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Stability of Time-Delay Systems with Delayed Impulses: Average Impulsive Estimation Approach
Abstract
This paper addresses the exponential stability for impulsive systems with distributed delay, where the delayed impulses are considered. Some sufficient conditions based on Lyapunov function for exponential stability are derived. Specially, the derived conditions do not impose any restriction on the magnitude relationship between the delay in continuous flow and the impulsive delay. It is interesting to show that the delay in continuous flow might have a potential effect on the stability of the systems. In most existing results, Lyapunov function was supposed to be increasing at impulses point by defining impulsive strength e δ with δ > 0 in the case of impulsive perturbation. Here, based on the proposed concepts of average impulsive estimation (AIE) and average positive impulsive estimation (APIE), impulsive estimation δ is not required to be positive and the information of impulsive delay can be integrated into impulsive estimation to guarantee the effect of impulsive perturbation. Furthermore, the results of stability analysis are applied to the synchronization of complex networks with distributed delay and impulses. Finally, some numerical examples are provided to illustrate the efficiency of the derived results.
... For instance, time delay may arise when network output information is transmitted over a digital communication network (see, e.g., prior studies [21,22]). Therefore, it is necessary to take into account the effect of delayed pulses on the considered systems (see, e.g., earlier research [23][24][25][26][27]). Due to the complexity of real-world situations, the delays of impulses are not always constant as they vary with the timing of the impulses. ...
... Consequently, (27) is valid for any k ∈ N 0 . Step 3. By using (27) and Definition 2, we can conclude that ...
In this paper, we study the stochastic weak local exponential stability of stochastic differential systems with state‐dependent delay (SDD) subject to average‐delay impulses by using Lyapunov–Krasovskii functional and stochastic analytical skill. Unlike time‐dependent delay discussed in the previous literature, we introduce SDD in impulsive stochastic differential systems, where SDD is a stochastic variable associated with system state, consequently leading to an indeterminate delay boundary. Our findings reveal that when destabilizing delayed impulses satisfying specific conditions are generated, the stability of stochastic differential systems with destabilizing delayed impulses and stable continuous stochastic dynamics can still be maintained. Additionally, if stable delayed impulses satisfy certain conditions, stability of the systems under consideration can also be achieved, irrespective of the stable status on the continuous stochastic dynamics. Finally, two examples are given to demonstrate the validity of theoretical results.
... From the structure matrix L and q[0], the discrete state of system (16) evolves in the following order: implies the discrete state eventually stays in a length-4 cycle ε = {q [3], q [4], q [5], q [6]}. Note that the continuous part of the system (16) is unstable. ...
... According to the definition of q[n], jump map I 1 = I s1 ∼ I q [1] = I δ 16 2 5 , jump map I 3 = I s2 ∼ I q [2] = I δ 24 2 5 , jump map I 4 = I s3 ∼ I q [3] = I δ 30 2 5 , and q[0] = I δ 15 2 5 , which implies the logical state is in a length-3 cycle ε = {q [1], q [2], q [3]}. Thus, we can obtain the structure matrix L as With L, we can construct nonlinear system (4). ...
This article investigates the global exponential stability of the nonlinear delayed systems with logically selected impulses. First, the mathematical model for a new kind of hybrid systems is proposed. Based on the semi-tensor product of matrices, several sufficient conditions for globally exponential stability are derived. The impulsive switching signals are generated by two types of logical dynamical systems of addressed systems. To be specific, one signal is given in advance, and the other one is determined by some impulsive cycle sequences. An algorithm is developed to design logical dynamical systems for determining impulsive signals. The effectiveness of the main results is verified by two examples.
... Actual nonlinear systems are subject to a wide variety of unknown disturbances, parameter uncertainties and delay phenomena [1,2]. Robust control and adaptive control methods have been proven successful at addressing these issues [3][4][5][6][7][8][9][10][11][12][13][14]. ...
... Here are some of the main contributions of this paper: (1) Contrary to the available results of infinite-time control as well as finite-time control, our predefined-time controller is capable of ensuring the CLS converges to zero within any given predefined constant number, and its convergence time cannot be affected by the initial state, making the system behavior better determined. (2) In contrast with the previous NTDS control results using the Hamiltonian approach (with the same power Hamiltonian form), this paper develops several more general results with different power Hamiltonian function forms, implying that these results have broader applications. (3) Compared with the recent predefined-time control result on the delay system in [29] (considering only the triangular form and unknown parameter), this paper examines a general class of NTDSs subject to uncertainties and disturbances and designs its predefined-time H ∞ controller and adaptive robust controller by applying different power Hamiltonian forms. ...
The article studies control as well as adaptive robust control issues on the predefined time of nonlinear time-delay systems with different power Hamiltonian functions. First, for such Hamiltonian systems with external disturbance and delay phenomenon, we construct the appropriate Lyapunov function and Hamiltonian function of different powers. Then, a predefined-time control approach is presented to stabilize the systems within a predefined time. Furthermore, when considering nonlinear Hamiltonian system with unidentified disturbance, parameter uncertainty and delay, we devise a predefined-time adaptive robust strategy to ensure that the systems reach equilibrium within one predefined time and have better resistance to disturbance and uncertainty. Finally, the validity of the results is verified with a river pollution control system example.
