Port-Hamiltonian systems: network modeling and control of ...

then it follows that (4) with inputs u = τ and outputs y = ˙q is a passive (in fact, lossless)state space system with storage function H(q, p) − C ≥ 0 (see e.g. [61, 20, 47] for the generaltheory of passive and dissipative systems). Since the energy is only defined up to a constant,we may as well as take as potential energy the function P (q) − C ≥ 0, in which case the totalenergy H(q, p) becomes nonnegative and thus itself is the storage function.System (4) is an example of a Hamiltonian system with collocated inputs and outputs,which more generally is given in the following form˙q = ∂H∂p (q, p) , (q, p) = (q 1, . . . , q k , p 1 , . . . , p k )ṗ = − ∂H∂q (q, p) + B(q)u, u ∈ Rm , (8)y = B T (q) ∂H∂p (q, p) (= BT (q) ˙q), y ∈ R m ,Here B(q) is the input force matrix, with B(q)u denoting the generalized forces resultingfrom the control inputs u ∈ R m . The state space of (8) with local coordinates (q, p) is usuallycalled the phase space. In case m < k we speak of an underactuated system. If m = k and thematrix B(q) is everywhere invertible, then the Hamiltonian system is called fully actuated.Because of the form of the output equations y = B T (q) ˙q we again obtain the energybalancedHdt (q(t), p(t)) = uT (t)y(t) (9)Hence if H is non-negative (or, bounded from below), any Hamiltonian system (8) is a losslessstate space system. For a system-theoretic treatment of the Hamiltonian systems (8),especially if the output y can be written as the time-derivative of a vector of generalizedconfiguration coordinates, we refer to e.g. [8, 43, 44, 10, 36].A major generalization of the class of Hamiltonian systems (8) is to consider systemswhich are described in local coordinates asẋ = J(x) ∂H∂x(x) + g(x)u, x∈ X , u ∈ Rmy = g T (x) ∂H∂x (x), y ∈ Rm (10)Here J(x) is an n × n matrix with entries depending smoothly on x, which is assumed to beskew-symmetricJ(x) = −J T (x), (11)and x = (x 1 , . . . , x n ) are local coordinates for an n-dimensional state space manifold X .Because of (11) we easily recover the energy-balance dH dt (x(t)) = uT (t)y(t), showing that (10)is lossless if H ≥ 0. We call (10) with J satisfying (11) a port-Hamiltonian system withstructure matrix J(x) and Hamiltonian H ([24, 30, 25]). Note that (8) (and hence (4)) is aparticular case [ of (10) ] with x = (q, p), [ and]J(x) being given by the constant skew-symmetricmatrix J = 0 Ik−I k 0, and g(q, p) = 0B(q) .As an important mathematical note, we remark that in many examples the structurematrix J will satisfy the “integrability” conditionsn∑[J lj (x) ∂J ik(x) + J li (x) ∂J kj(x) + J lk (x) ∂J ]ji(x) = 0, i, j, k = 1, . . . , n (12)∂x l ∂x l ∂x ll=14

In this case we may find, by Darboux’s theorem (see e.g. [60]) around any pointx 0 where the rank of the matrix J(x) is constant, local coordinates ˜x = (q, p, s) =(q 1 , . . . , q k , p 1 , . . . , p k , s 1 , . . . s l ), with 2k the rank of J and n = 2k + l, such that J in thesecoordinates takes the form⎡J = ⎣0 I k 0−I k 0 00 0 0⎤⎦ (13)The coordinates (q, p, s) are called canonical coordinates, and J satisfying (11) and (12) iscalled a Poisson structure matrix. In such canonical coordinates the equations (10) take theform˙q = ∂H∂p (q, p, s) + g q(q, p, s)uṗ = − ∂H∂q (q, p, s) + g p(q, p, s)uṡ = g s (q, p, s)u (14)y = g T q(q, p, s)∂H∂q (q, p, s) + gT p (q, p, s)∂H∂p (q, p, s) + gT s(q, p, s)∂H (q, p, s)∂swhich is, apart from the appearance of the variables s, very close to the standard Hamiltonianform (8). In particular, if g s = 0, then the variables s are merely an additional set of constantparameters.Although traditionally Hamiltonian systems arise from the Euler-Lagrange equations ofmotion (which are usually derived from variational principles) the point of departure for thetheory of port-Hamiltonian systems is different. Indeed, port-Hamiltonian systems arise systematicallyfrom network models of physical systems. In network models of complex physicalsystems the overall system is seen as the interconnection of energy-storing elements via basicinterconnection (balance) laws as Newton’s third law or Kirchhoff’s laws, as well as powerconservingelements like transformers, kinematic pairs and ideal constraints, together withenergy-dissipating elements. The basic point of departure for the theory of port-Hamiltoniansystems is to formalize the basic interconnection laws together with the power-conservingelements by a geometric structure, and to define the Hamiltonian as the total energy storedin the system. Indeed, for the (restricted) form of port-Hamiltonian systems given above thestructure matrix J(x) and the input matrix g(x) may be directly associated with the networkinterconnection structure, while the Hamiltonian H is just the sum of the energies of all theenergy-storing elements; see our papers [30, 24, 32, 31, 50, 52, 27, 46, 58]. In particular,network models of complex physical systems formalized within the (generalized) bond graphlanguage ([41, 7]) can be shown to immediately lead to port-Hamiltonian systems; see e.g.[19].Example 2.1 (LCTG circuits). Consider a controlled LC-circuit (see Figure 1) consistingof two inductors with magnetic energies H 1 (ϕ 1 ), H 2 (ϕ 2 ) (ϕ 1 and ϕ 2 being the magnetic fluxlinkages), and a capacitor with electric energy H 3 (Q) (Q being the charge). If the elementsare linear then H 1 (ϕ 1 ) = 12L 1ϕ 2 1 , H 2(ϕ 2 ) = 12L 2ϕ 2 2 and H 3(Q) = 12C Q2 . Furthermore let V = udenote a voltage source. Using Kirchhoff’s laws one immediately arrives at the dynamical5

