Robust boundary control of systems of conservation laws: theoretical study with numerical and experimental validations

Transcription

1 1 Robust boundary control of systems of conservation laws: theoretical study with numerical and experimental validations LAAS - CNRS Toulouse (France) Workshop on Irrigation Channels and Related Problems October 2-4, 2008 Maiori (Salerno, Italy)

2 Introduction on hydropower installations 2 Consider a hydropower installation. Longitudinal dimensions greater than transversal dimensions a one-dimensional approach x [0, L] the flow is normal to the cross-section A; the pressure p, the flow velocity C and the density ρ are uniform in a cross-section A. The momentum equation [Nicolet, 2007]: where t C + 1 ρ xp + C x C + g sin(α) + λc C 2D = 0 λ is the local loss coefficient; g is the gravity constant [m/s 2 ]; α is the slope and D is the pipe diameter [m].

3 Dynamical model 3 The continuity equation: where a is the wave speed [m/s]. t p + ρa 2 x C + C x p = 0 Denoting the discharge by Q [m 2 /s] and the piezometric head by h [m]: Q = CA, h = Z + p ρg where Z is the elevation [m] and A is the pipe cross-section [m 2 ]. Assuming high wave speed (large a) and low flow velocity (small C), we obtain the dynamical model: t Q + 0 ga x Q = λq Q 2DA. (1) h h 0 a 2 ga 0

4 Boundary conditions 4 The boundary conditions depend on the hydraulic installations. Here: a simple hydraulic circuit Z_0 Gallery Penstock Valve Given h 0, at x = 0, the piezometric line is h(x = 0, t) = h 0 (2) Given h L, at x = L, there is a valve with reference area A v h(x = L, t) h L = K v(s) Q 2 (x = L, t) (3) 2gA 2 v where K v (s) is the valve head loss coeff, s is the valve opening = control

5 Waterhammer effect 5 Hydraulic machines are increasingly subject to off-design operation, startup and shutdown sequences, quick set point changes, etc. The closure of the guide vanes induces a waterhammer effect in the penstock leading to head fluctuations (see [Nicolet, 07]). Waterhammer phenomenon = large value of the rate of variation of the flow. To overcome this waterhammer phenomenon, many hydraulic installations are equipped with surge tanks. The surge tank has as a free surface for wave reflection whose water level is function of the discharge time history.

6 A circuit with a surge tank 6 Surge tank Z_0 Gallery Penstock Valve Controlled surge tank, see e.g. [Crisafulli, Peirce, 99], [Vournas, Papaioannou, 95] Nonlinear approach in [Clark, Vinter, 03]. The controller should restrict peak values of the rate of the change of outflow.

7 Controlling the C 1 norm but without any surge tank 7 We may also control the outflow and guarantee a constraint on the norm of the differentiable function of the flow, and thus a constraint on the C 1 -norm of the flow. Justification to consider the norm of the rate of variation of the flow. We are interesting in controlling the norms Q(., t) C1 (0,L) and h(., t) C1 (0,L) directly (without any surge tank).

8 Control problem 8 Let us first compute the steady-state solution of (1). We compute h(x) = h 0 for all x in [0, L], and the Q function given by, for all x in [0, L], with the boundary condition (3): d x Q(x) = λg 2Da 2Q2 (x) h 0 h L = K v(s) 2gA 2 Q 2 (x = L). v If h 0 > h L then Q(x) > 0, for all x in [0, L], and thus we may remove the absolute value in (1) around the steady-state solution.

9 Control problem 9 Control problem: Compute the valve position s s.t. the control action depends only on the (measured) h(l, t) and Q(L, t) a solution of our model (PDE) state t + equilibrium solution stays close to the equilibrium in C 1 -norm

10 Outline 10 1 Basic ideas on the stability condition 1 1 Use of the Riemann coordinates 1 2 Choice of the control action 1 3 Review of the unperturbed case and basic ideas when perturbed 2 Theoretical result 2 1 More general context: non-homogeneous systems in R Statement of the main result Applications 3 Example of the hydropower installation 4 Boundary control of an open channel 4 1 Numerical simulations (physical parameters of the Sambre) 4 2 Experimental results (Valence micro-channel) Conclusion

11 1 Basic ideas on the stability condition 11 The system is written in matrix form as follows : t Q h + 0 ga a 2 ga 0 The eigenvalues of the matrix x Q h 0 ga a 2 ga 0 = They are called the characteristic velocities. λq Q 2DA 0 are a and a.. (4)

