Modeling and control of irreversible port-hamiltonian systems

Transcription

1 Modeling and control of irreversible port-hamiltonian systems - Preworkshop school LHMNLC Valparaiso, Chile Hector Ramirez FEMTO-ST, UMR CNRS 6174, Université de Bourgogne Franche-Comté, 26 chemin de l épitaphe, F Besançon, France. - 1 May 2018 FEMTO-ST / UFC / UBFC 1 / 51

2 Energy based modeling and control Motivations for adopting an energy-based perspective in modeling and control Physical system can be viewed as a set of simpler subsystems that exchange energy through ports, Energy is a concept common to all physical domains and is not restricted to linear or non-linear systems: non-linear approach, Energy can serve as a lingua franca to facilitate communication among scientists and engineers from different fields, Role of energy and the interconnections between subsystems provide the basis for various control techniques: Lyapunov based control. FEMTO-ST / UFC / UBFC 2 / 51

3 Conservative systems [Maschke and van der Schaft, 1992, van der Schaft, 2000] Port-Hamiltonian control systems J(x) = J(x) the interconnection matrix ẋ = J(x) U (x) + gu(t), y = g(x) U (x) Balance equations expressed by PHS: Conservation of the Hamiltonian and of Casimir s of the Poisson bracket du dt = U gu = u y, dc dt = C gu = u y C Poisson bracket: {Z, G} J = Z (x)j(x) G (x) FEMTO-ST / UFC / UBFC 3 / 51

4 Thermodynamic systems First and second principle of thermodynamics Consider a closed system, du dt = 0 and ds dt ( = σ x, U ) 0 for PHS ds dt = S ( J x, U ) U = σ 0, for any U(x) This is the reason to consider quasi Hamiltonian system: retain much of the PHS structure, but their structure matrices depend explicitly on the gradient of the Hamiltonian (GENERIC, quasi Hamiltonian systems, Brayton-Mooser formulation,..) [Grmela and Öttinger, 1997, Hangos et al., 2001, Otero-Muras et al., 2008, Eberard et al., 2007, Hoang et al., 2011, Favache and Dochain, 2010] FEMTO-ST / UFC / UBFC 4 / 51

5 This presentation is based on Ramirez, Maschke and Sbarbaro, Irreversible port-hamiltonian systems: A general formulation of irreversible processes with application to the CSTR, Chemical Engineering Science, Modeling Ramirez, Le Gorrec, Maschke and Couenne. On the passivity based control of irreversible processes: a port-hamiltonian approach. Automatica, Control FEMTO-ST / UFC / UBFC 5 / 51

6 Irreversible port Hamiltonian systems A non-linear extension of port Hamiltonian systems FEMTO-ST / UFC / UBFC 6 / 51

7 Irreversible port Hamiltonian systems (Ramirez et al., 2012) IPHS for closed system ( ) ẋ = R x, U, S J U (x) i. x R n, and U(x) and S(x) relates to the energy and entropy respectively. ii. The constant structure matrix J = J R n R n. The definition of R ( ) ( R x, U, S = γ x, U ) {S, U} J, γ 0 {S, U} J defines the thermodynamic driving force Entropy balance: S t ( = S RJ U = R{S, U} J = γ x, U ) {S, U} 2 J = σ 0. FEMTO-ST / UFC / UBFC 7 / 51

8 Irreversible port Hamiltonian systems (Ramirez et al., 2012) IPHS for open systems ( ) ẋ = R x, U, S J U ( (x) + g x, U ) u, i. x R n, and U(x), S(x) : R n R relates to the energy and entropy respectively. ii. The constant structure matrix J = J R n R n. iii. gu R n describes the interaction with the environment. Energy balance and entropy balance with irreversible entropy creation. du dt = U gu, ds dt = σ + S gu Now we have two conjugated outputs, one with respect to the energy and another with respect to the entropy y = U gu ys = S gu FEMTO-ST / UFC / UBFC 8 / 51

