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b
a
Fig. 9.2
( a ) Particles following reference trajectory (i), ( b ) associated control input
For non-constant acceleration, the generalized model of the harmonic oscillator
will be used and the results of Sect. 9.3 will be applied. Thus, the common control
input u .t/ can be selected as u
D Q.s/y, where y is the flat output, defined as
y D P kD1 c k z k C d k s z k .
9.5
Simulation Tests
The particles (interacting weights) were initialized at arbitrary positions on the 2D-
plane. Trajectory tracking under flatness-based control is examined. The following
reference trajectories have been tested:
x r .t/ D 5sin. 2 T /
y r .t/ D 5cos. 2 T /
(9.30)
x r .t/ D 5sin. 2 T /
y r .t/ D 1:5cos. 2 T / C 1:5sin. 4 T / C 1:5sin. 8 T /
(9.31)
Open-loop flatness-based control has been applied, which means that the control
signal consisted of the reference trajectory and its derivatives, while measurements
about the position and the velocity of the particles have not been used. It can be
observed that under flatness-based control the mean of the particle system can be
steered along a desirable path with infinite accuracy, while each individual particle
can track the trajectory within acceptable accuracy levels.
The top left diagram in Figs. 9.2 and 9.3 shows the trajectories followed by the
particles when a separate control law was designed for each particle, while the
bottom diagram shows the trajectories followed by the particles when the average
control input was applied to each one of them. The right plot in Figs. 9.2 and 9.3
 
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