Zhongxin Liu

Brief paper: Reaching a consensus via pinning control

The consensus problem in a multi-agent system with general nonlinear coupling is investigated in this paper. It is demonstrated that, under suitable conditions on communication, all agents approach a prescribed value if a small fraction of them are controlled by simple feedback control. The stability analysis is based on a blend of graph-theoretic and system-theoretic tools where the contraction analysis and multiple Lyapunov functions play central roles. Numerous numerical examples, which support the analytical results very well, are also included.

Event-triggered average-consensus of multi-agent systems with weighted and direct topology

This paper investigates the average-consensus problem of multi-agent systems with direct and weighted topologies. Event-triggered control laws are adopted so as to reduce the frequency of individual control updating since the agents may be resource-limited in many real systems. The discrete time instants where the events are triggered are determined by a trigger function with respect to a certain measurement error. A centralized average-consensus protocol is proposed first for networks with fixed interaction topology, the stability and influencing factors of which are also analyzed. The design of trigger functions for networks with variable topology is also discussed. Then the results are extended to the decentralized counterpart, in which agents require only the information of their neighbors. Numerical examples are also provided that demonstrate the effectiveness of the theoretical results.

Pinning control of complex dynamical networks with heterogeneous delays

Complex dynamical networks with heterogeneous delays in both continuous- and discrete-time domains are controlled by applying local feedback injections to a small fraction of nodes in the whole network. Some generic stability criteria ensuring delay-independent stability are derived for such controlled networks in terms of linear matrix inequalities (LMI), which guarantee that by placing a small number of feedback controllers on some nodes, the whole network can be pinned to its equilibrium. In some particular cases, a single controller can achieve the control objective. Numerical simulations of various representative networks, including a globally coupled network, a star-coupled network and an Extended Barabasi-Albert (EBA) scale-free network, are finally given for illustration and verification.

Pinning weighted complex networks with heterogeneous delays by a small number of feedback controllers

Weighted complex dynamical networks with heterogeneous delays in both continuous-time and discrete-time domains are controlled by applying local feedback injections to a small fraction of network nodes. Some generic stability criteria ensuring delay-independent stability are derived for such controlled networks in terms of linear matrix inequalities (LMIs), which guarantee that by placing a small number of feedback controllers on some nodes the whole network can be pinned to some desired homogeneous states. In some particular cases, a single controller can achieve the control objective. It is found that stabilization of such pinned networks is completely determined by the dynamics of the individual uncoupled node, the overall coupling strength, the inner-coupling matrix, and the smallest eigenvalue of the coupling and control matrix. Numerical simulations of a weighted network composing of a 3-dimensional nonlinear system are finally given for illustration and verification.

EPIDEMICS OF SIRS MODEL WITH NONUNIFORM TRANSMISSION ON SCALE-FREE NETWORKS

We investigate the effect of nonuniform transmission on the critical threshold of susceptible–infected–recovered–susceptible (SIRS) epidemic model on scale-free networks. Based on the mean-field theory, it is observed that the epidemic threshold is not only correlated with the topology of underlying networks, but also with the disease transmission mechanism (e.g., nonuniform transmission). The current findings will significantly help us to further understand the real epidemics taking place on social and technological networks.

Tracking control for multi-agent consensus with an active leader and directed topology

In this paper, we consider a multi-agent consensus problem of multi-agent systems with an active leader and directed topology. The state of the considered leader not only keeps changing but also may not be measured. Two different models with and without coupling time delays are presented. In each model, two different cases of coupling topologies are investigated. A sufficient condition is proved in the case when the interconnection topology is fixed and directed. Moreover, a necessary and sufficient condition was presented when the velocity can be measured. A sufficient condition is also proposed in the case when the coupling topology is switched and balanced.

Spreading behavior of SIS model with non-uniform transmission on scale-free networks

The non-uniform transmission and network topological structure are combined to investigate the spreading behavior of susceptible-infected-susceptible (SIS) epidemic model. Based on the mean-field theory, the analytical and numerical results indicate that the epidemic threshold is correlated with the topology of underlying networks, as well as the disease transmission mechanism. These discoveries can greatly help us to further understand the virus propagation on communication networks.

Semi-tensor product approach to controllability and stabilizability of finite automata

Using semi-tensor product of matrices, the controllability and stabilizability of finite automata are investigated. By expressing the states, inputs, and outputs in vector forms, the transition and output functions are represented in matrix forms.Based on this algebraic description, a necessary and sufficient condition is proposed for checking whether a state is controllable to another one. By this condition, an algorithm is established to find all the control sequences of an arbitrary length. Moreover, the stabilizability of finite automata is considered, and a necessary and sufficient condition is presented to examine whether some states can be stabilized. Finally, the study of illustrative examples verifies the correctness of the presented results/algorithms.

Semi-tensor product of matrices approach to reachability of finite automata with application to language recognition

This paper investigates the transition function and the reachability conditions of finite automata by using a semi-tensor product of matrices, which is a new powerful matrix analysis tool. The states and input symbols are first expressed in vector forms, then the transition function is described in an algebraic form. Using this algebraic representation, a sufficient and necessary condition of the reachability of any two states is proposed, based on which an algorithm is developed for discovering all the paths from one state to another. Furthermore, a mechanism is established to recognize the language acceptable by a finite automaton. Finally, illustrative examples show that the results/algorithms presented in this paper are suitable for both deterministic finite automata (DFA) and nondeterministic finite automata (NFA).

Stabilizing weighted complex networks

Real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections. In this paper, the V-stability problem is investigated for a class of asymmetric weighted coupled networks with nonidentical node dynamics, which includes the unweighted network as a special case. Pinning control is suggested to stabilize such a coupled network. The complicated stabilization problem is reduced to measuring the semi-negative property of the characteristic matrix which embodies not only the network topology, but also the node self-dynamics and the control gains. It is found that network stabilizability depends critically on the second largest eigenvalue of the characteristic matrix. The smaller the second largest eigenvalue is, the more the network is pinning controllable. Numerical simulations of two representative networks composed of non-chaotic systems and chaotic systems, respectively, are shown for illustration and verification.