9 The heat exchanger: interaction through two conducting walls Consider two simple thermodynamic systems with entropy balance equations ] [ T2 (S 2 ) T 1 (S 1 ) ] [ ] [Ṡ1 0 T = λ 1 (S 1 ) T Ṡ 1 (S 1 ) T 2 (S 2 ) + λ e u(t) T 2 (S 2 ) 2 T 2 (S 2 ) T 2 (S 2 ) with x = [S 1, S 2 ] the ( entropies, ) U = U 1 (x 1 ) + U 2 (x 2 ) the internal energy, U xi = T i (x i ) = T 0 exp where T i c 0 and c i are constants, u(t) external heat source i and λ, λ e > 0 Fourier s heat conduction coefficients of the wall. The process FEMTO-ST / UFC / UBFC 9 / 51

10 The heat exchanger: interaction through two conducting walls Consider two simple thermodynamic systems with entropy balance equations ] [Ṡ1 = λ Ṡ 2 [ T2 (S 2 ) T 1 (S 1 ) ] [ T 1 (S 1 ) T 1 (S 1 ) T 2 (S 2 ) + λ e T 2 (S 2 ) 0 u(t) T 2 (S 2 ) T 2 (S 2 ) ] The process may be written as IPHS ] [Ṡ1 = λ 1 1 [ ] 0 1 Ṡ U U }{{}}{{} ( R x, U ) J {S, U} J is the thermodynamic driving force {S, U} J = T 1 T 2 ( ) Hence, R(x, T ) = λ 1 1 T2 T1 = λ T 1 T 2 }{{} γ = S J U [ U ] [ 1 U + λ e 2 ] 0 u(t) T 2 (S 2 ) T 2 (S 2 ) } {{ } g(u,s 1,S 2 ) = [ ] [ ] [ ] T T 2 (T 1 T 2 ), with γ > 0 }{{} {S,U} J FEMTO-ST / UFC / UBFC 10 / 51

11 A single chemical reaction m r α i A i i=1 m β i A i with α i, β i the stoichiometric coefficients for species A i. The mass balance is i=1 ṅ i = r i V + F ei F si i = 1,..., m n i is the number of moles of the species i, n = (n 1,..., n m), r i = ν i r, where r (n, T ) = (r f r b ) is the reaction rate ν i is the signed stoichiometric coefficient: ν i = α i β i, F ei and F si are respectively the inlet and outlet molar flows, F e = (F e1,..., F em) The volume V in the reactor is assumed to be constant as well as the pressure The mass balance can be represented as ṅ = CrV + F e F i where C is a m 1 is called the stoichiometric vector FEMTO-ST / UFC / UBFC 11 / 51

12 The continuous stirred tank reactor (CSTR) The chemical reaction is denoted by ν 1 A ν m 1 A m 1 ν ma m, ν 1,..., ν m : A 1,..., A m : stoichiometric coefficients chemical species together with the definition of the reaction rate: r(a, T ) = r f (A f, T ) r r (A f, T ) with A the affinity of reaction. The mathematical model The balance equations [Aris, 1989], m ṅ i = r i V + F ei F }{{ si, Ṡ = (F } ei s ei F si s i ) + Q + σ, T w mass i=1 }{{} entropy FEMTO-ST / UFC / UBFC 12 / 51

13 A single chemical reaction ν 1 A ν m 1 A m 1 ν ma m, ν 1,..., ν m : A 1,..., A m : A stoichiometric coefficients chemical species affinity of reaction together with the definition of the reaction rate: r(a, T ) = r f (A, T ) r r (A, T ) U = the internal energy ν 1 0 [ ] ne n J = , g 1 = φ ( x, ν U ) 1, g 2 =., m }{{} 0 T e ν 1... ν m 0 Mass transfer 1 }{{}}{{} stoichiometric matrix Heat transfer [ u1 u 2 ] = [ F ] V Q m {S, U} J = A = ν i µ i i=1 γ = rv T A 0, µ 1,..., µ m : chemical potentials rv : molar flow FEMTO-ST / UFC / UBFC 13 / 51

