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Hamiltonian approach to the stabilization of systems of two conservation laws
8th IFAC Symposium on Nonlinear Control Systems University of Bologna, Italy, September 1-3, 2010
Hamiltonian approach to the stabilization of systems of two conservation laws V. Dos Santos Martins ∗ B. Maschke ∗ Y. Le Gorrec ∗∗ ∗ LAGEP, Universit´e de Lyon, Lyon, F-69003, France ; Universit´e Lyon 1, CNRS, UMR 5007, LAGEP, Villeurbanne, F-69622, France, ESCPE, Villeurbanne, F-69622 (e-mail: [email protected]) ∗∗ FEMTO-ST/AS2M, ENSMM Besan¸con, 24 rue Alain Savary, 25 000 Besan¸con (e-mail: [email protected])
Abstract: This paper aims at providing some synthesis between two alternative representations of systems of two conservation laws and interprets different conditions on stabilizing boundary control laws. The first is the representation in Riemann invariants coordinates whose representation has been applied successfully for the stabilization of linear and non-linear of such hyperbolic systems. The second representation is based on physical modelling and leads to port Hamiltonian systems which are extensions of infinite-dimensional Hamiltonian systems defined on Dirac structure. The stability conditions on the boundary feedback relations derived with respect to the Riemann invariants are interpreted in terms of the dissipation inequality of the Hamiltonian functional. Keywords: Boundary Port Hamiltonian systems, Distributed parameters systems, Stability. 1. INTRODUCTION In this paper we shall be concerned with the stabilization via boundary control of hyperbolic systems of two conservation laws mainly motivated by the control of the shallow water equations used as models of irrigation channels, but also the p-system (Serre, 1996). The stabilization of irrigation channels using boundary control has been intensively studied for instance in (Dos Santos et al., 2008) for both linear and non linear cases as the stability on a one-dimensional spatial domain for such systems (Bastin et al., 2004). One of the most often suggested approaches, uses Riemann invariants to derive a stabilizing boundary control (Greenberg and Ta, 1984) and some extensions based on Lyapunov function (Bastin et al., 2004; Coron et al., 2007). The use of physically motivated control Lyapunov function for the derivation of stabilizing control laws for non-linear finite-dimensional systems has proven to be very efficient and has lead to a great variety of results (Brogliato et al., 2007; van der Schaft, 2000). Very often, when the system stems from physical modelling, one may derive dissipation inequalities related to energy balance equations and energy dissipating phenomena (Willems, 1972). Using the dissipative port-Hamiltonian formulation for controlled physical systems, one may go one step further and assign in closed loop the dynamic behavior by the structure matrices of the Hamiltonian system in closed loop (Maschke et al., 2002). For infinite-dimensional systems very similar techniques, based on dissipation inequalities, which in terms of PDE’s amounts to consider some entropy function (Coron et al., 2007), have been used for the stabilization of boundary control systems (Curtain and Zwart, 1995; Luo et al., 1999). Recent works have used a boundary port-Hamiltonian formulation of systems of conservation laws (Maschke and van der Schaft, 2005) in order to derive stabilizing boundary control for a class of 978-3-902661-80-7/10/$20.00 © 2010 IFAC
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linear systems defined on one-dimensional spatial domains (Le Gorrec et al., 2005; Macchelli and Melchiorri, 2004; Villegas et al., 2006). The aim of this paper is twofold. Firstly, after a brief recall of the formulation of a system of two conservation laws in terms of port-Hamiltonian systems defined with respect to the canonical Stokes-Dirac structure (Maschke and van der Schaft, 2005) in Section 2, it derives the port-Hamiltonian formulation of these systems in terms of Riemann coordinates in Section 3. Secondly, in Section 4 the conditions expressed on the boundary values of the Riemann coordinates used in (Greenberg and Ta, 1984) are related to some conditions expressed on the boundary port variables of the Hamiltonian formulation. Using the fact that the Hamiltonian functional satisfies a conservation equation, that makes it become a natural candidate Lyapunov function, these conditions are then interpreted in terms of dissipativity of the Hamiltonian function and allow an easy physical interpretation. Conclusions end the paper in Section 5. 2. MOTIVATION THROUGH THE STABILIZATION OF THE P-SYSTEM USING THE RIEMANN INVARIANTS 2.1 Reminder on the stabilization of system of two conser-vations laws In this section, we recall the main result on the stabilization of a hyperbolic system of two conservation laws suggested by Greenberg & Li (Greenberg and Ta, 1984). Consider a spatial domain consisting of the finite interval [0, L] ∋ x with L ∈ IR∗+ 1 and time domain being the real interval [0, +∞) ∋ t. The state space is a non-empty 1
The set of positive and non nul reals
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connected open set in IR2 , denoted by Ω. Consider the system of two conservations laws: ∂t Y + ∂x f (Y ) = 0, (1) where • Y = (y1 y2 )T : [0, +∞) × [0, L] → Ω is the vector of the two dependent variables; • f : Ω → IR2 is a C 1 -function called the flux vector. Note that the system (1) may also be written: ∂t Y + F (Y )∂x Y = 0 (2) where F is the Jacobian of the flux vector f . Assuming that the system is hyperbolic, implies that this system can be diagonalised using the Riemann invariants (see for instance (Lax, 1973, pages 34 - 35)). This means that there exists a change of coordinates ξ(Y ) whose Jacobian matrix is denoted D(Y ), ∂ξ ,Y ∈ Ω (3) ∂Y and diagonalises F (Y ) in Ω: D(Y )F (Y ) = Λ(Y )D(Y ). In the coordinates ξ, the system (1) can then be rewritten in the following (diagonal) characteristic form: ∂t ξ + Λ(ξ)∂x ξ = 0 (4) 2 with ξ: [0, L] × [0, +∞) → IR , (x, t) 7→ ξ(x, t), and Λ(ξ) = diag(λ1 (ξ), λ2 (ξ)), assuming that λ1 (ξ), λ2 (ξ) satisfy the conditions: D(Y ) =
• the λi ’s are continuously differentiable functions on a neighborhood of the origin; • λ2 (0) < 0 < λ1 (0). In this paper we shall consider the following result of Greenberg and Li (Greenberg and Ta, 1984) which may be recalled as follows. Theorem 1. Consider the hyperbolic system of conservation laws in Riemannian coordinates (4) with the following relations on the boundary variables: ξ1 (0, t) = K1 (ξ2 (0, t)), ξ2 (L, t) = K2 (ξ1 (L, t)) (5) with the functions K1 and K2 being C 1 and satisfying 2 : K1 (0) = K2 (0) = 0 and |K1 ′ (0)K2 ′ (0)| < 1. Consider initial values: lim (ξ1 , ξ2 )(x, t) = (ξ1,0 , ξ2,0 )(x), 0 < x < L, t→0+ 1
(6) (7)
being C and satisfying the assumption that to be small in the C 1 norm and the compatibility conditions: ξ1,0 (0) = K1 (ξ2,0 (0)) and ξ2,0 (L) = K2 (ξ1,0 (L)) (8) λ1 (ξ1,0 , ξ2,0 )(0) ∂x ξ1,0 (0) = λ2 (ξ1,0 , ξ2,0 )(0) K1 ′ (0) ∂x ξ2,0 (0)
(9)
λ2 (ξ1,0 , ξ2,0 )(L) ∂x ξ2,0 (L) = λ1 (ξ1,0 , ξ2,0 )(L) K2 ′ (L) ∂x ξ1,0 (L) (10) Then the initial value problem, for this system, has a unique C 1 solution. Moreover, its solution decays to zero in the C 1 norm with an exponential rate. This result has been generalized to a greater class of non linear systems in infinite dimension, for n eigenvalues (Coron et al., 2007). 2
The prime is for the derivative of the function.
