Boundary Control of Systems of Conservation Laws: Lyapunov

Preprints of the 7th IFAC Symposium on Nonlinear Control Systems 21-24 August, 2007, Pretoria, South Africa

BOUNDARY CONTROL OF SYSTEMS OF CONSERVATION LAWS : LYAPUNOV STABILITY WITH INTEGRAL ACTIONS

Val´erieDos Santos ∗ Georges Bastin ∗∗ Jean-Michel Coron ∗∗∗ Brigitte d’Andr´ea-Novel ∗∗∗∗

∗ Universit´ede Lyon, Lyon, F-69003, France; Universit´ede Lyon 1, CNRS UMR 5007, LAGEP, ESCPE, Villeurbanne, F-69622, France

∗∗ CESAME, Universit´eCatholique de Louvain, 4, Avenue G. Lemaitre, 1348 Louvain-La-Neuve, Belgium.

∗∗∗ Department of Mathematics, Universit´eParis-Sud, Bˆatiment425, 91405 Orsay, France.

∗∗∗∗ Centre de Robotique (CAOR), ENSMP, 60, Boulevard Saint Michel, 75272 Paris Cedex 06, France.

Abstract: A boundary control law with integral actions is proposed for a generic class of two-by-two homogeneous systems of linear conservation laws. Suﬃcient conditions on the tuning parameters are stated that guarantee the asymptotic stability of the closed-loop system. The closed-loop stability is analysed with an appropriate Lyapunov function. The control design method is validated with an experimental application to the regulation of water depth and ﬂow rate in a pilot open-channel described by Saint-Venant equations. This hydraulic application shows that the control can be robustly implemented on nonhomogeneous systems of nonlinear conservation laws. Copyright c 2007 IFAC

Keywords: Lyapunov stability, Proportional-Integral control, Saint-Venant equations, Systems of conservation laws,

1. INTRODUCTION water in irrigation channels and waterways. This example will be presented in Section 4. Other typical examples include gas and fluid transporta- In this paper, we are concerned with two-by-two tion networks, packed bed and plug-flow reactors, systems of conservation laws that are described drawing processes in glass and polymer industries, by hyperbolic partial differential equations, with road traffic etc. For such systems, the boundary one independent time variable t ∈ [0, ∞) and control problem that we address is the problem of one independent space variable on a finite interval designing feedback control actions at the bound- x ∈ [0,L]. Such systems are used to model many aries (i.e. at x = 0 and x = L) in order to ensure physical situations and engineering problems. A that the smooth solution of the Cauchy problem famous example is that of Saint-Venant (or shal- converges to a desired steady-state. low water) equations which describe the flow of Preprints of the 7th IFAC Symposium on Nonlinear Control Systems 21-24 August, 2007, Pretoria, South Africa