14 Chemical reaction network Consider a chemical reaction network involving m chemical species, among which m r chemical reactions m r j m α ij A i β ij A i, j = 1,..., m r. i=1 i=1 The basic structure underlying the dynamics of the vector n of mole numbers of the chemical species is given by the mass balance law: ṅ = CrV + F e F s, where the m m r matrix C is called the stoichiometric matrix and whose columns are the stoichiometric vectors of each reaction: C = [C 1, C 2,..., C mr ], and r = [r 1, r 2,..., r mr ] is the vector whose elements are the reaction rates of each individual reaction. The energy and entropy balance are U = U in U out, Ṡ = S in S out + σ FEMTO-ST / UFC / UBFC 14 / 51

15 IPHS of chemical reaction networks Consider the chemical reaction network ṅ = CrV + F e F s, Ṡ = σ + S in S out, Define a vector containing the non-linear R j functions of each reaction: R R mr = [R 1,..., R m], then, [ ẋ = 0 m ] CR R C }{{ 0 } J R [ ] U + Fe F s S in S out The entropy balance is Ṡ = S ẋ = S J U R = R C µ = m r σ i where σ i is the entropy production due to the i-th chemical reaction. i=1 FEMTO-ST / UFC / UBFC 15 / 51

16 IPHS of chemical reaction networks Alternatively, The dynamic of the complete reaction is m k ẋ = X j + g(x, u) = i=1 ( mr ) R i J i } i=1 {{ } J R U + g(x, u). Notice that J i = with C i the m 1 stoichiometric vector. [ ] 0m C i Ci, 0 FEMTO-ST / UFC / UBFC 16 / 51

17 IPHS for coupled mechanical and thermodynamic systems A Reversible-Irreversible Port-Hamiltonian System (RIPHS) ( ) U ( ẋ = J ir x, U, S (x) + g x, U ) u, where the skew symmetric matrix J ir is defined as the sum : ( ) ( ) J ir x, U, S = J 0 (x) + R x, U, S J ( ) where J 0 (x) is the structure matrix of a Poisson bracket and R x, U, S and J are defined according of an IPHS. Furthermore the entropy function S (x) is a Casimir function of the Poisson structure matrix J 0 (x). FEMTO-ST / UFC / UBFC 17 / 51

18 RIPHS Consider for simplicity g = 0. First law is satisfied by skew-symmetry of J irr The entropy balance is given by ds dt = S U = U U (x)j ir (x) = 0 U J irr = S U ( J 0 + R ( = {S, U} J0 + γ ( = γ x, U x, U x, U, S ) {S, U} 2 J ) {S, U} 2 J = σ 0 ) ( ) S J U where we have used that S (x) is a Casimir function of the Poisson structure matrix J 0 (x), that is it satisfies {S, U} J0 = 0 for any Hamiltonian U (x). FEMTO-ST / UFC / UBFC 18 / 51

19 cause transformation of mechanical energy into heat. The sys cannot exchange heat with the environment, but a heat tran between the gas and the piston can take place. This example has already been used in order to illustrate GENERIC and Matrix formalism and to emphasize some relat with dissipative port-hamiltonian systems in Jongschaap The thermodynamic properties Öttinger (2004). It is a very simple example and hence the con GENERIC The piston in the gravitation field: H mec = 1 and Matrix models of this system are easy to b However2mit p2 is + sufficient mgz, where to illustrate z denotes the thedifferences and altitude of the piston and p its kinetic highlight momentum. the advantages of the contact model from the poin The properties of the perfect gas: its viewinternal of control. energy Indeed U(S, it is V an ) where open system S denotes and itthe can be seen entropy variable, V the volume variable complex andsystem the number composed of moles of twonsubsystems: is constant the piston and enclosed gas. The total energy of the system is: E(x) = The U((S, piston V )) is+ considered H mec(z, p), as awhere solid and its volume is consid x = [S, V, z, p] is the vector of state variables. to be constant. The Furthermore co-energy variables its mass is are constant definedas it is subje by the gradient of the total energy no mass exchange. Finally we assume that the tempera Example of the gas-piston system E S T E P V E = mg Fg z E p v where T is the temperature, P the pressure, F g the gravity Fig. 1. force, Study case: andthe v the adiabatic velocity piston. of the piston. FEMTO-ST / UFC / UBFC 19 / 51

