Neutral functional differential equations and systems of conservation laws

Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th9-14, World Congress Control The Federation of Toulouse, France, July 2017 Available online at www.sciencedirect.com The International International Federation of Automatic Automatic Control The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 13336–13341 Neutral functional differential equations and Neutral functional differential equations and Neutral functional differential and Neutral systems functional differential equations equations and of conservation laws systems of conservation laws systems of conservation laws systems of conservation laws

Daniela Danciu and Vladimir R˘asvan Daniela Danciu and Vladimir R˘ asvan Daniela Daniela Danciu Danciu and and Vladimir Vladimir R˘ R˘aasvan svan Department of Automation and Electronics, University of Craiova, A.I.Cuza, Department of Automation and Electronics, University Craiova, A.I.Cuza, 13, Craiova, RO-200585, Romaniaof Department and University of Department of of Automation Automation and Electronics, Electronics, University of Craiova, Craiova, A.I.Cuza, A.I.Cuza, Craiova, RO-200585, Romania (e-mail:13, {ddanciu,vrasvan}@automation.ucv.ro) 13, Craiova, RO-200585, Romania 13, Craiova, RO-200585, Romania (e-mail: {ddanciu,vrasvan}@automation.ucv.ro) (e-mail: (e-mail: {ddanciu,vrasvan}@automation.ucv.ro) {ddanciu,vrasvan}@automation.ucv.ro)

