Distributed PID Control for Consensus and Synchronization of Multi-agent Networks

Topology analysis of Lanzhou public transport network based on double-layer complex network theory

In this thesis the study of convergence of piecewise smooth systems and discontinuous dynamical networks is considered. The contributions of the thesis are related both in the investigation of synchronization of networks with discontinuities in the agents's dynamics or in the inter-agents communication and in the investigation of incremental stability of nonsmooth systems. In particular, a framework for the study of synchronization when the nodes' dynamics may be both piecewise smooth and/or nonidentical across the network is presented. The analysis is performed using nonsmooth analysis tools, in particular set-valued Lyapunov functions. In this way, sufficient conditions are derived in order to guarantee global bounded synchronization, with analytical expression of the estimates of the synchronization bound and of the minimum coupling strength required to achieve synchronization. Different hypotheses on the heterogeneity of the agents' dynamics is considered, and the analysis is conducted both for linear and nonlinear coupling protocols. Differently from the few results available in the literature, boundedness of trajectories of the coupled systems is not required a priori, as well as any condition of synchrony among switching signals. Referring to the networks where discontinuities are present at the level of the communication among the agents, in the thesis novel event-triggered control strategies are developed. Such distributed algorithms are able to synchronize networks of general Lipschitz nonlinear systems. The results represent an innovative contribution on the existing literature for two main reasons. The existing literature considers only synchronization of integrators or linear systems and the control signals and communication signals are not both simultaneously piecewise constant, which is instead guaranteed in the developed strategies. Furthermore, a strategy able to guarantee a lower bound for the inter-event times between consecutive updates of the control law is proposed. More in detail, the considered strategy is model based and each node uses the dynamical model of its neighbours to predict their evolution and evaluate the triggering events. Both the cases of exact and not exact information of the neighbours' dynamical model are addressed and synchronization is analytically proven in both the cases. Furthermore, Zeno behaviour are excluded. The focus on the convergence properties and on the incremental stability of nonsmooth dynamical systems leads in the thesis to extend contraction theory, which is a tool used in the literature to address incremental exponential stability of smooth systems. In particular, the extension of such tool to the class of time switching non differentiable dynamical systems which satisfy Caratheodory conditions for the existence and uniqueness of a solution is considered. More in detail, it is proved that infinitesimal contraction of each mode of a switched system of interest gives a sufficient condition for global exponential convergence of trajectories towards each other. Such extension can be related with analogous results already present in the literature for convergent systems, which instead make use of Lyapunov conditions. The results are then used to develop smooth and piecewise smooth distributed protocols to coordinate and synchronize networks of piecewise smooth agents. Furthermore, the open problem of proving incremental stability of Filippov systems is introduced and preliminary results are given for the case of planar systems. All the results in the thesis are analytically proven and illustrated via a set of representative numerical examples.

This paper investigates the synchronisation of nonlinear coupled complex dynamical networks with different nonlinear nodes and different orders by using decentralised dynamical compensation controllers. We propose a dynamical network mathematical model with similar nonlinear nodes, whose dimensions are different. For this kind of network model, the decentralised dynamical compensation controllers are designed for the state synchronisation of the coupled nodes. In addition, the synchronisation manifold is defined as an invariant manifold, which is regarded as the generalised case of dynamical networks with the same nodes’ dynamics. Furthermore, some stability criteria for the synchronisation are derived by means of rigorous theoretical analysis. Finally, numerical examples are presented to verify the effectiveness of the obtained theoretical results.

Stochastic synchronisation of complex networks has recently attracted considerable attention because of the ubiquitousness of random noise in real-world complex systems. Previous works have mainly considered noise-corrupted communication by ignoring intrinsic dynamics of nodes. This work provides a systematic study of complex networks by taking into account both additive noise and nonlinear dynamics of nodes, examining their stochastic synchronisation problems. We model the deterministic diffusive coupling and noisy additive coupling with the time-invariant strength and the time-varying strength, respectively. Using the stochastic Lyapunov technique, this framework enables an analytical treatment of the problem, ultimately leading to sufficient conditions for the onset of synchronisation. For symmetrical/asymmetrical connections, the synchronisation condition for the time-invariant coupling strength can be stated elegantly in terms of node dynamics and network topology, whereas that for the time-varying coupling strength is described as a robust condition. Finally, two simulation examples are carried out to illustrate the effectiveness of theoretical findings.

