Further results on asymptotic stability of linear neutral systems with multiple delays

The problem of asymptotic stability of linear neutral systems with multiple delays is addressed in this article. Using the characteristic equation approach, new delay-independent stability criteria are derived in terms of the spectral radius of modulus matrices. The structure information of the system matrices are taken into consideration in the proposed stability criteria, thus the conservatism found in the literature can be significantly reduced. Simple examples are given to demonstrate the validity of the criteria proposed and to compare them with the existing ones.     

New gain-scheduling control conditions for time-varying delayed LPV systems

This paper investigates the problem of stability and state-feedback control design for linear parameter-varying systems with time-varying delays. The uncertain parameters are assumed to belong to a polytope with bounded known variation rates. The new conditions are based on the Lyapunov theory and are expressed through Linear Matrix Inequalities. An alternative parameter-dependent Lyapunov-Krasovskii functional is employed and its time-derivative is handled using recent integral inequalities for quadratic functions proposed in the literature. As main results, a novel sufficient stability condition for delay-dependent systems as well as a new sufficient condition to design gain-scheduled state-feedback controllers are stated. In the new proposed methodology, the Lyapunov matrices and the system matrices are put separated making it suitable for supporting in a new way the design of the stabilization controller. An example, based on a model of a real-world problem, is provided to illustrate the effectiveness of the proposed method.     

New approaches to stability analysis for time-varying delay systems

In this paper, two new estimation approaches namely delay-dependent-matrix-based (DDMB) reciprocally convex inequality approach and DDMB estimation approach, are introduced for stability analysis of time-varying delay systems. Different from existing estimation techniques with constant matrices, the estimation approaches are with delay-dependent matrices, which can employ more free matrices and utilize more information of both time delay and its derivative. Based on the estimation approaches, less conservative stability criteria with lower computational complexity are derived in the form of linear matrix inequalities (LMIs). Finally, two numerical examples are given to illustrate the advantages of the proposed methods.     

Dynamics of linear delayed systems with decaying disturbances

Dynamical systems in the real world are always subject to various disturbances. This paper studies the dynamics of linear delayed systems with decaying disturbances, both discrete- and continuous-time cases are considered. It is first shown that if an unforced linear system is exponentially stable, then the disturbed system has a dynamical property like exponential stability provided that the disturbance decays at an exponential rate, and has a dynamical property like asymptotic stability provided that the disturbance asymptotically approaches zero. These results are then applied to block triangular systems in the presence of time-varying delays, leading to criteria for checking the stability properties of this class of systems by considering diagonal blocks of system matrices. Particularly, a block triangular system is exponentially stable if and only if each system described by the diagonal blocks of system matrices is exponentially stable. Finally, a numerical example is presented to illustrate the theoretical results.     

A modified Lyapunov functional with application to stability of neutral-type neural networks with time delays

This paper investigates the problem for stability of neutral-type dynamical neural networks involving delay parameters. Different form the previously reported results, the states of the neurons involve multiple delays and time derivative of states of neurons include discrete time delays. The stability of such neural systems has not been given much attention in the past literature due to the difficulty of finding Lyapunov functionals which are suitable for stability analysis of this type of neural networks. This paper constructs a generalized Lyapunov functional by introducing new terms into the well-known Lyapunov functional that enables us to conduct a theoretical investigation into stability analysis of delayed neutral-type neural systems. Based on this modified novel Lyapunov functional, sufficient criteria are derived, which guarantee the existence, uniqueness and global asymptotic stability of the equilibrium point of the neutral-type neural networks with multiple delays in the states and discrete delays in the time derivative of the states. The applicability of the proposed stability conditions rely on testing two basic matrix properties. The constraints impose on the system matrices are determined by using nonsingular M-matrix condition, and the constraints imposed on the coefficients of the time derivative of the delayed state variables are derived by exploiting the vector-matrix norms. We also note that the obtained stability conditions have no involvement with the delay parameters and expressed in terms of nonlinear Lipschitz activation functions. We present a constructive numerical example for this class of neural networks to give a systematic procedure for determining the imposed conditions on the whole system parameters of the delayed neutral-type neural systems.     