3 The present paper is in the direct continuation of with g0, gL : R → R. The functions u0, uL: our previous paper (Coron et al. (2007) ) where [0, +∞) → R represent the boundary control ac- a static feedback control law was presented and tions that can be manipulated by the operator. A the closed-loop stability analysed with an appro- concrete illustration of such boundary conditions priate Lyapunov function. But obviously, a static will be given in Section 4. control law may be subject to steady-state regulation errors in case of constant disturbances or In order to define the feedback control laws, it is model inaccuracies. In the present paper, we show convenient to introduce the Riemann coordinates how additional integral actions can be introduced (see e.g. Lax (1973)) defined by the following in the control law in order to cancel the static change of coordinates: errors and how the Lyapunov function can be a(t, x) = q(t, x) + βh(t, x) (4a) modified in order to prove the asymptotic stability b(t, x) = q(t, x) − αh(t, x). (4b) of the closed-loop system. The statement of the control law and the Lyapunov stability analysis With these coordinates, the system (1)-(2) is are developed in Sections 2 and 3 for a generic rewritten under the following diagonal form: homogeneous system of two linear conservation ∂ a(t, x) + α∂ a(t, x) = 0 (5a) laws. In Section 4, we present an experimental t x validation on a laboratory pilot plant. This hy- ∂tb(t, x) − β∂xb(t, x) = 0. (5b) draulic application clearly shows that the control The change of coordinates (4) is inverted as fol- can be robustly implemented on nonhomogeneous lows: systems of nonlinear conservation laws. a(t, x) − b(t, x) h(t, x) = (6a) α + β 2. STATEMENT OF THE CONTROL LAW αa(t, x) + βb(t, x) q(t, x) = . (6b) α + β We consider the class of two-by-two systems of In the Riemann coordinates, the control problem linear conservation laws of the general form: can be restated as the problem of designing the ∂th(t, x) + ∂xq(t, x) = 0 (1) control laws in such a way that the solutions a(t, x) and b(t, x) converge to zero. We shall show ∂tq(t, x) + αβ∂xh(t, x) + (α − β)∂xq(t, x) = 0 (2) that this problem can be solved by selecting the where boundary control laws u0(t) and uL(t) such that • t and x are the two independent variables : the Riemann coordinates satisfy linear boundary a time variable t ∈ [0, +∞) and a space conditions of the following form: variable x ∈ [0,L] on a finite interval; a(t, 0) + k b(t, 0) + m y (t) = 0 (7a) • (h, q) : [0, +∞) × [0,L] → R2 is the vector of 0 0 0 the two dependent variables (i.e. h(t, x) and b(t, L) + kLa(t, L) + mLyL(t) = 0 (7b) q(t, x) are the two states of the system); where k , k , m , m are constant tuning parame- • α and β are two real positive constants: 0 L 0 L ters while y0 and yL are integrals of the flow q(t, 0) α > β > 0. and the density h(t, L) respectively: The first equation (1) can be interpreted as a mass Z t Z t αa(s, 0) + βb(s, 0) conservation law with h the density and q the y0(t) = q(s, 0)ds = ds 0 0 α + β flux. The second equation can then be interpreted (8a) as a momentum conservation law. As usual in Z t Z t a(s, L) − b(s, L) control design, the model (1)-(2) must be viewed yL(t) = h(s, L)ds = ds. as a linear approximation of the system dynamics 0 0 α + β (8b) around a steady-state. This will be illustrated with the application of Section 4. Remarks We are concerned with the solutions of the Cauchy 1) Conditions (7) give only an implicit definition problem for system (1)-(2) over [0, +∞) × [0,L] of the control laws. The derivation of explicit ex- under an initial condition pressions obviously requires an explicit knowledge h(0, x), q(0, x) x ∈ [0,L]. of the functions g0 and gL in (3). In the special case where the boundary conditions (3) are lin- Furthermore, it is assumed that the system is sub- ear, u0 and uL reduce to standard Proportional- ject to physical boundary conditions that can be Integral (PI) control laws. This point will be fur- assigned by an external operator and are written ther illustrated in Section 4. in the following general abstract form: 2) In our previous paper (Coron et al. (2007)), we g (h(t, 0), q(t, 0), u (t)) = 0 t ∈ [0, +∞) (3a) 0 0 have dealt with the special case without integral gL(q(t, L), h(t, L), uL(t)) = 0 t ∈ [0, +∞) (3b) actions, i.e. m0=mL=0 in (7). We have shown Preprints of the 7th IFAC Symposium on Nonlinear Control Systems 21-24 August, 2007, Pretoria, South Africa

