The Geometry Underlying Port-Hamiltonian Systems

Transcription

1 The Geometry Underlying Port-Hamiltonian Systems Pre-LHMNC School, UTFSM Valparaiso, April 30 - May 1, 2018 Arjan van der Schaft Jan C. Willems Center for Systems and Control Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen, the Netherlands Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 1 / 66

2 Review on classical Hamiltonian systems Dirac structures on linear spaces Definition of port-hamiltonian systems on linear spaces Examples from different physical domains Dirac structures on manifolds (Almost) everything (and much more!) can be found in A.J. van der Schaft, D. Jeltsema, Port-Hamiltonian Systems Theory: An Introductory Overview, NOW Publishers, 2014 The pdf of the book is freely available from my home page arjan Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 2 / 66

3 Outline 1 Review on classical Hamiltonian systems 2 From network interconnection to geometric structure 3 Port-Hamiltonian systems 4 Definition of port-hamiltonian systems 5 Examples from different physical domains 6 Dirac structures on manifolds 7 Conclusions and Outlook Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 3 / 66

4 Review on classical Hamiltonian systems Euler-Lagrange equations: Consider a mechanical system with n degrees of freedom, and position coordinates q = (q 1,...,q n ) for the configuration manifold Q. Determine the kinetic energy K(q, q) = 1 2 qt M(q) q, M(q) > 0 and the potential energy P(q), where q T q Q. Define the Lagrangian function L : TQ R The equations of motion are d dt L(q, q) := K(q, q) P(q) ( L (q, q) q ) L (q, q) = τ q Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 4 / 66

5 Review on classical Hamiltonian systems Define the momenta p := L q (q, q) T qq and the Hamiltonian H : T Q R defined as H(q,p) := 1 2 pt M 1 (q)p +P(q) (kinetic energy plus potential energy = total energy). Then we obtain the classical Hamiltonian equations of motion q = H p (q,p) ṗ = H q (q,p)+τ Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 5 / 66

6 Review on classical Hamiltonian systems Geometrically (coordinate-free) this is described by the triple (T Q,ω,H) Q is the configuration manifold ω is canonical symplectic form on the cotangent bundle T Q the dynamics given by the Hamiltonian vector field X H satisfying ω(x H, ) = dh Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 6 / 66

7 The symplectic form ω on the cotangent bundle T Q is defined as follows. T Q is endowed with a canonical one-form θ (called Liouville one-form), defined as θ(α)(z) = α(π Z) where α T Q,Z T α (T Q) and π : T Q Q natural projection. Choosing any set of coordinates q = (q 1,,q n ) for Q, and natural induced coordinates for T Q, it follows that (q,p) = (q 1,,q n,p 1,,p n ) θ = n p i dq i i=1 Now define the symplectic form ω := dθ. In local coordinates n ω = dp i dq i i=1 Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 7 / 66

8 Review on classical Hamiltonian systems Equivalently, let {,} denote the Poisson bracket on T Q given as In natural coordinates for T Q {F,G} := ω(x F,X G ), F,G : T Q R {F,G} = n ( F G F G ) q i p i p i q i i=1 Then X H is determined by the requirement X H (F) = {F,H} for all F : T Q R. In an arbitrary set of local coordinates x the Hamiltonian dynamics takes the form ẋ = J(x) H x (x) where J(x) = J T (x) is the (invertible) Poisson structure matrix J ij = {x i,x j }, i,j = 1,,2n Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 8 / 66

9 Hamiltonian systems obtained by symmetry reduction On the other hand, it is well-known that many dynamical equations of physical interest are not precisely of this form. Typical example is formed by the Euler equations for rigid body motion ṗ x ṗ y ṗ z 0 p z p y = p z 0 p x p y p x 0 H p x H p y H p z with p = (p x,p y,p z ) the body angular momentum vector along the three ( principal axes, and H(p) = 1 p 2 x 2 I x + p2 y I y + p2 z the kinetic energy (with I x,i y,i z principal moments of inertia.) In general, many physical systems are of the Hamiltonian form I z ) ẋ = J(x) H x (x) with J(x) = J T (x), but not of full rank. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 9 / 66