Abstract: A natural way of introducing neutral functional differential equations is integration along the Abstract: A natural of introducing neutral differential equations isequations. integration the characteristics of theway Riemann invariants of thefunctional hyperbolic partial differential Upalong to now Abstract: way of neutral functional differential equations integration along the Abstract: A A natural natural way of introducing introducing neutral functional differential equations is isequations. integration along the characteristics of the Riemann invariants of the hyperbolic partial differential Up to now the problem has been discussedinvariants by A.D. of Myshkis, K.L. Cooke and their followers in the linear or characteristics of the Riemann the hyperbolic partial differential equations. Up to now characteristics of the Riemann invariants of the hyperbolic partial differential equations. Up to now the problem has been discussed by A.D. Cooke and their followers in the linear or quasilinear case. The conservation laws areMyshkis, nonlinearK.L. hyperbolic partial differential equations whose the problem has been discussed by A.D. Myshkis, K.L. Cooke and their followers in the linear or the problemcase. has The beenconservation discussed bylaws A.D.areMyshkis, K.L. Cooke and their followersequations in the linear or quasilinear nonlinear hyperbolic partial differential whose study goes back to the fifties of the previous century, being due to the pioneering papers of O.A. Oleinik quasilinear case. The conservation laws are nonlinear hyperbolic partial differential equations whose quasilinear case.toThe conservation laws arecentury, nonlinear hyperbolic partial differential equations whose study goes back the fifties of the previous being due to the pioneering papers of O.A. Oleinik and P.D. Lax. These equations describe important phenomena inthe Physics and Engineering, being also study goes back to fifties previous century, being pioneering papers studyP.D. goesLax. backThese to the theequations fifties of of the the previous century,phenomena being due due to to the pioneering papers of of O.A. O.A. Oleinik Oleinik and describe important in Physics and Engineering, subject toLax. control issues. The present paper is an attempt to extend the method of integratingbeing alongalso the and P.D. These equations describe important phenomena in Physics and Engineering, being also and P.D. Lax. These equations describe important phenomena in Physics and Engineering, being also subject to control issues. The present paper is an attempt to extend of integrating along the characteristics to the systems of conservation laws. The main purposethe ofmethod this attempt is to emphasize the subject to control issues. The present paper is an attempt to extend the method of integrating along subject to control issues. Theof present paper islaws. an attempt to extend the method of integrating along the the characteristics to the systems conservation The main purpose of this attempt is to emphasize occurring nonlinear neutral functional differential equations and to construct the augmented validation characteristics to the systems of conservation laws. The main purpose of this attempt is to emphasize the characteristics to theneutral systemsfunctional of conservation laws. equations The main and purpose of this attempt is to emphasize the occurring nonlinear differential to construct the augmented validation (existence, uniqueness, continuous data dependence, Stability Postulate) at the least at the level of the occurring neutral functional differential equations and to augmented validation occurring nonlinear nonlinear neutral functional differential equations and Postulate) to construct construct the augmented validation (existence, uniqueness, continuous data dependence, Stability at least at the level of the classical solutions. (existence, uniqueness, continuous data dependence, Stability Postulate) at least at the level (existence, uniqueness, continuous data dependence, Stability Postulate) at least at the level of of the the classical solutions. classical solutions. classical solutions. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Distributed parameter systems, Nonlinear systems, Time delay, Conservation laws Keywords: Distributed Distributed parameter systems, systems, Nonlinear systems, systems, Time delay, delay, Conservation laws laws Keywords: Keywords: Distributed parameter parameter systems, Nonlinear Nonlinear systems, Time Time delay, Conservation Conservation laws 1. MOTIVATING EXAMPLES where H is the water level and Q is the water flow, µ (t) repre1. MOTIVATING MOTIVATING EXAMPLES EXAMPLES (t) reprewhere H is the water level and Q is the water flow, sents the level control signal downstream, while theµ upstream 1. µ reprewhere H the water and Q flow, 1. MOTIVATING EXAMPLES µ (t) (t) reprewherethe H is is the control water level level anddownstream, Q is is the the water water flow, sents level signal while the upstream flow Q(0,t) is assumed not regulated. The control aim is to the level control signal downstream, while the upstream The conservation laws are basic in physics of the continuous sents sents the level control signal downstream, while the upstream flow Q(0,t) is assumed not regulated. The control aim is to ¯ maintain constant level H(t, L) ≡ H. The constant γ > 0 charThe conservation laws are basic in physics of the continuous flow Q(0,t) is assumed not regulated. The control aim is media. But this is laws not the place to study physics development The are basic physics of continuous flow Q(0,t) is assumed not L) regulated. controlγ > aim is to to ¯¯ TheThe The conservation conservation laws areplace basictoin instudy physics of the thedevelopment continuous maintain constant level H(t, ≡ H. constant 0 characterizes the down stream spillway. media. But this is not the physics maintain constant level H(t, L) ≡ H. The constant γ > 0 charor related mathematical aspects. We shallphysics rather focus on en- maintain constant level H(t, L) ≡ H. ¯ media. But this is not the place to study development The constant γ > 0 charmedia. Butmathematical this is not theaspects. place toWe study physics development the down stream spillway. or related shall rathertofocus focus on and en- acterizes acterizes the down stream spillway. gineering applications with particular reference control or mathematical aspects. We rather on enthe hand, down the stream spillway. On the other dynamics of hydraulic power plants also or related related applications mathematicalwith aspects. We shall shall rathertofocus on and en- acterizes gineering particular reference control modeling applications for control. To beparticular still morereference specific,tothere willand be On the other hand, the dynamics of hydraulic also gineering with control involves hydraulics butdynamics through galleries andpower pipes plants i.e. closed On the other hand, the of power plants also gineering applications with particular reference to control and the other hand, the dynamics of hydraulic hydraulic power plants also modeling for for control. To be be still still morewhich specific, therecommon will be be On considered here the boundary control is more involves hydraulics but through galleries and pipes i.e. closed modeling control. To more specific, there will media. The equations arethrough very similar to those of Saint Venant involves hydraulics but galleries and pipes i.e. closed modeling for control. To be still more specific, there will be involves hydraulics but through galleries and pipes i.e. closed considered here the boundary control which is more common in applications for engineering. With which respectistomore this common we shall Wylie media.and The equations are very similar to those of Saintthe Venant considered here the boundary Streeter (1978), Popescu (2008). However, disThe