Infinite dimensional modelling of open-channel hydraulic systems for control purposes Xavier Litrico

Vincent Fromion

Cemagref, UR IRMO

LASB, INRA Montpellier

B.P. 5095, 34033 Montpellier Cedex 1, France

2 place Viala, 34060 Montpellier, France

e-mail: [email protected]

e-mail: [email protected]

Abstract This paper provides a new computational method to obtain a frequency domain model of Saint-Venant equations, linearized around any stationary regime, including backwater curves. The obtained model is analyzed by characterizing the maximum achievable performance using inner-outer factorization, and a method to compute the poles of the system is indicated. The paper also provides a way to obtain an accurate rational approximation of the system. Keywords: Open-channel system, irrational system, H∞ control, rational approximation, Time Delay Systems. 1 Introduction Water is a precious resource, that should be managed efficiently. Irrigation is the main water consuming activity around the world, as it represents about 80% of the available fresh water consumption. It is widely accepted that automation could lead to a better efficiency of water management in many irrigation systems with open-channels, which are subject to large losses [14]. However, these systems are not easy to control, as they are large distributed systems, with complex dynamics. Hydraulic engineers use the so-called Saint-Venant equations to model the dynamics of water flowing in an open channel. These equations are known to provide a very accurate description of the dynamic behavior of a canal [5] with only one parameter that needs to be calibrated (friction coefficient). Saint-Venant equations are nonlinear hyperbolic partial differential equations, that are difficult to use directly for controller design. In order to bypass problems linked to the nonlinearity of those equations, a classical way is to only consider linear models which are associated to equilibrium regimes [9, 13, 15]. Even if this approach is heuristic, it has provided very efficient solutions to control irrigation canals. The aim of this paper is to provide an easy way to compute an accurate frequency response characteriza-

tion for any equilibrium regimes and any type of canal geometry. It may seem surprising that this problem has never been addressed before. For channels at uniform regime, corresponding to a specific case, where the geometry is uniform and the water depth is constant along the channel, an analytical solution exists in the frequency domain [4, 7, 2]. However, a real canal is seldom at uniform regime. This particular stationary flow condition is used for irrigation canal design, and usually corresponds to the maximum flow conditions, but necessitates a prismatic geometry, and no losses or intermediate water outlet. It is therefore necessary to be able to compute the transfer function corresponding to Saint-Venant equations for more realistic flow conditions. Very few models take into account explicitly the backwater curve (i.e. nonuniform stationary flow conditions) [15]. The paper adresses the three following questions: 1) How to obtain the transfer matrix of Saint-Venant equations for a non-uniform stationary regime? 2) How to characterize the obtained model? and 3) How to obtain a rational model for controller design? We will answer to these questions using various numerical tools based on theoretical results. 2 Saint-Venant model The Saint-Venant equations are widely used to model irrigation canals and rivers. These equations are nonlinear hyperbolic partial differential equations [5]: ∂A ∂Q + ∂t ∂x

=

ql

∂Y ∂Q ∂Q2 /A + + gA + gA(Sf − I) ∂t ∂x ∂x

=

kql V (2)

(1)

with A(x, t) is the wetted area [m2 ], Q(x, t) the discharge [m3 /s] across section A, V (x, t) the average velocity [m/s] in section A, ql (x) the lateral discharge [m2 /s] (ql > 0: inflow, k = 0 and ql < 0: outflow, k = 1), Y (x, t) the water depth [m], Sf (x, t) the friction slope, I the bed slope and g the gravitational acceleration [m/s2 ].

