Adaptive Bipartite Consensus of Linear Multi-agent Systems Under Jointly Connected Topologies

This paper considers bipartite consensus of linear multi-agent systems over signed directed graphs, where both collaboration and competition coexist. The agent dynamics are distinct and can all have different dimensions. The communication graph can be time-varying and need not to be connected at each time. A distributed control law is proposed which enables the group to split into two disjoint subgroups, and the outputs of all agents within each subgroup achieve consensus, but the two subgroups move oppositely towards each other. Rigorous analysis and a simulation example are provided.

This paper studies bipartite consensus problem for general linear multi-agent systems, where both collaboration and competition between agents coexist. We establish an equivalence between output feedback bipartite consensus problems and the conventional output feedback consensus problems, which enables a direct application of existing output feedback consensus algorithms to solve bipartite consensus problems. Moreover, we propose a systematic approach to design output feedback bipartite consensus controller.

In this work, stochastic consensus of linear multi-agent systems with multiplicative measurement noises is investigated under undirected graphs. Based on the algebraic graph theory and the matrix theory, the consensus problem is converted into the stochastic stability problem of stochastic differential equations (SDEs) driven by multiplicative noises. Then by stochastic stability theorem for SDEs, the stochastic consensus conditions are given for the multi-agent systems. For the general linear multi-agent systems, the sufficient conditions for the mean square and the almost sure consensus are obtained based on the solution to an algebraic Riccati equation, where the consentability condition on the algebraic connectivity and the channel uncertainties is revealed. For the case of second-order integrator dynamics, by choosing some appropriate Lyapunov functions, the sufficient conditions for the mean square and the almost sure consensus, and the necessary conditions for the mean square consensus are derived. Moreover, it is shown that for any bounded noise intensities, the mean square and the almost sure consensus can be achieved by carefully choosing the control gain.

SummaryIn this paper, a bipartite consensus problem is considered for a high‐order multiagent system with cooperative‐competitive interactions and unknown time‐varying disturbances. A signed graph is used to describe the interaction network associated with the multiagent system. The unknown disturbances are expressed by linearly parameterized models, and distributed adaptive laws are designed to estimate the unknown parameters in the models. For the case that there is no exogenous reference system, a fully distributed adaptive control law is proposed to ensure that all the agents reach a bipartite consensus. For the other case that there exists an exogenous reference system, another fully distributed adaptive control law is also developed to ensure that all the agents achieve bipartite consensus on the state of the exogenous system. The stability of the closed‐loop multiagent systems with the 2 proposed adaptive control laws are analyzed under an assumption that the interaction network is structurally balanced. Moreover, the convergence of the parameter estimation errors is guaranteed with a persistent excitation condition. Finally, simulation examples are provided to demonstrate the effectiveness of the proposed adaptive bipartite consensus control laws for the concerned multiagent system.

In this paper, the leader-following consensus problem of higher order multi-agent systems is considered. The dynamics of each agent is given in general form of linear system and the communication topology among the agents is assumed to be undirected and switching. To track the active leader, a distributed observer-based consensus protocol is proposed to each following agent, which is based on the relative outputs of neighboring agents. A sufficient consensus condition is established by used parameter-dependent lyapunov function method under switching communication topologies. Two gain matrices used in the proposed protocol can be constructed by solving the Riccati equation and Sylvester equation respectively. Finally, a simulation example is given to illustrate our obtained result.

In this brief, a novel distributed leader-following consensus controller for linear multi-agent systems (MASs) with one leader and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> followers is designed for optimal leader-following consensus control. A user-friendly global objective function is well designed in a quadratic form. The proper choice of the weighting matrices and necessary and sufficient conditions for optimizing the objective function are given. The proposed controller can guarantee the optimization of the objective function and the achievement of the leader-following consensus of the MASs. To deal with the coupling in the controllers, we propose a dynamic average consensus-based algorithm to estimate the controller distributedly, and the estimate errors converge to zero in finite time. A simulation example is provided to demonstrate the results.

This paper investigates the problem of leader-following consensus of a linear multi-agent system on a switching network. The input of each agent is subject to saturation. Low gain feedback based distributed consensus protocols are developed. It is established that, under the assumptions that each agent is asymptotically null controllable with bounded controls and that the network is connected or jointly connected, semi-global leader-following consensus of the multi-agent system can be achieved. Numerical examples are presented to illustrate this result.