equations are similar to of Venant considered herefor theengineering. boundary control control which media. TheStreeter equations are very very similar(2008). to those those of Saint Saintthe Venant in applications With respectisto tomore this common we shall shall media. mention traffic control described by conservation laws Kachroo Wylie and (1978), Popescu However, in applications for engineering. With respect this we tributed parameters are taken into account for waterhammer Wylie and Streeter (1978), Popescu (2008). However, the disin applications for engineering. With respect to laws this Kachroo we shall Wylie and Streeter (1978), Popescu (2008). However, the disdismention traffic control described by conservation (2007) and thecontrol applications to hydraulic and thermal power tributed parameters taken into account for waterhammer mention traffic described by conservation laws Kachroo computation only. are tributed parameters are taken into account for waterhammer mention traffic control described by conservation laws Kachroo tributed parameters are taken into account for waterhammer (2007) and the applications to hydraulic and thermal power engineering. only. (2007) computation (2007) and and the the applications applications to to hydraulic hydraulic and and thermal thermal power power computation computation only. engineering. Another classonly. of applications arises from thermal power engiengineering. engineering. A quite well studied application is modeling and stabilization Another class applications thermal power engineering. Here of it is interestingarises to citefrom three pioneering papers Another class of applications arises from thermal power engiclass of applications arises from thermal powerpapers engiA quite quite well studied application is basics modeling andmodeling stabilization of flows well in open canals where the of the are Another neering. Here it is interesting to cite three pioneering A studied application is modeling and stabilization in this field Sokolov (1946), Kabakov (1946), Kabakov and neering. Here it is interesting to cite three pioneering papers A quite well studied application is modeling and stabilization neering. Here it is interesting to cite three pioneering papers of flows in open canals where the basics of the modeling are given by the equations of where Saint Venant. Theof reader is sent e.g.are to in this field Sokolov Kabakov and of flows in open canals the basics the modeling Sokolov (1946) where (1946), there areKabakov used the (1946), equations of the isenin this field Sokolov (1946), Kabakov (1946), Kabakov and of flows in open canals where the basics of the modeling are in this field Sokolov (1946), Kabakov (1946), Kabakov and given by by the theand equations of(2002), Saint Venant. Venant. The reader reader is sent sent Coron e.g. to to Sokolov Leugering Schmidtof de Halleux et al. (2003), (1946) where there are used the equations of the isengiven equations Saint The is e.g. tropic (barotropic) flow under the form Sokolov (1946) where there are used the equations of the isengiven by theand equations of(2002), Saint Venant. The et reader is sent Coron e.g. to Sokolov (1946) where there are used the equations of the isenLeugering Schmidt de Halleux al. (2003), et al. (2007), Petre and (2002), R˘asvan de (2009). Theetliterature cited by tropic (barotropic) flow under the form Leugering and Schmidt Halleux al. (2003), Coron Leugering andPetre Schmidt (2002), de Halleux al. (2003), Coron tropic (barotropic) (barotropic) flow flow under under the the form form et al. (2007), (2007), andmore R˘aasvan svan (2009). Theet literature cited by tropic these papers can give insight to this problem which has et al. Petre and R˘ (2009). The literature cited by et al. (2007), Petre and R˘ a svan (2009). The literature cited by these papers can give more insight to this problem which has broad applications in irrigation control via control gates. ∂ρ ∂ these can more to problem which these papers papers can give give more insight insight to this this problem which has has + ∂ (ρ w) = 0 broad applications in irrigation irrigation control via control control gates. ∂∂ ρ broad applications in control via gates. ρ ∂ ∂ t broadshall applications in irrigation control gates. the + ∂∂l (ρ w) w) = 00 We give here, for the sake of via thecontrol completeness, ∂ρ + (2) ∂ tt + ∂∂ ll ((ρ ρ w) = =0 We shall give here, for the sake of the completeness, the ∂ equations of a controlled hydraulic channel Bastin et al. (2005) We shall give here, for the sake of the completeness, the (2) ∂ w ∂ w ∂p ∂ ∂ t l (2) We shall give here, for the sake of the completeness, the equations of of aa controlled controlled hydraulic hydraulic channel channel Bastin Bastin et et al. al. (2005) (2005) ρ ∂ w + ρ w ∂ w + ∂ p = 0, p = f (ρ ) (2) equations ∂ w ∂ w ∂ p ∂ t ∂ l ∂ l equations of a controlled hydraulic channel Bastin et al. (2005) + + = 0, p = f ( ρ ρ w ρ ) ∂ w ∂ w ∂ p + + = 0, p = f ( ρ ρ w ρ ) ∂ t ∂ l ∂ l ρ ∂ t + ρ w ∂ l + ∂ l = 0, p = f (ρ ) ∂H ∂Q ∂t ∂ l mass flow through the area with p – the pressure, ρ∂wl – the + ∂Q = 0 ∂∂ H H ∂ Q with p – the pressure, ρ w – the the area ∂ t ∂ x unit, (pρ ––the thepressure, mass density and wmass – theflow flowthrough velocity); here with ρ the flow through the area ∂H + ∂ Q = 00 with pρ ––the pressure, ρw w ––and the wmass mass flow through the here area ∂ tt +  + ∂∂ xx = = 02 unit, ( the mass density – the flow velocity); ∂ p = f ( ρ ) is the thermodynamic state equation for the barotropic 2 unit, ( ρ – the mass density and w – the flow velocity); here t ∂ x H ∂∂Q Q unit, ( ρ – the mass density and w – the flow velocity); here  = ff ((ρ ρ ) is the thefluid. thermodynamic state equation equation for the the barotropic + ∂  Q22 + g H 22  compressible In the aforementioned papers thebarotropic approach ppp = thermodynamic state for ∂∂ Q =0 = f (ρ )) is is thefluid. thermodynamic state equation for the barotropic t + ∂∂∂x  Q H2 + g H 0 compressible In the aforementioned papers the approach H22 = ∂∂Q Q Q is that of the brute force linearization in order to reduce the + + g = 0 compressible fluid. In the aforementioned papers the approach ∂∂ tt + ∂∂ xx H +g 2 =0 compressible fluid. Inforce the aforementioned papers thereduce approach H 2 is that of the brute linearization in order to the (1) model to athe boundary value linearization problem for linear hyperbolic partial Q(0,t) (t) is that of brute force in order to reduce the ∂=x Qd H ∂t 2 is that to ofathe brute force linearization in order to reduce the (1) model boundary value problem for linear hyperbolic partial Q(0,t) = Q (t) d differential equations (PDEs). (1) model to a boundary value problem for linear hyperbolic partial Q(0,t) = Q (t) d (1) model to a boundary value problem for linear hyperbolic partial Q(0,t) = Qd (t) differential equations equations (PDEs). (PDEs). differential Q(L,t) = γ (H(t, L) − µ (t))3/2 differential equations (PDEs). Equations (2) can however be given the form of a system of 3/2 3/2 Q(L,t) = γ (H(t, L) − µ (t)) Equations (2) can however be given the form of aa system of Q(L,t) conservation laws (we skip the manipulations) Equations (2) can however be given Q(L,t) = = γγ (H(t, (H(t, L) L) − −µ µ (t)) (t))3/2 Equations (2)laws can (we however be manipulations) given the the form form of of a system system of of dµ dσ conservation skip the conservation laws (we skip the manipulations) Ts dµ = σ , Tk dσ = −σ − ks µ + η (t) conservation laws (we skip the manipulations) d µ d σ −σ T , Tk ddtσ = − kks µ +η (t) µ = s ddt T =σ σ =− = = −σ Tss dt σ ,, T Tkk dt σ− − kss µ µ+ +η η (t) (t) dt dt dt dt Copyright © 2017 IFAC 13878 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 13878 Copyright 2017 responsibility IFAC 13878 Peer review© of International Federation of Automatic Control. Copyright ©under 2017 IFAC 13878 10.1016/j.ifacol.2017.08.1897

