Spectral, input-output and energetic properties of the reduced Hamiltonian formulation for the Shallow Water equations Boussad Hamroun *, Laurent Lef`evre ** and Eduardo Mendes * * LCIS - Laboratoire de conception et d’int´egration des syst`emes Grenoble INP, France ** LAGEP - Laboratoire d’Automatique et de GEnie des Proc´ed´es Universit´e Claude Bernard - Lyon 1, Lyon, France {boussad.hamroun, eduardo.mendes}@lcis.grenoble-inp.fr
[email protected]
Keywords: Shallow water equations, Port-Controlled Hamiltonian models, mixed finite elements method, irrational transfer functions, spectrum approximation. Abstract We present in this paper some important properties of a reduced port-controlled hamiltonian (PCH) model for the shallow water equations also called Saint-Venant equations used to model water dynamics in open channels and rivers. The convergence of the reduced finite dimensional spectrum to the original infinite dimensional one is established. The comparison of the reduced model transfer functions with the infinite dimensional ones exhibits the convergence of the reduced transfer functions to the infinite dimensional ones with respect to the resolution of the reduction scheme.
1
Introduction
We have proposed in [5] a reduced port hamiltonian model for the Saint-venant equations derived from its hamiltonian formulation, using a mixed finite element method based geometric reduction scheme. We aim in this paper at giving some important properties of this reduced port hamiltonian model which could be used for simulation or control purposes. This study allows to compare the performance of our reduction method (beyond the geometric interconnection structure and energetic properties conservation) with the many other existing reduction methods. We will first focus on the
spectral properties of the obtained reduced model. It will be shown that nice spectral properties for the reduced model result from the central features of our reduction method which is the reduction of the Stokes-Dirac interconnection structure. A proof of the convergence of the spectrum of the reduced geometric interconnection structure obtained by the reduction scheme to the infinite-dimensional one will be given. In the second part the input output properties of the reduced model are investigated. Irrational input-output transfer functions of the linearized infinite dimension model around a uniform equilibrium profile are derived. Then the reduced PCH model is linearized around the same equilibrium profile and its corresponding transfer functions are computed. Both transfer functions are compared using Bode diagrams.
2
A Reduced Port hamiltonian model for the shallow water equations
We consider a rectangular open channel with a single reach of slope I, with length L and width B. It is delimited by upstream and downstream gates and terminated by an hydraulic outfall. The flow dynamic within the reach is modelled by the well known shallow water equations. Its port-Hamiltonian model has been developed in [4] by choosing the elementary volume and kinetic momentum density along the spatial domain Z = [0, L] as energy (state) variables. The hydrodynamic pressure and fluid flow are derived as distributed efforts from the total energy of the flowing fluid. The reduced port-controlled Hamiltonian model has been developed in [5] using a mixed finite element method. It is based on a subdivision of the total length of the open channel into n elementary cells. The whole model is obtained by series interconnections of the adjacent cells and may be written in the following explicit form : ∂H ∂q ∂H ∂p
q˙ p˙
=
0 −M T
+ gu y1 y2
=
gu
T
−
0 0
0 G(q, p)
(1)
∂H ∂q ∂H ∂p
u1 u2
M 0
(2)
where q = [q1 . . . qn ]T and p = [p1 . . . pn ]T are, respectively, the state vector of cell volumes and kinetic momentums. The global interconnection structure sub-matrix
M (∈ 0
(4)
where K is the Manning-Strickler dissipation parameter. The total energy of the system is given as the sum of the individual energies of each cells: H(q, p) =
n X 1 qi (t)2 1 qi (t)pi (t)2 ( + − ρgIKab qi (t)) 2 Cab 2 Lab i=1
(5)
with the following numerical values for the reduced elements and reduced efforts: Cab =
b+a B (b − a), Lab = ρ(b − a)2 , Kab = ρg 2
∂H qi p2 (q, p) = + i − ρgIKab , ∂qi Cab 2Lab
3
∂H qi pi (q, p) = ∂pi Lab
(6) (7)
Spectral properties of the reduced port-Hamiltonian model
In this section we study the spectral properties for the reduced port-Hamiltonian model. The distributed parameters port-hamiltonian model is characterized by a geometric interconnection Stokes-Dirac structure (see [1]). This structure is reduced using a geometric method which leads to a finite-dimensional reduced Dirac structure. We study here the spectrum convergence of this reduced Dirac structure to the spectrum of the original Stokes-Dirac structure.