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2.2 Application to the p-system In their paper (Greenberg and Ta, 1984), the authors consider the p-system defined by the following system of conservation laws: u v ∂t − ∂x =0 (11) v σ(u)
where σ is a C 1 function satisfying: σ(0) = 0, σ ′ (u) > 0. (12) They consider the following relations on the boundary values: σ(u(0, t)) − rv(0, t) = 0, v(L, t) = 0 (13) with r ∈ IR∗+ . In (Greenberg and Ta, 1984), the authors construct a unique smooth solution using the properties of invariance of the Riemann coordinates along the Riemann forward characteristic ξ˙ = λ1 and the backward one ξ˙ = λ2 , where λi are the eigenvalues of the p-system. However, Greenberg and Li have made the remark that for the p-system example, the stability condition looks like the dissipativity condition of the energy of this system. Motivated by this remark, the following sections deal with the interpretation of the stabilization conditions in terms of dissipativity and Hamiltonian Structure. 2.3 Link with Hamiltonian Structure It is remarkable that, for this classical physical example, the authors use the physical variables (u being the volume, v the velocity of some isentropic gas for instance and σ denoting the opposite of the pressure) in order to express the boundary relations (13) and not the Riemann invariants. Furthermore they note that the condition on the parameter r in the boundary relations may be interpreted in terms of dissipativity properties. Indeed consider the energy of the system (sum of the kinetic and the internal energy of the gas): Z L 2 Z u v H(u, v) = σ(η) dη dx (14) + 2 0 0 and its balance equation: dH = v(t, L) σ(u(t, L)) − v(t, 0) σ(u(t, 0)). (15) dt The boundary relations (13) and the positivity of r imply then energy dissipation: dt H = −rv 2 (0, t) ≤ 0. (16) The relation between energy dissipation and the inequality (6) on the boundary relation of the theorem 1, conditions may be expressed as follows. Using the definition of the Riemann invariants and that σ(u) is a strictly increasing function, it follows that there exists a C 1 function u ¯ such that the inverse coordinate transformation may be written: u =u ¯(ξ2 − ξ1 ) (17) v = ξ2 + ξ1 Using the coordinate transformation and the implicit function theorem, there is a function K1 such that the boundary relations on the physical variables are translated as follows onthe Riemann invariants: ξ1 (0) = K1 (ξ2 (0)) (18) ξ2 (L) = K2 (ξ1 (L)) = −ξ1 (L)
NOLCOS 2010 Bologna, Italy, September 1-3, 2010
The inequality conditions (6) of the theorem 1 reduce in this case to: K1 (0) = 0 and |K1′ (0)| < 1. (19) Denoting the celerity of the system in Riemann coordinates by λ(ξ2 − ξ1 ) = λ1 (ξ1 , ξ2 ) = −λ2 (ξ1 , ξ2 ) > 0, a straightforward calculation on the boundary relations at x = 0, leads to the following relation: λ(ξ2 − ξ1 ) − r ′ . |K1 (0)| = (20) λ(ξ2 − ξ1 ) + r
This is clearly a Cayley transformation and hence the positivity of r which implies the energy dissipation, is equivalent with the circle condition (19) given by the theorem 1. This raised for us the question wether such an equivalence holds for some physical systems, in particular systems stemming from physical systems models and their Hamiltonian formulation (Maschke and van der Schaft, 2005).
Hence in terms of the Riemann invariants the system is written: ˜ −∂t ξ = (B∂x + C)δξ H(ξ), (22) 0 1 0 1 where B = Dξ DξT and C = Dξ ∂x (DξT ). 1 0 10 The following properties may be noted: firstly the matrix B is symmetric and secondly it is related with the matrix C by: ∂x B = C T + C. (23) 3.1.2. Boundary port variables: In this section we shall check the formal skew-symmetry of the differential operator (B∂x + C) and then define port boundary variables associated with it. Let us define the following bracket on smooth functions on the spatial domain [0, L]: {e1 , e2 } =
Z
L
eT1 (B∂x + C)e2 dx
(24)
0
and consider the symmetric product:
3. PORT HAMILTONIAN FORMULATION OF A HYPERBOLIC SYSTEM OF TWO CONSERVATION LAWS
{e1 , e2 } + {e2 , e1 } =
Boundary port Hamiltonian systems defined in the distributed parametrers framework in (Maschke and van der Schaft, 2005) allow the description of physical system which exchange a non-zero energy flow through its boundaries. 3.1 Boundary port Hamiltonian systems and Riemann coordinates 3.1.1. Hamiltonian operator expressed in the Riemann coordinates: In this section we shall consider a hyperbolic system of two conservation laws (1) which admits a Hamiltonian representation, that is such that the system is written: 0 ∂x −∂t Y = (δY H), (21) ∂x 0 ∂t Y and δY H belong to a Dirac structure. In the sequel we express the Hamiltonian system in terms of the Riemannian invariants and give the expression of the Hamiltonian operator as well as of the boundary port variables. Denote ˜ by H(ξ) the Hamiltonian expressed in the Riemann invari˜ ants: H(ξ) = H ◦ Y (ξ) where Y denotes, with an abuse of notation, the inverse change of coordinates to the Riemann coordinates. Recall that D denotes the Jacobian of change of coordinates to the Riemann invariants defined in (3) and define Dξ = D ◦ Y (ξ). One obtains by multiplying both terms of (21) by D: 0 1 ∂x (δY H) −D(Y )∂t Y = D(Y ) 10 0 1 ˜ ∂x (DξT )δξ H(ξ) ⇔ −∂t ξ = Dξ 10 0 1 ˜ DξT ∂x (δξ H(ξ)). +Dξ 1 0
+eT2 Ce1 +
Z
L
0 eT1
∂x (eT1 Be2 ) − eT1 ∂x Be2 + eT2 ∂x Be2 C + C T − ∂x B e2 dx.