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Daniela Danciu et al. / IFAC PapersOnLine 50-1 (2017) 13336–13341

∂ρ ∂ + (ρ w) = 0 ∂t ∂l   1 ∂ ∂ (ρ w) + f (ρ ) + (ρ w)2 = 0 ∂t ∂l ρ

(3)

Now the engineering approach to modeling is based on the use of the rated variables: the rating is done with respect to some reference (steady state, maximal) values. Introducing the rated variables (including the rated length of the pipe) the conservation laws (3) take the form

∂ ξρ ∂ ξw + = 0, t > 0, 0 < λ < 1 ∂t ∂λ   ∂ ξw ∂ ξ2 + ψc2 Tc ξρ + ψc2 w = 0 ∂t ∂λ ξρ Tc

(4)

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 (BCs) for (4) – in rated variables – are as follows

ξw (0,t) = πs (t)Φ(πs (t)/ξρ (0,t)) ξw (1,t) = ψs ξρ (1,t)

(5)

Here also some explanation are necessary and useful. The variable πs (t) is a rated pressure of the steam delivered by the steam turbine at the regulated steam extraction. The flow being sub-critical but also critical (especially during the transients), there was used the Saint Venant formula, with the Saint Venant function given by Φ(x) =



√ √ (1/x) x≤ e √ 2 ln x, 1 ≤ √ 1/ e, x≥ e

(6)

At the steam consumer (λ = 1) the flow is critical and ψs accounts for a coefficient that describes the consumer. We add to the aforementioned equations (4)–(6) the equations describing the steam turbine dynamics ds = απ1 + (1 − α )π2 − νg , 0 < α < 1 dt dπ1 T1 = µ1 − π 1 , 0 ≤ µ1 ≤ 1 dt (7) dπs Tp = π1 − βs µ2 πs − (1 − βs)ξw (0,t) dt dπ2 = µ2 πs − π2, 0 < µ2min ≤ µ2 ≤ 1. T2 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

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Looking at (5) – the first equation – and (7) – the third equation – one can observe some kind of internal feedback. As pointed out in Neymark (1978) an internal feedback may be the source of some instability. On the other hand, this feedback shows that (5) and (7) have to be considered together. Consequently, equations (4) – (5)) will define a boundary value problem for a system of conservation laws with some boundary conditions that are both non-standard and nonlinear. Under these circumstances the so-called augmented model validation R˘asvan (2014) that integrates existence of positive evolutions and inherent stability of equilibria (Stability Postulate ˇ of Cetaev (1931, 1936a,b)) to the usual well posedness in the sense of Hadamard appears as both necessary and useful. An additional remark concerns the small parameters. In the pioneering papers Sokolov (1946), Kabakov (1946), Kabakov and Sokolov (1946) some terms in (2) had been neglected from the beginning and the conservation law character of the equations had been lost (the PDEs had become linear). Under the rated variables in (4), one can observe the parameter ψc which might be small. Under the data of Kabakov and Sokolov (1946) ψc ≈ 0.1 hence ψc2 = 0.01 and the term containing it might be neglected. But in this case also ψc2 Tc has to be considered as small (while we do not know very well what are singular perturbations for such systems of partial differential equations); therefore ξρ (λ ,t) ≡ ξρ (t) – it is independent of λ . We integrate the first equation of (4) from 0 to 1 and take into account the boundary conditions to find d ξρ + ψs ξρ − πsΦ(πs /ξρ ) = 0 (8) dt to obtain, together with (7), a system of nonlinear ordinary differential equations. Tc

