Stability criteria for linear systems with multiple time-varying delays

New delay-independent and delay-dependent stability criteria for linear systems with multiple time-varying delays are established by using the time-domain method. The results are derived based on a new-type stability theorem for general retarded dynamical systems and new analysis techniques developed in the author's previous work. Unlike some results in the literature, all of the established results do not depend on the derivative of time-varying delays. Therefore, they are suitable for the case with very fast time-varying delays. In addition, some remarks are also given to explain the obtained results and to point out the limitations of the previous results in the literature. Keywords Stability - Delay-independent criteria - Delay-dependent criteria - Linear time-delay systems - Multiple time-varying delays This work was supported by NSFC Key-Project (No. 60334010) and Guangdong Province Natural Science Foundation of China (No. 31406).

Average Consensus in Networks of Multi-Agent with Multiple Time-Varying Delays

The average consensus in undirected networks of multi-agent with both fixed and switching topology coupling multiple time-varying delays is studied. By using orthogonal transformation techniques, the original system can be turned into a reduced dimensional system and then LMI-based method can be applied conveniently. Convergence analysis is conducted by constructing Lyapunov-Krasovskii function. Sufficient conditions on average consensus problem with multiple time-varying delays in undirected networks are obtained via linear matrix inequality (LMI) techniques. In particular, the maximal admissible upper bound of time-varying delays can be easily obtained by solving several simple and feasible LMIs. Finally, simulation examples are given to demonstrate the effectiveness of the theoretical results.

Robust stability analysis of switched uncertain systems with multiple time-varying delays and actuator failures

In this paper,the robust stability issue of switched uncertain multidelay systems resulting from actuator failures is considered.Based on the average dwell time approach,a set of suitable switching signals is designed by using the total activation time ratio between the stable subsystem and the unstable one.It is first proven that the resulting closed-loop system is robustly exponentially stable for some allowable upper bound of delays if the nominal system with zero delay is exponentially stable under these switching laws.Particularly,the maximal upper bound of delays can be obtained from the linear matrix inequalities.At last,the effectiveness of the proposed method is demonstrated by a simulation example.

STABILITY OF SINGULAR UNCERTAIN DIFFERENTIAL SYSTEMS WITH MULTIPLE TIME-VARYING DELAYS

The stability of singular uncertain differential systems with multiple time-varying delays is discussed, and many results on this are obtained.

Exact controllability of rational expectations model with multiplicative noise and input delay

This paper considers the rational expectations model with multiplicative noise and input delay,where the system dynamics rely on the conditional expectations of future states.The main contribution is to obtain a sufficient condition for the exact controllability of the rational expectations model.In particular,we derive a sufficient Gramian matrix condition and a rank condition for the delay-free case.The key is the solvability of the backward stochastic difference equations with input delay which is derived from the forward and backward stochastic system.

Delay effect in the Lasota-Wazewska model with multiple time-varying delays

In this paper, the effect of delay on the asymptotic behavior of a Lasota-Wazewska model with multiple time-varying delays is investigated. By employing the fluctuation lemma and some differential inequality techniques, delay-dependent criteria are obtained for the global attractivity of the considered model. An example is also given in the end of this paper to show the effectiveness of our results.

Leader-following coordination of multiple UUVs formation under two independent topologies and time-varying delays

The new method which uses the consensus algorithm to solve the coordinate control problems of multiple unmanned underwater vehicles （multi-UUVs） formation in the case of leader-following is adapted. As the communication between the UUVs is difficult and it is easy to be interfered under the water, time delay is assumed to be time-varying during the members communicate with each other. Meanwhile, the state feedback linearization method is used to transfer the nonlinear and coupling model of UUV into double-integrator dynamic. With this simplified double-integrator math model, the UUV formation coordinate control is regarded as consensus problem with time-varying communication delays. In addition, the position and velocity topologies are adapted to reduce the data volume in each data packet which is sent between members in formation. With two independent topologies designed, two cases of communication delay which are same and different are considered and the sufficient conditions are proposed and analyzed. The stability of the multi-UUVs formation is proven by using Lyapunov-Razumilkhin theorem. Finally, the simulation results are presented to confirm and illustrate the theoretical results.

Global exponential stability analysis of cellular neural networks with multiple time delays

Global exponential stability problems are investigated for cellular neural networks （CNN） with multiple time-varying delays. Several new criteria in linear matrix inequality form or in algebraic form are presented to ascertain the uniqueness and global exponential stability of the equilibrium point for CNN with multiple time-varying delays and with constant time delays. The proposed method has the advantage of considering the difference of neuronal excitatory and inhibitory effects, which is also computationally efficient as it can be solved numerically using the recently developed interior-point algorithm or be checked using simple algebraic calculation. In addition, the proposed results generalize and improve upon some previous works. Two numerical examples are used to show the effectiveness of the obtained results.