20 Example of the gas-piston system The gas in the cylinder under the piston undergo a non-reversible transformation due to friction: F r = νv. The entropy balance is then ds dt = 1 T νv 2 = σ int which is the irreversible entropy flow at the temperature T, induced by the heat flow νv 2 due to the friction of the piston. ds = 1 dt T νv 2 σ int dv =Av dt dz =v dt dp = F g + AP F r = mg + AP νv dt The coupling between the piston and the gas: [ ] [ ] [ ] f e V 0 A ( P) F e = A 0 v where A denotes the area of the piston. FEMTO-ST / UFC / UBFC 20 / 51

21 Example of the gas-piston system The dynamic equations are hence νv S T T d V dt z = A P F p νv T } A {{ 1 0 } v J irr (T,v) with A J 0 = J = 0 A 1 0 ( The modulating function R = γ x, U ( the driving force of the friction and γ ( ) J irr (T, v) = J 0 +R x, U, S J ) {S, U} J is composed by ] T P {S, U} J = [ ]J[ = v F v ) = γ (T ) = ν > 0. It may be easily T x, U checked that the entropy is a Casimir function of J 0 as its first row (and column) is zero. FEMTO-ST / UFC / UBFC 21 / 51

22 Example: the RLC circuit Consider the PHS formulation of a simple RLC circuit [ ] [ ] ] Q 0 1 = + λ 1 r ] [ Q C λ L [ 0 1] u e, y e = [ 0 1 ] [ Q C λ L The electrical energy and its derivative in time are given by H e(q, λ) = 1 Q 2 2 C + 1 λ 2 2 L, Ḣ e = r ( ) λ 2 + ye L ue. For the irreversible case, the internal energy is E e(q, λ, S) = 1 Q 2 2 C(S) + 1 λ 2 2 L(S) + Ue(S) The variation in time of the internal energy is given by (u e = 0) Ė e = Ee Q Ė e = r(s) Q + Ee λ λ + Ee Ṡ S ( ) λ 2 + Ee L(S) S ds dt = 0 FEMTO-ST / UFC / UBFC 22 / 51

23 Example: the RLC circuit From Gibb s relation we have that Ee S ds dt = T (S). Hence = r(s) ( ) λ 2 = σ r 0 T (S) L(S) where σ r is the internal entropy production. Hence an IPHS formulation of the irreversible RLC circuit Q ( λ = r λ ) Q C λ u e Ṡ }{{} T L L T 0 }{{} R e }{{} J e J r The irreversible driving force of the system is, {S, E e} Jr = λ, the electrical current. L Hence the irreversible RLC can be formulated as the RIPHS ẋ e = (J e + R r J r ) Ee + geue FEMTO-ST / UFC / UBFC 23 / 51

24 Example: the MSD system The MSD system [ [ ] [ q 0 1 kq = p ṗ] 1 b with the mechanical energy given by m ] [ 0 + 1] u m, y m = [ 0 1 ] [ ] kq p. m H m(q, p) = 1 2 kq2 + 1 p 2 ( p ) 2 2 m, Ḣ m = b + y m m u m. Analogous to the RLC circuit q ( ṗ = b Ṡ T }{{} J m p m }{{} R b ) kq p u m m T 0 }{{} J b The irreversible driving force is {S, E m} Jb = p, i.e., the velocity. Hence m ẋ m = (J m + R b J b ) Em + gmum FEMTO-ST / UFC / UBFC 24 / 51

25 Some remarks The benefit of the energy based formulation of IPHS (Hamiltonian given by the internal energy) is clearly emphasised in this case, since it allows to naturally perform the interconnection with conventional PHS. This is not the case for quasi-hamiltonian formulations of thermodynamic systems where for instance the entropy (or some function of the entropy) is used as Hamiltonian. The formalism has recently been extended to deal with irreversible infinite dimensional PHS and systems with hysteresis. FEMTO-ST / UFC / UBFC 25 / 51

