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.     

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.     

New finite-time stability for fractional-order time-varying time-delay linear systems: A Lyapunov approach

The primary goal of this paper is to examine the finite-time stability and finite-time contractive stability of the linear systems in fractional domain with time-varying delays. We develop some sufficient criteria for finite-time contractive stability and finite-time stability utilizing fractional-order Lyapunov-Razumikhin technique. To validate the proposed conditions, two different types of dynamical systems are taken into account, one is general time-delay fractional-order system and another one is fractional-order linear time-varying time-delay system, furthermore the efficacy of the stability conditions is demonstrated numerically.     

Finite-time stability analysis and stabilization for linear discrete-time system with time-varying delay

The problem of finite-time stability for linear discrete-time systems with time-varying delay is studied in this paper. In order to deal with the time delay, the original system is firstly transformed into two interconnected subsystems. By constructing a delay-dependent Lyapunov–Krasovskii functional and using a two-term approximation of the time-varying delay, sufficient conditions of finite-time stability are derived and expressed in terms of linear matrix inequalities (LMIs). The derived stability conditions can be applied into analyzing the finite-time stability and deriving the maximally tolerable delay. Compared with the existing results on finite-time stability, the derived stability conditions are less conservative. In addition, for the stabilization problem, we design the state-feedback controller. Finally, numerical examples are used to illustrate the effectiveness of the proposed method.     

Robust finite-time sliding mode control for discrete-time singular system with time-varying delays

This paper explores the finite-time bounded issue for discrete-time singular time-varying delay system via sliding mode control method. A suitable discrete-time sliding mode control law is constructed to drive the state trajectories onto the specified sliding surface in a given finite time interval. Meanwhile, sufficient conditions for finite-time bounded to the closed-loop delayed system are provided in both reaching phase and sliding motion phase. In addition, the finite-time sliding mode controller gain matrix can be solved by using the linear matrix inequalities approach. Finally, three numerical examples are illustrated to demonstrate the superiority and practicability of presented results.     

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.     

New results on robust finite-time stability of singular large-scale complex systems with interconnected delays

In this paper, we provide an efficient approach based on combination of singular value decomposition (SVD) and Lyapunov function methods to finite-time stability of linear singular large-scale complex systems with interconnected delays. By representing the singular large-scale system as a differential-algebraic system and using Lyapunov function technique, we provide new delay-dependent conditions for the system to be regular, impulse-free and robustly finite-time stable. The conditions are presented in the form of a feasibility problem involving linear matrix inequalities (LMIs). Finally, a numerical example is presented to show the validity of the proposed results.     

Stochastic finite-time stabilization for discrete-time positive Markov jump time-delay systems

In this paper, the problems of stochastic finite-time stability and stabilization of discrete-time positive Markov jump systems are investigated. To deal with time-varying delays and switching transition probability (STP), stochastic finite-time stability conditions are established under mode-dependent average dwell time (MDADT) switching signal by developing a stochastic copositive Lyapunov-Krasovskii functional approach. Then a dual-mode dependent output feedback controller is designed, thus stochastic finite-time stabilization is achieved based on linear programming technique. Finally, two examples are given to show the effectiveness of our results.     

New approaches to positive observer design for discrete-time positive linear systems

This paper aims at providing new design approaches for positive observers of discrete-time positive linear systems based on a construction method of linear copositive Lyapunov function for positive systems. First, an efficient positive observer design approach is proposed by using linear programming such that the observer error system is exponentially stable. Furthermore, an interval observer design is proposed for uncertain positive systems. Then, the results are extended to positive time delay systems. In contrast with the previous design approaches, the new design method provides a general observer design with lower computational burden. Finally, three comparison examples are given to show the merit of the new design approach.     

Practical stability of discrete-time systems

Theorems are stated and proved that provide necessary and sufficient conditions for practical stability of discrete-time systems.The first part of the paper deals with stability and instability with respect to time-varying sets, whereas the second part is devoted to the study of final and semi-final stability. The conditions obtained, which take the form of existence of discrete Lyapunov-like functions, generalize previous results.     

An improved result for the finite-time stability of the singular system with time delay

This paper pays attention to the finite-time stability problem for the singular system with time delay. Some improved inequalities are presented and by employing these inequalities along with the Lyapunov-Krasovskii functional method, an improved sufficient condition which ensures that the investigated system is regular, impulse-free and finite-time stable is derived. Finally, a numerical example is provided to demonstrate the effectiveness and superiority of the proposed approach.     

Uniform exponential stability of periodic discrete switched linear system

Let {Π_τ(m, n): m?≥?n?≥?0} be the family of periodic discrete transition matrices generated by bounded valued square matrices Λ_τ(n), where $\tau :[0,1,2,?)\to \Omega$ is an arbitrary switching signal. We prove that the family {Π_τ(m, n): m?≥?n?≥?0} of bounded linear operator is uniformly exponentially stable if and only if the sequence $n?{\sum }_{k=0}^n{e}^{i\alpha k}{\Pi }_{\tau }(n,k)w\left(k\right):{\mathbb{Z}}_{+}\to \mathbb{R}$ is bounded.     

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.     

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.     

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.     

Robust finite-time stability of singular nonlinear systems with interval time-varying delay

The problem of robust finite-time stability (RFTS) for singular nonlinear systems with interval time-varying delay is studied in this paper. Some delay-dependent sufficient conditions of RFTS are derived in the form of the linear matrix inequalities (LMIs) by using Lyapunov–Krasovskii functional (LKF) method and singular analysis technique. Two examples are provided to show the applications of the proposed criteria.     

New stability criteria for linear impulsive systems with interval impulse-delay

The robust stability problem for linear time-delay systems with general linear delayed impulses is investigated. Different from the previous results, the impulse-delays are allowed to be larger than the impulse period. An auxiliary state variable is introduced to construct an augmented model of the impulsive system, under which the discrete dynamics introduced by impulse-delays can be highlighted. A novel piecewise Lyapunov functional is introduced to analyze the stability of the augmented model. This functional is continuous along the trajectories of the augmented model, and is not required to be positive-definite at non-impulse instants. LMI-based exponential stability conditions are derived, which depend on both the impulse-dwell-time and the impulse-delay-interval. Numerical examples show that the obtained stability criteria are able to handle the benefit/harmful impulse-delays.     

Exponential stability of switched linear impulsive time-varying system and its application

This paper aims at the problem of exponential stability for switched linear impulsive time-varying system. By constructing two different switched discretized Lyapunov functions, some new sufficient conditions ensuring the global exponential stability of switched linear impulsive time-varying system are provided, which can be employed to the case when all subsystems are unstable. Furthermore, we apply theoretical results to the consensus of multi-agent system with switching topologies. Finally, numerical examples demonstrate the effectiveness of given results.     

Necessary conditions for exponential stability of linear neutral type systems with multiple time delays

In this paper, we will give necessary conditions for the exponential stability of linear neutral type systems with multiple time delays by employing the Lyapunov–Krasovskii functional approach. These conditions not only extend the existing results of the neutral-delay-free systems, but also provide a new tool for stability analysis of linear neutral type systems with multiple time delays by characterizing instability domains. As a medium step, we will investigate several crucial properties which are involved with both the fundamental matrix and Lyapunov matrix. Numerical examples illustrate the validity of the theoretical results.