Fixed-time Research Articles

Fixed-time leader-following consensus of multi-agent systems with intermittent control

In this article, the intermittent-based fixed-time (IBFT) consensus matter is researched for nonlinear multi-agent systems (MASs). By designing a novel IBFT control protocol, the follower can achieve consensus with the leader’s state within a settling time. To deal with the fixed-time (FT) consensus matter, a new differential inequality is proposed, which provides a way to analyze the IBFT stability. Some properties are given to ensure the FT consensus of the nonlinear MASs. Furthermore, we extend our results to the directed communication situation of agents. The FT leader-following consensus problem with directed topology via intermittent control is solved as well. The settling time achieved is relative to the protocol parameters and independent of the initial states. Three examples demonstrate the feasibility and effectiveness of the suggested control protocols, including two application examples, the Chua’s circuit system, a spring-mass-damper (SMD) system, and one numerical example.

Fixed-Time Stabilization of Delayed Discontinuous Fuzzy Neural Networks via Delayed Stability Conditions of Filippov Systems

This article aims to propose time-delayed and simplified inequality conditions and establish more novel fixed-time (FXT) stability principles of delayed Filippov systems. First, the sign function is used to unify the orders of the Lyapunov–Krasovskii functional, and the time-delayed term is introduced in the inequality conditions, then some new FXT stability theorems are established and new estimations of the settling time involving delayed parameter are given that are absolutely different from the previous ones. The traditional inequality conditions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\dot{V}(z)\leq -aV^\alpha (z)-bV^\beta (z)$</tex-math></inline-formula> is optimized and improved. For the purpose of illustrating the applicability, a class of delayed discontinuous fuzzy Cohen–Grossberg neural networks (DDFCGNNs) is studied. Some new criteria are derived, and the FXT stabilization of the addressed DDFCGNNs is achieved. Finally, a representative example and simulations are provided to examine the correctness.

Novel finite and fixed-time stability theorems for fractional-order impulsive discontinuous systems and their application to multi-agent systems

This article studies the finite-time (FNT) and fixed-time (FXT) stability theorems of general fractional-order impulsive discontinuous systems (FOIDSs) through an indefinite Lyapunov functional (LF) approach. Using differential inclusion theory and set-valued map concept, the general FOIDSs can be transformed into fractional-order impulsive differential inclusions (FOIDIs). Two new stability theorems on finite and FXT stability of FOIDSs with finite impulsive instants are derived using an indefinite LF and fractional derivatives of the LF on some impulsive instants, respectively. The settling time is explicitly calculated. The obtained theoretical findings are applied to fractional-order discontinuous multi-agent systems. Discrete control schemes and indefinite Lyapunov-Krasovskii functions (LKFs) are also developed for ensuring FXT synchronization in fractional-order discontinuous multiagent systems. To demonstrate the efficiency and feasibility of the proposed method, some numerical simulations are conducted.

Fixed/preassigned-time synchronization for impulsive complex networks with mismatched parameters

This paper is concerned with the issue of fixed-time (FT) and preassigned-time (PT) synchronization of impulsive complex networks (ICNs) with mismatched parameters. Firstly, several more succinct conditions are established to achieve FT stability of impulsive systems and an improved estimation for the stable time is given. Subsequently, based on the improved FT stability results, the FT synchronization criteria for ICNs with mismatched parameters are proposed by designing fixed-time controllers without the linear part. Correspondingly, the PT synchronization problem is also considered by developing several innovative control protocols, where the synchronized time can be arbitrary predefined according to task demands and the control gains are bounded. Ultimately, the presented synchronization criteria are verified by numerical simulation.

Fixed-time stochastic synchronization of Kuramoto oscillators subjected to noisy diffusion process

This investigation concerns the phase agreement (PA) for identical Kuramoto oscillators and frequency synchronization (FS) for nonidentical oscillators within fixed-time (FxT) in noisy environments. Comparing with the conventional FxT controllers, this investigation picks the smooth control protocol for stripping the signum function out, which could improve the synchronizing performance of networked systems. Furthermore, due to the ubiquity of stochastic perturbation, this investigation takes the noise into consideration. Additionally, the synchronizing criteria are deduced for realizing the FxT phase-FS in probability, and meanwhile the time estimation (TE) of realizing synchronization is given. At last, simulation results not only validate the correction of theoretical analysis, but also indicate that PA is more robust to stochastic disturbance, whereas FS of oscillators is relatively sensitive.

