Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

Distributed coordination control is the current trend in networked systems and finds prosperous applications across a variety of fields, such as smart grids and intelligent transportation systems. One fundamental issue in coordinating and controlling a large group of distributed and networked agents is the influence of intermittent interagent interactions caused by constrained communication resources. Event-triggered communication scheduling stands out as a promising enabler to strike a balance between the desired control performance and the satisfactory resource efficiency. What distinguishes dynamic event-triggered scheduling from traditional static event-triggered scheduling is that the triggering mechanism can be dynamically adjusted over time in accordance with both available system information and additional dynamic variables. This article provides an up-to-date overview of dynamic event-triggered distributed coordination control. The motivation of dynamic event-triggered scheduling is first introduced in the context of distributed coordination control. Then some techniques of dynamic event-triggered distributed coordination control are discussed in detail. Implementation and design issues are well addressed. Furthermore, this article exemplifies two applications of dynamic event-triggered distributed coordination control in the fields of microgrids and automated vehicles. Several challenges are suggested to direct the future research.

Dynamic Event-Triggered Distributed Coordination Control and its Applications: A Survey of Trends and Techniques

This tutorial clarifies the axiomatic definition of (v(α); i(β)) circuit elements via a lookup table dubbed an A-pad, of admissible (v; i) signals measured via Gedanken probing circuits. The (v(α); i(β)) elements are ordered via a complexity metric. Under this metric, the memristor emerges naturally as the fourth element, characterized by a state-dependent Ohm's law. A logical generalization to memristive devices reveals a common fingerprint consisting of a dense continuum of pinched hysteresis loops whose area decreases with the frequency ω and tends to a straight line as ω ~ ∞, for all bipolar periodic signals and for all initial conditions. This common fingerprint suggests that the term memristor be used hence-forth as a moniker for memristive devices.

The Fourth Element.

This paper addresses the finite-time event-triggered control problem for nonlinear semi-Markovian switching cyber-physical systems (S-MSCPSs) under false data injection (FDI) attacks. Compared with the traditional time-triggered mechanism, the proposed event-triggered scheme (ETS) can effectively avoid network resource waste. Considering the network-induced delay in the modeling, a closed-loop system model with time delay is established in the unified framework. By the use of a mode-dependent piecewise Lyapunov-Krasovskii functional (LKF), stochastic finite-time stability (SFTS) criteria are established for the resultant closed-loop system. Then, some solvability conditions are established for the desired finite-time controller in light of a linear matrix inequality framework. Finally, an application example of vertical take-off and landing helicopter model (VTOLHM) is provided to demonstrate the effectiveness of the theoretical findings.

Finite-Time Event-Triggered Control for Semi-Markovian Switching Cyber-Physical Systems With FDI Attacks and Applications

In this paper, the event-triggered consensus problem of linear multiagent systems with time-varying communication delays is addressed. Different from the existing event-triggered consensus results with communication delays, more general nonuniform time-varying communication delays are considered. To avoid the asynchronous phenomenon caused by nonuniform delays, a novel periodic switching controller is developed. Based on this controller, the resulting consensus error system can be modeled as a periodic switching system. Furthermore, the exponential stability of the consensus error system is derived by utilizing the Lyapunov approach and the dwell-time analysis method. Finally, an illustrative example is presented to demonstrate the effectiveness of the developed method.

Event-Triggered Consensus of Linear Multiagent Systems With Time-Varying Communication Delays

This paper formulates a class of complex-valued memristive neural networks as well as investigates the problem of anti-synchronization for complex-valued memristive neural networks. Under the concept of drive-response, several sufficient conditions for guaranteeing the anti-synchronization are given by employing suitable Lyapunov functional and some inequality techniques. The proposed results of this paper are less conservative than existing literatures due to the characteristics of memristive complex-valued neural networks. Moreover, the proposed results are easy to be validated with the parameters of system itself. Finally, two examples with numerical simulations are showed to demonstrate the efficiency of our theoretical results.

Global Anti-Synchronization of Complex-Valued Memristive Neural Networks With Time Delays

This article concentrates on the synchronization problem of memristive complex-valued neural networks (CVNNs) with time delays via the pinning control method. Different from general control schemes, the pinning control is beneficial to reduce the control cost by pinning the fractional nodes instead of all ones. By separating the complex-valued system into two equivalent real-valued systems and employing the Lyapunov functional as well as some inequality techniques, the asymptotic synchronization criterion is given to guarantee the realization of synchronization of memristive CVNNs. Meanwhile, sufficient conditions for exponential synchronization of the considered systems is also proposed. Finally, the validity of our proposed results is verified by a numerical example.

Synchronization of Memristive Complex-Valued Neural Networks With Time Delays via Pinning Control Method

Synchronization stability of memristor-based complex-valued neural networks with time delays

Robust event-triggered control for networked control systems

This article focuses on the leader-following consensus for linear multiagent systems under a directed communication topology. A novel dynamic event-triggered mechanism is put forward to schedule the interagent communication. It removes the assumption that each agent has to broadcast continuously, and therefore resource consumption is reduced significantly. Different from the existing event-triggered mechanisms, the proposed mechanism involves internal dynamic variables that play an important role in excluding Zeno behavior. Then, an event-based distributed control protocol is developed by using locally triggering data. It is proven that the leader-following consensus is achieved under our control strategy. Moreover, the criterion for designing a desired triggered mechanism and the control protocol is derived in terms of linear matrix inequalities. Numerical simulations demonstrate that the proposed strategy is capable of reducing communication more effectively than the existing results.

Dan Liu

Papers

Global Anti-Synchronization of Complex-Valued Memristive Neural Networks With Time Delays

Synchronization of Memristive Complex-Valued Neural Networks With Time Delays via Pinning Control Method

Synchronization stability of memristor-based complex-valued neural networks with time delays

Robust event-triggered control for networked control systems

A Dynamic Event-Triggered Control Approach to Leader-Following Consensus for Linear Multiagent Systems