An introduction to Port Hamiltonian Systems

Transcription

1 An introduction to Port Systems B.Maschke LAGEP UMR CNRS 5007, Université Claude Bernard, Lyon, France EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 1/127

2 Contact manifolds and Equilibrium Thermodynamics Lift of port Conservative system with ports on contact manifolds EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 2/127

3 Port-based modelling Contact manifolds and Equilibrium Thermodynamics Port-based or Bond graph modelling: main concepts: energy storage multiport C power continuous interconnection network pairs of conjugated power variables power dissipating element R Power continuous interconnection without irreversible transducer (RS-elements): leads to extension of port Lift of port Conservative system with ports on contact manifolds EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 3/127

4 Port Contact manifolds and Equilibrium Thermodynamics Lift of port formulation of port-based models power continuous interconnection geometric structure energy storage generating function: function dissipation additional (parasitic) geometry Main contribution to bond graph modelling: non-linear and integrability properties Conservative system with ports on contact manifolds EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 4/127

5 Sketch Contact manifolds and Equilibrium Thermodynamics Constrained Port Extension of port- to Irreversible Thermodynamics 5. Port conservative on contact manifolds Lift of port Conservative system with ports on contact manifolds EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 5/127

6 Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 6/127

7 Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation An electrical circuit without elements in exces Consider the following LC-circuit: with total energy: H 0 = Q2 C 2C + φ2 L 1 2L 1 + φ2 L 2 2L 2 and voltage source v s. The dynamical system is: Kirchhoff s laws {}} { q C = d dt φ L1 φ L i s = (0,1,0) }{{} Kirchhoff s laws Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 7/127 v C i L1 i L2 dh 0 { }} { v C i L1 i L2 + = i L1 Kirchhoff s laws { }} { v

8 Drift dynamics of the electrical circuit Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation The drift dynamics reduces to: d dt q C φ L1 φ L2 = Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 8/127 } 1 0 {{ 0 } J v C i L1 i L2 Virtual power for any pair of co-energy variables: bilinear product: Λ v C i L1 i L2, v C i L 1 i L 2 = (v C,i L1,i L2 ) J This is the foundation of geometric structure! v C i L 1 i L 2

9 Poisson bracket: definition Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Consider a differentiable manifold M of dimension n and the space of smooth real function C (M). A Poisson bracket is a mapping: satisfying: bilinearity {, } : C (M) C (M) C (M) (F,G) {F, G} skew-symmetry: {F, G} = {G, F } Jacobi identities: {F, {G,H}}+{G, {H,F }}+{H, {F,G}} = 0, F,G,H Leibniz rule: {F, GH} = {F, G}H + G{F, H} Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 9/127

10 Poisson bracket: associated tensor Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation With Poisson bracket is associated Poisson tensor (field): Λ : Ω 1 (M) Ω 1 (M) C (M) such that: Λ(dF,dG) = {F, G} is two times contravariant, skew-symmetric tensor field and satisfies the Jacobi identities. Using some coordinates: x 1,..,x n the Poisson tensor is defined by the structure matrix J: J ij = {x i, x j } which is skew-symmetric and satisfies the Jacobi identities, for any i, j, k = 1,..,n: n l=1 ( J lj (x) J ik x l + J li (x) J kj x l + J lk (x) J ji x l ) = 0 Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 10/127

11 Poisson morphism With the Poisson tensor, one may define a morphism of vector bundle: Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation such that: Λ : T M T M Λ (ω)(α) = Λ(α, ω), α Ω 1 (M) Using the structure matrix in some coordinates: x 1,..,x n and dual basis for T M and T M, this morphism is defined by the structure matrix: χ 1.. χ n = J(x) Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 11/ n where ω = n i=1 o i dx i and Λ (ω) = n i=1 χ i x i

12 Jacobi identities Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation The Jacobi identities are integrability conditions. The rank of the bracket at x M is rankj(x) and is even 2k. If n = 2k, then the bracket is symplectic. Then there exist canonical coordinates (q 1,..,q k,p 1,..,p k,r 1,..,r l ) with l = n 2k such that: J(x) = Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 12/127 0 I k 0 I k The coordinates (functions) r are a basis of Casimir functions: {r k, F } = 0, F C (M) If the Jacobi identities are not satisfied: {, } is a pseudo-poisson bracket.

13 Poisson bracket: electrical circuit Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Virtual power defines Poisson bracket: Λ v C i L1 i L2, v C i L 1 i L 2 = (v C,i L1,i L2 ) Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 13/ where the co-energy variables are the driving forces: v C i L1 i L2 = H 0 Q H 0 Φ 1 H 0 Φ 2 and H 0 is the total energy with energy variables (Q,Φ 1,Φ 2 ). Casimir function is total magnetic flux: r(x) = Φ 1 + Φ 2 v i L i L

14 Poisson bracket: Lie-Poisson bracket Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Consider as state variable the rotations in R 3 represented by rotation matrices in SO(3). The angular velocities are represented by skew-symmetric matrices in so(3) and endowed with Lie bracket: [ω 1,ω 2 ] = ω 1 ω 2 ω 2 ω 1. There is a canonical Lie-Poisson bracket on the momenta p so (3): {F, G}(p) = p, [df, dg] In Plücker coordinates, the structure matrix is: 0 p z p y J(p) = p z 0 p x p y p x 0 Casimir function is total momentum: r(x) = p 2 x + p 2 y + p 2 z Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 14/127

15 Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation A system is defined by: on a differentiable manifold M x with Poisson bracket {, } (and Λ the Poisson tensor) the internal function H 0 C (M) and the differential equations: Examples: ẋ = {x, H 0 } = Λ (dh 0 ) X H0 drift dynamics of the LC circuit Euler-Poinsot problem: rigid body spinning around its center of mass Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 15/127

16 Invariants of Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation There are two types of invariants due to the geometry (Poisson bracket): function due to the skew-symmetry of the bracket: dh 0 = {H 0, H 0 } = 0 dt the Casimir functions (non-symplectic case): dr dt = {r, H 0} = 0 For physical (lossless) : the energy is generating function H 0 and is conserved. Examples of other invariants than energy: total magnetic flux of the LC circuit total momentum of spinning body Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 16/127

17 Port Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation A port system is defined by: on a differentiable manifold M x with pseudo-poisson bracket {, } (and Λ the pseudo-poisson tensor) the internal function H 0 C (M) m inputs u i U and outputs y i U input vector fields g i and the system: { ẋ = X H0 + m i=1 Σ u ig i phs y i = g i, H 0 portconjugatedoutputs where X, ω denotes the pairing between vector fields and 1-forms. Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 17/127

18 Port in coordinates Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation In coordinates (x 1,..,x n ) a port system is written: a skew-symmetric structure matrix J(x) R n n the internal smooth function H 0 (x) m inputs u i R p and outputs y i R p input vector fields g i (x) R n and the system: ẋ = J(x) H 0 x + m i=1 u ig i (x) y i = g i (x) t H 0 x port conjugated outputs Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 18/127

19 Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Elementary storage of elastic energy Consider a (translational spring) q = f e = H q (q) 3,q,f,e R Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 19/127 (1) with: q the displacement vector, f the velocity, e the elastic force of the spring with potential energy H(q). It is a port system defined with respect to the Poisson structure matrix J = 0 and with g = 1.