For this system existence, uniqueness and data dependence have known standard issues. The proof of the existence of positive solutions goes as in R˘asvan (1981) while the inherent stability property requires association of a Liapunov function. In the bilinear case – with Φ(·) being a constant function – the aforementioned problem was considered in R˘asvan (1984).Also, in Danciu et al. (2015) a numerical solution is obtained by means of a computational procedure based on a “convergent” Method of Lines combined with the paradigm of cell-based neural networks. 2. THE FUNCTIONAL EQUATIONS OF THE CONTROLLED HYDRAULIC CHANNEL Among the approaches that are currently used in order to tackle the boundary value problems such as (1) or (4)–(7) is the integration of the Riemann invariants of the PDEs along the characteristics. The most general cases of this approach were considered in the papers of A.D. Myshkis and his co-workers Myshkis and Shlopak (1957), Abolinia and Myshkis (1960), Myshkis and Filimonov (1981, 2008). A simpler case, that of lossless/distorsionless systems can be found in the papers of Cooke and Krumme (1968), Cooke (1970) (the complete proof of their main result is to be found in R˘asvan (2014). The specific feature of the aforementioned papers is that they deal with linear partial differential equations or with quasi-linear at most, while containing hints for iterative approach of the nonlinear case.

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The conservation laws are highly nonlinear partial differential equations: they can be approached either by linearization or by other methods that can avoid this linearization. We shall refer for these techniques to Petre and R˘asvan (2009), R˘asvan (2015) for the hydraulic and for the thermal power case respectively. Consider first the equations (1) of the controlled hydraulic channel. The partial differential equations can be written as    1 ∂ H + =0 (9) Q2 Q ∂x Q gH − 2 2 H H for classical solutions at least. The eigenvalues of the system matrix are

∂ ∂t



H Q





0

Q  ± gH H √ and if the fluviality assumption Q < H gH holds then

λ1,2 =

(10)

Q  Q  − gH < 0 < + gH (11) H H and the system is hyperbolic. The matrix can be diagonalized as follows     Q ∂H ∂Q ± gH ∓ H ∂t ∂t        Q Q ∂H ∂Q ± =0 ± gH ± gH ∓ H H ∂x ∂x

   Q ∂ ± ∂ ± ξ (x,t) ± gH ± ξ (x,t) = 0 ∂t H ∂x

or

(13)

(14)

  ∂ + ∂ + ξ (x,t) + 3ξ + (x,t) − ξ − (x,t) ξ (x,t) = 0 ∂t ∂x (15)  −  ∂ − ∂ − + ξ (x,t) − 3ξ (x,t) − ξ (x,t) ξ (x,t) = 0 ∂t ∂x hence the partial differential equations are expressed in Riemann invariants only. The boundary conditions become 



  If (H(x,t), Q(x,t)) is a continuous classical solution, then (17) holds for these functions provided t > 0 is small enough. Based on (13) we can stress that

ξ0+ (x) = ξ+ (x, 0) > 0; ξ0− (x) = ξ− (x, 0) > 0

(18) ±  and ξ (x,t) > 0 provided t > 0 is small enough. Also, for t > 0 small enough we shall have 3ξ± (x,t) − ξ∓ (x,t) > 0

(19)

In order to emphasize the role of the initial conditions we make further notations ξ± (x,t) := ξ± (ξ0± (·); x,t) (20) The characteristics associated to this solution are defined by 1 dt  =  dx ± 3ξ±(ξ± (·); x,t) − ξ∓ (ξ∓ (·); x,t) 0 0

(21)

where 3ξ± (ξ0± (·); x,t) − ξ∓ (ξ0∓ (·); x,t) > 0 as a consequence of (17). For each (x,t) within the considered domain (0 < x < L and t > 0 sufficiently small) system (21) is defining an increasing characteristic curve t + (σ ; x,t) and a decreasing one t − (σ ; x,t). Consequently, t + (·; x,t) can be extended “to the right” up to x = L while t − (·; x,t) can be extended “to the left” up to x = 0. We write down the forward wave along the increasing characteristic and the backward wave along the decreasing characteristic (22) Φ± (σ ; x,t) := ξ± (σ ;t ± (σ ; x,t)) ± The functions Φ (·; x,t) are constant along the characteristics hence Φ+ (x; x,t) = Φ+ (L; x,t) ⇒ ξ+ (x,t) = ξ+ (L;t + (L; x,t))

Φ− (x; x,t) = Φ− (0; x,t)

(23)