Two boundary conditions are necessary for this partial differential system, for example Q(0, t) = Q0 (t) and Q(X, t) = QX (t), where X is the length of the considered channel. The initial conditions are given by Q(x, 0), Y (x, 0) and ql (x) for x ∈ [0, X]. The friction slope Sf is modelled with ManningStrickler formula: Sf =

Q2 n 2 A2 R4/3

Example: The article is illustrated with two trapezoidal prismatic channels, having different characteristics (see table 1, where X is the channel length [m], m the bank slope, B the bed width [m], I the bed slope, n the Manning coefficient [m−1/3 s], Yn the normal depth [m] corresponding to the maximum discharge Qmax [m3 s−1 ]). Canal 1 is a short flat canal, and canal 2 is a long sloping canal.

(3)

with n the Manning coefficient [sm−1/3 ] and R the hydraulic radius [m], defined by R = A/P , where P is the wetted perimeter [m]. These nonlinear partial differential equations are difficult to interpret directly. A classical approach is to study the linearized equations for particular regimes. For this purpose, a frequency domain approach is used, in order to understand the behavior of the linearized models.

Table 1: Parameters for the two canals canal 1 canal 2

X 3000 6000

m 1.5 1.5

B 7 8

I 0.0001 0.0008

Under these hypotheses, and denoting with an underscore zero the variables corresponding to the equilibrium regime (Q0 (x), Y0 (x), etc.), Saint-Venant equations become: dQ0 (x) = 0 (4) dx I − Sf 0 (x) dY0 (x) = (5) dx 1 − F0 (x)2 q gA0 V0 F0 is the Froude number F0 = C with C = 0 L0 , 0 0 V0 = Q A0 . Throughout the paper, the flow is assumed to be subcritical, i.e. F0 < 1.

These two equations define an equilibrium regime given by Q0 (x) = Q0 = QX and Y0 (x) solution of the ordinary differential equation (5), for a boundary condition in terms of downstream elevation. A particular solution is obtained when the depth is constant along the channel. In this case, the left side of equation (5) is equal to zero and then, given Q0 (x) = Q0 , the equilibrium solution Yn (also called normal depth) can be deduced by the resolution of the equation: Sf (Q0 , Yn ) = I This specific solution is classically called the uniform regime.

Yn 2.12 2.92

Qmax 14 80

Imposing a downstream boundary condition YX = Yn , the following backwater curves are obtained for Qmax , Qmax /2 and Qmax /8 for both canals (see figure 1). backwater curves, canal 1

2.5

backwater curves, canal 2

8

Qmax Q /2 max Qmax/8

7 6 5

1.5

elevation (m)

elevation (m)

2

2.1 Equilibrium regimes: backwater curves The method exposed below can be applied to any type of reaches (including variable geometry and a lateral discharge different from zero). In the sequel, in order to facilitate the exposition with reference to the wellknown uniform case, the lateral discharge ql is assumed to be equal to zero and the channel prismatic.

n 0.02 0.02

Qmax Qmax/2 Qmax/8

1

4 3 2

0.5

1 0 0

500

1000

1500 abscissa (m)

2000

2500

0 0

3000

1000

2000

3000 abscissa (m)

4000

5000

6000

Figure 1: Backwater curves of canal 1 and 2, for discharges equal to Qmax , Qmax /2 and Qmax /8

2.2 Linearized Saint-Venant model In order to obtain the linearized model around the equilibrium regime defined by equation (5), Q(x, t) = Q0 + q(x, t) and Y (x, t) = Y0 (x) + y(x, t) are replaced for in equations (1) and (2). Neglecting second order terms leads to the following equations: L0

∂q ∂y + =0 ∂t ∂x

(6)

with L0 the mirror width for the stationary regime, and ∂q ∂t

∂y ∂q + 2V0 ∂x − β0 q + (C02 − V02 )L0 ∂x − γ0 y = 0

Parameters γ0 and β0 are defined by: ¸ · ∂Y0 2 γ0 = gL0 (1 + κ)I − (1 + κ − F0 (κ − 2)) ∂x and β0 = −

2g V0

µ

I−

∂Y0 ∂x

¶

(7)

(8)

(9)

4A0 ∂P0 . The boundary conditions are with κ = 37 − 3L 0 P0 ∂Y then given by q(0, t) = q0 (t) and q(X, t) = qX (t).