Fixed-time synchronization of discontinuous competitive neural networks with time-varying delays

In this article, the fixed-time (FXT) synchronization of discontinuous competitive neural networks (CNNs) involving time-varying delays is investigated. Firstly, two kinds of discontinuous FXT control schemes are proposed and two forms of Lyapunov function are constructed based on p-norm and 1-norm to discuss the FXT synchronization of CNNs. By means of nonsmooth analysis and some inequality techniques, some simple criteria are obtained to achieve FXT synchronization and the upper bound of the settling time with less conservativeness is provided. Furthermore, the effect of time scale on FXT synchronization of CNNs is considered. Lastly, some numerical results for an example are provided to demonstrate the derived theoretical results.

Synchronization in fixed/preassigned-time of delayed fully quaternion-valued memristive neural networks via non-separation method

This paper mainly centers on investigating issues of synchronization of delayed memristive quaternion-valued neural networks (MQVNNs), which are first studied with an unconventional non-separation approach for the fully quaternion-valued (QV) parameters (i.e. state vectors, weights, inputs and outputs). First, QV sign function and norms are introduced and new properties of quaternions are established, which are the basis of the non-separation approach. Next, more relaxed conditions with a new pure lemma are explored to guarantee fixed-time (FXT) or preassigned-time (PST) synchronization and the settling times (STs) are measured. Specially, the ST of PST synchronization is pre-specified and irrelevant to any parameter of the systems or controllers, which is more independent. Subsequently, controller strategies are given to realize FXT and PST synchronization of the discussed systems. Lastly, the effectiveness and superiority of the achieved theoretical results are illustrated by numerical examples.

Fixed-time stabilization and synchronization for fuzzy inertial neural networks with bounded distributed delays and discontinuous activation functions

This paper mainly describes the fixed-time (FXT) stabilization and synchronization for fuzzy inertial neural networks (FINNs). Firstly, the discontinuous system is treated by using differential inclusion theory. Then, in order to guarantee the FXT stabilization and synchronization of FINNs, we use the fixed-time stability (FTS) theories of dynamic systems and design two efficient feedback controllers. Unlike most previous articles, the neural system model with bounded distributed delays and discontinuous activation functions is studied in this paper. In the end, we provide numerical examples showing the validity of the outcomes obtained from the four theorems proved in this paper. At the same time, the simulation results intuitively show the speed of upper settling-time (ST) estimation of different outcomes.

Improved Fixed-Time Stability Lemma of Discontinuous System and its Application

As is known to all that the Lyapunov-Krasovskii functional (LKF) has played a vital role in studying the fixed-time (FXT) stability of the time-delayed Filippov discontinuous systems. When investigating the FXT stability of the time-delayed discontinuous systems, the interesting topic is to provide the more relaxed conditions imposed on the LKF and obtain the more accurate settling-time (ST). This paper aims to investigate the new FXT stability lemma with more relaxed conditions for the time-delayed Filippov discontinuous systems. By using some novel inequality techniques and analytical methods, the new FXT stability lemma is established and more accurate ST is estimated. The traditional discussion about <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0 &lt; V &lt; 1$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V \geq 1$ </tex-math></inline-formula> is not required. For the purpose of validation, FXT stabilization of discontinuous inertial networks with mixed time delays is considered. By using differential inclusions theory and the new FXT stability lemma, the delay-dependent FXT stabilization criteria are derived. It is the first time to derive the delay-dependent FXT stabilization criteria for the discontinuous inertial networks. Finally, examples are given to show the effectiveness of the established results.

Quantized adaptive pinning control for fixed/preassigned-time cluster synchronization of multi-weighted complex networks with stochastic disturbances

The article studies preassigned-time (PAT) and fixed-time (FXT) cluster synchronization of multi-weighted complex networks (CNs) with stochastic disturbances through designing quantized adaptive pinning control scheme. Firstly, the controller can achieve lower control costs and save communication channels. By designing a novel and appropriate Lyapunov function, combining the characteristics of Wiener process and utilizing a method of comparison system, FXT synchronization criteria of CNs are proposed. The FXT criteria can be more widely applied in directed and undirected multi-weighted CNs. Besides, the PAT cluster synchronization is also investigated by utilizing a novel control scheme that the gains of controller are finite, where the synchronization time can be preassigned based on realistic situation. The effectiveness of theoretical results is illustrated via simulations.