20 Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Elementary storage of kinetic energy Consider ( point mass ) ṗ = f e = 1 m p, p,f,e R 3 (2) where p is the vector of momenta, m is the mass, f is the vector of external forces, and e denotes the velocity of the point mass. It is a port system with Poisson structure matrix J = 0, D = 0 and H(p) = 1 2m p 2 the kinetic energy. Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 20/127

21 LC circuit Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation The electrical circuit with source Poisson bracket {}} { q C d φ dt L1 = φ L i s = (0,1,0) }{{} g, Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 21/127 v C i L1 i L2 } {{ } dh 0 dh 0 { }} { v C i L1 i L }{{} input vector field

22 Spinning body Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Spinning body with actuating wrench Lie Poisson bracket { }} { p x 0 p z p y d p y = p z 0 p x dt p z p y p x 0 y = (0,1,0) }{{} g, Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 22/127 v x v y v z }{{} dh 0 dk { }} { v x v y v z where K(p) = 1 2 pt J 1 p and dk(p) is the velocity }{{} input vector fi

23 Port : skew-symmetry Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation The port is defined by: pseudo-poisson tensor and two dual input-output relations. This extends the skew-symmetric map Λ to the map T Λ,g i : ( Tx M R n ) T ( x M ) R n ( dh 0 (x) X = u y It is linear and skew-symmetric. Λ (dh 0 (x)) m i=1 u ig i L gi H 0 (x) Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 23/127 )

24 Power balance equation and invariants Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Feedback interconnection The skew-symmetry of the map T Λ,g i implies the power balance equation. The time derivative of the internal function is: dh 0 m = dh 0, X H0 + u j dh 0, g i dt EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 24/127 j=1 which by skew-symmetry of bracket and by definition of the port conjugated output becomes the power balance equation: dh 0 dt = m u j y j j=1 If H 0 is bounded from below, the system is lossless. For any Casimir function r: dc dt = m j=1 u j dr, X H0 extend energy-casimir methods to stabilizing control.

25 Input-ouput : introductio Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Input-output are the synthesis of: Analytical Mechanics: Poisson geometry and Systems theory and control References: 2 among many.. R.W.Brockett, Control theory and analytical mechanics, 1977 A.J. van der Schaft, Systems Theory and Mechanics, 1989 Main difference with port : use of true Poisson bracket use of interaction functions different definition of "conjugated" outputs Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 25/127

26 Input-output Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation An input-output system is defined by: on a differentiable manifold M x with Poisson bracket {, } (and Λ the Poisson tensor) the internal function H 0 C (M) m inputs u i and outputs y i m interaction functions H i C (M), i {1,...,m} and the system: { ẋ = X H0 m i=1 Σ u ix Hi io ỹ i = H i naturaloutputs Main difference: integrability assumptions! Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 26/127

27 Mass-spring with mixed boundary conditions Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation The state space is M = R 2 (q,p) with symplectic bracket. The internal is the total energy: H 0 (q,p) = 1 2m p2 + U(q) where m is the mass and U the potential energy of the system. Define the interaction s: H 1 (q,p) = p with the controlled velocity v of the basis H 2 (q,p) = q with the external force F applied on the mass The input-output system is: ( ) ( ) ( ) ( d q 0 1 U = q dt p 1 0 p v ( ) ( ) m p total momentum = q relative displacement ỹ 1 ỹ 1 Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 27/ ) + F ( 0 1 )

28 Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Power balance equation For any interaction H j, j {1,...,m}: ỹ j = dh j dt = {H j,h 0 } m u i {H j,h i } Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 28/127 i=1 Using the skew-symmetry of the Poisson bracket: dh 0 dt = {H 0,H 0 } m u j {H 0,H j } j=1 becomes the power balance equation: dh 0 dt = m u j ỹ j j=1 If the internal energy is bounded from below, the input-output is lossless

29 Comparison of input-output and port variable Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Consider a port system with input vector fields g j = X Hj, then: ỹ j = {H j,h 0 } m i=1 u i {H j,h i } = L Hj H 0 m i=1 u i {H j,h i } = y j m i=1 u i {H j,h i } Both outputs coincide if the interaction functions are in involution. Port variables are better suited for the interconnection (or composition) of port. On the example of the harmonic oscillator with mixed boundary conditions: ỹ 1 = kq + F and ỹ 2 = p m v. Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 29/127

30 Port with dissipation Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Consider Port system with port variables (u,y) and (u R,y R ): ẋ = X H0 + m i=1 u ig i + m i=1 ur i gr i y i = dh 0, g i y R i = dh 0, g R i and complete with dissipative closure equation: u R = R(x)(y R ) where R(x) is a linear, symmetric positive map. Then defining the Leibniz bracket: [F,G] Λ,gR,R = Λ(F,G) df,r(x) dg,g g with structure matrix: J(x) (g R ) t R(x)g(x). Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 30/127

31 Port with dissipation Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation The port system with dissipation: { ẋ = [x,h 0 ] Λ,gR,R + m i=1 u ig i y i = dh 0, g i satisfies the power balance equation: dh 0 dt = m u j y j [H 0,H 0 ] Λ,g,R j=1 which depends only on the symmetric part of the bracket, in coordinates: dh 0 m = u j y j H t 0 (g R ) t R(x)g R (x) H 0 dt x x j=1 Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 31/127

32 Feedback interconnection Consider two Port, k = 1,2: ẋ k = X H k 0 + m k i=1 uk i gk i + m i=1 vk i γk i y k i = dh k 0, gk i Motivating ι k i = dh0 k, γk i example Pseudo- γi 1 = ι 2 i Poisson with feedback interconnection: bracket γi 2 = ι 1 i Note that it is power continuous: γi 1 ι1 i + γ2 i ι2 i = 0 Skewsymmetry I/O versus port Input-output and port variables Dissipation Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 32/127

33 Feedback interconnection Motivating example Pseudo- Poisson bracket Skewsymmetry I/O versus port Input-output and port variables Dissipation Fedback interconnection leads to the port system: on product manifold M 1 M 2 endowed with pseudo-poisson bracket with structure matrix: ( ) J 1 g 1 g 2 t g 2 g 1 t Feedback EURON-GEOPLEX Summer School, July 2005 interconnection Port by B. Maschke p. 33/127 J 2 generated by total : H H2 0 with product input vector fields g 1 g 2

34 Dirac structures on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 34/127

35 Dirac structures: the geometry of interconnec In this section we will define a geometric representation of the interconnection structure appearing in port-based modelling. This key geometric structure is a so-called constant Dirac structure [Courant 1990]. In the sequel we will define Dirac structures on finite-dimensional vector spaces, although it generalizes Dirac structures to the infinite dimensional case, and will be also used for the description of distributed parameter on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 35/127

36 Basic vector spaces Dirac structures are a geometric object defined on the following vector spaces: a vector space V, called vector space of flow variables its dual vector space V, called vector space of effort variables Dirac structures and the bond space B = V V endowed with the symmetrical canonical pairing on Dirac structure with dissipation Interconnection (3) (v,v ), (ˆv, ˆv ) =< v ˆv > + < ˆv v > (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 36/127 endowed with the duality product < >

37 Dirac structure on a vector space Definition 1 [Courant 1990] 8 A constant Dirac structure on a linear space V is a linear subspace D B of the bond space B = V V with the property that: D = D (4) Dirac where D denotes the orthogonal with respect to, structures D = {(v,v ) V V (v,v ), (ˆv, ˆv ) = 0 for all on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 37/127 (ˆv, ˆv ) (5)