By using the fact that B is symmetric and the relation (23), the symmetric product reduces to: {e1 , e2 } + {e2 , e1 } =
Z
0
L
L ∂x (eT1 Be2 ) = (eT1 Be2 ) 0 .(25)
The product (25) corresponds to Stokes theorem applied to the equation for the differential operator (B∂x + C). Furthermore the second member of (25) vanishes for all functions e1 , e2 with compact support strictly included in the domain [0, L] and hence for these functions the bracket is skew-symmetric. One can also prove that Jacobi identities are satisfied. Then in this case the system (22) defines an infinite dimensional Hamiltonian system. However considering functions which do not vanish on the boundary of the domain, the bracket (24) is no more skew-symmetric. In this case the time variation of the Hamiltonian becomes: iL h ˜ dH(ξ) ˜ ˜ T B δξ H) . (26) = (δξ1 H 2 dt 0 Thus one may account for the energy flow through the boundary of the domain, however the structure of an infinite-dimensional Hamiltonian system is lost. Using the notion of Dirac structure one may still define a Hamiltonian system by extending the system with external variables associated with the boundary. Naturally the choice is not unique. The most natural choice related to the balance equation (26) consists in choosing the following pair of variables: δ H(0) ˜ ξ1 Dξ (0) 0 0 0 ˜ 0 w1 0 Dξ (L) 0 δξ1 H(L) = ˜ 0 0 Dξ (0) 0 w2 −δξ2 H(0) 0 0 0 Dξ (L) ˜ δξ2 H(L) (27)
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then the balance equation (26) is written in form of a dissipation equality (Willems, 1972): ˜ dH(ξ) Σ2 0 T = w1 w2 (28) 0 Σ2 dt
where B(0) = B(ξ(0)) and B(L) = B(ξ(L)). Remark 1. Following the definition of Dirac structure (Courant and Weinstein, 1988) and adapting the proofs in (Le Gorrec et al., 2005), one may prove that the differential operator (B∂x + C)combined with (27) generates a canonical Dirac structure. 3.2 Application to the shallow water equations We consider the special case of a reach of an open channel delimited by two underflow gates as represented in Figure 1.
general case (with slope and friction) in (Hamroun et al., 2006), see also (Pasumarthy and van der Schaft, 2006). The energy of the water flow in the channel is defined in terms of two state variables: • the momentum p(x, t) = ρv(x, t) where ρ the mass density of the water (constant as the water is assumed to be incompressible), • the section area of the water q(x, t) = W h(x, t). In the sequel we use the state vector defined by Y = T (q p) . The total energy of the systems (sum of kinetic and potential energies) is given by: H(Y ) =
1 2
Z
0
L
ρg 2 1 2 q + qp dx, W ρ
where g denotes the constant of gravity. The variational derivatives of the Hamiltonian functional define the two co-energy variables: ep the volumic flow and eq the hydrodynamic pressure as follows: p2 ρg 1 + q = ρv 2 + ρgh , 2ρ W 2 qp ep = δp H(Y ) = (= W hv) . ρ eq = δq H(Y ) =
Fig. 1. A reach of an open channel delimited by two adjustable underflow gates We assume that the channel is horizontal, prismatic with a constant rectangular section and a width, and that the friction effects are neglected. The flow in the canal may be described by the so-called shallow water equations or Saint-Venant equations which constitute a system of two conservation laws, actually the mass balance and the momentum balance equations: ∂t h + ∂x (hv) = 0, (29) 1 (30) ∂t v + ∂x ( v 2 + gh) = 0, 2 where v denotes the velocity of the water flow and the water level h. Each underflow gates imposes a boundary condition of the form: W h(0, t)v(0, t) = u0 (t)Ψ1 (h(0, t)),
(31)
W h(L, t)v(L, t) = uL (t)Ψ2 (h(L, t)), (32) where W the channel width and p Ψ1 (h(x, t)) = α0 2g(hup − h(x, t)), and p Ψ2 (h(x, t)) = αL 2g(h(x, t) − hdo ), hdo < h(L) and h(0) < hup where hup is the water level before the upstream gate, hdo is the water level after the downstream gate. α0 and αL are the product of the gate (or overflow) width and water-flow coefficient of the corresponding gate. U0 and UL are the control functions. 3.2.1. Hamiltonian formulation of the shallow water equations: In this section we shall present briefly the Hamiltonian formulation which is described in detail in the 584
(33)
(34) (35)
It may be shown (Hamroun et al., 2006; Pasumarthy and van der Schaft, 2006) that the shallow water equations may be expressed as a system of two conservation laws (1) in canonical interaction and admits a Hamiltonian formulation (21) by writing the flux vector: pq ρ δq H 01 (36) = ρg f (Y ) = p2 . δp H 10 q+ W 2ρ
Finally the Hamiltonian system may be completed with port boundary variables according to the section 3.1.2: gρ p2 (0) q(0) + 2ρ 0 W e∂ (t) gρ p2 (L) − q(L) − eL (t) W 2ρ ∂0 = f∂ (t) p(0)q(0) L f∂ (t) ρ p(L)q(L) ρ The energy balance is then expressed by:
dH 1 2 q(L)p(L) ρg T = e∂ f∂ = − q(L) + p (L) dt W 2ρ ρ ρg 1 2 q(0)p(0) + q(0) + p (0) W 2ρ ρ 3.2.2. Expression in Riemann coordinates: Using the Jacobian of the vector of flux variables p q ρ ρ ∇f (Y ) = F (Y ) = gρ p W ρ one obtains the following Riemann invariants:
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r gq p + 2 ρ W r ξ = ξ(Y ) = p gq −2 ρ W The Jacobian of this change of coordinates is: r 1 g rW q ρ D(Y ) = g 1 − Wq ρ and the celerities of the system are: r gq p 0 q + W λ1 0 r = Λ(ξ) = 0 λ2 gq p − 0 q W
(37)
(38)
According to the section 3.1.1, the Hamiltonian system is expressed in the Riemann invariants as (22) with: r 2 g 0 ρ Wq 01 r DT = B=D 10 g 2 0 − ρ Wq r r ′ g q 1 g q′ 1 − 2ρ W q q 2ρ W q q . r r C= 1 g q′ 1 g q′ − 2ρ W q q 2ρ W q q
boundary port variables(5) onthe boundary of the ′ values ξ1 (0) ξ1′ (L) . Finally = ∇K ′ Riemann coordinates: ξ2 (L) ξ2′ (0) let us note that, using the derivative of the functions Ki of the boundary conditions (5) can be expressed as: a(λ2 − λ1 ) (0) 0 −1 γλ1 ∇K = [Id − A] [Id + A] −2c (L) 0 λ2 γλ2 0 (0) c(λ2 − λ1 ) (42) λ 1 (L) 0 2a under the condition that [Id − A] is invertible and γ (L) 0 (α − λ1 ) A = ∇G −(α − λ ) . 1 (0) 0 γ Proposition 1. Consider that the assumption 1 is satisfied. If the sign of the spectral radius of ∇G is chosen such that the energy of the system is dissipated according to (41): ρ (∇G) > 0, G(0) = 0 −1
4. STABILITY AND DISSIPATION In this section we shall elaborate on the question posed by Greenberg and Li (Greenberg and Ta, 1984) about the interpretation of the inequality conditions they give in (6) in terms of dissipation of energy at the boundary using the port Hamiltonian formalism. It will secondly be applied on shallow water equations. 4.1 Port boundary variables, Riemann coordinates and stabilizing boundary relations
and then the spectral radius of [Id − A] [Id + A] satisfy −1 ρ([Id − A] [Id + A]) < 1, as it is exactly a Cayley transformation for matrix (Akhiezer and Glazman, 1963). If furthermore the compatibility conditions (9)-(10) are satisfied the conditions of theorem 1 are satisfied and the system is exponentially stable (see extension of this theorem given by (Coron et al., 2007)). ′ ′ ξ (0) ξ (0) Remark 2. If one defines ′2 = ∇K ′1 and ξ1 (L) ξ2 (L) L 0 L e0∂ = G1 (f∂0 , eL ∂ ), f∂ = G2 (f∂ , e∂ ) then the previous proposition is still realized. Furthermore in the case of the p-system treated in the section 2.3, using the change of coordinates (17) and assuming boundary relation to be dissipative, one can prove the exponential stability using Greenberg and Li theorem 1. 4.2 Shallow water equations, dissipativity and stabilization
Consider a hyperbolic system of two conservation laws (1) which admits a Hamiltonian representation (21) with port variables. And let us assume the following. Assumption 1. There exist two C 1 real functions L1 , L2 admitting an inverse and defining the following relations:
The shallow water equations (described in section 3.2) constitute an example to the preceding section. The main difference within the p-system example is that their Hamiltonian is not separated in the conserved quantities p and q. Let recall some elements of the shallow water equations:
eq = L1 (ξ1 − ξ2 , ξ1 + ξ2 )
(39)
ep = L2 (ξ1 − ξ2 , ξ1 + ξ2 ).