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

the semi-group theory [6], the second one is to use frequency domain tools [3]. A frequency domain approach is used in the sequel, in order to get the transfer function of the system in Laplace domain. This method allows in a first step to use classical controller design tools (Nyquist plot, Bode plot, Nichols chart, etc.).

ζ(x) = eAs x ζ0

3.1 ODE for transfer matrix representation of Saint-Venant model Applying Laplace transform to the linear partial differential equations (6) and (7), and reordering leads to an ordinary differential equation in the variable x, with a complex parameter s (the Laplace variable): µ ¶ µ ¶ d qs (x) q (x) = As (x) s (10) ys (x) dx ys (x) µ ¶ 0 −c(x)s with As (x) = and −d(x)s − e(x) a(x)s + b(x) 2V0 (x) 0 (x) b(x) = L0 (x)(C0γ(x) a(x) = C0 (x) 2 −V (x)2 , 2 −V (x)2 ) , 0 0 1 c(x) = L0 (x), d(x) = L0 (x)(C0 (x)2 −V0 (x)2 ) , e(x) = 0 (x) − L0 (x)(C0β(x) 2 −V (x)2 ) , with two boundary conditions: 0 q0 for x = 0 and qX for x = X.

The difficulties linked to the resolution of this two boundary problem can be easily avoided since (10) is a linear differential equation. For this purpose, let us consider the integration of this linear system (11)

with ζ(x) = (qs (x), ys (x))T and where the initial condition is defined at x = 0. The general solution of the previous system always exists and is given by µ ¶ γ11 (x, s) γ12 (x, s) ζ(x) = Γs (x, 0)ζ0 = ζ (12) γ21 (x, s) γ22 (x, s) 0 where Γs (x, 0) is the transition matrix and ζ0 the initial condition at x = 0. On this basis, the transfer matrix corresponding to the original ODE (10) is then given by : ¶ ¶µ ¶ µ µ qs (0) p11 (s) p12 (s) ys (0) (13) = qs (X) p21 (s) p22 (s) ys (X)

In general cases, since As (x) depends on x, the differential equation (11) has no closed-form solution, and it is necessary to use a numerical integration method1 . Unexpectedly, the use of classical numerical schemes is not possible from a practical point of view, because the obtention of good numerical approximations leads to prohibitive computational times. In a first analysis, one could think that this is linked to the variation of matrix As (x) with x, but in fact, the same difficulty appears even for the uniform case. In order to better understand the specific properties of equation (11), the computation for the uniform case is firstly recalled. Following [2], matrix As is diagonalized As = Ps−1 Ds Ps

1 γ12 (X,s) , p21 (s) = (X,s) p22 (s) = γγ22 , under 12 (X,s)

γ21 (X, s) − and the fact that γ12 (X, s) is not equal to zero. Let us note that the values of s such this quantity is equal to zero correspond to the poles of the Saint-Venant transfer matrix (see below). This is why system (10) will be solved as if the initial conditions were given in x = 0, which greatly simplifies the computation.

(15)

which allows to obtain the transition matrix Γs (x, 0) given by: Γs (x, 0) = Ps−1 eDs x Ps (16) with Ds =

µ

λ1 (s) 0

0 λ2 (s)

¶

Ps =

µ

λ2 (s) −λ1 (s)

cs −cs

¶

and where the eigenvalues are given by λi (s) =

√ as + b − (−1)i As2 + 2Bs + C , i = 1, 2 2

(17)

with A = (a2 + 4dc), B = (ab + 2ec), C = b2 . Straightforward manipulations lead to the classical result already obtained by [4], [7] and [2]. Let us return to the numerical problem analysis, with a more technical approach. For the uniform case, using (15) the integration of the ODE (11) can be restated as the integration of two complex ODEs given by

11 (X,s) with p11 (s) = − γγ12 (X,s) , p12 (s) =

γ22 (X,s)γ11 (X,s) γ12 (X,s)

(14)

In this case, the computation is done without any difficulty, for any value of s.