38 Alternative definition in finite dimension For finite dimensional vector spaces V of dimension n there is an alternative characterization of a constant Dirac structure. Proposition 1 [van der Schaft and Maschke, 1995] A Dirac structures linear subspace D B = V V is a constant Dirac structure on V if and only if on Dirac structure dimd = dim V (6) with and dissipation Interconnection (3) < v v >= 0, for all (v,v ) D (7) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 38/127

39 Image representation of a Dirac structure Proposition 2 (Courant, 1990) Consider a vector space V linear subspace of dimension n, D B = V V is a constant Dirac structure on V if and only if there exist two linear maps a : R n V and b : R n V satisfying (i) kera kerb = {0} Dirac structures (8) (ii) a b + b a = 0 on Dirac structure with dissipation Interconnection (3) such that D = {(v,v ) B/ λ R n ;(v,v ) = (a(λ),b(λ))} (9) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 39/127

40 Kernel representation of a Dirac structure Proposition 3 (van der Schaft and Maschke, 1995) Consider a vector space V linear subspace of dimension n, D B = V V is a constant Dirac structure on V if and only if there exist two linear maps F : V R n and E : V R n satisfying Dirac (i) rank [F + E] = n structures (10) on (ii) EF + FE = 0 Dirac structure such that with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 D = {(v,v ) B Fv + Ev = 0} Port by B. Maschke (11) p. 40/127

41 Effort constraint representation Proposition 4 (van der Schaft and Maschke, 1995) Consider a vector space V, a linear subspace of dimension n, D B = V V is a constant Dirac structure on V if and only if there exist a skew-symmetric map J : V V a linear map G : R k V with k N Dirac structures such that on Dirac structure with dissipation Interconnection (3) D = {(v,v ) B/ λ R k,v = Jv +Gλ and 0 = G v } (12) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 41/127

42 Graph of a Poisson map as a Dirac structure Consider a Dirac structure on a vector space V such that it admits a effort constraint representation where G = 0. Then a Dirac structure may be seen as the graph of a skew-symmetric map J : V V. Dirac It is a geometric representation of the Poisson tensor structures [Marle and Libermann, 1985] J defined for any pair (v1,v 2 ) V V by: on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 J (v1,v 2) = J (v1)(v 2) Port by B. Maschke (13) p. 42/127

43 Flow constraint representation Proposition 5 (van der Schaft and Maschke, 1995) Consider a vector space V, a linear subspace of dimension n, D B = V V is a constant Dirac structure on V if and only if there exist a skew-symmetric map Ω : V V a linear map H : R k V with k N Dirac structures such that on Dirac structure with dissipation Interconnection (3) D = {(v,v ) B/ λ R k,v = Jv+Hλ and 0 = H v} (14) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 43/127

44 Graph of a presymplectic map as special case Consider a Dirac structure on a vector space V such that it admits a flow constraint representation where H = 0. Then a Dirac structure may be seen as the graph of a skew-symmetric map Ω : V V. Dirac It is a geometric representation of the skew symmetric structures tensor associated with a presymplectic vector space [Marle and Libermann 1985] Ω defined for any pair on Dirac structure (v 1,v 2 ) V V by: with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Ω(v 1,v 2 ) = Ω (v 1 ) (v 2 ) Port by B. Maschke (15) p. 44/127

45 Input-output representation Proposition 6 (Bloch and Crouch, 1999) Consider a vector space V, a linear subspace of dimension n, D B = V V is a constant Dirac structure on V if and only if there exist a subspace P 1 V Dirac and a skew-symmetric map J : P1 P 1 structures such that ( on D = {(v,v ) = J(v ),v ),v P Dirac structure 1/} (16) with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 45/127

46 Dirac structure associated with a circuit (1) Consider a port connection graph G of a circuit containing n edges and choose a maximal tree T in G and denote by T complementary cotree. A basis ( of ) Kirchhoff cycle laws is given by: B v T v T = 0 (I T T B T ) where B = is the matrix of fundamental Dirac structures cycles associated with T. A basis ( of ) Kirchhoff cocycle laws is given by: on Dirac structure i T Q = 0 i T with dissipation where Q = ( B t ) T I T T is the matrix of fundamental Interconnection cocycles associated with the cotree T. (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 46/127

47 Dirac structure associated with a circuit (2) Proposition 7 Kirchhoff s laws define a constant Dirac structure D G on V I, the product space of cycle and cocycle variables of the circuit. It is immediate that: Dirac structures BQ t = 0 is skew symmetric and rank [B;Q] = n (17) on Dirac structure Hence the Kirchhoff s cycle and cocycle equations define a Dirac structure in V I. with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 47/127

48 Port The implicit system (X, F, D,H) is defined by: an n-dimensional vector space X the m-dimensional linear space F space of external flows, and the linear space of external efforts E = F a constant Dirac structure D on X F a H : X R Dirac and the implicit dynamic system: structures (ẋ,f, H on x (x), e) D, x X,f F, e F, (18) Dirac structure with H with x (x) the column-vector of partial derivatives of H dissipation regarded as an element of X. Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 48/127

49 Power balance equation Note that because of the power continuity property (7) one immediately obtains the power balance equation dh dt =< H x (x) ẋ >=< e f >, (19) Dirac structures on which expresses the fact that the increase in internal Dirac structure energy of an implicit system equals the with external power < e f > supplied to the system. dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 49/127

50 Coordinate representation of port Choose some basis of the vector space V and consider the dual basis for V and using the coordinate map, identify V and V with R n, and F and E with R m. By Proposition 1 there exist (n + m) n matrices F 1 and E 1, (n + m) m matrices F 2 and E 2 such that D = {(v,f,v, e) X F X F F 1 v+f 2 f+e 1 v E 2 e = 0} (20) with F 1,F 2,E 1,E 2 satisfying (21) Dirac structures on Dirac structure rank [F 1.F 2.E 1.E 2 ] = n + m with dissipation E 1 F1 T + E 2F2 T + F 1E1 T + F 2E2 T = 0 Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 50/127

51 Coordinate representation of port Then the port system (X, F, D,H) is written in coordinates: F 1 ẋ + F 2 f + E 1 H x (x) E 2e = 0 (22) where H x (x) denotes the gradient of the function H. Dirac It is not the usual input-state-output format considered in structures control theory: there is no a priori splitting of the external variables w = (f,e) into inputs and outputs. on Dirac structure It is a mixed set of differential and algebraic equations with (DAE s). dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 51/127

52 Linear port system If the H(x) is a quadratic function of x, that is H(x) = 1 2 xt Qx, Q = Q T, (23) then (22) reduces to the linear implicit system F 1 ẋ + F 2 f + E 1 Qx E 2 e = 0 (24) Dirac structures with state variables x, and external or port variables on Dirac structure w = (f,e). The (23) is in a so-called descriptor system with format. dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 52/127

53 Explicit linear port A particular subclass of (22) is formed by the (explicit) port ẋ = J H x (x) + Bf, J = JT Dirac structures with inputs f and outputs e. on It is easily checked that (25) is a special case of (22), with Dirac structure [ ] [ ] [ ] [ ] I n B J 0 with F 1 =, F 2 =, E 1 = dissipation 0 D B T, E 2 = I m Interconnection (26) (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 53/127 e = B T H x (x) + Df, D = DT (25)

