MULTIVARIABLE BOUNDARY PI CONTROL AND REGULATION OF A FLUID FLOW SYSTEM

. The paper is concerned with the control of a ﬂuid ﬂow system governed by nonlinear hyperbolic partial diﬀerential equations. The control and the output observation are located on the boundary. We study local stability of spatially heterogeneous equilibrium states by using Lyapunov approach. We prove that the linearized system is exponentially stable around each subcritical equilibrium state. A systematic design of proportional and integral controllers is proposed for the system based on the linearized model. Robust stabilization of the closed-loop system is proved by using a spectrum method.

1. Introduction. In the paper we study a fluid flow system governed by the Saint Venant equation ∂Q(x, t) ∂t + ∂ ∂x Q 2 (x, t) S(x, t) + gS(x, t) ∂Z(x, t) ∂x + gη 2 R −4/3 Q 2 (x, t) S(x, t) = 0, Q(0, t) = Q in (t), Z(l, t) = Z out (t), y(t) = (Z(0, t), Q(l, t)) (1) The partial differential equation (PDE) (1) models dynamic behaviour of the unsteady flow in a one-reach canal for shallow water, where x denotes the spatial domain (m), t the time (s), S(x, t) the flow cross-section (m 2 ), Z(x, t) the water level (m), Q(x, t) the flow discharge (m 2 /s), g the Newton acceleration constant (m/s 2 ), η the Manning coefficient for friction slope, R the hydraulic radius (sectional area/wetted perimeter), Q in (t) the upstream flow rate and Z out (t) the downstream level in the canal. The PDE (1) has three unknown functions S(x, t), Q(x, t) 502 CHENG-ZHONG XU AND GAUTHIER SALLET and Z(x, t). There is a relation between S(x, t) and Z(x, t) once the cross-section geometry is fixed : S(x, t) = S(Z(x, t)). ( Therefore we choose Z(x, t) and Q(x, t) as state, and Q in (t) and Z out (t) as control. The upstream level and the downstream flow rate are chosen as the output measurement : y(t) = (Z(0, t), Q(l, t)).
The objective is to control and regulate the output y(t) by using proportional and integral controllers (PI controllers) on the boundary. The one-reach canal system described by (1), (2) and (3) is a nonlinear multi-inputs and multi-outputs infinitedimensional system. As a first approach to tackling the control problem, we study the linearized model of the system around equilibrium states. Because of friction slope η > 0 each equilibrium state is spatially heterogeneous or function of space variable x. As a consequence the linearized model has for the coefficients functions of space variable x. Therefore studying local stability of spatially heterogeneous equilibrium states becomes more difficult than that of homogeneous one (see [1,2,25]) as the same as synthesis of stabilizing controllers [25]. Our work is motivated by the problem of controlling an open channel governed by the Saint Venant equation. The hyperbolic PDE describes the dynamics of open channel hydraulic systems, such as rivers, irrigation or drainage canals, sewers etc, assuming one dimensional flow. Description of recent developments in these aspects can be found in the published works [2,6,5].
Davison's robust PI controllers for finite dimensional systems [4] has been extended first by Pohjolainen [14] to a class of analytic semigroup systems and then to strongly continuous semigroup systems in [15,24] where either the control operator or the observation operator is bounded. The extended synthesis has found applications for systems such as counter flow heat exchanger system [24]. Synthesis of robust PI controllers has been exploited in [12] for a rather general class of infinite-dimensional systems called regular systems where both the control operator and the observation operator may be unbounded. In particular, if the input-output transfer matrix G(s) is invertible at s = 0, then it is possible to construct an integral controller which stabilizes the system and guarantees the regulation of the set point. Guided by the result we consider the linearized model of the Saint Venant system (1)-(3) where both the control and the observation are located on the boundary, and so represented by unbounded control operator and unbounded observation operator. We propose a systematic method of designing boundary PI controllers for the linearized system.