(40)
˜ dt H(ξ) = −eq (L)ep (L) + eq (0)ep (0) ρ ρ eq = (ξ1 + ξ2 )2 + (ξ1 − ξ2 )2 = L11 (ξ1 − ξ2 , ξ1 + ξ2 ) 8 16 W ep = (ξ1 + ξ2 )(ξ1 − ξ2 )2 = L2 (ξ1 − ξ2 , ξ1 + ξ2 ). 32g −ρ 0 q −λ2 (0) W q(L) 0 g q With A = and B = +λ1 (L) λ1 (0) W q(0) 0
Then consider relations on the boundary port variables defined by two C 1 functions G1 , G2 as follows: 0 L f∂0 = G1 (e0∂ , f∂L ), eL ∂ = G2 (e∂ , f∂ ), As a consequence the energy balance equation depends on G1 , G2 and becomes: dH = f∂L G2 (e0∂ , f∂L ) − e0∂ G1 (e0∂ , f∂L ) (41) dt Using the implicit function theorem, the relations (41) on the port boundary variables, may be expressed in terms of
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g
λ2 (L)
0
ρ
∇K = [Id − ∇GA]
−1
[Id + ∇GA] B
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As one can see the condition on ∇G is equivalent to ρ(∇G) > 0 ⇔ −G′12 G′21 > 0 and so G′21 and G′12 have opposite signs. For the shallow water equations (3.2), we can identify exactly the different elements of the equations above, with: r γ −L′ g = −ρ = ′ 1 < 0 and G′12 < 0, G′21 > 0. α − λ1 Wq L2
The proposition can be applied and the stability is ensured for all boundary conditions on the form (41). 5. CONCLUSIONS In the first part of this paper we have recalled the expression of the port Hamiltonian systems of two conservation laws and derived its expression in terms of Riemann invariant. This construction has been detailed for the examples of the p-system and the shallow water equations. In the second part of the paper, under the assumption that the port boundary variables may be expressed as a function of the Riemann invariants, we have expressed the stability conditions on the boundary values of the Riemann invariants with some conditions on the boundary constraints on the port variables. As a consequence we have given an interpretation of the stabilizing boundary relations in terms of the dissipation of the Hamiltonian function, in the case of physical systems identical with the total energy of the system, on the boundary of the system. The first remark is that the latter are more restrictive than the conditions of Greenberg and Li as they imply that the energy is dissipated at each time instant which appears to be a stronger than the requirement on the characteristics. This paper aimed to be a first step towards a physically motivated and interpretable construction of stabilizing boundary control. A first extension of this paper could be to relax the requirement on the instantaneous dissipation of the energy. A second extension could be to use the Riemann coordinate for assigning a closed-loop Hamiltonian functional in some analogy with the finite-dimensional case (Maschke et al., 2002). REFERENCES Akhiezer, N.I. and Glazman, I.M. (1963). Theory of linear operators in hilbert space. New York: Fredderick Ungar Publishing, [Book 7–116–1140226]. Bastin, G., Coron, J.M., and d’Andra Novel, B. (2004). A strict lyapunov function for boundary control of hyperbolic systems of conservation laws. 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas. Brogliato, B., Lozano, R., Maschke, B., and Egeland, O. (2007). Dissipative Systems Analysis and Control. Communications and Control Engineering Series, Springer Verlag, London, 2nd edition edition. Coron, J.M., d’Andrea Novel, B., and Bastin, G. (2007). A strict lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Automat. Control, 52, 2–11. Courant, T.J. and Weinstein, A. (1988). Beyond Poisson structures. S´eminaire Sud-Rhodanien de G´eom´etrie, Hermann, Paris, 27, 39–49. 586
Curtain, R.F. and Zwart, H.J. (1995). An introduction to infinite-dimensional linear system theory. SpringerVerlag, New York, xviii+698. Dos Santos, V., Bastin, G., Coron, J.M., and d’Andrea Novel, B. (2008). Boundary control with integral action for hyperbolic systems of conservation laws : Lyapunov stability analysis and experimental validation. Automatica, 44, 1310–1318. Greenberg, J.M. and Ta, T.L. (1984). The effect of boundary damping for the quasilinear wave equations. Journal of Differential Equations, 52, 66–75. Hamroun, H., Lefevre, L., and Mendes, E. (2006). Portbased modelling for open channel irrigation systems. WSEAS Trans. on Fluid Mechanics, 12, 995–1008. Lax, P.D. (1973). Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Conf. Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 11. Le Gorrec, Y., Zwart, H., and Maschke, B. (2005). Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM J. of Control and Optimization, 44, 1864–1892. Luo, Z.H., Guo, B.Z., and Morgul, O. (1999). Stability and stabilization of infinite dimensional systems with applications. Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, xiv+403. Macchelli, A. and Melchiorri, C. (2004). Modeling and control of the Timoshenko beam. the Distributed Port Hamiltonian approach. SIAM Journal On Control and Optimization, 43, 743–767. Maschke, B., Ortega, R., van der Schaft, A., and Escobar, G. (2002). Interconnection and damping assignment: passivity-based control of port-controlled Hamiltonian systems. Automatica J., 38, 585–596. Maschke, B. and van der Schaft, A.J. (2005). Advanced topics in control systems theory. Lecture Notes from FAP 2004, chapter Compositional modelling of distributed-parameter systems 311, Springer-Verlag London, Ltd., London, xviii+280. Pasumarthy, R. and van der Schaft, A.J. (2006). A porthamiltonian approach on modeling and interconnections of canal systems. Proceeding of the Mathematical Theory of Networks and Systems Conference, MTNS06, Kyoto, Japan, 1436–1443. Serre, D. (1996). Systems of conservation laws. Diderot Editeur, Paris, xii+300. van der Schaft, A.J. (2000). l2 -Gain and Passivity Techniques in Nonlinear control. Springer Communications and Control Engineering series, Springer-Verlag, London, 2nd revised and enlarged edition. Villegas, J.A., Le Gorrec, Y., Zwart, H., and Maschke, B. (2006). Boundary control for a class of dissipative differential operators including diffusion systems. Proc. 7th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, 297–304. Willems, J.C. (1972). Dissipative dynamical systems, part 1: General theory. Arch. Rational Mech. Anal., 45, 352– 393.
