Unstable Subsystems Research Articles

Event-Triggered and Self-Triggered L∞ Control for Markov Jump Stochastic Nonlinear Systems Under DoS Attacks.

This article investigates event-triggered and self-triggered L∞ control problems for the Markov jump stochastic nonlinear systems subject to denial-of-service (DoS) attacks. When attacks prevent system devices from obtaining valid information over networks, a new switched model with unstable subsystems is constructed to characterize the effect of DoS attacks. On the basis of the switched model, a multiple Lyapunov function method is utilized and a set of sufficient conditions incorporating the event-triggering scheme (ETS) and restriction of DoS attacks are provided to preserve L∞ performance. In particular, considering that ETS based on mathematical expectation is difficult to be implemented on a practical platform, a self-triggering scheme (STS) without mathematical expectation is presented. Meanwhile, to avoid the Zeno behavior resulted from general exogenous disturbance, a positive lower bound is fixed in STS in advance. In addition, the exponent parameters are designed in STS to reduce triggering frequency. Based on the STS, the mean-square asymptotical stability and almost sure exponential stability are both discussed when the system is in the absence of exogenous disturbance. Finally, two examples are given to substantiate the effectiveness of the proposed method.

Stability analysis of time-varying switched systems via indefinite difference Lyapunov functions

Asymptotic stability of time-varying switched systems is investigated in this paper. The less conservative sufficient criteria for asymptotic stability of time-varying discrete-time switched systems are proposed via common indefinite difference Lyapunov functions and multiple indefinite difference Lyapunov functions introduced in this note, respectively. Common indefinite difference Lyapunov functions can be used to analyze stability of a switched system with asymptotic stable subsystems and arbitrary switching signal. Multiple indefinite difference Lyapunov functions can be used to investigate stability of a switched system with unstable subsystems and a given switching signal. The difference of the proposed Lyapunov function may be positive at some instants for an asymptotically stable subsystem. We compare these main results and illustrate the effectiveness of the obtained theorems by three numerical examples.

The moment exponential stability of infinite-dimensional linear stochastic switched systems

<abstract><p>This paper studies the 2nd-moment exponential stability of a class of infinite-dimensional linear stochastic switched systems comprising two unstable subsystems. We first construct an algebraic sufficient condition on the existence of multiple Lyapunov functions. Then, two switching strategies are designed to stabilize infinite-dimensional linear stochastic switched systems in terms of the multiple Lyapunov function method. Moreover, the system possesses good robust stability of the switching time with our switching strategies.</p></abstract>

A Switching Predictive Control Method of Three Phase VSCs Based on Switching Affine Model

The working process of three-phase voltage source converters (VSCs) can be treated as the switching between the different subsystems, which is a typical switched system. The switching control strategies based on the switching model achieve satisfactory control performance of three-phase VSCs with very simple control structure and less control parameters. However, two drawbacks restrict the flexible applications of the present switching control strategies, i.e. the unfixed switching frequency problem and the unstable subsystem switching problem caused by the system delay. In this paper, a current switching predictive control (CSPC) method based on the current switching affine error model is proposed to the three-phase VSCs. The proposed method, by remaining the advantages of traditional switching control methods, realizes the fixed switching frequency of three-phase VSCs in the framework of switched system and achieves desired performance under system delay conditions. The simulation and experiment results verify the effectiveness of the proposed methodology.

Stability analysis of discrete-time switched stochastic non-autonomous systems with external inputs and time-varying delays under partially unstable subsystems

This paper addresses the globally stochastically exponentially input-to-state stability (ISS) and ISS of discrete-time switched stochastic systems with partially unstable subsystems under the influence of time-varying delays and external input by applying the mode-dependent average dwell time and the multiple Lyapunov-Krasovskii functional. Some main idea of the research is overcome the influences caused by partially unstable subsystems in the switching subsystems and time-varying delays. Some new stability criteria based on the related systems are derived, these criteria extend many models and relax some restrictions compared to previous work. Finally, we use two examples to verify the effectiveness of our study.

Finite-time stability of set switched systems with non-instantaneous impulses

In this paper, we discuss the finite-time stability of set switched systems with non-instantaneous impulses which consist of stable and unstable subsystems through introducing a revised mode-dependent average dwell time method. By designing time-dependent switching law and using the multiple Lyapunov-like functions method, the finite-time stability criteria of set switched systems with non-instantaneous impulses are given. Sufficient conditions which guarantee the finite-time boundedness of the switched systems with time-varying exogenous disturbances are also given. In our switching design strategy, slow switching and fast switching are, respectively, used among stable subsystems and unstable subsystems. The criteria obtained for the switched linear systems are all provided in terms of a set of linear matrix inequalities. Numerical examples are employed to verify the efficiency of the proposed method.

