Abstract
Extending the formulation of reversible thermodynamical transformations to the formulation of irreversible transformations of open thermodynamical systems different classes of nonlinear control systems has been defined in terms of control Hamiltonian systems defined on a contact manifold. In this paper we discuss the relation between the definition of variational control contact systems and the input-output contact systems. We have first 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.
Access provided by CONRICYT-eBooks. Download conference paper PDF
Similar content being viewed by others
Keywords
- Open irreversible thermodynamic systems
- Nonlinear control systems
- Hamiltonian systems on contact manifolds
1 Introduction
Extending the formulation of reversible thermodynamical transformations suggested in [11] to the formulation of irreversible transformations of open thermodynamical systems, a class of nonlinear control systems has been defined in terms of control Hamiltonian systems defined on a contact manifold [4, 5, 7, 14]. Their dynamic properties as well as their feedback invariance and stabilization properties have been studied in [2, 6, 13, 15]. An alternative definition, based on a variational formulation has been suggested in [10]. In this paper we shall discuss and compare this definition with the system-theoretic definition suggested in [13].
2 Control Hamiltonian Systems Defined on Contact Manifolds
Since Gibbs’ work, it has been established that the Thermodynamic Phase Space is intrinsically defined as a contact manifold, that is a differentiable manifold \(\mathcal {M}\ni \tilde{x}\) equipped with a contact form \(\theta \). In the sequel we shall denote by \(\left( x_{0},x,p{}^{\top }\right) \in \mathbb {R}\times \mathbb {R}^{n}\times \mathbb {R}^{n}\) a set of canonical coordinatesFootnote 1.
It has also been established that the dynamics of thermodynamic systems subject to reversible and irreversible processes may be formulated in terms of contact Hamiltonian vector fields [4, 5, 7, 11, 14].
For open thermodynamic systems, a class of nonlinear control systems [12] has been defined, where the drift vector field and the input vector fields are both contact Hamiltonian vector fields [4, 5, 14]. Deriving from condition on structure preserving state feedback control, the natural output functions have then be defined as the contact Hamiltonian functions defining the input vector field [13]. An alternative definition of control contact systems, derived from a variational formulation, has been suggested in [10].
In this section we shall recall these two different definitions of control Hamiltonian systems and formulate the variational contact systems in terms of nonlinear control systems [12].
2.1 Input - Output Contact Systems [13]
Let us first recall the definition of input-output contact systems.
Definition 1
[13]. An input - output contact system on the contact manifold \((\mathcal {M},\theta )\), with input variable belonging the trivial vector bundle \(F=\mathcal {M}\times \mathbb {R}^{m}\ni \left( \tilde{x},\,u\right) \) over \(\mathcal {M}\) and output variables being the dual vector bundle \(E=F^{*}\sim \mathcal {M}\times \mathbb {R}^{m}\ni \left( \tilde{x},\,y\right) \), is defined by the two functions \(K_{0}\in C^{\infty }(\mathcal {M})\), called the internal contact Hamiltonian, \(K_{c}\in C^{\infty }(\mathcal {M})\) called the interaction (or control) contact Hamiltonian, and the state and output equations
where \(X_{K_{0}}\) and \(X_{K_{i}}\) are the contact vector fieldsFootnote 2 of \((\mathcal {M},\theta )\) generated by the contact Hamiltonians \(K_{0}\) and \(K_{i}\) respectively.
Note that input - output contact system are the analogue of input-output Hamiltonian systems defined on symplectic manifolds for driven mechanical systems [3, 16, 17] but extended to contact manifolds.
The models of physical systems such as heat diffusion or the Continuous Stirred Tank Reactor belong to a subclass of contact systems [4, 5, 14], called conservative input-output contact systems.
Definition 2
[4]. A conservative input-output contact system with respect to the Legendre submanifold \(\mathcal {L}\) is an input-output contact system with the internal, respectively control, contact Hamiltonians \(K_{0}\), respectively \(K_{i}\), satisfying the two conditions:
(i) they are invariants of the Reeb vector field, satisfying
(ii) they satisfy the invariance condition
2.2 Control Contact System Arizing from a Variational Principle [10]
Arizing from the variational principle defined in [10] a more general class of contact systems has been defined which we briefly recall now.
