IEEE Circuits and Systems Magazine - Q3 2019 - 15

directions when zigzags or sudden turnings occur [93].
In 2015, Ozogány and Vicsek proposed a model which
simulated the emergence of modular leadership hierarchy of flight herds, and proved that the size distributions
of harems are close to a lognormal distribution  [94].
The leadership mechanisms are also promising in control area, most typical of all, in 2013, by pulling a virtual
leader into multi-agent systems with fixed and switching
topologies, respectively, Lu et al. investigated the finitetime distributed tracking control problems [95].
IV. Collective Decision Making in Cohesive
Bird Flocks and Control Theory
The collective decision making in bird flocks is the initial
stage of collective behaviors, and making movement decisions often depends on social interactions among group
members. Here, we present how group make consensus
decisions and what is the underlying mechanism in information transfer among birds in rapid turnings. At last, we
will bring in several models of consensus in control theory.
A. Consensus Decision Making in Bird Flocks
Note that consensus decision making of agents is a greatly
significant issue. By consensus, we mean individuals within a flock conclude an agreement with specific aims [96].
Bird flocks routinely make consensus decisions jointly
with one another. In particular, there are two interesting
questions for these phenomena:
1) What are the underlying theoretical mechanisms for
the consensus of flocks?
2) Which factors influence the consensus significantly?
To investigate these issues, Couzin et al. established
a novel model for consensus decision making of bird
flocks in 2005  [96]. At the beginning, they addressed
that the investigation was carried out in the absence of
signals and members within flocks did not know who
had the mobile information. Couzin et al. assumed that
there exist social interactions if the distance between
two agents is less than r, then they would be attracted
towards or, alignment  [96]. Constant s i represents the
speed of agent i. Hence the desired travel direction of
agent i (i ! I ) at time t + Tt was depicted by
d i (t + Tt ) =

/

j!i
j ! N i (t)

x j (t ) - x i (t )
v j (t )
+ /
.
; x j (t ) - x i (t ); j ! Ni (t) ;v j (t );

In practice, some individuals of group always have
the knowledge of which direction to move, for example,
the direction of food source or migration route. In this
work, Couzin et al. set proportion p of the total agents
as informed ones who were given information about
preferred direction denoted as g. Thus every bird will
be influenced by two factors: social interactions and the
THIRD QUARTER 2019

preferred direction, the new desired travel direction of
agent i at time t + Tt was recast by
d il =

d i (t + Tt/ ; d i (t + Tt ; ) + ~g i
,
; d i (t + Tt/ ; d i (t + Tt ; ) + ~g i ;

where ~ represents the balanced weighting term between the two factors. Obviously, if ~ = 0, then agent i
is a naive individual and has no desire to move in a specific direction. If 0 1 ~ # 1, the direction of agent i will
be balanced by the two aforementioned factors. While
as ~ exceeds 1, the preferred direction will seriously affect the desire moving direction of individual bird [96].
The information of informed agents may be uncertain, that is, they may not have perfect knowledge about
their preferred direction g. The authors assumed that
the uncertainty of information is subject to random influences, in the simulations, at each time step, rotating
d il by a random angle to obtain d im, the angle is taken
from a circular-wrapped Gaussian distribution, which is
centered on 0 and the standard deviation is 0.01. When
the angle between d im(t + Tt ) and v i (t ) is less than iTt,
they will achieve alignment, and one has
v i (t + Tt ) = d im(t + Tt ),
otherwise, agent i turns iTt towards d im(t + Tt ) as the
new v i (t + Tt ), at the meantime, the position of agent i
is given by
x i (t + Tt ) = x i (t ) + v i (t + Tt ) Tts i,
where i is a system parameter [96].
Because of the uncertainty of information, information transfer within flock will not be accurate any more.
The accuracy of the flock is quantified by the normalized angular deviation, namely, at time t + Tt, the accuracy of flock movement will be
accuracy =

v i (t + Tt) - g i
.
;v i (t + Tt) - g i ;

The accuracy of a flock describes the validity of consensus information transmission in the flock to some
extent. To avoid collisions, in simulations, one must maintain a personal space measured by a.
Employing analysis above methods, Couzin et al. found
that for a given size of bird flock, if the number of inform ed agents increased, the accuracy of flock movement
would also increase. Furthermore, they also demonstrated that the larger the bird flock was, the smaller the proportion of informed individuals in total flock would be
needed. Thereby, they revealed that the communication
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