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Next, it will be also shown that the kinematic model of each individual particle is
a differentially flat system and thus can be expressed using a flat output and its
derivatives. It will be proven that flatness-based control can compensate for the
effect of external potentials, and interaction forces, thus enabling the position of the
multi-particle formation to follow the reference path. When flatness-based control
is applied, the mean position of the formation of the N diffusing particles can be
steered along any desirable position in the 2D plane, while the i-th particle can track
this trajectory within acceptable accuracy levels.
9.3
Some Examples on Flatness-Based Control of Coupled
Oscillators
Flatness-based control of N linear coupled oscillators has been analyzed in [
172
].
The generalized coordinates
z
i
are considered and n oscillators are taken. The
oscillators can be coupled through an interaction term f
i
.
z
1
;
z
2
; ;
z
N
/ and through
the common control input
u
. This means that the general oscillator model can be
written as
d
2
dt
2
z
i
D.!
i
/
2
z
i
C f
i
.
z
1
;
z
2
; ;
z
N
/ C b
i
u
;
(9.9)
i D 1; ;N.Forf
i
.
z
1
;
z
2
; ;
z
N
/ D 0 one obtains
d
2
dt
2
z
i
D.!
i
/
2
z
i
C b
i
u
;iD 1; ;N
(9.10)
The terms !
i
>0and b
i
¤0 are constant parameters, while T>0and D¤0 are
also defined. The objective is to find open-loop control Œ0;T with t!
u
.t/ steering
the system from an initial to a final state. In [
172
] it has been shown that such control
can be obtained explicitly, according to the following procedure: using the Laplace
transform of Eq. (
9.10
) and the notation s D
d
dt
one has
.s
2
C .!
i
/
2
/
z
i
D b
i
u
;iD 1; ;N:
(9.11)
Then the system can be written in the form [
110
]:
z
i
D Q
i
.s/y;
u
D Q.s/y; with y D
P
kD1
c
k
z
k
Q
i
.s/ D
.!
i
/
2
Q
kD1
.1 C .
!
k
/
2
/ for k¤i;
b
i
(9.12)
Q.s/ D
Q
kD1
.1 C .
!
k
/
2
/; c
k
D
1
Q
k
.j!
k
/
2 R
The real coefficients q
k
and q
k
are defined as follows [
172
]:
Q
i
.s/ D
P
N1
kD0
q
k
s
2k
Q.s/ D
P
kD0
q
k
s
2k
(9.13)
