About the de nition of port variables for contact Hamiltonian systems

07/11/2017
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Extending the formulation of reversible thermodynamical transformations to the formulation of irreversible transformations of open thermodynamical systems di erent classes of nonlinear control systems has been de ned in terms of control Hamiltonian systems de ned on a contact manifold. In this paper we discuss the relation between the de nition of variational control contact systems and the input-output contact systems . We have rst given an expression of the variational control contact systems in terms of a nonlinear control systems. Secondly we have shown that the conservative input-output contact systems are a subclass of the contact variational systems with integrable output dynamics.

About the denition of port variables for contact Hamiltonian systems

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application/pdf About the de nition of port variables for contact Hamiltonian systems Bernhard Maschke, Arjan van der Schaft
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contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
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Extending the formulation of reversible thermodynamical transformations to the formulation of irreversible transformations of open thermodynamical systems di erent classes of nonlinear control systems has been de ned in terms of control Hamiltonian systems de ned on a contact manifold. In this paper we discuss the relation between the de nition of variational control contact systems and the input-output contact systems . We have rst given an expression of the variational control contact systems in terms of a nonlinear control systems. Secondly we have shown that the conservative input-output contact systems are a subclass of the contact variational systems with integrable output dynamics.
About the denition of port variables for contact Hamiltonian systems
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Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion About the definition of port variables for contact Hamiltonian systems B. Maschke* and A.J. van der Schaft ** * LAGEP, UMR 5007 CNRS-Université Lyon 1, France ** Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, the Netherlands 3rd Conference on GEOMETRIC SCIENCE OF INFORMATION, Paris, 7 - 9th Nov ; 2017 B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Introduction and motivation For electro-mechanical systems, there is a rich theory of control using invariants (Casimir functions ..) and Lagrangian and Hamiltonian systems defined Poisson manifolds augmented with input-output maps or Dirac structure on vector bundles including port variables. The objective is to develop a control theory for irreversible Thermodynamic systems : 1 for isolated systems : Hamiltonian systems on contact manifold [Eberard et al., 2007 ; Favache et al. 2010 ; Ramirez et al. 2013] 2 for open systems : input-output versus variational port Hamiltonian systems [Merker and Krüger, 2013] B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Legendre submanifolds Contact vector fields Thermodynamical properties: a geometric perspective A contact structure on a manifold M is determined by a 1-form θ of constant class (2n +1). The pair (M ,θ) is then called a contact manifold, and θ a contact form. According to Darboux’s theorem there exists a set of canonical coordinates (x0,x,p) ∈ R×Rn ×Rn of M where the contact form θ is given by : θ = dx0 −∑n i=1 pi dxi Définition The thermodynamic properties are given by Legendre submanifold L , the solution to a Pfaffian equation : θ|L = dx0 −pi dxi L = 0 (1) B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Legendre submanifolds Contact vector fields Contact vector fields: definition Definition A (smooth) vector field X on the contact manifold M is a contact vector field with respect to a contact form θ if and only if there exists a smooth function ρ ∈ C∞(M ) such that LX θ = ρθ, Alternative definition Definition A contact vector field XK generated by the Hamiltonian function K (x̃) is the unique vector field satisfying iX θ = K iX dθ = −dK    X −(iX θ)E | {z } =H (X)     = −(dK −(iE dK)θ) (2) B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Conservative control contact systems Feedback invariance of control contact systems Controlled conservative contact system Definition A conservative control contact system is defined by: a contact manifold R2n+1 with contact form θ (i) a Legendre submanifold L (ii) m +1 a contact Hamiltonians: K0 internal and Kj interaction Hamiltonian satisfying the invariance condition: Kj |L = 0, j = 0,...,m (iii) the differential equation: ˙ x̃ = XK0 +∑m j=1 uj XKj where u(t) ∈ F (R+) time dependent input function. Theorem [Mrugała, 1991] Then XK is tangent to L if and only if Kis identically zero on L : L ⊂ K−1 (0) B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Conservative control contact systems Feedback invariance of control contact systems Feedback equivalence with a strict contact vector field Theorem Define K0,Kc ,F ∈ C ∞(M ), satisfying iE . = 0, with closed-loop contact form θd = θ +dF and state-feedback α = ϕ ◦Kc where ϕ ∈ C∞(R), then X = XK0 +αXKc is contact vector field with respect to θd iff XK0 (F)+(ϕ ◦Kc )[Kc +XKc (F)]−Φ◦Kc = cF with ϕ (λ) = Φ0(λ). Furthermore the closed-loop Hamiltonian is K = K0 +Φ◦Kc Definition . The state-feedback may be interpreted as output feedback ϕ (y) with ouput y = Kc (x) B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Variational contact systems with ports [Merker et al. 2013] A variational control contact system on (M ,θ), is defined by (i) output variables defined by the vector bundle E 3 y overM with flat covariant derivative ∇ (ii) a bundle map A : T∗M → E with A(θ) = 0 (ii) conjugated input variables in the dual bundle E∗ 3 u overM (iii) input map defined by the adjoint bundle map A∗ : E∗ → TM (iv) internal contact Hamiltonian function K0 (x̃) and the dynamical system dx̃ dt = X (x̃, u, y) with the unique vector field X (x̃, u, y) satisfying i(X−A∗u)dθ +dK0 = 0 θ (X) = iX θ = K0+hu, yi (3) B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Variational contact systems as explicit nonlinear system (1) Using the decomposition of the tangent manifold TM = kerdθ ⊕kerθ, the variational contact system is dx̃ dt = X (x̃, u, y) = (iX θ)E | {z } ∈kerdθ + (X −(iX θ)E) | {z } =H (X)∈kerθ=C = (K0 +hu, yi)E +H (XK0 )+ A∗ u |{z} ∈kerθ=C = XK0 |{z} drift contact vect. field + A∗ u |{z} ∈kerθ=C +hu, yiE | {z } ∈kerdθ | {z } control vect. field control Hamiltonian system + external power flow (4) B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Variational contact systems as explicit nonlinear system (2) The output variable y defined by a dynamic equation d dt y = A◦dθ (X (x̃, u, y)) (5) But dθ (X (x̃, u, y)) = iX(x̃,u,y)dθ = iXK0 dθ +hu, yiiE dθ |{z} =0 +dθ (A∗ u) = −(dK0 −(iE dK0)θ)+dθ (A∗ u) hence the dynamics of the output becomes d dt y = −A(dK0)+(A◦dθ ◦A∗ )u | {z } feedthrough B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Relation with input-output contact systems The conservative contact input-output system with internal contact Hamiltonian K0 (x̃) and control contact Hamiltonians −Ki (x̃) is a variational control contact system with internal contact Hamiltonian K0 (x̃) and bundle map A : T∗M → Rn ×M defined by A(λ) = (Ai (λ))i=1,...,m = (hλ, H (XKi )i)i=1,...,m (6) B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Proof Decomposing the input contact vector field into its vertical and horizontal part and identifying with dx̃ dt = XK0 +hu, yiE +A∗u dx̃ dt = XK0 − m ∑ i=1 XKi ui = XK0 + m ∑ i=1 −Ki (x̃)ui | {z } =hu,yiE + m ∑ i=1 −H (XKi )ui | {z } =A∗u hence: the dual output bundle map A∗ (u) = −∑m i=1 H (XKi )ui and the outputs yi = −Ki (x̃) which satisfies dyj dt = [Kj , K0]θ −∑m i=1 ui [Kj , Ki ]θ = (A◦dθ ◦A∗)u −∑m i=1 ui LXj Ki asiE dKi = 0 (7) B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Conclusion: open problems find physically meaningfull applications of non-integrable outputs define interconnection of port contact systems consider the control design in order to : shape the losed-loop contact Hamiltonian function/the closed-loop Legendre submanifold control by interconnection apply to chemical reactor characterize the contact Hamiltonian functions: associated with irreversible transformation quasi-reversible transformations: lift of balance equations (bilinear) or alternatives ? far from equilibrium ???? B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion References : geometry of Equilibrium Thermodynamics some references 1 C.Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann., 1909 2 Gibbs, J.W.,Collected Works : I : Thermodynamics, Longmans, 1928 3 Herman, R., Geometry, Physics and Systems, Dekker, 1973 4 Arnold, V.I., 1989. Mathematical Methods of Classical Mechanics, 2nd ed.. Graduate Texts in Mathematics, vol. 60. Springer-Verlag, New York, USA 5 R. Mrugała, Geometrical formulation of equilibrium phenomenological Thermodynamics, Reports on Mathematical Physics, 1978 6 Grmela, M., Reciprocity relations in thermodynamics, Physica A, 2002 B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Some references 1 R. Balian and P. Valentin. Hamiltonian structure of thermodynamics with gauge. The European Physical Journal B-Condensed Matter and Complex Systems, 21(2) :269–282, 2001 2 D. Eberard, B.M. Maschke, and A.J. van der Schaft. An extension of pseudo-Hamiltonian systems to the thermodynamic space : towards a geometry of non-equilibrium thermodynamics. Reports on Mathematical Physics, 60(2) :175–198, 2007 3 A. Favache, D. Dochain, and Maschke B.M. An entropy-based formulation of irreversible processes based on contact structures,. Chemical Engineering Science, 65 :5204–5216, 2010 4 H. Ramırez Estay , B. Maschke and D. Sbarbaro, Irreversible port-Hamiltonian systems : A general formulation of irreversible processes with application to the CSTR, Chemical Engineering Science, Volume 89, pp. 223-234 15 February 2013 B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems Introduction Contact geometry and Thermodynamics Controlled contact systems Variational port Hamiltonian contact systems Conclusion Some references 1 J. Merker and M. Krüger. On a variational principle in thermodynamics. Continuum Mechanics and Thermodynamics, 25(6) :779–793, 2013 2 A. Bravetti, C. S. Lopez-Monsalvo, and H. Nettel, F.and Quevedo. Representation invariant geometrothermodynamics : Applications to ordinary thermodynamic systems. Journal of Geometry and Physics, 81 :1 – 9, 2014. 3 J.-Ch. Delvenne and Henrik Sandberg. Finite-time thermodynamics of port-Hamiltonian systems. Physica D : Non- linear Phenomena, 267(0) :123 – 132, 2014 4 D. Gromov and P.E. Caines. Stability of composite thermodynamic systems with interconnection constraints. IET Control Theory and Applications, 9(11) :1629–1636, 2015 B. Maschke* and A.J. van der Schaft ** Port variables for contact Hamiltonian systems