Discontinuous Lyapunov Function Research Articles

Exponential stability and non-weighted L 2-gain analysis for switched neutral systems with unstable subsystems

Exponential stability and non-weighted L 2 -gain are studied for switched neutral systems with unstable subsystems. Initially, a time scheduling strategy is introduced to formulate switching-signal-based multiple discontinuous Lyapunov functions. Multiple discontinuous Lyapunov functions can be applied to each subsystem, thereby relaxing the limitations of traditional Lyapunov functions. Meanwhile, mode-dependent average dwell times are utilised to design fast and slow switching signals, ensuring stability in switched neutral systems where the switching between unstable and stable subsystems is arbitrary. Moreover, in the presence of external disturbances, the non-weighted L 2 -gain performance is discussed. Sufficient conditions are proposed to achieve the stability of switched neutral systems and non-weighted L 2 -gain performance using the linear matrix inequality method. The validity of this study is ultimately demonstrated through simulation examples.

Improved event-based fault detection filter for networked fuzzy systems under DoS attacks

This paper investigates an improved event-based fault detection method for networked fuzzy systems under denial-of-service (DoS) attacks. In order to solve the bandwidth occupation problem of communication network, a resilient event-triggered transmission strategy is developed. Additionally, a fault detection filter is designed to estimate the time of fault occurrence by using the residual signal. Under this framework, a novel Lyapunov functional related to attack parameters is established to analyze the exponential convergence of the error signals, and the filter gains and event-triggered parameters are obtained by solving linear matrix inequalities. The designed functional reduces the conservatism of the stability criteria significantly in contrast with the previous discontinuous Lyapunov functionals. Finally, a simulation example is provided to verify the effectiveness of the proposed method.

Mean square exponential stability with l2−l∞ gain of discrete-time switched-Markovian jump systems

In this paper, the problem of mean square exponential stability with [Formula: see text] gain is studied for the discrete-time switched-Markovian jump systems, which are composed of a deterministic switched system and a Markovian jump system. The mode-dependent average dwell time method is adopted to design the switching signal for the deterministic switched system, and the Markovian jump property is presented by the Markov chain, which is dependent on the designed mode-dependent average dwell time switching signal. The multiple discontinuous Lyapunov function is constructed to guarantee the mean square exponential stability and [Formula: see text] performance. Furthermore, a novel inequality is established to show the relationship between multiple discontinuous Lyapunov functions and disturbances to derive the [Formula: see text] performance. Finally, two numerical examples and a circuit system are given to demonstrate the effectiveness of the obtained results.

Bipartite containment control of nonlinear multi‐agent systems with unknown inputs and state constraints

SummaryThis article addresses the issue of bipartite containment tracking for nonlinear strict feedback multi‐agent systems (MASs) with disturbances and measurement uncertainties. To provides a better performance against strong disturbance, a coordination transformation is applied such that all the nonlinear functions, controller disturbances and measurement uncertainties are lumped into a single disturbance signal in the last state equation. Then, the lumped disturbance is compensated through a novel disturbance rebuilding method. Moreover, in order to get better control performance, state constraints are also considered through a nonlinear transformation function (NTF). Different from traditional methods dealing with the state constraint problems, the NTF method can avoid using any discontinuous Lyapunov function. Finally, an adaptive finite‐time distributed controller is established via the disturbance estimates, backstepping technique and fuzzy logic system (FLS). The performance of the controller is illustrated by a simulation example in which the bipartite containment formation of followers is realized in a practical finite time.

A bumpless transfer piecewise filtering approach for fault detection of switched linear systems

AbstractThis study focuses on bumpless transfer (BT) fault detection in switched linear systems operating under mode‐dependent average dwell time (MDADT) switching. We propose a piecewise fault detection filter (FDF) inspired by the segmentation concept inherent in the multiple discontinuous Lyapunov function (MDLF) approach. This filter generates a residual signal that effectively minimizes the amplitude of residual bumps, surpassing the performance of traditional non‐piecewise FDFs. We introduce a new definition of residual BT performance, aimed at reducing residual bump amplitude during filter transitions. The goal is to design the piecewise FDF such that the augmented system maintains asymptotic stability with weighted performance, while also meeting the residual BT performance criteria. The feasibility of achieving BT fault detection filtering is established through linear matrix inequalities (LMIs), leveraging the MDLF and MDADT strategies. The efficacy and benefits of the proposed BT piecewise FDF design are demonstrated through simulations on a boost converter.

