Are thermodynamical systems port-hamiltonian?

Transcription

1 Are thermodynamical systems port-hamiltonian? Siep Weiland Department of Electrical Engineering Eindhoven University of Technology February 13, 2019 Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

2 Systems of systems: composition and decomposition Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

3 Outline Dissipation and port-hamiltonian systems 1 Dissipation and port-hamiltonian systems 2 Equilibrium thermodynamics 3 4 Conclusions Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

4 Dissipation and port-hamiltonian systems Port-Hamiltonian systems Definition A Port-Hamiltonian system is a (nonlinear) system of the form ẋ = [J(x) R(x)] H (x) + B(x)u x y = B(x) H x (x) H : R n R is the Hamiltonian, J(x) is skew symmetric and R(x) is positive semi-definite. ph systems are dissipative in the sense that H(x(t 1 )) H(x(t 0 )) + for all trajectories and for all t 0 t 1. t1 t 0 u(t), y(t) dt Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

5 Dissipation and port-hamiltonian systems Interconnections of ph systems u (1) y (2) Σ (1) Σ (2) y (1) u (2) Power preserving interconnections s(u (1), y (1) ) + s(u (2), y (2) ) = u (1), y (1) + u (2), y (2) = 0 Hamiltonian H = H (1) + H (2) defined on product space X = X (1) X (2) Power preserving interconnections of ph systems are ph Modularity or composition property. Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

6 Dissipation and port-hamiltonian systems Interconnections of ph systems u (1) y (2) Σ (1) Σ (2) y (1) u (2) Power preserving interconnections s(u (1), y (1) ) + s(u (2), y (2) ) = u (1), y (1) + u (2), y (2) = 0 Hamiltonian H = H (1) + H (2) defined on product space X = X (1) X (2) Power preserving interconnections of ph systems are ph Modularity or composition property. Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

7 Dissipation and port-hamiltonian systems Modular interconnection property Lossless 2-port transmission line V x = L(x) I t ; I x = C(x) V t Spatial discretization Hamiltonian evolution Σ 1 Σ n Lossless approximate model by neutral series interconnection of n port-hamiltonian systems. Voltage evolution Birgit van Huijgevoort, Hans Zwart, SW Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

8 Key questions Dissipation and port-hamiltonian systems Are thermodynamic systems port-hamiltonian?? If so, what are Hamiltonians (storage functions)? what are its generalized port variables? what is the equivalent of power (supply functions)? what is a neutral interconnection and how can modularity, composition and decomposition be defined? Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

9 Key questions Dissipation and port-hamiltonian systems Are thermodynamic systems port-hamiltonian?? If so, what are Hamiltonians (storage functions)? what are its generalized port variables? what is the equivalent of power (supply functions)? what is a neutral interconnection and how can modularity, composition and decomposition be defined? Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

10 Outline Equilibrium thermodynamics 1 Dissipation and port-hamiltonian systems 2 Equilibrium thermodynamics 3 4 Conclusions Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

11 Energy and Entropy Equilibrium thermodynamics Postulate (H. Callen) A thermodynamic system has equilibrium states determined by internal energy E, volume V and mole numbers N 1,..., N r of its chemical components. admits an entropy function S(E, V, N 1,..., N r ) defined for all equilibrium states, with the property that equilibria (E, V, N 1,..., N r ) of a composite maximize S. Energy representation E = E(S, V, N 1,..., N r ) Entropy representation S = S(E, V, N 1,..., N r ) Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

12 Equilibrium thermodynamics Energy representations Energy change expressed by partial derivatives of E = E(S, V, N 1,..., N r ): de = ( ) E S ( ) E ds + V dv + r ( ) E i=1 N i dn i Defines intensive variables Temperature T := ( ) E S Pressure P := ( ) E V ( ) Electro-chemical potential µ i := E N i V,N 1,...,N r S,N 1,...,N r S,V,N i Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

13 Equilibrium thermodynamics Energy representations Thus, de = T ds P dv + r T ds µ i dn i = P, dv µ }{{}} dn {{ } effort? flow? i=1 Chemical work dm µ dn T ds Heat flux dq Σ P dv Mechanical work dw Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

14 Equilibrium thermodynamics Entropy representations Similar fundamental relation for entropic representations ds = 1 T de + P r 1/T de T dv µ i T dn i = P/T, dv i=1 µ/t dn }{{}}{{} effort? flow? defines different intensive variables. µ/t dn 1/T de Σ P/T dv Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

15 Outline 1 Dissipation and port-hamiltonian systems 2 Equilibrium thermodynamics 3 4 Conclusions Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

