Boundary feedback control in networks of open channels

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1 Available online at Automatica 39 (3) Boundary feedback control in networks of open channels J. de Halleux a, C. Prieur b, J.-M. Coron c, B. d Andrea-Novel d, G. Bastin ad a Centre for Systems Engineering and Applied Mechanics (CESAME), Universite Catholique de Louvain, Bâtiment Euler, 4 6, Avenue Georges Lemaitre, 1348 Louvain-la-Neuve, Belgium b CNRS, Laboratoire SATIE, ENS de Cachan, 61, Avenue du President Wilson, 9435 Cachan, France c Departement de Mathematiques, Analyse numerique et EDP, Universite Paris-Sud, Bâtiment Orsay, France d Centre de Robotique, Ecole des mines de Paris, 6, Bvd Saint Michel 757 Paris, Cedex 6, France Received 5 February received in revised form 1 November accepted 17 March 3 Abstract This article deals with the regulation ofwater ow in open-channels modelled by Saint-Venant equations. By means ofa Riemann invariants approach, we deduce stabilizing control laws for a single horizontal reach without friction. The stability condition is extended to a general class ofhyperbolic systems which can describe canal networks with more general topologies. A control law design based on this condition is illustrated with a simple case study: two reaches in cascade. The proofofthe main stability theorem is based on a previous result from Li Ta-tsien concerning the existence and decay ofclassical solutions ofhyperbolic systems.? 3 Elsevier Ltd. All rights reserved. Keywords: Partial dierential equations Hyperbolic systems Stabilization Water management 1. Introduction The so-called Saint-Venant equations are the partial differential equations (PDE) that are commonly used in hydraulics to describe the ow ofwater in open channels (see e.g. the textbooks in Chow, 1954 or Graf, 1998). These equations are a standard tool for solving engineering problems regarding the dynamics ofcanals and rivers. In this paper we will focus our attention on canals made up of a cascade ofreaches delimited by underow gates. Such systems typically occur in canalized water-ways and irrigation networks. But we shall see that the results ofthe paper are directly applicable to more complicated networks ofcanals and other kinds ofcontrol gates. We address the problem ofregulating the water level and the water velocity in a channel by using the gate openings as control actions. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction ofeditor Hassan Khalil. Corresponding author. Fax: addresses: dehalleux@auto.ucl.ac.be (J. de Halleux), christophe.prieur@satie.ens-cachan.fr (C. Prieur), jean-michel.coron@ math.u-psud.fr(j.-m. Coron), andrea@caor.ensmp.fr (B. d Andrea- Novel), bastin@auto.ucl.ac.be (G. Bastin). This problem has been considered for a long time in the literature as reported in the survey paper Malaterre, Rogers and Schuurmans (1998) which involves a comprehensive bibliography. Starting from rudimentary and heuristic feedback control approaches, various advanced control methods where progressively investigated. Among other relevant references, we may mention for instance: LQ control methods which have been especially developed and studied in Balogun, Hubbard, and De Vries (1988), Garcia, Hubbard, and De Vries (199) and Malaterre (1998). On the basis ofnite-dimensional discrete linear approximations ofthe Saint-Venant equations. Robust H control design techniques which are developed in Litrico and Georges (1) and Litrico (1) on the basis ofa model approximation by a simple linear diusive wave equation. Boundary PI regulation which is analyzed in Xu and Sallet (1999) on the basis ofa linear PDE model around a steady state. In this paper, we go a step further since the control design is derived and analyzed directly from the nonlinear 5-198/3/$ - see front matter? 3 Elsevier Ltd. All rights reserved. doi:1.116/s5-198(3)19-

2 1366 J. de Halleux et al. / Automatica 39 (3) H up V( x, t) H( x, t) u 1 u x Fig. 1. An horizontal reach delimited by underow gates. Saint-Venant partial dierential equations without any model approximation, linearization or discretisation. This approach has also been used previously in Coron, d Andrea-Novel, and Bastin (1999) and Leugering and Schmidt (). In Section, we start our analysis with the ow modelling in the special case ofa single reach. Two dierent forms ofthe model, respectively, in terms ofow velocity and Riemann invariants are successively established. Sucient conditions for the system stability are then stated in Theorem 1. This theorem is due to Greenberg and Li (1984). Section 3 deals with the boundary control design in a single horizontal reach without friction. A control law is proposed on the basis ofthe Riemann invariants whose stabilizability is analyzed as an application oftheorem 1. Some illustrative simulation experiments ofthe control law are given in Section 4. The main result ofthe paper is presented in Section 5. The aim is to generalize the previous result to open channels made up ofseveral interconnected reaches in cascade. For the sake ofclarity, we treat the special case oftwo reaches in cascade. Our stability result is given in Theorem 4 which, as we will see in Appendix, is a consequence ofa theorem due to Li (1994). The theorem provides a sucient stability condition which can be applied to the stability analysis ofcanal networks having more general topologies (like for instance the star congurations considered in Leugering & Schmidt, ). Some conclusions are given in Section 6.. Modelling in open channels.1. Saint-Venant equations Let us consider a one-dimensional portion ofa canal delimited by two underow gates as depicted in Fig. 1 under the following modelling assumptions: the canal is horizontal, the canal is prismatic with a constant rectangular cross section and a unit width, the friction eects due to walls are neglected. L H do The dynamics ofthe system are then described by the Saint-Venant equations Saint-Venant (1871) (also called shallow water H + A(H V H = V where A(H V ) is the characteristic matrix V H A(H V )= g V x [L] is the space coordinate, L is the length ofthe reach, t is time, V (x t) and H(x t) are the water velocity and depth (at point x and time t) and g is the gravity constant. The control actions are provided by two underow gates located at the left end (x = ) and the right end (x = L) of the reach (see Fig. 1). The gate openings are denoted by u 1 and u. A standard discharge relationship for under ow gates is as follows: V (t) V (t) H (t)=u 1 (H up H(t)) () V (L t) V (L t) H (L t)=u (H(L t) H do ): (3) The left and right water levels outside the reach, denoted H up and H do, are supposed to be constant and satisfy the inequality H up H do. Eqs. () and (3) constitute the boundary conditions at x = and L, associated with the PDEs (1)... Steady states For given constant openings u 1 and u there exists a constant steady-state solution ( H V ) ofeq. (1) which satises, from () and (3), the following relations: H = u 1H up +u H do u 1 +u (4) V = 1 u 1 u (H up H do ): (5) H u 1 +u By inverting these relations, it is interesting to note that any arbitrary steady state ( H V ) satisfying the conditions H do H H up and V can be assigned by an appropriate choice ofu 1 and u..3. Characteristic velocities The eigenvalues c and c ofthe characteristic matrix A(H V ), c (H V )=V + gh and c (H V )=V gh (6) are called the characteristic velocities ofthe uid in the canal. The ow is said to be subcritical or uvial ifthe characteristic velocities are ofopposite sign: c (H V ) c (H V )

3 J. de Halleux et al. / Automatica 39 (3) β t α(,t ) = -k (,t ) t t α(x,t) = α(,t ) x β(x,t) = β(l,t 1 ) β(l,t ) = -k α(l,t ) t 1 L 1 L 1 Fig.. Following the invariants along the characteristic curves. which is equivalent to V gh:.4. Model in terms of Riemann invariants Let us now consider the following change of coordinates: = V V +( gh g H) = V V ( (7) gh g H): With these new coordinates (, ) system (1) is rewritten in the following diagonal + (8) where c and c are the characteristic velocities now expressed in terms of, : c = V + g H c = V g H: The solutions (x t) and (x t) of(8) are classically called Riemann invariants (see e.g. Serre, 1996, Vol. I, p. 96). The reason is that, for any given smooth solution H(x t), V (x t)of(1), the corresponding solutions (x t) and (x t) of(8) are easily shown to be constant along the characteristic curves (see Fig. in Section 3.) (x(t)t) which are dened as dx for dt = c ((x t)(x t)) dx for dt = c ((x t)(x t)) : Note that the change ofcoordinates (7) is a bijection..5. Stability analysis It is obvious that the equilibrium H, V expressed in the, coordinates is = and =: The stability ofthe ow in a neighborhood ofthis steady state in a single reach can be analyzed with the following theorem. The theorem is stated here in a rather general form because it will be used also later on for the control stability analysis. We consider the Saint-Venant equations (8), expressed in (,) coordinates, dened on the domain (x t) [L] [ ). The boundary conditions are supposed to be given in the following general form: f ( )= and f L ( L L )= where ( ):=((t)(t)) and ( L L ):=((L t) (L t)), and with the functions f and f L being ofclass C 1. By dierentiating these boundary conditions with respect to t and using Eq. (8), we have the following so-called boundary compatibility conditions on the initial state ((x )(x ))=( (x) (x)): f ( () ()) = f L ( (L) f ( () ())c ( () ())@ x () +@ f ( () ())c ( () ())@ x f L ( (L) (L))c ( (L) (L))@ x (L) +@ f L ( (L) (L))c ( (L) (L))@ x (L)=: (9) To state the following result, we need to dene the classical norms on C ([L]) and C 1 ([L]). Given continuous on [L] and dierentiable continuous on [L], we denote C ([L]) = max x [L] (x) C1 ([L]) = C ([L]) + C ([L]): Theorem 1. Assume that the initial conditions ( (x) (x)) C 1 ([L]) satisfy the boundary compatibility conditions (9) and that the following inequality holds: A A L 1 with A = (@f =@ ) (@f =@ ) and A L = (@f L =@ L ) (@f L =@ L ) : (1) Then, for all such that ( V g H)ln(A A L ) L (11) g H there exist positive constants, M, such that, if the initial condition is small enough: ( ) C 1 ([L]) + ( ) C 1 ([L]) 6

4 1368 J. de Halleux et al. / Automatica 39 (3) there is a unique solution (x t), (x t) of class C 1 on [L] [ ) which decays to zero with an exponential rate: t [ + [, ( t) C 1 ([L]) + ( t) C1 ([L]) 6 Me t : If the product A A L =, the right-hand side of (11) is equal to +. This theorem, which is a direct application oftheorem in Greenberg and Li (1984), can be used for instance for the stability analysis ofthe steady state in a single reach with constant gate openings u 1 and u. In that case, indeed, the functions f and f L are as follows: ( f ( )= ( f L ( L L )= V + + ) (( +4 g H) 16g ( u 1 H up ( +4 ) g H) 16g V + L + L ) (( L L +4 g H) 16g ( ( L L +4 ) g H) u H do : 16g It follows that A, A L are given by V H( H g V H) u 1 H A = V H( H g + V H)+ u 1 H V H( H g + V H) u H A L = V H( H g V H)+ u H : Ifthe gate openings u 1 and u are chosen such that A A L 1, the steady state is unstable and the trajectories diverge as it will be illustrated in Section 4.. In such a case, the stabilization ofthe steady state may be achieved with the feedback control techniques that are presented hereafter. 3. Feedback control design for a single reach 3.1. Statement of the control problem The control objective is to regulate system (1) at the set point ( H V ). The control actions are the two gate openings u 1 and u. The water levels H(t) and H(L t) are supposed to be measured online at each time instant t. The external constant water levels H up and H do are known. 3.. Control design based on Riemann invariants From (7), it is obvious that the set point ( H, V ) expressed in the (, ) coordinates is = and =: ) ) The control objective can thus be reformulated as the problem ofnding boundary controls able to regulate (x t) and (x t) at zero. Consider a solution (x t) along its characteristic curve, starting from (t ). By the invariance property and for ( ) C ([L]) + ( ) C ([L]) suciently small, there exist obviously a time instant t 1 t such that (L t 1 )=(t ). Suppose now that we are able to apply a boundary control at the right gate (x = L) such that (L t 1 )= k L (L t 1 ) with k L. Obviously, there exists a time instant t t 1 such that (t )=(L t 1 ). We now apply a boundary control at the left gate such that (t )= k (t ) with k and so on. This implies clearly that, for any arbitrary t there is a monotonically increasing sequence oftime instants t i i= 1 :::such that (t j )=(k k L ) j (t ) (L t j+1 )=(k k L ) j (L t 1 ) j =1 ::: : We choose k and k L such that 6 k k L 1, this allows to understand why this boundary control will guarantee the convergence of (x t) and (x t) to zero. The required boundary control is thus implicitly dened as (t)= k (t) and (L t)= k L (L t): (1) By using the change ofcoordinates (7), we get the following explicit expressions in the (H V ) coordinates: V = V ( gh g H) V L = V + L ( gh L g H) with = 1 k and L = 1 k L : 1+k 1+k L The feedback control actions (i.e. the gate openings) are then deduced from () and (3): u 1 = ( V ( gh g H)) H H up H u = ( V L ( gh L g H)) HL : (13) H L H do Obviously, these control laws are positive and well de- ned only if H up H and H L H do. The stability ofthe closed-loop system in a neighborhood ofthe set point is trivially analyzed with Theorem 1. Indeed, the functions f and f L representing the boundary conditions are immediately given by (1) f ( )= + k L f L ( L L )= L + k L L : It follows readily that A = k, A L = k L and the stability condition A A L 1 is satised for any positive constants k k L such that 6 k k L 1.