Observer-based controller design for switched systems with stable and unstable subsystems

Abstract This paper studies the problems of both state estimation and observer-based control for a class of switched linear systems with stable and unstable subsystems. By combining slow switching and fast switching mechanism, the observer design and observer-based controller are presented. Firstly, a state observer is developed to estimate the states of switched system by designing admissible edge-dependent switching signal, and a multiple discontinuous Lyapunov function is also constructed to obtain the existence condition of observer. Secondly, based on the stability characteristics of subsystem and the estimated state information, an observer-based feedback controller is proposed. The stability of the closed-loop system is also guaranteed by Lyapunov stability theory. Finally, the feasibility of the proposed method is verified by simulation.

Distributed hybrid control for heterogeneous multi-agent systems subject to deception attacks and its application to secondary control for DC microgrid

This paper addresses the mean-square quasi-consensus problem for a class of heterogeneous multi-agent systems (MASs) with cascade-type two-layer structure subject to discrete-time deception attacks. A two-layer distributed hybrid controller is proposed based on the structural characteristics of the considered MASs. The deception attacks are supposed to occur in the upper-layer communication channel and their occurrences are modeled by a Bernoulli process. The discrete-time deception signal is considered as a bounded external disturbance sequence, so the mean-square quasi-consensus problem can be reduced to a mean-square exponential input-to-state stability (MSEISS) problem. A novel MSEISS analysis method combined with the use of a weighted discontinuous Lyapunov function is developed to establish a MSEISS criterion of the consensus error system, where the robustness performance of the mean-square quasi-consensus with respect to the randomly occurring deception attacks is quantified by the MSEISS gain. The effectiveness of the proposed resilient distributed hybrid control algorithm is verified through numerical simulations. Specifically, the proposed resilient control algorithm is applied to the secondary control of DC microgrid with discrete interaction. Several case studies show that the resulting resilient distributed hybrid secondary control algorithm has a good robustness performance on resisting the deception attacks.

Resilient Sampled-Data Control for Stabilization of T--S Fuzzy Systems via Interval-Dependent Function Method: Handling DoS Attacks

This paper is concerned with the resilient sampled-data control for stabilization of T-S fuzzy systems in the presence of denial-of-service (DoS) attacks. To describe the effects of DoS attacks, an appropriate mathematical model is established to characterize the complicated dynamical behaviors of DoS attacks and periodic sampling mechanism. On this basis, a resilient sampled-data control scheme including communication protocols and security controller gains is proposed to account for the effects of DoS attacks. Meanwhile, a novel index is developed to measure the anti-attack ability and security level. By utilizing the information of DoS attacks and resilient sampling intervals, a continuous interval-dependent Lyapunov function is constructed which does not increase at the resilient sampling instants. Then, according to the designed interval-dependent function, several inequalities estimation techniques, convex combination approach, and together with discrete-time Lyapunov theory, two sufficient criteria are derived to guarantee that the closed-loop T-S fuzzy systems are globally asymptotically stable with prespecified security levels subject to DoS attacks. The advantages of designed function are well analyzed in contrast with the previous discontinuous Lyapunov functionals. At last, two simulation examples are given to demonstrate the validity and effectiveness of the obtained results.

Distributed event-based H∞ consensus filtering for 2-D T-S fuzzy systems over sensor networks subject to DoS attacks

This paper concentrates on the H∞ consensus filtering issue of a kind of nonlinear two-dimensional (2-D) systems over sensor networks, which are described by the Takagi-Sugeno fuzzy model. With consideration of the limited communication bandwidth and the malicious network attack behavior, a distributed event-triggered mechanism is considered for the sensor network under DoS attacks. Different from the implicit assumption of many previous results that detail information of the non-periodic DoS attacks is available, a switching-like distributed event-triggered mechanism is introduced under the 2-D framework, which could schedule the triggering strategy flexibly in the attack silent or active regions. By means of the switching system analysis method and the discontinuous Lyapunov functions, this paper aims to develop a proper distributed attack-resilient H∞ consensus filter based on the information from the local sensor and its neighbors. A set of sufficient conditions are presented so that the estimation error system can reach exponential stability and meet an expected H∞ performance. Finally, a numerical example is given to demonstrate feasibility of the filter designing method.