L 1 L 2Cϕ 1Qϕ 2VFigure 1: Controlled LC-circuitequations⎡ ⎤˙Q⎣ ˙ϕ 1⎦ =˙ϕ 2y = ∂H∂ϕ 1⎡⎡ ⎤0 1 −1⎣−1 0 0 ⎦ ⎢⎣1 0 0} {{ }J⎤∂H∂Q∂H ⎥∂ϕ 1∂H∂ϕ 2⎡ ⎤0⎦ + ⎣1⎦ u (15)0(= current through first inductor)with H(Q, ϕ 1 , ϕ 2 ) := H 1 (ϕ 1 )+H 2 (ϕ 2 )+H 3 (Q) the total energy. Clearly the matrix J is skewsymmetric,and since J is constant it trivially satisfies (12). In [31] it has been shown that inthis way every LC-circuit with independent elements can be modelled as a port-Hamiltoniansystem. Furthermore, also any LCTG-circuit with independent elements can be modelledas a port-Hamiltonian system, with J determined by Kirchhoff’s laws and the constitutiverelations of the transformers T and gyrators G.✷Example 2.2 (Actuated rigid body). Consider a rigid body spinning around its centerof mass in the absence of gravity. The energy variables are the three components of the bodyangular momentum p along the three principal axes: p = (p x , p y , p z ), and the energy is thekinetic energy()H(p) = 1 p 2 x+ p2 y+ p2 z,2 I x I y I zwhere I x , I y , I z are the principal moments of inertia. Euler’s equations describing the dynamicsare⎡ ⎤⎡ ⎤ ⎡⎤ ∂Hp˙x 0 −p z p y∂p x⎣p˙y⎦ = ⎣ p z 0 −p x⎦ ⎢ ∂H ⎥⎣ ∂p y ⎦ + g(p)u (16)p˙z −p y p x 0 ∂H} {{ } ∂p zJ(p)It can be checked that the skew-symmetric matrix J(p) satisfies (12). (In fact, J(p) is thecanonical Lie-Poisson structure matrix on the dual of the Lie algebra so(3) corresponding tothe configuration space SO(3) of the rigid body.) In the scalar input case the term g(p)u6

denotes the torque around an axis with coordinates g = (b x b y b z ) T , with correspondingcollocated output given asy = b xp xI x+ b yp yI y+ b zp zI z, (17)which is the velocity around the same axis (b x b y b z ) T .Example 2.3. A third important class of systems that naturally can be written as port-Hamiltonian systems, is constituted by mechanical systems with kinematic constraints. Consideras before a mechanical system with k degrees of freedom, locally described by k configurationvariables q = (q 1 , . . . , q k ). Suppose that there are constraints on the generalizedvelocities ˙q, described asA T (q) ˙q = 0, (18)with A(q) a r × k matrix of rank r everywhere (that is, there are r independent kinematicconstraints). Classically, the constraints (18) are called holonomic if it is possible to find newconfiguration coordinates q = (q 1 , . . . , q k ) such that the constraints are equivalently expressedas˙q k−r+1 = ˙q n−r+2 = · · · = ˙q k = 0 , (19)in which case one can eliminate the configuration variables q k−r+1 , . . . , q k , since the kinematicconstraints (19) are equivalent to the geometric constraintsq k−r+1 = c k−r+1 , . . . , q k = c k , (20)for certain constants c k−r+1 , . . . , c k determined by the initial conditions. Then the systemreduces to an unconstrained system in the remaining configuration coordinates (q 1 , . . . , q k−r ).If it is not possible to find coordinates q such that (19) holds (that is, if we are not able tointegrate the kinematic constraints as above), then the constraints are called nonholonomic.The equations of motion for the mechanical system with Lagrangian L(q, ˙q) and constraints(18) are given by the Euler-Lagrange equations [35]ddt( ∂L∂ ˙q)− ∂L∂q= A(q)λ + B(q)u, λ ∈ R r , u ∈ R mA T (q) ˙q = 0 (21)where B(q)u are the external forces (controls) applied to the system, for some k × m matrixB(q), while A(q)λ are the constraint forces. The Lagrange multipliers λ(t) are uniquelydetermined by the requirement that the constraints A T (q(t)) ˙q(t) = 0 have to be satisfied forall t.Defining as before (cf. (3)) the generalized momenta the constrained Euler-Lagrangeequations (21) transform into constrained Hamiltonian equations (compare with (8)),˙q = ∂H (q, p)∂pṗ = − ∂H (q, p) + A(q)λ + B(q)u∂qy = B T (q) ∂H (q, p)∂p(22)0 = A T (q) ∂H∂p (q, p) 7✷

with H(q, p) = 1 2 pT M −1 (q)p + P (q) the total energy. The constrained state space is thereforegiven as the following subset of the phase space:X c = {(q, p) | A T (q) ∂H (q, p) = 0} (23)∂pOne way of proceeding with these equations is to eliminate the constraint forces, and to reducethe equations of motion to the constrained state space. In [49] it has been shown that thisleads to a port-Hamiltonian system (10). Furthermore, the structure matrix J c of the port-Hamiltonian system satisfies the integrability conditions (12) if and only if the constraints(18) are holonomic. (In fact, if the constraints are holonomic then the coordinates s as in(13) can be taken to be equal to the “integrated constraint functions” q k−r+1 , . . . , q k of (20),and the matrix g s as in (14) is zero.)An alternative way of approaching the system (22) is to formalize it directly as an implicitport-Hamiltonian system, as will be sketched in Section 4.2.2 Basic properties of port-Hamiltonian systemsAs allude to above, port-Hamiltonian systems naturally arise from a network modeling ofphysical systems without dissipative elements, see our papers [24, 30, 25, 32, 31, 26, 50, 48,52, 27, 46]. Recall that a port-Hamiltonian system is defined by a state space manifold Xendowed with a triple (J, g, H). The pair (J(x), g(x)) , x ∈ X , captures the interconnectionstructure of the system, with g(x) modeling in particular the ports of the system. This isvery clear in Example 2.1, where the pair (J(x), g(x)) is determined by Kirchhoff’s laws, theparadigmatic example of a power-conserving interconnection structure, but it naturally holdsfor other physical systems without dissipation as well. Independently from the interconnectionstructure, the function H : X → R defines the total stored energy of the system. Furthermore,port-Hamiltonian systems are intrinsically modular in the sense that a power-conserving interconnectionof a number of port-Hamiltonian systems again defines a port-Hamiltonian system,with its overall interconnection structure determined by the interconnection structures of thecomposing individual systems together with their power-conserving interconnection, and theHamiltonian just the sum of the individual Hamiltonians (see [52, 46, 11]).As we have seen before, a basic property of port-Hamiltonian systems is the energybalancingproperty dH dt (x(t)) = uT (t)y(t). Physically this corresponds to the fact that theinternal interconnection structure is power-conserving (because of skew-symmetry of J(x)),while u and y are the power-variables of the ports defined by g(x), and thus u T y is theexternally supplied power.From the structure matrix J(x) of a port-Hamiltonian system one can directly extractuseful information about the dynamical properties of the system. Since the structure matrixis directly related to the modeling of the system (capturing the interconnection structure)this information usually has a direct physical interpretation.A very important property which may be directly inferred from the structure matrix is theexistence of dynamical invariants independent of the Hamiltonian H, called Casimir functions.Consider the set of p.d.e.’s∂ T C(x)J(x) = 0, x ∈ X , (24)∂x8