Definition 3
A variational control contact system [10] on the contact manifold \((\mathcal {M},\theta )\), is defined by
-
(i)
the set of output variables is defined by the vector bundle \(E\ni y\) over\(\mathcal {M}\) endowed with a (flat) covariant derivative \(\nabla \)
-
(ii)
a bundle map \(A:T^{*}\mathcal {M}\rightarrow E\) with \(A(\theta )=0\)
-
(ii)
the set of conjugated input variables is the dual bundle \(E^{*}\ni u\) over\(\mathcal {M}\)
-
(iii)
the input map defined by the adjoint bundle map \(A^{*}:E^{*}\rightarrow T\mathcal {M}\)
-
(iv)
a smooth real function \(K_{0}\left( \tilde{x}\right) \), called internal contact Hamiltonian function
and the dynamical system \(\frac{d\tilde{x}}{dt}=X\left( \tilde{x},\,u,\,y\right) \) associated with the unique vector field \(X\left( \tilde{x},\,u,\,y\right) \) satisfying
Let us write the system explicitely in the form of a nonlinear control system. Firstly, notice that the condition \(A(\theta )=0\) is equivalent to
that is, the image of \(A^{*}\) is contained in the field of contact elements \(\ker \theta =\mathcal {C}\) (or horizontal with respect to \(\theta \)).Footnote 3 Denoting by \(X_{K_{0}}\) the contact vector field generated by the internal contact Hamiltonian \(K_{0}\) and using the decomposition of the tangent manifold (8), the vector field X defined by (6) becomes
The second line of (10) shows the decomposition of the control vector field in terms of the vertical component which may be interpreted as the power balance term \(K_{0}+\left\langle u,\,y\right\rangle \) and the horizontal component which, using the tensor \(\theta ^{\sharp }\) mapping the semi-basic forms on the contact elementsFootnote 4 , may be interpreted as a Hamiltonian control system defined on the contact elements
Note that these properties are due to the assumption (7).
The third line of (10) shows the decomposition of the control vector field into an drift contact vector field \(X_{K_{0}}\) defined by the internal Hamiltonian function \(K_{0}\) and a control vector field decomposed into its vertical and horizontal parts.
The output variable y satisfies a dynamical equation on the output according to [10, p. 786–787]
Using the expression (10), one obtains
Using that \(A(\theta )=0\) hence the dynamics of the output (13) becomes
The Eq. (6) actually define the dynamical equations summarized in the following proposition.
Proposition 1
The Eq. (6) defining the dynamics of a variational control contact system of definition 3, are equivalent to the dynamical system
3 Relation Between Variational and Conservative Input-Output Contact Systems
In this section, we shall analyse the relations between conservative input-output contact systems of the definition 2 and the variational control contact systems of the definition 3. We shall give a direct proof that in this case the output dynamics (15) is integrable , that is when the output variable y may be expressed as a function of the state variable \(\widetilde{x}\) , as has been stated in [10, Sect. 4.1].
