Discrete Razumikhin-type stability theorems for stochastic discrete-time delay systems

By using the Razumikhin-type technique, for stochastic discrete-time delay systems, this paper establishes the discrete Razumikhin-type theorems on the pth moment stability, the global pth moment stability and the pth moment exponential stability, respectively. The almost sure exponential stability is also investigated by using the pth moment exponential stability and the Borel–Cantelli lemma. As the applications of t he established theorems, stability of a special class of stochastic discrete-time delay systems, synchronization of the stochastic discrete-time delay dynamical networks and stabilization of a stochastic discrete-time linear delay time invariant system are examined.     

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.     

Necessary and sufficient conditions to stability of discrete-time delay systems

This paper is concerned with the stability analysis of discrete-time linear systems with time-varying delays. The novelty of this paper lies in that a novel Lyapunov–Krasovskii functional that updates periodically along with the time is proposed to reduce the conservatism and eventually be able to achieve the non-conservativeness in stability analysis. It can be proved that the stability of a discrete-time linear delay system is equivalent to the existence of a periodic Lyapunov–Krasovskii functional. Two necessary and sufficient stability conditions in terms of linear matrix inequalities are proposed in this paper. Furthermore, the novel periodic Lyapunov–Krasovskii functional is employed to solve the ?₂-gain performance analysis problem when exogenous disturbance is considered. The effectiveness of the proposed results is illustrated by several numerical examples.     

Study on stability and stabilizability of discrete-time mean-field stochastic systems

This paper is to study the mean square stabilizability and regional stability of discrete-time mean-field stochastic systems. Firstly, a necessary and sufficient condition is presented via the spectrum of linear operator to illustrate the stabilizability of discrete-time mean-field stochastic systems. B(0, γ)-stabilizability is introduced and transformed into solving linear matrix inequalities (LMIs). Secondly, B_M-stability is characterized, especially, the stabilities of circular region, sector region and annulus regions are discussed extensively. Finally, as applications, it is shown that B(0, γ₁; γ₂)-stability has close relationship with the decay rate of the system state response and the Lyapunov exponent.     

New stability conditions for discrete-time switched positive linear time-varying systems

In this note, we will devote to investigate the stability of discrete-time switched positive linear time-varying systems (PLTVSs). Firstly, a new asymptotic stability criterion of discrete-time PLTVSs is obtained by using time-varying copositive Lyapunov functions (TVCLFs) and this criterion is then extended to the switched case based on the multiple TVCLFs. Furthermore, the sufficient conditions are derived for stability of discrete-time switched PLTVSs with stable subsystems by means of function-dependent average dwell time and function-dependent minimum dwell time. In addition, the stability sufficient conditions are drawn for the switched PLTVSs which contain unstable subsystems. It is worth noting that the difference of TVCLFs and multiple TVCLFs are both relaxed to indefinite in our work. The theoretical results obtained are verified by two numerical examples.     

Positivity and exponential stability of discrete-time coupled homogeneous systems with time-varying delays

This paper investigates the positivity and stability of discrete-time coupled homogeneous systems with time-varying delays. First, an explicit criterion is given for the positivity of discrete-time coupled homogeneous delay systems. Then, by using the properties of homogeneous functions, a sufficient condition is presented for ensuring the stability of the considered systems. Moreover, the obtained result is applied to study the stability of positive singular systems with time-varying delay. It should be noted that it is the first time that the stability result is given for discrete-time coupled homogeneous positive systems with time-varying delays. Two numerical examples are presented to demonstrate the effectiveness of the derived results.     

New criterion for finite-time stability of linear discrete-time systems with time-varying delay

This paper is concerned with the problem of finite-time stability analysis of linear discrete-time systems with time-varying delay. The time-varying delay has lower and upper bounds. By choosing a novel Lyapunov–Krasovskii-like functional, a new sufficient condition is derived to guarantee that the state of the system with time-varying delay does not exceed a given threshold during a fixed time interval. Then, the corresponding corollary is developed for the case of constant time delay. Numerical examples are provided to demonstrate the effectiveness and merits of the proposed method.     