10 Hamiltonian systems obtained by symmetry reduction The Euler equations can be regarded as the reduction of classical Hamiltonian equations on the cotangent bundle T Q where Q = SO(3). Reduced space is the orbit space of the action of a Lie group that leaves the Hamiltonian invariant. In fact, the cotangent bundle T SO(3) can be reduced by the action of SO(3) on T SO(3) into so(3), since the Hamiltonian (= kinetic energy) is invariant under this action. This holds in many situations, both in the finite- and infinite-dimensional case ( Marsden-Weinstein reduction by symmetry program ). Thus in these cases the Poisson structure is still derivable from a standard symplectic structure on a cotangent bundle. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 10 / 66

11 If we move away from mechanical systems the story becomes different. What is the Hamiltonian formulation of a general LC-circuit? Example (LC-circuits) Two inductors with magnetic energies H 1 (ϕ 1 ),H 2 (ϕ 2 ) (ϕ 1 and ϕ 2 magnetic flux linkages), and capacitor with electric energy H 3 (Q) (Q charge). $L_1$ $L_2 $\varphi_1$ $C$ $Q$ $ Figure: LC-circuit Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 11 / 66

12 Same question holds for multi-physics systems (e.g., electro-mechanical systems). Furthermore How to include energy-dissipation into a Hamiltonian formulation? How to interconnect systems? What if the interconnected system is a differential-algebraic equation (DAE) system? Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 12 / 66

13 Outline 1 Review on classical Hamiltonian systems 2 From network interconnection to geometric structure 3 Port-Hamiltonian systems 4 Definition of port-hamiltonian systems 5 Examples from different physical domains 6 Dirac structures on manifolds 7 Conclusions and Outlook Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 13 / 66

14 A paradigm-shift in modeling is needed, originating e.g. from electrical network theory; later on generalized to port-based modeling and bond-graphs. Example (The ubiquitous mass-spring system) Instead of starting with the position coordinate q of the mass, and its velocity q yielding the kinetic energy 1 2 m q2, together with the potential energy 1 2 kq2 of the spring, we consider the system as the network interconnection of two energy-storing elements corresponding to them: Spring Hamiltonian H s (q) = 1 2 kq2 (potential energy) q = f s = velocity (= e m ) e s = dhs dq (q) = kq = force Mass Hamiltonian H m (p) = 1 2m p2 (kinetic energy) ṗ = f m = force (= e s ) e m = dhm dp (p) = p m = velocity Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 14 / 66

15 A different starting point Network modeling of physical systems Prevailing trend in modeling and simulation of lumped-parameter systems (multi-body systems, electrical circuits, electro-mechanical systems, robotic systems, cell-biological systems, etc.). Main advantages of network modeling: Systematic modeling procedure, starting from simple components to complex systems. Re-usability of component models. Flexible adaptation. Design and control. Suited to multi-physics systems. Originates from electrical circuit theory (Kirchhoff) and mechanical engineering (Newtonian modeling). What is the underlying geometric structure? Not a cotangent bundle nor a reduced cotangent bundle! Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 15 / 66

16 Port-based network modeling Interaction between ideal system components is modeled by power-ports modeling the energy exchange between the components. Associated to every power-port there are conjugate pairs of (vectors of) variables, called flows f and efforts e, whose product e T f equals power. For example, voltages and currents forces and velocities pressure and volume change chemical potentials and concentration fluxes Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 16 / 66

17 Example (LC-circuits) Two inductors with magnetic energies H 1 (ϕ 1 ),H 2 (ϕ 2 ) (ϕ 1 and ϕ 2 magnetic flux linkages), and capacitor with electric energy H 3 (Q) (Q charge). $L_1$ $L_2 $\varphi_1$ $C$ $Q$ $ Figure: LC-circuit Question: How to write this LC-circuit as a Hamiltonian system? Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 17 / 66

18 Example (LC-circuit continued) Dynamical equations for the three components: Inductor 1 : ϕ 1 = f 1 (voltage) (current) e 1 = H 1 ϕ 1 Inductor 2 : ϕ 2 = f 2 (voltage) Capacitor : (current) e 2 = H 2 ϕ 2 Q = f 3 (voltage) e 3 = H 3 Q (current) If H i are quadratic, e.g., H 3 (Q) = 1 2C Q2, then the elements are linear. E.g., voltage over capacitor = H 3 Q = Q C, and similarly for the inductors. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 18 / 66