Proposition 2
The conservative contact input-output system of definition 2 with internal contact Hamiltonian \(K_{0}\left( \tilde{x}\right) \) and control contact Hamiltonians \(-K_{i}\left( \tilde{x}\right) \) is a variational control contact system defined in definition 3 with internal contact Hamiltonian \(K_{0}\left( \tilde{x}\right) \) and bundle map \(A:T^{*}\mathcal {M}\rightarrow \mathbb {R}^{n}\times \mathcal {M}\) defined by
Proof
Firstly, let us identify the dynamics Eqs. (14) and (1) by decomposing the input contact vector field into its vertical and horizontal part
Comparing this expression with third line of (10), leads to the natural identification of the dual output bundle map \(A^{*}\left( u\right) =\sum _{i=1}^{m}\mathcal {H}\left( X_{K_{i}}\right) u_{i}\) and the outputs \(y_{i}=K_{i}\left( \tilde{x}\right) \). The map \(A^{*}\) obviously satisfies the condition (7) and its dual is by definition (16). Let us now check that the defined output indeed satisfies the dynamic Eq. (15). Using that that the functions \(K_{i}\) are invariants of the Reeb vector field : \(i_{E}dK_{i}=0\), let us compute the j-th component of \(A\left( dK_{i}\right) \) Footnote 5
Compute now the control term of the output Eq. (2), using again that that the functions \(K_{i}\) are invariants of the Reeb vector field
Hence the second member of the dynamics (15) of the j-th component of output becomes
Using that, for functions \(K_{i}\) are invariants of the Reeb vector field \(\left[ K_{j},\,K_{i}\right] _{\theta }=L_{X_{j}}K_{i}\) , one obtains
Let us firstly notice that the output dynamics has a feedthrough term (depends explicitely on the input variables) which is linear in the Jacobi brackets of the control Hamiltonian functions. This resembles very much the situation for input-output Hamiltonian systems defined in symplectic or Poisson manifolds [9].
Let us discuss the example of integrable system given in [10, Sect. 4.1], for which the control contact Hamiltonians satisfy the conditions that they are in involution with respect to the Jacobi bracket. Indeed a contact manifold may be identified with the 1-jet of some manifold Q, (called configuration manifold in [10] and manifold of independent extensive variables in the context of Thermodynamics [4]). This 1-jet manifold may be identified with \(\mathbb {R}\times T^{*}Q\) and equiped with the canonical contact structure. As the control Hamiltonian functions are chosen to be function of the configuration manifold only, they are in involution. If \(\left[ K_{j},\,K_{i}\right] _{\theta }=0\,,i,\,j=1,\ldots ,\,m\) , then the output dynamics (17) does not depend on the control variables. It may be noticed that this condition is not fullfilled for the models of physical systems given in [4, 5, 14], except for the single input case of course.
4 Conclusion
In this paper we have discussed the relation between the definition of variational control contact systems suggested in [10] and the input-output contact systems defined in [13]. We have first given an expression of the variational control contact systems of [10] 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 defined in [10] with integrable output dynamics.
Notes
- 1.
- 2.
Recall that a contact vector field \(X_{K}\) generated by the Hamiltonian function \(K\left( \tilde{x}\right) \) is the unique vector field satisfying
$$\begin{aligned} \begin{array}{rcl} i_{X}\theta &{} = &{} K\\ i_{X}d\theta &{} = &{} -dK\left( \mathcal {H}\left( X\right) \right) . \end{array} \end{aligned}$$(3) - 3.
The tangent bundle \(T\mathcal {M}\) may be decomposed into
$$\begin{aligned} T\mathcal {M}=\ker d\theta \oplus \ker \theta \end{aligned}$$(8)where \(\ker d\theta \) , called vertical bundle , is of rank 1 and is generated by the Reeb vector field and \(\ker \theta \) , called horizontal bundle, is of rank 2n . Every vector field X on \(\mathcal {M}\) may be decomposed in a unique way into
$$\begin{aligned} X=\left( i_{X}\theta \right) E+\left( X-\left( i_{X}\theta \right) E\right) \end{aligned}$$(9)where \(\left( i_{X}\theta \right) E\in \ker d\theta \) is vertical and \(\left( X-\left( i_{X}\theta \right) E\right) =\mathcal {H}\left( X\right) \in \ker \theta =\mathcal {C}\) is horizontal with respect to the contact form \(\theta \).
- 4.
Any contact vector fields may be decomposed into
$$\begin{aligned} X_{K\ }=K\,E+\theta ^{\sharp }\left( dK-\left( i_{E}dK\right) \theta \right) \end{aligned}$$(13)where \(K\,E\) is the vertical and \(\theta ^{\sharp }\left( dK-\left( i_{E}dK\right) \theta \right) \) is the horizontal components of the contact vector field where \(\theta ^{\sharp }\) denotes the inverse of the isomorphism \(\left. \theta ^{\flat }\right| _{\mathcal {C}}\) from the vector space \(\mathcal {C}\) of horizontal vector fields onto the space \(\mathcal {F}\) of semi-basic 1-forms induced by the map \(\theta ^{\flat }\left( X\right) =-i_{X}d\theta \). [8, p. 293].