In this paper, the consensus problem is studied for a class of multi-agent systems with sampled data and packet losses, where random and deterministic packet losses are considered, respectively. For random packet losses, a Bernoulli-distributed white sequence is used to describe packet dropouts among agents in a stochastic way. For deterministic packet losses, a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way. The purpose of this paper is to derive consensus criteria, such that linear multi-agent systems with sampled-data and packet losses can reach consensus. By means of the Lyapunov function approach and the decomposition method, the design problem of a distributed controller is solved in terms of convex optimization. The interplay among the allowable bound of the sampling interval, the probability of random packet losses, and the rate of deterministic packet losses are explicitly derived to characterize consensus conditions. The obtained criteria are closely related to the maximum eigenvalue of the Laplacian matrix versus the second minimum eigenvalue of the Laplacian matrix, which reveals the intrinsic effect of communication topologies on consensus performance. Finally, simulations are given to show the effectiveness of the proposed results.In this paper, the consensus problem is studied for a class of multi-agent systems with sampled data and packet losses, where random and deterministic packet losses are considered, respectively. For random packet losses, a Bernoulli-distributed white sequence is used to describe packet dropouts among agents in a stochastic way. For deterministic packet losses, a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way. The purpose of this paper is to derive consensus criteria, such that linear multi-agent systems with sampled-data and packet losses can reach consensus. By means of the Lyapunov function approach and the decomposition method, the design problem of a distributed controller is solved in terms of convex optimization. The interplay among the allowable bound of the sampling interval, the probability of random packet losses, and the rate of deterministic packet losses are explicitly derived to characterize consensus conditions. The obtained criteria are closely related to the maximum eigenvalue of the Laplacian matrix versus the second minimum eigenvalue of the Laplacian matrix, which reveals the intrinsic effect of communication topologies on consensus performance. Finally, simulations are given to show the effectiveness of the proposed results.

This article systematically studies consensus of linear multiagent systems (MASs) on directed graphs through adaptive event-triggered control. It presents innovative adaptive event-triggered state-feedback protocols with novel composite event-triggering conditions. Two specific designs in terms of different event-triggering conditions and laws of adaption are first discussed for linear MASs on strongly connected directed graphs, which are then extended to general directed graphs that contain a spanning tree. Moreover, another adaptive event-triggered protocol is proposed for solving leader-follower consensus that tracks a leader of a bounded control input. The protocols inherit the merits of both adaptive control and event-triggered control: the protocols can be implemented in a fully distributed way, since the Laplacian is avoided in design, and each agent only needs to know the relative information between neighbors at discrete instants determined by event-triggering conditions. Compared with the existing related results, the proposed protocols are applicable for linear MASs on general directed graphs, and moreover, the time-dependent term in the event-triggering conditions is allowed to be a class of positive L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> functions. Two numerical examples clearly verify the effectiveness of the proposed protocols.

Leader–follower consensus of linear multi-agent systems with unknown external disturbances

This paper proposes a robust distributed event-triggered approach for consensus in linear multi-agent systems (MAS) with uncertain network topologies. To achieve consensus, each agent transmits its information only when a certain event-triggering condition is fulfilled. The connection weights in the network are uncertain and hence the information received by each agent is unreliable. In such an uncertain topology, the objective is to co-design robust consensus parameters (namely, the state transmission threshold and local control gains) that collectively ensure an exponential rate for consensus convergence. An objective function incorporating the transmission load and control effort is minimized to compute the design parameters. Numerical simulations quantify the effectiveness of the proposed event-triggered consensus approach in a second-order MAS.

Event-triggered bipartite consensus for high-order multi-agent systems with input saturation

Collective behaviors of multi-agent systems over signed graphs find applications in a variety of scenarios including social networks, predator-prey dynamics, which however have not been adequately addressed as their counterparts with nonnegative graphs. This paper studies bipartite consensus problem of general linear multi-agent systems over signed digraphs. First, we show that for general linear agents, bipartite consensus over signed graphs and ordinary consensus over nonnegative graphs are equivalent. This indicates that prevailing consensus controllers for nonnegative graphs can be adopted to solve bipartite consensus problems. Based on this observation, an existing Riccati equation based cooperative tracking controller is extended to solve the bipartite consensus problem for general linear systems.

This note further studies the previously proposed consensus protocol for linear multi-agent systems with communication noises in [15], [16]. Each agent is allowed to have its own time-varying gain to attenuate the effect of communication noises. Therefore, the common assumption in most references that all agents have the same noise-attenuation gain is not necessary. It has been proved that if all noise-attenuation gains are infinitesimal of the same order, then the mean square leader-following consensus can be reached. Furthermore, the convergence rate of the multi-agent system has been investigated. If the noise-attenuation gains belong to a class of functions which are bounded above and below by $t^{-\beta}$ $(\beta\in(0,1))$ asymptotically, then the states of all follower agents are convergent in mean square to the leader's state with the rate characterized by a function bounded above by $t^{-\beta}$ asymptotically.

In this article, we study the problems of bipartite and cooperative consensus with a strictly dissipative performance for fuzzy multiagent systems (MASs) in a unified framework. First, we prove that bipartite consensus over a structurally balanced signed graph is equivalent to cooperative consensus over the corresponding unsigned graph by leveraging the gauge transformation for a class of nonlinear MASs. Then, a polynomial fuzzy model is constructed to describe the nonlinear MAS formed by one leader and followers. For mitigating communication and computational load, a mode-dependent event-triggered transmission strategy is proposed. By establishing the switching topologies through Markovian process, a new sampled-data event-triggered consensus protocol is designed. With a mode-dependent Lyapunov–Krasovskii function, a novel relaxed dissipative criterion is obtained. The criterion guarantees that all agents can achieve both event-triggered cooperative consensus and event-triggered bipartite consensus with the same magnitude but opposite signs for MASs over structurally balanced signed directed graphs and Markovian switching topologies. Moreover, the event-triggered parameters and consensus control gains can be numerically solved via the sum-of-squares method. Simulation results are given to show the effectiveness of the proposed design method.