Consensus of multiple autonomous underwater vehicles with double independent Markovian switching topologies and timevarying delays

A new method in which the consensus algorithm is used to solve the coordinate control problems of leaderless multiple autonomous underwater vehicles（multi-AUVs） with double independent Markovian switching communication topologies and time-varying delays among the underwater sensors is investigated.This is accomplished by first dividing the communication topology into two different switching parts,i.e.,velocity and position,to reduce the data capacity per data package sent between the multi-AUVs in the ocean.Then,the state feedback linearization is used to simplify and rewrite the complex nonlinear and coupled mathematical model of the AUVs into a double-integrator dynamic model.Consequently,coordinate control of the multi-AUVs is regarded as an approximating consensus problem with various time-varying delays and velocity and position topologies.Considering these factors,sufficient conditions of consensus control are proposed and analyzed and the stability of the multi-AUVs is proven by Lyapunov-Krasovskii theorem.Finally,simulation results that validate the theoretical results are presented.

Position Measurement Based Slave Torque Feedback Control for Teleoperation Systems With Time-Varying Communication Delays

Bilateral teleoperation system is referred to as a promising technology to extend human actions and intelligence to manipulating objects remotely.For the tracking control of teleoperation systems,velocity measurements are necessary to provide feedback information.However,due to hardware technology and cost constraints,the velocity measurements are not always available.In addition,the time-varying communication delay makes it challenging to achieve tracking task.This paper provides a solution to the issue of real-time tracking for teleoperation systems,subjected to unavailable velocity signals and time-varying communication delays.In order to estimate the velocity information,immersion and invariance(I&I)technique is employed to develop an exponential stability velocity observer.For the proposed velocity observer,a linear relationship between position and observation state is constructed,through which the need of solving partial differential and certain integral equations can be avoided.Meanwhile,the mean value theorem is exploited to separate the observation error terms,and hence,all functions in our observer can be analytically expressed.With the estimated velocity information,a slave-torque feedback control law is presented.A novel Lyapunov-Krasovskii functional is constructed to establish asymptotic tracking conditions.In particular,the relationship between the controller design parameters and the allowable maximum delay values is provided.Finally,simulation and experimental results reveal that the proposed velocity observer and controller can guarantee that the observation errors and tracking error converge to zero.

Exponential Stability of Large-Scale Non-Linear Systems with Multiple The Delays

The problem of stabilizing a class of large-scale non-linear multiple delay systems is considered.The complicated system is decomposed into several subsystems; each function of them is expressed by a set of components of the overall state vector,with interconnections between them, and the subsystems are coupled by the delayed state. In this paper, a method is devised to be a suitable choice of state feedback controls of every subsystems, moreover, it is proved that the large-scale system is exponential stable.

Analysis for Cohen-Grossberg neural networks with multiple delays

The stability analysis of Cohen-Grossberg neural networks with multiple delays is given. An approach combining the Lyapunov functional with the linear matrix inequality （LMI） is taken to obtain the sufficient conditions for the globally asymptotic stability of equilibrium point. By using the properties of matrix norm, a practical corollary is derived. All results are established without assuming the differentiability and monotonicity of activation functions. The simulation samples have proved the effectiveness of the conclusions.

Minimizing the Impact of Long Propagation Delay on Multiple Access for Wireless Networks

A multiple access protocol is proposed to greatly improve multiple access performance of wireless networks with long propagation delay. All the nodes with packets to send can make rapid successful reservation in access reservation mini-slots, which is adaptively adjusted according to current traffic load and idle channel resources. A Central Control Node (CCN) coordinates channel reservation and allocates on-demand channel resources to the successfully accessed nodes on two channels. Each node can employ only one handshake to accomplish each communication session, and transmit one or multiple data packets piggybacked with acknowledgment (ACK) information to one or multiple destination nodes in each frame until the end of their communication sessions, which greatly minimizes the impact of long propagation delay caused by handshakes and improves channel efficiency. Simulation results show that the proposed protocol obviously outperforms the Centralized Scheduling-based Medium Access Control (CSMAC) protocol, especially in the presence of long propagation delay.

ACTIVE CONTROL OF A FLEXIBLE CANTILEVER PLATE WITH MULTIPLE TIME DELAYS

Active control of a flexible cantilever plate with multiple time delays is investigated using the discrete optimal control method. A controller with multiple time delays is presented. In this controller, time delay effect is incorporated in the mathematical model of the dynamic system throughout the control design and no approximations and assumptions are made in the controller derivation, so the system stability is easily guaranteed. Furthermore, this controller is available for both small time delays and large time delays. The feasibility and efficiency of the proposed controller are verified through numerical simulations in the end of this paper.