Fixed/Preassigned-Time Synchronization Control of Complex Networks With Time Varying Delay

The problem of fixed-time (FXT) and preassigned-time (PAT) synchronization for delayed dynamic complex networks via developing control schemes is concerned. First, by comparing with the existing results, some new results for delayed networks model to ensure the synchronization of FXT and PAT are obtained. Second, some looser conditions are obtained for FXT synchronization, and several more accurate settling time (ST) estimates are established through several special functions. Third, by using some non-trivial control strategy, the PAT synchronization is investigated with limited control gains, and the synchronization time is able to specified in advance on the basis of practical need and is not relate to any parameter and any initial value. In particular, as a distinct form, the essential conditions for the FXT and PAT synchronization are obtained for the coupled neural networks. last, the improved the FXT and PAT synchronization results of dynamic delayed networks are demonstrated by one numerical example.

Some Further Results on Fixed-Time Synchronization of Neural Networks with Stochastic Perturbations

In this paper, fixed-time (FXT) synchronization issue of a type of neural networks (NNs) with stochastic perturbations is considered. First, we obtained some novel sufficient criteria to guarantee the FXT synchronization of considered networks via introducing two types of controllers and employing some inequality techniques. Lastly, our theoretical results are verified via giving two numerical examples with their Matlab simulations.

Fixed/predefined-time synchronization of fuzzy neural networks with stochastic perturbations

In this work, fixed-time (FXT) and predefined-time (PDT) synchronization issue of a type of fuzzy cellular neural networks (FCNNs) with stochastic perturbations is concerned via using some improved FXT stability results. First, we introduced some generalized FXT and PTD stability lemmas for a class of nonlinear stochastic differential equation. Then, based on these novel FXT stability results, we investigated the FXT synchronization of a type of FCNNs with stochastic fluctuations by designing two kinds of simple controllers, which not use a linear term −kiei(t) and provide more accurate settling time (ST) estimations compared to early published literatures. Furthermore, we have also analyzed the PDT synchronization of considered stochastic fuzzy networks via developing several nontrivial PDT control protocols and applying some inequality methods. Finally, we provided one numerical example with its Matlab simulations to show the feasibility of our developed results. We believe that methodology used in this paper can provide some new guidance to FXT and PDT synchronization studies of other nonlinear stochastic systems with or without fuzzy templates.

New criteria on periodicity and stabilization of discontinuous uncertain inertial Cohen-Grossberg neural networks with proportional delays

This paper is concerned with the periodicity and stabilization of a class of discontinuous uncertain inertial Cohen-Grossberg neural networks with proportional delays. First of all, based on the differential inclusions and coincidence theorem, some new proportional delay-dependent criteria are obtained for ensuring the existence of periodic solutions. It is the first result on the proportional delay-dependent criteria of delayed inertial neural networks (INNs). Secondly, in light of the definition of fixed-time (FXT) stable, the zero solution of the considered inertial neural model can achieve the FXT stabilization with the help of the designed discontinuous control laws. Finally, for the purpose of examining the correctness of the established results, one numerical example and illustrative simulations are provided.

Fixed-time Synchronization of Coupled Memristive Complex-valued Neural Networks

By directly proposing discontinuous complex-valued controllers and constructing Lyapunov functions for complex-valued networks, this article is devoted to fixed-time (F-T) synchronization of coupled memristive complex-valued neural networks (MCNNs) with time-varying delays. Above all, a new theorem for F-T stability is established to reduce the conservatism of the existing work in terms of conditions and the estimate of stability time. Subsequently, under the framework of the complex-valued sign function, several simplified complex-valued F-T control laws are proposed by excluding the linear part in previous F-T design. Furthermore, without splitting the addressed coupled MCNNs into real-valued submodels in the theoretical analysis, several sets of conditions of F-T synchronization are obtained for coupled MCNNs by means of the theory of complex-variable functions and the improved F-T stability. Lastly, the developed control strategies and the results on F-T synchronization are verified by the presence of a numerical example.

