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According to Step 3, by substituting the expressions of (19) into (20), the
boundary feedback control laws are obtained by
⎧
⎨
√
2
g
(
H
up
−H
(0
,t
))
H
(
H
(0
,t
)
H
)]
H
(0
,t
)
[
V
1+
m
0
1
u
0
(
t
)=
−
−
−
m
0
√
2
g
(
H
(
L,t
)
−H
do
)
H
(
H
(
L,t
)
H
)]
,
|
m
0
m
1
|
<
1
.
(21)
⎩
H
(
L,t
)
[
V
+
1+
m
1
1
u
L
(
t
)=
−
−
m
1
6Con lu on
This paper considered the boundary feedback control of canals with trapezoidal
cross section, which is the common case in open canals. In such a case, the results
achieved so far in the literature don't work well. Starting from the nonlinear
Saint-Venant equations, this work investigates the boundary feedback control
for a single canal as well as canal networks via a Riemann invariants approach.
The main contribution is to propose a unified boundary feedback control design,
which extends the results of the literature [6][7][8]. For a single canal, a set of
boundary conditions in terms of Riemann invariants to guarantee the closed-
loop system exponentially stable is developed. Based on these conditions, the
stabilizing boundary control laws are derived. Then the stability condition is
generalized to the cascaded case. The boundary control laws thus derived are
elementary functions of the water levels at the gate boundaries. The advantage
of our proposed approach lies in that the boundary feedback control laws are
always elementary functions of the boundary water levels. Thus the boundary
feedback control laws can be implemented only by taking the boundary water
levels as the feedback.
Acknowledgments.
This work was supported in part by the National Nat-
ural Science Foundation of China under Grant 61104078, 61074060, 61272028
and 61073079, the Foundation of Key Laboratory of System Control and In-
formation Processing under Grant SCIP2011009, Ministry of Education, the
Specialized Research Fund for the Doctoral Program of Higher Education under
Grant 20110162120045 and the Fundamental Research Fund of Central South
University under Grant 201012200159.
References
1. Chow, V.T.: Open Channel Hydraulic. Mac-Graw Hill Book Company, New York
(1985)
2. Schuurmans, J., Clemmens, A.J., Dijkstra, S., Hof, A., Brouwer, R.: Modelling of
Irrigation and Drainage Canals for Controller Design. J. Irrig. Drain. Eng. 125(6),
338-344 (1999)
3. Sawadogo, S., Fayer, R.M., Mora-Camino, F.: Decentralized Adaptive Predictive
Control of Multireach Irrigation Canal. Int. J. Syst. Sci. 32(10), 1287-1296 (2001)
4. Gomez, M., Rodellar, J., Mantecon, J.A.: Predictive Control Method for Decentral-
ized Operation of Irrigation Canals. Appl. Math. Model. 26(1), 1039-1056 (2002)
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