Asynchronous Control of Switched Discrete-Time Positive Systems With Delay

This article considers the stabilization issue of switched discrete-time positive systems (SDPSs) with delay by using asynchronous control. Combining transition probability (TP) with mode-dependent average dwell time (MDADT), called TP-based MDADT switching, the SDPSs are more practical than classical models with average dwell-time (ADT) switching. With the aid of a co-positive Lyapunov–Krasovskii functional (CLKF), sufficient conditions ensuring the exponential stability almost surely (ES a.s.) of the SDPSs without control is studied, where the mode is not necessary to be stable. After that, a mode-dependent controller with switching delay is designed to stabilize the SDPSs in the case that there are unstable subsystems. An algorithm is provided to design the control gains. It is discovered that the mode in the closed-loop SDPSs is not required to be stable on synchronous and asynchronous switching intervals. Numerical simulations verify the merits of the new results.

Stability of networked switched systems in the presence of denial‐of‐service attacks

AbstractThis article is concerned with the stability issue of networked switched systems which consist of only unstable subsystems subject to Denial of Service (DoS) attacks. The stability analysis of networked switched systems in the presence of DoS attacks under state‐dependent switching is first presented. It is shown that the derived restrictions which differ from those in the existing work are imposed on DoS attack model. Specifically, the proposed conditions to characterize the duration and frequency of DoS attacks are state‐dependent and mode‐dependent. As a result, the ratio of total duration of DoS attack to the operating time of switched systems is higher than the previous literature. Furthermore, our result is extended to the stability of networked switched affine systems in the presence of DoS attacks. In the end, two simulation examples are given to demonstrate the effectiveness of our work.

Stability analysis of semi-Markov switching stochastic mode-dependent delay systems with unstable subsystems

This paper is dedicated to studying the stability of semi-Markov switching stochastic delay systems with unstable subsystems, where delay depends on another semi-Markovian process and transition rates are uncertain. Based on the stochastic analysis theory and the mode-dependent multi-Lyapunov function method, relevant sufficient conditions are established to ensure the asymptotic stability in mean square of the system. In addition, contrary to the existing works, the system under consider is more general and our works are not only applicable to the system with partial unstable subsystems, but also applicable to the system with all unstable subsystems. Finally, one numerical example is provided to illustrate the correctness of the method described in the paper.

Stability analysis for positive switched systems having stable and unstable subsystems based on a weighted average dwell time scheme

By developing two novel switching schemes — the slow and fast weighted average dwell time, the stability problem for positive switched systems with subsystems stable or partially unstable is investigated. It can be seen that the new schemes cover both the average dwell time and mode-dependent average dwell time schemes under some mild conditions, and stability results based on our proposed approach have lower bound and are more flexible than the existing results. Three numerical examples comparing the relevant results are analyzed to illustrate the efficiency of the new findings.

Exponential Stability of Switched Neural Networks with Partial State Reset and Time-Varying Delays

This paper mainly investigates the exponential stability of switched neural networks (SNNs) with partial state reset and time-varying delays, in which partial state reset means that only a fraction of the states can be reset at each switching instant. Moreover, both stable and unstable subsystems are also taken into account and therefore, switched systems under consideration can take several switched systems as special cases. The comparison principle, the Halanay-like inequality, and the time-dependent switched Lyapunov function approach are used to obtain sufficient conditions to ensure that the considered SNNs with delays and partial state reset are exponentially stable. Numerical examples are provided to demonstrate the reliability of the developed results.

Stability of multi-dimensional switched systems with an application to open multi-agent systems

A multi-dimensional switched system or multi-mode multi-dimensional (M3D) system extends the classic switched system by allowing different subsystem dimensions. The stability problem of the M3D system, whose state transitions at switching instants can be discontinuous due to the dimension-varying feature, is studied. The discontinuous state transition is formulated by an affine map that captures both the dimension variations and the state impulses, with no extra constraint imposed. In the presence of unstable subsystems, the general criteria featuring a series of Lyapunov-like conditions for the practical and asymptotic stability properties of the M3D system are provided under the proposed slow/fast transition-dependent average dwell time framework. Then, by considering linear subsystems, we propose a class of parametric multiple Lyapunov functions to verify the obtained Lyapunov-like stability conditions and explicitly reveal a connection between the practical/asymptotic stability property and the non-vanishing/vanishing property of the impulsive effects in the state transition process. Further, the obtained stability results for the M3D system are applied to the consensus problem of the open multi-agent system (MAS), whose network topology can be switching and size-varying due to the migrations of agents. It shows that through a proper transformation, the seeking of the (practical) consensus performance of the open MAS with disconnected digraphs boils down to that of the (practical) stability property of an M3D system with unstable subsystems.

Generic Stability Criteria for Switched Nonlinear Systems With Switching-Signal-Based Lyapunov Functions Using Takagi–Sugeno Fuzzy Model

The aim of this article is to establish generic stability conditions for switched nonlinear systems with mode-dependent average dwell-time (MDADT) switching rules via Takagi–Sugeno (T–S) fuzzy modeling, which cover all unstable modes, all stable modes, and partially unstable modes as special cases. Different from traditional multiple and multiple discontinuous Lyapunov function (MDLF) methods, the proposed novel switching-signal-based multiple discontinuous Lyapunov function (SMDLF) approach divides each mode-running interval dynamically according to the actual switching of the system. The developed approach can not only yield less conservative bounds on the MDADT but can also handle all unstable subsystems, which cannot be done by means of the traditional MDLF. A class of SMDLF is constructed for continuous-time switched T–S fuzzy systems, and relaxed stability conditions are presented for the system with both stable and unstable subsystems, using a larger switching signal space. Then, novel stability conditions are deduced for all stable and unstable subsystems in terms of slow and fast MDADT switching signals, respectively. Finally, comparative simulation examples are provided to verify the advantages and effectiveness of the proposed approach.