Improved stability criteria for linear systems with time-varying delays

This paper is concerned with the stability analysis of linear systems with time-varying delays. First, by introducing the quadratic terms of time-varying delays and some integral vectors, a more suitable Lyapunov-Krasovskii functional (LKF) is constructed. Second, two new delay-dependent estimation methods are developed in the stability analysis of linear system with time-varying delays, which include a reciprocally convex matrix inequality and an integral inequality. More information about time-varying delays and more free matrices are introduced into the two estimation approaches, which play a key role for obtaining an accurate upper bound of the integral terms in time derivative of LKFs. Third, based on the novel LKFs and new estimation approaches, some less conservative criteria are derived in the form of linear matrix inequality (LMI). Finally, three numerical examples are applied to verify the advantages and effectiveness of the newly proposed methods.     

Robust stability of uncertain second-order linear time-varying systems

This paper is concerned with robust stability analysis of second-order linear time-varying (SLTV) systems with time-varying uncertainties (perturbations). With the specific Lyapunov functions, a simple and neat algebraic criterion for testing uniformly asymptotic stability of SLTV systems are derived. Without transformation to a system of first-order equations, the new conditions are imposed directly on the time-varying coefficient matrices of the system. The main feature of the proposed algebraic criterion is that the uncertain coefficient matrices are time-varying and not necessarily symmetric. Finally, the proposed stability conditions are used to design the extending space structures system of the spacecraft. Simulation results are provided to illustrate the convenience and effectiveness of the proposed method.     

Quadratic stability analysis and robust distributed controllers design for uncertain spatially interconnected systems

This paper is concerned with the quadratic stability analysis and robust distributed controllers design of both continuous-time and discrete-time uncertain spatially interconnected systems (USISs), where uncertainties are modeled by linear fractional transformation (LFT). The well-posedness, quadratic stability, and contractiveness of USISs are properly defined for the first time. A sufficient condition employing the given system matrices is established to check the well-posedness, quadratic stability and contractiveness. This condition is simpler than the existing conditions based on the decomposition of system matrices. Based on the new condition derived, a sufficient condition is given for the existence of robust distributed controllers and a constructive method is then presented for the design of robust distributed controllers. The advantage of the proposed constructive approach is that it employs the given system matrices while the existing methods conduct the bilinear transformation on these matrices when design controllers, and consequently, the constructive approach in this paper is computationally more efficient than the existing methods. Several examples are included to demonstrate the simplicity, efficiency and applicability of the derived theoretical results.     

Stability criteria of a class of nonlinear impulsive switching systems with time-varying delays

This paper investigates stability problems of a class of nonlinear impulsive switching systems with time-varying delays. Based on the common Lyapunov function method and Razumikhin technique, several stability criteria are established for nonlinear impulsive switching systems with time-varying delays. Our results show that switching systems can be stabilized by impulsive switching signals even if the system matrices are all unstable. In the absence of impulses, some of our results reduce to similar stability criteria for nonimpulsive switching systems in some recent research articles. Several examples with simulations are given to illustrate the efficiency of our results.     

Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities

This paper addresses the problem of the delay-dependent stability for neutral Markovian jump systems with partial information on transition probability. The time delays discussed in this paper are time-varying delays. Combined the new constructed Lyapunov functional with the introduced free matrices, and using the analysis technique of matrix inequalities, the delay-dependent stability conditions are obtained. The obtained results are formulated in terms of LMIs, which can be easily checked in practice by Matlab LMI control toolbox. Three numerical examples are given to show the validity and potential of the developed criteria.     

Another sufficient condition for the stability of grey discrete-time systems

In this paper, the stability of grey discrete-time systems is discussed whose state matrices are interval matrices. A new approach is obtained which guarantee the stability of grey discrete-time systems. The sufficient condition for robust stability of grey time delay systems subjected to interval systems is also derived. By mathematical analysis, the stability criterion is less conservative than those in previous results. Examples are given to compare the proposed method with reported recently.     

Improved exponential stability analysis for delayed recurrent neural networks

In this paper, we investigate the problem of global exponential stability analysis for a class of delayed recurrent neural networks. This class includes Hopfield neural networks and cellular neural networks with interval time-delays. Improved exponential stability condition is derived by employing new Lyapunov-Krasovskii functional and the integral inequality. The developed stability criteria are delay dependent and characterized by linear matrix inequalities (LMIs). The developed results are less conservative than previous published ones in the literature, which are illustrated by representative numerical examples.     

