Spectral Decomposition of Open
Channel Flow
Xavier Litrico (Cemagref, UMR G-EAU, Montpellier)
with Vincent Fromion (INRA MIG, Jouy-en-Josas)
Motivation
• Agriculture = 70% of fresh water world consumption
• Irrigated agriculture = 17% of agricultural area, 40% of food
production
• Water for agriculture
• Large operational losses: 20% to 70%
• Strong incentives to limit them: save water in summer and users
requiring a better service
• Towards automatic management
• Improve water resource management
• Improve service to user
• Facilitate irrigation canal operational management
Objective
•
•
•
•
Canal dynamics are complex
Represented by nonlinear PDE: Saint-Venant equations
Linear approach leads to effective results
But no existing classification for canal dynamics
• Objective: understand the dynamics of linearized SaintVenant equations
•
•
•
•
Frequency domain approach
Poles
Spectral decomposition
From horizontal frictionless canal to uniform and non uniform
cases
Different views of irrigation canals
Outline
• Introduction
• Modeling of open channel flow
• Spectral decomposition
• Time domain response
• Illustrations
• Horizontal frictionless case
• Uniform flow case
• Non uniform flow case
• Analysis of Preissmann discretization scheme
• A link between Riemann invariants and frequency
domain approaches
• Conclusion
Main irrigation canal
• Series of canal pools
• We consider a single pool of length L
Modeling of open-channel flow
• Saint-Venant equations
• Mass conservation
• Momentum conservation
Friction slope:
• Initial condition
• Boundary conditions
Linearized Saint-Venant equations
• Linearized around (non uniform) steady flow
Frequency response
• Laplace transform leads to a distributed transfer
matrix:
• Poles pk in the horizontal frictionless case
Spatial Bode plot (horizontal frictionless case)
Uniform flow case
• Poles
0.05
0.04
0.03
0.02
imag
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-3
-2
-1
0
real
1
2
3
x 10
-3
Spatial Bode plot (Uniform flow case)
Non uniform flow case
• Compute the distributed transfer function using an
efficient numerical procedure (Litrico & Fromion, J. of Hydraulic
Engineering 2004)
• Compute the poles using this numerical method
• Conclusion: non uniform flow is qualitatively similar to
uniform flow
• Question: can we decompose the system along the
poles?
• Answer: Yes!
Main result: spectral decomposition
• The elements gij(x,s) of the distributed transfer matrix
G(x,s) can be decomposed as follows:
aij(k)(x) is the residue of gij(x,s) at the pole pk
Spatio-temporal representation of gij(x,s)
Sketch of proof
Define
Apply the Cauchy residues theorem to
on a series of nested contours CN
gives
Implications
• SV transfer matrix belongs to the Callier-Desoer class
of transfer functions
• Nyquist criterion provides a necessary and sufficient
condition for input-output stability
• Link with exponential stability using dissipativity
approach (see Litrico & Fromion, Automatica, 2009, in press)
Horizontal frictionless case
Residues aij(k)(x)
Residues aij(k)(x)
3
x 10
|a (k)(x)|
-5
11
k=0
k=1
k=2
k=3
2.5
2
1.5
1
0.5
0
0
500
1000
1500
abscissa (m)
2000
2500
3000
Coeff aij(k)(x), Non uniform case, canal 1
x 10
3
canal 1
k=0
-5
3.40.04
3
3.2
x 10
k=1
-5
accelerating
uniform
decelerating
2.5
3
0.03
2.5
2
2.8
2.6
1.5
0.02
2.4
elevation
(m)
imag
2.2
1
2
0.5
20.01
1.8
0
500
1.5
1000
1500
2000
2500
0
0
3000
500
1000
1500
2000
2500
3000
0
3
x 10
k=2
-5
3
-0.01
1
2.5
x 10
k=3
-5
2.5
2
-0.02
accelerating
uniform
decelerating
2
0.5
1.5
1.5
-0.03
1
0.5
1
0
0
-0.04
0 -2.5
0
500
1000
-2
500
1000 1500 2000
abscissa (m)
2500
1500 0.5 2000
abscissa
(m)0 -1
-1.5
3000
real
0
500
2500
3000
-0.5
1000 1500 2000
abscissa (m)
0
2500
-3
3000
x 10
Bode plot: approximations with different numbers
of poles
-10
-15
-15
-20
-20
Gain (dB)
-10
-25
-30
-25
-30
-35
-35
-40
-40
-45 -4
10
Phase (deg)
p 22
-5
10
-3
10
-2
10
-45 -4
10
-1
20
300
0
250
-20
200
Phase (deg)
Gain (dB)
p 21
-5
-40
-60
-80
-100 -4
10
10
-3
10
-2
10
-1
150
100
50
10
-3
10
Freq. (rad/s)
-2
10
-1
0 -4
10
10
-3
10
Freq. (rad/s)
-2
10
-1
Time response
• Rational approximations
• Unit step response
Step response (horizontal frictionless case)
y11
y12
0.08
0
0.06
-0.02
0.04
-0.04
0.02
-0.06
0
-0.08
-0.02
0
500
1000
1500
time (s)
2000
2500
3000
