1. Spectral Decomposition of Open Channel Flow Xavier Litrico (Cemagref, UMR G-EAU, Montpellier) with Vincent Fromion (INRA MIG, Jouy-en-Josas)

2. 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

3. 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

4. Different views of irrigation canals

5. 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

6. Main irrigation canal Series of canal pools We consider a single pool of length L

7. Modeling of open-channel flow Saint-Venant equations Mass conservation Momentum conservation Initial condition Boundary conditions Friction slope:

8. Linearized Saint-Venant equations Linearized around (non uniform) steady flow

9. Frequency response Laplace transform leads to a distributed transfer matrix: Poles p k in the horizontal frictionless case

10. Spatial Bode plot (horizontal frictionless case)

11. Uniform flow case Poles

12. Spatial Bode plot (Uniform flow case)

13. 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!

14. Main result: spectral decomposition The elements g ij (x,s) of the distributed transfer matrix G(x,s) can be decomposed as follows: a ij (k) (x) is the residue of g ij (x,s) at the pole p k

15. Spatio-temporal representation of g ij (x,s)

16. Sketch of proof Define gives Apply the Cauchy residues theorem to on a series of nested contours C N

17. 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)

18. Horizontal frictionless case Residues a ij (k) (x)

19. 050010001500200025003000 0 0.5 1 1.5 2 2.5 3 x 10 -5 abscissa (m) |a 11 (k) (x)| k=0 k=1 k=2 k=3

20. Coeff a ij (k) (x), Non uniform case, canal 1

21. Bode plot: approximations with different numbers of poles 10 -4 10 -3 10 -2 10 -45 -40 -35 -30 -25 -20 -15 -10 -5 Gain (dB) p 21 10 -4 10 -3 10 -2 10 -100 -80 -60 -40 -20 0 20 Freq. (rad/s) Phase (deg) 10 -4 10 -3 10 -2 10 -45 -40 -35 -30 -25 -20 -15 -10 -5 Gain (dB) p 22 10 -4 10 -3 10 -2 10 0 50 100 150 200 250 300 Freq. (rad/s) Phase (deg)

22. Time response Rational approximations Unit step response

23. Step response (horizontal frictionless case) 050010001500200025003000 -0.02 0 0.02 0.04 0.06 0.08 time (s) y 11 050010001500200025003000 0 0.02 0.04 0.06 0.08 0.1 time (s) y 21 050010001500200025003000 -0.1 -0.08 -0.06 -0.04 -0.02 0 time (s) y 12 050010001500200025003000 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 time (s) y 22

24. Step response (uniform flow) 050010001500200025003000 -0.02 0 0.02 0.04 0.06 0.08 time (s) p 11 050010001500200025003000 0 0.02 0.04 0.06 0.08 0.1 time (s) p 21 050010001500200025003000 -0.05 -0.04 -0.03 -0.02 -0.01 0 time (s) p 12 050010001500200025003000 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 time (s) p 22

25. Spatial Bode plot (Uniform flow case, canal 2)

26. Coeff a ij (k) (x), Non uniform case, canal 2

27. 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

28. Preissmann scheme

29. 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

30. 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

31. Example: effect of spatial discretization Horizontal frictionless case

32. Effect of parameter theta

33. Bode plot of discretized system

34. 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…

35. Riemann invariants (horizontal frictionless case) SV equations Diagonalize matrix A Laplace transform This is a delay system!

36. Uniform flow case (with slope and friction) Laplace transform + diagonalize matrix A -1 (sI+B): SV equations New variables:

37. Uniform flow case (cont’d) Solution in the Laplace domain: « generalized » delay system We have: with and

38. Solution in the time domain Time evolution of generalized characteristics with Change of variables

39. Solution in the time domain (cont’d) Change of variables Inverse Laplace transform

40. Solution in the time domain Inverse transform (time)

41. Application: motion planning We want to find the controls steering the system from 0 to a desired state in a given time T r. The evolution equation leads to:

42. Feedback control Controller or with System Closed-loop system Sufficient stability condition

43. Control of SV oscillating modes: root locus

44. Control of SV oscillating modes

45. 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.