19 Example (LC-circuit continued) Kirchhoff s voltage and current laws are f e 1 f 2 = e 2 f e 3 Substitution of eqns. of components yields port-hamiltonian system H ϕ ϕ 1 ϕ 2 = H ϕ 2 Q H Q with H(ϕ 1,ϕ 2,Q) := H 1 (ϕ 1 )+H 2 (ϕ 2 )+H 3 (Q) total energy. Can be directly extended to RLC-circuits. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 19 / 66

20 Preliminary conclusions The structure matrix J is completely determined by the interconnection structure of the circuit (in this case, Kirchhoff s current and voltage laws). Skew-symmetry of J corresponds to the interconnection being power-conserving (Tellegen s theorem derived from Kirchhoff s laws). There is no clear underlying cotangent bundle nor symplectic manifold! Building blocks are open dynamical systems (systems point of view). Not yet general enough: how do we formulate the LC-circuit with the same topology, but with capacitors and inductors swapped (corresponding to an algebraic constraint)? Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 20 / 66

21 Outline 1 Review on classical Hamiltonian systems 2 From network interconnection to geometric structure 3 Port-Hamiltonian systems 4 Definition of port-hamiltonian systems 5 Examples from different physical domains 6 Dirac structures on manifolds 7 Conclusions and Outlook Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 21 / 66

22 The basic picture storage e S D e R dissipation f S f R e P f P Figure: Port-Hamiltonian system Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 22 / 66

23 The basic elements Port-based modeling is based on viewing the physical system as the interconnection of ideal energy processing elements, all expressed in (vector) pairs of flow variables f F, and effort variables e E, where F and E are linear spaces of equal dimension. Furthermore, there is a pairing between F and E defining the power < e f > Canonical choice: E = F with < e f >= e T f, but more involved in e.g. multi-body systems. Energy-storing elements: Purely energy-dissipating elements: ẋ = f e = H x (x) R(f,e) = 0, e T f 0 Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 23 / 66

24 The basic elements Energy-routing elements: generalized transformers, gyrators: f 1 = Mf 2, e 2 = M T e 1, f = Je, J = J T They are power-conserving: Ideal interconnection constraints Also power-conserving: e T f = 0 0-junction : e 1 = e 2 = = e k, f 1 +f 2 + +f k = 0 1-junction : f 1 = f 2 = = f k, e 1 +e 2 + +e k = 0 Ideal flow or effort constraints : f = 0, or e = 0 e 1 f 1 +e 2 f 2 + +e k f k = 0 Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 24 / 66

25 From power-conserving elements to Dirac structures All power-conserving elements/interconnection constraints have the following properties in common. They are described by linear equations in f,e R k satisfying e T f = e 1 f 1 +e 2 f 2 + +e k f k = 0, while furthermore the number of independent equations is equal to k. All power-conserving elements/interconnection constraints will be grouped into one geometric object: the Dirac structure. (Note: the linearity of the relations between efforts and flows breaks down in thermodynamic systems.) Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 25 / 66

26 Dirac structures on linear spaces Definition A (constant) Dirac structure (on a linear space) is a subspace such that (i) e T f = 0 for all (f,e) D, (ii) dimd = dimf. D F E For any skew-symmetric map J : E F its graph given as {(f,e) F E f = Je} is a Dirac structure. Similarly, the graph of any skew-symmetric map ω : F E is a Dirac structure. But not all Dirac structures are of this type! E.g, for any subspace V F, the product V V F E is also a Dirac structure. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 26 / 66

27 Alternative definition of Dirac structure Symmetrized form of power < e f >= e T f, (f,e) F E. Symmetrization leads to the indefinite bilinear form, on F E: (f a,e a ),(f b,e b ) := < e a f b > + < e b f a >, (f a,e a ),(f b,e b ) F E. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 27 / 66

28 Alternative definition of Dirac structure Definition A (constant) Dirac structure is a subspace D F E such that D = D, where denotes orthogonal companion with respect to the bilinear form,. Due to Weinstein & Courant, Dorfman; with a connection to the Dirac bracket of constrained Hamiltonian systems. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 28 / 66