Irrigation systems have received considerable attention since ten years and numerous interesting results on the feedback stabilization have been obtained by the Lyapunov approach [3] and by the transfer function approach [11]. However there are still interesting and open questions as suggested in [1]. Specifically, when friction slope is taken into account in the Saint Venant model, i.e. η > 0 in (1), feedback stabilizability of heterogeneous equilibrium states has been established by solving some nonlinear ordinary differential equation. Indeed it is not easy to say if the ordinary differential equation has a bounded solution (see [1]). The present paper is a further endeavour towards understanding the nonlinear fluid flow system described by the Saint Venant equation. We have for the first objective to find easily checkable conditions for local stability of heterogeneous equilibrium states.
We assume the presence of friction slope in the model and study local stability of heterogeneous equilibrium states. Under general conditions exponential stability of the linearized system is proved by using a Lyapunov direct approach. Roughly speaking, if the Manning coefficient of friction slope η is small, then the linearized system is exponentially stable. Then stabilizing PI controllers are designed based on the linearized system. Exponential stability of the closed-loop system is shown by a spectral analysis method so that the output regulation is guaranteed. The contribution of the paper is twofold: proof of exponential stability of the linearized Saint Venant system and synthesis of multivariable boundary PI controllers for the hyperbolic system.
Although our study is carried out directly on the PDE describing the system, concepts such as operator semigroups, admissibility of observation and control operators, and well-posedness and regularity of linear systems are both helpful and important for the clarity of reasoning throughout the paper. We shall give a brief presentation of these notions in the next section. The interested reader is referred to Pazy [13], Weiss [22] and Tucsnak and Weiss [21] for more details.
The paper is organized as follows : Section 2 is devoted to show that integral stabilization implies regulation for regular systems; Section 3 is devoted to computing equilibrium states which are spatially heterogeneous; In Section 4 is proposed some necessary and sufficient condition for exponential stability of two coupled hyperbolic equations; The open-loop exponential stability is proved in Section 5 by taking into account friction slope; The closed-loop stability of the linearized system by the designed PI controllers is proved in Section 6 with a spectral analysis method; Section 7 contains our conclusions and the Appendix is added for proofs of technical results.

2.
Integral stabilization implies regulation. Let X be a Hilbert space and let A : D(A) → X is the generator of an exponentially stable C 0 semigroup e tA on X. The Hilbert space X 1 is D(A) with the norm z 1 = Az , where 0 ∈ ρ(A), the resolvent set of A. The Hilbert space X −1 is the completion of X with respect to the norm z −1 = A −1 z . This space is isomorphic to D(A * ) , the dual space of D(A * ) and densely and with continuous embeddings. The semigroup e tA extends to a semigroup on X −1 , denoted by the same symbol. The generator of the extended semigroup is an extension of A, whose domain is X, so that A : X → X −1 . Without loss of generality for our applications it is assumed that the input space is the same as the output space and equal to a finite dimensional real Euclidian space R m . We consider a well-posed linear system Σ with input space U = R m , state space X, output space Y = R m , semigroup generator A, control operator B, observation operator C and transfer function G. Thus, B ∈ L(R m , X −1 ) is an admissible control operator for e tA and C ∈ L(X 1 , R m ) is an admissible observation operator for e tA . The control operator B is called bounded if B ∈ L(U, X), and C is called bounded if it can be extended such that C ∈ L(X, Y ). An important subclass of the well-posed linear systems are the regular linear systems [22]. The system Σ is called regular if for each v ∈ R m , the following limit exists (in R m ) :

CHENG-ZHONG XU AND GAUTHIER SALLET