54 LC circuit with elements in excess Consider an LC circuit with port connection network G. containing n C (possibly nonlinear) capacitors and n L inductors as well as n p free ports. Denote the vector of voltages and currents corresponding to the capacitors by v C V C and i C I C, and those corresponding to the inductors by v L V L and i L IL. Furthermore, let f = v p Vp and e = i p I p denote the Dirac structures voltages, respectively currents, at the ports of the circuit. on Dirac structure Recall that according to proposition 7 Kirchhoff s laws define a constant Dirac structure D G on V C V L Vp. with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 54/127

55 LC-circuits: continued Denote the charge variables q C R n C and flux linkage variables ϕ L R n L. The constitutive relations of the capacitors, respectively inductors, are q = i C, v C = H C q (q) (27) Dirac ϕ = v L, i C = H L structures ϕ (ϕ) on and the dynamics of the LC-circuit is a port Dirac structure system with defined with respect to the Dirac structure D LC and dissipation generated by the function Interconnection H(q,ϕ) = H C (q) + H L (ϕ) being the total electromagnetic energy of the circuit [Maschke and van der Schaft 1995, 1996 ]. (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 55/127

56 Port system with dissipation One extracts the dissipative elements from the overall physical system under consideration, thereby leaving an implicit system (22), with external flows f R and efforts e R F 1 ẋ + F 2 f + F R f R + E 1 H x (x) E 2e + E R e R = 0 Dirac rank [F 1 F 2 F R E 1 E 2 E R ] = n + m + dim f R structures on E 1 F Dirac structure 1 T + E 2F2 T + E RFR T + F 1E1 T + F 2E2 T + F RER T = 0 (28) with dissipation where the additional ports are terminated by setting Interconnection (3) f R = R (e R ) e R for some dissipation function R. (29) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 56/127

57 Power balance equation For port with dissipation, the following generalization of the power balance (19) dh dt =< e f > + < e R f R >=< e f > < e R R (e R ) > e R (30) The second principle of Thermodynamics implies that Dirac the dissipation function R(e structures R ) will be such that the last term in (30) is always non-positive (for example, on R(e R ) = 1 2 et R Re R for some matrix R = R T 0), Dirac structure implying that the increase in internal energy is less than with dissipation or equal to the externally supplied power. Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 57/127

58 Interconnection of Port We show that the class of port is closed with respect to interconnection by Dirac structures A power-conserving interconnection of implicit with constant geometries defines a system which is again an implicit system with constant geometry, where the constant Dirac structure (constant geometry) Dirac structures is the constant Dirac structures of the individual sub composed by the interconnection constraints. on Dirac structure The interconnection of implicit with corresponds to network modelling and reticulation of dissipation (energy-conserving) physical. Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 58/127

59 Constant power-conserving interconnection Consider k implicit with constant geometry (X j = R n j, F j = R m j, D j,h j ), j = 1,,k. Suppose F 1 F k = F i F p (31) with the subspace F i denoting the external flows to be Dirac interconnected, and F p the remaining (free) external structures flows. on By defining E i := (F p ) and E p := (F i ) we obtain the Dirac structure dual direct sum decomposition with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 E 1 E k = E i E p Port by B. Maschke (32) p. 59/127

60 Definition constant power-conserving interco A constant power-conserving interconnection is a subspace I F i E i, with dim I = dim F i, having the property (f i,e i ) I < e i f i >= 0 (33) Note that by Proposition 1 the subspace I defines a constant Dirac structure on F i. Dirac structures on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 60/127

61 Interconnexion Theorem 1 Consider k implicit with constant geometry (X j, F j, D j,h j ), j = 1,,k, and a constant power-conserving interconnection I as above. Then the interconnected system is an implicit system with constant geometry, with state space X := X 1 X k, space of external flows F p, H(x 1,,x k ) := H 1 (x 1 ) + + H k (x k ), and constant Dirac structure D on X F p given as Dirac structures (v 1,,v k,f p,v1,,v k, ep ) D on Dirac structure (f 1,,f k,e 1,,e k ) F 1 F k E 1 E k with dissipation Interconnection (v j,f j,vj, e j) D j, j = 1,,k, while (f 1,,f k,e 1,,e k ) = (f i,f p,e i,e p ) with (f i,e i ) I (3) (34) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 61/127 such

62 Linear Port The rest of this section will be devoted to linear implicit (24). We will briefly indicate that in this case the structure actually follows from the energy-conservation property, i.e., the energy balance (19). However, for implicit with non-quadratic s or, a fortiori, for Dirac structures with non-constant geometry as dealt later, this is not the case. on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 62/127

63 Linear Port (2) Consider a linear (implicit) port system (24) with constant Dirac structure on X F = R n R m. By the definition of Dirac structures (see (52)) < v ˆv > + < ˆv v > < e ˆf > < ê f >= 0 (35) for all 4-tuples (v,v,f, e),(ˆv, ˆv, ˆf, ê) D. Substitution of v = ẋ,v = Qx and ˆv = ˆv, ˆv = Qˆx yields Dirac structures on < Qx ˆv > + < Qˆx ẋ > < e ˆf > < ê f >= 0 or equivalently Dirac structure d with dt xt Qˆx =< e ˆf > + < ê f > dissipation (36) Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 63/127

64 Linear Port (3) Let us now assume that (24) is controllable and observable (in the sense of behavioral system theory [Willems, 1972]). By restricting to flow-effort time functions (f( ),e( )),( ˆf( ),ê( )) of compact support, it follows from (36) that the compact support behavior B c of (24) satisfies the property Dirac B c = Bc (37) structures, where on Dirac structure Bc = {(f( ),e( )) : R R m R m, with compact support with dissipation + Interconnection < e(t) ˆf(t) > + < ê(t) f(t) > dt = 0, ˆf( ),ê( ) B c } (38) (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 64/127

65 Linear Port (5) It can be proved that we can split the external variables (f 1,,f m,e 1,,e m ) into an input vector u R m and an output vector y R m in the following manner. There exists a subset K of M := {1,,m} such that u j = f j, j K, y j = e j, j K (39) u j = e j, j M\K, y j = f j, j M\K Dirac structures Then the system may be described by a proper transfer on matrix G(s) (from u to y)which satisfies, by Laplace Dirac structure transformation of (38), the property with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 G(s) = G T ( s) Port by B. Maschke (40) p. 65/127

66 Linear Port Consider an input-output behaviour given by a transfer function satifying the property G(s) = G T ( s) (41) Thus by classical realization theory [Willems, 1972, Maschke and van der Schaft 1992] a controllable and observable realization of G(s) Dirac ẋ = Ax + Bu structures (42) on Dirac structure with y = Cx + Du satisfies for a unique invertible Q dissipation Interconnection (3) A T Q + QA = 0, B T Q = C, D = D T (43) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 66/127

67 Linear Port Properties (43) are readily seen to express energy-conservation d 1 dt 2 xt Qx = y T u. The structure follows by defining the skew-symmetric matrix J := AQ 1, and then rewriting (42) as ẋ = JQx + Bu, J = J T (44) Dirac y = B T Qx + Du, D = D T structures which is easily seen to be a special case of a linear on implicit port system (24) Dirac structure (Note that for u = f, y = e it is exactly an explicit port with dissipation controlled system(25) with H(x) = 1 2 xt Qx.) Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 67/127