... Definition 4 [41] Define the AIE in the form of limit as x during the impulsive sequence ft k g k2P þ , if ...
In this paper, the issue of exponential synchronization in chaotic neural networks with system time-varying delay is investigated by employing delayed impulsive control. In order to tackle the challenge posed by the flexible delay existing in impulsive controller, the concept of average impulsive delay (AID) is adopted, which considers time delay in impulse from a holistic perspective. Then, some Lyapunov-based relaxed synchronization conditions are established by applying average impulsive interval and AID methods, where such delay is no longer constrained by some strong conditions. Meanwhile, considering different functionalities of impulsive signal, the convergence rates are precisely calculated, which indicates that the system delay may either promote or impede network synchronization. In addition, the results also clearly show that the time delay in impulse plays either a positive or a negative role in the synchronization of chaotic neural networks. Moreover, allow for the controller involving synchronizing and desynchronizing delayed impulses, the notion of average impulsive estimation serves to address time-varying impulsive effects. Ultimately, three examples are simulated to exemplify the rightness of the theoretical results.
... A numerical example that is given demonstrates how the primary theorem, and the outcomes are consistent. In future works, we can consider delayed impulses in the same scenario, as in [34,35]. ...
The exponential stability of hybrid event‐triggered singular time‐delay (STD) systems is established in this paper. To ensure their exponential stability, we give sufficient conditions for linear matrix inequalities (LMIs) for STD systems with average impulsive gain. Furthermore, zeno behavior is avoided by carefully designing the settings for event‐triggering conditions. A few numerical examples demonstrate the usefulness of the suggested findings.
This article investigates fixed-time synchronization (FxTS) for neural networks with delayed impulses. There are two main challenges associated with the delay in this area. One is that it causes the networks to oscillate close to the equilibrium. The other is that it causes the synchronization criterion to be very conservative, particularly when the impulses contain large (exceeding impulsive intervals) delays. To overcome these challenges, we propose the concept of equivalent impulsive sequence and the method of delayed impulsive sequence decomposition, respectively. Meanwhile, we establish a framework that equivalently transforms the networks with large delays into a collection of networks with small (shorter than impulsive intervals) delays. Particularly, this transformation is a sufficient and necessary condition, and thus does not induce any conservatism. Following this, we delineate the FxTS criteria for the networks containing synchronizing and desynchronizing impulses. Interestingly, it is shown that under some conditions, the large delay has no effect on the FxTS criterion of networks with synchronizing impulses, but destroys those of networks with desynchronizing impulses. Furthermore, we prove the necessity and rationality of the FxTS criterion. Finally, the conclusions are substantiated through numerical demonstrations.
This paper focuses on the Lyapunov-Razumikhin method for finite-time stability (FTS) in probability of impulsive stochastic delayed systems (ISDSs). To establish these properties, several Lyapunov-based Razumikhin conditions on FTS for ISDSs are derived by employing the comparison principle, which explicitly incorporate delay-dependent factors. Therefore, the conservativeness is less. Then, the theoretical results are utilized to address the problem of finite-time consensus in a class of stochastic delayed multi-agent systems by a designed hybrid controller, which includes an impulsive controller and a finite-time controller. It is noted that the finite-time controller contains no sign function that can avoid the chattering phenomenon caused by control signals. At last, a numerical example is presented to demonstrate the validity of the results obtained.
In this article, we investigate globally exponential synchronization problems in delayed complex dynamic networks (DCDNs) characterized by both time‐varying impulsive delay and gain (TIDG). Our research is grounded on the Halanay inequality, which serves as the keystone of our analysis. Adopting the method of average impulsive delay‐gain (AIDG), we formulate criteria for globally exponential synchronization dependent on the overall impulsive disturbances. Our criteria reveal the negative effect of AIDG on synchronization, which hinders the synchronization process. Additionally, we refine the concept of average impulsive gain to enhance its applicability. Furthermore, our results demonstrate that even in the simultaneous presence of desynchronizing and synchronizing impulses, along with time‐varying impulsive delays, DCDNs are able to maintain the original synchronization under appropriate conditions, irrespective of whether the average impulsive interval is finite or not. Finally, we validate our theoretical findings by applying them to network examples.
The paper studies the robust stability of linear uncertain systems under periodic event-triggered impulsive control (ETIC). Based on sliding variables, some sufficient conditions are derived to ensure the robust stability of the addressed impulsive control systems, and the effects of both external disturbance and uncertainty on the stability are revealed. Interestingly, it is verified that the proposed ETIC can ensure the robustness of the addressed systems with any bounded external disturbance. While, it is shown that the uncertainty may bring destabilizing impact to the stability. Further, compared with continuous event-triggered mechanism that often requires high hardware accuracy and sensor complexity in previous results, the proposed periodic event-triggered mechanism is more relaxed and realistic. It is also shown that the interevent time generated by the proposed periodic event-triggered mechanism has a unified positive lower bound, which cannot be derived by previous continuous event-triggered mechanism. Two illustrative examples are given to show the excellence of the main results.