Hamiltonian approach to the stabilization of systems of two conservation laws V. Dos Santos Martins ∗ B. Maschke ∗ Y. Le Gorrec ∗∗ ∗ LAGEP, Universit´e de Lyon, Lyon, F-69003, France ; Universit´e Lyon 1, CNRS, UMR 5007, LAGEP, Villeurbanne, F-69622, France, ESCPE, Villeurbanne, F-69622 (e-mail: [email protected]) ∗∗ FEMTO-ST/AS2M, ENSMM Besan¸con, 24 rue Alain Savary, 25 000 Besan¸con (e-mail: [email protected])
Abstract: This paper aims at providing some synthesis between two alternative representations of systems of two conservation laws and interprets different conditions on stabilizing boundary control laws. The first is the representation in Riemann invariants coordinates whose representation has been applied successfully for the stabilization of linear and non-linear of such hyperbolic systems. The second representation is based on physical modelling and leads to port Hamiltonian systems which are extensions of infinite-dimensional Hamiltonian systems defined on Dirac structure. The stability conditions on the boundary feedback relations derived with respect to the Riemann invariants are interpreted in terms of the dissipation inequality of the Hamiltonian functional. Keywords: Boundary Port Hamiltonian systems, Distributed parameters systems, Stability. 1. INTRODUCTION In this paper we shall be concerned with the stabilization via boundary control of hyperbolic systems of two conservation laws mainly motivated by the control of the shallow water equations used as models of irrigation channels, but also the p-system (Serre, 1996). The stabilization of irrigation channels using boundary control has been intensively studied for instance in (Dos Santos et al., 2008) for both linear and non linear cases as the stability on a one-dimensional spatial domain for such systems (Bastin et al., 2004). One of the most often suggested approaches, uses Riemann invariants to derive a stabilizing boundary control (Greenberg and Ta, 1984) and some extensions based on Lyapunov function (Bastin et al., 2004; Coron et al., 2007). The use of physically motivated control Lyapunov function for the derivation of stabilizing control laws for non-linear finite-dimensional systems has proven to be very efficient and has lead to a great variety of results (Brogliato et al., 2007; van der Schaft, 2000). Very often, when the system stems from physical modelling, one may derive dissipation inequalities related to energy balance equations and energy dissipating phenomena (Willems, 1972). Using the dissipative port-Hamiltonian formulation for controlled physical systems, one may go one step further and assign in closed loop the dynamic behavior by the structure matrices of the Hamiltonian system in closed loop (Maschke et al., 2002). For infinite-dimensional systems very similar techniques, based on dissipation inequalities, which in terms of PDE’s amounts to consider some entropy function (Coron et al., 2007), have been used for the stabilization of boundary control systems (Curtain and Zwart, 1995; Luo et al., 1999). Recent works have used a boundary port-Hamiltonian formulation of systems of conservation laws (Maschke and van der Schaft, 2005) in order to derive stabilizing boundary control for a class of 978-3-902661-80-7/10/$20.00 © 2010 IFAC
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linear systems defined on one-dimensional spatial domains (Le Gorrec et al., 2005; Macchelli and Melchiorri, 2004; Villegas et al., 2006). The aim of this paper is twofold. Firstly, after a brief recall of the formulation of a system of two conservation laws in terms of port-Hamiltonian systems defined with respect to the canonical Stokes-Dirac structure (Maschke and van der Schaft, 2005) in Section 2, it derives the port-Hamiltonian formulation of these systems in terms of Riemann coordinates in Section 3. Secondly, in Section 4 the conditions expressed on the boundary values of the Riemann coordinates used in (Greenberg and Ta, 1984) are related to some conditions expressed on the boundary port variables of the Hamiltonian formulation. Using the fact that the Hamiltonian functional satisfies a conservation equation, that makes it become a natural candidate Lyapunov function, these conditions are then interpreted in terms of dissipativity of the Hamiltonian function and allow an easy physical interpretation. Conclusions end the paper in Section 5. 2. MOTIVATION THROUGH THE STABILIZATION OF THE P-SYSTEM USING THE RIEMANN INVARIANTS 2.1 Reminder on the stabilization of system of two conser-vations laws In this section, we recall the main result on the stabilization of a hyperbolic system of two conservation laws suggested by Greenberg & Li (Greenberg and Ta, 1984). Consider a spatial domain consisting of the finite interval [0, L] ∋ x with L ∈ IR∗+ 1 and time domain being the real interval [0, +∞) ∋ t. The state space is a non-empty 1
The set of positive and non nul reals
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connected open set in IR2 , denoted by Ω. Consider the system of two conservations laws: ∂t Y + ∂x f (Y ) = 0, (1) where • Y = (y1 y2 )T : [0, +∞) × [0, L] → Ω is the vector of the two dependent variables; • f : Ω → IR2 is a C 1 -function called the flux vector. Note that the system (1) may also be written: ∂t Y + F (Y )∂x Y = 0 (2) where F is the Jacobian of the flux vector f . Assuming that the system is hyperbolic, implies that this system can be diagonalised using the Riemann invariants (see for instance (Lax, 1973, pages 34 - 35)). This means that there exists a change of coordinates ξ(Y ) whose Jacobian matrix is denoted D(Y ), ∂ξ ,Y ∈ Ω (3) ∂Y and diagonalises F (Y ) in Ω: D(Y )F (Y ) = Λ(Y )D(Y ). In the coordinates ξ, the system (1) can then be rewritten in the following (diagonal) characteristic form: ∂t ξ + Λ(ξ)∂x ξ = 0 (4) 2 with ξ: [0, L] × [0, +∞) → IR , (x, t) 7→ ξ(x, t), and Λ(ξ) = diag(λ1 (ξ), λ2 (ξ)), assuming that λ1 (ξ), λ2 (ξ) satisfy the conditions: D(Y ) =
• the λi ’s are continuously differentiable functions on a neighborhood of the origin; • λ2 (0) < 0 < λ1 (0). In this paper we shall consider the following result of Greenberg and Li (Greenberg and Ta, 1984) which may be recalled as follows. Theorem 1. Consider the hyperbolic system of conservation laws in Riemannian coordinates (4) with the following relations on the boundary variables: ξ1 (0, t) = K1 (ξ2 (0, t)), ξ2 (L, t) = K2 (ξ1 (L, t)) (5) with the functions K1 and K2 being C 1 and satisfying 2 : K1 (0) = K2 (0) = 0 and |K1 ′ (0)K2 ′ (0)| < 1. Consider initial values: lim (ξ1 , ξ2 )(x, t) = (ξ1,0 , ξ2,0 )(x), 0 < x < L, t→0+ 1
(6) (7)
being C and satisfying the assumption that to be small in the C 1 norm and the compatibility conditions: ξ1,0 (0) = K1 (ξ2,0 (0)) and ξ2,0 (L) = K2 (ξ1,0 (L)) (8) λ1 (ξ1,0 , ξ2,0 )(0) ∂x ξ1,0 (0) = λ2 (ξ1,0 , ξ2,0 )(0) K1 ′ (0) ∂x ξ2,0 (0)
(9)
λ2 (ξ1,0 , ξ2,0 )(L) ∂x ξ2,0 (L) = λ1 (ξ1,0 , ξ2,0 )(L) K2 ′ (L) ∂x ξ1,0 (L) (10) Then the initial value problem, for this system, has a unique C 1 solution. Moreover, its solution decays to zero in the C 1 norm with an exponential rate. This result has been generalized to a greater class of non linear systems in infinite dimension, for n eigenvalues (Coron et al., 2007). 2
The prime is for the derivative of the function.