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.     

A delay-variation-dependent stability criterion for discrete-time systems via a bivariate quadratic function negative-determination lemma

This article concerns with stability analysis of discrete linear systems with time-varying delays. Firstly, we extend a quadratic function negative-determination lemma for a single variable to the bivariate case. Secondly, we construct a novel Lyapunov-Krasovskii functional (LKF) with a quadratically delay-dependent matrix to investigate the stability of discrete-time systems with time-varying delays. Based on the proposed lemma, a new delay-variation-dependent stability criterion is derived. Finally, numerical examples are given to illustrate the theoretical result and the proposed criterion is shown to be less conservative than some previous ones.     

Practical stability of impulsive stochastic delayed systems driven by G-Brownian motion

This paper investigates practical stability problem for nonlinear impulsive stochastic delayed systems driven by G-Brownian motion (IGSDSs). Practical stability can describe quantitative properties and qualitative behavior in contrast to traditional Lyapunov stability theory. Based on G-Lyapunov function, Razumikhin-type theorem, G-Itô formula, Burkholder–Davis–Gundy (B-D-G) inequalities I & II and stochastic analysis technique, some new criteria for moment and quasi sure global practical uniform exponential stability of IGSDSs are proposed. Finally, two examples are presented to verify validity of our theoretical results.     

On controllability of discrete-time bilinear systems

In this paper, necessary and sufficient conditions for the controllability of a class of discrete-time bilinear systems are proposed, which extend the existing results and show that the controllability counterexample in [1] that is derived by Euler discretization of an uncontrollable bilinear system, is a special case of the necessary and sufficient conditions.     

Inverse control of discrete-time multivariable systems

The inverse control of discrete-time single-input/single-output linear systems is extended to multivariable systems. The extension considered is far from being trivial when a multivariable system possesses interactions and multiple delays, leading to ill-conditioning or even singularity of the problem. A feedback control algorithm suggested is based on decoupling and compensating the plant by preceding it with the inverse controller. The inverse controller is derived by using the regularization of the pseudoinverse of the plant convolution matrix. In this way a regularized multivariable control with a minimum norm is provided. A numerical example of the inverse control of a distillation column is presented.     

Mean-square exponential stability for a class of discrete-time nonlinear singular Markovian jump systems with time-varying delay

This paper investigates the problem of mean-square exponential stability for a class of discrete-time nonlinear singular Markovian jump systems with time-varying delay. The considered systems are with mode-dependent singular matrices _Er(k)$E_{r\left(k\right)}$. By using the free-weighting matrix method and the Lyapunov functional method, delay-dependent sufficient conditions which guarantee the considered systems to be mean-square exponentially stable are presented. Finally, some numerical examples are employed to demonstrate the effectiveness of the proposed methods.     

Dissipative observers for discrete-time nonlinear systems

This paper addresses the problem of designing a state observer for a class of nonlinear discrete-time systems using the dissipativity theory. We show that the dissipative observation methodology, originally proposed by one of the authors for continuous-time nonlinear systems, can be extended to the discrete-time case. For constructing a convergent observer, the methodology is applied to the nonlinear estimation error dynamics, which is decomposed into a discrete-time Linear Time-Invariant (LTI) subsystem in the forward loop, connected to a time-varying static nonlinearity in the feedback loop. In order to assure asymptotic stability of the closed-loop, complementary dissipativity conditions are imposed on each of the subsystems: (i) the static nonlinearity is required to be dissipative with respect to a quadratic supply rate, and (ii) the observer gains are designed such that the LTI system is dissipative with respect to a complementary supply rate. As in the continuous time framework, the proposed method includes as special cases, unifies and generalizes some observer design methods proposed previously in the literature. A great advantage of the Dissipative Observer Design Method proposed here is that it leads to Matrix Inequalities for the design of the observer gains, and these can be usually converted into Linear Matrix Inequalities (LMI’s). The results are illustrated using Chua’s Chaotic system.     