Note to Practitioners
—It is known that the industrial plant emulator setup serves as a scaled-down representation of various industrial plant mechanisms, such as conveyor belts, drives, servomechanisms, and assembly machines. This paper is motivated by the problem of impulsive control for an industrial plant emulator setup, which has been studied by using continuous-time control or intermittent control in the previous results. Note that impulsive control only needs to implement at some discrete instants, which can reduce control cost and communication pressure compared with the continuous-time control and intermittent control. In this study, periodic ETIC, which contains the advantages of both periodic event-triggered control and impulsive control, is proposed to achieve the robust stability of linear multi-input uncertain systems with external disturbances. Some novel sliding-variables-based conditions have been obtained to guarantee the robust stability of addressed systems and the impacts of uncertainty and external disturbances on the stability are revealed simultaneously. It has been verified that the uncertainty may have destabilizing impact on the stability and under some conditions, the addressed periodic ETIC systems are robust with unknown bounded external disturbances. In addition, there exists a unified minimum time between arbitrary two consecutive triggered instants. In fact, the periodic ETIC can be also applied to some practical systems such as the second-order relay system and buck converter system, as these systems can be represented as linear uncertain systems with external disturbances. Furthermore, it is an intriguing future research topic to extend the derived results to the case of nonlinear systems.
How to deal with the effect of infinite-time delay and maximize rest width is the main difficulty for intermittent control techniques. This paper considers asymptotic synchronization of coupled neural networks (CNNs) with bounded time-varying discrete delay and infinite-time distributed delay (mixed delays). A quantized intermittent pinning control scheme is designed to save both channel resources and control cost and reduce both the amount of transmitted information and channel blocking. Two weighted integral inequalities are first established to deal with the infinite-time distributed delay. Based on weighted double-integral inequalities, novel Lyapunov–Krasovskii functionals with negative terms are designed, which can reduce the conservativeness of the results. Some sufficient conditions in the form of linear matrix inequalities are obtained to ensure that the CNNs asymptotically synchronize to an isolated system. Moreover, the relationships between the control width, rest width, and convergence rate are explicitly given. Furthermore, an optimal algorithm is provided to increase the rest width as large as possible. As special cases, the synchronization of the CNNs with quantized pinning control and quantized intermittent control are also considered, respectively. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical analysis.
This paper focuses on the hybrid effects of parameter uncertainty, stochastic perturbation, and impulses on global stability of delayed neural networks. By using the Ito formula, Lyapunov function, and Halanay inequality, we established several mean-square stability criteria from which we can estimate the feasible bounds of impulses, provided that parameter uncertainty and stochastic perturbations are well-constrained. Moreover, the present method can also be applied to general differential systems with stochastic perturbation and impulses.
Stability analysis and control of linear impulsive systems is addressed in a
hybrid framework, through the use of continuous-time time-varying discontinuous
Lyapunov functions. Necessary and sufficient conditions for stability of
impulsive systems with periodic impulses are first provided in order to set up
the main ideas. Extensions to stability of aperiodic systems under minimum,
maximum and ranged dwell-times are then derived. By exploiting further the
particular structure of the stability conditions, the results are
non-conservatively extended to quadratic stability analysis of linear uncertain
impulsive systems. These stability criteria are, in turn, losslessly extended
to stabilization using a particular, yet broad enough, class of state-feedback
controllers, providing then a convex solution to the open problem of robust
dwell-time stabilization of impulsive systems using hybrid stability criteria.
Relying finally on the representability of sampled-data systems as impulsive
systems, the problems of robust stability analysis and robust stabilization of
periodic and aperiodic uncertain sampled-data systems are straightforwardly
solved using the same ideas. Several examples are discussed in order to show
the effectiveness and reduced complexity of the proposed approach.
In this paper, a general framework is presented for analyzing the synchronization stability of Linearly Coupled Ordinary Differential Equations (LCODEs). The uncoupled dynamical behavior at each node is general, and can be chaotic or otherwise; the coupling configuration is also general, with the coupling matrix not assumed to be symmetric or irreducible. On the basis of geometrical analysis of the synchronization manifold, a new approach is proposed for investigating the stability of the synchronization manifold of coupled oscillators. In this way, criteria are obtained for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles in the stability analysis of the synchronization manifold. Furthermore, the roles of the uncoupled dynamical behavior on each node and the coupling configuration in the synchronization process are also studied.
In the paper, the exponential stability of impulsive control systems with time delay is studied. By using the average impulsive interval method, some sufficient Lyapunov-based conditions are established for the stability of impulsive time-delay systems, and the impacts of delay on the stability analysis method are further revealed. It is interesting to show that some unstable impulsive time-delay systems may be stabilized by increasing the time delay in continuous dynamics. More interestingly, it is proved that along with the increase of the delay within a certain range, the convergence rate of such impulsive timedelay systems also increases correspondingly. Further, a strict comparison principle for impulsive control systems with delay is established. Then by utilizing this comparison principle, it can be shown that for some stable impulsive systems with delay, under certain conditions, the stability is robust against any large but bounded delay. Compared with the previous results on delay-free impulsive systems, some potential impacts of delay on the stability are investigated. Particularly, the obtained results are extended to the case of impulsive control systems with hybrid im
A new type of discretized Lyapunov functional is introduced for stability and hybrid L2×l2-gain analysis of linear impulsive delay systems. This functional consists of three parts. The first part is the conventional discretized functional for delay-independent stability analysis. The second part is the looped-functionals-like functional for exploiting the internal structure inside impulse intervals, in which the decision matrix functions are approximated by piecewise linear matrix functions. The third part is a discretized adjustive factor, which makes the proposed Lyapunov functional continuous along the system trajectories. Thanks to this continuity, the resultant criteria for exponential stability and finite hybrid L2×l2-gain are expressed in terms of a combination of the continuous and discrete parts of the system. It is shown through numerical examples that the accuracy of the stability test increases with the increase of the partition number on impulse intervals, and the convergence rate is faster than that of the previous discretized functionals-based approach.