3.1
Eigenvalues of the Stokes-Dirac structure
First, we need to compute the eigenvalues of the infinite-dimensional Stokes-Dirac structure. It has been shown in [4] that an equivalent Hamiltonian formulation for our
shallow water problem may be givent as ∂ q(x, t) 0 d eq (x, t) eq (0, t) u1 (t) = and = (8) p(x, t) d 0 ep (x, t) ep (L, t) u2 (t) ∂t where q and p denotes respectively elementary volume and momentum densities (differential 1-forms), eq and ep are respectively the hydrodynamic pressure and water volumetric flow (functions) and d denotes the spatial exterior derivative. This Hamiltonian formulation exhibits a canonical inter-domain interconnection structure whose corresponding eigenvalues are defined as the solution of d ψq (x) ψq (x) 0 dx ψq (0) 0 = λ and = (9) d ψp (x) ψp (x) ψp (L) 0 0 dx Short calculations lead to imaginary pairs of eigenvalues λk =
3.2
2k + 1 πi 2L
∀k ∈ Z
(10)
Eigenvalues of the reduced Dirac structure
The reduced interconnection structure used in the reduced defined by the skew-symmetric matrix: −1 0 1 −1 0 M T .. J = −J = where M = 0 . −M T 0 . . . . . . 0 ...
PCH formulation (1) is ... .. . .. . .. . 0
... .. ..
.
. 1
0 .. . .. .
(11) 0 −1
The matrix J being skew symmetric, its eigenvalues are purely imaginary pairs λ satisfying det(λ2 I + M T M ) = 0. Let σ (λ2 = −σ 2 ) be the singular values of M . Since AM = M > M has the simple tri-diagonal form −2 1 0 0 0 0 1 −2 1 0 0 0 0 1 −2 1 0 0 . .. .. .. .. .. AM = (12) .. . . . . . .. .. 0 . . 1 −2 1 0 0 ... 0 1 −1 the characteristic polynomial Pn (σ 2 ) = det(σ 2 I − AM ) satisfies the 3-terms recurrence relation PN = (σ 2 + 2)PN −1 − PN −2 , ∀N ∈ {1, . . . , n} with P0 = 1 and
P1 = 1 + σ 2 . This recurrence relation defines a scaled n-th. order Chebyshev polynomial whose roots are σk2 = 2 cos(
2k + 1 π) − 2 2n + 1
∀k ∈ {0, . . . , (n − 1)}
(13)
For large n σk2 = −(
2k + 1 2 π) + o 2n + 1
1 n3
(14)
The eigenvalues associated with the initial problem are then: n 2k + 1 n 2k + 1 1 1 1 =± λk = ± πi + o πi 1 − +o , ∀k ∈ Z 2 L L 2n + 1 n 2L 2n n (15) Thus the reduced Dirac structure spectrum converge to the infinite-dimensional one. Figures 1 and 2 show some numerical results on this convergence. N=5
N=10
N=20
N=50
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
0
0
0
0
−1
−1
−1
−1
−2
−2
−2
−2
−3
−3
−3
−3
−4
−4
−4
−4
0.35 First structure pole relative error Second structure pole relative error Third structure pole relative error
0.3
0.25
0.2
0.15
0.1
0.05
0
−5 −1 0
10
20
30
40 50 60 Number of cells
70
80
90
4.1
0
−5 1 −1
0
1
−5 −1
0
100
Figure 1: Relative error of the three first eigenvalues
4
0
−5 1 −1
Figure 2: ”+” Spectrum of the reduced Dirac structure, ”o” Spectrum of Stokes-Dirac structure
Input-Output Properties Infinite dimensional transfer functions:
The Saint-Venant equations may be linearized around an uniform equilibrium profile if the equilibrium water flow Qe and water level he are chosen such that friction forces compensate exactly the gravity acceleration due to the canal slope (see [3], otherwise linearization around a non uniform water level equilibrium profile has to be
1
performed). The obtained linearized model, expressed in water level h(x, t) and water flow Q(x, t) is # " 1 ∂ ∂ h(x, t) h(x, t) 0 0 h(x, t) 0 = + (16) B Q(x, t) Q(x, t) a3 −a4 Q(x, t) ∂t a1 −a2 ∂x Q2
with a1 = ( Bhe2 − gBhe ), a2 = e
2Qe Bhe ,
Bhe e ,he ) a3 = gB 4J(Q ( B+2h ) and a4 = 9he e
2gBhe J(Qe ,he ) . Qe
In the Laplace domain, equations (16) tansform into a system of linear differential equations in the Laplace transforms h(x, s) and Q(x, s) whose general solution is λ1 (s)α λ1 (s)x λ2 (s)β λ2 (s)x e e + h(x, s) Bs = Bs (17) Q(x, s) λ1 (s)x λ2 (s)x α e +β e 2 Bs) 2 Bs) + r1 , λ2 = − (a3 +a + r2 and where β, α are arbitrary with λ1 = − (a3 +a a1 a1 integration constants. The linearized power conjugate port variables are the fluid flow and the hydrodynamic pressure. The hydrodynamic pressure is defined as a nonlinear function of fluid flow and level. We thus linearize this hydrodynamic pressure around the same uniform equilibrium profile (ve , he ) and obtain
Q(x, s) P (x, s)
=
=
with c1 = ρ g −
ve2 he
1 h(x, s) ρve Q(x, s) Bhe
0 2 v ρ g− e he
α eλ1 x + β eλ2 x
(18)
(19) c1 λ1 c λ 1 2 λ1 x λ2 x α − + c2 e + c2 e +β − Bs Bs
and c2 =
ρve Bhe .
