Razumikhin Approach Research Articles

Stability of switched singular time delay systems with switching induced state jumps

This study addresses the problem of exponential stability for switched singular state-delayed systems with switching induced state jumps, which has not been studied up to now. The considered state delay varies in a time-varying interval. On the basis of equivalent dynamics decomposition, a model of state jump at switching instants is firstly established under an assumption that the time length between arbitrary two adjacent switches is larger than the upper bound of the state delay. Then, a sufficient condition on exponential stability of the system under the reranged dwell-time switching constraint is presented. The key idea is the design of a dwell-time-dependent generalised Lyapunov function as well as a dwell-time-dependent function with respect to algebraic variables and application of the Razumikhin approach. The obtained stability condition exploits the lower bound and the upper bound of the dwell time. In addition, it is independent of the size of the state delay and allows the delay to be a fast time-varying function. Finally, two numerical examples are given to show the efficacy of the derived result.

Stability Analysis for Homogeneous Hybrid Systems With Delays

The stability problem is studied for hybrid systems with delays in this paper. Based on Lyapunov–Razumikhin approach, a novel theorem is presented for such system with the characteristic of homogeneity so that the system is globally preasymptotically stable. In particular, under the homogeneous assumption, we are able to obtain some rather weak conditions compared with general nonhomogeneous hybrid systems in this paper. Finally, two illustrative numerical examples are presented to demonstrate the applicability and the effectiveness of our theorems.

On estimation of rates of convergence in Lyapunov–Razumikhin approach

Considering a retarded nonlinear system, this note proposes several modifications of the Lyapunov–Razumikhin approach guaranteeing the existence of an upper estimate on convergence rate of the system solutions. The cases of exponential, finite-time and fixed-time (with respect to a ball) convergences are studied. The proposed approach is illustrated by simulation of academic examples.

Some Improved Razumikhin Stability Criteria for Impulsive Stochastic Delay Differential Systems

This paper is devoted to study the Razumikhin stability theorem for a class of impulsive stochastic delay differential systems. By developing a new lemma, stochastic analysis technique, and Razumikhin approach, several novel criteria of the $p$ th moment exponential stability are derived for the related systems. The key feature of the criteria is that time-derivatives of the Razumikhin functions are allowed to be indefinite, which loosens the constraints of the existing results greatly. Finally, two examples are given to illustrate the usefulness and significance of the theoretical results.

Analysis and Design of a Wide-Area Damping Controller for Inter-Area Oscillation With Artificially Induced Time Delay

One of the major issues in an interconnected power system is the low damping of inter-area oscillations. A proportional-derivative (PD) compensator is a popular control method which is extensively being used as an active damping control in various domains such as robotics, smart structure, etc. However, existence of noise in wide-area signal makes an ideal PD compensator unrealisable in power systems. In this endeavour, a new methodology christened as time-delayed control is proposed where artificial delay is used to provide derivative action which acts as a damping component of wide-area controller. The delay margin is computed for the closed loop system using Razumikhin approach. It is observed that the proposed controller is able to damp the oscillations adequately with reduced sensitivity to the noise in the signals. Further, the proposed controller is able to provide robustness against uncertainties arising from the system dynamics and inherent delay in communication channel.

Sufficient Conditions for the Existence of Asymptotic Quiescent Position for One Class of Differential-Difference Systems

The time-delay systems are considered and the limiting behavior of their solutions is investigated. The case in which the solutions have the trivial equilibrium that may not be an invariant set of the system is studied. The notion of an asymptotic quiescent position for the trajectories of delayed systems is introduced. Its stability is analyzed by the method of Lyapunov functions using the Razumikhin approach. Sufficient conditions for the existence of an asymptotic quiescent position for one class of the systems of differential-difference equations are established. Some illustrative examples of nonlinear differential equations with delay that have an asymptotic quiescent position are given and the sufficient conditions are applied to them.

Stability of neutral type delay systems: A joint Lyapunov–Krasovskii and Razumikhin approach

In this paper, a new stability criterion for linear time-delay systems of neutral type is presented. By virtue of the well-known Krasovskii theorem, the system is exponentially stable, if and only if there exists a functional which (i) has a negative definite derivative along the solutions of the system, and (ii) admits a quadratic lower bound, of course under an assumption that the difference operator corresponding to the system is stable. We modify the second condition in the following manner: The functional is required to admit a quadratic lower bound only on the special set of initial functions in our stability result instead of the set of all appropriate functions in the Krasovskii theorem. This special set consists of the functions satisfying a Razumikhin-type inequality and a similar inequality on the derivative. Basing on such modification and staying within the framework of the functionals with a given derivative, we suggest a methodology for the stability analysis, which is described and tested on the examples in the case of a scalar equation. Unlike most of recent results, we use the functional with a derivative prescribed just as a negative definite quadratic form of the “current” state of a system, what does not impose restrictions on the approach.