Example 2.5 (Example 2.2 continued). The quantity 1 2 p2 x + 1 2 p2 y + 1 2 p2 z (total angularmomentum) is a Casimir function.For a further discussion of the dynamical properties of Hamiltonian systems (especially ifJ satisfies the integrability conditions (12)) we refer to the extensive literature on this topic,see e.g. [1, 23].2.3 Port-Hamiltonian systems with dissipationEnergy-dissipation is included in the framework of port-Hamiltonian systems (10) by terminatingsome of the ports by resistive elements. Indeed, consider instead of g(x)u in (10) aterm[g(x) gR (x) ] [ ]u= g(x)u + gu R (x)u R (28)Rand extend correspondingly the output equations y = g T (x) ∂H∂x(x) to⎡[ ] g T (x) ∂Hy∂x (x) ⎤= ⎣⎦ (29)y RgR T ∂H(x)∂x (x)Here u R , y R ∈ R mr denote the power variables at the ports which are terminated by staticresistive elementsu R = −F (y R ) (30)where the resistive characteristic F : R mr → R mrsatisfiesy T RF (y R ) ≥ 0, y R ∈ R mr (31)(In many cases, F will be derivable from a so-called Rayleigh dissipation function R : R mr → Rin the sense that F (y R ) = ∂R∂y R(y R ).) In the sequel we concentrate on port-Hamiltonian systemswith ports terminated by linear resistive elementsu R = −Sy R (32)for some positive semi-definite symmetric matric S = S T ≥ 0. Substitution of (32) into (28)leads to a model of the formẋ = [J(x) − R(x)] ∂H∂x(x) + g(x)uy = g T (x) ∂H∂x (x) (33)where R(x) := g R (x)SgR T (x) is a positive semi-definite symmetric matrix, depending smoothlyon x. In this case the energy-balancing property (8) takes the formdHdt (x(t)) = uT (t)y(t) − ∂T H∂H∂x(x(t))R(x(t))∂x (x(t))≤u T (t)y(t).(34)showing that a port-Hamiltonian system is passive if the Hamiltonian H is bounded frombelow. We call (33) a port-Hamiltonian system with dissipation. Note that in this case two10

Furthermore, E is a voltage source. The dynamical equations of motion can be written asthe port-Hamiltonian system with dissipation⎡ ⎤⎡⎣˙qṗ˙Q⎤⎛⎡⎦ = ⎝⎣y 1 = ∂H∂p = ˙qy 2 = 1 ∂HR0 1 0−1 0 00 0 0⎤⎡⎦ − ⎣0 0 00 c 00 0 1/R⎤⎞⎦⎠⎢⎣(39)⎡ ⎤0⎡0⎤∂Q = I + ⎣ 1 ⎦ F + ⎣001/R⎦ Ewith p the momentum, R the resistance of the resistor, I the current through the voltagesource, and the Hamiltonian H being the total energyH(q, p, Q) = 12m p2 + 1 2 k(q − ¯q)2 + 12C(q) Q2 , (40)with ¯q denoting the equilibrium position of the spring. Note that F ˙q is the mechanical power,and EI the electrical power applied to the system. In the application as a microphone thevoltage over the resistor will be used (after amplification) as a measure for the mechanicalforce F .Example 2.7. ([Ortega et al. [39]]) A permanent magnet synchronous motor can be writtenas a port-Hamiltonian system with dissipation (in a rotating reference, i.e. the dq frame) forthe state vector⎡ ⎤⎡⎤i dL d 0 0x = M ⎣ i q⎦ , M = ⎣ 0 L q 0 ⎦ (41)ω0 0jn pthe magnetic flux linkages and mechanical momentum (i d , i q being the currents, and ω theangular velocity), L d , L q stator inductances, j the moment of inertia, and n p the numberof pole pairs. The Hamiltonian H(x) is given as H(x) = 1 2 xT M −1 x (total energy), whilefurthermore J(x), R(x) and g(x) are determined as⎡⎤0 L 0 x 3 0J(x) = ⎣ −L 0 x 3 0 −Φ q0⎦ ,0 Φ q0 0⎡⎤ ⎡R S 0 01 0 0R(x) = ⎣ 0 R S 0 ⎦ , g(x) = ⎣ 0 1 00 0 00 0 − 1n pwith R S the stator winding resistance, Φ q0 a constant term due to interaction of the permanentmagnet and the magnetic material in the stator, and L 0 := L d n p /j. The three inputs are thestator voltage (v d , v q ) T and the (constant) load torque. Outputs are i d , i q and ω.12⎤⎦∂H∂q∂H∂p∂H∂Q⎥⎦(42)

