Lyapunov-Razumikhin Functions Research Articles

Stabilization of nonlinear time-delay systems: Flexible delayed impulsive control

This paper studies the stabilization of nonlinear time-delay systems under flexible delayed impulsive control. Some sufficient conditions are provided for establishing stability property in terms of exponential Lyapunov-Razumikhin functions. It is shown that the size of delay in continuous dynamics can be flexible. Specially, it can be smaller or larger than the impulsive intervals, and there is no magnitude relationship between the delay in continuous flow and impulsive delay. In most existing results, from the impulsive control point of view, the Lyapunov functions were based on the assumption that there was a common threshold at every impulse point. In this study, utilizing the proposed method of average impulsive estimation (AIE), the rate coefficients are flexible, and the impulsive delay can be integrated to guarantee the effect of stabilization of impulses. As an application, the theoretical results are applied to the synchronization of a chaotic neural network, and the impulsive control input is formalized in terms of linear matrix inequalities (LMIs). The efficiency of the derived results is illustrated by two numerical examples.

Finite-Time Input/Output-to-State Stability of Impulsive Switched Nonlinear Time-Delay Systems

In this brief, we investigate the stability of impulsive switched nonlinear time-delay systems (ISNTDSs) with two asynchronous switching (TAS) phenomena. This brief extends the well-known concepts of input/output-to-state stability and integral input/output-to-state stability to finite-time control problems. In other words, two new concepts, finite-time input/output-to-state stability (FTIOSS) and finite-time integral input/output-to-state stability (FTiIOSS), are proposed for the first time, which play an important role in many practical applications. TAS phenomena refer to that the switching modes of candidate controllers does not coincide with the corresponding subsystems modes, and system impulsive instants, system switching instants are allowed to be inconsistent each other. By utilizing multiple Lyapunov-Razumikhin functions and two new switching strategies, some stability criteria on FTIOSS and FTiIOSS are presented for ISNTDSs under TAS effects. A simulation example is provided to verify the effectiveness of the developed approach.

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.

Event-Triggered Control of Nonlinear Systems With Time-Varying State Delays

Nonlinear systems with time delays have been object of intense investigation in the last decades for their ability of capturing the behavior of many natural and engineering systems, including ecological, biomedical, industrial, robotic, and networked systems. In this article, a method for the event-triggered stabilization in the sample-and-hold sense of nonlinear time-delay systems with time-varying state delays is presented. The approach exploits control Lyapunov-Razumikhin functions and a spline-based approximation of the functional state feedback, resulting in a finite-memory controller that avoids (by construction) continuous-state monitoring while ensuring also that the minimal interevent time is strictly positive. Examples of application of the illustrated technique highlight the potential of the approach taken.

The Method of Lyapunov-Razumikhin Functions in Stability Analysis of Systems with Delay

The history of development and the current state-of-the-art in the stability theory of systems with delay based on an effective generalization of the direct Lyapunov method are presented. This method uses “classical” Lyapunov functions in combination with the Razumikhin condition.

Stability of the SNIS epidemic spreading model with contagious incubation period over heterogeneous networks

In this paper, a novel susceptible–infected–susceptible epidemic model with contagious incubation period is proposed by using the N-intertwined mean field approximation on directed and heterogeneous networks. Both continuous and impulsive strategies that consider the detection and identification of individuals in incubation period are studied. The impacts of the incubation period incorporating with the network topology and the effects of the strategies on the epidemic spreading process are discussed. We analyze the stability of disease-free equilibrium and obtain the corresponding sufficient conditions by means of Lyapunov–Razumikhin functions and inequality techniques. Numerical simulations are presented to illustrate and validate the theoretical analysis. These results provide some meaningful clues for the public health department to conduct appropriate and effective defensive measures to eradicate the epidemic in the crowd.

Lyapunov Conditions for Input-to-State Stability of Hybrid Systems With Memory

This paper studies input-to-state stability for hybrid systems with memory, which models hybrid dynamics affected by time delays. Using both Lyapunov–Razumikhin functions and Lyapunov–Krasovskii functionals, Lyapunov-based sufficient conditions are established for input-to-state stability. In addition, further extensions and relaxations are proposed for special cases, such as the stable flow/jump cases and the cases that Lyapunov functions do not decrease strictly during flow/jumps. Finally, two examples are used to illustrate the developed results

On Control Lyapunov–Razumikhin Functions, Nonconstant Delays, Nonsmooth Feedbacks, and Nonlinear Sampled-Data Stabilization