Recently, there has been a growing interest in network controllability for determining the number and placement of controllers. In the framework of linear dynamics, this paper regarding the network controllability studies both the formulation of the control input matrix and the influence of the linear nodal dynamics, with precise guidelines derived on how to design the input matrix. This design strategy takes the column vectors of the input matrix as the basis to maximize the controllable subspace. In the one-dimensional case, it is found that nodal dynamics with high degrees of heterogeneity dominate the network topology, but identical nodal dynamics have no effect on the network controllability. In the higher dimensional case, it reveals why controllable network topology and controllable nodal dynamics together are still insufficient to ensure the controllability of the whole network, with a feasible solution to the problem presented based on a suitably designed input matrix. All characteristics of the network topology and nodal dynamics are integrated into the input matrix formulation, which plays an essential role in determining the network controllability.

We study the synchronizability of weighted aging scale-free networks with non-normalized and asymmetrical coupling matrices. We found that the synchronizability of such networks is improved when the couplings from older to younger nodes become dominant, where the out degrees of the nodes are heterogeneous and their in degrees are homogeneous, and that the synchronizabilty of the networks is seriously weakened or even lost when the couplings from younger to older nodes become dominant, where the out degrees of the nodes are homogeneous and their in degrees are heterogeneous. We also found that both the heterogeneity of nodes and a smaller average degree can improve the synchronizability of the networks with some appropriately chosen weighting parameter values. We finally show an example of the coupled Lorenz systems for illustration and verification of the theoretical analysis.

We present a framework for solving simultaneously the problems of facility location and path optimization in static and dynamic spatial networks. In the static setting, the objective is to determine facility locations and transportation paths from each node to the destination via the network of facilities such that the total cost of commodity transportation is minimized. This is an NP-hard problem. We propose a novel stage-wise viewpoint of the paths which is instrumental in designing the decision variable space in our framework. We use the maximum entropy principle to solve the resulting optimization problem. In the dynamic setting, nodes and destinations are dynamic. We design an appropriate control Lyapunov function to determine the time evolution of facilities and paths such that the transportation cost at each time instant is minimized. Our framework enables quantifying attributes of the facilities and transportation links in terms of the decision variables. Consequently, it becomes possible to incorporate application specific constraints on individual facilities, links, and network topology. We demonstrate the efficacy of our proposed framework through extensive simulations.

In this chapter, we will present two different approaches to solve the problem of synchronizing networks of interacting dynamical systems. The former will be based on making the coupling between agents in the network adaptive and evolving so that synchronization can emerge asymptotically. The latter will be using recent results from contraction theory to give conditions on the node dynamics and the network topology that result into the desired synchronized motion. The theoretical results will be illustrated by means of some representative examples, including networks of neural oscillators.

Exploring synchronizability of complex dynamical networks from edges perspective

: In this paper, we extend the Push Pull Communication Model to provide end to end quality of service (QoS) for clients located in distributed and heterogeneous nodes. Push pull communications is a middleware service that has been implemented on top of a Resource Kernel. It is a many to many communication model, which can easily and quickly disseminate information across heterogeneous nodes with flexible communication patterns. It supports both push (data transfer initiated by a sender) and pull (data transfer initiated by a receiver) communications. Nodes with widely differing processing power and networking bandwidth can coordinate and co-exist by the provision of appropriate and automatic support for transformation on data communication frequencies. In particular, different information sources and sinks can operate at different frequencies and also can choose another (intermediate) node to act as their proxy and deliver data at the desired frequency. In this paper, we specifically address the timeliness and bandwidth guarantees of the push pull model. The location of a proxy, the network topology and the underlying network support can impact the timeliness of data. We formally analyze the problem of choosing an optimal proxy location within a network. We obtain the somewhat counter intuitive result that if slightly longer end to end latencies can be tolerated and unicast protocols are used, locating the proxy at the publisher node is the best. The situation turns complex if multicast protocols are used. We show that this problem of optimal proxy allocation can be formulated as a mixed integer programming problem that can be solved efficiently. As an example, we solve the proxy location problem for a high speed vBNS network configuration. We obtain our end to end timeliness and bandwidth guarantees by using a resource kernel offering CPU guarantees at the end points and the use of a guaranteed bandwidth network between push pull clients.