In the cases when the increasing characteristic curve can be extended also backwards up to x = 0 and the decreasing one forwards up to x = L we can obtain from (23)

 2 ξ + (L,t) − ξ − (L,t) ξ + (L,t) + ξ − (L,t)

 3/2 2 √ ξ + (L,t) + ξ − (L,t) − 4 gµ (t)

(17)

⇒ ξ− (x,t) = ξ− (0;t − (0; x,t))

 2 ξ + (0,t) − ξ −(0,t) ξ + (0,t) + ξ −(0,t) = 32gQd (t)

= 4g3/4γ

Q0 (x)  Q0 (x)  − gH0 (x) < 0 < + gH0 (x) H0 (x) H0 (x)

(12)

Solving a differential equation with exact differential by finding an integrating factor, the Riemann invariants are obtained:  Q ξ ± (x,t) = 2 gH ± H The integrating factor being 1/H, (12) becomes

solution of (1) defined by some given u(t) and by the initial  0) = Q 0 (x), defined on [0, L]  0) = H 0 (x), Q(x, conditions H(x, 0 (x) > 0, as sufficiently smooth functions. We assume also H  Q0 (x) > 0 and such that (11) holds for these functions i.e.

ξ+ (0,t) = ξ+ (L;t + (L; 0,t))

(16)

We shall now illustrate the integration along the characteristics R˘asvan (2014). The problem is here nonlinear, hence all results   will have a local character. Therefore let (H(x,t), Q(x,t)) be a

ξ− (L,t) = ξ− (0;t − (0; L,t)).

We can now define the propagation times of the two waves

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(24)

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Daniela Danciu et al. / IFAC PapersOnLine 50-1 (2017) 13336–13341

T + (ξ�0± (·),t) = t + (L; 0,t) − t

(25)

mathematical background can be proved a posteriori for the closed loop system.

T − (ξ�0± (·),t) = t − (0; L,t) − t

Here we made use of (20) to emphasize the dependence of the propagation times on the initial conditions. Now, we re-write (24) as

ξ�+ (0,t) = ξ�+ (L;t + T + ), ξ�− (L,t) = ξ�− (0;t + T − )

and introduce the functions

y�± (t)

(26)

by

y�+ (t) := ξ�+ (L,t) ⇒ ξ�+ (0,t) = y�+ (t + T + )

(27)

y� (t) := ξ�− (0,t) ⇒ ξ�− (L,t) = y�− (t + T − ). −

If the boundary conditions (16) are taken into account, it is found that y�± (t) are subject to the nonlinear difference equations

3. THE FUNCTIONAL DIFFERENTIAL EQUATIONS ASSOCIATED TO THE CO-GENERATION MODEL We shall consider here the co-generation model defined by (3) – (7) including a steam turbine with one regulated steam pipe with distributed parameters connecting the turbine steam extraction with the steam consumer. This model is highly nonlinear: the partial differential equations are nonlinear systems of conservation laws for the isentropic steam, the boundary conditions at the steam extraction chamber is nonlinear since the steam flow is sub-critical subject to the Saint Venant nonlinear dependence (6) and the turbine model is bilinear since the controlled flow within the turbine is critical. Since previous studies dealt with linearized model only, we shall consider the nonlinear conservation laws (4) and look for the Riemann invariants. We write down these equations in a vector-matrix form (assuming the solutions to be smooth enough)

�

�� �2 y�+ (t + T + ) − y�−(t) y�+ (t + T + ) + y�− (t) = 32gQd (t)

∂ ∂t

� + �� �2 y� (t) − y�− (t + T − ) y�+ (t) + y�− (t + T − ) ��

�3/2 �2 √ y�+ (t) + y�−(t + T − ) − 4 gµ (t) (28) with time varying T ± – also dependent on the initial conditions of the considered solution of (15) – (16). The initial conditions for (28) can be determined by considering those characteristics which cannot be extended up to x = 0 (the increasing one) or to x = L (the decreasing one) because they cross the abscissa of the definition rectangle [0, L] × [0, tˆ) before reaching the verticals x = 0 or x = L. = 4g3/4γ

The problem is now to construct by steps the solution of (28) where the time delays T ± (t) are time dependent (the initial conditions ξ�0± (·) are fixed for a given solution). Moreover, the difference equations are non-linear.

An additional problem would be to obtain from the construction by steps positiveness of the solutions, knowing that the initial conditions were assumed to be positive.