with an appropriate Lyapunov function that in- We first consider the special case where µ = 0 and equality |k0kL| < 1 is a sufficient condition to (10)-(11) reduce to: guarantee the closed-loop stability and the expo- 2 2 Ak0 − B b0 + [2Ak0m0 + N0(β − αk0)] b0y0 nential convergence of a(t, x) and b(t, x) to zero. 2 2 Our contribution in the present paper is to extend + Am0 − N0m0α y0, (12) this Lyapunov stability analysis to the case where Bk2 − A a2 + [2Bk m + N (1 + k )] a y integral terms are introduced in the control law L L L L L L L L 2 2 and to validate the methodology with experimen- + BmL + NLmL yL. (13) tal results. We are going to prove that there exist A > 0, B > 0, N0 > 0, NL > 0 such that (12)-(13) are negative definite quadratic (NDQ) forms in b0, y0 3. LYAPUNOV STABILITY ANALYSIS and aL, yL respectively. We set B = 1. Then (12) is a NDQ form if (and only if) Let us define the following candidate Lyapunov 1 function: A < 2 (14) k0 A Z L U(t) = a2(t, x)e−(µ/α)xdx and α 0 P = 4Am2+4N (Ak β−α)m +N 2(β−αk )2 < 0. B Z L 0 0 0 0 0 0 0 + b2(t, x)e+(µ/β)xdx β P0 is a polynomial of degree 2 in N0 with discrim- 0 inant α + β 2 2 + N0y0(t) + NLyL(t) (9) 2 2 2 2 2 ∆0 = 16(α − Aβ )(1 − Ak0)m0. and the following norm: In view of (14), ∆0 > 0 if Z L 2 2 2 β ψ(t) = a (t, x) + b (t, x) dx A < 2 . (15) 0 α 2 2 + |y0(t)| + |yL(t)| . Then, if inequalities (14)-(15) hold, P0 has two real roots and we will have N0 > 0 and P0 < 0 if We have the following stability result. the greatest root is positive, that is if (and only if) the following inequality holds: Theorem. If the four constant tuning parameters q k , k , m , m satisfy the following inequalities: 2 2 2 2 0 L 0 L (Ak0β − α)m0 < (α − Aβ )(1 − Ak0)m0. α |k | < 1 |k | < |k k | < 1 Under conditions (14)-(15), it can be shown that 0 L β 0 L this inequality holds for any m0 > 0. Let us m0 > 0 mL < 0, summarize : there exist A > 0 and N0 > 0 such there exist six positive constants A, B, µ, N0, NL, that P0 is an NDQ form if A can be selected such C such that that 1 β2 −µt 0 < A < 0 < A < . (16) U˙ −µU and ψ(t) Cψ(0)e 2 2 6 6 k0 α for every solution a(t, x), b(t, x), t > 0, x ∈ [0,L] Similarly (13) is a NDQ form if (and only if) of (5)-(7). 2 A > kL Proof : The time derivative of the function U(t) and along the solutions of the system (5)-(7) is 2 2 2 PL = 4AmL +4NL(kL +A)mL +NL(1+kL) < 0. ˙ ˙ ˙ U = −µU + U0 + UL With a similar line of reasoning, we can show that with there exist A > 0 and NL > 0 such that PL is an ˙ 2 2 NDQ form if A can be selected such that U0 = Ak0 − B b0 k2 < A 1 < A. (17) + [2Ak0m0 + N0(β − αk0)] b0y0 L α + β Hence, using conditions (16)-(17) together, a suf- + Am2 + µN − N m α y2 (10) 0 0 2 0 0 0 ficient condition to have A > 0, B > 0, N0 > 0, NL > 0 such that (12)-(13) are NDQ forms is and 1 α2 ˙ h ˜ 2 ˜i 2 max k2 , 1 < min , UL = BkL − A aL L 2 2 k0 β h i + 2Bk˜ LmL + NL(1 + kL) aLyL which is equivalent to α α + β |k0| < 1 |kL| < |k0kL| < 1. + Bm˜ 2 + N m + µ N y2 (11) β L L L 2 L L By continuity with respect to µ, under the same with A˜ = Ae−µL/α, B˜ = BeµL/β. conditions, there exist µ > 0, A > 0, B > 0, Preprints of the 7th IFAC Symposium on Nonlinear Control Systems 21-24 August, 2007, Pretoria, South Africa

N0 > 0, NL > 0 such that (10) and (11) are quadratic negative deﬁnite forms. This implies that U is a Lyapunov function for the system (5)- ˙ (7) such that U 6 −µU along the system solutions with U˙ = 0 if and only if U = 0. The inequality −µt ψ(t) 6 Cψ(0)e follows readily. QED.