The operator D ∈ L(U, Y ) is then called the feedthrough operator of Σ. We define the following extension of C by where D(C Λ ) is the space of those x ∈ X for which the above limit exists. The system Σ is regular if and only if (sI −A) −1 BU ⊂ D(C Λ ) for some (hence, for every) s ∈ ρ(A) and, if this is the case, then Moreover, for a regular system, y(t) = C Λ x(t) + Du(t) holds for almost every t ≥ 0 (for every initial state x(0) ∈ X and every input signal u ∈ L 2 loc (R + , R m )). On the state space X the system is written under the form of evolution equations as follows The following result can be found in the literature (cf. [22] or [12]). Proposition 1. Assume that the system (6) is regular and that the semigroup e tA is exponentially stable on X. Then the following properties hold true : We consider the closed-loop system (6 ) controlled by the following integral controller :ξ(t) = y(t)−y r and u(t) = K I ξ where K I ∈ R m×m is the integral controller gain and y r ∈ R m is the set point. It is easy to see that the closed-loop system is governed by the following evolution equation where w c and w o added here represent control and output disturbances, respectively. It is well known that the closed-loop system (7 ) is also regular (see [12]). Stabilization of the closed-loop system by the integral controller implies the regulation guaranteed, independently of unknown constant disturbances w c , w o ∈ R m . Here we prove a little more on the regulation result. We say that the disturbances w c (t) and w o (t) are quadratically close to constant if there are constantsw c ,w o ∈ R m such that c (t) = (w c (t)−w c ) and o (t) = (w o (t)−w o ) belong to L 2 (R + , R m ). If the disturbances w c (t) and w o (t), instead of being constants, are quadratically close to constant, then the regulation result is still true if stabilization is fulfilled. However the regulation must be understood in a broad sense as follows.
The following general result holds true for regular linear systems having possibly unbounded control and observation operators. It means that regulation is automatically deduced from stabilization. For the reader's convenience a simple proof of the following proposition is given in the Appendix.
Remark 1. If the control and output disturbances are null, i.e., w c (t) ≡ 0 and w o (t) ≡ 0, then we recover the regulation in the classical sense : 3. Equilibrium solutions. In the paper the synthesis of boundary PI-controllers will be carried out based on the linearized Saint Venant model. For the sake of simplicity it is assumed that the canal has a rectangular cross section. As a consequence we have the relations : where B is the base width. LetQ in andZ out be positive constants. The equilibrium solution of the Saint Venant equation (1) is given by Definition 3.1. We call subcritical equilibrium state for the Saint Venant system (1) each solution (Q(x),Z(x)) of (8) satisfying the subcritical hydraulic condition : As the reader will see each subcritical equilibrium solution satisfies the inequality on the interval Lemma 3.2. Assume that the upstream flow rateQ in and the downstream levelZ out satisfy the subcritical condition (9). Then the system (8) has a unique equilibrium is a decreasing function of x obtained by solving the following differential equation : where (and hereafter)Z x denotes the partial derivative ofZ(x) w.r.t. x.
Remark 2. The right member of the equation (10) is a C 1 function ofZ. It has a unique local solution. The subcritical boundary condition being satisfied the unique solutionZ(x) is bounded on the interval [0, l] whatever l > 0.
Remark 3. If the Manning coefficient η = 0, then the equilibrium state is given by two positive constants (Q,Z) independent of x. In this case each term a k , The following lemmas are easy to prove by direct computation.
Lemma 3.3. The linearized system around the equilibrium solution is governed by Let the matrices A(x) and B(x) be defined by Lemma 3.4. The matrix A(x) defined in (14) has two different eigenvalues under the subcritical condition: the one is negative and the other is positive, given by By substituting the expressions a k (x) into the eigenvalues (15) we get the following relation : In the next section we present a result of stability concerned with two hyperbolic PDE. To the best of our knowledge the result is new in the sense that linear PDE of spatially heterogeneous coefficients are dealt with.
4. Exponential stability of coupled hyperbolic PDE. Consider the linear hyperbolic system governed by the following PDE where λ i (x) is C 1 and α i is a real constant ∀ i = 1, 2. Assume that λ 1 (x) < 0 and λ 2 (x) > 0 ∀ x ∈ [0, l]. Consider for the state space the Hilbert space X = (L 2 (0, 1)) 2 equipped with the usual quadratic norm.
