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Two boundary conditions are necessary for this partialdifferential system, for example Q(0, t) = Q0(t) andQ(X, t) = QX(t), where X is the length of the con-sidered channel. The initial conditions are given byQ(x, 0), Y(x, 0) and ql(x) forx [0, X].

The friction slope Sf is modelled with Manning-Strickler formula:

Sf = Q2n2

A2R4/3 (3)

withn the Manning coefficient [sm1/3] andR the hy-draulic radius [m], defined byR = A/P, wherePis thewetted perimeter [m].

These nonlinear partial differential equations are dif-ficult to interpret directly. A classical approach is tostudy the linearized equations for particular regimes.For this purpose, a frequency domain approach is used,in order to understand the behavior of the linearizedmodels.

2.1 Equilibrium regimes: backwater curves

The method exposed below can be applied to any typeof reaches (including variable geometry and a lateraldischarge different from zero). In the sequel, in orderto facilitate the exposition with reference to the well-known uniform case, the lateral dischargeqlis assumedto be equal to zero and the channel prismatic.

Under these hypotheses, and denoting with an under-

score zero the variables corresponding to the equilib-rium regime (Q0(x), Y0(x), etc.), Saint-Venant equa-tions become:

dQ0(x)

dx = 0 (4)

dY0(x)

dx =

I Sf0(x)

1 F0(x)2 (5)

F0 is the Froude number F0 = V0C0

with C0 =

gA0L0

,

V0 = Q0A0

. Throughout the paper, the flow is assumedto be subcritical, i.e. F0< 1.

These two equations define an equilibrium regime givenby Q0(x) = Q0 = QX and Y0(x) solution of the ordi-nary differential equation (5), for a boundary conditionin terms of downstream elevation.

A particular solution is obtained when the depth isconstant along the channel. In this case, the leftside of equation (5) is equal to zero and then, givenQ0(x) = Q0, the equilibrium solution Yn (also callednormal depth) can be deduced by the resolution of theequation:

Sf(Q0, Yn) =I

This specific solution is classically called the uniformregime.

Example: The article is illustrated with two trape-zoidal prismatic channels, having different characteris-tics (see table 1, whereX is the channel length [m],m the bank slope, B the bed width [m], I the bedslope, n the Manning coefficient [m1/3s], Yn the nor-mal depth [m] corresponding to the maximum dischargeQmax[m3s1]). Canal 1 is a short flat canal, and canal2 is a long sloping canal.

Table 1: Parameters for the two canalsX m B I n Y n Qmax

canal 1 3000 1.5 7 0.0001 0.02 2.12 14canal 2 6000 1.5 8 0.0008 0.02 2.92 80

Imposing a downstream boundary conditionYX = Yn,the following backwater curves are obtained forQmax,Qmax/2 andQmax/8 for both canals (see figure 1).

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5backwater curves, canal 1

abscissa (m)

elevation(m)

Qmax

Qmax

/2

Qmax

/8

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

8backwater curves, canal 2

abscissa (m)

elevation(m)

Qmax

Qmax

/2

Qmax

/8

Figure 1: Backwater curves of canal 1 and 2, for dis-charges equal toQmax, Qmax/2 andQmax/8

2.2 Linearized Saint-Venant modelIn order to obtain the linearized model around theequilibrium regime defined by equation (5), Q(x, t) =Q0+ q(x, t) and Y(x, t) =Y0(x) +y(x, t) are replacedfor in equations (1) and (2). Neglecting second orderterms leads to the following equations:

L0y

t +

q

x = 0 (6)

withL0the mirror width for the stationary regime, and

qt + 2V0

qx 0q+ (C

20 V

20)L0

yx 0y= 0 (7)

Parameters 0 and0 are defined by:

0= gL0

(1 + )I (1 + F20 ( 2))

Y0x

(8)

and

0= 2g

V0

I

Y0x

(9)

with = 73 4A0

3L0P0P0Y . The boundary conditions are

then given by q(0, t) =q0(t) and q(X, t) =qX(t).

The model of the system is therefore given by two lin-ear partial differential equations. There are two waysto study this type of system: the first one is to use

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For s= j, which corresponds to frequencies of inter-est with respect to Bode plot, the exact solution of (11)is unstable and highly oscillatory. Moreover, the oscil-lation frequency and instability drastically increase forhigh values ofs = jw.In this case, for high values of, Re(i(j)) remainssmall while Im(i(j)) is proportional to . Thenthe qualitative behavior of the solution (19) is closeto a sinusoidal response of frequencyi = |i(j)| (i.e.i(x) e

Re(i)xcos(ix)). This is the reason why clas-sical numerical methods necessitate a very small stepsize to solve ODE (18) with a good precision.

A way to avoid this difficulty is to use an exponential-type method (see e.g. [11]). Rather surprisingly, thissolution exactly corresponds in case of constant matrixAs to the above computation in the uniform case.We propose in the following an efficient method to com-

pute the solution in more realistic cases, i.e. when ma-trix As(x) varies with x.

3.3 A new numerical schemeLet xk be a space discretization of interval [0, X] inton subintervals:

0 = x0< x1

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In this case, the delays can be obtained by integratingthe characteristics lines along the channel:

1=

X0

dx

V0(x) + C0(x) (30)

for the delay of the positive characteristic line, and

2=

X0

dx

C0(x) V0(x) (31)

for the delay of the negative characteristic line.

5 Rational approximation

In order to use advanced automatic controller designtools (such as H control design), it is useful to havea rational model of the system. We will now use the

frequency domain model obtained numerically for nonuniform flow conditions, in order to get an approximaterational model that would fit this numerical frequencyresponse.

Since we know thenppoles of the transfer function thatwe want to approximate, the problem can be put in aconvex optimization form (not presented here for lackof space, only graphical results are depicted). The ra-tional approximation obtained using np = 8 poles andn = 5 frequency points is shown in figure 3 for factorp21o(s) of canal 1 and 2. The maximum absolute erroris about -50 dB for canal 1, and -100 dB for canal 2.

105

104

103

102

101

40

30

20

10

0

10

freq.(rad/s)

105

104

103

102

101

100

80

60

40

20

0

20

freq.(rad/s)

Bode plot, canal 1

freq.(rad/s)

gain

(dB)

phase

(dg)

105

104

103

102

101

60

40

20

0

20

40

freq.(rad/s)

gain

(dB)

105

104

103

102

101

120

100

80

60

40

20

0

phase

(dg)

freq.(rad/s)

Bode plot, canal 2

Figure 3: Bode plot of p21o(s) : rational approximation( ) and irrational system () for canal 1 and

2 around Q0= Qmax/2

6 Conclusion

This paper considers the Saint-Venant equations,widely used by hydraulic and automatic control en-gineers to model the dynamics of water flowing in achannel. Only specific regimes (namely the uniformregime) are considered in the literature. The main con-tribution of the paper is to provide a way to obtain anexact model, i.e. as accurate as required on any fre-quency range, for general regimes (including backwatercurves). This issue has never been considered beforeand is very important from a practical point of view,

since uniform regimes are almost never encountered inreal situations. The model has been analyzed and char-acterized so that we can have an a priori knowledge ofthe performance limitations induced by the inner part.Using the poles calculated by a numerical procedure,the transfer matrix can then be approximated by a ra-tional model of low order that fits very well the fre-quency response.

Acknowledgements

This work was partially supported by the joint research pro-gram INRA/Cemagref ASS AQUAE n 02 on the controlof delayed hydraulic systems.

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