed there could not be applied to hybrid SDDEs with non-differentiable time delays, or whose drift coefficients miss the key decomposition in (Fei et al., 2019) (see Assumption 1 below). This paper therefore is to deal with these two challenging problems so that the delay-dependent stability criteria derived in (Fei et al., 2019) could be improved. The decomposition scheme is modified in order to include more general hybrid SDDEs. The differentiability assumption on time-varying delays is replaced by a relatively weaker one. Also the Lyapunov functional used in this paper is modulated to adapt to these new changes. Finally, two interesting examples, an application to mosquito model, and design of nonlinear delay feedback control, respectively, are given to demonstrate the effectiveness of our new theory.

On the Existence of the Exact Solution of Quaternion-Valued Neural Networks Based on a Sequence of Approximate Solutions.

In many practical applications, it is difficult or impossible to obtain the exact solution of the mathematical model due to the limitations of solving methods and the complexity of the neural network itself. A natural problem is given as follows: does the exact solution of quaternion-valued neural networks (QVNNs) exist when successively improved approximate solutions can be obtained? Fortunately, the Hyers-Ulam stability happens to be one of the important means to deal with this problem. In this article, the issue of Hyers-Ulam stability of QVNNs with time-varying delays is addressed. First, inspired by the Hyers-Ulam stability of general functional equations, the concept of the Hyers-Ulam stability of QVNNs is proposed along with the QVNNs model. Then, by utilizing the successive approximation method, both delay-dependent and delay-independent Hyers-Ulam stability criteria are obtained to ensure the Hyers-Ulam stability of the QVNNs considered. Finally, a simulation example is given to verify the effectiveness of the derived results.

Spatiotemporal Tipping Induced by Turing Instability and Hopf Bifurcation in a Population Ecosystem Model with the Fear Factor

Population ecosystems can display the tipping points at which extinctions of species happens. To predict the appearance of tipping points and to understand their evolution mechanism are of uttermost importance for ecological balance. Using techniques from bifurcation theory, we can predict the emergence of tipping points based on a spatiotemporal predator-prey system having a fear effect before an instability is encountered. In the case of no time delay, the tipping induced by Turing instability is studied. The conditions for Turing instability and local asymptotic stability of coexisting equilibrium points are given. It is ascertained that the introduction of diffusion causes the ecosystem to change from stable to unstable, and then the tipping occurs. Then, we investigate another tipping due to Hopf bifurcation. The delay-dependent stability criterion and Hopf bifurcation condition are derived, and the onset of Hopf bifurcations (tipping point) is also determined. In order to further probe into the mechanism of tipping evolution, explicit formulae are derived to ascertain the stability of bifurcated oscillations and the direction of bifurcation via the center manifold reduction. It is revealed that many tipping points may exist in ecological competition systems, and the tipping occurs many times as the fear delay increases. Finally, several simulation examples are provided to substantiate the analytical results.

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.

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.

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 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.

The effect of communication delays on the frequency stability of power systems integrated with inverter air conditioners

Massive inverter air conditioners (IACs) are widely integrated into the smart grid, which can be employed to provide frequency support. However, the participation of IACs may inevitably introduce communication delays in the process of data detection and transmission, which easily leads to oscillation and instability of power systems. In this paper, we explore a delay-dependent stability of the power system with the participation of IACs and further reveal the impact of delay in the IACs control loop during the frequency regulation. Initially, a linear time-delay system based on the state-space model is established, making it possible to investigate the effect of delays by a time-domain method. Then, applying Lyapunov theory and linear matrix inequality technique, a delay-dependent stability criterion is proposed, considering the tradeoff between accuracy of delay margins and computational efficiency. Finally, the interactive effects among delay margin and PI controller gains are explored in different scenarios and the validity of the proposed criterion is verified, with a less computational burden that can achieve nearly accurate delay margins. The proposed criterion supplies reference for determining the delay requirement and selecting controller parameters of IACs participating in frequency regulation.