Exponential stability of PI control for Saint-Venant equations

Transcription

1 Exponential stability of PI control for Saint-Venant equations Georges Bastin and Jean-Michel Coron February 28, 218 Abstract We consider open channels represented by Saint-Venant equations that are monitored and controlled at the downstream boundary and subject to unmeasured flow disturbances at the upstream boundary. We address the issue of feedback stabilization and disturbance rejection under Proportional-Integral PI boundary control. For channels with uniform steady states, the analysis has been carried out preiously in the literature with spectral methods as well as with Lyapuno functions in Riemann coordinates. In this article, our main contribution is to show how the analysis can be extended to channels with non-uniform steady states with a Lyapuno function in physical coordinates. Introduction The hyperbolic Saint-Venant equations are commonly used for the description of water flow dynamics in open channels and for the design of management and control systems in irrigation networks and naigable riers. In particular, the exponential stabilization of Saint-Venant equations by boundary feedback control has been a recurring research topic in the literature for more than twenty years. The earlier results dealt with static proportional control. In the simplest case of horizontal channels with negligible friction, the stability analysis was carried out in [1] with an entropy Lyapuno function, in [2, 3] with the method of characteristics, and in [4, Section VI] with a Lyapuno function in Riemann coordinates. The stability analysis was then extended to channels with slope and friction. In the special case of a uniform steady state, the stability analysis was carried out with a spectral method for linearized equations in [5, Section 6]. Howeer the linearized system stability does not directly imply the stabiliy of the steady state for the nonlinear Saint- Venant equations see e.g. [6]. For this nonlinear case, the stability analysis is done in [7, 8] with a Lyapuno function in Riemann coordinates. More recently, the case of channels with friction and non-uniform steady state was considered in [9] and [1] with dedicated Lyapuno functions expressed in physical coordinates. The boundary feedback stabilization of Saint-Venant equations by Proportional-Integral PI control has receied much less attention in the literature. It has been analyzed for channels with uniform steady states in [11] with a spectral method and in [12, Section 4], [13, Section 5.5] with Lyapuno functions in Riemann coordinates. In the present article, our main contribution is to show how the analysis of [9] can be extended to channels with non-uniform steady states under PI control, using a Lyapuno function in physical coordinates. Obiously, in principle, stabilization is also possible with more sophisticated control laws. In particular, the recent backstepping method for 2 2 hyperbolic systems, see e.g. [14, 15, 16], allows to design stabilizing boundary output feedbacks in obserer-controller form for Saint-Venant Department of Mathematical Engineering, ICTEAM, Uniersity of Louain, Louain-La-Neue, Belgium. Laboratoire Jacques-Louis Lions, Uniersité Pierre et Marie Curie, Paris, France. 1

2 equations. Howeer, it is clear that such adanced solutions are far from being used in practice and that PI controllers are the only regulators that are really implemented in the ast majority of field applications. The reason is obiously that PI regulators, besides their great ease of implementation, are the simplest solution to cancel off-set errors and attenuate load disturbances. In a PI regulator, the parameter k i is a measure of the disturbance attenuation efficiency, but too large alues may produce instability. The analysis of the stability of a closed-loop sytem under PI control, as we present in this article, is therefore an important and releant issue. Saint-Venant equations We consider a pool of a prismatic horizontal open channel with a rectangular cross section, as shown in Fig.1. The dynamics of the system are described by the Saint-Venant equations H t + HV x =, 1a V t + gh + 12 V 2 + S f H, V =, 1b with the state ariables Ht, x = water depth and V t, x = horizontal water elocity at the time instant t and the location x along the channel. L is the length of the pool and g is the graity acceleration. S f H, V is the friction term for which arious empirical models are aailable in the engineering literature. In this article, we adopt the simple model x with C a constant friction coefficient. S f H, V C V 2 H 2 Q t Ht, x V t, x Ht, L Ut L x Figure 1: Pool of an open channel with an oerflow gate at the downstream side. The system is subject to the following boundary conditions: Ht, V t, = Q t, Ht, LV t, L = υ G Ht, L Ut. 3a 3b The first boundary condition 3a imposes the alue of the canal inflow rate which is an unknown disturbance denoted Q t. The second boundary condition 3b is a simple linear model of a spillway outflow gate with Ut the gate eleation used as control input and υ G a constant gate parameter. When Ut is an exogenous command signal, the system 1, 2, 3 is an open loop boundary control system. 2