On stabilization for systems with two additive time-varying input delays arising from networked control systems

This paper is concerned with the stability and stabilization for systems with two additive time-varying input delays arising from networked control systems. A new Lyapunov functional is constructed and a tighter upper bound of the derivative of the Lyapunov functional is derived by applying a convex polyhedron method. The resulting stability criteria are of fewer matrix variables and less conservative than some existing ones. Based on the stability criteria, a state feedback controller is designed such that the closed-loop system is asymptotically stable. Numerical examples are given to show the less conservatism of the stability criteria and the effectiveness of the designed method.     

Stability of discrete-time systems with time-varying delay based on switching technique

In this paper, the stability problem of discrete-time systems with time-varying delay is considered. Some new stability criteria are derived by using a switching technique. Compared with the Lyapunov–Krasovskii functional (LKF) approach, the method used in this paper has two features. First, a switched model, which is equivalent to the original system and contains more delay information, is introduced. It means that the criteria obtained by using the LKF method can be regarded as stability criteria for the switched system under arbitrary switching. Second, when the switching signal is known, the stability problem for the switched model under constrained switching is considered and piecewise LKFs are adopted to obtain stability criteria. Since constrained switching is less conservative than arbitrary switching if the switching signal is known, one can know that the obtained results in this paper are less conservative than some existing ones. Two examples are given to illustrate the effectiveness of the obtained results.     

Global exponential stability of discrete-time higher-order Cohen–Grossberg neural networks with time-varying delays,connection weights and impulses

In this paper, an auxiliary model-based nonsingular M-matrix approach is used to establish the global exponential stability of the zero equilibrium, for a class of discrete-time high-order Cohen–Grossberg neural networks (HOCGNNs) with time-varying delays, connection weights and impulses. A new impulse-free discrete-time HOCGNN with time-varying delays and connection weights is firstly constructed, and the relationship between the solutions of original systems and new HOCGNNs is indicated by a technical lemma. From which, the global exponential stability criteria for the zero equilibrium are derived by using an inductive idea and the properties of nonsingular M-matrices. The effectiveness of the obtained stability criteria is illustrated by numerical examples. Compared with the previous results, this paper has three advantages: (i) The Lyapunov–Krasovskii functional is not required; (ii) The obtained global exponential stability criteria are applied to check whether a matrix is a nonsingular M-matrix, which can be conveniently tested; (iii) The proposed approach applies to most of discrete-time system models with impulses and delays.     

A further note on stability criterion of linear neutral delay-differential systems

This paper is a further note which corrects an error of the paper by Park and Won. A new stability criteria which is less conservative than those in the literature is presented.     

Controller design for discrete-time hybrid linear parameter-varying systems with semi-Markov mode switching

The paper is concerned with the stability and stabilization problems for a family of hybrid linear parameter-varying systems with stochastic mode switching. The switching phenomenon is modeled by a semi-Markov stochastic process which is more generalized than a Markov stochastic process. With the construction of a Lyapunov function that depends on both the parameter variation and operating mode, numerical testable stability and stabilization criteria are established in the sense of σ-error mean square stability with the aid of some mathematical techniques that can eliminate the terms containing products of matrices. To test the effectiveness of the designed stabilizing controller, we apply the developed theoretical results to a numerical example.     

Controller synthesis for Markovian jump systems with incomplete knowledge of transition probabilities and actuator saturation

This paper is concerned with Markovian jump systems subject to incomplete knowledge of transition probabilities and actuator saturation. The system under consideration is more general, which covers the systems with completely known and completely unknown transition probabilities. By introducing some free-connection weighting matrices to handle the inaccessible elements of transition probabilities, a new criterion is established to guarantee the stochastic stability of the closed-loop system. An optimization problem with LMI constraints is then formulated to determine the largest contractively invariant set in mean square sense. Finally, two numerical examples are provided to illustrate the merits of our method.     

Stability analysis of linear continuous-time delay-difference systems with multiple time-delays

This paper is concerned with the stability analysis of linear continuous-time delay-difference systems with multiple delays. Firstly, a new method for testing the L₂-exponential stability of the considered system is proposed, which is easier to use than the one in the existing literature. In view of the conservatism and the complexity of the obtained stability conditions in the existing literature, a complete Lyapunov–Krasovskii functional (LKF) is constructed by analyzing the relationship among the multiple delays. Sufficient conditions for both L₂-exponential stability and exponential stability are then derived based on the constructed LKFs, which are delay-independent. Exponential convergence rate for the considered system is also investigated by a new method, which is shown to be equivalent to the existing approach by using weighted LKFs. Robust stability under parameter uncertainties is also investigated. Numerical examples are provided to demonstrate the effectiveness and less conservativeness of the proposed method.