In some cases the interconnection structure J(x) may be actually varying, depending onthe mode of operation of the system, as exemplified by the following simple dc-to-dc powerconverter with a single switch. See for a further treatment of power converters in this context[39].Example 2.8. Consider the ideal boost converter given in Figure 3.Ls = 1Es = 0CRFigure 3: Ideal boost converterThe system equations are given as] [ẋ1ẋ 2=([ ] [ ]) [ ∂H0 −s 0 0 ∂x−1s 0 0 1/R ∂H∂x 2][ 1+ E0]y = ∂H∂x 1(43)with x 1 the magnetic flux linkage of the inductor, x 2 the charge of the capacitor, andH(x 1 , x 2 ) = 12L x2 1 + 12C x2 2 the total stored energy. The internal interconnection structure[ ] [ ]0 0 0 −1matrix J is either or, depending on the ideal switch being in position0 0 1 0s = 0 or s = 1.3 Control of port-Hamiltonian systemsThe aim of this section is to discuss a general methodology for port-Hamiltonian systems (withor without dissipation) which exploits their Hamiltonian properties in an intrinsic way, seee.g. [33, 34, 39, 47]. An expected benefit of such a methodology is that it leads to physicallyinterpretable controllers, which possess inherent robustness properties. Future research isaimed at corroborating these claims.We have already seen that port-Hamiltonian systems are passive if the Hamiltonian H isbounded from below. Hence in this case we can use all the results from the theory of passivesystems, such as asymptotic stabilization by the insertion of damping by negative output feedback,see e.g. [47]. The emphasis in this section is however on the somewhat complementaryaspect of shaping the energy of the system, which directly involves the Hamiltonian structureof the system, as opposed to the more general passivity structure.13

3.1 Control by interconnectionConsider a port-Hamiltonian system (33)P :ẋ = [J(x) − R(x)] ∂H∂x(x) + g(x)uy = g T (x) ∂H∂x (x) x ∈ X (44)regarded as a plant system to be controlled. In the previous section we have seen that manyphysical systems can be modelled in this way, and that the defining entities J(x), g(x), R(x)and H have a concrete physical interpretation. Furthermore, if H is bounded from belowthen (44) is a passive system.Recall the well-known result that the standard feedback interconnection of two passive systemsagain is a passive system; a basic fact which can be used for various stability and controlpurposes. In the same vein we can consider the interconnection of the plant (44) with anotherport-Hamiltonian systemC :˙ξ = [J C (ξ) − R C (ξ)] ∂H C∂ξ(ξ) + g C(ξ)u Cy C = gC T (ξ) ∂H C∂ξ (ξ) ξ ∈ X C (45)regarded as the controller system, via the standard feedback interconnectionu = −y C + eu C = y + e C(46)with e, e C external signals inserted in the feedback loop. The closed-loop system takes theform[ẋ ] [J(x) −g(x)gC T = ((ξ)] [ ]R(x) 0−)˙ξ g C (ξ)g T (x) J C (ξ) 0 R C (ξ)} {{ } } {{ }J cl (x,ξ)R cl (x,ξ)[ ∂H∂x (x)] [ ] [ ]g(x) 0 e(47)∂H C∂ξ (ξ) +0 g C (ξ) e C[ ] [ ] [y g(x) 0∂H∂x=(x) ]y ∂H C 0 g C (ξ) C∂ξ (ξ)which again is a port-Hamiltonian system, with state space given by the product space X ×X C ,total Hamiltonian H(x) + H C (ξ), inputs (e, e C ) and outputs (y, y C ). Hence the feedbackinterconnection of any two port-Hamiltonian systems results in another port-Hamiltoniansystem; just as in the case of passivity.It is of interest to investigate the Casimir functions of the closed-loop system, especiallythose relating the state variables ξ of the controller system to the state variables x of theplant system. Indeed, from a control point of view the Hamiltonian H is given while H Ccan be assigned. Thus if we can find Casimir functions C i (ξ, x), i = 1, · · · , r, relating ξ to xthen by the Energy-Casimir method the Hamiltonian H + H C of the closed-loop system maybe replaced by the Hamiltonian H + H C + H a (C 1 , · · · , C r ), thus creating the possibility of14

obtaining a suitable Lyapunov function for the closed-loop system.In particular, let us consider Casimir functions of the formξ i − G i (x) , i = 1, . . . , dim X C = n C (48)That means (see (35)) that we are looking for solutions of the p.d.e.’s (with e i denoting thei-th basis vector)[] ⎡ J(x) − R(x) −g(x)g− ∂T C T G (ξ) ⎤i∂x (x) eT ⎣⎦ = 0ig C (ξ)g T (x) J C (ξ) − R C (ξ)or written out∂ T G i∂x (x) [J(x) − R(x)] − gi C (ξ)gT (x) = 0∂ T G iwith ∂T G i∂x∂x (x)g(x)gT C (ξ) + J i C (ξ) − Ri C (ξ) = 0 (49)denoting as before the gradient vector(∂Gi∂x 1, . . . , ∂G i∂x n), and g i C , J i C , Ri C denotingthe i-th row of g C , J C , respectively R C .Suppose we want to solve (49) for i = 1, . . . , ¯n , with ¯n ≤ n C (possibly after permutationof ξ 1 , . . . , ξ nC ). with ¯J C (ξ), ¯R C (ξ) the ¯n × ¯n left-upper submatrices of J C , respectively R C .The following proposition has been shown in [47].Proposition 3.1. The functions ξ i − G i (x), i = 1, . . . , ¯n ≤ n C , satisfy (49) (and thus areCasimirs of the closed-loop port-controlled Hamiltonian system (47) for e = 0, e c = 0) if andonly if G = (G 1 , . . . , G¯n ) T satisfies∂ T G ∂G∂x(x)J(x)∂x (x) = ¯J C (ξ)R(x) ∂G∂x (x) = 0¯R C (ξ) = 0(50)∂ T G∂x (x)J(x) = ˜g C (ξ) g T (x)with ¯J C (ξ), ¯R C (ξ) the ¯n × ¯n left-upper submatrices of J C , respectively R C .In particular, we conclude that the functions ξ i −G i (x), i = 1, . . . ¯n, are Casimirs of (47) fore = 0, e C = 0, if and only if they are Casimirs for both the internal interconnection structureJ cl (x, ξ) as well as for the dissipation structure R cl (x, ξ). Hence, as in (38), it follows directlyhow the closed-loop port-controlled Hamiltonian system with dissipation (47) for e = 0, e C = 0reduces to a system any multi-level set {(x, ξ) |ξ i = G i (x) + c i , i = 1, . . . , ¯n}, by restrictingboth J cl and R cl to this multi-level set.Example 3.2. [56] Consider the “plant” system[ ] [ ] ⎡ ⎤∂H[ ]˙q 0 1∂q⎢ ⎥ 0=⎣ ⎦ + uṗ −1 0 ∂H 1⎡ ⎤ ∂py = [ 0 1 ] ∂H∂q⎢ ⎥⎣ ⎦∂H∂p15(51)