3 A new computational method for Saint-Venant transfer matrix

dζ(x) = As (x)ζ(x) dx

3.2 Some numerical aspects for the resolution In a specific case (which corresponds to the well-known uniform case), matrix As does not depend on x and the solution of equation (11) is given by:

dνi = λi (s)νi , dx

i = 1, 2

(18)

with ν(x) := Ps ζ(x) = (ν1 (x), ν2 (x))T . The general solution of the complex first order ODE (18) is given by νi (x) = eλi x νi (0) 1 The

(19)

resolution of equation (11) necessitates to split real and imaginary parts, which leads to four differential equations.

For s = jω, which corresponds to frequencies of interest with respect to Bode plot, the exact solution of (11) is unstable and highly oscillatory. Moreover, the oscillation frequency and instability drastically increase for high values of s = jw. In this case, for high values of ω, Re(λi (jω)) remains small while Im(λi (jω)) is proportional to ω. Then the qualitative behavior of the solution (19) is close to a sinusoidal response of frequency ωi = |λi (jω)| (i.e. νi (x) ≈ eRe(λi )x cos(ωi x)). This is the reason why classical numerical methods necessitate a very small step size to solve ODE (18) with a good precision. A way to avoid this difficulty is to use an exponentialtype method (see e.g. [11]). Rather surprisingly, this solution exactly corresponds in case of constant matrix As to the above computation in the uniform case. We propose in the following an efficient method to compute the solution in more realistic cases, i.e. when matrix As (x) varies with x. 3.3 A new numerical scheme Let xk be a space discretization of interval [0, X] into n subintervals: 0 = x0 < x1 < · · · < xk < · · · < xn = X,

Runge-Kutta method, with a good precision in both cases. For high frequencies, the time needed for the Runge-Kutta method to compute the solution with a small relative error becomes incompatible with practical applications (296 s for canal 1 and 807 s for canal 2 at ω = 1 rad/s). In this case, the oscillatory behavior of the system leads to more iterations and therefore a large computational time. In fact, the error also increases a lot, which is not the case for the proposed method, which is therefore more precise and needs less computation time than the Runge-Kutta method. In order to compute a transfer function response with 500 frequency points, the needed computational time rises up to 5 h (resp. 13 h 50 min) for canal 1 (resp. canal 2) using Runge-Kutta method, which is clearly not suited for practical applications. Whereas the proposed method gives very small computational times (see table 2). Table 2: Computational time for 500 frequency points canal 1 canal 2

xk+1 = xk + hk

proposed method 10 s 26 s

Runge-Kutta method 5h 13 h 50 min

We propose the following one step method:

Remark 1 The numerical scheme leads to the following approximation of transition matrix of (11) eAs (xk )hk

(20)

k=0

which can be interpreted as an approximation of nonuniform regime by a series of constant water depth regimes associated to the value of As at xk .

Bode plot canal 1

20

0

max

−20 −30 −4

10

−3

10 freq.(rad/s)

−2

10

−4

10 freq.(rad/s)

−4

10 freq.(rad/s)

10

−3

10

−2

10

−3

10

−1

−2

10

0 −1000

−1000

−3000

The proposed method compares favorably with the fourth order Runge-Kutta method. For low frequencies, the computational time is acceptable for the

−40

−80 −5 10

−1

10

0

−4000 −5 10

max

−20

−60

−2000

3.4 Numerical efficiency For both example canals at uniform maximum flow, the results given by two methods are compared to the analytical solution: the method developed in this paper and a Runge-Kutta method, classically used to solve differential equations. All computations are done on the same computer.

