Balanced Digraphs Research Articles

Stochastic consensus of single-integrator multi-agent systems under relative state-dependent measurement noises and time delays

Summary This paper proposes a consensus algorithm for continuous-time single-integrator multi-agent systems with relative state-dependent measurement noises and time delays in directed fixed and switching topologies. Each agent's control input relies on its own information state and its neighbors' information states, which are delayed and corrupted by measurement noises whose intensities are considered a function of the agents' relative states. The time delays are considered time-varying and uniform. For directed fixed topologies, condition to ensure mean square linear χ-consensus (average consensus, respectively) are derived for digraphs having spanning tree (balanced digraphs having spanning tree, respectively). For directed switching topologies, conditions on both time delays and dwell time have been given to extend the mean square linear χ-consensus (average consensus, respectively) of fixed topologies to switching topologies. Copyright © 2016 John Wiley & Sons, Ltd.

Consensus control of discrete-time multi-agent systems with time-delays and multiplicative measurement noises

This work is concerned with stochastic consensus conditions of discrete-time multi-agent systems with time-delays and multiplicative measurement noises. By the algebraic graph theory and the matrix theory, the stochastic consensus problem is converted into the stochastic stability problem of discrete-time stochastic delayed systems driven by multiplicative noises. Then by establishing the stochastic stability criteria for discrete-time stochastic delayed systems, the stochastic consensus conditions are deduced. For the case with first-order dynamics, the sufficient conditions for the mean square and the almost sure strong consensus are deduced under balanced digraphs. For the case with second-order dynamics, the consensus analysis is given under undirected graphs, and it is proved that for any given bounded time-delay and noise intensity, the stochastic weak consensus for the position and stochastic strong consensus for the velocity can be achieved by carefully choosing the control gains. These consensus results are further extended to the leader-following case.

Consensus seeking in multi-agent systems with noisy and delayed communication in digraphs having spanning tree

This paper studies the consensus problem of continuous-time single-integrator multi-agent systems with measurement noises and time delays under directed fixed topologies. Each agent in the team receives imprecise and delayed information from its neighbours. The noises are considered white noises, and time delays are assumed to be uniform for all the received information states. An analysis framework based on graph theory and stochastic tools is followed to derive conditions under which the asymptotic unbiased mean square linear χ-consensus is achieved in directed fixed topologies having a spanning tree. Then, conditions to achieve asymptotic unbiased mean square average consensus are deduced for fixed balanced digraphs having a spanning tree. The effectiveness of the proposed algorithms is proved through some simulations.

Distances of Centroid Sets in a Graph-Based Construction for Information Security Applications

The aim of this paper is to prove that, for every balanced digraph, in every incidence semiring over a semifield, each centroid set J of the largest distance also has the largest weight, and the distance of J is equal to its weight. This result is surprising and unexpected, because examples show that distances of arbitrary centroid sets in incidence semirings may be strictly less than their weights. The investigation of the distances of centroid sets in incidence semirings of digraphs has been motivated by the information security applications of centroid sets.

Non-Ergodicity of Uniform Quadratic Stochastic Operators

In this paper we consider a uniform extremal Volterra quadratic stochastic operator defined on an even-dimensional unit simplex, which corresponds to a balanced digraph, and show that such operator has the non-ergodicity property. We also reinterpret this result in the framework of zero-sum games obtaining that these operators correspond to rock-paper-scissors games.

Semi-global output consensus of a group of linear systems in the presence of external disturbances and actuator saturation: An output regulation approach

Summary This paper studies the problem of semi-global leader-following output consensus of a multi-agent system. The output of each follower agent in the system, described by a same general linear system subject to external disturbances and actuator saturation, is to track the output of the leader, described by a linear system, which also generates disturbances as the exosystem does in the classical output regulation problem. Conditions on the agent dynamics are identified, under which a low-gain feedback-based linear state-control algorithm is constructed for each follower agent such that the output consensus is achieved when the communication topology among the agents is a digraph containing no loop, and the leader is reachable from any follower agent. We also extend the results to the non-identical disturbance case. In this case, conditions based on both the agent dynamics and the communication topology are identified, under which a low-gain feedback-based linear state-control algorithm is constructed for each follower agent such that the leader-following output consensus is achieved when the communication topology among the follower agents is a strongly connected and detailed balanced digraph, and the leader is a neighbor of at least one follower. In addition, under some further conditions on the agent dynamics, the control algorithm is adapted so as to achieve semi-global leader-following output consensus for a jointly connected undirected graph and the leader reachable from at least one follower. Copyright © 2015 John Wiley & Sons, Ltd.

