Discrete Time-delay Systems Research Articles

Input-to-State Stability of Discrete-Time Delay Systems with Delayed Impulses

This paper investigates the input-to-state stability (ISS) of discrete-time delay systems with delayed impulses. By employing Lyapunov functions together with Razumikhin technique, a number of ISS criteria are obtained. It is shown that if the original impulse-free system has no ISS property, it can become input-to-state stable by adding certain impulsive controls which are stabilizing. Correspondingly, the original system without impulses can retain its ISS property with appropriate impulsive perturbations which are destabilizing. Then, robust ISS property for a class of uncertain discrete-time delay systems with delayed impulses is considered. Finally, some numerical examples are given to illustrate the effectiveness and the superiority of the results.

Input‐to‐state exponents and related ISS for delayed discrete‐time systems with application to impulsive effects

SummaryThis paper aims to investigate the input‐to‐state exponents (IS‐e) and the related input‐to‐state stability (ISS) for delayed discrete‐time systems (DDSs). By using the method of variation of parameters and introducing notions of uniform and weak uniform M‐matrix, the estimates for 3 kinds of IS‐e are derived for time‐varying DDSs. The exponential ISS conditions with parts suitable for infinite delays are thus established, by which the difference from the time‐invariant case is shown. The exponential stability of a time‐varying DDS with zero external input cannot guarantee its ISS. Moreover, based on the IS‐e estimates for DDSs, the exponential ISS under events criteria for DDSs with impulsive effects are obtained. The results are then applied in 1 example to test synchronization in the sense of ISS for a delayed discrete‐time network, where the impulsive control is designed to stabilize such an asynchronous network to the synchronization.

Improved stability criterion and output feedback control for discrete time-delay systems

• A new finite-sum inequality is provided. • By applying the above inequality, new stability criteria are derived. • New algorithms are established to solve the SOF controller design problems. This paper investigates the stability and stabilization for a class of linear systems with time-varying delay. We provide a new finite-sum inequality which is a powerful tool for stability analysis of time-delay systems. Applying the inequality, a new stability criterion is proposed in terms of linear matrix inequalities (LMIs). We also design a method for static output feedback (SOF) control problems which contains two parts. The first part is to find an initial values of the matrix variables. By utilizing the initial values, the condition for SOF control problems can be solved by an improved path-following method. Numerical examples demonstrate the effectiveness of the stability criterion and the SOF stabilization method.

Robust model predictive control of discrete nonlinear systems with time delays and disturbances via T–S fuzzy approach

In this paper, robust fuzzy model predictive control of a class of nonlinear discrete systems subjected to time delays and persistent disturbances is investigated. Based on the modeling method of delay difference inclusions, nonlinear discrete time-delay systems can be represented by T–S fuzzy systems comprised of piecewise linear delay difference equations. Moreover, Lyapunov–Razumikhin function (LRF), instead of Lyapunov–Krasovskii functional (LKF), is employed for time-delay systems due to its ability to reflect system original state space and its advantages in controller synthesis and computation. The robust positive invariance and input-to-state stability with respect to disturbance under such circumstances are investigated. A robust constraint set is adopted that the system state is converged to this set round the desired point. In addition, the controller synthesis conditions are derived by solving a set of matrix inequalities. Simulation results show that the proposed approach can be successfully applied to the well-known continuous stirred tank reactor (CSTR) systems subjected to time delay.

Wirtinger-like Lyapunov–Krasovskii functionals for discrete-time delay systems

Time-dependent Lyapunov functionals appeared to be very efficient for sampled-data systems. In [14], new Lyapunov functionals were constructed for sampled-data control in the presence of a constant input delay. The construction of these functionals was based on Wirtinger's inequality leading to simplified and efficient stability conditions in terms of Linear Matrix Inequalities (LMIs). In the present paper we extend the latter results to the discrete-time sampled-data systems. We show that the proposed approach is less conservative on some examples with a lower number of decision variables.

Distributed recursive filtering for stochastic systems under uniform quantizations and deception attacks through sensor networks

This paper is concerned with the distributed recursive filtering problem for a class of discrete time-delayed stochastic systems subject to both uniform quantization and deception attack effects on the measurement outputs. The target plant is disturbed by the multiplicative as well as additive white noises. A novel distributed filter is designed where the available innovations are from not only the individual sensor but also its neighboring ones according to the given topology. Attention is focused on the design of a distributed recursive filter such that, in the simultaneous presence of time-delays, deception attacks and uniform quantization effects, an upper bound for the filtering error covariance is guaranteed and subsequently minimized by properly designing the filter parameters via a gradient-based method at each sampling instant. Furthermore, by utilizing the mathematical induction, a sufficient condition is established to ensure the asymptotic boundedness of the sequence of the error covariance. Finally, a simulation example is utilized to illustrate the usefulness of the proposed design scheme of distributed filters.