5 J. de Halleux et al. / Automatica 39 (3) E (t) 15 1 H (x,) [m] t [s] Fig. 3. Evolution ofthe entropy. Legend: solid line for Riemann control, dashed for open-loop control. It must be emphasized that the practical implementation ofthe feedback control law (13) is very simple since it involves on-line measurements ofthe water level H (t) and H L (t) at the gates only. Thus, neither water level measurements inside the reach nor any water velocity measurements (which are much more dicult to carry on in practice) are needed. Furthermore, the control laws are totally decentralized: the control u 1 at the left gate depends only on the local measurement H at the same gate but not on the measurement H L at the other gate, and vice versa. 4. Simulation experiments 4.1. Comparison with a unit-step open-loop control law The control design method is illustrated with some realistic simulation experiments. In this section, we consider a small channel which is typical in local irrigation networks. Simulations for larger waterways will be given in the next section. The simulation parameters are L = 1 m, width =1m,H up =m,h do =:5 m,h(x )=1:4 m, Q(x )=3m 3 s 1, H =:7 m, Q =1m 3 s 1. The Saint-Venant equations are integrated numerically using a standard Preissman scheme (see e.g. Graf, 1998, Chapter 5) with a spatial step size x = 1 m, a time step t = 1 s and a weighting coecient =:57. Along the solutions, we will keep a subcritical ow with a Froude number around.35. The Riemann control is implemented with gain values k =:1 and k L =:45: These gain values have been roughly optimized in order to get reasonably smooth control actions. The deviation ofthe water state with respect to the equilibrium is measured by the entropy ofthe uid (see Coron et al., 1999): E = L H (V V ) + g (H H) dx: (14) In Fig. 3, the evolution ofthe entropy E is represented for the closed-loop system and compared with the open-loop system where unit-step gate openings are applied at the initial time instant (i.e. without any on-line feedback). In this H (x,5) [m] H (x,1) [m] x [m] x [m] x [m] Fig. 4. Water depth curves H(x t) fort = 5 1 s. Legend: solid line for Riemann control, dashed for open-loop control. gure, we can rst observe that the Riemann control law effectively stabilizes the system at the desired set point. More important, we can appreciate the acceleration ofthe convergence compared with the open-loop behavior. In Fig. 4, the proles ofthe water levels in the channel are displayed at time instants t = 5 and t =1 s. It can be observed that the closed-loop control strategy provides a fast convergence compared to the open-loop control with waves ofrelatively small amplitudes. In Fig. 5, the evolutions ofthe control actions u 1 (t) and u (t) are displayed. 4.. Simulation with an unstable open-loop We now consider simulation conditions such that A A L 1 for the open-loop control: H up = m, H do =:1 m, L =1m, H =1:93 m, V =1:76 ms 1. For these settings, the product A A L =1:5. The open-loop is compared with a closed-loop control with tuning parameters k =: and k L =:8. The evolution ofthe entropy is depicted in Fig. 6. One can see that the feedback control loop eectively stabilizes the canal while the open loop diverges.

6 137 J. de Halleux et al. / Automatica 39 (3) u (t) u1 (t) t [s] t [s] Fig. 5. Gate openings. Up: u 1, Down: u. Legend: solid line for Riemann control, dashed for open-loop control. E (t) t [s] Fig. 6. Evolution ofentropy. Legend: solid line for Riemann control, dashed for open-loop control. H up V 1( x, t) H 1,L H 1, H 1( x, t) H, u 1 u V ( x, t) H ( x, t) x L x L Reach 1 Reach H,L H u do 3 Fig. 7. A canal with two reaches and three underow gates. 5. Control of multireach canals The aim ofthis section is to generalize the previous sucient stability condition to open channels made up ofseveral interconnected reaches and to show how this condition can be used for control law design. However, for the sake of clarity, we shall treat explicitly the special case oftwo reaches in cascade separated by three underow gates as depicted in Fig Modelling The following notations are introduced (see Fig. 7 and Section.1 above). For the sake ofsimplicity, we assume without loss ofgenerality that the two reaches have the same length L. H i (x t) is the water level in the ith reach (i =1 ): H i = H i (t) and H il = H i (L t): V i (x t) is the water velocity in the ith reach: V i = V i (t) and V il = V i (L t): H up and H do ( H up ) are the left and right water levels outside the canal. The Saint-Venant equations (1) are written for each reach i =1 Hi + A(H i V i Hi = : V V i The ows are sub-critical in each reach: V i gh i : (16) The control actions are provided by the three gate openings denoted u 1 for the left gate, u for the intermediate gate and u 3 for the right gate. The discharge relationships for the three gates are similar to, 3. The discharge relationships for the three gates are written as V 1 V 1 H1 = u 1 (H up H 1 ) V 1L V 1L H 1L = u (H 1L H ) V L V L H L = u 3 (H L H do ): (17) Note that an equation ofow conservation is involved at the intermediate gate: H 1L V 1L = H V : (18) Eqs. (17) and (18) are the four boundary conditions associated with PDEs (15). 5.. Steady states As for the single reach case, for given constant gate openings u 1, u, u 3 there exists a steady-state solution ( H 1 H V 1 ) ofthe Saint-Venant equations. Any arbitrary steady state ( H 1 H V 1 ) satisfying the conditions H do H H 1 H up and H 1 V 1 can be assigned by an appropriate choice ofu 1,u and u Statement of the control problem The control objective is to stabilize the water levels H 1 and H and the velocity V 1 and V at given set points ( H 1 H V 1 ). The control actions are the three gate openings u 1, u and u 3. The levels H 1 (t), H 1L (t), H (t) and H L (t) are supposed to be measured on-line at each time instant t. The external water levels H up and H do are known.