26 Ongoing work IPHS formulation of distributed irreversible processes Example: the heat equation s t = R sj s u s + z ( ) u R s s where ( ) R s = γ s s, u s, z {S, U} Js ( ) with J s = z, γs = 1 T λ s, u 2 s, z and {S, U} Js = T. The input and output of the z system are defined, respectively as ( ) [ ] u R s (t, b) u s (t, b) v = ( R s u s ), y = (t, a) s u s (t, a) FEMTO-ST / UFC / UBFC 26 / 51

27 Ongoing work It is direct to verify that the entropy production is equal to R s{s, U} 2 J s and that ( ) ( ) f s = λ T = u R z T z z s. The overall energy balance is s du dt b du b = a dt dz = a b ( u = a s ( u s ) ( ) s dz t ) ( R s(t, z)j u s + z an integrating by parts this can be written as ( du b ( ) ) u u = R s dz dt a z s s [ ] b ] b = y v = fs T [f = Q a a ( )) u R s dz, s FEMTO-ST / UFC / UBFC 27 / 51

28 Ongoing work The hysteron [Karnopp, 1983] b kh k m u d q z q t Total force applied to the hysteron is and the the dynamic of the hysteron is F h = F k + F d q z = d(f d ) = d(f h F k ) = d(f h k h q z), FEMTO-ST / UFC / UBFC 28 / 51

29 Ongoing work Irreversible entropy produced by the hysteron q z. q z q z max Fdm Energy lost Fdm FdM Fd FdM Fd From the second law the energy dissipated through the dissipative element produces entropy q zf d = d(f d )F d = σ d 0 The irreversible entropy production corresponds to the surface in between the hysteresis curve of the system. FEMTO-ST / UFC / UBFC 29 / 51

30 Ongoing work q t ṗ q z = b p T m Ṡ T d(f d ) k(q t q z) p m k h q z k(q t q z) T u m 0 where q t = q + q z, and with F d = k(q t q z) k h q z. R d = γ d {S, U} Jd = 1 T d (F d ) with γ d = 1 T d( ) > 0 and {S, U} J d = k(q t q z) k h q z = F d. Consider u m = 0, then the entropy balance Ṡ = b ( p ) T m T d (F d ) F d = σ b + σ d 0, FEMTO-ST / UFC / UBFC 30 / 51

31 Control of irreversible port-hamiltonian systems FEMTO-ST / UFC / UBFC 31 / 51

32 Energy based modeling and control Motivations for adopting an energy-based perspective in modeling and control Physical system can be viewed as a set of simpler subsystems that exchange energy through ports, Energy is a concept common to all physical domains and is not restricted to linear or non-linear systems: non-linear approach, Energy can serve as a lingua franca to facilitate communication among scientists and engineers from different fields, Role of energy and the interconnections between subsystems provide the basis for various control techniques: Lyapunov based control. FEMTO-ST / UFC / UBFC 32 / 51

33 Control of PHS [Ortega et al., 2001] Interconnection and damping assignment passivity based control IDA-PBC objective Find a static state-feedback control u(x) = β(x) such that the closed-loop dynamics is a PH system with interconnection and dissipation of the form ẋ = (J d (x) M d (x)) U d (x), U d (x), has a strict local minimum at x, J d (x, u) = J d (x, u) T, the desired interconnection matrix, M d (x, u) = M d (x, u) T 0, the desired dissipation matrix, FEMTO-ST / UFC / UBFC 33 / 51

34 IDA-PBC of PHS The procedure consists in the matching of the open and desired closed-loop vector fields J(x) U (x) + g(x)β(x) = ( J d (x) M d (x) ) U d (x) with u = β(x) a state modulated source. If this quasi-linear PDE is satisfied then and ẋ = (J d (x) M d (x)) U d (x), U d = U d M U d d < 0, x x, Ud (x ) = 0. FEMTO-ST / UFC / UBFC 34 / 51