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2.2 Application to the p-system In their paper (Greenberg and Ta, 1984), the authors consider the p-system defined by the following system of conservation laws: u v ∂t − ∂x =0 (11) v σ(u)
where σ is a C 1 function satisfying: σ(0) = 0, σ ′ (u) > 0. (12) They consider the following relations on the boundary values: σ(u(0, t)) − rv(0, t) = 0, v(L, t) = 0 (13) with r ∈ IR∗+ . In (Greenberg and Ta, 1984), the authors construct a unique smooth solution using the properties of invariance of the Riemann coordinates along the Riemann forward characteristic ξ˙ = λ1 and the backward one ξ˙ = λ2 , where λi are the eigenvalues of the p-system. However, Greenberg and Li have made the remark that for the p-system example, the stability condition looks like the dissipativity condition of the energy of this system. Motivated by this remark, the following sections deal with the interpretation of the stabilization conditions in terms of dissipativity and Hamiltonian Structure. 2.3 Link with Hamiltonian Structure It is remarkable that, for this classical physical example, the authors use the physical variables (u being the volume, v the velocity of some isentropic gas for instance and σ denoting the opposite of the pressure) in order to express the boundary relations (13) and not the Riemann invariants. Furthermore they note that the condition on the parameter r in the boundary relations may be interpreted in terms of dissipativity properties. Indeed consider the energy of the system (sum of the kinetic and the internal energy of the gas): Z L 2 Z u v H(u, v) = σ(η) dη dx (14) + 2 0 0 and its balance equation: dH = v(t, L) σ(u(t, L)) − v(t, 0) σ(u(t, 0)). (15) dt The boundary relations (13) and the positivity of r imply then energy dissipation: dt H = −rv 2 (0, t) ≤ 0. (16) The relation between energy dissipation and the inequality (6) on the boundary relation of the theorem 1, conditions may be expressed as follows. Using the definition of the Riemann invariants and that σ(u) is a strictly increasing function, it follows that there exists a C 1 function u ¯ such that the inverse coordinate transformation may be written: u =u ¯(ξ2 − ξ1 ) (17) v = ξ2 + ξ1 Using the coordinate transformation and the implicit function theorem, there is a function K1 such that the boundary relations on the physical variables are translated as follows onthe Riemann invariants: ξ1 (0) = K1 (ξ2 (0)) (18) ξ2 (L) = K2 (ξ1 (L)) = −ξ1 (L)
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The inequality conditions (6) of the theorem 1 reduce in this case to: K1 (0) = 0 and |K1′ (0)| < 1. (19) Denoting the celerity of the system in Riemann coordinates by λ(ξ2 − ξ1 ) = λ1 (ξ1 , ξ2 ) = −λ2 (ξ1 , ξ2 ) > 0, a straightforward calculation on the boundary relations at x = 0, leads to the following relation: λ(ξ2 − ξ1 ) − r ′ . |K1 (0)| = (20) λ(ξ2 − ξ1 ) + r
This is clearly a Cayley transformation and hence the positivity of r which implies the energy dissipation, is equivalent with the circle condition (19) given by the theorem 1. This raised for us the question wether such an equivalence holds for some physical systems, in particular systems stemming from physical systems models and their Hamiltonian formulation (Maschke and van der Schaft, 2005).
Hence in terms of the Riemann invariants the system is written: ˜ −∂t ξ = (B∂x + C)δξ H(ξ), (22) 0 1 0 1 where B = Dξ DξT and C = Dξ ∂x (DξT ). 1 0 10 The following properties may be noted: firstly the matrix B is symmetric and secondly it is related with the matrix C by: ∂x B = C T + C. (23) 3.1.2. Boundary port variables: In this section we shall check the formal skew-symmetry of the differential operator (B∂x + C) and then define port boundary variables associated with it. Let us define the following bracket on smooth functions on the spatial domain [0, L]: {e1 , e2 } =
Z
L
eT1 (B∂x + C)e2 dx
(24)
0
and consider the symmetric product:
3. PORT HAMILTONIAN FORMULATION OF A HYPERBOLIC SYSTEM OF TWO CONSERVATION LAWS
{e1 , e2 } + {e2 , e1 } =
Boundary port Hamiltonian systems defined in the distributed parametrers framework in (Maschke and van der Schaft, 2005) allow the description of physical system which exchange a non-zero energy flow through its boundaries. 3.1 Boundary port Hamiltonian systems and Riemann coordinates 3.1.1. Hamiltonian operator expressed in the Riemann coordinates: In this section we shall consider a hyperbolic system of two conservation laws (1) which admits a Hamiltonian representation, that is such that the system is written: 0 ∂x −∂t Y = (δY H), (21) ∂x 0 ∂t Y and δY H belong to a Dirac structure. In the sequel we express the Hamiltonian system in terms of the Riemannian invariants and give the expression of the Hamiltonian operator as well as of the boundary port variables. Denote ˜ by H(ξ) the Hamiltonian expressed in the Riemann invari˜ ants: H(ξ) = H ◦ Y (ξ) where Y denotes, with an abuse of notation, the inverse change of coordinates to the Riemann coordinates. Recall that D denotes the Jacobian of change of coordinates to the Riemann invariants defined in (3) and define Dξ = D ◦ Y (ξ). One obtains by multiplying both terms of (21) by D: 0 1 ∂x (δY H) −D(Y )∂t Y = D(Y ) 10 0 1 ˜ ∂x (DξT )δξ H(ξ) ⇔ −∂t ξ = Dξ 10 0 1 ˜ DξT ∂x (δξ H(ξ)). +Dξ 1 0
+eT2 Ce1 +
Z
L
0 eT1
∂x (eT1 Be2 ) − eT1 ∂x Be2 + eT2 ∂x Be2 C + C T − ∂x B e2 dx.