68 with non-constant geom Until now we have dealt with (energy-conserving) physical with a constant underlying geometry: (multi-)graphical interconnection structure canonical coordinates defined by a constant Dirac structure defined on a vector space. Now we consider extension to nonconstant geometry: Dirac kinestatic models of multibody structures no canonical coordinates on Dirac structure The geometric structure is generalized to a Dirac bundle Courant [1990] Dorfman [1993] or with dissipation (generalized) Dirac structure on a differentiable manifold. Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 68/127

69 Generalized Dirac structure on a manifold Let X be an n-dimensional manifold with tangent bundle T X and cotangent bundle T X. We define T X T X as the (smooth) vector bundle over X with fiber at each x X given by T x X T x X. Let throughout X be a smooth (that is, C ) vector field, and α be a smooth one-form on X. We say that (X,α) belongs to a smooth vector subbundle D T X T X (denoted (X,α) D) if (X(x),α(x)) D(x) for every x X. Furthermore for a smooth vector subbundle D T X T X we define the smooth vector subbundle D T X T X as Dirac structures on Dirac structure with D = {(X,α) < α ˆX dissipation > + < ˆα X >= 0, ( ˆX, ˆα) D} Interconnection (45) with < > denoting the natural pairing between smooth one-forms and smooth vector fields (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 69/127

70 Definition of generalized Dirac structures Definition 2 Courant 1990, Dorfman 1993 A generalized Dirac structure on an n-dimensional manifold X is a smooth vector subbundle D T X T X such that D = D. By taking ˆX = X, ˆα = α in (45) it immediately follows that a generalized Dirac structure D satisfies the power continuity property Dirac structures on Dirac structure with dissipation Interconnection (3) < α X >= 0, for all (X,α) D (46) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 70/127

71 Relation with Dirac structures defined on vec Given a generalized Dirac structure D on an n-dimensional manifold X for every x X, the subspace D(x) T x X T x X defines a constant Dirac structure on the tangent space T x X. Dirac In particular (cf. Proposition 1), dim D(x) = n for every structures x X. on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 71/127

72 Coordinate representation of Dirac bundles Similarly to the case of constant Dirac structures (see (20) (22) locally on X there exist matrices F 1 (x),f 2 (x),e 1 (x),e 2 (x), smoothly depending on x, such that D(x) = {(v,f,v, e) T x X F T x X F F 1 (x)v + F 2 (x)f = E 1 (x)v E 2 (x)e} Dirac (47) structures with F 1 (x),f 2 (x),e 1 (x),e 2 (x) satisfying on [ ] Dirac structure rank F 1 (x).f 2 (x).e 1 (x).e 2 (x) = n + m, x X with dissipation Interconnection E 1 (x)f1 T(x) + E 2(x)F2 T(x) + F 1(x)E1 T(x) + F 2(x)E2 T (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 72/127 (x) = 0, x (48)

73 Integrability condition The Dirac structures have been called generalized because in the original definition [Courant 1990] the following integrability, or closedness, condition is imposed. Definition 3 [?] A generalized Dirac structure D on X is closed (or simply a Dirac structure) if for arbitrary smooth (X 1,α 1 ),(X 2,α 2 ),(X 3,α 3 ) D there holds Dirac < L X1 α 2 X 3 > + < L X2 α 3 X 1 > + < L X3 α 1 X 2 >= 0 structures (49) with L on Xi denoting the Lie-derivative. Dirac structure The above closedness condition generalizes properly the closedness of symplectic forms and the Jacobi identity with dissipation for Poisson brackets, and has important consequences Interconnection for the structural properties of the implicit system. (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 73/127

74 Closed, regular Dirac structures A constant Dirac structure trivially satisfies the closedness condition, and conversely a generalized Dirac structure satisfying the closedness condition (and an additional constant rank condition, can be represented locally as a constant Dirac structure, by an appropriate choice of local coordinates on X F! Although in many examples the closedness condition is satisfied, there are important examples (such as mechanical (58) with nonholonomic kinematic constraints) where it is not. Checkable necessary and sufficient conditions for closedness of various representations of implicit generalized have been derived in [Dalsmo and van der Schaft 1999]. Dirac structures on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 74/127

75 Basic vector spaces Actually Dirac bundles corresponds to the following generalization of the basic spaces: a vector space F, called vector space of flow variables its dual vector space E, called vector space of effort variables equipped with a pairing < >, that is, a non-degenerated bilinear map F E L with L a linear space and the bond space B = F E endowed with the Dirac structures on symmetrical canonical pairing Dirac structure ( ) (f,e), ˆf,ê =< e ˆf > + < ê f > (50) with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 75/127

76 Dirac structure: generalization Definition 4 A constant Dirac structure on a linear space V is a linear subspace D B of the bond space B = F E with the property that: D = D (51) Dirac where D denotes the orthogonal with respect to, structures D = {(v,v ) V V (v,v ), (ˆv, ˆv ) = 0 for all on (52) Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 76/127 (ˆv, ˆv )

77 Canonical examples Let M be a finite-dimensional manifold. Let F = V (M) denote the Lie algebra of smooth vector fields on M, and let E = Ω 1 (M) be the linear space of smooth 1-forms on M. The usual pairing < α X >= i X α between 1-forms α and Dirac vectorfields X defines a non-generated pairing in L the structures linear space of smooth functions on M. on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 77/127

78 Canonical examples continued (a) Let J be a Poisson structure on M, with Poisson fiber morphism J : Ω 1 (M) V (M). Then graph J V (M) Ω 1 (M) is a Dirac structure. (b) Let ω be a (pre-)symplectic structure on M, defining the fiber morphism ω : V (M) Ω 1 (M). Then graph ω V (M) Ω 1 (M) is a Dirac structure. (c) Let V be a constant-dimensional distribution on M, and Dirac let annv be its annihilating co-distribution. Then structures V annv is a Dirac structure. on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 78/127

79 Definition 5 [van der Schaft and Maschke, 1995; Dalsmo and van der Schaft 1999 ] Let X be an n-dimensional manifold of energy variables, let H : X R be a (internal energy), and let F be the linear space R m of external flows f, with dual the space E := F of external efforts e. Consider a generalized Dirac structure D on X F, only depending on x. The implicit generalized system (X, F, D, H) is defined by Dirac structures on Dirac structure with dissipation Interconnection (3) (ẋ,f, H (x), e) D(x), x x X (53) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 79/127

80 We use the identification T y F F for every y F = R m. The minus sign in front of e again comes from the identification (α, e) T x X F (α,e) (T x X F). By (46) it follows that < α X > < e f >= 0 for all (X,f,α, e) D, and thus an implicit generalized system satisfies as in (19) Dirac structures the power balance on dh Dirac structure dt = et f (54) with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 80/127

81 Coordinate representation of Port Hence the implicit generalized system (X, F, D,H) takes, at least locally, the form F 1 (x)ẋ + F 2 (x)f = E 1 (x) H x (x) E 2(x)e, (55) which is a nonlinear set of DAE s in the state variables x with external variables (f, e). An explicit input-state-output specialization of (55) is as follows Dirac structures on Dirac structure ẋ = J(x) H x (x) + g(x)f, J(x) = JT (x), with e = g T (x) H x (x) + D(x)f, D(x) = DT (x), (56) dissipation Interconnection which was called port-controlled generalized system, with inputs f and outputs e [Maschke and van der Schaft 1992, 1994, 1995 ]. (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 81/127