Q max Qmax/2 Q /8

0

−10

−40 −5 10

Bode plot canal 2

20

Qmax Q /2 max Q /8

10 gain (dB)

n−1 Y

phase (dg)

Γs (X, 0) ≈

gain (dB)

It is possible to prove that this numerical scheme is at least of order 1 when the derivative of As (x) with respect to x is bounded.

3.5 Frequency responses: Bode plots With the example canals defined above, the following Bode plots are obtained when the discharge varies (see figure 2) for transfer function p21 (s). One can notice the behavior of both canals, which is completely different: canal 1 is oscillating, with resonant modes, and canal 2 exhibits a long time-delay. Their behavior changes with the discharge: the oscillating modes of canal 1 seem to stay at the same frequency, only their damping changes (it decreases with the discharge). The delay of canal 2 tends to augment when the discharges diminishes.

phase (dg)

η0 := ζ0 ; for k = 0, 1, · · · : ηk+1 := eAs (xk )hk ηk , xk+1 := xk + hk

−2000 −3000 −4000 −5000

−4

10

−3

10 freq.(rad/s)

−2

10

−1

10

−6000 −5 10

10

−1

Figure 2: Bode plots of p21 (s) for canal 1 and 2 for various discharges

4 Transfer matrix analysis We first study the poles of the model obtained. The poles are useful to: 1) characterize the behavior of the

oscillating modes of the model, and 2) obtain a good rational approximation, since the rational approximation problem can then be formulated as a convex optimization problem. We will then characterize the inner part of the transfer matrix, that is important for control purposes, as it limits the available performance of the closed-loop system [1, 12]. 4.1 Poles of Saint-Venant model Uniform case: Saint-Venant model with upstream and downstream discharges as the inputs is given by transfer matrix P (s) of equation (13). The poles of this transfer matrix are solutions of equation γ12 (s) = 0, with γ12 (s) given by equation (12).

Once obtained the outer factor phase, the inner factor can be computed using: Gi (jω) =

pk = 2

with ∆(k) = B − A(C +

p ∆(k) A

4k2 π 2 X 2 ).

This computation of the poles proved to be numerically difficult, but we managed to track the low frequency poles (the one obtained for small values of k in the uniform case), by computing successive linear models for decreasing discharges Q0 . 4.2 Inner-outer factorization As already shown by the authors [12], the structural performance of open-channel system is linked to the inner factor of the system transfer function. We will in the sequel exhibit the inner part of the system, first for the uniform regime, then in the general case. Review of classical results in complex analysis: Any function G(s) analytic in the right half-plane can be factorized into two terms: G(s) = Go (s)Gi (s)

(22)

where Go (s) is the outer factor, and Gi (s) the inner factor. An inner function belongs to H∞ and is such that |Gi (jω)| = 1 for all ω ∈ R. An outer function belongs to H∞ and has no zeros in the right half-plane. The outer factor of G(s) analytic in the right half-plane is obtained by [10]: Go (s1 ) = exp

·

1 π

Z

+∞ −∞

dω ωs1 + j log |G(jω)| ω + js1 1 + ω2

p11 (X, s)

=

p12 (X, s)

=

p21 (X, s)

=

p22 (X, s)

=

λ 2 e λ1 X − λ 1 e λ2 X cs (eλ2 X − eλ1 X ) λ1 − λ2 cs (eλ2 X − eλ1 X )

(21)

Backwater case: In this case, poles are obtained numerically, by finding the zeros of equation γ12 (s) = 0. Since the transfer function can obtained numerically, we know how to compute γ12 (s) for s ∈ C, and we can look for values of s such that γ12 (s) = 0.