with q the position and p being the momentum of the mass m, in feedback interconnection(u = −y C + e, u C = y) with the controller system (see Figure 4)k cm ckmebFigure 4: Controlled mass⎡⎣˙∆q cṗ c˙∆q⎤⎡⎦ = ⎣0 1 0−1 −b 10 −1 0⎡⎤⎦⎢⎣∂H C∂∆q c∂H C∂p c⎤⎡+ ⎣⎥⎦001⎤⎦ u C(52)y C = ∂H C∂∆q∂H C∂∆qwhere ∆q c is the displacement of the spring k c , ∆q is the displacement of the spring k, and p cis the momentum of the mass m c . The plant Hamiltonian is H(p) = 12m p2 , and the controllerHamiltonian is given as H C (∆q c , p c , ∆q) = 1 2 ( p2 cm c+ k(∆q) 2 + k c (∆q c ) 2 ). The variable b > 0is the damping constant, and e is an external force. The closed-loop system possesses theCasimir functionC(q, ∆q c , ∆q) = ∆q − (q − ∆q c ), (53)implying that along the solutions of the closed-loop system∆q = q − ∆q c + c (54)with c a constant depending on the initial conditions. With the help of LaSalle’s Invarianceprinciple it can be shown that restricted to the invariant manifolds (54) the system is asymptoticallystable for the equilibria q = ∆q c = p = p c = 0.✷Let us next consider the special case ¯n = n C , in which case we wish to relate all thecontroller state variables ξ 1 , . . . , ξ nC to the plant state variables x via Casimir functionsξ 1 − G 1 (x), . . . , ξ nC − G nC (x). Denoting G = (G 1 , . . . , G nC ) T this means that G should satisfy(see (50))∂ T G ∂G∂x(x)J(x)∂x (x) = J C (ξ)R(x) ∂G∂x (x) = 0 = R C(ξ)(55)∂ T G∂x (x)J(x) = g C (ξ) g T (x)In this case the reduced dynamics on any multi-level setL C = {(x, ξ)|ξ i = G i (x) + c i , i = 1, . . . n C } (56)16

can be immediately recognized. Indeed, the x-coordinates also serve as coordinates for L C .Furthermore, the x-dynamics of (47) with e = 0, e C = 0 is given asẋ = [J(x) − R(x)] ∂H∂x (x) − g(x)gT C (ξ)∂H C(ξ). (57)∂ξUsing the second and the third equality of (55) this can be rewritten as( )∂H ∂Gẋ = [J(x) − R(x)] (x) +∂x ∂x (x)∂H C∂ξ (ξ) . (58)and by the chain-rule property for differentiation this reduces to the port-Hamiltonian systemẋ = [J(x) − R(x)] ∂H s(x), (59)∂xwith the same interconnection and dissipation structure as before, but with shaped HamiltonianH s given byH s (x) = H(x) + H C (G(x) + c). (60)An interpretation of the shaped Hamiltonian H s in terms of energy-balancing is the following.Since R C (ξ) = 0 by (55) the controller Hamiltonian H C satisfies dH Cdt= u T C y C. Hence alongany multi-level set L C given by (56), invariant for the closed loop port-Hamiltonian system(47) for e = 0, e C = 0dH sdt= dH dt + dH Cdt= dH dt − uT y (61)since u = −y C and u C = y. Therefore, up to a constant,H s (x(t)) = H(x(t)) −∫ t0u T (τ)y(τ)dτ, (62)and the shaped Hamiltonian H s is the original Hamiltonian H minus the energy suppliedto the plant system (44) by the controller system (45) (modulo a constant; depending on theinitial states of the plant and controller).Remark 3.3. Note that from a stability analysis point of view (62) can be regarded asan effective way of generating candidate Lyapunov functions H s from the Hamiltonian H.(Compare with the classical construction of Lur’e functions.)Example 3.4. A mechanical system with damping and actuated by external forces u ∈ R mis described as a port-Hamiltonian system[ ˙q=ṗ]([ ] [ ] [0 I k 0 0 ) ∂H] [ ]∂q 0−+ u−I k 0 0 D(q)B(q)(63)y = B T (q) ∂H∂p∂H∂pwith x = [ q p ], where q ∈ R k are the generalized configuration coordinates, p ∈ R k thegeneralized momenta, and D(q) = D T (q) ≥ 0 is the damping matrix. In most cases theHamiltonian H(q, p) takes the formH(q, p) = 1 2 pT M −1 (q)p + P (q) (64)17

where M(q) = M T (q) > 0 is the generalized inertia matrix,12 pT M −1 (q)p = 1 2 ˙qT M(q) ˙q isthe kinetic energy, and P (q) is the potential energy of the system. Now consider a generalport-Hamiltonian controller system (45), with state space R m . Then the equations (55) forG = (G 1 (q, p), . . . , G m (q, p)) T take the form∂ T G ∂G∂q∂p − ∂T G∂p∂G∂q = J C (ξ)D(q) ∂G∂p = 0(65)∂ T G∂por equivalentlyJ C = 0,= 0, ∂T G∂q= g C (ξ)B T (q)∂G∂p = 0,gT C(ξ)B(q) = ∂G (q) (66)∂qNow let g C (ξ) be the m × m identity matrix. Then there exists a solution G =(G 1 (q), . . . , G m (q)) to (66) if and only if the columns of the input force matrix B(q) satisfythe integrability conditions∂B il∂q j(q) = ∂B jl∂q i(q), i, j = 1, . . . k, l = 1, . . . m (67)Hence, if B(q) satisfies (67), then the closed-loop port-Hamiltonian system (47) for the controllersystem (45) with J C = 0 admits Casimirs ξ i − G i (q), i = 1, . . . , m, leading to a reducedport-Hamiltonian system[ [ ] [ ]˙q 0 Ik 0 0= ( − )ṗ]−I k 0 0 D(q)y = B T (q) ∂Hs∂pfor the shaped Hamiltonian⎡⎢⎣∂H s∂q∂H s∂p⎤⎥⎦ +[ ] 0eB(q)H s (q, p) = H(q, p) + H C (G 1 (q) + c 1 , . . . , G m (q) + c m ) (69)If H(q, p) is as given in (64), thenH s (q, p) = 1 2 pT M −1 (q)p + [P (q) + H C (G 1 (q) + c 1 , · · · , G m (q) + c m )] (70)(68)and the control amounts to shaping the potential energy of the system, see [59, 38].✷3.2 Passivity-based control of port-Hamiltonian systemsIn the previous section we have seen how under certain conditions the feedback interconnectionof a port-Hamiltonian system having Hamiltonian H (the “plant”) with another port-Hamiltonian system with Hamiltonian H C (the “controller”) leads to a reduced dynamicsgiven by (see (59))ẋ = [J(x) − R(x)] ∂H s(x) (71)∂x18