12 1 1 Use of the Riemann coordinates 12 We introduce the corresponding Riemann coordinates ξ 1 = Q Q ga a (h h) and ξ 2 = Q Q + ga a (h h), and the model (4) rewrites locally around the steady-state: t ξ 1 ξ 2 + diag( a, a) x ξ 1 ξ 2 = = λ 2DA ( ξ 1+ξ Q) 2 λgq2 2Da λ 2DA ( ξ 1+ξ 2 h 1(ξ 1, ξ 2 ) h 2 (ξ 1, ξ 2 ) 2 + Q) 2 + λgq2 2Da for suitable h 1 and h 2.

13 1 1 Use of the Riemann coordinates 13 Let us describe the boundary conditions in the Riemann coordinates. At x = 0, (2) rewrites ξ 2 (x = 0, t) = ξ 1 (x = 0, t). (5) Now for any k R, the BC (3) at x = L holds as soon as the boundary condition is satisfied together with a 2gA (1 k)ξ 2(L, t) + h h L = K v(s) 2gA 2 v ξ 1 (x = L, t) = kξ 2 (x = L, t) (6) ( 1 + k 2 ξ 2 (L, t) + Q(x = L)).

14 1 2 Choice of the control action 14 This latter condition allows us to define (locally around the equilibrium) the controller K v (s) = 2aA 2 v(1 k)ξ 2 (L, t) A((1 + k)ξ 2 (L, t) + 2Q(x = L)) + 4gA 2 v (1 + k)ξ 2 (L, t) + 2Q(x = L) (h 0 h L ) where ξ 2 (L, t) = Q(L, t) Q(L, t) + ga a (h(l, t) h 0). The valve position s may be computed by inverting the function K v (see the graph in [Nicolet, 07]). Note that this controller is a state-feedback. However to compute it we need only the values of the state at x = L, i.e. at the end where the control is implemented.

15 1 3 Review of the unperturbed case 15 If h 1 = h 2 = 0, then along the trajectories ξ i (x i (t), t) = ξ i (x i (0), 0) ẋ 1 (t) = a, ẋ 2 (t) = a The Riemann coordinates are invariant along the characteristics. Recall the boundary conditions: ξ 1 (L, t) = kξ 2 (L, t) and ξ 2 (0, t) = ξ 1 (0, t) Sufficient condition for the stability in the unperturbed case (h 1 = h 2 = 0): k < 1 See [de Halleux et al, 03] and [Li, 1994].

16 Idea for the stability in the perturbed case 16 Along the trajectories ẋ 1 (t) = a, ẋ 2 (t) = a we have ξ i (x i (t), t) = ξ i (x i (0), 0) + t 0 h i (ξ 1 (x i (s), s), ξ 1 (x i (s), s))ds The Riemann coordinates are NOT invariant along the characteristics. BUT if k < 1 and if the non-homogenous term h i are sufficiently small then stability.

17 Idea for the stability in the perturbed case 17 Moreover on numerical simulations or on real applications, to cancel an offset (due to possible actuator imperfection e.g.) it may be necessary to add an integral action More generally instead of the BC (5) and (6) we may consider the perturbed boundary conditions: ξ 1(L, t) = 0 k ξ 1(0, t) + e p ξ 1(0, t) ξ 2 (0, t) 1 0 ξ 2 (L, t) ξ 2 (L, t) t +e i 0 ξ 1(0, s) ds t 0 ξ 2(L, s) ds We may prove that if k < 1 and if the errors e p and e i are sufficiently small then stability.

18 2 1 More general context: non-homogeneous systems in R 2 18 We study a system of two conservation laws. Let us consider ξ: [0, L] [0, + ) R 2 such that: t ξ + Λ(ξ) x ξ = h(ξ) (7) where Λ: B(ε 0 ) R 2 2 is a continuously differentiable function such that Λ = diag(λ 1, λ 2 ), with λ 1 (0) < 0 < λ 2 (0), (8) and h = (h 1, h 2 ): B(ε 0 ) R 2 is a C 1 -function such that h(0) = 0. (9)

