Stability Of Time-delay Systems Research Articles

Stabilization of time delay systems with saturations via PDE predictor boundary control design

This paper addresses the stabilization issue of linear time delay system with input saturation and distinct input delays via predictor feedback boundary control algorithm by employing transport partial differential equations (PDEs). First, the addressed ordinary differential equation (ODE) system with input delay is equivalently represented as a cascade of an ODE and transport PDEs. Second, by employing the backstepping Volterra integral transformation technique, the equivalent cascade system is transformed into a stable target system, whose kernels are solved by the constraints satisfying transport PDEs. Third, based on the boundary conditions of the obtained invertible transformation, the proposed feedback control law can be formulated. Fourth, by applying semigroup operator theory, the well-posedness of the resulting system is proved and consequently, novel exponential stability conditions of the addressed system are established. Then, the domain of attraction region under the given actuator saturation constraints is estimated with the help of the solution of obtained stability conditions. Finally, a demonstrative simulation example is offered to verify the feasibility and usefulness of the results.

Input-to-state stability for discrete hybrid time-delay systems with admissible edge-dependent average dwell time

The input-to-state stability (ISS) for a class of discrete hybrid time-delay systems with admissible edge-dependent average dwell time (AED-ADT) is investigated in this study. By using the Lyapunov-like functions, the conditions for ISS are derived under the following situations: all subsystems are stable, all subsystems are unstable, and some subsystems are unstable. When all the continuous subsystems are ISS, although the impulsive effects are destabilizing, the system is ISS with respect to a lower bound of AED-ADT. Moreover, when all the subsystems are not ISS, the impulsive effects can still successfully stabilize the system for an upper bound of AED-ADT. When some continuous subsystems are not ISS, the impulsive effects can also successfully stabilize the system with a lower bound of AED-ADT under specific conditions. For the special case in which the Lyapunov-like function is allowed to decrease and increase during the operation time of a certain mode, a fresh transformation scheme of the Lyapunov-like functions is used to address the difficulty caused by the external input. A connection is established among AED-ADT, impulses magnitude, the interval length of increase of the Lyapunov function, and decaying/increasing rates of the Lyapunov function, such that the impulsive switched delayed system is ISS. The conclusions can be applied to a large class of systems because the transformation technique fills the gap between special and general cases. Finally, an example is provided, and simulations are performed to confirm the validity of the main results.

Delay-Dependent Stabilization of Time-Delay Systems with Nonlinear Perturbations

Delay-Dependent Stabilization of Time-Delay Systems with Nonlinear Perturbations

Research on Fault Tolerance Consistency of Multi - Agent Based on Event Triggering Mechanism

For the multi-agent system with directed topology, considering the possible actuator faults of any agent, the robust H ∞ fault-tolerant consensus controller design problem is studied when the system is disturbed by external finite energy. Firstly, a distributed event-triggered fault-tolerant consistency protocol is proposed, which is used to transform the fault-tolerant consistency of distributed multi-agent systems into the stability of time-delay systems. Secondly, by using Lyapunov stability theory, a sufficient condition for the fault-tolerant consensus of the multi-agent system is given for the actuator faults that may occur in any agent. A cooperative design method that saves network communication resources and realizes fault-tolerant consensus control is obtained, which makes all states of the fault multi-agent system asymptotically consistent and has a certain disturbance rejection performance. Finally, the feasibility and effectiveness of the proposed method are verified by simulation.

Generalized polynomial chaos-informed efficient stochastic Kriging

Stochastic kriging (SK) offers an explicit way to characterize heterogeneous noise variance in stochastic computer simulations and has gained considerable traction recently as a surrogate model. Nevertheless, SK relies on tedious Monte Carlo (MC) method to estimate the intrinsic variance at each design input. For computationally expensive simulations, the substantial replication effort has essentially rendered SK intractable. To this end, we develop generalized polynomial chaos (gPC)-informed efficient stochastic kriging (gPC-SK) to ameliorate the computational cost. At its core, gPC supplants the tedious repetitive MC simulations, instead resting on a much smaller set of sampling points to estimate the intrinsic uncertainty, thus applicable to those prohibitively expensive simulations. We present the gPC-SK in sequential optimal design on the borehole function and stability of time-delay dynamic systems.