JSH transformation and its application to stability analysis for linear discrete time-delay systems

This paper is concerned with stability analysis of linear discrete time-delay systems. Firstly, by JSH transformation, we developed a new necessary and sufficient condition for the asymptotic delay-dependent stability. Secondly, in order to make the designed results of this work more applicable, an unified matrix polynomial approach and numerical radius are used to get a sufficient delay-independent asymptotic stability criterion. The effectiveness and the conservatism reduction of the proposed asymptotic stability results are shown by means of three numerical examples.

On Finite frequency H∞ performance for discrete linear time delay systems

ABSTRACTThis paper is concerned with the finite frequency H∞ performance of discrete linear systems with time delay. The disturbance of the system is considered to be restricted in low/middle/high frequency. A novel summation inequality is proposed by including more free weighed matrices. Then by using generalised Kalman–Yakubovich–Popov Lemma and a Lyapunov–Krasovskii functional like method, a new bounded real lemma is presented for the finite frequency H∞ performance analysis. The advantages and reduced conservatism of the new criterion are illustrated by a classical example from the literature.

Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances

In this paper, a sliding mode control (SMC) of uncertain discrete singular systems with external disturbances and time-varying delays is under consideration. By use of the free weighting matrices and the Lyapunov–Krasovskii functional, a delay-dependent sufficient condition is given in strict linear matrix inequality (LMI) format to guarantee the sliding mode dynamics to be admissible (regular, causal and stable). Furthermore, a proposed SMC law and an adaptive SMC law are synthesized to make sure that the trajectories of system can be driven to a region near equilibrium point in finite time. Finally, a numerical example is designed to display the effectiveness of the control scheme. All these results are expected to propose a new approach for the research on SMC of discrete time-delay singular systems.

Multiple summation inequalities and their application to stability analysis of discrete-time delay systems

This paper is devoted to stability analysis of discrete-time delay systems based on a set of Lyapunov–Krasovskii functionals. New multiple summation inequalities are derived that involve the famous discrete Jensen׳s and Wirtinger׳s inequalities, as well as the recently presented inequalities for single and double summation in [16]. The present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of LMIs establishing a rigorous theoretical basis for the comparison of conservatism of the investigated methods. Numerical examples illustrate the efficiency of the method.

Delay-Variation-Dependent Stability of Delayed Discrete-Time Systems

This note is concerned with the stability analysis of linear discrete-time system with a time-varying delay. A generalized free-weighting-matrix (GFWM) approach is proposed to estimate summation terms in the forward difference of Lyapunov functional, and theoretical study shows that the GFWM approach encompasses several frequently used estimation approaches as special cases. Moreover, an augmented Lyapunov functional with a delay-product type term is constructed to take into account delay changing information. As a result, the proposed GFWM approach, together with the augmented Lyapunov functional, leads to a less conservative delay-variation-dependent stability criterion. Finally, numerical examples are given to illustrate the advantages of the proposed criterion.

New stability analysis for discrete time-delay systems via auxiliary-function-based summation inequalities

For the stability analysis of discrete time-delay systems, Jensen inequality has been widely used as the method supporting inequalities for summation quadratic functions. It not only requires a smaller number of decision variables than other approaches but also achieves identical or comparable performance behavior. Based on the analysis for the conservatism of Jensen inequality, however, this paper suggests a new summation inequality say an auxiliary-function-based summation inequality. It is verified that the proposed inequality is a generalized form of the novel summation inequality reported recently. Also, an application to stability analysis for discrete time-delay systems is provided.

Security-guaranteed filtering for discrete-time stochastic delayed systems with randomly occurring sensor saturations and deception attacks

In this paper, the security-guaranteed filtering problem is studied for a class of nonlinear stochastic discrete time-delay systems with randomly occurring sensor saturations (ROSSs) and randomly occurring deception attacks (RODAs). The nonlinearities in systems satisfy the sector-bounded conditions, and the time-varying delays are unknown with given lower and upper bounds. A novel measurement output model is proposed to reflect both the ROSSs and the RODAs. A new definition is put forward on the security level with respect to the noise intensity, the energy bound of the false signals, the energy of the initial system state, and the desired security degree. We aim at designing a filter such that, in the presence of ROSSs and RODAs, the filtering error dynamics achieves the prescribed level of security. By using the stochastic analysis techniques, a sufficient condition is first derived under which the filtering error system is guaranteed to have the desired security level, and then, the filter gain is designed by solving a linear matrix inequality with nonlinear constraints. Finally, a numerical example is provided to demonstrate the feasibility of the proposed filtering scheme. Copyright © 2016 John Wiley & Sons, Ltd.