7 J. de Halleux et al. / Automatica 39 (3) Control analysis from Riemann invariants The Riemann invariants for the ith reach (i =1 ) are denoted i (x i t) and i (x i t) with Reach 1 Reach α 1 β 1 α 1 β i = i (t) and il = i (L t): In terms ofriemann invariants, model (15) is written i = i (t) i L = i (L t) i = i (t) i L= i (L t) (19) with c i ( i i ) c i ( i i ). As we have seen above, the control objective is to stabilize the system at the origin. We are looking for decentralized boundary control laws where the control action at a given gate is a feedback function of the state variables at the same gate only. Let us assume that such decentralized feedback control laws are selected for the three gate openings: u 1 ( 1 1 ) u ( 1L 1L ) u 3 ( L L ): Introducing these expressions in (13) and using the change ofcoordinates ( i i ) (H i Q i ), the four boundary conditions (17) may be formally written as f 1 ( 1 1 )= f ( 1L 1L )= f 3 ( 1L 1L )= f 4 ( L L )=: () We dene the following vectors (x t), + (x t), (x t) by 1 (x t) 1 (x t) (x t)= + (x t)= (x t) (x t) and (x t) (x t)= : + (x t) Model (19) is @ +( ) = where ( ) is a diagonal matrix in R 4 4 dened by ( ) = diag(c 1 ( )c ( )c 1 ( )c ( )): We dene the vector ofboundary functions f f( (t) (L t) + (t) + (L t)) =(f 1 f f 3 f 4 ) T : () Note that due to (17), we have f()=: (3) t t β 1 (a) x x (c) α 1 x x β α Fig. 8. Illustration of(5). t (b) x x t (d) x x According to the Inverse Function Theorem (see e.g. Cartan, 1967, Theorem 4.7.1), these four equations are solvable with respect to ( (L t) + (t)) in a neighborhood ofthe origin ifthe functions f are continuously dierentiable and satisfy the following conditions: det [ (Lt) +(t)]f() (4) where [ (Lt) +(t)]f denotes the Jacobian of f with respect to the vector [ (L t) T + (t) T ] T. Then the boundary conditions () may be rewritten as follows in a neighborhood of the origin: (L t) (t) = g (5) + (t) + (L t) where g : R 4 R 4 is a suitable function. The Jacobian of g is dened at the equilibrium by g() = ( [ (Lt) +(t)]f()) 1 [ (t) +(Lt)]f() where g denotes the Jacobian of g with respect to the vector [ (t) T + (L t) T ] T. Remark. A natural justication ofthe form ofeq. (5) is that, at each boundary, the outgoing invariants (L t)= ( i (L t)) and + (t)=( i (t)) are expressed in terms of the incoming invariants (t)=( i (t)) and + (L t)= ( i (L t)). This is illustrated in Fig. 8 where the arrows denote the characteristic curves and the direction ofthe invariants. Dierentiating (5) with respect to t and using (1), it can be shown that ( (L t))@ x (L t) + ( (t))@ x + (t) (t) ( (t))@ x (t) = g (6) + (L t) + ( (L t))@ x + (L t) where ( ) = diag(c 1 ( )c ( )) and + ( )= diag(c 1 ( )c ( )). α β

8 137 J. de Halleux et al. / Automatica 39 (3) For a matrix A R m p, A=(a ij ) 1 6 i 6 m 1 6 j 6 p, we dene the following norms: p A := max a ij : i {1:::m} j=1 abs(a) R m p such that (abs(a)) ij = a ij (A):=lim A l 1=l = max{ z : z C and det(a zi)=}: Eqs. (5) and (6) lead to the denition ofthe compatibility condition (C): Denition 3. The function C 1 ([L] R 4 ) satises the compatibility condition (C) if (L) () = g +() +(L) ( ( (L))@ x (L) ) () = g + ( ())@ x +() +(L) ( ( ())@ x ) () : + ( (L))@ x +(L) The main result ofthis section is the following. Theorem 4. If (abs( g()) 1 then there exists, and C such that, for all C 1 ([L] R 4 ) satisfying condition (C) and such that C 1 ([L]) 6 there exists one and only one function C 1 ([L] [ + [ R 4 ) satisfying (1) (5) and (x ) = (x) x [L]: (7) Moreover, this function decays to zero with an exponential rate (: t) C1 ([L]) 6 Ce t C1 ([L]) t : Proof. This theorem is a special case ofthe more general Theorem 6 which is proved in Appendix A. As explained in the appendix, this theorem is a consequence ofa previous result of Li (1994) on the stability ofhyperbolic systems Example of application Our purpose in this section is to illustrate how the above theorem can be used to analyze the stability ofa particular control structure for a channel with two reaches in cascade. We