Information Technology Reference
In-Depth Information
u
i
D !x
r
Cx
r
C
u
i
c
;
(9.24)
where x
i
is the reference trajectory for the i-th particle, and x
i
is the derivative of
the i-th desirable trajectory. Moreover
u
i
c
D
i
stands for an additional control
term which compensates for the effect of the noise
i
on the i-th particle. Thus, if
the disturbance
i
that affects the ith-particle is adequately approximated it suffices
to set
u
i
c
D
i
. The application of the control law of Eq. (
9.24
) to the model of
Eq. (
9.23
) results in the error dynamics
x
i
Dx
r
!x
i
C !x
r
C
i
u
i
c
)
x
i
x
r
C !.x
i
x
i
/ D
i
(9.25)
C
u
c
)
e
i
C !e
i
D
i
C
u
c
:
Thus, if
u
c
D
i
, then lim
t!1
D 0.
Next, the case of the N interacting particles will be examined. The control law
that makes the mean of the multi-particle system follow a desirable trajectory Efx
r
g
can be derived. The kinematic model of the mean of the multi-particle system is
given by
Efx
i
gD!Efx
i
gCEf
u
i
gCEf
i
g
(9.26)
i D 1; ;N, where Efx
i
is the mean value of the particles' position, Efx
i
g
g
is the mean velocity of the multi-particle system, Ef
i
g
is the average of the
disturbance signal, and Ef
u
i
g is the control input that is expected to steer the mean
of the multi-particle formation along a desirable path. The open-loop controller is
selected as:
g
r
C Ef x
i
Ef
u
i
gD!Efx
i
g
r
Ef
i
g
(9.27)
where Efx
i
g
r
is the desirable trajectory of the mean. From Eq. (
9.27
) it can be
seen that the particle's control consists of two parts: (1) a part that compensates for
the interaction between the particles and, (2) a part that compensates for the forces
which are due to the harmonic external potential (slowing down of the particles'
motion).
Assuming that for the mean of the particles' system holds Ef
i
gD0, then the
control law of Eq. (
9.27
) results in the error dynamics for the mean position of the
particles
Efe
i
gC!Efe
i
gD0;
(9.28)
which assures that the mean position of the particles will track the desirable
trajectory, i.e.
lim
t!1
Efe
i
gD0:
(9.29)