4. EXPERIMENTAL VALIDATION

4.1 Saint-Venant equations

The Saint-Venant equations are among the most frequent application examples of two-by-two hyperbolic systems of conservation laws in engi- Fig. 1. The ESISAR pilot channel. neering. They are used for a long time for sim- r ∂H Qe ulation and design of model-based controllers in α = gAe (Ae) + ∂A Ae open-channels (see e.g. the recent publications by r de Halleux et al. (2003), Litrico et al. (2005) and ∂H Qe β = gAe (Ae) − Litrico and Fromion (2006) and the references ∂A Ae therein). ∂J γ = gA (A ,Q ) e ∂A e e For a pool of a prismatic open channel with a ∂J constant slope, the flow dynamics are described δ = gAe (Ae,Qe). ∂Q as follows by the Saint-Venant equations: In the special case where the channel is horizontal ∂ A + ∂ Q = 0 (18a) t x (S = 0) and the friction slope is negligible (n ≈ 2 Q 0), we observe that γ = δ = 0 and that this ∂tQ + ∂x + gA∂xH(A) = gA [S − J(A, Q)] A linearized system is exactly in the form of the (18b) linear hyperbolic system we have considered in Sections 2 and 3. Moreover, the flow in the channel with L the pool length, A(t, x) the wet area at is fluvial (as opposed to torrential) when α > β > time t and abscissa x ∈ [0,L], Q(t, x) the flow 0. It is therefore legitimate to apply the control rate at time t and abscissa x ∈ [0,L], g the gravity with integral actions that we have analysed above constant and H(A) the water depth. Furthermore to open channels having small bottom and friction S is the bottom slope and slopes under fluvial flow conditions. In the next Q2n2 section, we shall illustrate the efficiency of the J(A, Q) = 4/3 control with experimental results on a real life A2 [R(A)] laboratory plant. is the friction slope with n the Manning friction coefficient and R(A) the hydraulic radius. 4.2 Experimental setup A steady-state regime (Ae,Qe) is a constant solution of equations (18), i.e. A(t, x) = A , Q(t, x) = e An experimental validation has been performed Q ∀t and ∀x ∈ [0,L], which satisfies the relation e with the Valence micro-channel (see Fig. 1). This

J(Ae,Qe) = S. (19) pilot channel is located at ESISAR/INPG engineering school in Valence (France). It is operated A linearized model is used to describe the varia- under the responsability of the LCIS laboratory. tions about this equilibrium. The following nota- The channel (overall length = 8 meters) has an tions are introduced: adjustable slope and a rectangular cross-section. The water depth in the channel is denoted H(t, x) h(t, x) = A(t, x) − A q(t, x) = Q(t, x) − Q . e e such that: The linearized model is then written as A(t, x) = wH(t, x)(w = channel width).

∂th(t, x) + ∂xq(t, x) = 0 The channel is furnished with three underflow ∂tq(t, x) + αβ∂xh(t, x) + (α − β)∂xq(t, x) control gates. Ultrasound sensors provide water = −γh(t, x) − δq(t, x) level measurements at different locations of the channel. For the experiments reported hereafter with the configuration is a single pool (length L = Preprints of the 7th IFAC Symposium on Nonlinear Control Systems 21-24 August, 2007, Pretoria, South Africa

7 meters, width w=0.1 meter, slope S= 0.0016) bounded by two underﬂow gates. The ﬂow rates at the gates are supposed to be related to the water depths by the gate characteristics expressed as q Q(t, 0) = c0U0(t) g(Hup − H(t, 0)) (21a) p Q(t, L) = cLUL(t) g(H(t, L) − Hdo) (21b)

where c0 and cL are constant coeﬃcients while U0 and UL denote the control signals at the upstream and downstream gates respectively. Hup is the water depth at the upstream of the upstream gate, Hdo is the water depth at the downstream of the downstream gate.

4.3 PI control laws

In order to explicit the control laws, the gate characteristics (21) are linearised about the steady- state (Ae = wHe,Qe):