By using the fact that B is symmetric and the relation (23), the symmetric product reduces to: {e1 , e2 } + {e2 , e1 } =
Z
0
L
L ∂x (eT1 Be2 ) = (eT1 Be2 ) 0 .(25)
The product (25) corresponds to Stokes theorem applied to the equation for the differential operator (B∂x + C). Furthermore the second member of (25) vanishes for all functions e1 , e2 with compact support strictly included in the domain [0, L] and hence for these functions the bracket is skew-symmetric. One can also prove that Jacobi identities are satisfied. Then in this case the system (22) defines an infinite dimensional Hamiltonian system. However considering functions which do not vanish on the boundary of the domain, the bracket (24) is no more skew-symmetric. In this case the time variation of the Hamiltonian becomes: iL h ˜ dH(ξ) ˜ ˜ T B δξ H) . (26) = (δξ1 H 2 dt 0 Thus one may account for the energy flow through the boundary of the domain, however the structure of an infinite-dimensional Hamiltonian system is lost. Using the notion of Dirac structure one may still define a Hamiltonian system by extending the system with external variables associated with the boundary. Naturally the choice is not unique. The most natural choice related to the balance equation (26) consists in choosing the following pair of variables: δ H(0) ˜ ξ1 Dξ (0) 0 0 0 ˜ 0 w1 0 Dξ (L) 0 δξ1 H(L) = ˜ 0 0 Dξ (0) 0 w2 −δξ2 H(0) 0 0 0 Dξ (L) ˜ δξ2 H(L) (27)
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then the balance equation (26) is written in form of a dissipation equality (Willems, 1972): ˜ dH(ξ) Σ2 0 T = w1 w2 (28) 0 Σ2 dt
where B(0) = B(ξ(0)) and B(L) = B(ξ(L)). Remark 1. Following the definition of Dirac structure (Courant and Weinstein, 1988) and adapting the proofs in (Le Gorrec et al., 2005), one may prove that the differential operator (B∂x + C)combined with (27) generates a canonical Dirac structure. 3.2 Application to the shallow water equations We consider the special case of a reach of an open channel delimited by two underflow gates as represented in Figure 1.
general case (with slope and friction) in (Hamroun et al., 2006), see also (Pasumarthy and van der Schaft, 2006). The energy of the water flow in the channel is defined in terms of two state variables: • the momentum p(x, t) = ρv(x, t) where ρ the mass density of the water (constant as the water is assumed to be incompressible), • the section area of the water q(x, t) = W h(x, t). In the sequel we use the state vector defined by Y = T (q p) . The total energy of the systems (sum of kinetic and potential energies) is given by: H(Y ) =
1 2
Z
0
L
ρg 2 1 2 q + qp dx, W ρ
where g denotes the constant of gravity. The variational derivatives of the Hamiltonian functional define the two co-energy variables: ep the volumic flow and eq the hydrodynamic pressure as follows: p2 ρg 1 + q = ρv 2 + ρgh , 2ρ W 2 qp ep = δp H(Y ) = (= W hv) . ρ eq = δq H(Y ) =
Fig. 1. A reach of an open channel delimited by two adjustable underflow gates We assume that the channel is horizontal, prismatic with a constant rectangular section and a width, and that the friction effects are neglected. The flow in the canal may be described by the so-called shallow water equations or Saint-Venant equations which constitute a system of two conservation laws, actually the mass balance and the momentum balance equations: ∂t h + ∂x (hv) = 0, (29) 1 (30) ∂t v + ∂x ( v 2 + gh) = 0, 2 where v denotes the velocity of the water flow and the water level h. Each underflow gates imposes a boundary condition of the form: W h(0, t)v(0, t) = u0 (t)Ψ1 (h(0, t)),
(31)
W h(L, t)v(L, t) = uL (t)Ψ2 (h(L, t)), (32) where W the channel width and p Ψ1 (h(x, t)) = α0 2g(hup − h(x, t)), and p Ψ2 (h(x, t)) = αL 2g(h(x, t) − hdo ), hdo < h(L) and h(0) < hup where hup is the water level before the upstream gate, hdo is the water level after the downstream gate. α0 and αL are the product of the gate (or overflow) width and water-flow coefficient of the corresponding gate. U0 and UL are the control functions. 3.2.1. Hamiltonian formulation of the shallow water equations: In this section we shall present briefly the Hamiltonian formulation which is described in detail in the 584
(33)
(34) (35)
It may be shown (Hamroun et al., 2006; Pasumarthy and van der Schaft, 2006) that the shallow water equations may be expressed as a system of two conservation laws (1) in canonical interaction and admits a Hamiltonian formulation (21) by writing the flux vector: pq ρ δq H 01 (36) = ρg f (Y ) = p2 . δp H 10 q+ W 2ρ
Finally the Hamiltonian system may be completed with port boundary variables according to the section 3.1.2: gρ p2 (0) q(0) + 2ρ 0 W e∂ (t) gρ p2 (L) − q(L) − eL (t) W 2ρ ∂0 = f∂ (t) p(0)q(0) L f∂ (t) ρ p(L)q(L) ρ The energy balance is then expressed by:
dH 1 2 q(L)p(L) ρg T = e∂ f∂ = − q(L) + p (L) dt W 2ρ ρ ρg 1 2 q(0)p(0) + q(0) + p (0) W 2ρ ρ 3.2.2. Expression in Riemann coordinates: Using the Jacobian of the vector of flux variables p q ρ ρ ∇f (Y ) = F (Y ) = gρ p W ρ one obtains the following Riemann invariants:
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r gq p + 2 ρ W r ξ = ξ(Y ) = p gq −2 ρ W The Jacobian of this change of coordinates is: r 1 g rW q ρ D(Y ) = g 1 − Wq ρ and the celerities of the system are: r gq p 0 q + W λ1 0 r = Λ(ξ) = 0 λ2 gq p − 0 q W
(37)
(38)
According to the section 3.1.1, the Hamiltonian system is expressed in the Riemann invariants as (22) with: r 2 g 0 ρ Wq 01 r DT = B=D 10 g 2 0 − ρ Wq r r ′ g q 1 g q′ 1 − 2ρ W q q 2ρ W q q . r r C= 1 g q′ 1 g q′ − 2ρ W q q 2ρ W q q
boundary port variables(5) onthe boundary of the ′ values ξ1 (0) ξ1′ (L) . Finally = ∇K ′ Riemann coordinates: ξ2 (L) ξ2′ (0) let us note that, using the derivative of the functions Ki of the boundary conditions (5) can be expressed as: a(λ2 − λ1 ) (0) 0 −1 γλ1 ∇K = [Id − A] [Id + A] −2c (L) 0 λ2 γλ2 0 (0) c(λ2 − λ1 ) (42) λ 1 (L) 0 2a under the condition that [Id − A] is invertible and γ (L) 0 (α − λ1 ) A = ∇G −(α − λ ) . 1 (0) 0 γ Proposition 1. Consider that the assumption 1 is satisfied. If the sign of the spectral radius of ∇G is chosen such that the energy of the system is dissipated according to (41): ρ (∇G) > 0, G(0) = 0 −1
4. STABILITY AND DISSIPATION In this section we shall elaborate on the question posed by Greenberg and Li (Greenberg and Ta, 1984) about the interpretation of the inequality conditions they give in (6) in terms of dissipation of energy at the boundary using the port Hamiltonian formalism. It will secondly be applied on shallow water equations. 4.1 Port boundary variables, Riemann coordinates and stabilizing boundary relations
and then the spectral radius of [Id − A] [Id + A] satisfy −1 ρ([Id − A] [Id + A]) < 1, as it is exactly a Cayley transformation for matrix (Akhiezer and Glazman, 1963). If furthermore the compatibility conditions (9)-(10) are satisfied the conditions of theorem 1 are satisfied and the system is exponentially stable (see extension of this theorem given by (Coron et al., 2007)). ′ ′ ξ (0) ξ (0) Remark 2. If one defines ′2 = ∇K ′1 and ξ1 (L) ξ2 (L) L 0 L e0∂ = G1 (f∂0 , eL ∂ ), f∂ = G2 (f∂ , e∂ ) then the previous proposition is still realized. Furthermore in the case of the p-system treated in the section 2.3, using the change of coordinates (17) and assuming boundary relation to be dissipative, one can prove the exponential stability using Greenberg and Li theorem 1. 4.2 Shallow water equations, dissipativity and stabilization
Consider a hyperbolic system of two conservation laws (1) which admits a Hamiltonian representation (21) with port variables. And let us assume the following. Assumption 1. There exist two C 1 real functions L1 , L2 admitting an inverse and defining the following relations:
The shallow water equations (described in section 3.2) constitute an example to the preceding section. The main difference within the p-system example is that their Hamiltonian is not separated in the conserved quantities p and q. Let recall some elements of the shallow water equations:
eq = L1 (ξ1 − ξ2 , ξ1 + ξ2 )
(39)
ep = L2 (ξ1 − ξ2 , ξ1 + ξ2 ).
