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 Saint-Venant 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

• Initial condition

• Boundary conditions

Friction slope:

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

-3 -2 -1 0 1 2 3

x 10-3

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

real

imag

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

gives

Apply the Cauchy residues theorem to on a series of nested contours CN

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)

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3x 10

-5

abscissa (m)

|a11(k)(x)|

k=0

k=1k=2

k=3

Coeff aij(k)(x), Non uniform case, canal 1

0 500 1000 1500 2000 2500 30001.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4x 10

-5 k=0

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3x 10

-5 k=1

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3x 10

-5

abscissa (m)

k=2

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3x 10

-5 k=3

abscissa (m)

-2.5 -2 -1.5 -1 -0.5 0

x 10-3

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

real

imag

acceleratinguniformdecelerating

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3canal 1

abscissa (m)

elev

atio

n (m

)

acceleratinguniformdecelerating

Bode plot: approximations with different numbers of poles

10-4

10-3

10-2

10-1

-45

-40

-35

-30

-25

-20

-15

-10

-5G

ain

(dB

)

p21

10-4

10-3

10-2

10-1

-100

-80

-60

-40

-20

0

20

Freq. (rad/s)

Pha

se (

deg)

10-4

10-3

10-2

10-1

-45

-40

-35

-30

-25

-20

-15

-10

-5

Gai

n (d

B)

p22

10-4

10-3

10-2

10-1

0

50

100

150

200

250

300

Freq. (rad/s)

Pha

se (

deg)

Time response

• Rational approximations

• Unit step response

Step response (horizontal frictionless case)

0 500 1000 1500 2000 2500 3000-0.02

0

0.02

0.04

0.06

0.08

time (s)

y11

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

time (s)

y21

0 500 1000 1500 2000 2500 3000-0.1

-0.08

-0.06

-0.04

-0.02

0

time (s)

y12

0 500 1000 1500 2000 2500 3000-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

time (s)

y22

Step response (uniform flow)

0 500 1000 1500 2000 2500 3000-0.02

0

0.02

0.04

0.06

0.08

time (s)

p11

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

time (s)

p21

0 500 1000 1500 2000 2500 3000-0.05

-0.04

-0.03

-0.02

-0.01

0

time (s)

p12

0 500 1000 1500 2000 2500 3000-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

time (s)

p22

Spatial Bode plot (Uniform flow case, canal 2)

Coeff aij(k)(x), Non uniform case, canal 2

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

8canal 2

abscissa (m)

elev

atio

n (m

)

acceleratinguniformdecelerating

0 1000 2000 3000 4000 5000 60000

1

2

x 10-4 k=0

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

8x 10

-4 k=1

0 1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

-4

abscissa (m)

k=2

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

8x 10

-5 k=3

abscissa (m)

-7 -6 -5 -4 -3 -2 -1 0

x 10-3

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

real

imag

acceleratinguniformdecelerating

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 t , x and

• Root locus with a downstream controller

Preissmann scheme

Study of the discretized system

• The linearized SV equations discretized with this scheme give

Assuming , the condition for the existence of a non trivial solution is

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

1 2 3 4 5 6 7 8 90

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

mode

ima

g

Effect of parameter theta

-0.03 -0.02 -0.01 0 0.01 0.02 0.03-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

real

ima

g

=0.1=0.5=0.9

Bode plot of discretized system

10-5

10-4

10-3

10-2

10-1

-50

-40

-20

0

20Bode plot canal 1

freq.(rad/s)

ga

in (

dB

)

10-5

10-4

10-3

10-2

10-1

-1500

-1000

-500

0

freq.(rad/s)

ph

ase

(d

g)

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)

Laplace transform + diagonalize matrix A-1(sI+B):

SV equations

New variables:

Uniform flow case (cont’d)

Solution in the Laplace domain:

« generalized » delay system

We have:

with

and

Solution in the time domain

Time evolution of generalized characteristics

with

Change of variables

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.