A methodology for the design of stabilizers in the sample-and-hold sense for nonlinear retarded systems is provided. The methodology is based on control Lyapunov–Razumikhin functions. Fully nonlinear retarded systems with an arbitrary number of arbitrary time-varying time-delays of discrete and distributed type are covered by the theory developed here, as long as the map describing the dynamics is uniformly in time Lipschitz on bounded sets of the Banach state space and the Euclidean input space. The standard assumption that all involved time-delay signals are globally Lipschitz is introduced. It is assumed, for the system at hand, that there exist a control Lyapunov–Razumikhin function and a suitably defined induced steepest descent state feedback. Moreover, such state feedback has to satisfy a suitably weak Lipschitz property, uniformly in time and in any bounded subset of the Banach state space. Such weak Lipschitz property has the advantage to allow for discontinuities in the feedback, which arise, for instance, in sliding-mode control methodologies. The following fact is shown: the above steepest descent state feedback is a stabilizer in the sample-and-hold sense, that is, such feedback, if applied by suitably fast sampling and holding, guarantees practical semiglobal stability of the closed-loop system, with arbitrary small final target ball of the origin. The problem of nonavailability in the buffer device of past values of the system internal variables is solved by spline approximation methods, even in the case discontinuities arise in the state feedback.

Stability analysis of some classes of nonlinear switched systems with time delay

ABSTRACTStability of certain classes of nonlinear time-delay switched systems is studied. For the corresponding families of subsystems, conditions of the existence of common Lyapunov–Razumikhin functions are found. The fulfilment of these conditions provides the asymptotic stability of the zero solutions of the considered hybrid systems for any switching law and for any nonnegative delay. Some examples are presented to demonstrate the effectiveness of the obtained results.

Lyapunov-Based Sufficient Conditions for Stability of Hybrid Systems With Memory

Hybrid systems with memory are dynamical systems exhibiting both hybrid and delay phenomena. In this note, we study the asymptotic stability of hybrid systems with memory using generalized concepts of solutions. These generalized solutions, motivated by studying robustness and well-posedness of such systems, are defined on hybrid time domains and parameterized by both continuous and discrete time. We establish Lyapunov-based sufficient conditions for asymptotic stability using both Lyapunov-Razumikhin functions and Lyapunov-Krasovskii functionals. Examples are provided to illustrate these conditions.

Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids

ABSTRACTMotivated by the problem of phase-locking in droop-controlled inverter-based microgrids with delays, the recently developed theory of input-to-state stability (ISS) for multistable systems is extended to the case of multistable systems with delayed dynamics. Sufficient conditions for ISS of delayed systems are presented using Lyapunov–Razumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in a feedback. The derived theory is applied to two examples. First, the ISS property is established for the model of a nonlinear pendulum and delay-dependent robustness conditions are derived. Second, it is shown that, under certain assumptions, the problem of phase-locking analysis in droop-controlled inverter-based microgrids with delays can be reduced to the stability investigation of the nonlinear pendulum. For this case, corresponding delay-dependent conditions for asymptotic phase-locking are given.

Stability criteria of nonlinear impulsive differential equations with infinite delays

In this paper, we consider the uniform stability and uniformly asymptotical stability of nonlinear impulsive infinite delay differential equations. Instead of putting all components of the state variable x in one Lyapunov function, several Lyapunov-Razumikhin functions of partial components of the state variable x are used so that the conditions ensuring that stability are simpler and less restrictive; moreover, examples are discussed to illustrate the advantage of the results obtained.

Study and Application of State Estimator for Time-Delay Systems with State Saturation Nonlinearities

The design and application of state estimator for a class of time-delay systems with state saturation nonlinearities are firstly explored. Based on the Lyapunov-Razumikhin functions, delay-dependent sufficient conditions are established to guarantee the asymptotic stability of the error between the state estimate and the true state. Besides, an upper bound of arbitrary time-varying delays is derived to assure that the resulting error can converge asymptotically to zero. Finally, applications to the secure communication and simulation results are given to demonstrate the feasibility and effectiveness of the main results.

Design, Analysis, and Experimental Validation of a Distributed Protocol for Platooning in the Presence of Time-Varying Heterogeneous Delays

This paper presents a novel control design framework for vehicle platooning together with its experimental validation. The problem of controlling the vehicles within a platoon, so that they converge to their desired velocities and intervehicle distances, is formulated as a high-order network consensus problem. By means of Lyapunov–Razumikhin functions, convergence is proven of the platoon to the desired consensus speed and intervehicle spacing under both fixed and switching communication network topologies, thus confirming the capability of the proposed approach to cope with maneuvers where vehicles join or leave the platoon and communication failures. Tuning criteria for the control gains are provided to guarantee string stability under the proposed control law. Finally, results of numerical simulations and in-vehicle experiments demonstrate the effectiveness of the proposed approach in a three-vehicle platoon.