Ideal basis in constructions defined by directed graphs

The present article continues the investigation of visible ideal bases in constructions defined using directed graphs. This notion is motivated by its applications for the design of classication systems. Our main theorem establishes that, for every balanced digraph and each idempotent semiring with identity element, the incidence semiring of the digraph has a convenient visible ideal basis. It also shows that the elements of the basis can always be used to generate ideals with the largest possible weight among the weights of all ideals in the incidence semiring.

Cooperative control of multiple heterogeneous agents with unknown high-frequency-gain signs

In this paper, we investigate the cooperative control of networked agents with unknown high-frequency-gain signs. A Nussbaum-type adaptive controller is designed for each agent such that consensus of the network can be achieved while all signals in the overall system maintain bounded. The distributed controller for each agent has two parts: neighborhood error between itself and the neighbors and a Nussbaum-type item for seeking control direction adaptively. The argument of the Nussbaum-type function is tuned on line via an appropriately designed update law. It is proved that when the undirected graph is connected or the balanced digraph is weakly connected, consensus of the network can be realized. Furthermore, a distributed asymptotic regulator is proposed to regulate the overall system to the equilibrium. Simulation results are presented to verify the effectiveness of the proposed controllers.

Lyapunov, Adaptive, and Optimal Design Techniques for Cooperative Systems on Directed Communication Graphs

This paper presents three design techniques for cooperative control of multiagent systems on directed graphs, namely, Lyapunov design, neural adaptive design, and linear quadratic regulator (LQR)-based optimal design. Using a carefully constructed Lyapunov equation for digraphs, it is shown that many results of cooperative control on undirected graphs or balanced digraphs can be extended to strongly connected digraphs. Neural adaptive control technique is adopted to solve the cooperative tracking problems of networked nonlinear systems with unknown dynamics and disturbances. Results for both first-order and high-order nonlinear systems are given. Two examples, i.e., cooperative tracking control of coupled Lagrangian systems and modified FitzHugh–Nagumo models, justify the feasibility of the proposed neural adaptive control technique. For cooperative tracking control of the general linear systems, which include integrator dynamics as special cases, it is shown that the control gain design can be decoupled from the topology of the graphs, by using the LQR-based optimal control technique. Moreover, the synchronization region is unbounded, which is a desired property of the controller. The proposed optimal control method is applied to cooperative tracking control of two-mass–spring systems, which are well-known models for vibration in many mechanical systems.

On arc reversal in balanced digraphs

In this note we consider closed walks, which are cycles that are not necessarily elementary. We prove that any arc reversal in a balanced multidigraph without loops decreases the number of closed walks. This also proves that arc reversal in a simple balanced digraph decreases the number of closed walks.

Super Connectivity of Line Graphs and Digraphs

The h-super connectivity κ h and the h-super edge-connectivity λ h are more refined network reliability indices than the connectivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ1(L) = 2λ1(D), and that for a connected graph G and its line graph L, if one of κ1(L) and λ2(G) exists, then κ1(L) = λ2(G). This paper determines that κ1(B(d, n)) is equal to 4d -8 for n = 2 and d ≥ 4, and to 4d -4 for n ≥ 3 and d ≥ 3, and that κ1(K(d, n)) is equal to 4d - 4 for d ≥ 2 and n ≥ 2 except K(2, 2). It then follows that B(d, n) and K(d, n) are both super connected for any d ≥ 2 and n ≥ 1.

Orientations of Hamiltonian cycles in bipartite digraphs

The main result of the paper is the following Theorem. Let D=(X,Y,A) be a bipartite, balanced digraph of order 2n and size at least 2n2−2n+3. Then D contains an almost symmetric Hamiltonian cycle (i.e. a Hamiltonian cycle in which at least 2n−1 arcs are symmetric edges), unless D has a vertex which is not incident to any symmetric edge of D. This theorem implies a number of results on cycles in bipartite, balanced digraphs, including some recent results of N. Chakroun, M. Manoussakis and Y. Manoussakis.

An algorithm for identifying Morishima and anti-Morishima matrices and balanced digraphs

We present an algorithm for identifying Morishima and anti-Morishima matrices. Since Morishima matrices have signed diagraphs that are balanced, our algorithm also identifies such diagraphs. We actually work with a signed graph and our algorithm is a depth first traversal with certain markers added to keep track of parsing of nodes and signs of edges. The algorithm is of course, linear in the amount of input.