Stability of two‐dimensional Roesser systems with time‐varying delays via novel 2D finite‐sum inequalities

This study considers the problem of stability analysis of discrete-time two-dimensional (2D) Roesser systems with interval time-varying delays. New 2D finite-sum inequalities, which provide a tighter lower bound than the existing ones based on 2D Jensen-type inequalities, are first developed. Based on an improved Lyapunov–Krasovskii functional, the newly derived inequalities are then utilised to establish delay-range-dependent linear matrix inequality-based stability conditions for a class of discrete time-delay 2D systems. The effectiveness of the obtained results is demonstrated by numerical examples.

Finite-time stability for a linear discrete-time delay systems by using discrete convolution: An LMI approach

The problem of finite-time stability for a class of linear discrete-time systems with state delay is studied. The Lyapunov-Krasovskii-like functional which is based on discrete convolutions of delayed state vector and time-dependent discrete vector functions is used. In order to obtain much less conservative results, new sufficient condition is derived in form of linear matrix inequalities. Numerical example is given to demonstrate the effectiveness of the proposed stability criterion. It was shown that the obtained results are less conservative than some existing ones in the literature. A computer simulation was performed for the analysis of the dynamical behaviour of this system.

Componentwise ultimate bounds for positive discrete time-delay systems perturbed by interval disturbances

This paper presents a method to derive componentwise ultimate upper bounds and componentwise ultimate lower bounds for linear positive systems with time-varying delays and bounded disturbances. The disturbance vector is assumed to vary within a known interval whose lower bound may be different from zero. We first derive a sufficient condition for the existence of componentwise ultimate bounds. This condition is given in terms of the spectral radius of the system matrices which is easy to check and allows us to compute directly both the smallest componentwise ultimate upper bound and the largest componentwise ultimate lower bound. Then, by using the comparison method, we extend the obtained result to a class of nonlinear time-delay systems which has linear positive bounds. Two numerical examples are given to illustrate the effectiveness of the obtained results.

Stability analysis for discrete time-delay systems based on new finite-sum inequalities

A new finite-sum inequality is derived that includes the discrete Jensen’s inequality and the Abel lemma-based finite-sum inequality as special cases. Another new inequality, which needs fewer decision variables than the first one and provides a tighter lower bound than the Abel lemma-based finite-sum inequality, is also given. Applying these new inequalities yields new results on stability analysis for discrete time-delay systems. Two numerical examples demonstrate the effectiveness and superiority of the proposed methods.

Dissipativity and passivity analysis for discrete-time T–S fuzzy stochastic neural networks with leakage time-varying delays based on Abel lemma approach

In this paper, the problem of dissipativity and passivity analysis for discrete-time T–S fuzzy stochastic neural networks with leakage time-varying delays is investigated based on Abel lemma approach. In order to obtain less conservative results, Jensen inequality, free-weighting matrix approach and Wirtinger-based inequality have been intensively used in the context of time delay systems. In parallel, the above-mentioned approaches have also been applied to discrete time-delay systems. However, it is well-known that these inequalities may introduce an undesirable conservatism in the dissipativity and passivity conditions in the existing available literature. In this paper, we propose an alternative inequality based on Abel lemma, more precisely on the Abel lemma-based finite sum inequalities. By constructing suitable Lyapunov–Krasovskii functional and using the stochastic analysis technique, strictly (Q,S,R)−γ-dissipativity and passivity conditions are derived to the concerned neural networks. The proposed criterion that depends on the upper bounds of the leakage time-varying delay is given in terms of linear matrix inequalities, which can be solved by MATLAB LMI Control Toolbox. Finally, numerical examples are shown to demonstrate the usefulness and effectiveness of the proposed methods.

Subspace-based identification of discrete time-delay system

We investigate the identification problems of a class of linear stochastic time-delay systems with unknown delayed states in this study. A time-delay system is expressed as a delay differential equation with a single delay in the state vector. We first derive an equivalent linear time-invariant (LTI) system for the time-delay system using a state augmentation technique. Then a conventional subspace identification method is used to estimate augmented system matrices and Kalman state sequences up to a similarity transformation. To obtain a state-space model for the time-delay system, an alternate convex search (ACS) algorithm is presented to find a similarity transformation that takes the identified augmented system back to a form so that the time-delay system can be recovered. Finally, we reconstruct the Kalman state sequences based on the similarity transformation. The time-delay system matrices under the same state-space basis can be recovered from the Kalman state sequences and input-output data by solving two least squares problems. Numerical examples are to show the effectiveness of the proposed method.

New finite-sum inequalities with applications to stability of discrete time-delay systems

This paper is concerned with the problem of stability analysis of discrete time-delay systems. New finite-sum inequalities, which encompass the ones based on Abel lemma or Wirtinger type inequality, are first proposed. The potential capability of the newly derived inequalities is then demonstrated by establishing less conservative stability conditions for some classes of linear discrete-time systems with delay. The derived stability criteria are theoretically and numerically proved to be less conservative than existing results.