Stability analysis of fractional delayed equations and its applications on consensus of multi-agent systems

This article considers stability of fractional delayed equations and its applications on consensus of multi-agent systems. A modified fractional Razumikhin theorem is presented and some simple algebraic criteria on asymptotic stability of fractional equations with discrete delays and with mixed delays are proposed, respectively. Several sufficient conditions on leader-following and leaderless consensus are proposed by employing graph theory and fractional Razumikhin approach, respectively. The proposed conditions on stability and consensus are easily to be checked, illustrative examples show further validity and conveniences of our derived results.

Efficient Robust Fuzzy Model Predictive Control of Discrete Nonlinear Time-Delay Systems via Razumikhin Approach

In this paper, two efficient robust fuzzy model predictive control algorithms are investigated for discrete nonlinear systems with multiple time delays and bounded disturbances. The famous Takagi–Sugeno (T–S) fuzzy systems are utilized to represent nonlinear systems. Instead of the Lyapunov–Krasovskii functional, the Lyapunov–Razumikhin function is adopted to deal with time delays because it involves invariant sets in the original state space of the system. A sequence of explicit control laws corresponding to a sequence of constraint sets are computed offline so that the online computational burden associated with the classical model predictive control algorithms is significantly reduced. In particular, the set invariance theory behind the Razumikhin approach, which is more complicated than the one for nondelayed systems, is directly observed. Additionally, it is proved that all (delayed) states can enter the terminal set in finite time. Moreover, robust positive invariance and input-to-state stability for time-delay systems concerning disturbances are realized. Additionally, an online optimization algorithm is also provided based on the offline computed ellipsoidal sets. Therefore, the conservatism induced by the Razumikhin approach is relaxed, while the computational cost is not significantly increased.

On the Stability Analysis of Equations with Bounded Time-Varying Delay

The paper is devoted to a generalization of the Myshkis’s 3/2 stability theorem. This theorem gives an exact stability boundary for a scalar equation with a real parameter and an arbitrary delay, which is bounded by a prescribed value. In our paper, we consider the equation with complex parameter that opens up a direct opportunity for the analysis of systems of several equations. Via the Razumikhin approach, a stability region (not necessarily exact) is obtained, it is shown that its boundary can be expressed in radicals, and in the limit case the result coincides with the one of Myshkis.

Krasovskii and Razumikhin Stability Theorems for Stochastic Switched Nonlinear Time-Delay Systems

This paper studies stability properties of stochastic switched nonlinear time-delay systems. The stability analysis is based on two extensions of the Lyapunov-based method: the Krasovskii approach and the Razumikhin approach. In terms of the Krasovskii approach, Krasovskii-type stability conditions are derived based on Lyapunov--Krasovskii functions and average dwell-time condition. In terms of the Razumikhin approach, Razumikhin-type stability conditions are obtained via Lyapunov--Razumikhin functions, the small gain condition, and the fixed dwell-time condition. Furthermore, as a widespread phenomenon in switched systems, the asynchronous switching case is studied. Both Krasovskii-type and Razumikhin-type stability conditions are established for the asynchronous switching case. Finally, the developed results are illustrated via two examples from the mechanical rotational cutting process and networked switched control systems.

Razumikhin and Krasovskii methods for asymptotic stability of nonlinear delay impulsive systems on time scales

In this paper, asymptotic stability problems of nonlinear delay impulsive systems on time scales are considered based on the Razumikhin and Krasovskii methods. First, a Krasovskii-type theorem is provided for uniform asymptotic stability and uniform exponential stability of delay impulsive systems on time scales. It is shown that this Krasovskii stability criterion does not require the time derivative of the Lyapunov functional to be necessarily nonpositive on each impulsive interval. Second, asymptotic stability problems of delay impulsive systems on time scales is investigated based on the Razumikhin approach. The advantage of this method is that the length of each impulse interval does not depend on the time delay, which results in the fact that the state trajectory may not decrease instantly and sharply at each impulsive point. Moreover, the idea of this paper provides a unified approach to study continuous system and its discrete counterpart simultaneously. An example is given for illustrating the effectiveness of the proposed results.

Consensus analysis of fractional-order nonlinear multi-agent systems with distributed and input delays

Fractional-order nonlinear multi-agent systems with distributed and input delays are considered, in which the topology of multi-agent system corresponds to a directed weighted graph or an undirected weighted graph. Several algebraic conditions on leader-following consensus are presented by using fractional-order Razumikhin approach and algebraic graph theory. Finally, we present two numerical examples to illustrate those conditions proposed are feasible and convenient to evaluate consensus.