Identification of time-varying system and energy-based optimization of adaptive control in seismically excited structure

The combination of structural health monitoring and vibration control is of great importance to provide components of smart structures.While synthetic algorithms have been proposed,adaptive control that is compatible with changing conditions still needs to be used,and time-varying systems are required to be simultaneously estimated with the application of adaptive control.In this research,the identification of structural time-varying dynamic characteristics and optimized simple adaptive control are integrated.First,reduced variations of physical parameters are estimated online using the multiple forgetting factor recursive least squares(MFRLS)method.Then,the energy from the structural vibration is simultaneously specified to optimize the control force with the identified parameters to be operational.Optimization is also performed based on the probability density function of the energy under the seismic excitation at any time.Finally,the optimal control force is obtained by the simple adaptive control(SAC)algorithm and energy coefficient.A numerical example and benchmark structure are employed to investigate the efficiency of the proposed approach.The simulation results revealed the effectiveness of the integrated online identification and optimal adaptive control in systems.

Robust stability analysis for Markovian jumping stochastic neural networks with mode-dependent time-varying interval delay and multiplicative noise

This paper is concerned with the problem of robust stability for a class of Markovian jumping stochastic neural networks （MJSNNs） subject to mode-dependent time-varying interval delay and state-multiplicative noise. Based on the Lyapunov-Krasovskii functional and a stochastic analysis approach, some new delay-dependent sufficient conditions are obtained in the linear matrix inequality （LMI） format such that delayed MJSNNs are globally asymptotically stable in the mean-square sense for all admissible uncertainties. An important feature of the results is that the stability criteria are dependent on not only the lower bound and upper bound of delay for all modes but also the covariance matrix consisting of the correlation coefficient. Numerical examples are given to illustrate the effectiveness.

Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen-Grossberg neural networks

This paper addresses the coexistence and local stability of multiple equilibrium points for fractional-order Cohen-Grossberg neural networks(FOCGNNs)with time delays.Based on Brouwer's fixed point theorem,sufficient conditions are established to ensure the existence of Πi=1^n(2Ki+1)equilibrium points for FOCGNNs.Through the use of Hardy inequality,fractional Halanay inequality,and Lyapunov theory,some criteria are established to ensure the local Lagrange stability and the local Lyapunov asymptotical stability of Πi=1^n(Ki+1)equilibrium points for FOCGNNs.The obtained results encompass those of integer-order Hopfield neural networks with or without delay as special cases.The activation functions are nonlinear and nonmonotonic.There could be many corner points in this general class of activation functions.The structure of activation functions makes FOCGNNs could have a lot of stable equilibrium points.Coexistence of multiple stable equilibrium points is necessary when neural networks come to pattern recognition and associative memories.Finally,two numerical examples are provided to illustrate the effectiveness of the obtained results.

Design method of multiple time-delay controller for active structural vibration control

An optimal control method for seismic-excited building structures with multiple time delays is investigated in this paper. The system state equation with multiple time delays is discretized and transformed into a standard discrete form without any explicit time delay by a particular augmenting for state variables. A time-delay controller is then designed based on this standard equation using the discrete optimal control method. Effectiveness of the proposed controller is demonstrated by numerical simulations. Simulation results indicate that a very small time delay may result in the instability of the control system if it is not compensated in the control design. Time delay may be compensated effectively by the proposed controller, in the mean time, an effective control may be obtained. The proposed controller is valid for both small and large time delays.

Stability and attractive basins of multiple equilibria in delayed two-neuron networks

Multiple stability for two-dimensional delayed recurrent neural networks with piecewise linear activation flmctions of 2r （r 〉 1） corner points is studied. Sufficient conditions are established for checking the existence of （2r ＋ 1）2 equilibria in delayed recurrent neural networks. Under these conditions, （r ＋ 1）2 equilibria are locally exponentially stable, and （2r＋ 1）2 -（r ＋ 1）2 -r2 equilibria are unstable. Attractive basins of stable equilibria are estimated, which are larger than invariant sets derived by decomposing state space. One example is provided to illustrate the effectiveness of our results.

Dynamic modeling and aperiodically intermittent strategy for adaptive finite-time synchronization control of the multi-weighted complex transportation networks with multiple delays

The idea of network splitting according to time delay and weight is introduced.Based on the cyber physical systems(CPS),a class of multi-weighted complex transportation networks with multiple delays is modeled.The finite-time synchronization of the proposed complex transportation networks model is studied systematically.On the basis of the theory of stability,the technique of adaptive control,aperiodically intermittent control and finite-time control,the aperiodically intermittent adaptive finite-time synchronization controller is designed.The controller designed in this paper is beneficial for understanding the synchronization in multi-weighted complex transportation networks with multiple delays.In addition,the conditions for the existence of finite time synchronization have been discussed in detail.And the specific value of the settling finite time for synchronization is obtained.Moreover,the outer coupling configuration matrices are not required to be irreducible or symmetric.Finally,simulation results of the finite-time synchronization problem are given to illustrate the correctness of the results obtained.