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.
In the brief, delayed impulsive control is investigated for the synchronization of chaotic neural networks. In order to overcome the difficulty that the delays in impulsive control input can be flexible, we utilize the concept of average impulsive delay (AID). To be specific, we relax the restriction on the upper/lower bound of such delays, which is not well addressed in most existing results. Then, by using the methods of average impulsive interval (AII) and AID, we establish a Lyapunov-based relaxed condition for the synchronization of chaotic neural networks. It is shown that the time delay in impulsive control input may bring a synchronizing effect to the chaos synchronization. Furthermore, we use the method of linear matrix inequality (LMI) for designing average-delay impulsive control, in which the delays satisfy the AID condition. Finally, an illustrative example is given to show the validity of the derived results.
The paper deals with the global asymptotic stability of general nonlinear time-delay systems with delay-dependent impulses through the Lyapunov-Krasovskii method. We derive a unified stability criterion which can be applied to a variety of impulsive systems. The cases when each of the continuous dynamics and the impulsive component is either stabilizing or destabilizing are investigated. Both theoretically and numerically, we demonstrate that the obtained result is more general than those existing in the literature.
This paper studies the stability of time-delay systems with impulsive control involving stabilizing delays. A Razumikhin-type inequality with delayed impulses is proposed, where the information of time delay existing in impulses is effectively fetched and then integrated to the stabilization of time-delay systems. Some sufficient conditions ensuring global exponential stability (GES) for a class of nonlinear time-delay systems are obtained. The proposed conditions do not impose any restriction on the upper bounds of two consecutive impulse signals and on the size relation between time delays in system and in impulses, which is improved in the existing results on this topic. Moreover, our proposed results show that the time delays in impulses can contribute to the stability of the time-delay systems. Two numerical examples are given to demonstrate the applicability of our proposed results.
This paper studies the input/output-to-state stability (IOSS) of impulsive switched systems. With the help of Lyapunov and average dwell-time (ADT) methods, some sufficient conditions for IOSS of impulsive switched systems are obtained, where both types of impulses, stabilizing impulses and destabilizing impulses, are considered. It is shown that when all of the modes are IOSS, a switched system under an ADT scheme is IOSS even if there exist destabilizing impulses, and when none of the modes is IOSS, IOSS can still be achieved under the designed ADT scheme coupled with stabilizing impulses. In particular, for a special case in which an impulsive switched system is composed of some IOSS modes and some non-IOSS modes, a relationship is established among the ADT scheme, impulses, and the total dwell time between non-IOSS and IOSS modes such that the impulsive switched system is IOSS. Two examples are provided to illustrate the applications of our results.
In the paper, we investigate the exponential stability of nonlinear delayed systems with destabilizing and stabilizing delayed impulses, respectively. Specifically, the study can be divided into two cases: (1) stability of delayed systems with destabilizing delayed impulses, where the time delays in impulses can be flexible and even larger than the length of impulsive interval, and (2) stability of delayed systems with stabilizing delayed impulses, where the time delays in impulses are flexible between two consecutive impulsive instants. In order to address the time-delay term in impulses, the concept of average impulsive delay (AID) is proposed. Using the ideas of average impulsive interval and AID, we present some Lyapunov-based exponential stability criteria for delayed systems with average-delay impulses, where the delays in impulses satisfy the proposed AID condition. It is shown that time delay in impulse has double effects, namely, it may destabilize a stable system or stabilize an unstable system. Interestingly, it is also shown that for some stable delayed systems with stabilizing delayed impulses, under certain conditions, the stability can be ensured regardless of the size of delay in continuous dynamics. Further, we apply the theoretical results to the impulsive synchronization control of Chua's circuits with both transmission delay and sampling delay. Finally, some examples are given to illustrate the validity of the derived results.
For time-invariant (nonimpulsive) systems, it is already well-known that the input-to-state stability (ISS) property is strictly stronger than integral input-to-state stability (iISS). Very recently, we have shown that under suitable uniform boundedness and continuity assumptions on the function defining system dynamics, ISS implies iISS also for time-varying systems. In this paper, we show that this implication remains true for impulsive systems, provided that asymptotic stability is understood in a sense stronger than usual for impulsive systems.