Theorem 4.1. The null solution R(·, t) ≡ 0 is exponentially stable for the system (17) if and only if α 2 1 α 2 2 < 1. Remark 4. Greenberg and Li [7] have proved that, in the quasilinear case and under the form of two Riemann invariants, the above condition was a sufficient condition for the null equilibrium solution to be exponentially stable. Our condition is necessary and sufficient for exponential stability of spatially heterogeneous linear PDE. It is concerned with two different situations.
Proof of Theorem 4.1. We prove the sufficient condition first. Consider the candidate of Lyapunov functional V W : X → R + such that where the weight matrix W is a diagonal matrix defined by andW i (x), i = 1, 2, bỹ Notice that the constant ε is fixed, small and positive such that α 2 1 + ε α 2 2 < 1. Computing the time derivative of the Lyapunov functional V W (R(·, t)) along the smooth trajectories of the system (17) gives us the followinġ Hence we havė where Λ(x) = diag (λ 1 (x), λ 2 (x)) and Using the boundary condition in (17) we get . From the fact that F (θ) is a decreasing function of θ having limit equal to 1 as θ → ∞ and by the hypothesis (α 2 1 + ε)α 2 2 < 1, there exists a positive constant θ 1 > 0 such that, for every θ ≥ θ 1 ,

CHENG-ZHONG XU AND GAUTHIER SALLET
LetW be the diagonal matrix defined bỹ and let VW be the functional defined as in (18) with the weightW . It is easy to find positive constants K 1 , K 2 ,K 1 andK 2 such that (25) For every smooth initial condition R 0 ∈ X and by substituting (23) and (25) into (21), the following inequality is obtained : So we have proved exponential stability of the system (18).

5.
Open-loop stability of the Saint Venant system. We investigate first local stability of the Saint Venant system around each equilibrium solution. More precisely we are interested to know if the linearized system (13) is exponentially stable. If the viscous friction slope is null, our Theorem 4.1 is applied to prove exponential stability of the linearized system. If the friction slope is small, exponential stability still holds true for the linearized system.
The following useful identities may be proved by direct computation.
Proof of Corollary 1. If the viscous friction slope is null, each a i is constant and positive ∀ i = 1, 2, 3 and a 4 = a 5 = 0. Thus the matrix A(x) is equal to constant matrix A A = 0 −a 1 −a 2 −a 3 which has two real eigenvalues of opposite sign λ 1 and λ 2 . Consider the invertible linear transformation T 1 : L 2 (0, l) × L 2 (0, l) → L 2 (0, l) × L 2 (0, l) such that

Remark 5. It is not difficult to show that the open-loop system
If the viscous friction slope is positive, we have the following result.
Theorem 5.2. The null solution R(·, t) ≡ 0 is exponentially stable for the linearized system (13) around each subcritical equilibrium solution, provided that the viscous friction slope is small. More precisely there exists some positive constant η * > 0 such that the linearized system (13) is exponentially stable ∀ η ∈ [0, η * ].
Proof of Theorem 5.2. Consider the same transformation T 1 as the previous one in (26). Without ambiguity we set Note that T 1 (x) is a smooth functions of x because λ 1 and λ 2 are. The same transformation carried out on the linearized system (13) gives us where Direct computation gives usB(x) (λ 1 denotes the derivative of λ 1 (x) w.r.t. x.) MoreoverB(x) can be written as followsB where b ij (x) can be computed explicitly from the equilibrium state. From the equilibrium equation (10) there exists some positive constant β > 0 such that In (29) Let V W (R) be the same Lyapunov functional as in the proof of Theorem 4.1 but with ε = 0. By computing the time derivative of the Lyapunov functional V W (R) along the smooth solutions of (29), we get the followinġ where the constantsK 1 , K 2 and θ 1 are chosen as in the proof of Theorem 4.1. We have used the fact that By (25) and (33) we have some positive constant K 5 such that the following inequality holds Substituting (36) into (34) allows to find some constant η * > 0 such that for every With the same notation as in the proof of Theorem 4.1, we get So is proved exponential stability of the linearized system (13) for η > 0.
