Time delay and wave propagation in controlling

Time delay and wave propagation in controlling systems of conservation laws ∗ Dept.

Vladimir R˘asvan∗ , Daniela Danciu∗ and Dan Popescu∗

of Automation,Electronics and Mechatronics, University of Craiova, Craiova, Romania RO-200585 Email: {vrasvan,ddanciu,dpopescu}@automation.ucv.ro

Abstract—In this paper there are considered some applications arising from power engineering and involving hyperbolic equations of conservation laws. Focusing on the control of the flow in open channels, there is considered the linearized model and a Lyapunov functional is associated. A linear multivariable controller with saturation is synthesized to make the Lyapunov functional exponentially decreasing along the solutions of the closed loop system. By integrating the Riemann invariants along the characteristics there is associated a linear system of difference equations. On this system the basic theory is constructed and the stability verified. Finally a research program for analyzing nonlinear cases is sketched. Keywords—control design; distributed parameter systems; delay systems

I. INTRODUCTION. SOME CONTROLLED SYSTEMS INVOLVING CONSERVATION LAWS In a challenging paper [1], the author points out that “the important class of hyperbolic conservation laws that includes the Euler equations for gas dynamics gathers all the difficulties... ” (poor regularity, necessity of an “entropy” criterion to select the relevant solution a.s.o.). In this paper we want to focus on some applications of the conservation laws which are relevant in energy production and distribution and whose control is performed at the boundaries. The applications arise from thermal and hydraulic engineering. We give below the mathematical models displaying systems of conservation laws which are controlled at the boundaries. Two of them describe systems occurring in hydraulics 1.1 The controlled equations of a hydroelectric power plant without surge tank but taking into account the waterhammer are as follows [2], [3], [4], [5]

∂x (gh + v2 /2) + ∂t v + i(v) = 0 , ∂t h + (a2 /g)∂x v = 0 √ h(0,t) = H0 , v(L,t) = k0 z(t) h(L,t) JΩ0

dΩ γ = η Ω0 f v3 (L,t) − Ng , f = fmax (1 − z/zmax ) dt 2g

(1) The notations are explained in the aforementioned references and are standard in the domain. We have to mention however the following aspects: the partial differential equations describe conservation laws (which are lossless only if the losses term i(v) (usually described by the formula of Darcy) is neglected. Also the dynamics of the turbine penstock and of its air-dome are neglected. There are introduced the dynamics of the actuator, electro-hydraulic valve and the PI corrector. The loop has to be closed via the speed controller of the turbine.

The state variables are rated to some significant steady state values. Some new results concerning stability and control of such systems are in progress [6]. 1.2 An application which “ignited” control studies for conservation laws arose from the domain of the open canal flow: the basic model is given here by the equations of Saint Venant; the reader can consult published work such as [7], [8], [9], [10], [11], [12]. The problem has quite broad applications in irrigation control through control gates. For the present paper we shall consider the models of [9], [10], [11] under the following assumptions: horizontal prismatic channel with constant rectangular section; negligible friction. Consequently the equations are the following version of the Saint Venant equations

∂t H + ∂x Q = 0 , ∂t Q + ∂x (Q2 /H + gH 2 /2) = 0

(2)

Concerning the boundary conditions, there exist several versions in the aforementioned references. For instance, in [10] the boundary conditions are as follows Q(0,t) ≡ Q0 ; Q(L,t) = γ (H(L,t) − uL (t))3/2

(3)

where γ > 0 is a constant of the spillway, uL (t) - its position is the control input. The quantities denoted by H account for the water levels and those denoted by Q - for the water flows. In [11] there are two control spillways (upstream and downstream) and the boundary conditions are Q(0,t) = γ (Z0 − u0 (t))3/2 ; Q(L,t) = γ (H(L,t) − uL (t))3/2 (4) with Z0 - the water level above the reach; the spillways are considered identical - the same γ . In [9], where networks of open channels are considered, the boundary conditions - the discharge relationships - have the following form Q(0,t)|Q(0,t)| = u0 (t)(H0 − H(0,t)) Q(L,t)|Q(L,t)| = uL (t)(H(L,t) − HL )