82 Constrained port An important class of implicit generalized is provided by of the form ( constrained port-controlled generalized ) ẋ = J(x) H x (x) + g(x)f + b(x)λ, J(x) = JT (x) e = g T (x) H x (x) + D(x)f, D(x) = DT (x) Dirac structures 0 = b T (x) H x (x) (57) on Dirac structure involving a vector λ of Lagrange multipliers, associated with the algebraic constraints 0 = b T (x) H x (x), with b(x) with dissipation of constant rank. Interconnection By pre-multiplication of the first equation of (57) with a matrix S(x) of maximal rank satisfying S(x)b(x) = 0 one may reduce the system to an explicit port system. (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 82/127

83 Spatial mechanism Example 1 An actuated mechanical system with n degrees of freedom q = (q 1,,q n ), subject to kinematic constraints A T (q) q = 0, can be represented as [ q ṗ ] = [ 0 I n I n 0 ][ H q (q,p) H p (q,p) Dirac e = B T (q) H p (q,p) = BT (q) q structures on 0 = A T (q) H p (q,p) = AT (q) q Dirac structure (58) with which is of the form (57), with f being the vector of dissipation controlled external forces, e the corresponding vector of Interconnection generalized velocities, and λ the constraint forces corresponding to the constraints A T (q) q = 0. (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 83/127 ] + [ 0 B(q) ] f + [ 0 A(q) ]

84 Dissipation Dissipation can be brought into the framework in the same way as for port defined on Dirac structures, by terminating some of the ports of an implicit generalized system by dissipative elements (29). Dirac structures on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 84/127

85 Interconnection of Port The definition of a power-conserving interconnection is the same as in the constant case, but it is now allowed that the subspace I F i E i may depend on the states x 1,,x k of the to be interconnected. Dirac structures on Dirac structure with dissipation Interconnection (3) Thus we require instead of (33) that for all x 1,,x k (f i,e i ) I(x 1,,x k ) < e i f i >= 0 (59) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 85/127

86 Interconnection of Port Theorem 2 Consider k implicit generalized (X j, F j, D j,h j ),j = 1,,k, subject to a power-conserving interconnection I. Then the interconnected system is an implicit generalized system with state space X := X 1 X k, flow space F p, H(x 1,,x k ) := H 1 (x 1 ) + + H k (x k ), and generalized Dirac structure D on X F p given as Dirac structures on Dirac structure (X(x j ),f j,α j (x j ), e j ) D j (x j ), with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 86/127 (X,α) = (X 1,,X k,α 1,,α k ) D f 1,,f k,e 1,, x j X j, j = 1,,k, while (f 1,,f k,e 1,,e k ) = (f i,f p,e i,e p ) with (f i,e i ) I(x 1,,x (60)

87 Conservation laws and contraints The Dirac structure D of an implicit port system (X, F, D,H) characterizes some important structural properties of the system: constraints and dynamical invariants. In a first instance, consider an implicit generalized system without any external port variables ( F is void, and D is a generalized Dirac structure defined solely on X ). In some coordinates (x 1,...x n ), the autonomous implicit generalized system take the form Dirac structures on Dirac structure with dissipation Interconnection (3) F(x)ẋ = E(x) H (x) x with rank [F(x).E(x)] = dimx and E(x)F T (x) + F(x)E T (x) = 0. (61) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 87/127

88 Characteristic distributions and co-distributio With the generalized Dirac structure D on X we may associate two smooth distributions on X G 0 = {X T X (X,0) D} (62) Dirac G 1 = {X T X α T X s.t. (X,α) D} structures and two smooth co-distributions on X on P Dirac structure 0 = {α T X (0,α) D} (63) with dissipation Interconnection (3) P 1 = {α T X X T X s.t. (X,α) D} EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 88/127

89 Characteristic distributions and co-distributio By definition of a Dirac structure these are immediately linked as follows: G 0 = kerp 1 P 0 = anng 1 (64) (KerP 1 is the distribution spanned by the vector fields X Dirac such that < α X >= 0 for all α P 1, and dually, anng 1 structures is the co-distribution spanned by the one-forms α such that < α X >= 0 for all X G on 1.) Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 89/127

90 Geometrical characterization of the constrain The co-distribution P 1 determines the (primary) algebraic constraints appearing in the implicit generalized system. Indeed, if H is the (total stored energy) then necessarily the system should satisfy the following algebraic (effort) constraints on the state variables x H x (x) P 1(x), Dirac x X (65) structures on It is determined by the effort constraint representation of Dirac bundles Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 90/127

91 Geometrical characterization of the invariants The distribution G 1 determines the admissible state flows ẋ of the implicit generalized system since necessarily ẋ G 1 (x), x X (66) If the distribution G 1 is involutive ( this is e.g. guaranteed by closedness of the Dirac structure) and if G 1 is constant-dimensional then by Frobenius theorem we may locally find independent functions C 1,,C k such that Dirac structures on Dirac structure P 0 = span{dc 1,,dC k }, k = codimg 1 (67) and necessarily by (66) dc i dt = 0, i = 1,,k, along the with dissipation trajectories of the implicit system. Interconnection Thus C 1,,C k are independent conserved quantities for any function.. (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 91/127

92 Non integrable case If G 1 is not involutive, then we may consider G 1, the smallest involutive distribution containing G 1, and apply Frobenius theorem to G 1 in order to obtain k independent conserved quantities, where k = codimg 1 (possibly zero). Dirac structures on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 92/127

93 Conclusion Port are defined with respect to a Dirac structures and generated by functions. This corresponds to network models of physical system: Dirac structure = geometry of power continuous interconnection function = total energy Dirac structures Dirac structures expresses constraints, dynamical invariants and relations with external, i.e. port variables. on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 93/127

94 Conclusion cont d Port are closed with respect to interconnection: total function is the sum of the functions total Dirac structure is the composition, through an interconnection structure, of the Dirac structures Dirac Paradigm: spatial multibody defined with structures respect to the Lie group of rigid body displacements on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 94/127

95 Conclusion cont d Port : do not have an a priori input-output structure in coordinates lead to DAE s (differential-algebraic equations) New synthesis methods based on the shaping of the energy and interconnection and damping structure Dirac structures on Dirac structure with dissipation Interconnection (3) EURON-GEOPLEX Dirac bundlessummer School, July 2005 Port by B. Maschke p. 95/127

96 Contact manifolds and Equilibrium Thermodynamics Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 96/127

97 A counter-example: heat conduction Consider closed system composed of 2 phases exchanging a heat flux Q and denote: the entropies of the phases by S 1 and S 2 the temperatures are: T 1 = U S 1 and T 2 = U S 2 where U is the sum of the internal energies. Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations The continuity of heat flow due to the conduction leads to: ds 1 dt T 1 = ds 2 dt T 2 = Q = λ(t 2 T 1 ) leading to the nonlinear relation on the conjugated power variables: (ds1) ( dt 1 ds 2 = λ 1 ) ( ) ( ) 0 1 T 1 T dt 1 T T 2 }{{}}{{} heat conduction du Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 97/127

98 Sketch recall use of contact forms for Equilibrium Thermodynamics and contact vector fields lift port as input-output contact vector fields define conservative with ports as an extension of port to open arising from Irreversible Thermodynamics Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 98/127

