We further define a region ?±
which includes both equilibrium points E??? and E??????, as shown in Figure 5. Based on
the definition of the candidate Lyapunov function (eq. 3) and the controller ui in (eq.
4), it is seen that
@VA???
@t
< 0,
@VA??????
@t
< 0; ??? pi ??? ? ??? ???\?±. (8)
???
???
A
A
A???
E???
A
E
D D B
C C
?„¦ ?„¦
B
Chapter 3 A Scalable Graph Model and Coordination Algorithms 73
In other words, for pi, pj ??? ? ??? ???
\?±, if potential energy at point pi is smaller than
the potential energy at point pj for topology A???, i.e., VA??? (pi, t) < VA??? (pj , t), then it
is also true for topology A??????, i.e., VA?????? (pi, t) < VA?????? (pj , t).
(a) Both equilibria inside ?± (b) Both equilibria outside ?±
Fig. 5. Convergence analysis with topological events.
Fig. 6. Candidate Lyapunov functions in a switching system.
The location of the equilibrium is partially determined by the desired distance
between neighboring nodes. Figure 5 shows two possible situations: both equilibria
are inside or outside of the circular region ?. For the situation shown in Figure 5 (a),
we assume a candidate trajectory of the node which incurs topological events three
74 Jindong Tan
times at time instance t1, t2 and t3 respectively. VA??? and VA?????? govern the motion of
the node alternatively during the topological changes, as shown in Figure 6. It can
be concluded that VA?????? (t??’
2 ) < VA??????+(t1) and VA??? (t+
2 ) < VA??? (t??’
1 ).
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