6. PI-controllers and closed-loop stability. We have shown that the linearized model (13) can be transformed into the form (29). In this section our Theorem 5.2 is applied to prove exponential stability of the linearized model controlled by the proportional output feedback law. Moreover the closed-loop system is stabilized and regulated by PI-controllers of the form : where K p , K I ∈ R 2×2 , k I ∈ R and y r ∈ R 2 is the control setpoint. In other words we give a systematic design method for tuning the controller matrices K P and K I such that the following properties hold true : (a) the closed-loop system (13) by the PI output feedback law (38) is well-posed and the associated semigroup is exponentially stable; (b) the output regulation is achieved in the sense of Definition 2.2 : independently of known or unknown disturbances quadratically close to constant.
Recall that the matrix 0 −a 1 −a 2 −a 3 in (13) has two real eigenvalues of opposite sign. Denote by λ 1 the negative one and by λ 2 the positive one Let us set and where the constants K P,1 and K P,2 are defined in (41) and Notice that the matrix K I is invertible.
Remark 6. The integral controller matrix K I in (42) is obtained by computing where G 0 is the mapping v → y defined by the following ordinary differential equation : where A(x) and B(x) are defined in (14). Notice that G 0 is just the transfer matrix G(s) of the system (13) with u = K P y + v evaluated at s = 0. If K P = 0, we recover a pure integral controller which also works for the considered system.
Our main result is summarized in the coming theorem.
(b) There exists a constant k * I > 0 such that the PI-controller (38) and (41)-(42) stabilizes exponentially the linearized Saint Venant system (13) with the output regulation guaranteed, ∀ 0 < k I < k * I and ∀ 0 ≤ η ≤ η * . In the following, we show how the linearized Saint Venant system (13) controlled by the PI controller (38) is transformed into the form (29) and prove Theorem 6.1 by a spectral method. The natural state space for the closed-loop system is the Hilbert space H = L 2 (0, l) × L 2 (0, l) × R 2 . 6.1. P-controller design. Synthesis of the proportional controller is to keep satisfied the following basic requirements : 1) On each boundary point the outgoing information should be determined by the incoming information for existence and uniqueness of the PDE solutions (see [18]); 2) The dissipation conditions are made satisfied "at the very most" ; 3) The exponential decay rate is made as large as possible for the underlying semigroup.
After the proportional output feedback u = K P (y − y r ) + v and by the transformation (z, q) τ = T 1 R in (26), the closed-loop system (13) or (29) is governed by the following PDE where the matricesB(x) and K P are defined in (31) and (41), respectively. Lemma 6.2. Let y r = 0. The P-controlled system in (45) is well-posed and exponentially stable ∀ η ∈ [0, η * ] for some η * > 0.
Proof of Lemma 6.2 . Substituting y into the boundary condition from (45) leads to the following PDE where C 3 is a constant matrix given by (56). The well-posedness of the system has been proved in [26]. Let us prove just exponential stability of the system. Let v = 0 and y r = 0. Define the unbounded linear operatorÃ : D(Ã) → X as follows : and for every f ∈ D(Ã),Ã It is easy to see thatÃ is the generator of a C 0 semigroup on X. By using (31) and the same argument as in the proof of Theorem 5.2 we prove exponential stability of the semigroup e tÃ and hence exponential stability of the P-controlled system. 6.2. I-controller design. By adding the integral controller the closed-loop system (46) is described by the following PDE ∂ ∂t Consider the homogeneous PDE corresponding to the above closed-loop system (49)-(50): where the linear operators C 1 : (H 1 (0, 1)) 2 → R 2 and C 2 , C 3 : R 2 → R 2 , defined by and The first objective is to prove exponential stability of the system governed by the PDE (52)-(53). For each k I > 0 we define the unbounded operator A 1 : and for every f ∈ D(A 1 ), which is exactly the right hand member of (52). Then we have the following result whose proof will be presented at the end of the section. Theorem 6.3. (i) There exists some constant k * I > 0 such that A 1 is the generator of an exponentially stable C 0 semigroup on H whatever 0 < k I < k * I . (ii) The output regulation is guaranteed ∀ 0 < k I < k * I : For all control and output disturbances quadratically close to constant the output converges to the set point in the sense that Since the integral part is applied on the boundary and the output observation is also on the boundary, the existing theory in [14,15] and [24] is not directly useful to prove exponential stability of the semigroup e tA1 . The reason is that its domain D(A 1 ) depends on K I and k I . It makes incongruous to assign the spectrum of A 1 by using the perturbation theory [9]. Our idea is to look for a transformation such that the domain of the new generator is independent of K I and k I . Then the technical methods in [14], [24] and [9] can be used for our transformed system. Since the transformation is continuous and invertible, the transformed system is exponentially stable if and only if the original one is.