Nonlinear systems with time delays have been object of intense investigation in the last decades for their ability of capturing the behavior of many natural and engineering systems, including ecological, biomedical, industrial, robotic and networked systems. In this work, a method for the event-triggered stabilization in the sample-and-hold sense of nonlinear time-delay systems with time-varying state delays is presented. The approach exploits Control Lyapunov-Razumikhin Functions and a spline-based approximation of the functional state feedback, resulting in a finite-memory controller which avoids (by construction) continuous-state monitoring while ensuring also that the minimal inter-event time is strictly positive. Examples of application of the illustrated technique highlight the potential of the approach taken.
In this paper, a nonlinear time-delay system with delayed impulses and external disturbances is considered. The robustness problem of Lp-stability with respect to impulse-delays is addressed. In order to circumvent the technical difficulties caused by delayed impulses, the impulses-delayed system is modeled as an impulse-delay free system with switching impulses by means of an augmentation method. Based on the switching structure of the augmented system, a time-dependent Lyapunov function is constructed from the original Lyapunov function for the impulse-delay free case. Then by employing the new designed Lyapunov function combined with the use of Razumikhin-type arguments, sufficient conditions for exponential stability and Lp-stability are derived. The developed results characterize the relationship among the impulse intervals, the size of impulse-delays, and the level of Lp-gain. Two numerical examples are presented to illustrate the effectiveness of the developed methodology.
This paper studies the exponential stability of nonlinear delay systems by means of event-triggered impulsive control (ETIC) approach, where impulsive instants are determined by a Lyapunov-based event-triggered mechanism (ETM). Based on the ETM, sufficient conditions are presented to exclude Zeno behavior and guarantee the exponential stability in the framework of Lyapunov–Razumikhin method. Different from time-triggered impulsive control in which the triggered time is determined artificially, ETIC is activated only when some well-designed events occur. Moreover, control input is only needed at triggered instants and there is no any control input during two consecutive triggered instants. As an application, the theoretical result is applied to nonlinear delay multi-agent systems. A class of ETIC strategies is designed to achieve consensus of the addressed systems. Finally, two numerical examples are presented to illustrate the effectiveness of the developed approach.
This article deals with the exponential synchronization problem for complex dynamical networks (CDNs) with coupling delay by means of the event-triggered delayed impulsive control (ETDIC) strategy. This novel ETDIC strategy combining delayed impulsive control with the event-triggering mechanism is formulated based on the
quadratic
Lyapunov function. Among them, the event-triggering instants are generated whenever the ETDIC strategy is violated and delayed impulsive control is implemented only at event-triggering instants, which allows the existence of some network problems, such as packet loss, misordering, and retransmission. By employing the Lyapunov–Razumikhin (L–R) technique and impulsive control theory, some sufficient conditions with less conservatism are proposed in terms of linear matrix inequalities (LMIs), which indicates that the ETDIC strategy can guarantee the achievement of the exponential synchronization and eliminate the Zeno phenomenon. Finally, a numerical example and its simulations are presented to verify the effectiveness of the proposed ETDIC strategy.
In this study, the problem of input-to-state stability (ISS) is systematically investigated for nonlinear systems with stochastic impulses. First, the ISS problem is considered for systems, where impulsive strengths are assumed to be stochastic and impulsive intervals are confined by the average impulsive interval. Then, the ISS problem is analyzed for nonlinear systems with both stochastic impulsive intensity and density. Several different cases are considered, i.e., cases where the impulsive instants are satisfied by a renewal process, Poisson process, semi-Markov chain and Markov chain. An example of coordination of multi-agent systems is provided to illustrate the effectiveness of the proposed new stability criteria.
In the paper, synchronization of coupled neural networks with delayed impulses is investigated. In order to overcome the difficulty that time delays can be flexible and even larger than impulsive interval, we propose a new method of average impulsive delay (AID). By the methods of average impulsive interval (AII) and AID, some sufficient synchronization criteria for coupled neural networks with delayed impulses are obtained. We prove that the time delay in impulses can play double roles, namely, it may desynchronize a synchronous network or synchronize a nonsynchronized network. Moreover, a unified relationship is established among AII, AID and rate coefficients of the impulsive dynamical network such that the network is globally exponentially synchronized (GES). Further, we discuss the case that time delays in impulses may be unbounded, which has not been considered in existing results. Finally, two examples are presented to demonstrate the validity of the derived results.
In this paper two important structural properties, i.e. strong observability and strong detectability, are introduced for linear hybrid systems with periodic jumps. These properties are characterized in terms of geometric and algebraic conditions over the system matrices. The concepts and characterizations of the weakly unobservable subspace and the hybrid invariant zeros are also introduced, respectively. An algorithm to compute the weakly unobservable subspace is provided. In addition, it is shown that there exists a close relationship between the hybrid invariant zeros and the properties of strong observability and strong detectability. Some examples illustrate the proposed properties.