29 Outline 1 Review on classical Hamiltonian systems 2 From network interconnection to geometric structure 3 Port-Hamiltonian systems 4 Definition of port-hamiltonian systems 5 Examples from different physical domains 6 Dirac structures on manifolds 7 Conclusions and Outlook Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 29 / 66

30 The basic model Consider a Dirac structure D F E, where f = (f S,f R,f P ), e = (e S,e R,e P ) storage e S D e R dissipation f S f R e P f P Figure: Port-Hamiltonian system Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 30 / 66

31 The port-hamiltonian system is defined by closing the energy-storing and energy-dissipating ports of the Dirac structure D by their constitutive relations ẋ = f S, H x (x) = e S respectively R(f R,e R ) = 0 This leads to, in principle, a mixture of differential and algebraic equations (DAE systems) ( ẋ(t),f R (t),f P (t), H x (x(t)),e R(t),e P (t)) D R(f R (t),e R (t)) = 0 t R Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 31 / 66

32 Energy-balance Power-conservation implies the energy-balance e T S f S +e T R f R +e T P f P = 0 dh dt (x(t)) = T H x (x(t))ẋ(t) = e T R (t)f R(t)+e T P (t)f P(t) e T P (t)f Pt) (showing passivity if H is bounded from below). Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 32 / 66

33 DAE example Consider the same LC-circuit as before, but now the inductors and capacitors are swapped. C 1 C 2 Q 1 + Q 2 ϕ L Figure: LC circuit. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 33 / 66

34 The capacitors (assumed to be linear) are described by Q i = I i, V i = Q i C i, for i = 1,2. Here I i and V i are the currents through, respectively voltages across, the two capacitors. C i are their capacitances. Q i are the charges of the capacitors. Similarly, the linear inductor is described by ϕ = V L, I L = ϕ L, where I L is the current through the inductor, and V L is the voltage across the inductor. ϕ is the flux-linkage of the inductor, and L its inductance. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 34 / 66

35 Parallel interconnection of these three subsystems by Kirchhoff s laws amounts to the same interconnection equations V 1 = V 2 = V L, I 1 +I 2 +I L = 0, and thus to the same Dirac structure as before, where however some efforts are replaced by flows and conversely. The equation V 1 = V 2 gives rise to the algebraic constraint Q 1 C 1 = Q 2 C 2, relating the two state variables Q 1,Q 2. The resulting system is a port-hamiltonian DAE system F(ẋ, H x (x)) = 0. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 35 / 66

36 Outline 1 Review on classical Hamiltonian systems 2 From network interconnection to geometric structure 3 Port-Hamiltonian systems 4 Definition of port-hamiltonian systems 5 Examples from different physical domains 6 Dirac structures on manifolds 7 Conclusions and Outlook Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 36 / 66

37 Example (The mass-spring system revisited) Two storage elements: Spring Hamiltonian H s (q) = 1 2 kq2 (potential energy) q = f s = velocity e s = dhs dq (q) = kq = force Mass Hamiltonian H m (p) = 1 2m p2 (kinetic energy) ṗ = f m = force e m = dhm dp (p) = p m = velocity Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 37 / 66

38 Example Dirac structure linking f s,e s,f m,e m,f P,e P as f s = e m = f P, f m = e s e P (power-conserving since f s e s +f m e m +uy = 0) yields the port-hamiltonian system ] [ ][ [ q 0 1 H q = (q,p) ] [ ] 0 H ṗ 1 0 p (q,p) + e P 1 with f P = [ 0 1 ][ H q (q,p) ] H p (q,p) H(q,p) = H s (q)+h m (p) Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 38 / 66

39 Example (Electro-mechanical systems) H q q ṗ = (q,p,φ) 0 H p (q,p,φ) + 0 V, I = H ϕ 0 0 R H ϕ (q,p,φ) ϕ (q,p,φ) 1 Coupling electrical/mechanical domain via Hamiltonian H(q,p,ϕ) = mgq + p2 2m + ϕ2 2L(q) Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 39 / 66

40 DC motor R L + ω V K J _ I b τ Figure: DC motor. 6 interconnected subsystems: 2 energy-storing elements: inductor L with state ϕ (flux), and rotational inertia J with state p (angular momentum); 2 energy-dissipating elements: resistor R and friction b; gyrator K; voltage source V. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 40 / 66