Pole Placement for Delay Differential Equations With Time-Periodic Delays Using Galerkin Approximations

Abstract Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems (TDS). In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

A partial solution operator discretization-based method for small signal stability analysis of time-delay power systems

To analyze the small signal stability of large-scale time-delay power systems (TDPSs), critical oscillation modes are required to be reliably computed. The solution operator discretization with pseudo-spectral collocation (SOD-PS)-based method has been proposed to indirectly calculate critical eigenvalues from the solution operator’s discretization matrix. However, dimension of the discretization matrix is dozens of times the number of system state variables, and thus limits the efficiency of SOD-PS. To resolve this, SOD-PS is evolved into partial SOD-PS (PSOD-PS) in this paper by applying the idea of partial spectral discretization (PSD). Specifically, only the discretization of retarded states is reserved instead of discretizing all system states in delay interval. The resultant partial discretization matrix exhibits very low order, which is close to the number of system state variables. In addition, the rotation-and-multiplication preconditioning technique is studied to efficiently capture oscillation modes with damping ratios less than the given threshold. Specifically, the preconditioning technique is implemented by two ways and their consistency is derived in detail. Both theoretical analyses and intensive tests on the 16-machine 68-bus test system, a real-life 516-bus provincial transmission system of China and the 33028-bus ultra-high-voltage (UHV) interconnected system of China verify the accuracy and efficiency of the presented PSOD-PS.

Improvement of Stability of Time-Delayed Linear Systems via New Constrained Quadratic Matrix Inequality

Improvement of Stability of Time-Delayed Linear Systems via New Constrained Quadratic Matrix Inequality

On the time-varying Halanay inequality with applications to stability analysis of time-delay systems

The main results of the paper are improvements on the stability analysis of Halanay inequalities with time-varying coefficients in both continuous-time and discrete-time setting. Three classes of improved conditions are established to ensure that the solution to the Halanay inequality is uniformly exponentially stable. The merit of the proposed new conditions is that the coefficients of the Halanay inequality can be unbounded and sign indefinite. This is achieved by using the notion and properties of uniformly asymptotic stable (UAS) functions. Based on the improved stability conditions for the Halanay inequality and the Lyapunov Razumikhin approach, three classes of sufficient conditions are established for testing the stability of time-varying time-delay systems. Finally, the advantages of the proposed methods are illustrated by some numerical examples with some of them borrowed from the literature.

Indefinite Lyapunov–Razumikhin Functions-Based Stability and Event-Triggered Control of Switched Nonlinear Time-Delay Systems

This brief studies stability and event-triggered stabilization of switched nonlinear time-varying systems by virtue of Lyapunov-Razumikhin functions (LRFs), when the requirement on derivative of LRFs to non-positive is relaxed. A class of such LRFs is first constructed for the input-to-state stability (ISS) of nonlinear time-delay systems. Using multiple LRFs with indefinite derivatives, the ISS of switched nonlinear time-delay systems is presented under average dwell time switching. Then, a nonlinear event-triggered controller is designed for the underlying systems, and it is pointed out that the Zeno phenomenon can be avoided under the proposed controller. Moreover, an event-triggered control is introduced for the systems subject to disturbances to reach the ISS. Finally, one example is given to verify the validity of the proposed approaches.

Adaptive event-triggered consensus control of linear multi-agent systems with cyber attacks

This paper investigates the adaptive event-triggered consensus control problem for a class of linear multi-agent systems subject to cyber attacks and communication delays. To effectively alleviate unnecessary signals transmission among the agents and achieve the reasonable allocation of resources, an adaptive event-triggering scheme, whose threshold parameters are adaptively adjusted based on system performance needs, is proposed. The event-triggering scheme can achieve the discontinuous communication and significantly reduce the number of signals transmission while ensuring the desired control performance. By considering the communication scheme and consensus control protocol in a unified framework, the linear multi-agent system is reformulated as a time-delay error system. Sufficient conditions are developed to guarantee the asymptotical stability and security of time-delay error system. Moreover, a co-design for the gain of the controller and the parameters of the adaptive event-triggering scheme is provided. Finally, an example on the tunnel diode circuit system is given to show the effectiveness and advantages of the proposed method.

Event-Triggered Control for Nonlinear Time-Delay Systems

This article studies the event-triggered control problem of general nonlinear systems with time delay. A novel event-triggering scheme is presented with two tunable design parameters, based on a Lyapunov functional result for the input-to-state stability of time-delay systems. The proposed event-triggered control algorithm guarantees the resulting closed-loop systems to be globally asymptotically stable, uniformly bounded, and/or globally attractive for different choices of these parameters. Sufficient conditions on the parameters are derived to exclude Zeno behavior. Two illustrative examples are studied to demonstrate our theoretical results.