16 Scope and challenges aims to describe physics beyond or away from thermodynamic equilibrium needs to incorporate time-courses of intensive variables T (t), P(t), µ(t) requires extended concept of entropy requires definition of non-equilibrium state variables needs to recover properties of equilibrium thermodynamics in steady state equilibria. enable non-uniform spatial densities for generalized extensive variables... First contributions by Onsager (1931) related to concept of dissipation. Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

17 Historic context Lars Onsager, 1968 Nobel prize Ilya Prigogine, 1977 Nobel prize on non-equilibrium thermodynamics Sybren de Groot, 1962 book publication on linear irreversible thermodynamics Peter Mazur, founder Lorentz Institute Theoretical Physics Josef Meixner, 1965, Rheology and non-equilibrium thermodynamics Michal Pavelka, Václav Klika, Miroslav Grmela, Ryszard Mrugala, Hans Öttinger, Bernard Maschke, Arjan van der Schaft,... Lars Onsager Ilya Prigogine Sybren de Groot Peter Mazur Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

18 Proposed dynamic model 2 dx dt = J(x) E x (x) + R(x) S }{{} x (x) }{{} reversible part irreversible part E and S energy and entropy functions, S concave Symmetry requirements Degeneracy requirements J(x) + J(x) = 0, R(x) = R(x) 0 J(x) S (x) = 0, x E R(x) x (x) = 0 entropy and energy cannot be affected by reversible and irreversible dynamics, respectively. 2 Grmela and Öttinger, Beyond Equilibrium Thermodynamics, Wiley, 2005 Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

19 Proposed dynamic model 2 dx dt = J(x) E x (x) + R(x) S }{{} x (x) }{{} reversible part irreversible part E and S energy and entropy functions, S concave Symmetry requirements Degeneracy requirements J(x) + J(x) = 0, R(x) = R(x) 0 J(x) S (x) = 0, x E R(x) x (x) = 0 entropy and energy cannot be affected by reversible and irreversible dynamics, respectively. 2 Grmela and Öttinger, Beyond Equilibrium Thermodynamics, Wiley, 2005 Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

20 Proposed dynamic model 2 dx dt = J(x) E x (x) + R(x) S }{{} x (x) }{{} reversible part irreversible part E and S energy and entropy functions, S concave Symmetry requirements Degeneracy requirements J(x) + J(x) = 0, R(x) = R(x) 0 J(x) S (x) = 0, x E R(x) x (x) = 0 entropy and energy cannot be affected by reversible and irreversible dynamics, respectively. 2 Grmela and Öttinger, Beyond Equilibrium Thermodynamics, Wiley, 2005 Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

21 Proposed dynamic model 2 dx dt = J(x) E x (x) + R(x) S }{{} x (x) }{{} reversible part irreversible part E and S energy and entropy functions, S concave Symmetry requirements Degeneracy requirements J(x) + J(x) = 0, R(x) = R(x) 0 J(x) S (x) = 0, x E R(x) x (x) = 0 entropy and energy cannot be affected by reversible and irreversible dynamics, respectively. 2 Grmela and Öttinger, Beyond Equilibrium Thermodynamics, Wiley, 2005 Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

22 dx dt E S = J(x) x (x) + R(x) x (x) For isolated systems this leads to: strong form of energy conservation de dt = E E x(t) J(x(t)) x x x(t) = 0 strong first law of thermodynamics. strong form of entropy conservation ds dt = S x strong second law of thermodynamics. x(t) R(x(t)) S x x(t) 0 Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

23 What about its steady state properties? Equilibrium state maximizes S subject to constraint on E. Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

24 What about its steady state properties? Equilibrium state maximizes S subject to constraint on E. Define Lagrangian function L(x, λ E ) := S(x) λ E E(x) Then equilibrium states x (λ E ) satisfy with E = E(x ) and S = S(x ). L x (x (λ E ), λ E ) = 0 Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

25 What about its steady state properties? Equilibrium state maximizes S subject to constraint on E. Define Lagrangian function L(x, λ E ) := S(x) λ E E(x) Then equilibrium states x (λ E ) satisfy L x (x (λ E ), λ E ) = 0 with E = E(x ) and S = S(x ). Moreover, ( ẋ = R(x) 1 ) L J(x) λ E x (x), dl(x(t)) dt 0 is a Hamiltonian system and shows that x is fixed point, L is Lyapunov function. Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