intend to apply the Riemann invariant approach of Section 3.. We choose to select the gate opening u to stabilize the water velocity ofthe upstream reach: V 1L = V k 1+k ( gh 1L or, expressed in Riemann invariants, f 3 ( 1L 1L )= 1L + k 1L g H 1 ) for the positive gain k in [ 1[ that can be used to tune the control sensitivity. The other control laws are identical to the single reach case f 1 ( 1 1 )= 1 + k 1 1 f 4 ( L L )= L + k 3 L : The boundary condition f is given by the ow conservation condition (18): f ( 1L 1L ) =( 1L + 1L +V 1 )( 1L 1L +4 g H 1 ) ( + +V )( +4 g H ) : For these boundary conditions, we can compute g: [ (Lt) +(t)]f() = 16 gh 1 ( [ (t) +(Lt)]f() 1 gh 1 V 1 ) 16 gh ( gh + V ) 1 1 k 1 16 gh ( gh V ) 16 gh 1 ( gh 1 + V 1 ) = k k 3 g() k k 3 = k 1 g H V g H 1 ( gh 1 (1 k )+ V 1 (1+k )) g H ( gh + V ) g H + V and, nally, k1 k (abs( g())) = max g k 3 H V gh + V :

9 J. de Halleux et al. / Automatica 39 (3) x1e3.6 E (t) H (x,5)-he(x) [m] e+3 1.5e+3 e+3.5e+3 t [s] Fig. 9. Evolution ofthe entropy. Legend: solid line for Riemann control, dashed for open-loop control. This control design is illustrated with a realistic simulation experiment for a channel with a constant rectangular cross section (unit width) and two reaches oflength L = 1 m. The two external water levels are H up = m and H do =: m. The set points are selected as ( H 1 H )=(1 :8) m Q =:5 m 3 s 1 : The initial conditions are, for x [L] (H 1 (x )H (x ))=(1:5 1:) m Q(x )=1:5 m 3 s 1 : The control gains k 1, k and k 3 are set to.. With these numerical values, (abs( g()))= :89 for the open loop (abs( g()))= :35 for the closed loop (8) which satises the required inequality oftheorem 4. The Saint-Venant equations are integrated numerically with a spatial step x = 1 m, a time step t = 5 s and a weighting coecient = :6. In Fig. 9 the evolution ofthe entropy function L1 (V 1 V 1 ) E = [H 1 + g (H 1 H 1 ) ] dx L (V V ) + [H + g (H H ) ] dx: is represented for the closed-loop system and compared with the open-loop system where unit-step gate openings are applied at the initial time instant. In this gure, we can observe the acceleration ofthe convergence with respect to the open-loop behavior since the response time in closed-loop is much smaller than in open-loop: around 5 s (8 min s) for the closed-loop simulations and more than 5 s (41 min 4 s) for the open-loop. In Fig. 1, the deviations ofthe water levels in the channel are displayed at time instants t = 5 1 and 15 s. 6. Conclusions In this paper, a general sucient stability condition for water velocities and water levels in open channels has been H (x,1)-he(x) [m] H (x,15)-he(x) [m] 5 1e+3 1.5e+3 e+3 x [m] e+3 1.5e+3 e+3 x [m] e+3 1.5e+3 e+3 x [m] Fig. 1. Deviation ofthe water depth, H(x t) H(x), at time instants 5, 1 and 15 s. Legend: solid line for Riemann control, dashed for open-loop control. described and analyzed. A control law design based on this stability condition has been proposed and applied to reaches in cascade. The main theoretical result ofthe paper is an application ofa previous result ofli Ta-tsien given in Theorem 6. For the sake ofsimplicity, the theorem has been applied to a prototype canal made up oftwo horizontal reaches in cascade with a rectangular cross section and under ow gates. Theorem 6 as such does not give a fully systematic method for the design of stabilizing boundary control laws for general canal networks. It is, however, worth to emphasize that Theorem 6 is valid for any hyperbolic PDE system (A.1) with boundary conditions ofthe form (A.) as long as the damping condition (A.3) is satised. Hence, as it is clearly illustrated in this paper, Theorem 6 provides a very ecient tool for verifying the stabilizability properties of boundary control laws for canal networks with more general topologies, with reaches having non-rectangular cross sections and other kinds ofhydraulic gates like mobile spillways (see e.g. de Halleux & Bastin, ), provided they can be cast in the form (A.1) with boundary conditions ofthe form (A.).