41 The energy-storing elements (here assumed to be linear) are given by ϕ = V L Inductor: I = d ( ) 1 dϕ 2L ϕ2 = ϕ L, ṗ = τ J Inertia: ω = d dp ( 1 2J p2 ) = p J. Hence, the corresponding total Hamiltonian reads H(p,φ) = 1 The energy-dissipating relations (also assumed to be linear) are V R = RI, τ b = bω, 2L φ J p2. with R,b > 0, where τ b is a damping torque. The equations of the gyrator (converting magnetic power into mechanical, and conversely) are with K the gyrator constant. V K = Kω, τ K = KI. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 41 / 66

42 The subsystems are interconnected by the equations V L +V R +V K +V = 0, τ J +τ b +τ K +τ = 0. The Dirac structure is defined by the interconnection equation, together with the equations for the gyrator. This results in the port-hamiltonian model [ ] [ ] ϕ ϕ R K L = ṗ K b p + J [ ] [ ] ϕ I 1 0 L =. ω 0 1 p J [ of the form ẋ = [J R] H H x (x)+gu, y = GT x (x). ][ V τ ], Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 42 / 66

43 Synchronous machine Classical 8-dimensional model of the Synchronous Generator (SG) can be put into port-hamiltonian form SG ω τ θ ψ r p ψ s V s I s mechanical V f I f electrical power power 0 power excitation system Same model for synchronous motors. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 43 / 66

44 ψ s ψ r ṗ θ I s = I f = ω R s R r d H ψ s H ψ r H p H θ I 3 [ 0 3 ] I V s V f τ H ψ s H ψ r H p H θ r s 0 0 r f 0 0 where R s = 0 r s 0, R r = 0 r kd 0, d 0 0 r s 0 0 r kq are the stator resistances, rotor resistances, mechanical friction constants. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 44 / 66

45 ψ s R 3 are stator fluxes ψ r R 3 are rotor fluxes: field winding and two damper windings p is angular momentum of rotor θ is the angle of the rotor V s R 3,I s R 3 are the three-phase stator terminal voltages and currents V f,i f are the rotor field winding voltage and current τ,ω are the mechanical torque and angular velocity The Hamiltonian (total stored energy) is H(ψ s,ψ r,p,θ) = 1 2 [ ψ T s ψ T r where L(θ) is the 6 6 inductance matrix. ] ] L (θ)[ 1 ψs + 1 ψ 2J p2 r = magnetic energy + kinetic energy Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 45 / 66

46 In the round rotor case [ ] Lss L L(θ) = sr (θ) L T sr(θ) L rr where L aa L ab L ab L ffd L akd 0 L ss = L ab L aa L ab, L rr = L akd L kkd 0 L ab L ab L aa 0 0 L kkq while cosθ cosθ sinθ L afd 0 0 L sr (θ) = cos(θ 2π 3 ) cos(θ 2π 3 ) sin(θ 2π 3 ) 0 L akd 0 cos(θ + 2π 3 ) cos(θ + 2π 3 ) sin(θ+ 2π 3 ) 0 0 L akq Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 46 / 66

47 Outline 1 Review on classical Hamiltonian systems 2 From network interconnection to geometric structure 3 Port-Hamiltonian systems 4 Definition of port-hamiltonian systems 5 Examples from different physical domains 6 Dirac structures on manifolds 7 Conclusions and Outlook Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 47 / 66

48 Recall of the basic picture of port-hamiltonian systems Consider a Dirac structure D, linking the flow and effort variables f = (f S,f R,f P ), e = (e S,e R,e P ) storage e S D e R dissipation f S f R e P f P leading to the port-hamiltonian system ( ẋ(t),f R (t),f P (t), H x (x(t)),e R(t),e P (t)) D R(f R (t),e R (t)) = 0 t R Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 48 / 66

49 For many systems, especially those with 3-D mechanical components, the interconnection structure will be modulated by the energy or geometric variables. This leads to the notion of (non-constant) Dirac structures on manifolds. Definition Consider a smooth manifold X. A Dirac structure on X is a vector subbundle D TX T X such that for every x X the vector space is a Dirac structure as before. D(x) T x X T xx Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 49 / 66

