Boundary Feedback Control Design and Stability of Open-Channel Networks - Intelligent Computing for Sustainable Energy and Environment - page 346

Without loss of generality, we take the simple case: two reaches in cascade. These
reaches are assumed to have the same length L . As depicted in Fig. 2, x =0and
x = L corresponds to the left and right side of each reach respectively.

Fig. 2. A canal system consisting of 2 reaches

The Saint-Venant equations are written for each reach i ,( i =1 , 2)

a i
v i

+ ∂

∂x

a i
v i

= 0
0

,

V i + v i A i + a i
g/b i ( A i + a i ) V i + v i

∂
∂t

(11)

( i =1 , 2) , ( x,t ) ∈ [0 ,L ] × [0 , + ∞ ) .

Equation (11) has to be complemented by the initial conditions

a ( x, 0) = a 0 ( x ) , v ( x, 0) = v 0 ( x ) (12)

where a ( x,t )=( a 1 ( x,t ) ,a 2 ( x,t )) T , v ( x,t )=( v 1 ( x,t ) ,v 2 ( x,t )) T .
The boundary conditions are as follows.

F 1 ( v 1 (0 ,t ) ,a 1 (0 ,t )) = v 1 (0 ,t ) −f 1 ( a 1 (0 ,t )) = 0 ,
F 2 ( v 1 ( L,t ) ,a 1 ( L,t )) = v 1 ( L,t ) −f 1 ( a 1 ( L,t )) = 0 ,
F 4 ( v 2 ( L,t ) ,a 2 ( L,t )) = v 2 ( L,t ) −f 2 ( a 2 ( L,t )) = 0 ,
F 3 ( v 1 ( L,t ) ,a 1 ( L,t ) ,v 2 (0 ,t ) ,a 2 (0 ,t )) = ( A 1 + a 1 ( L,t ))
( V 1 + v 1 ( L,t ))

(13)

( A 2 + a 2 (0 ,t ))( V 2 + v 2 (0 ,t )) = 0 ,

−

where f i ( u )and f i ( u ), ( i =1 , 2) are continuously differentiable about u in a
neighborhood of 0 R .Theysatisfy f i (0) = 0 and f i (0) = 0.

The Riemann invariants are computed as ξ i
±

∈

( a i ,v i )= 2 ( v i + a i

p i ( α ) dα ),

where p i ( α )= g

0

( A i + α ) b i ( A i + α ) . For the sake of simplicity, we denote the fol-
lowing vectors ξ − ( x,t ), ξ + ( x,t )and ξ ( x,t )by

ξ − ( x,t ) =( ξ 1

( x,t ) ,ξ 2
−

( x,t )) T ,

−

ξ + ( x,t ) =( ξ + ( x,t ) ,ξ + ( x,t )) T ,