for the shaped Hamiltonian H s (x) = H(x) + H C (G(x) + c), with G(x) a solution of (55).From a state feedback point of view the dynamics (71) could have been directly obtained bya state feedback u = α(x) such thatg(x)α(x) = [J(x) − R(x)] ∂H C(G(x) + c) (72)∂xIndeed, such an α(x) is given in explicit form asα(x) = −gC(G(x) T + c) ∂H C(G(x) + c) (73)∂ξA state feedback u = α(x) satisfying (72) is customarily called a passivity-based control law,since it is based on the passivity properties of the original plant system (44) and transforms(44) into another passive system with shaped storage function (in this case H s ).Seen from this perspective we have shown in the previous section that the passivity-basedstate feedback u = α(x) satisfying (72) can be derived from the interconnection of the port-Hamiltonian plant system (44) with a port-Hamiltonian controller system (45). This fact hassome favorable consequences. Indeed, it implies that the passivity-based control law definedby (72) can be equivalently generated as the feedback interconnection of the passive system(44) with another passive system (45). In particular, this implies an inherent invarianceproperty of the controlled system: the plant system (71), the controller system (60), as wellas any other passive system interconnected to (71) in a power-conserving fashion, may changein any way as long as they remain passive, and for any perturbation of this kind the controlledsystem will remain stable.The implementation of the resulting passivity-based control u = α(x) is a somewhat complexissue. In cases of analog controller design the interconnection of the plant port-Hamiltoniansystem (44) with the port-Hamiltonian controller system (45) seems to be the logical option.Furthermore, in general it may be favorable to avoid an explicit state feedback, but insteadto use the dynamic output feedback controller (45). On the other hand, in some applicationsthe measurement of the passive output y may pose some problems, while the state feedbacku = α(x) is in fact easier to implement, as illustrated in the next example.Example 3.5 (Example 3.4 continued). The passivity-based control u = α(x) resultingfrom (66) is given by (assuming g C to be the identity matrix and B(q) to satisfy (67))u i = − ∂H C∂ξ i(G(q) + c), i = 1, . . . m (74)This follows from (73), and can be directly checked by substituting (74) into (63) and using theequality B(q) = ∂G∂q(q). Comparing the implementation of the state feedback controller (74)with the implementation of the port-Hamiltonian controller system based on the measurementofy = B T (q) ∂H∂p (q, p) = BT (q) ˙q (75)one may note that the measurement of the generalized velocities (75) is in some cases (forexample in a robotics context) more problematic than the measurement of the generalizedpositions (74).✷19

Remark 3.6. On the other hand, the control of the port-Hamiltonian plant system (44) byinterconnection with the port-Hamiltonian controller system (45) allows for the possibility ofinserting an asymptotically stabilizing damping not directly in the plant but instead in thecontroller system, cf. Example 3.2.In the rest of this section we concentrate on the passivity-based (state feedback) controlu = α(x). The purpose is to more systematically indicate how a port-Hamiltonian system withdissipation (44) may be asymptotically stabilized around a desired equilibrium x ∗ in two steps:I Shape by passivity-based control the Hamiltonian in such a way that it has a strictminimum at x = x ∗ . Then x ∗ is a (marginally) stable equilibrium of the controlled system.II Add damping to the system in such a way that x ∗ becomes an asymptotically stableequilibrium of the controlled system.As before, we shall concentrate on Step I. Therefore, let us consider a port-Hamiltoniansystem with dissipation (44) with X the n-dimensional state space manifold. Suppose wewish to stabilize the system around a desired equilibrium x ∗ , assigning a closed-loop energyfunction H d (x) to the system which has a strict minimum at x ∗ (that is, H d (x) > H d (x ∗ ) forall x ≠ x ∗ in a neighbourhood of x ∗ ). DenoteH d (x) = H(x) + H a (x), (76)where the to be defined function H a is the energy added to the system (by the control action).We have the followingProposition 3.7. [39, 34] Assume we can find a feedback u = α(x) and a vector functionK(x) satisfyingsuch that[J(x) − R(x)] K(x) = g(x)α(x) (77)(i)∂K i∂x j(x) = ∂K j∂x i(x),i, j = 1, . . . , n(ii) K(x ∗ ) = − ∂H∂x (x∗ )(78)(iii)∂K∂x (x∗ ) > − ∂2 H(x ∗ )∂x 2with ∂K∂x the n×n matrix with i-th column given by ∂K i∂x (x), and ∂2 H(x ∗ ) denoting the Hessian∂x 2matrix of H at x ∗ . Then the closed-loop system is a Hamiltonian system with dissipationẋ = [J(x) − R(x)] ∂H d(x) (79)∂xwhere H d is given by (76), with H a such thatK(x) = ∂H a(x) (80)∂xFurthermore, x ∗ is a stable equilibrium of (79).20