RobustH∞Control for Switched Nonlinear System with Multiple Delays

This paper focuses on the problem of robustH∞control for a class of switched nonlinear systems with multiple time delays which are allowed to appear not only in state but also in input. We will design a controller and a switching law under which the system is “robust stable” and has performanceL2-gainγ. Besides, multiple Lyapunov-Razumikhin functions which are relatively less conservative are introduced to investigate the stability of the candidate switched system. In detail, firstly, for uncertain nonlinear switched system with time-varying delays, sufficient condition is given by inequalities which are equivalent to LMIs, so it is easy to be solved; then, as a special case, nonlinear switched system with constant time delay is also considered. Finally, by providing numerical examples, the feasibility of the proposed approach is demonstrated.

Direct and converse Lyapunov theorems for functional difference systems

New Lyapunov criteria for asymptotic stability and input-to-state stability of infinite dimensional systems described by functional difference equations are provided. Conditions in terms of both Lyapunov–Razumikhin functions defined on Euclidean spaces and of Lyapunov–Krasovskii functionals defined on infinite dimensional spaces are found. For the case of Lyapunov–Krasovskii functionals, necessary and sufficient conditions are provided for the asymptotic stability, in both the local and the global case, and for the input-to-state stability. This is the first time in the literature that converse Lyapunov theorems are provided for the class of nonlinear systems here studied.

Stability of a General Class of Distributed Algorithms for Power Control in Time-Varying Wireless Networks

In order for a wireless network to function effectively, the signal power of each user's transmitter must be sufficiently large to ensure a reliable uplink connection to the receiver, but not so large as to cause interference with neighboring users. We consider a general class of distributed algorithms for the control of transmitter power allocations in wireless networks with a general form of interference nonlinearity. In particular, we allow this interference to have explicit time-dependence, allowing our analysis to remain valid for network configurations that vary with time. We employ appropriately constructed Lyapunov functions to show that any bounded power distribution obtained from these algorithms is uniformly asymptotically stable. Further, we use Lyapunov-Razumikhin functions to show that, even when the system incorporates heterogeneous, time-varying delays, any solution along which the generalized system nonlinearity is bounded must also be uniformly asymptotically stable. Moreover, in both of these cases this stability is shown to be global, meaning that every power distribution has the same asymptotic behavior. These results are also used in the paper to derive time-invariant asymptotic bounds for the trajectories when the system nonlinearities are appropriately bounded.

Hybrid Systems with Memory: Modelling and Stability Analysis via Generalized Solutions

Hybrid systems with memory are dynamical systems exhibiting both hybrid and delay phenomena. We present a general modelling framework for such systems using hybrid functional inclusions, whose generalized solutions are defined on hybrid time domains and evolve in the phase space of hybrid memory arcs equipped with the graphical convergence topology. We prove a general existence result based on some basic conditions on the data of hybrid systems. We further establish sufficient conditions for the stability analysis of hybrid systems with memory using Lyapunov-Razumikhin (L-R) functions. We demonstrate the stability results on a general nonlinear system with switching dynamics and state jumps.

Necessary and Sufficient Razumikhin-Type Conditions for Stability of Delay Difference Equations

This technical note considers stability analysis of time-delay systems described by delay difference equations (DDEs). All existing analysis methods for DDEs that rely on the Razumikhin approach provide sufficient, but not necessary conditions for asymptotic stability. Nevertheless, Lyapunov-Razumikhin functions are of interest because they induce invariant sets in the underlying state space of the dynamics. Therefore, we propose a relaxation of the Razumikhin conditions and prove that the relaxed conditions are necessary and sufficient for asymptotic stability of DDEs. For linear DDEs, it is shown that the developed conditions can be verified by solving a linear matrix inequality. Moreover, it is indicated that the proposed relaxation of Lyapunov-Razumikhin functions has an important implication for the construction of invariant sets for linear DDEs.

The design of digital stabilizing regulators for continuous systems based on the Lyapunov function approach

New results regarding asymptotic stability of continuous systems with piecewise constant control are derived. The results are expressed in terms of the Lyapunov-Razumikhin functions, taking into account the specifics of the selected type of control. Certain illustrative examples are provided.