(5)

where u0 , uL account for gate opening and H0 , HL are the water levels outside the reach (upstream and downstream, respectively); these levels are considered to be constant (if a single reach is to be considered, then see [9]). 1.3 Another application where conservation laws are present arises from co-generation i.e. combined heat electricity generation. A most studied model is that of the co-generation based on a steam turbine with a single regulated steam extraction:

this steam extraction is connected to the relatively far away thermal consumer by a steam pipe filled up with a barotrop fluid - the steam. The basics of this model appeared in [13]; they were further developed by introducing rated variables and rated parameters in order to make the model as independent as possible with respect to the operating regimes [14], [15], [16], [17]. The partial differential equations describing the dynamics of a compressible fluid are as follows

ψc Tc ∂t ξρ + ∂λ ξw = 0, t > 0, 0 < λ < 1 ( ) ξw2 ψc Tc ∂t ξw + ∂λ ξρ + =0 ξρ

(6)

where ξρ and ξw account for the rated variables of the steam pressure (for isothermal flow) and steam flow respectively, Tc = L/wr – the propagation time constant for the pipe of length L under the so-called reduced steam velocity wr – a rather constant reference under various steady states. The coefficient ψc = wr /c0 denotes the ratio of the aforementioned reduced velocity to the sound velocity at maximal flow; normally 0 < ψc < 1. The boundary conditions for (6) - in rated variables are deduced from the subcritical/critical flow at the steam extraction (λ = 0) - the Saint Venant formula (not to mix up with the shallow water a.k.a. Saint Venant equations) - and the critical flow at the steam consumer (λ = 1)

ξw (0,t) = πs (t)Φ(πs (t)/ξρ (0,t)) ; ξw (1,t) = ψs ξρ (1,t) (7) In (7) we denoted by πs (t) the rated steam pressure delivered by the turbine at the steam extraction. The Saint Venant function for isothermal processes reads { √ √ (1/x) 2 ln x, 1 ≤ x ≤ e Φ(x) = (8) √ √ x≥ e 1/ e, At λ = 1 the coefficient ψs describes the steam consumer. To the aforementioned equations one has to add the equations of the steam turbine dynamics ds = απ1 + (1 − α )π2 − νg , 0 < α < 1 dt dπ1 T1 = µ1 − π1 , 0 ≤ µ1 ≤ 1 dt (9) dπs Tp = π1 − βs µ2 πs − (1 − βs )ξw (0,t) dt dπ2 T2 = µ2 πs − π2 , 0 < µ2min ≤ µ2 ≤ 1 dt where s is the rated rotating speed deviation with respect to the synchronous speed, νg is the mechanical load, π1 and π2 are the steam pressures in the high pressure (HP) and low pressure (LP) turbine cylinders respectively and µ1 , µ2 are the control signals (rated values). Ta

II. STABILITY AND CONTROL FOR CHANNEL FLOWS We have examined in the previous section three models where conservation laws are integrated. Two of them those

connected with energy generation - are easily linearized by neglecting the terms v2 /2 in (1) and ξw2 /ξρ in (6); these terms account for the kinetic energy of the flow according to Bernoulli equation (principle). It is stated in the basic reference texts [4], [13] that these terms (more precisely their space variations) are negligible. This decision of neglecting the kinetic term in comparison to the pressure term is acceptable in high pressure systems (as the aforementioned ones). It will be no longer the case for hydraulic channels, operated at relatively low pressure, and, besides, this neglecting will not lead to a linearized system of hyperbolic equations. According to the theoretical engineering philosophy stated elsewhere [18] one can perform the control system synthesis remaining at a formal level and rigorously analyze the basics on the closed loop system. Consequently we shall take a similar approach to the proposals in [9], [10], [11] and start with the control synthesis using a suitable Lyapunov functional for the linearized version of the system and of the boundary conditions. The analysis of the closed loop system will appear as a separate problem. A. The fully linearized case We shall consider equations (2) - the Saint Venant equations a.k.a. the shallow water equations - under the more convenient form

∂t H + ∂x (HV ) = 0 , ∂t V + ∂x (gH +V 2 /2) = 0

(10)

(The nonlinear functions being polynomial are easier to linearize). We choose the boundary conditions (4) as having “average” difficulty among the three aforementioned ones. They read as below if the new variables are to be introduced V (0,t)H(0,t) = γ (Z0 − u0 (t))3/2 V (L,t)H(L,t) = γ (H(L,t) − uL (t))3/2