- 5.
The Jacobi bracket \(\left[ f,\,g\right] _{\theta }\) of two differentiable functions f and g , defined by \(\begin{array}{rcl} \left[ f,\,g\right] _{\theta }= & {} i_{E}\left( \left[ X_{f},\,X_{g}\right] \right) \end{array}\) where \(\left[ \,\,,\right] \)denotes the Lie bracket on vector fields. We shall use the following identities \(\left[ f,\,g\right] _{\theta }=i_{X_{f}}dg-g\,i_{E}df=-i_{X_{g}}df+f\,i_{E}dg\).
References
Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989). 2 edition, ISBN 0-387-96890-3
Bravetti, A., Lopez-Monsalvo, C.S., Nettel, F.: Contact symmetries and Hamiltonian thermodynamics. Ann. Phys. 361, 377–400 (2015)
Brockett, R.W.: Geometric control theory, volume 7 of lie groups: history, frontiers and applications. In: Martin, C., Herman, R. (eds.) Control Theory and Analytical Mechanics, pp. 1–46. Math. Sci. Press, Brookline (1977)
Eberard, D., Maschke, B.M., van der Schaft, A.J.: An extension of pseudo-Hamiltonian systems to the thermodynamic space: towards a geometry of non-equilibrium thermodynamics. Rep. Math. Phys. 60(2), 175–198 (2007)
Favache, A., Dochain, D., Maschke, B.M.: An entropy-based formulation of irreversible processes based on contact structures. Chem. Eng. Sci. 65, 5204–5216 (2010)
Favache, A., Dos Santos Martins, V., Maschke, B., Dochain, D.: Some properties of conservative control systems. IEEE trans. Autom. Control 54(10), 2341–2351 (2009)
Grmela, M.: Reciprocity relations in Thermodynamics. Phys. A 309, 304–328 (2002)
Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987)
Maschke, B.M., van der Schaft, A.J.: Port controlled Hamiltonian systems: modeling origins and system theoretic properties. In: Proceedings of 3rd International IFAC Conference on Nonlinear Systems’ Theory and Control, NOLCOS 1992, pp. 282–288, Bordeaux, June 1992
Merker, J., Krüger, M.: On a variational principle in Thermodynamics. Continuum Mech. Thermodyn. 25(6), 779–793 (2013)
Mrugała, R.: On a special family of thermodynamic processes and their invariants. Rep. Math. Phys. 46(3), 461–468 (2000)
Nijmeijer, H., van der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer, New York (1990). 1st edition, ISBN: 0-387-97234-X
Ramirez, H., Maschke, B., Sbarbaro, D.: Feedback equivalence of input-output contact systems. Syst. Control Lett. 62(6), 475–481 (2013)
Ramirez, H., Maschke, B., Sbarbaro, D.: Irreversible port-Hamiltonian systems: a general formulation of irreversible processes with application to the CSTR. Chem. Eng. Sci. 89, 223–234 (2013)
Ramirez, H., Maschke, B., Sbarbaro, D.: Partial stabilization of input-output contact systems on a Legendre submanifold. IEEE Trans. Autom. Control 62(3), 1431–1437 (2017)
van der Schaft, A.: Three decades of mathematical system theory. System Theory and Mechanics. LNCIS, vol. 135, pp. 426–452. Springer, Heidelberg (1989). doi:10.1007/BFb0008472
van der Schaft, A., Crouch, P.E.: Hamiltonian and self-adjoint control systems. Syst. Control Lett. 8, 289–295 (1987)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Maschke, B., van der Schaft, A. (2017). About the Definition of Port Variables for Contact Hamiltonian Systems. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_49
Download citation
DOI: https://doi.org/10.1007/978-3-319-68445-1_49
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68444-4
Online ISBN: 978-3-319-68445-1
eBook Packages: Computer ScienceComputer Science (R0)