A key component of distributed cooperative decision making involves performing consensus amongst agents, which is defined as reaching an agreement on quantities of interest, such as plans, situational awareness, or other desired parameters. Most distributed planning ap- proaches employ consensus algorithms, which are sets of rules, or protocols, that determine how information is exchanged between agents to ensure that the team will convergence on the parameters of interest.
As a simple illustrative example, the following linear consensus protocol can be used to
2For example, Section 4.3 proposes an extension to CBBA to handle communication-limited environ-
ments, which involves combining (local) task space partitioning and task consensus to achieve conflict-free assignments in the presence of network disconnects.
3
converge on a continuous parameter z, ˙ xi(t) = X j∈Ni (xj(t) − xi(t)), ∀i (3.1) xi(0) = zi, zi ∈ R
where each agent i computes errors with its set of neighbors Ni and uses these to correct
its parameter estimate [161]. Collectively the team dynamics for n agents can be written
as an nth order linear system,
˙x(t) = −Lx(t) (3.2)
where L = D − A is known as the graph Laplacian, which is computed using an adjacency
matrix A describing connectivity between agents (the elements aij are 1 if j is a neighbor
of i and 0 otherwise), and a degree matrix D = diag(d1, . . . , dn), with elements di = n
X
j=1
aij
(number of connections for agent i). The maximum degree, denoted as ∆ = maxidi, is
useful in bounding the eigenvalues of L, which for an undirected connected network can be ordered sequentially as
0 = λ1≤ λ2 ≤ · · · ≤ λn≤ 2∆. (3.3)
The eigenvalues of L can be used to predict convergence rates and stability properties of
these linear consensus algorithms (in particular, λ2is related to speed of convergence and λn
provides stability bounds in time-delayed networks [161]). As shown above, the nontrivial
eigenvalues are all positive (all except λ1), and since Eq. (3.2) describes a linear system,
the consensus algorithm is globally asymptotically stable and converges exponentially to an
equilibrium with rate given by λ2 [161]. Furthermore, for the system described in Eq. (3.1),
the algorithm is guaranteed to achieve a unique equilibrium, ¯z = 1
n
n
X
i=1
zi, where ¯z is the
average of all the agents’ initial values.
Recent research has explored the effects of more realistic mission environments on these types of linear consensus algorithms for multi-agent teams. Some examples include analyz- ing the impact of time-delayed messages and dynamic network topologies on convergence and stability properties of the consensus algorithms. The work in [161] shows that global exponential convergence guarantees can be extended to dynamic networks as long as the network remains connected at each time t. The agents are guaranteed to reach consensus
with convergence rate greater than or equal to λ?2 = min
t λ2(G(t)), where λ2(G(t)) is the
second eigenvalue of the Laplacian for the graph at time t, G(t). Similar guarantees can be made for time-delayed networks, where messages are received after a delay τ instead of instantaneously. The system dynamics can be modified as follows,
˙x(t) = −Lx(t − τ ) (3.4)
and global exponential convergence guarantees are retained for delays within the range
τ < π/2λn. Note that convergence rates and robustness to time-delays can be improved by
actively controlling the network structure (modifying G and L), which is an active area of research [46, 103, 161, 184, 186].
Consensus algorithms have been applied to a wide variety of distributed decision making applications, ranging from flocking to rendezvous [20, 76, 103, 135, 161, 183, 184]. Most of these consensus algorithms are computationally inexpensive and guarantee convergence of team situational awareness, even over large, complex, and dynamic network topolo- gies [98, 211, 222]. A common issue with classical consensus algorithms is that agents’ observations are often treated with equal weight, whereas in reality some agents may have more precise information than others. Extending classical consensus algorithms to account for this uncertainty in local information, Kalman consensus approaches have been devel- oped that approximate the inherent uncertainty in each agent’s observations using Gaussian distributions [8, 185]. These algorithms produce consensus results that are more heavily influenced by agents with smaller covariance (therefore higher certainty) in their estimates. A limitation of Kalman consensus approaches is that Gaussian approximations may not be well-suited to model systems with arbitrary noise characteristics, and applying Kalman filter based consensus methods to the mean and covariance of other distributions can sometimes produce biased steady-state estimation results [82].
Other Bayesian decentralized data and sensor fusion methods have been explored to determine the best combined Bayesian parameter estimates given a set of observations [95, 139, 218]. A major challenge, however, is that these decentralized data fusion approaches require channel filters to handle common or repeated information in messages between neighboring nodes. These channel filters are difficult to design for arbitrary network struc- tures, and generic channel filter algorithms have not been developed other than for simple
network structures (e.g. fully connected and tree networks), thus limiting the applicability of decentralized data fusion methods [95]. Recent work has addressed this issue by show- ing that, through a combination of traditional consensus-based communication protocols and decentralized data fusion information updates, scalable representative information fu- sion results can be achieved, without requiring complex channel filters or specific network topologies [82, 83, 160, 223]. In particular, the work in [160] utilized dynamic-average con- sensus filters to achieve an approximate distributed Kalman filter, while [223] implemented a linear consensus protocol on the parameters of the information form of the Kalman filter, permitting agents to execute a Bayesian fusion of normally-distributed random variables. However, as previously noted, these Kalman based methods are derived specifically for normally-distributed uncertainties [160, 223], and thus can produce biased results if the lo- cal distributions are non-Gaussian. Recent work has extended these combined filtering and data fusion approaches to allow networked agents to agree on the Bayesian fusion of their local uncertain estimates under a range of non-Gaussian distributions [83]. In particular, the approach exploits conjugacy of probability distributions [87], and can handle several different types of conjugate distributions including members of the exponential family (e.g. Dirichlet, gamma, and normal distributions) [82, 83]. The approach in [83] is termed hyper- parameter consensus, and has demonstrated flexibility in handling several realistic scenarios, including ongoing measurements and a broad range of network topologies, without the need for complex channel filters.