Information Technology Reference
In-Depth Information
9.4
Flatness-Based Control for the Multi-Particle System
First, the motion of particles is considered, i.e. it is assumed that no external control
affects the particle's motion. The particles move on the 2D-plane only under the
influence of an harmonic potential. The kinematic model of the particles is as
described in Sect. 8.2 .
x i
D F i
2 P iD1 P jD1;j¤i ŒV a .jjx i
1
F i
Dr x i fV i .x i / C
x j
jj C V r .jjx i
x j
jj/g
(9.20)
The interaction between the i-th and the j-th particle is
g.x i
x j / D.x i
x j /Œg a .jjx i
x j
jj/ g r .jjx i
x j
jj/
(9.21)
where g a ./ denotes the attraction term and is dominant for large values of jjx i
x j
jj,
while g r ./ denotes the repulsion term and is dominant for small values of jjx i
x j
jj.
From Eq. ( 9.21 ) it can be seen that g.x i
x i /, i.e. g./ is an odd
function. Therefore, for the center of the multi-particle system holds that
x j / Dg.x j
M . P jD1;j¤i g.x i
1
x j // D 0 and
(9.22)
M P iD1 Œr x i V i .x i /
x D
1
According to the Lyapunov stability analysis and application of LaSalle's
theorem given in Chap. 8 , it has been shown that the mean of the multi-particle
system will converge exactly to the goal position Œx ;y D Œ0;0 while each
individual particle will remain in a small bounded area that encircles the goal
position [ 64 , 92 , 150 , 152 ].
Next, flatness-based control for the multi-particle system will be analyzed. The
equivalent kinematic model of the i-th particle is given in Eq. ( 9.20 ) and can be
written in the form:
x i
D!x i
C u i
C i
(9.23)
where !x i is the drift term due to the harmonic potential, u i is the external control,
and i is a disturbance term due to interaction with the rest N 1 particles, or due
to the existence of noise. Then it can be easily shown that the system of Eq. ( 9.23 )
is differentially flat, while an appropriate flat output can be chosen to be y D x i .
Indeed all system variables, i.e. the elements of the state vector and the control input
can be written as functions of the flat output y, and thus the model that describes
the i-th particle is differentially flat.
An open-loop control input that makes the i-th particle track the reference
trajectory y r is given by
 
Search WWH ::




Custom Search