This paper studies input-to-state stability (ISS) of general nonlinear time-delay systems subject to delay-dependent impulse effects. Sufficient conditions for ISS are constructed by using the method of Lyapunov functionals. It is shown that, when the continuous dynamics are ISS but the discrete dynamics governing the delay-dependent impulses are not, the impulsive system as a whole is ISS provided the destabilizing impulses do not occur too frequently. On the contrary, when the discrete dynamics are ISS but the continuous dynamics are not, the delayed impulses must occur frequently enough to overcome the destabilizing effects of the continuous dynamics so that the ISS can be achieved for the impulsive system. Particularly, when the discrete dynamics are ISS and the continuous dynamics are also ISS or just stable for the zero input, the impulsive system is ISS for arbitrary impulse time sequences. Compared with the existing results on impulsive time-delay systems, the obtained ISS criteria are more general in the sense that these results are applicable to systems with delay-dependent impulses while the existing ones are not. Moreover, when considering time-delay systems with delay-free impulses, our result for systems with unstable continuous dynamics and stabilizing impulses is less conservative than the existing ones, as a weaker condition on the upper bound of impulsive intervals is obtained. To demonstrate the theoretical results, we provide two examples with numerical simulations, in which distributed delays and discrete delays in the impulses are considered respectively.
The paper studies global asymptotic stability property of the trivial solution to nonlinear impulsive differential equations. Robustness of the global asymptotic stability with respect to the perturbations of the moments of jumps is investigated. Less conservative estimates on the admissible magnitude of perturbations preserving stability compared to the existing ones in the literature are proposed. A relation between stability properties of the perturbed and unperturbed systems is studied. Finally, utilizing Wintner–Conti theorem, we present two new Lyapunov-like theorems ensuring the global asymptotic stability of the impulsive system with unstable continuous dynamics that is being stabilized by impulsive jumps and discuss their relation to the previously known results.
We consider a class of nonlinear impulsive systems with delayed impulses, where the time delays in impulses exist between two consecutive impulse instants. Based on impulsive control theory and the ideas of average dwell-time (ADT), a set of Lyapunov-based sufficient conditions for globally exponential stability are obtained. It is shown that the time delay in impulses exhibits double effects (i.e., negative or positive effects) on system dynamics, namely, it may destroy the stability and lead to unde- sired performance, and conversely, it may stabilize an unstable system and achieve better performance. Then we apply the the- oretical results to synchronization control of chaotic systems and design some impulsive controllers that are formalized in terms of linear matrix inequality (LMI) and ADT-like conditions. Two numerical examples including chaotic cellular neural network and Chua's oscillator are given to illustrate the applicability of the presented impulsive control schemes. Our examples show that, under some conditions, the impulsive synchronization of chaotic systems can be achieved via the input delays in impulses.
This monograph discusses the issues of stability and the control of impulsive systems on hybrid time domains, with systems presented on discrete-time domains, continuous-time domains, and hybrid-time domains (time scales). Research on impulsive systems has recently attracted increased interest around the globe, and significant progress has been made in the theory and application of these systems. This book introduces recent developments in impulsive systems and the fundamentals of various types of differential and difference equations. It also covers studies in stability related to time delays and other various control applications on the different impulsive systems. In addition to the analyses presented on dynamical systems that are with or without delays or impulses, this book concludes with possible future directions pertaining to this research.
In this paper, the stability and
-gain properties of linear impulsive delay systems with delayed impulses are studied. Commonly employed techniques in which the delayed impulses are treated using Newton-Leibniz formula may not be applicable to
-gain analysis since they makes the disturbance input appear in the impulse part. In order to circumvent the difficulty, we first augment the considered system to a time-delay system with switching nondelayed impulses. Due to the absence of delayed impulses, this new approach has advantages in constructing Lyapunov functions and handling the effects of impulse delays on the system performance. Switching-based time-dependent Lyapunov functions are introduced to deal with the resultant switching impulses of the augmented system. Sufficient conditions for exponential stability and
-gain properties are derived in terms of linear matrix inequalities. Numerical examples are provided to illustrate the efficiency of the new approach.
This paper studies stability for impulsive switched time-delay systems with state-dependent impulses. Since the impulses and the switches are not necessary synchronous, we start from stability analysis of impulsive switched time-delay systems with time-dependent impulses. Sufficient conditions are derived to guarantee the stability properties, which extends the previous results for synchronous switch and impulse case and lays the foundation for the proceeding analysis. For the state-dependent impulse case, using the B-equivalent method, impulsive switched time-delay systems with state-dependent impulses are transformed into impulsive switched time-delay systems with time-dependent impulses. The equivalence between the original system and the transformed system is established, and stability conditions are obtained for impulsive switched time-delay systems with state-dependent impulses. Finally, a numerical example is given to demonstrate the obtained results.
A novel technique is provided for the global stability analysis of Boolean control networks (BCNs) under aperiodic sampled-data control (ASDC). The sampling period is allowed to be taken from a limited number of values. Using the semitensor product of matrices, a BCN under ASDC can be converted into a switched Boolean network (BN). Here, the switched BN can only switch at sampling instants, which does not mean that the switches occur at each sampling instant. For the switched BN, we consider two cases: (i) switched BN with all stable subsystems; (ii) switched BN containing both stable subsystems and unstable subsystems. For these two cases, the techniques of switching-based Lyapunov function and the average dwell time method are used to derive sufcient conditions for global stability of BCNs under ASDC, respectively. The upper bounds of the cost function, which is defned to guarantee that the considered BCN is not only globally stabilized but also can assurance the performance at an appropriate level, are determined, respectively. Moreover, an algorithm is presented to construct the ASDCs. Finally, the obtained results are demonstrated by several examples.