Global synchronization of fractional coupled networks with discrete and distributed delays

This article considers fractional coupled networks with discrete and distributed delays, an algebraic criterion on global synchronization is derived by using fractional Razumikhin approach and matrix inequalities. Furthermore, two types of special fractional delayed networks are also investigated and several simple sufficient conditions on global synchronization are presented, respectively. Finally, two numerical examples are provided to demonstrate the validity of our theoretical results.

Fuzzy Model Predictive Control of Discrete-Time Systems with Time-Varying Delay and Disturbances

In this paper, model predictive control (MPC) of discrete T–S fuzzy systems subjected to bounded time-varying delay and persistent disturbances is investigated. The Razumikhin approach is adopted for time-delay systems because it involves a Lyapunov function associated with the original nonaugmented state space of system dynamics when compared to the Krasovskii approach. As such, the Razumikhin approach has a good potential to avoid the inherent complexity of the Krasovskii approach especially in the presence of large delays and disturbances. Based on which, both online and offline MPC approaches for systems with time-varying delay are provided. In addition, persistent disturbances are considered that robust positive invariance and input-to-state stability under such circumstances are realized. In the offline approach, a sequence of explicit control laws that correspond to a sequence of robust constraints sets are computed offline. And it is proved that system states including all possibly delayed states can be steered to the terminal constraint set in finite time. Moreover, it allows the exact time delay to be unknown in the proposed two approaches. In particular, for systems with time-varying delay, the special positively invariant set theory and finite-time control theory based on the Razumikhin approach are directly revealed via the proposed offline approach.

A remark on stochastic stability of a novel hybrid system

A stochastic switching Susceptible–Exposed–Infective–Removed ( S E I R ) epidemic model with continuous and impulsive control schemes is proposed and investigated, in which a general incidence function is considered. The existence and uniqueness of the global positive solution is achieved. By using the Lyapunov–Razumikhin approach, sufficient conditions are developed which guarantee the hybrid system is stochastically stable. The results of simulating our new S E I R model indicate that the hybrid control schemes could be an effective strategy to eradicate the disease.

Razumikhin-Type Theorems on General Decay Stability of Impulsive Stochastic Functional Differential Systems with Markovian Switching

In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switching. Based on the Lyapunov-Razumikhin methods, some sufficient conditions are derived to check the stability of impulsive stochastic functional differential systems with Markovian switching. One numerical example is provided to demonstrate the effectiveness of the results.

Observer-based stabilisation of a class of nonlinear systems in the presence of measurement delay

ABSTRACTIn this paper, the stabilising control problem for a class of nonlinear system in the presence of measurement delay is addressed. A full-order observer is designed to eliminate the effect of variable output time delay, which is bounded and known. Then, the estimated states are utilised for the state feedback control law to stabilise the considered control system. Lyapunov–Razumikhin approach is employed to analyse the stability of the closed-loop system. Unlike the previous work, the exponential convergence of the estimation error is ensured, rather than asymptotic convergence, by designing a delay-dependent gain of the observer. Moreover, comparison with similar work is also made in simulation to illustrate the effectiveness of the proposed strategy.

Decentralised variable structure observation of nonlinear interconnected systems with time delay

Decentralised local variable structure observers are proposed for a class of nonlinear interconnected systems in which the system states experience time delay while the inputs and outputs are delay free. The interconnections are nonlinear, unknown and time delayed, which provides particular challenge for the case of decentralised observation. It is not required that the bounds on the uncertainties in the isolated subsystems are functions of the system outputs. Unlike much of the existing work where the designed observer is an interconnected system, the observers designed in this paper are a set of decoupled dynamics and thus decentralised, which is more reliable in real implementation. A set of sufficient conditions are derived such that in the considered domain the corresponding error dynamics are asymptotically stable where the Lyapunov Razumikhin approach is employed to deal with the time delay. The delays are time varying, and there is no limitation on the rate of change of the time delays. A numerical example is presented to show the effectiveness of the results obtained.

Time-delayed feedback stabilisation of nonlinear potential systems

Mechanical systems with nonlinear potential forces and delayed feedback are studied. It is assumed that, in the absence of control, the trivial equilibrium positions of considered systems are stable, but they are not attracting ones. An approach for the constructing of nonlinear controllers providing the asymptotic stability of the equilibrium positions is proposed. By the use of the Lyapunov direct method and the Razumikhin approach, it is proved that for the corresponding closed-loop systems the asymptotic stability can be guaranteed even in the cases when delay is unknown and time-varying. Moreover, estimates for solutions of closed-loop systems are found. An example and the results of a computer simulation are presented to demonstrate the effectiveness of the proposed approach.