35 How does IDA-PBC look like? IDA-PBC design gives a possibly non-linear PDE ( J x, U ) ( U (x) + g(x)β(x) = (J d x, U ) ) d Ud M d (x) (x) Hence, there are two big reasons to consider an particular structure for irreversible systems Thermodynamically coherent models Non-linear control design FEMTO-ST / UFC / UBFC 35 / 51

36 This presentation is based on Ramirez, Maschke and Sbarbaro, Irreversible port-hamiltonian systems: A general formulation of irreversible processes with application to the CSTR, Chemical Engineering Science, Modeling Ramirez, Le Gorrec, Maschke and Couenne. On the passivity based control of irreversible processes: a port-hamiltonian approach. Automatica, Control FEMTO-ST / UFC / UBFC 36 / 51

37 Interconnection and entropy rate assignment The idea look for the conditions to derive a stabilizing state feedback which renders the closed-loop system in the form ( ) with M x, U, A > 0. The time variation of A is in this case ( ) A ẋ = M x, U, A (x), Ȧ = A ( ) A (x)m x, U, A (x) 0 and under some additional properness conditions the closed-loop system becomes asymptotically stable. FEMTO-ST / UFC / UBFC 37 / 51

38 A class of Lyapunov function The availability function Use the negative of the total entropy as a convex extension to construct the availability function using thermodynamic considerations [Ydstie and Alonso, 1997, Alonso and Ydstie, 2001]. Consider Gibb s relation N 2 du = TdS PdV + µ i dn i = w(z) dz. With z = [S, V, n 1,..., n N 2 ] and w(z) = [T (z), P(z), µ 1 (z),..., µ N 2 (z)]. Since U is a homogeneous function of degree 1, from Euler s Theorem U = w(z) z, i=1 So we define an energy based availability function w(z) = U z (z) [ A(x, x ) = U(x) U(x ) + U ] (x ) (x x ) 0 FEMTO-ST / UFC / UBFC 38 / 51

39 The energy based availability function FEMTO-ST / UFC / UBFC 39 / 51

40 Stabilization condition The closed-loop equilibrium x of a controlled IPHS is asymptotically stable if A is strictly convex and x satisfies: γ {S, U} J {A, U} J + ỹ u = s, ỹ = g A. where s > 0, with strict equality only at x, u, where u is the steady state value of the control-input at the desired equilibrium. Proof (sketch): da dt = A ( U = = R (x) dx dt, (x) U (x ) ( U (x ) J U (x) ) RJ U (x) + ) + ( U U (x) ( U U (x) (x ) = γ {S, U} J {A, U} J + ỹ u, Def. Poisson Br ) (x ) gu, ) gu, Def. IPHS FEMTO-ST / UFC / UBFC 40 / 51

41 A globally stabilizing solution Consider the following solution ỹ u = R{A, U} J σ d [A, A] M + R d {A, A} Jd with R d = γ d {S, A} Jd and σ d = γ d {S, A} 2 J d with γ d > 0. where σ d is a closed-loop entropy rate. The condition is equivalently written in terms of the control u = β(x) ( ) ( ) β(x) = g U U (x) R d J d σ d M (x) (x ) g (x)rj U (x) and the matching condition ( ) ( ) g U U (x) R d J d σ d M (x) (x ) g (x)rj U (x) = 0 FEMTO-ST / UFC / UBFC 41 / 51

42 Proposition - Ramirez et al. Automatica, 2016 Assume A is strictly convex and that there exist matrices M(x) 0 and J d (x) = J d (x), scalar functions γ d > 0 such that σ d = γ d {S, A} 2 J d and R d = γ d {S, A} Jd, and a full-rank left annihilator g (x) of g(x) that verify the matching equation. Then the control u = β(x) globally asymptotically stabilizes the closed-loop equilibrium x. Furthermore, the closed-loop system is ( ) A ẋ = σ d M + R d J d. A The closed-loop system is IPHS with dissipation s = σ d M A. ) ( β(x) = g (x) R d J d σ d M matching condition ) ( U U (x) (x ) g (x)rj U (x) ( ) ( ) g U U (x) R d J d σ d M (x) (x ) g (x)rj U (x) = 0 FEMTO-ST / UFC / UBFC 42 / 51