¸

(23)

(24)

Uniform case: It can be shown that the inner parts of the Saint-Venant transfer functions are pure delays. The two delays correspond to the upstream and downstream hydraulic propagation times. In fact, these two time-delays correspond exactly to well-known quantities, i.e. hydraulic characteristics. This shows that the characteristics give the exact time-delays of the system, and that there is no unstable zeros. The Saint-Venant transfer matrix P (s) is given by

There is a pole in zero, the other poles being given by −B ±

G(jω) exp[−j arg(Go (jω))] |Go (jω)|

(λ2 − λ1 )e(λ1 +λ2 )X cs (eλ2 X − eλ1 X ) λ 1 e λ1 X − λ 2 e λ2 X cs (eλ2 X − eλ1 X )

(25) (26) (27) (28)

After technical developments, the inner-outer factorization of the elements of P (s) is:

with τ1 =

p11 (X, s) p12 (X, s)

= =

p11o (X, s) p12o (X, s)e−τ2 s

p21 (X, s) p22 (X, s)

= =

p21o (X, s)e−τ1 s p22o (X, s)

X C0 +V0 ,

τ2 =

X C0 −V0 .

The characteristics lines of Saint-Venant equations are given by [5]: (

dY 1 dQ 1 ± C(x,t) dt + gA dt + Sf − I = 0 dx dt = V (x, t) ± C(x, t)

(29)

Integrating the characteristic lines along the channel length gives the time delay for downstream propagation (positive characteristic, corresponding to V + C) and upstream propagation (negative one, corresponding to V − C). Backwater case: In the non uniform case, the innerouter factorization can be computed numerically, by integrating equation (23), as proposed by [8]. In fact, this method computes the integral (23) on a given frequency range ω ∈ [−M, M ]. The frequency bound M has to be very large in order to get an accurate result, which increases the necessary computational effort. This is why we use the method of characteristics, which provides the exact delays present in the system (which correspond to the inner parts of the system).

In this case, the delays can be obtained by integrating the characteristics lines along the channel: τ1 =

Z

X 0

dx V0 (x) + C0 (x)

(30)

for the delay of the positive characteristic line, and τ2 =

Z

X 0

dx C0 (x) − V0 (x)

(31)

for the delay of the negative characteristic line.

In order to use advanced automatic controller design tools (such as H∞ control design), it is useful to have a rational model of the system. We will now use the frequency domain model obtained numerically for non uniform flow conditions, in order to get an approximate rational model that would fit this numerical frequency response. Since we know the np poles of the transfer function that we want to approximate, the problem can be put in a convex optimization form (not presented here for lack of space, only graphical results are depicted). The rational approximation obtained using np = 8 poles and n = 5 frequency points is shown in figure 3 for factor p21o (s) of canal 1 and 2. The maximum absolute error is about -50 dB for canal 1, and -100 dB for canal 2. Bode plot, canal 2 40

Bode plot, canal 1

20 gain (dB)

gain (dB)

0 −10

−40

−30 −40 −5 10

−60 −5 10 −4

−3

10

20

10 freq.(rad/s)

−2

10

−3

10

−3

10

10 freq.(rad/s)

−2

10

−1

−2

10

0

freq.(rad/s)

−20 phase (dg)

0 phase (dg)

−4

10

−1

10

−20 −40 −60

−40 −60 −80

−100

−80 −100 −5 10

0

−20

−20

−4

10

−3

10

−2

10

−1

10

−120 −5 10

freq.(rad/s)

−4

10

10 freq.(rad/s)

Acknowledgements This work was partially supported by the joint research program INRA/Cemagref ASS AQUAE n◦ 02 on the control of delayed hydraulic systems.

5 Rational approximation

10

since uniform regimes are almost never encountered in real situations. The model has been analyzed and characterized so that we can have an a priori knowledge of the performance limitations induced by the inner part. Using the poles calculated by a numerical procedure, the transfer matrix can then be approximated by a rational model of low order that fits very well the frequency response.

−1

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 engineers to model the dynamics of water flowing in a channel. Only specific regimes (namely the uniform regime) are considered in the literature. The main contribution of the paper is to provide a way to obtain an “exact” model, i.e. as accurate as required on any frequency range, for general regimes (including backwater curves). This issue has never been considered before and is very important from a practical point of view,

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