26 What about its steady state properties? Equilibrium state maximizes S subject to constraint on E. Define Lagrangian function L(x, λ E ) := S(x) λ E E(x) Then equilibrium states x (λ E ) satisfy L x (x (λ E ), λ E ) = 0 with E = E(x ) and S = S(x ). Moreover, ( ẋ = R(x) 1 ) L J(x) λ E x (x), dl(x(t)) dt 0 is a Hamiltonian system and shows that x is fixed point, L is Lyapunov function. Structure of equilibria thermodynamics is recovered from stationary solutions of non-equilibrium setting! Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

27 Non-isolated or controlled thermodynamic systems Theorem The non-isolated/controlled thermodynamic system ẋ = J(x) E x y rev = B(x) E x (x), (subject to same conditions) is (x) + R(x) S (x) + B(x)u x y irrev = B(x) S x (x) conservative with respect to supply u, y rev dissipative with respect to supply u, y irrev. Proof through dissipation inequalities: de dt = u, y rev, ds dt u, y irrev Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

28 Non-isolated or controlled thermodynamic systems Theorem The non-isolated/controlled thermodynamic system ẋ = J(x) E x y rev = B(x) E x (x), (subject to same conditions) is (x) + R(x) S (x) + B(x)u x y irrev = B(x) S x (x) conservative with respect to supply u, y rev dissipative with respect to supply u, y irrev. Proof through dissipation inequalities: de dt = u, y rev, ds dt u, y irrev Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

29 Interconnections of non-isolated thermodynamical systems This result paths the way to define interconnected thermodynamic systems: u (1) Σ (1) y rev (1) y rev (2) Σ (2) y (1) irrev u (2) y (2) irrev Neutral interconnections defined by joint equations u (1), y (1) rev + u (2), y (2) rev = 0, u (1), y (1) irrev + u(2), y (2) irrev = 0 Energy and Entropy E = E (1) + E (2), S = S (1) + S (2) defined on product space X = X (1) X (2) Neutral interconnection of thermodynamic systems will be thermodynamic! Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

30 Interconnections of non-isolated thermodynamical systems This result paths the way to define interconnected thermodynamic systems: u (1) Σ (1) y rev (1) y rev (2) Σ (2) y (1) irrev u (2) y (2) irrev Neutral interconnections defined by joint equations u (1), y (1) rev + u (2), y (2) rev = 0, u (1), y (1) irrev + u(2), y (2) irrev = 0 Energy and Entropy E = E (1) + E (2), S = S (1) + S (2) defined on product space X = X (1) X (2) Neutral interconnection of thermodynamic systems will be thermodynamic! Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

31 Interconnections of non-isolated thermodynamical systems This result paths the way to define interconnected thermodynamic systems: u (1) Σ (1) y rev (1) y rev (2) Σ (2) y (1) irrev u (2) y (2) irrev Neutral interconnections defined by joint equations u (1), y (1) rev + u (2), y (2) rev = 0, u (1), y (1) irrev + u(2), y (2) irrev = 0 Energy and Entropy E = E (1) + E (2), S = S (1) + S (2) defined on product space X = X (1) X (2) Neutral interconnection of thermodynamic systems will be thermodynamic! Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

32 Interconnections of non-isolated thermodynamical systems This result paths the way to define interconnected thermodynamic systems: u (1) Σ (1) y rev (1) y rev (2) Σ (2) y (1) irrev u (2) y (2) irrev Neutral interconnections defined by joint equations u (1), y (1) rev + u (2), y (2) rev = 0, u (1), y (1) irrev + u(2), y (2) irrev = 0 Energy and Entropy E = E (1) + E (2), S = S (1) + S (2) defined on product space X = X (1) X (2) Neutral interconnection of thermodynamic systems will be thermodynamic! Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

33 Outline Conclusions 1 Dissipation and port-hamiltonian systems 2 Equilibrium thermodynamics 3 4 Conclusions Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22

34 Conclusion Conclusions Conclusions Many conceptual demands on non-equilibrium thermodynamics. No consensus on unifying mathematical structure of dynamics in thermodynamics Introduced natural notion of dissipation and generalized notion of power for non-isolated thermodynamic systems. Classical equilibrium thermodynamics entirely contained as stationary solutions of this framework. Implications for modeling, model reduction, simulation, identification, control,... Thanks to: Amritam Das, Daming Lou, David van den Hurk, Ruben Merks, Thomas Meijer, Birgit van Huijgevoort, Cristi Iacob,... Siep Weiland (TUE) Are thermodynamical systems port-hamiltonian? February 13, / 22