3 Proportional-Integral control In this article we are concerned with the case where the outflow gate is proided with a Proportional- Integral PI control law t Ut U r + k p H sp Ht, L + k i H sp Hτ, Ldτ 4 where H sp denotes the set-point for the downstream leel Ht, L which is assumed to be measured on line. The first term U r is an arbitrary constant alue for the gate eleation. The second term is the proportional correction action with the tuning parameter k p. The last term is the integral action with the tuning parameter k i. With this control law, defining Zt Ut + k p Ht, L, the boundary conditions are written in differential form as follows: Ht, V t, = Q t, Ht, LV t, L = υ G [ 1 + kp Ht, L Zt ], dz dt = k ih sp Ht, L. 5a 5b 5c When Ut is the feedback command signal 4, the system 1, 5 is a closed loop boundary control system. In this article, our main purpose is to analyze the exponential stability of this closed loop control system. Fluial steady state In case of a constant positie disturbance Q > and a constant positie set point H sp >, a steady state of the closed loop control system is a time-inariant solution H x, V x, Z, x [, L], gien by: H x solution of gh 3 Q 2 H x + CQ 2 =, H L = H sp, V x = Q H x, Z = 1 + k p H sp Q υ G. 6a 6b 6c The existence of a solution to 6a requires that gh 3 sp Q 2. If gh 3 sp > Q 2, then the steady state flow is subcritical or fluial. In such case, from 6a and 6b, according to the physical eidence, the state H, V is positie : and satisfies the following inequality: H x >, V x >, for all x [, L], 7 < gh x V 2 x = gh 3 x Q 2, x [, L]. 8 In the case where gh 3 sp < Q 2, the steady state, if it exists, is said to be supercritical or torrential. We do not consider that case in the present article. 3

4 Linearization In order to linearize the model, we define the deiations of the states Ht, x, V t, x and Zt with respect to the steady states H x, V x and Z : ht, x Ht, x H x, t, x V t, x V x, z = Zt Z. 9 Then the linearized Saint-Venant equations around the steady-state are ht t V H hx Vx + + g V x C V 2 H 2 and the linearized boundary conditions are Hx h Vx + 2C V =, 1 H t, = b ht, with b = V H, 11a t, L = b L ht, L b z zt, z t = k i ht, L. b L = υ G1 + k p V L H, b z = υ G L H L, 11b 11c Exponential stability of the linearized system Let us consider the linearized system 1, 11 under an initial condition such that h, x = h o x,, x = o x, z = z o, 12 h o, o L 2, L; R 2, z o R. 13 The Cauchy problem is well-posed see [13, Appendix A]. Our concern is to analyze the exponential stability of the system 1-11 according to the following definition. Definition 1. The system 1-11 is exponentially stable for the L 2 -norm if there exist ν > and C o > such that, for eery initial condition h o, o L 2, L; R 2, z o R, the solution to the Cauchy problem 1, 11, 12 satisfies ht,, t, L 2 + zt C o e νt[ ] h o, o L 2 + z o. 14 We now proe that the linearized control system 1-11 is exponentially stable if the steady state is subcritical and the control tuning parameters are positie: k p > and k i >. For this stability analysis, the following candidate Lyapuno function is considered: Vh,, z = gh 2 + H 2 dx + qz The time deriatie of this function along the C 1 solutions of the Cauchy problem 1, 11, 12 is dv L dt = 2 ghh t + H t dx + qzz t. 16 4

5 Using the system equation 1 and the boundary condition 4c,we hae dv dt = 2 ghv h x + H x + V x h + H x Then, using integration by parts, we hae with and + H gh x + V x C V 2 H 2 h + V x + 2C V H dx 2qk i zht, L. 17 [ ] L dv h dt = h Mx We introduce the notations Mx = h Nx h gv x gh x gh x gcv 3 H gh V 2 Nx = 2 CV H H xv x dx 2qk i zht, L, 18, 19 2 CV H 2CV 3 gh V 2 + 4CV. 2 h = ht,, h L = ht, L, = t,, L = t, L, 21 H = H, V = V, H L = H L, V L = V L. 22 Then, using the boundary conditions 11a, 11b, we hae [ ] L h h Mx = gv L h 2 L + 2gH L h L L + Q L 2 gv h 2 2gH h Q 2 23 = gv L h 2 L + 2gH L h L b L h L b z z + Q b L h L b z z 2 gv h 2 + 2gH h b h Q b h 2 24 = gv L + 2gb L H L + Q b 2 Lh 2 L + gv + 2gb H Q b 2 h 2 + Q b 2 zz 2 + 2gb z H L 2Q b L b z h L z. 25 Consequently dv dt = m h 2 h L z M hl z h Nx h dx 2aqk i z 2, 26 with m = gv + 2gb H Q b 2, 27 gvl + 2gb L H L + Q b 2 L gb z H L Q b L b z + qk i M =, 28 gb z H L Q b L b z + qk i Q b 2 z 2aqk i and a is a real positie constant to be determined. Under the subcritical flow condition 8, using the definition of b 11a, we hae that m = gv + 2gb H H V b 2 = b gh V 2 > 29 5