(40)
˜ dt H(ξ) = −eq (L)ep (L) + eq (0)ep (0) ρ ρ eq = (ξ1 + ξ2 )2 + (ξ1 − ξ2 )2 = L11 (ξ1 − ξ2 , ξ1 + ξ2 ) 8 16 W ep = (ξ1 + ξ2 )(ξ1 − ξ2 )2 = L2 (ξ1 − ξ2 , ξ1 + ξ2 ). 32g −ρ 0 q −λ2 (0) W q(L) 0 g q With A = and B = +λ1 (L) λ1 (0) W q(0) 0
Then consider relations on the boundary port variables defined by two C 1 functions G1 , G2 as follows: 0 L f∂0 = G1 (e0∂ , f∂L ), eL ∂ = G2 (e∂ , f∂ ), As a consequence the energy balance equation depends on G1 , G2 and becomes: dH = f∂L G2 (e0∂ , f∂L ) − e0∂ G1 (e0∂ , f∂L ) (41) dt Using the implicit function theorem, the relations (41) on the port boundary variables, may be expressed in terms of
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g
λ2 (L)
0
ρ
∇K = [Id − ∇GA]
−1
[Id + ∇GA] B
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As one can see the condition on ∇G is equivalent to ρ(∇G) > 0 ⇔ −G′12 G′21 > 0 and so G′21 and G′12 have opposite signs. For the shallow water equations (3.2), we can identify exactly the different elements of the equations above, with: r γ −L′ g = −ρ = ′ 1 < 0 and G′12 < 0, G′21 > 0. α − λ1 Wq L2
The proposition can be applied and the stability is ensured for all boundary conditions on the form (41). 5. CONCLUSIONS In the first part of this paper we have recalled the expression of the port Hamiltonian systems of two conservation laws and derived its expression in terms of Riemann invariant. This construction has been detailed for the examples of the p-system and the shallow water equations. In the second part of the paper, under the assumption that the port boundary variables may be expressed as a function of the Riemann invariants, we have expressed the stability conditions on the boundary values of the Riemann invariants with some conditions on the boundary constraints on the port variables. As a consequence we have given an interpretation of the stabilizing boundary relations in terms of the dissipation of the Hamiltonian function, in the case of physical systems identical with the total energy of the system, on the boundary of the system. The first remark is that the latter are more restrictive than the conditions of Greenberg and Li as they imply that the energy is dissipated at each time instant which appears to be a stronger than the requirement on the characteristics. This paper aimed to be a first step towards a physically motivated and interpretable construction of stabilizing boundary control. A first extension of this paper could be to relax the requirement on the instantaneous dissipation of the energy. A second extension could be to use the Riemann coordinate for assigning a closed-loop Hamiltonian functional in some analogy with the finite-dimensional case (Maschke et al., 2002). REFERENCES Akhiezer, N.I. and Glazman, I.M. (1963). Theory of linear operators in hilbert space. New York: Fredderick Ungar Publishing, [Book 7–116–1140226]. Bastin, G., Coron, J.M., and d’Andra Novel, B. (2004). A strict lyapunov function for boundary control of hyperbolic systems of conservation laws. 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas. Brogliato, B., Lozano, R., Maschke, B., and Egeland, O. (2007). Dissipative Systems Analysis and Control. Communications and Control Engineering Series, Springer Verlag, London, 2nd edition edition. Coron, J.M., d’Andrea Novel, B., and Bastin, G. (2007). A strict lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Automat. Control, 52, 2–11. Courant, T.J. and Weinstein, A. (1988). Beyond Poisson structures. S´eminaire Sud-Rhodanien de G´eom´etrie, Hermann, Paris, 27, 39–49. 586
Curtain, R.F. and Zwart, H.J. (1995). An introduction to infinite-dimensional linear system theory. SpringerVerlag, New York, xviii+698. Dos Santos, V., Bastin, G., Coron, J.M., and d’Andrea Novel, B. (2008). Boundary control with integral action for hyperbolic systems of conservation laws : Lyapunov stability analysis and experimental validation. Automatica, 44, 1310–1318. Greenberg, J.M. and Ta, T.L. (1984). The effect of boundary damping for the quasilinear wave equations. Journal of Differential Equations, 52, 66–75. Hamroun, H., Lefevre, L., and Mendes, E. (2006). Portbased modelling for open channel irrigation systems. WSEAS Trans. on Fluid Mechanics, 12, 995–1008. Lax, P.D. (1973). Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Conf. Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 11. Le Gorrec, Y., Zwart, H., and Maschke, B. (2005). Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM J. of Control and Optimization, 44, 1864–1892. Luo, Z.H., Guo, B.Z., and Morgul, O. (1999). Stability and stabilization of infinite dimensional systems with applications. Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, xiv+403. Macchelli, A. and Melchiorri, C. (2004). Modeling and control of the Timoshenko beam. the Distributed Port Hamiltonian approach. SIAM Journal On Control and Optimization, 43, 743–767. Maschke, B., Ortega, R., van der Schaft, A., and Escobar, G. (2002). Interconnection and damping assignment: passivity-based control of port-controlled Hamiltonian systems. Automatica J., 38, 585–596. Maschke, B. and van der Schaft, A.J. (2005). Advanced topics in control systems theory. Lecture Notes from FAP 2004, chapter Compositional modelling of distributed-parameter systems 311, Springer-Verlag London, Ltd., London, xviii+280. Pasumarthy, R. and van der Schaft, A.J. (2006). A porthamiltonian approach on modeling and interconnections of canal systems. Proceeding of the Mathematical Theory of Networks and Systems Conference, MTNS06, Kyoto, Japan, 1436–1443. Serre, D. (1996). Systems of conservation laws. Diderot Editeur, Paris, xii+300. van der Schaft, A.J. (2000). l2 -Gain and Passivity Techniques in Nonlinear control. Springer Communications and Control Engineering series, Springer-Verlag, London, 2nd revised and enlarged edition. Villegas, J.A., Le Gorrec, Y., Zwart, H., and Maschke, B. (2006). Boundary control for a class of dissipative differential operators including diffusion systems. Proc. 7th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, 297–304. Willems, J.C. (1972). Dissipative dynamical systems, part 1: General theory. Arch. Rational Mech. Anal., 45, 352– 393.