43 A useful corollary Corollary x is asymptotically stable with u = β(x) if g J = 0, g J d = 0, g M = 0. Remark Using IDA-PBC is not obvious when dealing with irreversible processes. The present result can be interpreted as a thermodynamic equivalent, with structure and dissipation matrices R d J d and σ d M (interconnection and entropy assignment), and energy function A (energy shaping) FEMTO-ST / UFC / UBFC 43 / 51

44 Control of CRN Assumptions The reactor operates in liquid phase, The molar volumes of each species are identical and the total volume is constant, The initial number of moles is equal to the number of moles of the inlet of the same species, For a given temperature T there is only one possible steady state for the mass. The last assumption doesn t imply that the chemical reaction network doesn t admit multiple stable or unstable equilibrium points. Remark The assumption on constant volume assures that the availability function A is strictly convex. FEMTO-ST / UFC / UBFC 44 / 51

45 Recall the IPHS formulation of the CRN [ ẋ = 0 m ] CR R C }{{ 0 } J R U + [ ne ( n ) 0 φ x, U 1 T e ] [ F ] ( mr V = Q i=1 R i J i ) } {{ } J R U + gu ñ 2 ñ One possible annihilator: g 0 ñ 3 ñ (x) = ñ m ñ m 1 0 The matching equation (using the Corollary with J d = J R ) g J R = g CR where g denotes the matrix obtained by removing the last column of g. Hence the Corollary is satisfied if g C = g C 1 = g C 2 =... = g C mr = 0 FEMTO-ST / UFC / UBFC 45 / 51

46 Solving the matching equation it suffices to check that for each individual reaction ν i 1 ñ 2 ν i 2 ñ 1 g C ν i 2 ñ 3 ν i 3 ñ 2 i = = ν i m 1 ñ m ν i m ñ m 1 which is true if ñ 1 ν i 1 = ñ2 ν i 2 = = ñm 1 ν i m 1 = ñm ν i m. Since by assumption the initial numbers of moles of each species equals the numbers of moles at the inlet, i.e., n(t = 0) = n 0 = n e we have n 0i n i ν i i = ξ, De Donder s extent of reaction (always holds) The second condition in the Corollary is solved by any matrix M(x) = M (x) 0 for which the first m rows and columns forms a null submatrix. This comes from the fact that the last column of g is zero. FEMTO-ST / UFC / UBFC 46 / 51

47 A classical benchmark: the van der Vusse reactor A non-isothermal CSTR with series/parallel reactions, The reactor model is C 5 H 6 k 1 /+H 2 O 2C 5 H 6 k 3 C 10 H 12 C 5 H 7 OH k 2 /+H 2 O C 5 H 8 (OH) 2 ċ 1 = k 1 (T )c 1 k 3 (T )c (c 10 c 1 )u 1 ċ 2 = k 1 (T )c 1 k 2 (T )c 2 c 2 u 1 Ṫ = ϑ(c, T ) + u 2 ρc p + (T 0 T )u 1, with a thermodynamic linearized energy balance ϑ(c, T ) = H 1k 1 (T )c 1 + H 2 k 2 (T )c 2 + H 3 k 3 (T )c 2 1 ρc p. The rate coefficients k i are dependent on the reactor temperature via the Arrhenius equation k i (T ) = k i0 exp E i, i = 1, 2, 3. RT FEMTO-ST / UFC / UBFC 47 / 51