(11)

As follows from the applications description, the steady state ¯ to is imposed by requesting H¯ and V¯ (i.e. H¯ and Q¯ = V¯ H) have some prescribed values. We shall obtain immediately the prescribed values of the control gate positions at the spillways ¯ γ )2/3 , u¯L = H¯ − (V¯ H/ ¯ γ )2/3 u¯0 = Z0 − (V¯ H/

(12)

Some comments seem necessary: clearly we must have H¯ < Z0 . Therefore u¯0 − u¯L = Z0 − H¯ > 0. Since u¯0 > 0, u¯L > 0 (again physical reasons)√we must check u¯L > 0. Therefore ¯ γ i.e. V¯ < γ H. ¯ This last inequality is clearly a H¯ 3/2 > V¯ H/ standard steady state hydrodynamics condition with the significance that the flow (velocity) and water level are connected through the constructive characteristics of the spillways. To these considerations we add the following: for sufficiently smooth solutions (10) can be written as ( ) ( ) ( ) H V H H ∂t + ∂x =0 (13) V g V V √ and the eigenvalues of the matrix in (13) are V ± gH; the hyperbolic character of (10) - and of (13) - is given by the opposite sign of the eigenvalues. This is the so called fluvialness assumption ensuring a smooth hydraulic flow. Therefore

the spillways are to be designed to fulfil the fluvialness i.e. √ γ < g. Introducing the deviations of the variables h(x,t) := H(x,t) − H¯ , v(x,t) := V (x,t) − V¯

µi (t) := ui (t) − u¯i , i = 0, L

¯ w+ (0,t) = w+ (L,t + cL) ¯ , w− (L,t) = w− (0,t + dL) (14)

we deduce the linearized equations - both partial differential equations and boundary conditions

∂t h + V¯ ∂x h + H¯ ∂x v = 0 , ∂t v + g∂x h + V¯ ∂x v = 0 ¯ V¯ h(0,t) + Hv(0,t) = −(γ˜/2)µ0 (t)

(15)

¯ (V¯ − (γ˜/2))h(L,t) + Hv(L,t) = −(γ˜/2)µL (t) √ √ γ˜ = 3γ Z0 − u¯0 = 3γ H¯ − u¯L

(16)

(17)

and deduce the Riemann invariants √ w± (x,t) = v(x,t) ± g/H¯ h(x,t)

(18)

Therefore the linearized equations can be written in the form of the Riemann invariants ∂t w+ + c¯∂x w+ = 0 , ∂t w− − d¯∂x w− = 0 ( ) ( ) V¯ V¯ + 1+ √ w (0,t) + 1 − √ w− (0,t) gH¯ gH¯ ¯ µ0 (t) = −(γ˜/H) V¯ − γ˜/2 1+ √ gH¯

The boundary conditions of (19) are thus written as ¯ µ0 (t) w+ (L,t + cL) ¯ + ρ1 w− (0,t) = −(1 + ρ1 )(γ˜/(2H)) ¯ + ρ2 w+ (L,t) = −(1 + ρ2 )(γ˜/(2H)) ¯ µL (t) w− (0,t + dL) (23) where we denoted √ √ 1 − V¯ / gH¯ 1 + (V¯ − γ˜)/ gH¯ √ √ , ρ2 = ρ1 = 1 + V¯ / gH¯ 1 − (V¯ − γ˜)/ gH¯

(19) (

V¯ − γ˜/2 w+ (L,t) + 1 − √ gH¯

) w− (L,t)

¯ µL (t) = −(γ˜/H) The structure (19) allows the development of a basic theory (well posedness in the sense of Hadamard) by integrating along the characteristics the Riemann invariants and associating a system of functional equations with deviated arguments as follows. The differential equations of the characteristics are dt dt = c¯ , = −d¯ dx dx

¯ y+ (t) := w+ (L,t + cL) ¯ , y− (t) := w− (0,t + dL) the following difference system is obtained ¯ = −(1 + ρ1 )(γ˜/(2H)) ¯ µ0 (t) y+ (t) + ρ1 y− (t − dL)

We denote the eigenvalues in the linearized case as √ √ c¯ = V¯ + gH¯ > 0 , d¯ = gH¯ − V¯ > 0

)

(22)