6 and that the matrix Nx is positie definite for all x [, L] since CV 2 det[nx] = H On the other hand, M is positie definite if 2 2gH V 2 gh V gH gh V 2 1 >. 3 a gv L + 2gb L H L + Q b 2 L >, 31 b detm = gv L + 2gb L H L + Q b 2 LQ b 2 z 2aqk i gb z H L + Q b L b z qk i 2 >. 32 The parameter υ G can be considered as an approximate alue of the water elocity oer the spillway at the steady state. Therefore, as it follows from 3b, we hae that υ G > V L. Hence, since k p >, we hae from 11b and Condition a is satisfied. b L = υ G V L + υ G k p H L Regarding condition b, using the definition of b L, we hae with and > 33 detm = α + 2βk i q k 2 i q 2 = Pq, 34 α = gb 2 zh L gh L V 2 L 35 β = gb z H L + Q b L b z agv L + 2gb L H L + Q b 2 L. 36 Pq is a degree-2 polynomial in q with discriminant = 4k 2 i β 2 α. 37 We obsere that α > under the subcritical flow condition 8. Moreoer, it is easy to check that the positie parameter a can be selected sufficiently small so that β > and β 2 α >. Hence, if k i >, Pq has two positie real roots and there exists a positie alue of q depending on k i such that detm > and condition b is satisfied. Then, it follows directly from the definition 15 of V and from 26 that there exists a positie real constant µ such that dv dt µv 38 along the C 1 -solutions of the system. Howeer, since the C 1 -solutions are dense in the set of L 2 -solutions, inequality 38 is also satisfied in the sense of distributions for L 2 -solutions see [13] for details. Consequently, V is an exponentially decaying Lyapuno function for the L 2 -norm and the system 1-11 is exponentially stable in the sense of Definition 1. Exponential stability of the steady state of the Saint-Venant equations In the preious section, we hae shown that the PI controller 4 stabilizes the linearized Saint- Venant equations if the steady state is subcritical and the control tuning parameters are positie: k p > and k i >. In this section, we briefly explain how it can be shown that the same PI controller is also sufficient to guarantee the local exponential stability in H 2 of the steady state H x, V x of the nonlinear system of Saint-Venant equations 1, 2 under the nonlinear boundary conditions 5. 6

7 Let us rewrite the Saint-Venant equations in the h, coordinates, ht t + V x + H x + h hx g V x + x Vx x + V 2 x C H xh x + h with the boundary conditions using the notations 21 and 22 H xx V x x + C 2V x + H x + h h =, 39 = b h + V H H + h h2, 4a L = b L h L b z z + V L υ G1 + k P h 2 L + υ Gh L z, 4b H L H L + h L z t = k i h L. 4c Then, we transform the system into Riemann coordinates which are defined as follows: R + + 2ηh R = = with ηh = gh + h gh. 41 2ηh R With these coordinates, the system 39 is written in the following characteristic form: R t + ΛR, xr x + BR, x =, 42 with the diagonal matrix λ ΛR, x = + R, x λ R, x with λ ± R, x = V ± gh + ± ηh, 43 and an appropriate definition of BR, x. The goal is to proe the H 2 exponential stability of the zero steady state for the system 43 under the boundary conditions 4 and under an initial condition according to the following definition. R, x = R o x, z = z o, 44 Definition 2. The steady state Rt, x of the system 43, 4, 44, is exponentially stable for the H 2 norm if there exist δ >, ν > and C > such that, for eery initial condition R o H 2, L; R n satisfying R o H2,L;R n δ and compatibility 1 conditions of order 1, the solution R of the Cauchy problem 43, 4, 44 is defined on [, + [, L] and satisfies Rt,. H2,L;R n + zt C e νt[ ] R o H2,L;R n + z o. 45 The proof can be build in a way ery similar to the proof gien in [13, Chapter 6] for a general class of quasi-linear hyperbolic systems with static boundary conditions. Here we limit ourseles to the key points of the proof and we refer the reader to [13, Section 6.2] for a comprehensie deelopment. 1 For an explanation of the concept of compatibility of initial conditions, see [13, Section 4.5.2] 7