We select now the input and output variables
used to define the reduced port hamiltonian model (1) u1 (s) = Q(x = 0, s), u2 (s) = P (x = L, s), y2 (s) = P (x = 0, s) and y1 (s) = Q(x = L, s) to obtain the following transfer functions y1 (s) G11 (s) G12 (s) u1 (s) = (20) y2 (s) G21 (s) G22 (s) u2 (s) " # (d2 −d1 )e(λ1 +λ2 )L −eλ1 L +eλ2 L u1 (s) λ2 L −d eλ1 L λ2 L −d eλ1 L d e d e 2 1 2 1 = λ2 L λ1 L d1 d2 (e −e ) −d1 +d2 u2 (s) λ L λ L λ L λ L d2 e
2
−d1 e
1
d2 e
1 λ1 1 λ2 with d1 = − cBs + c2 and d2 = − cBs + c2 .
2
−d1 e
1
4.2
Transfer functions of linearized reduced Port-Hamiltoian model
The linear PCH model is obtained by linearizing the vector of efforts around the uniform equilibrium point xe corresponding to uniform equilibrium profiles qe and he . This guarantees that the linearized model remains in PCH form. One obtains x˙ y
=
[J(xe ) − R(xe )]M (xe )x + gu (xe )u = Ax + Bu
(21)
=
guT (xe )M (xe )x
(22)
= Cx
The reduced transfer functions of this model are now compared with the infinite dimensional ones. We give in figures 3,4,5 and 6 the bode diagrams of G12 and G21 which are transfers between spatially opposite port variables. They are characterized by some delay between input and output. As we can see, the transfer functions of the reduced model recover the infinite dimensional ones (gain and phase) with respect to the resolution of the reduction scheme. Gain: Upstream flow to Downstream flow
Gain: Downstream Pressure to Upstream Pressure 50
0
0
−50
−50
−100
Magnitude (dB)
Magnitude (dB)
−150 −100
N=2
−200
N=2 Infinite Dimension Transfer
−250
N=10 N=20
Infinite Dimension Transfer
−150
N=10 −300
N=20
−350 −200 −400
−2
10
−1
10
Frequency (rad/sec)
Figure 3:
5
0
10
1
10
−450 −2 10
−1
10
0
10 Frequency (rad/sec)
1
10
2
10
Figure 4:
Conclusion
We have demonstrated in this paper important spectral and input-output properties of the reduced port hamiltonian model for the Saint-Venant equations. A proof for the convergence of the reduced Dirac structure spectrum to the linearized infinite dimensional Stokes-Dirac structure spectrum. It implies the convergence for the dynamical model since closure relations are algebraic constitutive equations (linearized in the linear case) thus trivially bounded. We have shown in the second part that, in addition of the structural properties preservations (conserved energy, power product and resulting interconnection subspace) the reduced PCH model has an interesting inputoutput spectral accuracy. This has been shown by observing that the reduced transfer functions recover the infinite dimensional ones on a large spectrum range.
Phase: Upstream flow to Downstream flow
Phase: Downstream Pressure to Upstream Pressure
500
0
0
−1000
−500
−2000
−1000
−3000
N=2 Infinite Dimension Transfer N=10 N=20
−1500
−4000
−2000
−5000
−2500
−6000
−3000 −3 10
−2
−1
10
10
Figure 5:
0
10
1
10
−7000 −3 10
N=2 Infinite Dimension Transfer N=10 N=10
−2
10
−1
10
0
10
1
10
Figure 6:
References [1] Van der Schaft.A and Maschke.B, Hamiltonian formulation of distributedparameter systems with boundary energy flow, Journal of Geometry and Physics, vol. 42, 2002, pp 166-194. [2] Litrico.X and Fromion.V, Frequency modeling of open-channel flow, Journal of Hydraulic Engineering, Vol. 130, 2004, No. 8, pp. 806-815 [3] Ouarit.H, Lef`evre.L and Georges.D, Robust Optimal Control of one-reach openchannels, in Proc. of European Control Conference ECC’2003, Cambridge, United Kingdom, 2003. [4] Hamroun.B, Lef`evre.L and Mendes.E, Port-Based Modelling for Open Channel Irrigation Systems WSEAS Transactions on Fluid Mechnaics 1,(2006) 995– 1008. [5] Hamroun.B, Lef`evre.L and Mendes.E, Port-Based Modelling and Geometric Reduction for Open Channel Irrigation Systems Proceedings of the 46th Conference on Decision and Control CDC’07, New orleans,USA,(2007) 1578–1583.