A further generalization is to use state feedback in order to change the interconnectionstructure and the resistive structure of the plant system, and thereby to create more flexibilityto shape the storage function for the (modified) port-controlled Hamiltonian systemto a desired form. This methodology has been called Interconnection-Damping AssignmentPassivity-Based Control (IDA-PBC) in [39, 40], and has been succesfully applied to a numberof applications. The method is especially attractive if the newly assigned interconnectionand resistive structures are judiciously chosen on the basis of physical considerations, andrepresent some “ideal” interconnection and resistive structures for the physical plant. For anextensive treatment of IDA-PBC we refer to [39, 40].4 Implicit port-Hamiltonian systemsFrom a general modeling point of view physical systems are, at least in first instance, oftendescribed as DAE’s, that is, a mixed set of differential and algebraic equations. This stemsfrom the fact that in many modeling approaches the system under consideration is naturallyregarded as obtained from interconnecting simpler sub-systems. These interconnections ingeneral, give rise to algebraic constraints between the state space variables of the sub-systems;thus leading to implicit systems. While in the linear case one may argue that it is oftenrelatively straightforward to eliminate the algebraic constraints, and thus to reduce the systemto an explicit form, in the nonlinear case such a conversion from implicit to explicit form isusually fraught with difficulties. Indeed, if the algebraic constraints are nonlinear they neednot be analytically solvable (locally or globally). More importantly perhaps, even if they areanalytically solvable, then often one would prefer not to eliminate the algebraic constraints,because of the complicated and physically not easily interpretable expressions for the reducedsystem which may arise.Therefore it is important to extend the framework of port-Hamiltonian systems, assketched in the previous sections, to the context of implicit systems. In order to give thedefinition of an implicit port-Hamiltonian system (with dissipation) we first consider the notionof a Dirac structure, formalizing the concept of a power-conserving interconnection, andgeneralizing the notion of a structure matrix J(x) as encountered before.4.1 Power-conserving interconnectionsLet us return to the basic setting of passivity, starting with a finite-dimensional linear spaceand its dual, in order to define power. Thus, let F be an l-dimensional linear space, anddenote its dual (the space of linear functions on F) by F ∗ . The product space F × F ∗ isconsidered to be the space of power variables, with power defined byP =< f ∗ |f >, (f, f ∗ ) ∈ F × F ∗ , (81)where < f ∗ |f > denotes the duality product, that is, the linear function f ∗ ∈ F ∗ acting onf ∈ F. Often we call F the space of flows f, and F ∗ the space of efforts e, with the powerof an element (f, e) ∈ F × F ∗ denoted as < e|f >.Remark 4.1. If F is endowed with an inner product structure , then F ∗ can be naturallyidentified with F in such a way that < e|f >=< e, f >, f ∈ F, e ∈ F ∗ ≃ F.21

Example 4.2. Let F be the space of generalized velocities, and F ∗ be the space of generalizedforces, then < e|f > is mechanical power. Similarly, let F be the space of currents, and F ∗be the space of voltages, then < e|f > is electrical power.There exists on F × F ∗ a canonically defined symmetric bilinear form< (f 1 , e 1 ), (f 2 , e 2 ) > F×F ∗:=< e 1 |f 2 > + < e 2 |f 1 > (82)for f i ∈ F, e i ∈ F ∗ , i = 1, 2. Now consider a linear subspaceS ⊂ F × F ∗ (83)and its orthogonal complement with respect to the bilinear form F×F ∗ on F ×F ∗ , denotedasS ⊥ ⊂ F × F ∗ . (84)Clearly, if S has dimension d, then the subspace S ⊥ has dimension 2l − d.(F × F ∗ ) = 2l, and F×F ∗ is a non-degenerate form.)(Since dimDefinition 4.3. [9, 12, 11] A constant Dirac structure on F is a linear subspace D ⊂ F × F ∗such thatD = D ⊥ (85)It immediately follows that the dimension of any Dirac structure D on an l-dimensionallinear space is equal to l. Furthermore, let (f, e) ∈ D = D ⊥ . Then by (82)0 =< (f, e), (f, e) > F×F ∗= 2 < e|f > . (86)Thus for all (f, e) ∈ D we obtain< e | f >= 0. (87)Hence a Dirac structure D on F defines a power-conserving relation between the powervariables (f, e) ∈ F × F ∗ .Remark 4.4. The condition dim D = dim F is intimately related to the usually expressedstatement that a physical interconnection can not determine at the same time both the flowand effort (e.g. current and voltage, or velocity and force).Constant Dirac structures admit different matrix representations. Here we just list anumber of them, without giving proofs and algorithms to convert one representation intoanother, see e.g. [11].Let D ⊂ F × F ∗ , with dim F = l, be a constant Dirac structure. Then D can be representedas1. (Kernel and Image representation, [11, 50]).D = {(f, e) ∈ F × F ∗ |F f + Ee = 0} (88)for l × l matrices F and E satisfying(i) EF T + F E T = 0(ii)rank [F .E] = l(89)22

Equivalently,D = {(f, e) ∈ F × F ∗ |f = E T λ, e = F T λ, λ ∈ R l } (90)2. (Constrained input-output representation, [11]).D = {(f, e) ∈ F × F ∗ |f = Je + Gλ, G T e = 0} (91)for an l × l skew-symmetric matrix J, and a matrix G such that ImG ={f|(f, 0) ∈ D}. Furthermore, KerJ = {e|(0, e) ∈ D}.3. (Hybrid input-output representation, [6]).Let D be given as in (88). Suppose rank F = l 1 (≤ l). Select l 1 independent columns ofF , and group them into a matrix F 1 . Write[(possibly] [after]permutations) F = [F 1 .F 2 ]fand, correspondingly E = [E 1 .E 2 1 e1], f =f 2 , e =e 2 . Then the matrix [F 1 .E 2 ]can be shown to be invertible, andD ={( ) ( ) ∣( ) ( )}f1 e1 ∣∣∣ f1 e1f 2 ,e 2 e 2 = Jf 2] −1 ]with J := −[F 1 .E[F 2 2 .E 1 skew-symmetric.4. (Canonical coordinate representation, [9]).There exist linear coordinates (q, p, r, s) for F such in these coordinates and dual coordinatesfor F ∗ , (f, e) = (f q , f p , f r , f s , e q , e p , e r , e s ) ∈ D if and only if⎧⎨⎩f q = e p , f p = −e qf r = 0, e s = 0Example 4.5. Kirchhoff’s laws are an example of (88), taking F the space of currents andF ∗ the space of voltages.Given a Dirac structure D on F, the following subspaces of F, respectively F ∗ , are of importance(92)(93)G 1 := {f ∈ F | ∃e ∈ F ∗ s.t. (f, e) ∈ D}P 1 := {e ∈ F ∗ | ∃f ∈ F s.t. (f, e) ∈ D}(94)The subspace G 1 expresses the set of admissible flows, and P 1 the set of admissible efforts. Itfollows from the image representation (90) thatG 1 = Im E TP 1 = Im F T (95)23