This technical note studies impulsive stabilization of general nonlinear systems with time-delay. Distributed time-delay is considered in the proposed nonlinear impulsive controller. Using Lyapunov–Razumikhin method, an exponential stability criterion is constructed, which is then applied to investigate stabilization of a linear time-delay system under linear distributed-delay dependent impulsive control. Sufficient conditions on the system parameters, impulsive control gains, impulsive instants and distributed delays are obtained in the form of an inequality for global exponential stability. In these results, it is shown that an unstable time-delay system can be successfully stabilized by distributed-delay dependent impulses. It is worth noting that the proposed impulsive controllers are independent of the system states at each impulsive instant, and the states with distributed delays play the key role in the stabilization process. A numerical example is provided to demonstrate the efficiency of the main results.
This paper focuses on the pinning synchronization problem of impulsive Lur’e networks with nonlinear and asymmetrical coupling. In order to study the situation where synchronizing impulses and desynchronizing impulses are allowed to occur simultaneously, a single pinning impulsive controller is designed to investigate the synchronization of Lur’e networks with hybrid impulses based on the methods of average impulsive interval and average impulsive gain. By employing the Lyapunov method, some sufficient conditions are derived to guarantee the exponential synchronization of Lur’e networks with synchronizing impulses and desynchronizing impulses simultaneously. Two numerical examples are given to illustrate the results.
The problems of exponential stability and La-gain for a class of time-delay systems with impulsive effects are studied. The main tool used is the construction of an impulse-time-dependent complete Lyapunov functional. By dividing the impulse interval and delay interval into several segments, the matrix functions of this functional are chosen to be continuous piecewise linear. Moreover, an impulse-time-dependent weighting factor is introduced to coordinate the dynamical behavior of the nondelayed and integral terms of this functional along the trajectories of the system. By applying this functional, delay-dependent sufficient conditions for exponential stability and L-2-gain are derived in terms of linear matrix inequalities. As by-products, new delay-independent sufficient conditions for the same problems are also derived. The efficiency of the proposed results is illustrated by numerical examples.
Dealing with impulsive effects is one of the most challenging problems in the field of fixed-time control. In this paper we solve this challenging problem by considering fixedtime synchronization of complex networks (CNs) with impulsive effects. By designing a new Lyapunov function and constructing comparison systems, a sufficient condition formulated by matrix inequalities is given to ensure that all the dynamical subsystems in the CNs are synchronized with an isolated system in a settling time which is independent of the initial values of both the CNs and the isolated system. Then, by partitioning impulse interval and using convex combination technique, sufficient conditions in terms of linear matrix inequalities are provided. Our synchronization criteria unify synchronizing and desynchronizing impulses. Compared with existing controllers for fixed-time and finite-time techniques, the designed controller is continuous and does not include any sign function, and hence the chattering phenomenon in most existing results is overcome. An optimal algorithm is proposed for the estimation of the settling time. Numerical examples are given to show the effectiveness of our new results.
This article investigates the asymptotic stability of impulsive delay dynamical systems (IDDS) by using the Lyapunov–Krasovskii method and looped-functionals. The proposed conditions reduce the conservatism of the results found in the literature by allowing the functionals to grow during both the continuous dynamics and the discrete dynamics. Sufficient conditions for asymptotic stability in the form of linear matrix inequalities (LMI) are provided for the case of impulsive delay dynamical systems with linear and time-invariant (LTI) base systems (non-impulsive actions). Several numerical examples illustrate the effectiveness of the method.
This paper studies the input-to-state stability (ISS) and integral input-to-state stability (iISS) of nonlinear systems with delayed impulses. By using Lyapunov method and the analysis technique proposed by Hespanha et al. (2005), some sufficient conditions ensuring ISS/iISS of the addressed systems are obtained. Those conditions establish the relationship between impulsive frequency and the time delay existing in impulses, and reveal the effect of delayed impulses on ISS/iISS. An example is provided to illustrate the efficiency of the obtained results.
In this paper we study the delayed impulsive control of nonlinear differential systems, where the impulsive control involves the delayed state of the system for which the delay is state-dependent. Since the state dependence of the delay makes the impulsive transients dependent on the historical information of the states, which means that it is hard to know exactly a priori how far in the history the information is needed, the main challenge is how to determine the historical states. We resolve this challenge and establish some sufficient conditions for local stability of nonlinear differential systems with state-dependent delayed impulsive control based on impulsive control theory. Two examples are given to show the effectiveness of the proposed approach.
In this paper, the input-to-state stability (ISS), integral-ISS (iISS) and stochastic-ISS (SISS) are investigated for impulsive stochastic delayed systems. By means of the Lyapunov–Krasovskii function and the average impulsive interval approach, the conditions for ISS-type properties are derived under linear assumptions, respectively, for destabilizing and stabilizing impulses. It is shown that if the continuous stochastic delayed system is ISS and the impulsive effects are destabilizing, then the hybrid system is ISS with respect to a lower bound of the average impulsive interval. Moreover, it is unveiled that if the continuous stochastic delayed system is not ISS, the impulsive effects can successfully stabilize the system for a given upper bound of the average impulsive interval. An improved comparison principle is developed for impulsive stochastic delayed systems, which facilitates the derivations of our results for ISS/iISS/SISS. An example of networked control systems is provided to illustrate the effectiveness of the proposed results.