50 Special cases (a) Let J be a Poisson structure on X, defining a skew-symmetric mapping J : T X TX. Then graph J T X TX is a Dirac structure. (b) Let ω be a (pre-)symplectic structure on X, defining a skew-symmetric mapping ω : TX T X. Then graph ω TX T X is a Dirac structure. (c) Let K be a constant-dimensional distribution on X, and let annk be its annihilating co-distribution. Then K annk TX T X is a Dirac structure. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 50 / 66

51 Mechanical systems with kinematic constraints Consider a mechanical system with n degrees of freedom. Kinematic constraints are constraints on the n-dimensional vector of generalized velocities q: A T (q) q = 0 with A(q) some n k matrix (k the number of kinematic constraints). This leads to constrained Hamiltonian equations q = H p (q,p) ṗ = H q (q,p)+a(q)λ 0 = A T (q) H p (q,p) with H(q, p) total energy, and A(q)λ the constraint forces. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 51 / 66

52 The resulting Dirac structure D on X = T Q is defined by the canonical Poisson structure on T Q together with the constraints A T (q) q = 0: D(q,p) = {(f S,e S ) T (q,p) X T(q,p) X λ Rk s.t. [ ] [ ] 0 In 0 f S = e S λ, [ 0 A T (q) ] e S = 0} I n 0 A(q) Or in more geometric form D(q,p) = {(f S,e S ) T (q,p) X T(q,p) X λ Rk s.t. [ ] 0 f S = J(q,p)e S + λ, [ 0 A T (q) ] e S = 0} A(q) with J the structure matrix of the canonical Poisson structure on T Q. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 52 / 66

53 Example (Rolling coin) y ϕ θ (x,y) x Figure: The geometry of the rolling peso Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 53 / 66

54 Example Let x,y be the Cartesian coordinates of the point of contact of the coin with the plane. Furthermore, ϕ denotes the heading angle, and θ the angle of the coin. The rolling constraints (rolling without slipping) are (set all parameters equal to 1) The total energy is ẋ = θcosϕ, ẏ = θsinϕ H = 1 2 p2 x p2 y p2 θ p2 ϕ and the constraints thus can be rewritten in the form A T (q) H p (q,p) = 0 as p x p θ cosϕ = 0, p y p θ sinϕ = 0. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 54 / 66

55 Now consider a Dirac structure on X F R F P, with X the state space manifold, and F R,F P linear spaces, which is independent of the position in F R F P. (Geometric definition can be given using symmetries of Dirac structures.) This leads to the definition of a port-hamiltonian system with state space manifold X as ( ẋ(t),f R (t),f P (t), H x (x(t)),e R(t),e P (t)) D(x(t)) R(f R (t),e R (t)) = 0 t R Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 55 / 66

56 Special case: Consider Dirac structure D on X F R F P given as graph of skew-symmetric map modulated by x X f S J(x) G R (x) G(x) e S f R = GR T(x) 0 0 e R, f P G T (x) 0 0 e P together with a linear resistive relation Writing out e R = Rf R, R = RT 0 ẋ = J(x) H x (x)+g R(x)f R +G(x)e P, f R = G T R (x) H x (x) and denoting R(x) = G R (x) RG R T (x) 0, this leads to ph systems ẋ = [J(x) R(x)] H x (x)+g(x)u y = G T (x) H x (x) with inputs u = e P and outputs y = f P. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 56 / 66

57 Integrability of Dirac structures There is an important notion of integrability of a Dirac structure on a manifold. A Dirac structure is integrable if there exist coordinates in which the Dirac structure is a constant Dirac structure. Theorem (Dorfman, Courant) A Dirac structure D on a manifold X is called integrable < L X1 α 2 X 3 > + < L X2 α 3 X 1 > + < L X3 α 1 X 2 >= 0 for all (X 1,α 1 ),(X 2,α 2 ),(X 3,α 3 ) D. Indeed, for constant Dirac structures the integrability condition is automatically satisfied. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 57 / 66