Model-based event-triggered control of switched discrete-time systems

In this paper, we aim to study model-based event-triggered control for a class of uncertain switched discrete-time systems composed of stabilizable and unstabilizable subsystems. A nominal model is introduced at the controller side to form a dynamic controller so that it can provide a kind of approximate estimate of the system state for system input even the overall switched discrete-time system is running in open-loop during any two consecutive event-triggered instants. By using multi-Lyapunov function method and the average dwell time switching strategy, stability conditions given in linear matrix inequality form for the closed-loop switched discrete-time system are derived. The design of control gains is also given. Finally, a numerical example and a physical example are provided to verify the effectiveness and usefulness of the proposed method.     

Output reversibility in linear discrete-time dynamical systems

Output reversibility involves dynamical systems where for every initial condition and the corresponding output there exists another initial condition such that the output generated by this initial condition is a time-reversed image of the original output with the time running forward. Through a series of necessary and sufficient conditions, we characterize output reversibility in linear discrete-time dynamical systems in terms of the geometric symmetry of its eigenvalue set with respect to the unit circle in the complex plane. Furthermore, we establish that output reversibility of a linear continuous-time system implies output reversibility of its discretization. In addition, we present a control framework that allows to alter the system dynamics in such a way that a discrete-time system, otherwise not output reversible, can be made output reversible. Finally, we present numerical examples involving a discretization of a Hamiltonian system that exhibits output reversibility and an example of a controlled system that is rendered output reversible.     

New conditions on finite-time stability of linear discrete-time system

This paper is concerned with the problems of set-based finite-time stability (SFTS) and set-based finite-time boundedness (SFTB) for both certain and uncertain linear time-varying systems. The concepts of SFTS and SFTB are defined. Different from existing results, sufficient conditions for SFTS and SFTB are directly derived from the basic definitions of finite-time stability (FTS) and finite-time boundedness (FTB) by using the convex hull technique rather than utilizing the weighted quadratic functions. Thus, more practical constraints on the system states can be dealt with. Furthermore, intervals, zonotopes and polytopes are employed to describe the typical compact convex sets. For linear uncertain systems, the uncertain time-varying state sets are assumed to be represented by interval matrices and matrix zonotopes, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the main results.     

Chaotic behavior of discrete-time linear inclusion dynamical systems

Given any finite family of real d-by-d nonsingular matrices $\{S_1,\dots ,S_l\},$ by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:

$\begin{array}{c}\hfill x_n\in \{S_k{x}_{n?1};1\le k\le l\},x_0\in {\mathbb{R}}^d\mathrm{and}n\ge 1,\end{array}$

we study the chaotic dynamics of the state trajectory (x_n(x₀, σ))_{n ≥ 1} with initial state $x_0\in {\mathbb{R}}^d,$ governed by a switching law $\sigma :\mathbb{N}\to \{1,\dots ,l\}$. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all $x_0,y_0\in {\mathbb{R}}^d,x_0\ne y_0,$

$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; inf}\parallel x_n(x_0,\sigma )?x_n(y_0,\sigma )\parallel =0\mathrm{and}\underset{n\to +\infty }{lim\; sup}\parallel x_n(x_0,\sigma )?x_n(y_0,\sigma )\parallel =\infty .\end{array}$

This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (x_n(x₀, σ))_{n ≥ 1} are extremely sensitive to the initial states $x_0\in {\mathbb{R}}^d$. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.     

Practical exponential stability of switched homogeneous positive nonlinear systems with stable and unstable modes

In this article, we primarily investigate practical exponential stability of switched homogeneous positive nonlinear systems (SHPNSs) that have partial unstable modes and perturbation. First of all, the max-separable Lyapunov function (MSLF) technique and a special pre-setting switching sequence are used to define a number of significant stability conditions that ensure the state trajectories of the system converge to a confined region in continuous-time and discrete-time domains. In addition, the key results include several earlier findings as special situations and can be directly applied to general switched systems, not necessarily positive. Finally, two important instances are provided to further illustrate the validity of the theoretical findings.