Figure 3: Pilot channel of Valence.
One experimentation has been chosen to illustrate
this approach.
Note that water ﬂow is deduced from the gate
equations, and has not been measured directly. The
data pictured below have been ﬁltered to get a better
idea of the experimentation results.
In each experiment, the system is initially in open
loop at a steady state:
Q
e
(0) = 2.5dm
3
.s
−1
, H
e
(0) = 1.1dm, H
e
(L) = 1.26dm.
The loop is closed at time t = 50sec with a new set
point given by:
¯
Q(0) = 2dm
3
.s
−1
,
¯
H(0) = 1.3dm,
¯
H(L) = 1.43dm.
Two experimentations are pictured in Fig. (4), with
maxk
0
k
L
= 0.888 and the following values:
1. k
0
k
L
= 0.247 & m
0
= 0, m
L
= 0,
2. k
0
k
L
= 0.247 & m
0
= −0.002, m
L
= 0.001.
0 100 200 300 400 500
1.5
2
2.5
3
Water flow at upstream
Time (s)
(dm
3
.s
−1
)
without Integral Action
with Integral action
reference
0 100 200 300 400 500
1
1.2
1.4
1.6
1.8
2
Downstream water
Time (s)
(dm)
without integral action
with integral action
reference
Figure 4: Water ﬂow at upstream and level at downstream.
To conclude this part, let notice that for the micro-
channel, both tests (27)-(28) are quiet equivalent (is
not the case for rivers like the Sambre in Belgium). In
all the cases, one conclusion is the same, the stability
of the system is insured if both tests (27)-(28) and the
condition (31) are realized.
Exact convergence is ensured if the integral part of the
control is added even if it is applied to the real and so
the non linear system.
6 CONCLUSION
In this paper, a boundary control law with integral ac-
tion is proposed, as a new stability condition depend-
ing on the Riemann coordinates. Simulations and ex-
perimentations realized strengthen on the fact that the
stability conditions (16)-(18) can be developed to ﬁt
to the non linear case. Improvements will be the de-
velopment of the works on (Dos Santos et al., 2007)
to non linear system of conservation laws, and/or to
couple both previous results and generalize them to
greater dimension systems.
ACKNOWLEDGEMENTS
The authors would like to thank professor E. Mendes
and the LCIS to have allowed us to realize our exper-
imentations on the micro-channel. In the same way,
thanks to the Cemagref for the use of the software
SIC.
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