99 The Thermodynamic Properties Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations They are described in the Thermodynamic Phase space: energy ǫ R extensive variables x R n intensive variables e x R n They are given by a (potential) function U(x) defining the constitutive relations which obey Gibb s relation: du = n e x i dx i Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 99/127 i=1 For a simple Thermodynamic System: extensive variables: energy U, entropy S, volume V, number of moles N intensive variables: temperature T, pressure P, chemical potential µ Gibbs relation du = TdS + ( P)dV + µdn

100 A differential-geometric approach References: 2 among many.. C.Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann., 1909 Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations R. Mrugala, Geometrical formulation of equilibrium phenomenological Thermodynamics, Reports on Mathematical Physics, 1978 Use Pfaffian forms and contact forms P.Libermann and C.-M.Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, 1987 Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 100/127

101 Contact manifolds and contact forms Let M be a 2n + 1-dimensional differentiable manifold A Pfaffian equation on M is a vector subbundle E of rank 1 of T M. Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations A contact structure on M is defined by specifying a Pfaffian equation E of constant class 2n + 1. (M, E) is called a contact manifold A Pfaffian structure on a strictly contact manifold (M, E) is the structure defined by the choice of a form θ that determines E globally. Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 101/127

102 Darboux s theorem a) Let E be a Pfaffian equation of constant class 2n + 1 on M. For every point x of M there exists a neighborhood in which E is by a Pfaffian form θ of constant class 2n + 1. b) Let θ be a Pfaffian form of constant class 2n + 1 in an open set U of M. For every x of M there exists a family (x 0,x 1,...,x n,p 1,...,p n ) of independents functions in a neighborhood V of x such that Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations Example: Pure monoatomic ideal gas The thermodynamical space: energy x 0 = U extensive variables: x i = S,V,N θ V = dx 0 p i dx i (68) intensive variables: p i = T,P,µ Gibbs form: θ = du TdS + PdV µdn Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 102/127

103 Legendre submanifold Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations A Legendre submanifold of a 2n + 1-dimensional contact manifold (M, E) is an n-dimensional submanifold of M that is an integral manifold of E. A Legendre submanifold is locally defined, in a given set of canonical coordinates (x 0,x 1,...,x n,p 1,...,p n ) by: a partition I J of the set of indices {1,...,n} a differentiable function F(x I,p J ) of n variables, i I,j J and the equations: x 0 = S p J F p J, x J = F p J, p I = F x I Note: it is the analogue of Lagrangian submanifolds of symplectic manifolds Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 103/127

104 Thermodynamic properties Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations The properties of thermodynamic (Equilibrium Thermodynamics) may be described by: a thermodynamic phase space composed of: energy x 0 pairs of extensive and intensive variables ( x i,p i ) a contact form (Gibbs form): θ V = dx 0 p i dx i a Legendre submanifolds L of the thermodynamic phase space Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 104/127

105 Pure monoatomic ideal gas the thermodynamical space: energy x 0 = U extensive variables: x i = S,V,N intensive variables: p i = T,P,µ Gibbs form θ = du T ds + P dv µdn the Legendre submanifold L defined by the map ψ: U(T, P,N) = 3 2 NRT Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations S(T, P,N) = Ns = Ns NRln( T T 0 ) NRln( P P 0 ) V (T, P,N) = NRT P µ(t, P,N) = 5 RT Ts 2 On L, the first principle of thermodynamics is satisfied: ψ θ = 0. Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 105/127

106 Contact transformations and vector fields A contact transformation on a contact manifold (M, E) is an automorphism of the equation E. An infinitesimal automorphism of a contact structure is called a contact vector field. A vector field X on (M, E) is an contact vector field if and only if there exists a differentiable function ρ such that L X θ = ρθ. Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations When ρ vanishes, X is an infinitesimal automorphism of the Pfaffian structure. The Reeb vector field is the unique vector field E such that: i E θ = 1, i E dθ = 0 and can be express locally as: E = x 0 Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 106/127

107 Contact vector fields and contact Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations There is an isomorphism Φ from the vector space L X of the contact vector fields onto the vector space L f of the differentiable real-valued function on M: Φ(X) = i X θ = f where f is called the contact. Denote: X f = Φ 1 (f) then the isomorphism Φ induces the following bracket on L : {f,g} = i [Xf,X g ]θ The function ρ such that L Xf θ = ρθ is equal to ρ = i E df. Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 107/127

108 Decomposition of contact vector fields A contact vector field X f is also defined by: i Xf θ = f, i Xf dθ = Df, where Df denotes the covariant derivative: Df = df ( i E df ) θ Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations It corresponds to the decomposition: X f = fe + θ #( df ( i }{{} E df ) ) θ }{{} kerdθ :vertical component kerθ :horizontal component Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 108/127

109 Contact vector fields in coordinates The contact vector field X f = Φ 1 (f), with contact f is written in canonical coordinates: X f = ( f n k=1 p k f p k ) x 0 + f x 0 ( n k=1 p k p k ) Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations = f x 0 + +Df n k=1 } ( f x k f p k p k {{ symplectic bracket x k Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 109/127

110 Invariance of Legendre submanifolds Let (M, E) be a strictly contact manifold and with θ the contact form θ. Consider a Legendre submanifold L of (M, E). Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations Then X f is tangent to L if and only if f is identically zero on L: L f 1 (0) This characterizes contact fields which leave invariant some thermodynamical properties. For instance: reversible transformations. Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 110/127

111 Example of the ideal gas Consider the contact : Contact manifolds and Equilibrium Thermodynamics Heat conduction Thermodynamic Phase Space Thermodynamic properties Reversible transformations f(u,s,v,b, T,P, µ) = U (3/2)NRT The associated contact vector field is : X f = U U +T T +P ( P + µ 3 ) 2 RT µ +3 2 NR S +0 V +0 Thus the integral curves of X f are U(t) = U 0 e t, T(t) = T 0 e t, P(t) = P 0 e t, µ(t) = µ 0 e t 3 2 RT S(t) = S N 0Rt, V (t) = V 0, N(t) = N 0 Since L f 1 (0), X f is tangent to L: the thermodynamical properties of the ideal gas are preserved along the integral curves. Lift of port EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 111/127

112 Lift of port Contact manifolds and Equilibrium Thermodynamics Lift of port Lift on the cotangent bundle Lift on the Thermodynamic Phase Space EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 112/127

113 To prepare the definition of associated with open thermodynamical : express the lift of port on the Thermodynamic Phase Space as a contact vector field with inputs and outputs Contact manifolds and Equilibrium Thermodynamics use Legendre submanifold as geometric definition of the function of the system Lift of port exhibit the generating of the contact vector Lift on the cotangent field bundle Lift on the Thermodynamic Phase Space EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 113/127

114 Reminder: port system Contact manifolds and Equilibrium Thermodynamics Let N be a differential manifold endowed with a pseudo-poisson bracket denoted by {.,.} (Λ is the pseudo-poisson tensor). A port system is defined by a function H 0 (x) C (N), an input vector u(t) = (u 1,...,u m ) T function of t, m input vector fields g 1,...,g m on N, and the equations : { ẋ = Λ # (d x H 0 (x)) + m y j p = L gj.h 0 (x) i=1 u i(t)g i (x) (69) Lift of port Lift on the Lift on the cotangent cotangent bundle T N using the adjoint variational bundle system Lift on the Thermodynamic Phase thermodynamic phase space:r T N Space EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 114/127