Stabilization of Switched Linear Neutral Systems With Time-Scheduled Feedback Control Strategy

This article is concerned with the stabilization of continuous-time switched linear neutral systems with tighter bounds on mode-dependent average dwell time. By introducing a novel time-scheduled control strategy, a switching-signal-based multiple discontinuous Lyapunov function (SMDLF) is constructed for continuous-time switched linear neutral systems, where the discontinuous points of the Lyapunov function during each mode-running interval are dynamically adjusted according to the actual system switching signal. First, sufficient criteria are established for global uniform exponential stability (GUES) of switched nonlinear systems. Then, based on the constructed SMDLF, we derive less conservative stability conditions for switched linear neutral systems. Furthermore, a set of time-dependent state feedback controllers are designed to guarantee the GUES of the underlying system. In the end, a simulation example with comparison results is provided to verify the effectiveness and advantages of the presented approach.

Asynchronously finite‐time fault detection for switched systems with unstable modes and intermittent measurements

This study proposes a novel fault detection (FD) filter design approach for switched systems with unstable modes and intermittent measurement; the FD filter design approach considers asynchronous switching and ensures finite-time stability (FTS). First, a model of switched systems with undesirable communication and unstable modes is established. The FD filter is developed using the observer technique, and the conditions of FTS and H∞ performance for the above systems are proposed and verified. The multiple discontinuous Lyapunov function (MDLF) method and the mode-dependent average dwell time (DT) method are used to obtain the switching strategy, which includes fast and slow switching to address the stability of unstable and stable modes, respectively. Compared with conventional approaches, the proposed method can realize less conservative stability conditions and a more flexible design framework, providing an efficient approach for improving system performance. Finally, based on the determined FTS criteria and H∞ performance, the FD filter for switched systems is presented. Simulation results are provided to verify the effectiveness of the proposed method.

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.

Guaranteed performance impulsive tracking control of multi-agents systems under discrete-time deception attacks

This paper is concerned with the impulsive consensus tracking problem of Lipschitz nonlinear multi-agent systems subject to deception attacks. The proposed distributed impulsive tracking algorithm allows only partial state variables of every agent to be governed by consensus impulses. The considered deception attacks are supposed to take place in the controller–actuator channel, and can be modelled by a set of Bernoulli processes. By considering the discrete-time deception signal as bounded external disturbance, the consensus tracking problem reduces to the exponential input-to-state stable (EISS) problem of the tracking error system. The EISS-gain is used as an assessment index in measuring the capacity of consensus control laws for attenuating attacks. Specifically, a novel EISS analysis method combined with the use of a weighted discontinuous Lyapunov function is proposed to establish the consensus tracking criterion. A convex optimization problem with linear matrix inequalities-based constraints is formulated to design the guaranteed performance distributed impulsive controller, which makes the EISS-gain as small as possible. Two illustrated examples quantify the effectiveness of the proposed analysis and design approach.

Leader-following consensus considering effects of agents on each other via impulsive control: A topology dependent average dwell time approach

This paper intends to investigate the consensus problem of a nonlinear multi-agent system with new nonlinear terms added to the dynamics of each agent in the leader-following framework with impulsive control. The main contribution of this paper is introducing these new terms expressing the effect of each agent on neighbor agents. The new terms called effect terms (ETs) are considered with time-varying delay. Moreover, the communication interactions among all agents are addressed by a set of consensusable and unconsensusable switching topologies. In particular, the topology-dependent average dwell time (TDADT), one of the significant practical analysis methods for switched systems, has been calculated for each topology. The globally uniformly exponentially stability (GUES) for the consensus error dynamics is analyzed by employing algebraic graph theory and a multiple discontinuous Lyapunov function approach (MDLF) regarding separate Lyapunov functions for impulse instants. Furthermore, sufficient conditions in terms of linear matrix inequalities (LMIs) are derived to ensure that consensus can be achieved. Finally, the effectiveness of the theoretical analysis is corroborated by a numerical example.

Admissible edge-dependent finite-time ℋ∞ filtering for discrete-time switched systems with average dwell time

Considering that the property of finite-time convergence is of great significance for practical filters, this article is focused on the finite-time ℋ∞ filtering for discrete-time switched systems with average dwell time. Different from the case of the general mode-dependent switching, the information of the admissible switching edge is fully exploited to derive more preferable filtering scheme. In order to further reduce the design conservatism, the multiple discontinuous Lyapunov function approach, which incorporates the concept of admissible edge-dependent average dwell time, is used to analyze the finite-time stability and ℋ∞ performance of the filtering error system. This approach assigns each subsystem of the error system with a discontinuous function rather than a continuous one, so it is less conservative compared to the conventional multiple Lyapunov function approach. First, the finite-time boundedness and ℋ∞ performance is analyzed for general switched nonlinear systems with admissible edge-dependent average dwell time. Then, a sufficient condition of the finite-time boundedness and ℋ∞ performance is derived for the filtering error system based on the obtained criterion. Subsequently, an admissible edge-dependent finite-time filtering scheme is proposed for the underlying system in terms of linear matrix inequalities. Finally, the proposed filter design scheme is applied to a DC–DC power converter circuit, and its effectiveness and superiority are verified by the simulation results.