We consider nonlinear differential systems with state-dependent delayed impulses (impulses which involve the delayed state of the system for which the delay is state-dependent). Such systems arise naturally from a number of applications and the stability issue is complex due to the state-dependence of the delay. We establish general and applicable results for uniform stability, uniform asymptotic stability and exponential stability of the systems by using the impulsive control theory and some comparison arguments. We show how restrictions on the change rates of states and impulses should be imposed to achieve system’s stability, in comparison with general impulsive delay differential systems with state-dependent delay in the nonlinearity, or the differential systems with constant delays. In our approach, the boundedness of the state-dependent delay is not required but derives from the stability result obtained. Examples are given to demonstrate the sharpness and applicability of our general results and the proposed approach.
Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mech-anisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
This note considers globally finite-time synchronization of coupled networks with Markovian topology and distributed impulsive effects. The impulses can be synchronizing or desynchronizing with certain average impulsive interval. By using M-matrix technique and designing new Lyapunov functions and controllers, sufficient conditions are derived to ensure the synchronization within a setting time, and the conditions do not contain any uncertain parameter. It is demonstrated theoretically and numerically that the number of consecutive impulses with minimum impulsive interval of the desynchronizing impulsive sequence should not be too large. It is interesting to discover that the setting time is related to initial values of both the network and the Markov chain. Numerical simulations are provided to illustrate the effectiveness of the theoretical analysis.
In this paper, global exponential synchronization stability in an array of linearly diffusively coupled reaction-diffusion neural networks with time-varying delays is investigated by adding impulsive controller to a small fraction of nodes (pinning-impulsive controller). In order to overcome the difficulty resulting from the fact that the impulsive controller affects only the dynamical behaviors of the controlled nodes, a new analysis method is developed. By using the developed method, two known lemmas on stability of delayed functional differential equation with and without impulses, and Lyapunov stability theory, several novel and easily verified synchronization criteria guaranteeing the whole network will be pinned to a homogenous solution are derived. Moreover, the effects of the pinning-impulsive controller and the dynamics of the uncontrolled nodes and the diffusive couplings on the synchronization process are explicitly expressed in the obtained criteria. Our results also show that we can always design an appropriate pinning-impulsive controller to realize the synchronization goal as long as a conventional state feedback pinning controller or an adaptive pinning controller can achieve the synchronization goal by controlling the same nodes. Furthermore, the function extreme value theory is utilized to reduce the conservativeness of the synchronization criteria. Some existing results are improved and extended. Numerical simulations including an asymmetric coupling network and BA (Barabási-Albert) scale-free network are given to show the effectiveness of the theoretical results.
This book brings together two emerging research areas: synchronization in coupled nonlinear systems and complex networks, and study conditions under which a complex network of dynamical systems synchronizes. While there are many texts that study synchronization in chaotic systems or properties of complex networks, there are few texts that consider the intersection of these two very active and interdisciplinary research areas. The main theme of this book is that synchronization conditions can be related to graph theoretical properties of the underlying coupling topology. The book introduces ideas from systems theory, linear algebra and graph theory and the synergy between them that are necessary to derive synchronization conditions. Many of the results, which have been obtained fairly recently and have until now not appeared in textbook form, are presented with complete proofs. This text is suitable for graduate-level study or for researchers who would like to be better acquainted with the latest research in this area. © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
This paper considers general impulsive delay differential equations. By utilizing a non-classical approach, the theory of existence and uniqueness of solutions are developed. Criteria on boundedness of solutions are also established through the use of Lyapunov functionals.
In this paper, the global exponential synchronization of coupled connected neural networks with both discrete and distributed delays is investigated under mild condition, assuming neither the differentiability and strict monotonicity for the activation functions nor the diagonal for the inner coupling matrices. By employing a new Lyapunov–Krasovskii functional, applying the theory of Kronecker product of matrices and the linear matrix inequality (LMI) technique, several delay-dependent sufficient conditions in LMI form are obtained for global exponential synchronization of such systems. Moreover, the decay rate is estimated. The proposed LMI approach has the advantage of considering the difference of neuronal excitatory and inhibitory efforts, which is also computationally efficient as it can be solved numerically using efficient Matlab LMI toolbox, and no tuning of parameters is required. In addition, the proposed results generalize and improve the earlier publications. An example with simulation is given to show the effectiveness of the obtained results.
This paper focuses on the problem of globally exponential synchronization of impulsive dynamical networks. Two types of impulses are considered: synchronizing impulses and desynchronizing impulses. In previous literature, all of the results are devoted to investigating these two kinds of impulses separately. Thus a natural question arises: Is there any unified synchronization criterion which is simultaneously effective for synchronizing impulses and desynchronizing impulses? In this paper, a unified synchronization criterion is derived for directed impulsive dynamical networks by proposing a concept named “average impulsive interval”. The derived criterion is theoretically and numerically proved to be less conservative than existing results. Numerical examples including scale-free and small-world structures are given to show that our results are applicable to large-scale networks.
This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS, but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS, but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS, and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case, we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network.