19 2 1 More general context: non-homogeneous systems in R 2 19 The boundary conditions of (7) are ξ 1(L, t) = g ξ 1(0, t) +e p ξ 1(0, t) ξ 2 (0, t) ξ 2 (L, t) ξ 2 (L, t) +e i where g, e p and e i : B(ε 0 ) R 2 is a C 1 -function satisfying g(0) = e p (0) = e i (0) = 0. t 0 ξ 1(0, s) ds t 0 ξ 2(L, s) ds (10),

20 2 1 More general context: non-homogeneous systems in R 2 20 Definition A function ξ # C 1 (0, L; R 2 ) satisfies the compatibility condition C if ξ# 1 (L) ξ # 2 (0) = g ξ# 1 (0), ξ # 2 (L) and λ 1(ξ # (L)) x ξ # 1 (L) h 1(ξ # (L)) λ 2 (ξ # (0)) x ξ # 2 (0) h 2(ξ # (0)) = ( g + e p ) ξ# 1 (0) ξ # 2 (L) λ 1(ξ # (0)) x ξ # 1 (0) h 1(ξ # (0)) λ 2 (ξ # (L)) x ξ # 2 (L) h 2(ξ # (L)). We denote by B C (ε 0 ) the set of C 1 -functions ξ # : [0, L] B(ε 0 ) satisfying the compatibility assumption C.

21 2 1 More general context: non-homogeneous systems in R 2 21 Given Φ continuous on [0, L] and Ψ continuously differentiable on [0, L], we denote Given matrix A = (a ij ), Φ C 0 (0,L) = max x [0,L] Φ(x), Ψ C1 (0,L) = Ψ C0 (0,L) + Ψ C0 (0,L) ρ(a) is its spectral radius abs(a) is the matrix defined by abs(a) = ( a ij )

22 2 2 Main result 22 Theorem[CP, Winkin, Bastin, 08] + [CP, preprint-08] If ρ(abs( g(0)) < 1, (11) then there exist ε > 0, and E > 0 such that, for all C 1 -functions h, e p and e i : B(ε 1 ) R 2 such that h(0) = 0 and h(0) + e p (0) + e i (0) E, (12) for all ξ # B C (ε), there exists one and only one function ξ C 1 ([0, L] [0, + ) ; R 2 ) satisfying (7), (10) and ξ(x, 0) = ξ # (x), x [0, L]. (13) Moreover, there exist µ > 0 and C > 0 such that ξ(., t) C 1 (0,L) Ce µt ξ # C 1 (0,L), t 0.

23 2 2 Main result 23 Proof: estimation of the influence of the non-homogeneous terms on the evolutions of the Riemann coordinates. the damping condition (11) is strong enough to manage the non-homogeneous terms, and the boundary errors. Their value and their derivative are assumed to be small at the origin due to (12).

24 2 2 Main result 24 Remark We assumed that the non-homogeneous terms and the errors are sufficiently small. We didn t prescribe any sign on the proportional term. We may try to use its sign for the stability Open question Can we impose the sign of the integral part, and the sign of 1 g(0) so that the boundary condition is like a PI controller? A first answer is given in [Dos Santos, Bastin, Coron, and d Andréa-Novel, 07], where it is proven that : For the linearized Saint-Venant equations, the existence of a decreasing Lyapunov function (with a given structure) is equivalent to the signs conditions of the PI.

25 3 Example of the hydropower installation 25 If the local loss coefficient λ is sufficient small, Then the output feedback law K v (s) = 2aA 2 v (1 k)ξ 2(L,t) A((1+k)ξ 2 (L,t)+2Q(x=L)) + 4gA 2 v (1+k)ξ 2 (L,t)+2Q(x=L) (h 0 h L ) where ξ 2 (L, t) = Q(L, t) Q(L, t) + ga a (h(l, t) h 0), is a stabilizing feedback. An estimation on the rate of change of the flow is guaranted (may reduce the waterhammer effect). This is only a preliminary application on hydropower installation. Further developments are needed: with other hydraulic components (Francis turbines); more complete model

26 0 L L u 4 Application to the control of an open channel 26 Level and flow control in an horizontal reach of an open channel Control = two overflow spillways: Q L ( t 0( t H L ( t H ) 0 u 0 ( t ) ) Q ) H(x, t) Q(x, t) where H(x, t) is the water level and Q(x, t) the water flow rate in the reach.