-0.1
0
500
1000
y21
1500
time (s)
2000
2500
3000
2000
2500
3000
y22
0.1
0
-0.01
0.08
-0.02
-0.03
0.06
-0.04
0.04
-0.05
-0.06
0.02
-0.07
0
0
500
1000
1500
time (s)
2000
2500
3000
-0.08
0
500
1000
1500
time (s)
Step response (uniform flow)
p 11
p 12
0.08
0
0.06
-0.01
0.04
-0.02
0.02
-0.03
0
-0.04
-0.02
0
500
1000
1500
time (s)
2000
2500
3000
-0.05
0
500
1000
p 21
1500
time (s)
2000
2500
3000
2000
2500
3000
p 22
0.1
0
-0.02
0.08
-0.04
0.06
-0.06
0.04
-0.08
0.02
0
0
-0.1
500
1000
1500
time (s)
2000
2500
3000
-0.12
0
500
1000
1500
time (s)
Spatial Bode plot (Uniform flow case, canal 2)
Coeff aij(k)(x), Non uniform case, canal 2
canal 2
8 x 10
k=0
-4
8
0.015
x 10
k=1
-4
accelerating
accelerating
uniform
uniform
decelerating
decelerating
7
7
6
2
5
0.01
6
4
3
elevation (m)
1
2
5
imag
0.005
1
0
0
1000
4
2000
3000
4000
5000
6000
0
0
1000
2000
3000
4000
5000
6000
0
3 x 10
k=2
-4
1.6
8
1.4
-0.005
7
1.2
2
6
1
5
0.8
-0.01
4
1
0.6
3
0.4
2
0
0
0
0.2
-0.015
-7
0
1000
-6
1000
-5
2000
2000 3000 4000
abscissa (m)
1
-5
k=3
-3
-2
-1
3000
4000
5000 -30 6000
0
real
6000
0
1000 2000 3000 4000x 105000 6000
abscissa
(m)
abscissa (m)
-4
5000
x 10
Applications
• Preissmann scheme = Classical numerical scheme
used to solve the equations
• We study the scheme on the linearized equations
• We relate the discretized poles to the continuous ones
• Poles location as a function of Dt , Dx and q
• Root locus with a downstream controller
Preissmann scheme
Study of the discretized system
• The linearized SV equations discretized with this scheme give
Assuming
non trivial solution is
, the condition for the existence of a
with
Study of the discretized system (cont’d)
or
with
And finally
This equation is formally identical to the one obtained in the
continuous time case!!!
One may show that the poles can be computed in two steps:
• first compute the continuous time poles obtained due to the
spatial discretization
• then compute the discrete time poles
Example: effect of spatial discretization
• Horizontal frictionless case
0.18
0.16
0.14
imag
0.12
0.1
0.08
0.06
0.04
0.02
0
1
2
3
4
5
mode
6
7
8
9
Effect of parameter theta
0.05
0.04
0.03
0.02
imag
0.01
q=0.1
q=0.5
q=0.9
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.03
-0.02
-0.01
0
real
0.01
0.02
0.03
Bode plot of discretized system
Bode plot canal 1
20
gain (dB)
0
-20
-40
-50
-5
10
10
-4
-3
10
freq.(rad/s)
10
-2
10
-1
phase (dg)
0
-500
-1000
-1500
-5
10
10
-4
-3
10
freq.(rad/s)
10
-2
10
-1
A link between frequency domain and Riemann
invariants methods
• For horizontal frictionless canals, Riemann invariants
and frequency domain methods lead to the same
result:
• Open-channel flow can be represented by a delay system
• How to extend this to the case of nonzero slope and
friction?
• For nonzero slope and friction, Riemann coordinates
are no longer invariants!
• But frequency domain methods enable to diagonalize
the system…
Riemann invariants (horizontal frictionless case)
SV equations
Diagonalize matrix A
Laplace transform
This is a delay system!
Uniform flow case (with slope and friction)
SV equations
Laplace transform + diagonalize matrix A-1(sI+B):
New variables:
Uniform flow case (cont’d)
Solution in the Laplace domain:
« generalized » delay system
We have:
and
with
Solution in the time domain
Time evolution of generalized characteristics
Change of variables
with
Solution in the time domain (cont’d)
Change of variables
Inverse Laplace transform
Solution in the time domain
Inverse transform (time)
Application: motion planning
We want to find the controls steering the system from 0
to a desired state in a given time Tr. The evolution equation leads to:
Feedback control
Controller
or
with
System
Closed-loop system
Sufficient stability condition
Control of SV oscillating modes: root locus
Control of SV oscillating modes
Conclusion
• Analysis of linearized Saint-Venant equations
• Poles and spectral decomposition
• Analytical results in horizontal frictionless and uniform cases
• Numerical method in non uniform cases
•
•
•
•
Rational models
Complete characterization of the flow dynamics
Analysis of Preissmann discretization scheme
Generalized characteristics (using Bessel functions)
• More details and applications in the book « Modeling
and control of hydrosystems », Springer, to appear in
2009.