With the additional notations

where we denoted

(

now let (x,t) be such that the aforementioned characteristics crossing it can be extended “to the left” - the increasing one - and “to the right” - the decreasing one

(20)

Let (x,t) be some point in the strip {(x,t)|0 ≤ x ≤ L , t > 0}, crossed by the characteristics ¯ ξ − x) t + (ξ ; x,t) = t + c( ¯ ξ − x) , t − (ξ ; x,t) = t − d( We integrate w+ (ξ ,t + c( ¯ ξ − x)) from ξ = x to ξ = L and ¯ ξ − x)) from ξ = 0 to ξ = x; the following w− (ξ ,t − d( representation for the Riemann invariants is obtained ¯ (21) w+ (x,t) = w+ (L,t + c(L ¯ − x)) , w− (x,t) = w− (0,t + dx)

¯ µL (t) y− (t) + ρ2 y+ (t − cL) ¯ = −(1 + ρ2 )(γ˜/(2H))

(24)

Its solution can be constructed by steps; the initial conditions ¯ y+ ¯ 0) and y− 0 (t), t ∈ (−cL, 0 (t), t ∈ (−dL, 0) can be obtained also by integrating along those characteristics which cannot be extended on the entire segment 0 ≤ x ≤ L since they cross the axis t = 0 somewhere on 0 ≤ x ≤ L: for instance t + (ξ ; x,t) = t + c( ¯ ξ − x) equals 0 for ξ = x − t/c¯ and this can happen for those characteristics satisfying 0 ≤ x − t/c¯ ≤ L. We do not insist on this aspect. Note that the solutions of (24) have finite jumps at t = k1 c¯ + k2 d¯ where ki are positive integers. Taking into account the definition of y± (t), the representation formulae (21) become ¯ − x)) w+ (x,t) = y+ (t − cx) ¯ , w− (x,t) = y− (t − d(L

(25)

showing once more that w + (x,t) is a forward wave while w− (x,t) is a backward wave. Using the solutions of (24) and the representation formulae (25) the entire basic theory (existence, uniqueness, data dependence) can be constructed for (15). B. Synthesis of the stabilizing controller in the linearized case We shall follow the line of [11] by defining the control Lyapunov functional candidate (along the solutions of the system (24)) U(t) = U0 (t) +UL (t) =

1A 2 c¯

∫ L

¯ dx+ w+ (x,t)2 e−(α /c)x

0

(26) ∫ 1B L − ¯ + ¯ w (x,t)2 e(α /d)x dx 2d 0 where A > 0, B > 0, α > 0 are free parameters. Differentiating along the solutions we shall have, after some manipulation dU 1 = −α U − (Bw− (0,t)2 − Aw+ (0,t)2 )− dt 2 ( ) (27) 1 ¯ ¯ w+ (L,t)2 − Be(α /d)L w− (L,t)2 − Ae−(α /c)L 2

The choice of µi (t) will be concerned with the quadratic terms of the boundary conditions. Taking into account that (23) can be re-written as follows ¯ µ0 (t) w+ (0,t) = −ρ1 w− (0,t) − (1 + ρ1 )(γ˜/(2H)) (28) ¯ µL (t) w− (L,t) = −ρ2 w+ (L,t) − (1 + ρ2 )(γ˜/(2H)) and making the choices ¯ γ˜)δL w+ (L,t) (29) ¯ γ˜)δ0 w− (0,t) ; µL (t) = (2H/ µ0 (t) = (2H/ with δ0 > 0, δL > 0, the equality (27) will take the form [ ] dU 1 B 2 = −α U − A − (ρ1 + (1 + ρ1 )δ0 ) w− (0,t)2 − dt 2 A ] [ 1 B ¯ ¯ − A e−(α /c)L − e(α /d)L (ρ2 + (1 + ρ2 )δL )2 w+ (L,t)2 2 A (30) In order to have dU/dt < 0 the following choice of B/A has to be feasible ¯ B exp{−α L(1/c¯ + 1/d)} (ρ1 + (1 + ρ1 )δ0 )2 < < (31) A (ρ2 + (1 + ρ2 )δL )2 For this it is necessary that (ρ1 + (1 + ρ1 )δ0 )(ρ2 + (1 + ρ2 )δL ) < ¯ < exp{−α (L/2)(1/c¯ + 1/d)}

(32)

which imposes the small gain choice

δ0