A Discontinuous Lyapunov Function Approach for Hybrid Event-Triggered Control of T–S Fuzzy Systems

This article addresses the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> control issue for Takagi–Sugeno (T–S) fuzzy systems via an event-triggered control mechanism. First, a hybrid event-triggered control strategy combined with an adaptively adjusted approach is proposed, by which the data transmission can be effectively reduced, and meanwhile, the Zeno behavior is excluded. Then, a discontinuous Lyapunov function is constructed for the resulting hybrid event-triggered control system, and a novel lemma is developed for studying the exponential stability and disturbance attenuation performance. By utilizing a time-varying variable to manage the error during the interevent intervals together with using the newly proposed lemma, sufficient criteria are established for the stability and control design of the event-triggered T–S fuzzy system with the imperfect premise matching. Finally, numerical simulations are presented to illustrate the usefulness and advantages of the obtained theorems.

Stability analysis for nonlinear switched singular systems via T-S fuzzy modeling

The stability for discrete nonlinear switched singular systems with unstable subsystems is investigated. First, by constructing an appropriate multiple discontinuous Lyapunov function, and utilizing the characteristics of mode-dependent average dwell time switching signals, new stability results for nonlinear switched singular system are established. Then, we adopt the T-S fuzzy modeling method to approximate the nonlinear switched singular systems and get general stability conditions in forms of linear matrix inequalities. Compared to the current results, our technique is more flexible and we also get tighter dwell time boundaries. Furthermore, a numerical example demonstrates the effectiveness of the proposed method.

Impulsive consensus of nonlinear fuzzy multi-agent systems under DoS attack

In this paper, the impulsive consensus of fuzzy nonlinear multi-agent systems is investigated under denial-of-service (DoS) attack. By constructing discontinuous Lyapunov function, some sufficient conditions are derived to guarantee the consensus of the nonlinear T–S fuzzy multi-agent systems in the presence of DoS attack. Different from the existing results of impulsive consensus of multi-agent systems under DoS attack, the method proposed in this paper can flexibly deal with both the problems of communication interruption and topology switching. Finally, two numerical examples are given to verify the effectiveness of the main results.

Novel asynchronous premise reconstruction based adaptive event-triggered control for fuzzy systems with actuator failure

This paper explores the adaptive event-triggered (AET) based variable gains controller design problem for a category of discrete-time nonlinear networked control systems (NCSs) with actuator failure. An AET communication scheme is applied for NCSs to reduce the utilization of limited network bandwidth. For overcoming the asynchronous issue of premise variable caused by AET scheme, a novel asynchronous premise reconstruction technique is introduced. Meanwhile, a new variable gains controller is designed subject to AET approach, where the controller gains will be automatically updated by executing the online iteration algorithm. According to a discontinuous Lyapunov function, sufficient conditions are given to ensure that the closed-loop system is asymptotically stable and satisfies H∞ performance. Finally, in the simulation results, the validity of the proposed control method is verified and it is shown that the proposed scheme can achieve better system performances than the existing design technique with fixed threshold.

Stability analysis for a class of mechanical systems with piecewise constant coefficients

A nonlinear mechanical system described by a vector equation of the Lienard type is considered. It is assumed that the forces acting on the mechanical system are given by functions that are piecewise constant with respect to time. With the aid of a special variable substitution, the equation is reduced to a system that can be considered as an impulsive switched system with infinite numbers of operating modes. The stability problem for the trivial equilibrium position of the obtained system is studied. The stability analysis is carried out using a specially constructed discontinuous Lyapunov function. This function can be understood as a multiple Lyapunov function, consisting of an infinite number of partial Lyapunov functions, each of which is used for a certain time interval. Differentiating the chosen Lyapunov function with respect to solutions of the given system on the corresponding time intervals and applying the theory of differential inequalities, sufficient conditions of the asymptotic stability of the equilibrium position are established.