Finite-time and fixed-time bipartite synchronization of complex networks with signed graphs

In this paper, we study finite-time (FET) and fixed-time (FDT) bipartite synchronization of complex networks (CNs) with signed graphs. Under the framework of signed graphs, the interactions between nodes can be either collaborative or antagonistic, which are different from the traditional CNs. Two types of control schemes without the sign function are designed to realize FET and FDT bipartite synchronization of CNs, respectively. By 1-norm analytical techniques and Lyapunov functional method, FET bipartite synchronization criterion is established. By means of the constructed comparison system, FDT bipartite synchronization is proved. In addition, the settling times of both FET synchronization and FDT synchronization are estimated. It is worth noting that the settling time of FDT bipartite synchronization is not related to the initial values. Moreover, numerical simulations are given to illustrate the new results.

Fixed-time control of competitive complex networks

This paper aims to establish several synchronization criteria of competitive complex networks (CCNs) by using fixed-time (FDT) control. In CCNs, the variations of different nodes are diverse if they are influenced by external environment. Here, we design two types of controller to deal with the different variations of nodes. Meanwhile, these designed controllers guarantee the synchronization of the CCNs in a given time. The estimated settling time improves corresponding results in the literature. Furthermore, based on rigorous mathematical proof and the structured comparison system, several FDT synchronization criteria are obtained. Some comparisons are presented to show the advantages of these new theoretical results. The validity of our theoretical results is illustrated by numerical simulations.

Training Petting Zoo Sheep to Act Like Petting Zoo Sheep: An Empirical Evaluation of Response-Independent Schedules and Shaping with Negative Reinforcement.

Simple SummaryThe present study examines the use of response-independent food schedules (food delivered independent of what an animal is doing) and negative reinforcement in the form of a trainer removing themselves from the presence of an animal to effectively train approach and contact behaviors in petting zoo sheep. All sheep had previously been reported to avoid/escape the presence of zoo staff and caretakers, as well as refuse to eat in the presence of any persons. In Experiment 1, we were able to effectively use response-independent schedules to produce both approach and contact behaviors in a Hampshire sheep. In Experiment 2, a Jacob sheep was trained for approach/contact via a shaping procedure that initially rewarded approach behaviors by removing the presence of the trainer (negative reinforcement), and was later switched to rewarding responses directly with food (positive reinforcement) once the sheep began to regularly eat in the trainer’s presence. The results are discussed with respect to how both procedures are effective in examining the applied approach/avoidance behaviors with zoo animals, as well as the greater ethical considerations involved in using negative vs. positive reinforcement to train animals.Shaping through differential reinforcement of successive approximations to a target response has been a cornerstone procedure for the training of novel behavior. However, much of how it has traditionally been implemented occurs through informal observation, rather than any direct, systematic measurement. In the present study, we examine the use of response-independent food schedules and shaping for increasing approach and contact behaviors in petting zoo sheep. In Experiment 1, a fixed-time (FT) 15 s food schedule was used to effectively increase approach and contact behaviors in one sheep. In Experiment 2, negative reinforcement in the form of removal of the presence of a trainer was made contingent on the successful completion of approximations within a shaping procedure and later switched to food rewards. A changing-criterion design was used to empirically examine the effects of the shaping procedure during each step of the program. The result is one of the first studies to demonstrate the utility of using negative reinforcement within a shaping procedure to successfully intervene on approach/avoidance behaviors in an applied animal setting.

Fixed/Preassigned-Time Synchronization of Complex Networks via Improving Fixed-Time Stability.

This article is concerned with the problem of fixed-time (FXT) and preassigned-time (PAT) synchronization for discontinuous dynamic networks by improving FXT stability and developing simple control schemes. First, some more relaxed conditions for FXT stability are established and several more accurate estimates for the settling time (ST) are obtained by means of some special functions. Based on the improved FXT stability, FXT synchronization for discontinuous networks is discussed by designing a simple controller without a linear feedback term. Besides, the PAT synchronization is also explored by developing several nontrivial control protocols with finite control gains, where the synchronized time can be prespecified according to actual needs and is irrelevant with any initial value and any parameter. Finally, the improved FXT stability and the synchronization for complex networks are confirmed by two numerical examples.

Cluster stochastic synchronization of complex dynamical networks via fixed-time control scheme

By means of fixed-time (FDT) control technique, cluster stochastic synchronization of complex networks (CNs) is investigated. Quantized controller is designed to realize the synchronization of CNs within a settling time. FDT synchronization criteria are established with the help of Lyapunov functional and comparison system methods. It should be noted that the convergence of synchronization is further improved by comparing with existing FDT synchronization results. Numerical simulations are given to illustrate our results.