58 Excursion to generalized geometry This is strictly related to the Courant bracket on TX T X given as [[(X 1,α 1 ),(X 2,α 2 )]] = ([X 1,X 2 ],L X1 α 2 L X2 α d(α 1(X 2 ) α 2 (X 1 )) In fact, the Dirac structure is integrable if and only if the Courant bracket of any two elements in D is again in D. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 58 / 66

59 Integrability of the Dirac structure is equivalent to the existence of canonical coordinates: If the Dirac structure D on X is integrable then there exist coordinates (q,p,r,s) for X such that D(x) = {(f q,f p,f r,f s,e q,e p,e r,e s ) T x X T x X} f q = e p, f p = e q f r = 0, 0 = e s Hence the Hamiltonian system corresponding to D and H : X R is q = H p (q,p,r,s) ṗ = H q (q,p,r,s) r = 0 0 = H s (q,p,r,s) Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 59 / 66

60 Special cases of integrability First case Let the Dirac structure D be given for every x X as the graph of a skew-symmetric mapping J(x) from the co-tangent space Tx X to the tangent space T x X. Integrability in this case means that J(x) satisfies the conditions n l=1 [ J lj (x) J ik (x)+j li (x) J kj (x)+j lk (x) J ] ji (x) = 0, i,j,k = 1,...,n x l x l x l Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 60 / 66

61 In this case we may find, by Darboux s theorem around any point x 0 where the rank of the matrix J(x) is constant, local canonical coordinates x = (q,p,r) in which the matrix J(x) becomes the constant skew-symmetric matrix 0 I k 0 I k Then J(x) defines a Poisson bracket on X, given for every F,G : X R as {F,G} = T F x J(x) G x Indeed, by the integrability condition the Jacobi-identity holds: for all functions F,G,K. {F,{G,K}}+{G,{K,F}}+{K,{F,G}} = 0 Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 61 / 66

62 Second case A similar story holds for a Dirac structure given as the graph of a skew-symmetric mapping ω(x) from the tangent space T x X to the co-tangent space TxX. In this case the integrability conditions take the form ω ij (x)+ ω ki (x)+ ω jk (x) = 0, i,j,k = 1,...,n x k x j x i The skew-symmetric matrix ω(x) can be regarded as the coordinate representation of a differential two-form ω on X, that is ω = n i=1,j=1 dx i dx j, and the integrability condition corresponds to the closedness of this two-form (dω = 0). Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 62 / 66

63 The differential two-form ω is called a pre-symplectic structure, and a symplectic structure if the rank of ω(x) is equal to the dimension of X. By a version of Darboux s theorem we may find, around any point x 0 where the rank of the matrix ω(x) is constant, local coordinates x = (q,p,s) in which the matrix ω(x) becomes the constant skew-symmetric matrix 0 I k 0 I k Third case: Let K be a constant-dimensional distribution on X, and let ann K be its annihilating co-distribution. Then the Dirac structure K annk TX T X is integrable if and only if the distribution K is involutive. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 63 / 66

64 The Dirac structure corresponding to mechanical systems with kinematic constraints D(q,p) = {(f S,e S ) T (q,p) X T(q,p) X λ Rk s.t. [ ] 0 f S = J(q,p)e S + λ, [ 0 A T (q) ] e S = 0} A(q) is integrable if and only if the kinematic constraints A T (q) q = 0 are holonomic, which means that it is possible to find configuration coordinates q = (q 1,...,q n ) such that the constraints are equivalently expressed as q n k+1 = q n k+2 = = q n = 0, In this case one may eliminate the configuration variables q n k+1,...,q n, since the kinematic constraints are equivalent to the geometric constraints q n k+1 = c n k+1,...,q n = c n, for certain constants c n k+1,...,c n determined by the initial conditions. Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 64 / 66

65 Outline 1 Review on classical Hamiltonian systems 2 From network interconnection to geometric structure 3 Port-Hamiltonian systems 4 Definition of port-hamiltonian systems 5 Examples from different physical domains 6 Dirac structures on manifolds 7 Conclusions and Outlook Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 65 / 66

66 Paradigm shift from cotangent bundle to network structure Dirac structure determined by network structure Port-Hamiltonian systems Dirac structures on manifolds and integrability Arjan van der Schaft (Univ. of Groningen) Geometry of ph Systems 66 / 66