For the purpose we consider the invertible transformation T 2 : H → H such that where the matrix C 3 : R 2 → R 2 is defined in (56). Applying the transformation (R, ξ) τ = T 2 (R,ξ) τ on the homogeneous equation (52)-(54) leads us to the following PDE : where Now we define the unbounded operator A 2 : D(A 2 ) → H by and for every f ∈ D(A 2 ), which is exactly the right-hand member of (60) and whereÃ has been defined in (47)-(48). Obviously A 1 is the generator of an exponentially stable semigroup on H if and only if A 2 is. If the resolvent R(λ, A 2 ) = (λI − A 2 ) −1 is bounded in some right half plane e(λ) ≥ −α for some α > 0, it follows from the result of Huang [8] or Prüss [16] that the semigroup e tA2 is exponentially stable.
Remark 7. The operator A 2 is the generator of a C 0 semigroup on H and its domain D(A 2 ) is independent of K I and k I . However they appear in the perturbation term.
Let α > 0 and let C + −α denote the closed right half plane : C + −α = {λ ∈ C | e(λ) ≥ −α}. The complementary set of C + −α is the open left half plane noted as C − −α = {λ ∈ C | e(λ) < −α}. We consider also the closed set Ω −α defined in C by (see Figure 1) : Define the unbounded operator Then the operator A 2 can be written as follows where the perturbation term P is given by The resolvent R(λ, A 2 ) is written by (P R(λ, A 3 )) n (68) exponential stability of the closed-loop system resulted from the designed PI controllers. It is our future work to study the closed-loop stability by the PI controllers from the nonlinear models.
where C 5 = B 1 C 3 − C 4 C 3 k I K I C 2 and B 1 , C 3 , K I and C 2 are defined in (61), (56), (42) and (55), respectively. Denote by P λ the operator defined by the matrix in (71). By Lemma 6.4-(ii), we have sup λ∈Ω−α k I P λ < 1, ∀ 0 < k I < k * I Thus the following sum converges normally on Ω −α . Similarly σ(A 2 ) ∩ D α contains only eigenvalues of A 2 . More precisely it contains two eigenvalues only. Indeed the dimension of the eigen-space encircled by Γ α is still equal to two. Consider the projector P 1 From (70) and (73), we have P 0 −P 1 L(H) < 1 for every k I small enough, which implies that the two projectors have the same dimension (cf. [9, p.34]). The assertion (i) is proved. It is sufficient to assign by means of K I the two eigenvalues λ 1 and λ 2 in D α to the left half part. From (72), where C 6 = (C 2 − C 1Ã We claim that the following identity holds true By the claim and (74), one finds some k * I > 0 such that max{ e(λ 1 ), e(λ 2 )} = −k I + O(k 2 I ) < 0, ∀ k I ∈ (0, k * I ). The rest of the spectrum σ(A 2 ) satisfies e(λ) ≤ −α for λ = λ 1 or λ 2 . So we prove that sup{ e(λ) | λ ∈ σ(A 2 )} < −k I /2.
To finish we prove the claim. Notice that the matrix C 6 is the mapping v → y defined by the following ordinary differential equation 0 =ÃR + B 1 C 3 ṽ R 1 (0) =R 2 (l) = 0 y = C 1R + C 2 v.