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Introduction and Mathematical Background 5
team of agents reaches an agreement on a common value by interacting with
each other via a sensing or communication networ k.
Consensus problems have a long history in computer sci en ce and form the
foundation of distributed computing. Pioneering works on the consensus prob-
lems for distributed decision mak i ng and parallel computing includ e [14, 170].
The emergi n g interest in consensus control in the last decade is mainly stim-
ulated by [65] and [121]. In particul ar, a theoretical explanation was provided
in [65] for the alignment behavior observed in the Vicsek model [174] and a
general framework of the consensus problem for networks of integrators was
proposed in [121]. Since then, a large body of works has been reported on the
consensus prob l em s; refer to the surveys [4, 20, 120, 134].
In the following, we briefly introduce some recent works on consensus. In
[133], a sufficient condition was deri ved to achieve consensus for first-order
integrator multi-agent systems with jointly connected communication graphs.
Consensus of networks of second- and high-order integrators was studied in
[68, 97, 130, 132, 135, 202]. The consensus problem of multi-agent systems with
general linear dynamics was addressed in [83, 86, 93, 106, 146, 173, 185, 195].
Consensus of multiple rigid bodies or spacecraft described as Euler-Lagrange
systems was investigated in [7, 24, 25, 79, 109, 142, 176]. In [90, 156, 189],
conditions were derived for achieving consensus for multi-agent systems with
Lipschitz-type nonlinearity. Consensus algorithms were designed in [22, 75]
for multi-agent systems wi t h quantized communication li n ks . The effect of
various communication delays an d input delays on consensu s was considered
in [13, 97, 114, 122, 166]. Sampled-data control protocols were proposed to
achieve consensus for fixed and switching agent networks in [17, 47]. Distribut-
ed consensus protocols with event-triggered c ommunications were designed in
[34, 149]. Distributed H
∞
consensus and control problems were investigated
in [85, 99] for networks of agents subject to external disturbances.
Existing consensus algorithms can be roughly categorized into two cl asses,
namely, consensus without a leade r (i.e., leaderless consensus) and consensus
with a leader. The latter is also called leader-follower consensus or distributed
tracking. The authors in [56, 57] designed a distributed neighbor-based esti ma-
tor to track an active leade r. Di st r i but e d tracking algorithms were proposed in
[131] for a network of agents with first-order dynamics. In [21], discontinuous
controllers were studied in the absence of velocity or acceleration measure-
ments. The authors in [109] addressed the distributed coordinated tracking
problem for multiple Euler-Lagrange systems with a dynami c leader. The dis-
tributed tracking of general linear multi-agent systems with a leader of nonzero
control input was addressed in [92, 95].
The dist r i bu t ed tracking problem deals with only one leader. However, in
some practical applicat i ons , there might exist more than one leader in agent
networks. In the presence of multiple leaders, the containment control prob-
lem arises, where the followers are to be driven into a given geometric space
spanned by the leaders [67]. The st u dy of containment control has been mo-
tivated by many potential applications. For instance, a group of auton omou s