115 reminder: variational adjoint system Consider ( s ) the following non-linear system (with inputs and outputs) : { ẋ = f(x,u) y = h(x) Contact manifolds and Equilibrium Thermodynamics The variational adjoint system ( p s ) of ( s ), along the trajectory (x 0 (t),u 0 (t),y 0 (t)), is : e x = f T (x 0,u 0 )e x h T (x 0 )u x x Lift of port Lift on the cotangent y bundle v = f T (x 0,u 0 )e x u Lift on the Thermodynamic Phase Space EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 115/127

116 Lift on the cotangent bundle Contact manifolds and Equilibrium Thermodynamics Using its variational adjoint system, the port system ( ) on (N, Λ) can be lifted to the cotangent bundle (T N, Λ s ) d dt ( ) ( ) x 0 Id = e x Id 0 }{{} Λ # s ( H ) a x H a e x = Λ # s (d x Ha ) = X H a where the generating H a is given by the duality product: H a (x,e x,u i,u i ) = e x,λ # (d x H 0 )(x) }{{} + u i e x,g i (x) }{{} X H0 Lift of port H v,i Lift on the + u cotangent i d x H 0,g i (x) }{{} bundle Lift on the Thermodynamic H = v,i yi p Phase Space EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 116/127

117 Lift on the Thermodynamic Phase Space Contact manifolds and Equilibrium Thermodynamics Define the thermodynamic state space T associated with the base manifold N by: T = R T N endowed with the canonical contact form written in the canonical coordinates (ε,x i,e i x) : θ = dε n e i xdx i (70) Lift of port Lift on the cotangent bundle Lift on the. Thermodynamic Phase Space EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 117/127 i=1 Define the lift on the Thermodynamic Phase Space by the input-output contact field X H a with contact : H a (x,e x,u i,u i )

118 In canonical coordinates Contact manifolds and Equilibrium Thermodynamics The contact field on the thermodynamic phase space T with contact H a (x,e x,u i,u i ) is written : ε 0 e T x ( ) H a d dt x = 0 Id H a H T a x e + x 0 Id 0 0 e x = H a e x, H a X H a e x = m i=1 u v,i L g i.h 0 X H a Lift of port Lift on the cotangent bundle Lift on the Thermodynamic Phase Space EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 118/127

119 Invariance of the Legendre submanifold L H0 Contact manifolds and Equilibrium Thermodynamics A sufficient condition for the contact field X H a to preserve the Legendre submanifold L H0 generated by the internal energy function H 0, is that Indeed: u i = u i for all i in {1,...,m}. H a LH0 ( x) = d x H 0 (x),λ # J (x)d xh 0 (x) + m i=1 u i d x H 0 (x),g i (x) m i=1 u i d xh 0,g i (x) Then the contact becomes: Lift of port H a (x,e x,u i ) = e x,λ # (d x H 0 )(x) + u }{{} i e x d x H 0,g i (x) }{{} Lift on the cotangent bundle Lift on the Thermodynamic Phase Space X H0 and may be interpreted as a virtual power. H i EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 119/127

120 Power balance equation Write the restriction of the contact H a (x,e x,u i ) to L H0 as follows : H a LH0 ( x) = dh 0 /dt {}}{ d x H 0 (x),λ # J (x)d xh 0 (x) + u i d x H 0 (x),g i (x) m i=1 u i d x H 0,g i (x) Contact manifolds and Equilibrium Thermodynamics Hence the invariance condition H a LH0 0 may be interpreted as the power balance equation : Lift of port dh 0 /dt u i L g i.h 0 = 0 Lift on the }{{} i=1 cotangent yp i bundle Lift on the Thermodynamic Phase Space EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 120/127 m

121 Conservative system with ports on contact manifolds Contact manifolds and Equilibrium Thermodynamics Lift of port Conservative system with ports on contact manifolds Closed EURON-GEOPLEX Summer School, July 2005 Open Port by B. Maschke p. 121/127

122 Conservative system on a contact manifold Contact manifolds and Equilibrium Thermodynamics Lift of port A conservative system on a contact manifold is defined by: a strictly contact manifold (M, E) with contact form θ (the Thermodynamic Phase Space) a Legendre submanifold L of (M, E) (the Thermodynamic properties) a contact K 0 (the potential generating the fluxes) and satisfying the invariance condition: K 0 L = 0 (the potential generating the fluxes) Conservative system with ports the differential equation: x = X K0 on contact manifolds Closed EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 122/127 Open

123 Example : heat conduction Contact manifolds and Equilibrium Thermodynamics Thermodynamic phase space: energy ǫ entropies S i N = R 2 temperature T i T S N R2 with contact form θ = dǫ T i ds i a Legendre submanifold L generated by U(S) the contact K 0 = R(S,T)Λ s (T,dU) where R = λ(t)(1/t 1 1/T 2 ), with (T i 0) satisfies the definition. Lift of port Conservative system with ports on contact manifolds Closed Open Generates a contact vector field which restricted to L is : ( ) ds1 dt ds 2 dt ( 1 = λ 1 T 1 T 2 ) ( ) ( EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 123/127 T 1 T 2 )

124 Conservative system with ports Contact manifolds and Equilibrium Thermodynamics Lift of port A conservative system on a contact manifold is defined by: a strictly contact manifold (M, E) with contact form θ a Legendre submanifold L of (M, E) m + 1 a contact s: K 0 internal and K j interaction satisfying the invariance condition: K j L = 0, j = 0,...,m (the potential generating the fluxes) the differential equation: x = X K0 + m j=1 u j X Kj Conservative system with ports on contact manifolds Closed EURON-GEOPLEX Summer School, July 2005 Open Port by B. Maschke p. 124/127

125 Example : heat conduction Thermodynamic phase space: energy ǫ entropies S i N = R 2 temperature T i T S N R2 with contact form θ = dǫ T i ds i Contact manifolds and Equilibrium Thermodynamics Lift of port Conservative system with ports on contact manifolds Closed Open a Legendre submanifold L generated by U(S) the internal contact K 0 = R(S,T)Λ s (T,dU) where R = λ(t)(1/t 1 1/T 2 ), with (T i 0) and the interaction is K 1 = T 1 u satisfies the definition. Generates a contact vector field which restricted to L is : ) ( 1 = λ 1 ) ( ) ( ) ( ) 0 1 T f ext T 1 T ( ds1 dt ds 2 dt EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 125/127 T 2

126 State feedback of conservative with Consider state feedback: X := X K0 + The vertical component is: m α j (x)x Kj j=1 Contact manifolds and Equilibrium Thermodynamics Lift of port i X θ = i XK0 + αj X Kj θ = K 0 + α j K j =: f and gives the possible contact! Hence it must satisfy: L(X)θ = ( i(e)df ) θ L(X K0 )θ + j (α j X Kj )θ ( i(e)df ) θ = 0 Conservative system with ports K j i(x αj )dθ = 0 on contact manifolds j Closed EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 126/127 Open

127 Conclusion extension of port to Irreversible Thermodynamics extension of contact vector fields to input-output contact fields Contact manifolds and Equilibrium Thermodynamics Lift of port Conservative system with ports on contact manifolds Closed Open contact associated with irreversible phenomena thermodynamic properties: invariance of Legendre submanifold interconnection of conservative with ports and constraints feedback of conservative with ports: passive controllers EURON-GEOPLEX Summer School, July 2005 Port by B. Maschke p. 127/127