10 1374 J. de Halleux et al. / Automatica 39 (3) The control laws analyzed in this paper are based on Riemann invariants. An alternative Lyapunov approach is studied in Coron et al. (1999), de Halleux, d Andrea-Novel, Coron, and Bastin (1) with the entropy function introduced in Section 4 ofthis paper as Lyapunov function. Acknowledgements This paper present research results ofthe Belgian Programme on Interuniversity Poles ofattraction, initiated by the Belgian State Prime Minister s Oce for Science, Technology and Culture. The scientic responsibility rests with its authors. Appendix A. Proof of Theorem 4 As mentioned after the statement of Theorem 4, we give in this section a more general result than Theorem 4 and explain how this general result is a direct application of Theorem 1.3 in Li (1994, Chap. 5). Let 6 m 6 n be two integers. Let : R n R n n be a continuously dierentiable function in a neighborhood of R n such that = diag( 1 ::: n ) with i () i {1:::m} and i () i {m + 1:::n}. Let : R n R m m and + : R n R (n m) (n m) be the two functions dened by = diag( 1 ::: m ) = diag( m+1 ::: n ): For all =( 1 ::: n ) T R n, let =( 1 ::: m ) T R m and + =( m+1 ::: n ) T R n m. We consider, for x [L] and t R, the +( = (A.1) together with the boundary condition (L t) (t) = g (A.) + (t) + (L t) where g : R n R n is a continuously dierentiable function in a neighborhood of R n satisfying g()=. Similarly as in Denition 3 for (1) and (5), we dene the compatibility condition for (A.1) and (A.): Denition 5. The function C 1 ([L] R n ) satises the compatibility condition (C) if (L) () = g +() +(L) (L) = g () + ( () +(L) () : + ( (L) Theorem 4 is a consequence ofthe following result by choosing n = 4 and m =. Theorem 6. If (abs( g()) 1 (A.3) then there exists, and C such that, for every C 1 ([L] R n ) satisfying condition (C) and such that C1 ([L]) 6 there exists one and only one function C 1 ([L] [ + ) R n ) satisfying (A.1), (A.) and (x ) = (x) x [L]: (A.4) Moreover, this function satises (: t) C1 ([L]) 6 Ce t C1 ([L]) t : (A.5) This theorem is a special case of(the proofof) Theorem 1.3 in Li (1994, Chapter 5) ifthe boundary condition (A.) has the following particular form: (L t) g1 ( + (L t)) = (A.6) + (t) g ( (t)) where g 1 : R n m R n and g : R m R n are ofclass C 1 on neighborhoods of R n m and of R m respectively. 1 But one can use Li (1994) to prove Theorem 6 even if(a.6) does not hold by doubling the size ofthe state as follows. Consider the + = (A.7) with =( 1 T T 1+ T +) T T where 1 R m, R n m, 1+ R n m, + R m and :R n R n n is dened by (( 1 T 1+) T T ) ( ) + (( + T ) T T ) = diag : + (( 1 T 1+) T T ) (( + T ) T T ) 1 Let us point out that with the denition of A min given in Li (1994, p. 17) one has A min = abs(a) min = (abs(a)). See Lemma.4 in Li (1994, p. 146).

11 J. de Halleux et al. / Automatica 39 (3) The boundary condition for (A.7) is dened by () 1 (L t) 1 1+ (L t) = g (L t) 1 + (L t) 1+ (t) 1 1 (t) = g : + (t) 1 (t) This boundary condition can be written in the following form: ( (L t) + (L t) g1 ( ) + (L t)) = g = + (t) (t) g ( (A.8) (t)) with =( T 1 T ) T + =( T 1+ T +) T (A.9) ( ) 1 ( ) 1 g 1 = g 1 g = 1 g: (A.1) In particular, the boundary condition for has the special form required by Theorem 1.3 in Li (1994, Chap. 5) and (abs( g())) = (abs( g())) : (A.11) Let C 1 ([L] R n ) satisfying condition (C) and such that C 1 ([L]) is small enough. We choose as initial condition for at t =, 1 (x)= (x) (x)= +(L x) 1+(x)= +(x) +(x)= (L x): (A.1) One easily sees that := ( 1 T T T 1+ T + )T satises the compatibility condition associated to (A.7) and (A.8). Hence there exists a unique C 1 -solution of(a.7) and (A.8) such that (x ) = (x): Let + (L x t) T 1+ (L x t) T (x t)= : (L x t) T 1 (L x t) T (A.13) Then, as one easily checks, satises as hyperbolic system (A.7), boundary condition (A.8) and initial condition (A.13). Hence by the uniqueness ofthe C 1 -solution ofthe Cauchy problem associated to (A.7) and (A.8), one has = : In particular, 1 (x t)= + (L x t) 1+ (x t)= (L x t): Hence, if (x t) := 1 (x t) + (x t) := 1+ (x t) (A.14) then =( T T +) T satises (A.1), (A.) and (A.4). Conversely, if =( T T +) T satises (A.1), (A.) and (A.4), then dened by 1 (x t) := (x t) + (x t) := (L x t) 1+ (x t) := + (x t) (x t) := + (L x t) satises hyperbolic system (A.7), boundary condition (A.8) and initial condition (A.13). Hence, see also (A.11), Theorem 6 for the hyperbolic system (A.1) and the boundary condition (A.) is a consequence of(the proofof) Theorem 1.3 in Li (1994, Chap. 5) (see also Qin, 1985 Zhao, 1986) for the hyperbolic system (A.7) and boundary condition (A.8). References Balogun, O., Hubbard, M., & De Vries, J. (1988). Automatic control of canal ow using linear quadratic regulator theory. Journal of Hydraulic Engineering, 119(4), Cartan, H. (1967). Calcul dierentiel. Paris: Hermann. Chow, V. (1954). Open-channel hydraulics (International Student ed.). New York: McGraw-Hill. Coron, J.