27 4 Application to the control of an open channel 27 Model [Chow, 54] or [Graf, 98]: mass conservation momentum conservation t H(x, t) + x ( t Q(x, t)+ x ( Q2 (x, t) BH(x, t) + gbh2 (x, t) 2 where g denotes the gravitation constant q(x) the water supply/removal function B is the channel width I is the bottom slope Q(x, t) ) = q(x) (14) B ) = gbh(i J)+kq Q BH (15) J(Q, H) = n 2 M Q2 S(H) 2 R(H) 4/3, is the slope s friction

28 Related work 28 About the level and flow control in a reach For a survey, see [Malaterre, Rogers, and Schuurmans, 98]. Discrete linear approximations of the perturbed Saint-Venant equations: [Garcia, Hubbard, and De Vries, 92], and [Malaterre, 98]. H control design is developed in [Litrico, and Georges, 99]. Spectral decomposition [Litrico, Fromion, 06] Lyapunov methods [Coron, d Andréa-Novel, and G. Bastin, 07], [Dos Santos, Bastin, Coron, and d Andréa-Novel, 07]

29 Restatement of the control problem 29 The system is written in matrix form as follows : t H + A(H, Q) x H q(x) = Q Q gbh(i J) + kq Q BH with the matrix A(H, Q) defined as : 0 1/B A(H, Q) = gbh (Q 2 /BH 2 ) 2Q/BH. The eigenvalues of the matrix A(H, Q) : λ 1 (H, V ) = (Q/H) gh λ 2 (H, V ) = (Q/H) + gh are the characteristic velocities. The flow is said to be fluvial (or subcritical) when λ 1 (H, Q) < 0 < λ 2 (H, Q).

30 Steady-state solution 30 Under constant boundary conditions Q(0, t) = Q 0 and H(L, t) = H L t, there exists a steady-state solution : H(x, t) = H(x) and Q(x, t) = Q(x) x [0, L] t which satisfies the differential equations : x Q(x) x H(x) = Bq(x) I J Q = g H q λ 1 λ2 B 2 H (k 2B) λ 1 λ2, Control problem: Make this equilibrium locally asymptotically stable

31 The Riemann coordinates 31 Definition ξ 1 = (Q/H) 2 gh ( Q/ H) + 2 g H ξ 2 = (Q/H) + 2 gh ( Q/ H) 2 g H The model can be written as t ξ 1 + λ 1 (ξ 1, ξ 2 ) x ξ 1 = h 1 (ξ 1, ξ 2 ) t ξ 2 + λ 2 (ξ 1, ξ 2 ) x ξ 2 = h 2 (ξ 1, ξ 2 ) (16) with h 1 (ξ) =... and h 2 (ξ) =... The control laws are equivalent to the following boundary conditions: and ξ 1 (L, t) = k L ξ 2 (L, t) with k L = 1 α L 1 + α L ξ 2 (0, t) = k 0 ξ 1 (0, t) with k 0 = 1 α α 0

32 Stabilizing output feedback law 32 Applying the previous result, we get that if k 0 k L < 1 and if the non-homogenous term h i are sufficiently small then stability. We obtain a stabilizing output feedback law. We may state a little more precise result by computing the damping condition taking the slope and the friction into account.

33 Stabilizing output feedback law 33 Let t 1 be the time instant defined by x 1 (t 1 ) = L, where x 1 is the solution of ẋ 1 (t) = λ 1 ( H, Q), x 1 (0) = 0. Similarly, let t 2 be the time instant defined by x 2 (t 2 ) = 0, where x 2 is the solution of ẋ 2 (t) = λ 2 ( H, Q), x 2 (L) = 0. The damping condition is max ( k 0 k L +t 2 k 0 l 2 + t 1 l 1, k 0 k L +t 1 k L l 1 + t 2 l 2 ) < 1 (17) where l 1 =... and l 2 =... Theorem[Dos Santos, CP, 08] If the bottom slope function I, the slope s friction function J and the supply function q are sufficiently small in C 1 norm, then we have max(t 1 l 1, t 2 l 2 ) < 1, (18) In that case, for all (k 0, k L ) R 2 such that (17) holds,