Protocol-Based Stability Analysis of Stochastic Hybrid Systems Under DoS Attacks

The stability issue of stochastic hybrid systems against energy-constrained denial-of-service (DoS) attacks is investigated in this article. A sampled-data-based Round-Robin protocol, which sends measurement data cyclically according to the predetermined transmission sequence, is introduced to avoid communication network congestion. Additionally, a new switched time-delay stochastic closed-loop system with an unstable subsystem is established by discussing the effect of DoS attacks. Then, the stability of the closed-loop system is discussed in light of the piecewise Lyapunov–Krasovskii functional method and stochastic analysis technique. Finally, two examples are included to illustrate the correctness and applicability of the proposed method.

Resilient fixed-time synchronization of neural networks under DoS attacks

In this paper, the fixed-time synchronization of neural networks (NNs) with denial-of-service (DoS) attacks is investigated. A switching system with a stable subsystem and an unstable subsystem is utilized to describe the error dynamic behaviours between the master and slave NNs with DoS attacks. By virtue of the comparison principle, a novel fixed-time stability lemma for the switching system is established. The estimation of upper bound of setting time is provided. We further apply this lemma to investigate the resilient fixed-time synchronization of NNs subject to DoS attacks. By designing an appropriate controller, some sufficient criteria are proposed to guarantee the synchronization within a prearranged fixed time. The effects of DoS attacks on the settling time are discussed as well. Two numerical examples are given to show the effectiveness of main results and a potential application in the secure communication is provided.

Input-to-state stability for impulsive switched systems with delays based on a class of generalised impulsive and switching signals

ABSTRACT The input-to-state stability (ISS) for discrete-time impulsive switched systems with delays is investigated, which contains both stable and unstable subsystems. By utilising the multiple Lyapunov–Krasovskii functionals (MLKFs), the delay effects, which are divided into delay-independent part and delay-dependent part, have been processed. Some sufficient conditions are presented to ensure the ISS of system, provided that the switchings and the impulses do not occur too frequently and the activation time of unstable subsystems is comparatively short. Meanwhile, a relationship is built up among admissible edge-dependent average dwell time (AED-ADT), admissible edge-dependent average impulsive interval (AED-AII), impulses magnitude, and decaying/increasing rates of the Lyapunov function. Contrast to previous works, some improvements are made: the effect of delays are fully explored, and the resulting ISS criteria are available with different impulses; AED-AII is first proposed , and combined with AED-ADT to handle the stability problem. Two numerical examples are given to illustrate the theoretical results.

Construction of a Lyapunov function for a linear large-scale periodic system with possibly unstable subsystems

This article proposes an approach to construct a Lyapunov function for a linear large-scale periodic system. In this case, in contrast to various variants of small-gain stability conditions for large-scale systems, the presence of the asymptotic stability property of independent subsystems is not assumed. To analyze the asymptotic stability of a large-scale system, the direct Lyapunov method is used in combination with the discretization method and identities of the commutator calculus. The main results are illustrated by means of examples.

Stability of Switched Systems with Time-Varying Delays under State-Dependent Switching

This paper studies the stability of linear switched systems with time-varying delays and all unstable subsystems. According to the largest region function strategy, the state-dependent switching rule is designed. By bringing in integral inequality and multiple Lyapunov-Krasovskii functionals, the stability results of delayed switched systems with or without sliding motions under the designed state-dependent switching rule are derived for different assumptions on time delay. Several numerical examples are employed to show the effectiveness and superiority of the proposed results.

Free-quasi-alternative switching signal for switched systems with unstable subsystems

This paper is concerned with the stability of switched systems with unstable subsystems under a novel switching signal. By establishing the free-quasi-alternative switching signal where switches are admissible between any two subsystems, the global exponential stability of the switched system is discussed. Compared with stability criteria under the quasi-alternative switching signal, switches among unstable subsystems are allowable, and the restriction that all unstable subsystems should reside for short is relaxed. The stability results are more general and less conservative in this paper, which is illustrated with a numerical example.

Exponential stability of totally positive switched linear systems with both stable and unstable subsystems

AbstractThis article discusses the exponential stability of totally positive switched linear systems (TPSLSs) for continuous‐time and discrete‐time cases. By employing the multiple piecewise‐continuous linear copositive Lyapunov function (MPLCLF) and mode‐dependent average dwell time (MDADT) methods, TPSLSs containing both stable and unstable subsystems are firstly considered and two kinds of switching strategies are proposed, which reduce the conservativeness greatly and provide a more relaxed constraints on parameters compared with the previous results. Finally, two examples are given to illustrate the effectiveness of our results.