-M., d Andrea-Novel, B., Bastin, G. (1999). A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations. In CD-Rom proceedings, Paper F18-5. ECC 99, Karlsruhe, Germany. de Halleux, J., Bastin, G. (). Stabilization ofsaint-venant equations using Riemann invariants: Application to waterways with mobile spillways. In CD-Rom proceedings, Barcelona, Spain. de Halleux, J., d Andrea-Novel, B., Coron, J.-M., Bastin, G. (1). A Lyapunov approach for the control of multi reach channels modelled by Saint-Venant equations. In CD-Rom proceedings NOLCOS 1, St-Petersburg, Russia (pp ). Garcia, A., Hubbard, M., & De Vries, J. (199). Open channel transient ow control by discrete time LQR methods. Automatica, 8(), Graf, W. (1998). Fluvial hydraulics. New York: Wiley. Greenberg, J.-M., & Li, T.-t. (1984). The eect ofboundary damping for the quasilinear wave equations. Journal of Dierential Equations, 5, Leugering, G., & Schmidt, J.-P. G. (). On the modelling and stabilisation ofows in networks ofopen canals. SIAM Journal of Control Optimization, 41(1), Li, T.-t. (1994). Global classical solutions for quasilinear hyperbolic systems. Research in applied mathematics. Paris, Milan, Barcelona: Masson and Wiley. Litrico, X. (1). Robust ow control ofsingle input multiple output regulated rivers. Journal of Irrigation and Drainage Engineering, 17(5), Litrico, X., & Georges, D. (1). Robust LQG control ofsingle input multiple output dam river systems. International Journal of Systems Science, 3(6),

12 1376 J. de Halleux et al. / Automatica 39 (3) Malaterre, P. (1998). Pilot: A linear quadratic optimal controller for irrigation canals. Journal of Irrigation and Drainage Engineering, 14(3), Malaterre, P., Rogers, D., & Schuurmans, J. (1998). Classication ofcanal control algorithms. Journal of Irrigation and Drainage Engineering, 14(1), 3 1. Qin, T.-h. (1985). Global smooth solutions ofdissipative boundary value problems for rst order quasilinear hyperbolic systems. Chinese Annals of Mathematics, 6B, Saint-Venant, B. d. (1871). Theorie du mouvement non-permanent des eaux avec applications aux crues des rivieres et a l introduction des marees dans leur lit. Comptes-rendus de l Academie des Sciences, 73, , Serre, D. (1996). Systemes de lois de conservations I and II. Diderot editeur, Art et Sciences. Xu, C., Sallet, G. (1999). Proportional and integral regulation ofirrigation canal systems governed by the Saint-Venant equation. In Proceedings of the 14th world control congress, Beijing, China. Zhao, Y.-c. (1986). Classical solutions for hyperbolic systems. Ph.D. thesis, Fudan University (in Chinese). Jonathan de Halleux received the Applied Mathematics engineering degree in from Universite Catholique de Louvain, Louvain-la-Neuve, Belgium. He is presently doing a Ph.D. in the same institution on the stabilization ofnonlinear partial dierential equations. Christophe Prieur was born in Essey-les- Nancy, France, in He graduated in Mathematics from the Ecole Normale Superieure, France in. He received the Ph.D. degree in 1 in Applied Mathematics from the Universite Paris-Sud, France. Since he is research associate CNRS at the Laboratoire SATIE, Ecole Normale Superieure de Cachan, France. His current research interest include nonlinear control theory, robust control and control of nonlinear partial dierential equations. Jean-Michel Coron obtained the diploma of engineer from Ecole polytechnique in 1978 and from the Corps des Mines in He got his These d Etat in 198. He was researcher at Ecole Nationale Superieure des Mines de Paris, associate professor at Ecole polytechnique. He is currently Professor at Universite Paris-Sud. His research interests include nonlinear partial dierential equations, calculus ofvariations and nonlinear control theory. Brigitte d Andrea-Novel graduated from Ecole Superieure d Informatique Electronique Automatique in She received the Doctorate degree from Ecole Nationale Superieure des Mines de Paris in 1987 and her Habilitation degree from Universite Paris-Sud in She is currently Professor of Systems Control Theory and responsible for a research group on Advanced Controlled Systems at the Centre de Robotique Ecole des Mines de Paris. Her current research interests include nonlinear control theory and applications to underactuated mechanical systems, control ofwheeled vehicles with application to automated highways and boundary control ofexible mechanical systems coupling ODEs and PDEs. Georges Bastin received the electrical engineering degree and the Ph.D. degree, both from Universite Catholique de Louvain, Louvain-la-Neuve, Belgium. He is presently Professor in the Center for Systems Engineering and Applied Mechanics (CESAME) at the Universite Catholique de Louvain and Associate Professor at the Ecole des Mines de Paris. His main research interests are in system identication and nonlinear control theory with applications to mechanical systems and robotics, biological and chemical processes, environmental problems and communication networks.