Input-to-state stabilization of time-delay systems: An event-triggered hybrid approach with delay-dependent impulses

This paper studies the input-to-state stabilization problem of nonlinear time-delay systems. A novel event-triggered hybrid controller is proposed, where feedback controller and distributed-delayed impulsive controller are taken into account. By using the Lyapunov-Krasovskii method, sufficient conditions for input-to-state stability are constructed under the designed event-triggered hybrid controller, the relation among control parameters, threshold parameter of the event-triggered mechanism and time delay in the impulsive signals is derived. Compared with the existing results, the obtained input-to-state stability criteria are applicable to time-delay systems with stabilizing delay-dependent impulsive effects and destabilizing ones. Numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Exponential stability of time delay systems based on intermediate polynomial-based weighted functions

We investigate exponential stability of time-varying delay systems in this article. By considering intermediate polynomials and introducing slack variables, we develop new intermediate polynomial-based weighted functions (IPWFs). Then an exponential stability criterion, in terms of linear matrix inequalities (LMIs), is obtained by choosing an appropriate Lyapunov–Krasovskii functional (LKF) and some novel techniques on IPWFs. Numerical results show that the criterion obtained in this article is less conservative than some previous results.

A novel descriptor redundancy method based on delay partition for exponential stability of time delay systems

This paper investigates the exponential stability of uncertain time delay systems using a novel descriptor redundancy approach based on delay partitioning. First, the original system is casted into an equivalent descriptor singular state–space representation by introducing redundant state variables so that the resulting delay is progressively reduced. From the equivalent model and applying Lyapunov Functional method, a sufficient condition based on Linear Matrix Inequalities (LMIs) for exponential stability with guaranteed decay rate performance is obtained. As a result, the inherent conservatism of Lyapunov–Krasovskii functional techniques can arbitrarily be reduced by increasing the number of delay partition intervals including decay rate performance and model uncertainties in polytopic form. Various benchmark examples are provided to validate the effectiveness of the proposed method, showing better trade-off between conservatism and performance in comparison to previous approaches.

Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay

This work exploits structural properties of a class of functional Vandermonde matrices to emphasize some qualitative properties of a class of linear autonomous n−th order differential equation with forcing term consisting in the delayed dependent-variable. More precisely, it deals with the stabilizing effect of delay parameter coupled with the coexistence of the maximal number of real spectral values. The derived conditions are necessary and sufficient and represent a novelty in the litterature. Under appropriate conditions, such a configuration characterizes the spectral abscissa corresponding to the studied equation. A new stability criterion is proposed. This criterion extends recent results in factorizing quasipolynomial functions. The applicative potential of the proposed method is illustrated through the stabilization of coupled oscillators.

Time-delay systems with delayed impulses: A unified criterion on asymptotic stability

The paper deals with the global asymptotic stability of general nonlinear time-delay systems with delay-dependent impulses through the Lyapunov–Krasovskii method. We derive a unified stability criterion which can be applied to a variety of impulsive systems. The cases when each of the continuous dynamics and the impulsive component is either stabilizing or destabilizing are investigated. Both theoretically and numerically, we demonstrate that the obtained result is more general than those existing in the literature.

Stability and control of fuzzy time delay systems with unmatched preconditions

The stability and control of nonlinear time-delay systems of Takagi-Sugeno (T-S) fuzzy model are studied in this paper. The integral inequality of a free weight matrix is chosen to give a less conservative delay-dependent stability criterion in the form of linear matrix inequalities (LMIs). The premise mismatch strategy is applied, it is combined with Finsler lemma, a more flexible design method of fuzzy state feedback controller is proposed. This method does not require the controller and system to share the common premise membership function and the number of rules. The controller design strategy proposed in this paper can effectively solve the control problem of fuzzy systems when the number of state variables is not equal to the number of input variables (r≠c), or mi⁢(x⁢(t))≠hi⁢(x⁢(t)),i=1,2,…,r. Finally, two simulation examples are given to prove the advancement and effectiveness of the proposed theory.

Remarks on Lyapunov-Krasovskii Functionals for Integral Input-to-Output Stability Properties

Abstract In this work, we consider the Lyapunov method for integral input-to-output stability (iios) of time-delay systems. We particularly focus on the output-Lagrange iios property, and show that for this property, the existence of a Lyapunov-Krasovskii functional with a dissipation rate depending only on the present values of the output function (in contrast to the norm of the output function over the delay interval) is sufficient. In the special case when the output variables represent the full set of state variables, the result recovers a recent result on integral input-to-state stability (iiss).

Weak and strong stabilization for time-delay semi-linear systems governed by constrained feedback control

This paper is concerned with the issue of weak and strong stabilization for distributed semi-linear systems with time delay using a constrained feedback control. The results of the semi-linear systems without delay are generalized for strong and weak stabilization cases. Illustrating applications to hyperbolic and parabolic equations are considered.