48 The IPHS model We complete the model with the balance equations of C 5 H 8 (OH) 2 and C 10 H 12, ċ 3 = k 2 (T )c 2 ċ 4 = 1 2 k 3(T )c 2 1 The stoichiometric matrices are J 1 = , J = , J = / /2 0 Assumption The approximated internal energy and energy availability function satisfy µ V 1 κ 1 (c 1 c 1 U c = µ V 2 A µ, V 3 c = ) κ 2 (c 2 c 2 ) κ 3 (c 3 c 3 ) µ V 4 κ 4 (c 4 c 4 ) FEMTO-ST / UFC / UBFC 48 / 51

49 The IPHS model The CRN can then be written as µ V 1 (c 10 c 1 ) 0 µ V 2 ẋ = (R 1 J 1 + R 2 J 2 + R 3 J 3 ) µ V 3 + c µ V u T ρcp T (T 1 0 T ) T where x = [c 1, c 2, c 3, c 4, S], u = [u 1, u 2 ], R 1 = k 1(T ) c T 1, R 2 = k 2(T ) c T 2, R 3 = k 3(T ) c T 1 2 and such that H 1 = µ V 1 µ V 2, H 2 = µ V 2 µ V 3 and H 3 = µ V µ V 4. Notice that c 3 and c 4 don t affect the computation of the controller. C A0 5.0 mol/l k h T K k h C p 3.01 kj/(kg K) k l/(mol h) ρ kg/l E 1 /R K H kj/mol E 2 /R K H kj/mol E 3 /R K H kj/mol FEMTO-ST / UFC / UBFC 49 / 51

50 Numerical simulations Figure: Molar concentration c 2 and temperature FEMTO-ST / UFC / UBFC 50 / 51

51 Some final remarks IPHS are thermodynamically coherent models which retain passivity features of PHS and satisfy the second principle. Thermodynamically coherent systems cannot be in PHS form, hence standard PBC techniques such as IDA-PBC cannot be applied directly. In this work the closed-loop system is interpreted as a thermodynamic system, hence the control parameters are related with thermodynamic quantities, such as the reaction rates in the case of chemical reactions. FEMTO-ST / UFC / UBFC 51 / 51

52 Alonso, A. A. and Ydstie, B. E. (2001). Stabilization of distributed systems using irreversible thermodynamics. Automatica, 37: Aris, R. (1989). Elementary chemical reactor analysis. Chemical Engineering. Butterworths, Stoneham, USA. Eberard, D., Maschke, B. M., and van der Schaft, A. J. (2007). An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics, 60: Favache, A. and Dochain, D. (2010). Power-shaping control of reaction systems: The CSTR case. Automatica, 46(11): Grmela, M. and Öttinger, H. (1997). Dynamics and thermodynamics of complex fluids. i. development of a general formalism. Physical Review E, 56(6): Hangos, K. M., Bokor, J., and Szederkényi, G. (2001). Hamiltonian view on process systems. AIChE Journal, 47: Hoang, H., Couenne, F., Jallut, C., and Le Gorrec, Y. (2011). The port Hamiltonian approach to modelling and control of continuous stirred tank reactors. FEMTO-ST / UFC / UBFC 51 / 51

53 Journal of Process Control, 21(10): Karnopp, D. (1983). Computer models of hysteresis in mechanical and magnetic components. Journal of the Franklin Institute, 316(5): Maschke, B. and van der Schaft, A. (1992). Port controlled Hamiltonian systems: modeling origins and system theoretic properties. In Proceedings of the 3rd IFAC Symposium on Nonlinear Control Systems, NOLCOS 92, pages , Bordeaux, France. Ortega, R., van der Schaft, A., Mareels, I., and Maschke, B. (2001). Putting energy back in control. IEEE Control Systems Magazine, 21(2): Otero-Muras, I., Szederkényi, G., Alonso, A. A., and Hangos, K. M. (2008). Local dissipative Hamiltonian description of reversible reaction networks. Systems & Control Letters, 57: van der Schaft, A. J. (2000). L2-Gain and Passivity Techniques in Nonlinear Control. Ydstie, B. E. and Alonso, A. A. (1997). Process systems and passivity via the clausius-planck inequality. Systems & Control Letters, 30(5): FEMTO-ST / UFC / UBFC 51 / 51