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continuously differential solutions and the closed-loop stability based on the
Saint-Venant equations. Some results also arise in recent years. [6] discussed the
boundary feedback control of a single canal and a multi-reach canal in cascade
with rectangular cross section by means of Riemann invariants. Coron [7], Li
[8] and Cen and Xi [9] proposed a Lyapunov approach to study the boundary
feedback control of a single canal and its stability. Litrico proposed a frequency
domain approach to investigate boundary control of the linear Saint-Venant
equations [10].
Boundary feedback control indicates that the control laws are elementary
functions of the boundary water levels as the feedback. However, all the re-
sults mentioned above are limited to canals with rectangular cross sections or
linearized Saint-Venant equations. For open canals with some particular cross
sections such as the trapezoidal one that is the mostly common case, the above
results fails to give the boundary feedback controller by only taking the bound-
ary water levels as the feedback. This paper solves this problem. We start from
the nonlinear Saint-Vennant equations without any approximation, linearization
and discretisation. The aim is to propose a unified framework for the boundary
feedback control design not only for a single canal but also for canal networks
with multireaches in cascade.
2 Modelling in Open Channels
2.1 Saint-Venant Equations
Let us consider one horizontal canal with trapezoidal cross section delimited by
two gates without friction, as shown in Fig. 1.
Fig. 1. A canal delimited by two underflow gates
In the following, regardless of a single canal or multi-reach canal we always
assume that [11]: (A) The water flow satisfies the sub-critical condition. (B)
The water levels at the gate boundaries can be measured online. (C) The gate
openings, as the physical control actions, are elementary functions of the water
heights ( H (0 ,t ) ,H ( L,t )) at the gate boundaries.
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