34 Stabilizing output feedback law 34 the following boundary output feedback controller U 0 = H 0 Q 0 B H 0 2 gα 0 H0 U L = H L h s H 0 µ 0 2g(zup H(0,t)) [ h Q H L L B H +2 HL i 2 gα L H L L, 2gµ 2 L ] 1/3, makes the closed loop system locally exponentially stable, i.e. there exist ε > 0, C > 0 and µ > 0 such that, for all initial conditions (H #, Q # ) : [0, L] (0, + ) C 1, satisfying some compatibility conditions, and the inequality (H #, Q # ) ( H, Q) C1 (0,L) ε, there exists a unique C 1 solution of the Saint-Venant equations, with the boundary conditions, and the initial condition, defined for all (x, t) [0, L] [0, + ). Moreover it satisfies, t 0, (H, Q) ( H, Q) C1 (0,L) Ce µt (H #, Q # ) C1 (0,L)

35 4 1 Numerical simulations 35 The Sambre river in Belgium Study between Charleroi and Namur parameters B L slope I µ 0 n M (m) (m) (m 1.s 1 ) = µ L (s.m 1/3 ) values e The Preismann numerical scheme has been used (analogous results with the the Chang and Coper scheme)

36 4 1 Numerical simulations 36 Three simulations have been computed, with the following values: (S1) k 0 = 0.25, and k L = 0.16 (k 0 k L = 0.041). The stability condition (17) is satisfied, (S2) k 0 = 0.92, and k L = 0.59 (k 0 k L = 0.55). The stability condition (17) doesnot hold, but we have stability. (S3) k 0 = 1.05, and k L = 0.68 (k 0 k L = 0.71). The stability condition (17) doesnot hold, and we do not have the stability

37 4 1 Numerical simulations 37 (m 3.s 1 ) (m) Water flow at upstream 10 equilibrium t (s) (S1) (S2) Water level at downstream (S3) t (s) (m) (m) (S1) t (s) (S2) Downstream gate 1.8 (S3) Upstream gate t (s) Our condition is sufficient but is not necessary for the stability Speed of convergence increases when k 0 k L gets small Due to the numerical approximation, there is some offset...

38 with and without an integral part 38 Stability with a integral part is also guaranteed. And it allows to remove the offset on numerical simulations. 20 Water flow at upstream 16 Water flow at upstream (m 3.s 1 ) (m 3.s 1 ) (m) equilibrium t (s) (S1) Water level at downstream (SI1) t (s) (m) equilibrium t (s) (S2) Water level at downstream (SI2) t (s)

39 with and without closing the loop 39 Comparison of (S1) with constant control values equivalent to the equilibrium 20 Water flow at upstream 15 (m 3.s 1 ) equilibrium t (s) (S1CL) Water level at downstream (S1OL) (m) t (s)

40 3 2 Experimental results 40 Valence micro-channel parameters B(m) L (m) n M (s.m 1/3 ) µ 0 µ L slope (m.m 1 ) values / 00

41 3 2 Experimental results 41

42 3 2 Experimental results 42 Let us consider the following parameters (E1) k 0 = , k L = 0.463, (k 0 k L = ); (E2) k 0 = , k L = , (k 0 k L = 0.247); (E3) k 0 = , k L = 1.852, (k 0 k L = ). In each case, the stability condition (17) holds

43 3 2 Experimental results 43 3 Upstream water flow (E1) (E2) (E3) equilibrium Downstream water level (dm 3.s 1 ) (dm) t (s) t (s) Upstream gate Downstream gate (E1) (E2) (E3) (dm) 0.4 (dm) t (s) t (s)

44 with and without an integral part for (E2) only 44 (EI1) no integral action (EI1=E2), (EI2) small integral action (EI3) larger integral action, but overshoot Upstream water flow Downstream water level equilibrium (EI1) (EI2) (EI3) (dm 3.s 1 ) (dm) (EI1) (EI2) (EI3) equilibrium t(s) t (s)

45 Conclusion 45 We have considered non-homogeneous systems of n conservation laws. The control problem is the boundary damping of the solutions. Main result states that If the damping condition is sufficiently large If the perturbations of the homogeneous system are sufficiently small Then the solution converges (exponentially) to the (perturbed) equilibrium. We applied this result to the level and flow control in a reach where control actions = the both spillways = the boundary conditions Illustrations on numerical simulations for a real river on an experimental setup

46 Conclusion 46 Under actual investigation Consider other damping condition e.g. using the condition of [Coron, Bastin, d Andréa-Novel, 08] Consider other applications e.g. on fluid models for road traffic networks [Haut, Bastin, 05] or on hydraulic installations with pipe filled with water (some preliminary results have been done in that direction)