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�

ξρ ξw

�

 0 1  ξ2 1 + − w2 Tc 2 ψc ξ ρ

 � � 1 ∂ ξρ ξw  = 0 (29) 2 ξ w ∂λ ξρ

The eigenvalues of the matrix in (29) are

1 ξw ± (30) ξ ρ ψc and if we want λmin < 0 < λmax , some kind of fluviality assumption is also required here. We compute the diagonalization matrix and obtain the equations

λ1,2 =

�

� 1 ξw ∂ ξ ρ ∂ ξ w ± ∓ ψc ξ ρ ∂t ∂t (31) � � �� � � 1 ξw 1 ξw ∂ ξ ρ ∂ ξ w 1 + = 0. ± ∓ ± Tc ξρ ψc ψc ξ ρ ∂ λ ∂λ

Computing the integrating factor we find the Riemann invariants

To end this section we send again to Petre and R˘asvan (2009) where the same problem has been discussed under more general assumptions on the hydraulics. The variables in the conservation laws had been chosen the flow and the cross-section area. The associated functional differential equations were somehow more tractable. The explanation is the nonlinear dependence of cross-section area versus water level.

deducing that

The aforementioned paper as well as the results of this section deal nevertheless with nonlinear models unlike almost all cited literature where the models are linearized around a steady state.

Therefore, the Riemann invariants are subject to the equations

Two final remarks are useful here. First, the control is a boundary control and by control synthesis it is possible to linearize the controlled boundary condition. Consequently, the associated difference equations may be simpler. The second remark is that the aforementioned control synthesis based e.g. on the energy integral identity can be accomplished at the formal level. The

ξ ± (ξρ , ξw ) =

1 ξw ln ξρ ± ψc ξρ

ξw 1 + = (ξ − ξ −). ξρ 2

� � 1 2 ∂ξ− ∂ξ+ + − + ξ −ξ + =0 ∂t 2Tc ψc ∂ λ � � 1 2 ∂ξ− ∂ξ− + ξ+ −ξ−− =0 ∂t 2Tc ψc ∂ λ and the boundary conditions

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(32)

(33)

(34)

Proceedings of the 20th IFAC World Congress 13340 Daniela Danciu et al. / IFAC PapersOnLine 50-1 (2017) 13336–13341 Toulouse, France, July 9-14, 2017

 1 + ξ (0,t) − ξ −(0,t) 2 + − 1 = √ πs (t)e−(ψc /2)(ξ (0,t)+ξ (0,t)) e

(35)

ξ + (1,t) − ξ −(1,t) = 2ψs (we assumed, for simplicity, that the flow at the boundary x = 0 is only critical).

These equations have to be coupled to (7) by substituting ξw (0,t) from the boundary conditions. √ However, in the critical flow case it will result ξw (0,t) = (1/ eπs (t)) hence the turbine differential equations will be decoupled from (41) that might be discussed independently, starting form a construction by steps. Equations (41) can be given the form y + (t + T + )e(ψc /2)y

The state variable πs (t) is obtained from (7) where ξw (0,t) has to be obtained from the Riemann invariants

ξw (λ ,t)  + − 1 + ξ (λ ,t) − ξ −(λ ,t) e(ψc /2)(ξ (λ ,t)−ξ (λ ,t)) . = 2

(36)

As in the previous section, the results concerning the associated functional equations have a local character. Therefore, let ( s(t), π1 (t), πs (t), π2 (t), ξρ (λ ,t), ξw (λ ,t)) be a solution of 2 (t) and by the ini1 (t), µ (4) – (7) defined by some given µ tial conditions ( s(0), π1 (0), πs (0), π2 (0), ξρ (λ , 0), ξw (λ , 0)) where the initial conditions are sufficiently smooth functions. Let ξρ (λ , 0) > 0, ξw (λ , 0) > 0 and also such that ψc ξw (λ , 0) < ξρ (λ , 0). If (ξρ (λ ,t) < ξw (λ ,t)) is a continuous classical solution, then the aforementioned inequalities hold for these functions provided t > 0 is small enough. This will make sense for the Riemann invariants (32). We can define here the characteristic equations associated to this solution 2ψc Tc dt . =  dλ ξ + (λ ,t) − ξ−(λ ,t) ± 2

(37)

Taking into account the aforementioned inequalities, for t > 0 sufficiently small, an increasing characteristic curve t + (σ ; x,t) is defined, also a decreasing t − (σ ; x,t). Proceeding as in Section 2 we obtain the analogue of (24)

also of (26)

ξ+ (x,t) = ξ+ (1;t + (1; x,t))

(38)

ξ+ (0,t) = ξ+ (1;t + (1; 0,t))

(39)

ξ− (x,t) = ξ− (0;t − (0; x,t))

ξ− (1,t) = ξ− (1;t − (0; 1,t))

Introducing the propagation times as in (25) and introducing y + (t) := ξ+ (1,t), y − (t) := ξ+ (0,t) we substitute the boundary conditions to obtain  1 + y (t + T + ) − y − (t) 2 + + − 1 = √ πs (t)e−(ψc /2)(y (t+T )−y (t)) e y + (t) − y − (t + T − ) = 2ψs