This paper aims at investigating the topology identification problem of complex dynamical networks with varying node dynamics parameters and fixed inner coupling matrices. In particular, by employing the unified chaotic system as node dynamics, this work further explores the influence of continuously changing node dynamics parameters on topology identification of complex dynamical networks with different coupling strengths. Results show that for sufficiently small or large coupling strengths, the performance of topology identification is not affected by the change of node parameters. Specifically, for small enough coupling strengths, the topological structure can be completely identified regardless of the change of node parameters, while for sufficiently large coupling strengths, the connectivity (presence and absence of connections) cannot be successfully identified. Furthermore, for certain coupling strengths, with the increase of node dynamics parameters, the topology identification varies from completely unidentifiable to partially or event completely identifiable. Therefore, the synchronization-based topology identification depends on node dynamics. Even for the same node dynamical model, different parameters can have a significant impact on identification results. Furthermore, for networks consisting of chaotic oscillators defining node dynamics, small coupling strengths are conducive to topology identification. A broader conclusion is that projective synchronization, rather than just complete synchronization, is an obstacle to the network topology identification. The findings in this paper will add to our understanding of conditions for identifying topologies of complex networks.

Adaptive synchronization of drive-response fractional-order complex dynamical networks with uncertain parameters

Vehicular digital twin networks (VDTN) is an emerging paradigm integrating physical vehicular networks with their virtual digital twins (DT) mirror, enabling real-time mapping, simulation, and optimization of complex systems. However, constrained resources, high data synchronization costs, and dynamic network conditions in vehicular networks may degrade the performance of DT. We consider the interaction between task performance guarantees and node resource constraints to adaptively determine collaborative task computation transfer optimization in VDTN. In this paper, we design a semantic-aware multi-task vehicular digital twin network model, where vehicles extract semantic representations to achieve lightweight data transmission and efficient DT synchronization. We formulate a problem of maximizing the average utility of DT tasks by jointly considering the synchronization performance of DT tasks and the resource consumption among heterogeneous nodes. To solve the formulated problem, we develop a dynamic collaborative task computation transfer algorithm involving the high mobility of vehicles and heterogeneous resources of nodes. The algorithm is optimized in two phases to maximize average utility. A coarse-grained policy space is first obtained through an adaptive multi-agent deep reinforcement learning approach, aiming to alleviate the policy space explosion caused by dynamic task requirements and heterogeneous node collaboration. Subsequently, a fine-grained policy is derived via a resource-aware refinement mechanism. Numerical results validate the effectiveness and robustness of our proposed algorithm.

Complex dynamical networks consist of a large set of interconnected nodes with each node being a fundamental unit with detailed contents. A great number of natural and man-made complex networks such as social networks, food networks, neural networks, the World Wide Web, computer networks, electrical power grid and so on, can be effectively modelled by complex dynamical networks. Synchronization of a group of dynamical nodes in a complex network topology is one of the most interesting and significant phenomenon in complex dynamical networks. In this section, we shall discuss impulsive synchronization of the discrete delay networks. By employing the impulsive control method and the Razumikhin-type results obtained in previous chapter, two criteria for exponential synchronization will be derived.

Based on adaptive-impulsive control method, synchronization of uncertain complex networks with heterogeneous nodes is considered in this paper. A novel control protocol is derived for the proposed uncertain complex networks model, which guarantee the achievement of synchronization and identification of uncertain node’s parameters. Further, a simple corollary is given for adaptive-impulsive synchronization of networks with homogeneous nodes. Finally, take a scale-free complex network as an example, some numerical experiments illustrate the validity and feasibility of our proposed results.KeywordsAdaptive-impulsive synchronizationComplex networkHeterogeneous nodes