gQe q(t, 0) = − 2 h(t, 0) + c0K0u0(t) (22a) 2K0 gQe Fig. 2. Experimental results q(t, L) = 2 h(t, L) + cLKLuL(t) (22b) 2KL with with 1 gQe µL KpL = λL + 2 KiL = . q p cLKL 2KL cLKL K0 = g(Hup − He) KL = g(He − Hdo) Hence the control law u0 is a PI dynamic feedback Qe Qe and: u = U − u = U − . of the flow rate q(t, 0) = Q(t, 0) − Qe and the 0 0 c K L L c K 0 0 L L control law uL is a PI dynamic feedback of the Moreover, using the definition of the Riemann water depth h(t, L) = w(H(t, L) − He). These coordinates (4), the boundary conditions (7) are control laws are implemented with direct on- rewritten as line measurements of the water levels Hup, Hdo, Z t H(t, 0), H(t, L)). The flow rates at the gates are q(t, 0) + λ0h(t, 0) + µ0 q(s, 0)ds = 0 (23a) not directly measured but are computed on-line 0 with the gate characteristics (21). Z t q(t, L) + λLh(t, L) + µL h(s, L)ds = 0 (23b) 0 with: 4.4 Experimental results β − k0α kLβ − α The experimental results are shown in Fig.2. λ0 = λL = 1 + k0 1 + kL Three experiments are presented with increasing m0 mL µ0 = µL = . values of the integral gains m0 and mL indicated 1 + k0 1 + kL in the figure captions while the parameters k0 and Then, by eliminating h(t, 0) between (22a) and kL are set to (23a), we get the following PI control law for u0: k0 = −0.213, kL = −1.157, k0kL = 0.246. Z t In each experiment, the system is initially in open u0(t) = Kpoq(t, 0) − Kio q(s, 0)ds 0 loop at a steady state with Q(0, x) ≈ 2.35 lit/sec,H(0,L) ≈ 0.125 m. 1 gQe gQeµ0 The loop is closed at time t = 50sec with a new Kpo = 1 − 2 Kio = 3 . c0K0 2λ0K0 2λ0c0K0 set point given by:

Similarly, by eliminating q(t, L) between (22b) Qe = 2 lit/sec,He(L) = 0.143 m. and (23b), we get the following PI control law for In the ﬁrst experiment without integral actions uL: (m0=mL=0), there is clearly an oﬀset of about Z t 4 cm on the level H(t, L). In the second exper- uL(t) = −KpLh(t, L) − KiL h(s, L)ds 0 iment with m0 = 0.002 and mL = −0.001 the Preprints of the 7th IFAC Symposium on Nonlinear Control Systems 21-24 August, 2007, Pretoria, South Africa

static error is eﬃciently cancelled by the integral X. Litrico and V. Fromion. Boundary control actions. Finally a third experiment illustrates the of linearized Saint-Venant equations oscillating sensitivity of the transient behaviour with respect modes. Automatica, 42:967–972, 2006. to the choice of the gain values. For largest gain X. Litrico, V. Fromion, J-P. Baume, C. Arranja, values (m0 = 0.005, mL = −0.001), the closed and M. Rijo. Experimental validation of a loop system starts to oscillate and becomes even methodology to control irrigation canals based unstable if the gains are further increased. on Saint-Venant equations. Control Engineer- ing Practice, 13:1425–1437, 2005.

5. CONCLUSION

In this paper, we have been concerned with the boundary control of two-by-two systems of conservation laws that are described by hyperbolic partial differential equations, with one independent time variable and one independent space variable. A boundary control law with integral actions has been proposed for a generic class of homogeneous systems of two linear conservation laws. Sufficient conditions on the tuning parameters have been stated in order to guarantee the asymptotic stability of the closed-loop system which has been proved with an appropriate Lyapunov stability analysis. The control design method has been validated with an experimental application to the regulation of water depth and flow in a pilot open- channel described by Saint-Venant equations.

ACKNOWLEDGEMENTS

This paper presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Oﬃce. The scientiﬁc responsibility rests with its author(s). This work was supported in part by National ICT Australia and by the Department of Electrical and Electronic Engineering, The University of Melbourne, Australia where the second author (G. Bastin) was a Visiting Professor from August 2006 to January 2007.

REFERENCES J-M. Coron, B. d’Andr´ea-Novel, and G. Bastin. A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Transactions on Automatic Control, 52 (1):2–11, January 2007. J. de Halleux, C. Prieur, J-M. Coron, B. d’Andr´ea Novel, and G. Bastin. Boundary feedback control in networks of open-channels. Automatica, 39:1365–1376, 2003. P.D. Lax. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. In Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathe- matics, N ◦11, Philadelphia, 1973. SIAM.