(40)

(41)

+ (t+T + )

− 2 = y − (t) + √ πs (t)e−(ψc /2)y (t) e

(42)

y − (t + T − ) = y + (t) − 2ψs

We have to take into account however that the propagation times are time dependent also solution dependent. 4. SOME CONCLUSIONS AND HINTS ON FUTURE RESEARCH We have sketched in this paper some connection between the description by conservation laws of the controlled systems with distributed parameters and some functional equations associated by integration of the Riemann invariants along the characteristics. It is a common knowledge fact that for linear hyperbolic partial differential equations this association of functional equations by integration along the characteristics represents the most natural way of introducing equations with deviated arguments of retarded, neutral, even advanced type. The neutral equations (including difference equations with continuous time) are most frequent, as a consequence of the character of the boundary conditions; moreover, if these boundary conditions are nonlinear, the resulting equations are also nonlinear. The systems of conservation laws are nonlinear hyperbolic partial differential equations and display such phenomena as shock and rarefaction waves. Integration along the characteristics in this case leaves aside generalized solutions – only classical, at most discontinuous solutions are tractable by this method – and the characteristics themselves are solution dependent, the deviating argument of the functional equations being thus time varying. Since these functional equations appear also as highly nonlinear – see e.g. the applications tackled in the present paper – their mathematical tractability appears as quite doubtful. Consequently we stress that the basic theory for model validation (well posedness in the sense of Hadamard) would be better served by the standard results of the field, incorporated in some widely cited monographs - see Bressan (2000), Dafermos (2010), Godunov and Romenskii (2003), Lax (1987, 2006), Li (1994), Li and Wang (2009), Rozhdestvensky and Yanenko (1983), Serre (2000). We have to add here a remark made throughout the paper concerning the approach to be taken in control synthesis: one may proceed in a formal (mathematically speaking) way in order to synthesize the controller and obtain the structure and the model of the closed loop system. In this way some subsystems might result simplified e.g. linearized – for instance the boundary conditions since the control is mainly boundary control. The mathematical theory can thus start afterwards and concern the closed loop system (where obviously stability is a fundamental issue due to the feedback structure. Finally a remark about numerics: it is useful to obtain a sound mathematical basis for the Method of Lines (MoL) for conservation laws Danciu et al. (2015) in the nonlinear case. At their

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turn, the numerical results can be useful to see if consideration of nonlinear partial differential equations makes indeed a significant difference with respect to the linearized versions. Summarizing – quite a program to follow. ACKNOWLEDGEMENT This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CCCDI UEFISCDI, project number 78 BM. REFERENCES Abolinia, V. and Myshkis, A. (1960). Mixed problem for an almost linear hyperbolic system in the plane (in russian). Matem. Sbornik, 50(92), 423–442. Bastin, G., Coron, J.M., dAndr´ea Novel, B., and Moens, L. (2005). Boundary control for exact cancellation of boundary disturbances in hyperbolic systems of conservation laws. In Proc. 44th IEEE Conf. Decision and Control and the Eur. Contr. Conf. IEEE Press, NJ USA. Bressan, A. (2000). Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem. Oxford University Press, Oxford UK. Cooke, K. (1970). A linear mixed problem with derivative boundary conditions. In D. Sweet and J. Yorke (eds.), Seminar on Differential Equations and Dynamical Systems (III), volume 51, 11–17. University of Maryland, College Park, lecture series edition. Cooke, K. and Krumme, D. (1968). Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations. J. Math. Anal. Appl., 24, 372–387. Coron, J.M., dAndr´ea Novel, B., and Bastin, G. (2007). A strict lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Aut. Contr., 52, 2–11. Dafermos, C. (2010). Hyperbolic Conservation Laws in Continuum Physics. Springer, New York. Danciu, D., Popescu, D., and Bobas¸u, E. (2015). Computational issues based on neural networks for a class of systems of conservation laws. In Proc. 16th Int. Carpathian Control Conf. ICCC2015. IEEE Hungary Section. de Halleux, J., Prieur, C., Coron, J.M., dAndr´ea Novel, B., and Bastin, G. (2003). Boundary feedback control in networks of open-channels. Automatica, 39, 1365–1376. Godunov, S. and Romenskii, E. (2003). Elements of Continuum Mechanics and Conservation Laws. Springer, Berlin Heidelberg. Kabakov, I. (1946). Concerning the control process for the steam pressure (in russian). Inzh. sbornik, 2, 27–60. Kabakov, I. and Sokolov, A. (1946). Influence of the hydraulic shock on the process of steam turbine speed control (in russian). Inzh. sbornik, 2, 61–76. Kachroo, P. (2007). Optimal and Feedback Control for Hyperbolic Conservation Laws. Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg. Lax, P. (1987). Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM Publications, Philadelphia.

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