8 First, we consider an augmented system with state R, R t, R tt where the dynamics of R t and R tt are simply obtained by taking partial time deriaties of the system equation 43 and the boundary conditions 4. Then the candidate Lyapuno function is defined as with V NL = V 1 R, z + V 2 R t, z t + V 3 R tt, z tt, 46 V 1 R, z = V 2 R t, z t = V 3 R tt, z tt = 1 2 H R T Rdx + qz 2, H R T t R t dx + qz 2 t, H R T ttr tt dx + qz 2 tt. 49 We remark that, for small h, the function V 1 R, z can be iewed as a perturbation of the Lyapuno function Vh,, z of the linearized system see equation 15: V 1 R, z = = = 1 2 H R T Rdx + qz 2 5 4H η 2 h + H 2 dx + qz 2 gh 2 + H 2 + Oh 3 dx + qz 2 = Vh,, z + Similar expressions of V 2 and V 3 are obtained as follows: V 2 R t, z t = Vh t, t, z t +.V 3 R tt, z tt = Vh tt, tt, z tt + Oh 3 dx. 51 Ohh 2 t dx, 52 Oh 2 t h tt + hh 2 ttdx. 53 Let us now introduce a notation to deal with higher order terms in the time deriatie of the Lyapun function. We denote by OX 1 ; X 2, with X 1 and X 2, quantities for which there exist C > and ε > independent of R, R t and R tt, such that X 2 ε = OX 1 ; X 2 C X 1. It follows that the time deriaties of V 1, V 2 and V 3 along the system solutions can be expressed in the following form 2 dv 1 dt [ ] L h h L h = Mx h Nx dx qzz t + O Rt, 3 + Rt, L 3 ; Rt, + Rt, L + O R 3 + R t R 2 dx; Rt,., 54 2 For a ector ξ = ξ 1,..., ξ n T R n, we denote ξ = max{ ξ j ; j {1,..., n}}. For a map f C [, L]; R n, we denote f = max{ fx ; x [, L]}. 8

9 [ ] L dv 2 ht ht = t Mx dt dv 3 dt t [ ] L ht ht = t Mx ht ht t Nx dx qz t z tt + O R t t, 3 + R t t, L 3 ; R t t, + R t t, L t + O t R t 2 R t + R dx; Rt,., 55 ht ht t Nx dx qz tt z ttt + O R tt t, 2 R t t, + R tt t, R t t, 2 + R t t, 4 + R tt t, L 2 R t t, L + R tt t, L R t t, L 2 + R t t, L 4 ; R t t, + R t t, L + O Rtt 2 R t + R + R tt R 2 t dx; Rt,. + R t t,.. 56 We obsere that, in each case, we recoer the quadratic formula of the linear case augmented with at least cubic terms that are negligible for small Rt,. + R t t,.. It is therefore not surprising that the local H 2 stability of the nonlinear steady state can be deduced from the global L 2 stability of the linear system. By proceeding analogously to [13, Chapter 6], it can be shown that there exist positie constants α and δ such that, for eery R such that Rt,. + R t t,. < δ, we hae dv NL dt t αv NL 57 along the system solutions. It follows that the system steady-state is locally exponentially stable for the H 2 -norm in the sense of Definition 2. Conclusion In this article, our main contribution was to exhibit a Lyapuno function which allows to study the exponential stability of Saint-Venant equations with nonuniform steady-states under boundary feedback PI control. References [1] J.-M. Coron, B. d Andréa-Noel, and G. Bastin, A Lyapuno approach to control irrigation canals modeled by Saint-Venant equations, in Proceedings European Control Conference, Karlsruhe, Germany, September [2] G. Leugering and J.-P. Schmidt, On the modelling and stabilisation of flows in networks of open canals, SIAM Journal of Control and Optimization, ol. 41, no. 1, pp , 22. [3] J. de Halleux, C. Prieur, J.-M. Coron, B. d Andréa-Noel, and G. Bastin, Boundary feedback control in networks of open-channels, Automatica, ol. 39, no. 8, pp , 23. [4] J.-M. Coron, B. d Andréa-Noel, and G. Bastin, A strict Lyapuno function for boundary control of hyperbolic systems of conseration laws, IEEE Transactions on Automatic Control, ol. 52, pp. 2 11, January 27. [5] X. Litrico and V. Fromion, Boundary control of hyperbolic conseration laws using a frequency domain approach, Automatica, ol. 45, pp , 29. 9

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