4.2 Implicit port-Hamiltonian systemsFrom a network modeling perspective a (lumped-parameter) physical system is naturallydescribed by a set of (possibly multi-dimensional) energy-storing elements, a set of energydissipatingor resistive elements, and a set of ports (by which interaction with the environmentcan take place), interconnected to each other by a power-conserving interconnection, see Figure5.resistiveelementsenergystoringelementspowerconservinginterconnectionportsFigure 5: Implicit port-Hamiltonian system with dissipationHere the power-conserving interconnection also includes power-conserving elements like(in the electrical domain) transformers, gyrators, or (in the mechanical domain) transformers,kinematic pairs and kinematic constraints.Associated with the energy-storing elements are energy-variables x 1 , · · · , x n , being coordinatesfor some n-dimensional state space manifold X , and a total energy H : X → R. The powerconservinginterconnection is formalized in first instance (see later on for the non-constantcase) by a constant Dirac structure D on the finite-dimensional linear space F := F S ×F R ×F P ,with F S denoting the space of flows f S connected to the energy-storing elements, F R denotingthe space of flows f R connected to the dissipative (resistive) elements, and F P the spaceof external flows f P which can be connected to the environment. Dually, we write F ∗ =FS ∗ × F R ∗ × F P ∗ , with e S ∈ FS ∗ the efforts connected to the energy-storing elements, e R ∈ FR∗the efforts connected to the resistive elements, and e P ∈ FP ∗ the efforts to be connected tothe environment of the system.The flow variables of the energy-storing elements are given as ẋ(t) = dxdteffort variables of the energy-storing elements as ∂H(t), t ∈ R, and the∂H∂x(x(t)) (implying that =dHdt(x(t)) is the increase in energy). In order to have a consistent sign convention for energyflow we putf S = −ẋe S = ∂H∂x (x) (96)Similarly, restricting to linear resistive elements as in (32), the flow and effort variablesconnected to the resistive elements are related asf R = −Re R (97)24

for some matrix R = R T ≥ 0.Substitution of (96) and (97) into the Dirac structure D leads to the following geometricdescription of the dynamics(f S = −ẋ, f R = −Re R , f P , e S = ∂H∂x (x), e R, e P ) ∈ D (98)We call (98) an implicit port-Hamiltonian system (with dissipation), defined with respect tothe constant Dirac structure D, the Hamiltonian H and the resistive structure R.An equational representation of an implicit port-Hamiltonian system is obtained by takinga matrix representation of the Dirac structure D as discussed in the previous subsection. Forexample, in kernel representation the Dirac structure on F = F S × F R × F P may be given asD = {(f S , f R , f P , e S , e R , e P ) |F S f S + E S e S + F R f R + E R e R + F P f P + E P e P = 0}(99)for certain matrices F S , E S , F R , E R , F P , E P satisfying(i) E S F T S + F SE T S + E RF T R + F RE T R + E P F T P + F P E T P = 0(ii)rank][F S .F R .F P .E S .E R .E P = dim F(100)Then substitution of (96) and (97) into (99) yields the following set of differential-algebraicequations for the implicit port-Hamiltonian systemF S ẋ(t) = E S∂H∂x (x(t)) − F RRe R + E R e R + F P f P + E P e P , (101)Different representations of the Dirac structure D lead to different representations of theimplicit port-Hamiltonian system, and this freedom may be exploited for simulation andanalysis.Actually, for many purposes this definition of port-Hamiltonian system is not generalenough, since often the Dirac structure is not constant, but modulated by the state variablesx. In this case the matrices F S , E S , F R , E R , F P , E P in the kernel representation depend(smoothly) on x, leading to the implicit port-Hamiltonian systemF S (x(t))ẋ(t) = E S (x(t)) ∂H∂x (x(t)) − F R(x(t))Re R (t)+E R (x(t))e R (t) + F P (x(t))f P (t) + E P (x(t))e P (t), t ∈ R(102)withE S (x)F T S (x) + F S(x)E T S (x) + E R(x)F T R (x) + F R(x)E T R (x)+ E P (x)FP T (x) + F P (x)EP T (x) = 0, ∀x ∈ X[]rank F S (x).F R (x).F P (x).E S (x).E R (x).E P (x) = dim F(103)25

Remark 4.6. Strictly speaking the flow and effort variables ẋ(t) = −f S (t), respectively∂H∂x (x(t)) = e S(t), are not living in a constant linear space F S , respectively FS ∗ , but instead inthe tangent spaces T x(t) X , respectively co-tangent spaces Tx(t) ∗ X , to the state space manifoldX . This is formalized in the definition of a non-constant Dirac structure on a manifold; seethe references [9, 12, 11, 47].It can be checked that the definition of a port-Hamiltonian system as given in (33) is a specialcase of (102), see [47]. By the power-conservation property of a Dirac structure (cf. (87)) itfollows directly that any implicit port-Hamiltonian system satisfies the energy-balancedHdt∂H(x(t)) = == −e T R (t)Re R(t) + e T P (t)f P (t),(104)as was the case for an (explicit) port-Hamiltonian system (33).The algebraic constraints that are present in the implicit system (102) are expressed bythe subspace P 1 , and the Hamiltonian H. In fact, since the Dirac structure D is modulatedby the x-variables, also the subspace P 1 is modulated by the x-variables, and thus the effortvariables e S , e R and e P necessarily satisfy(e S , e R , e P ) ∈ P 1 (x), x ∈ X , (105)or, because of (95),e S ∈ Im FS T (x), e R ∈ Im FR T (x), e P ∈ Im FP T (x). (106)The second and third inclusions entail the expression of e R and e P in terms of the othervariables, while the first inclusion determines, since e S = ∂H∂x(x), the following algebraicconstraints on the state variables∂H∂x (x) ∈ Im F S T (x). (107)Remark 4.7. Under certain non-degeneracy conditions the elimination of the algebraic constraints(107) for an implicit port-Hamiltonian system (98) can be shown to result in anexplicit port-Hamiltonian system.The Casimir functions C : X → R of the implicit system (102) are determined by the subspaceG 1 (x). Indeed, necessarily (f S , f R , f P ) ∈ G 1 (x), and thus by (95)f S ∈ Im E T S (x), f R ∈ Im E T R(x), f P ∈ Im E T P (x). (108)Since f S = ẋ(t), the first inclusion yields the flow constraintsẋ(t) ∈ Im E T S (x(t)), t ∈ R. (109)dCThus C : X → R is a Casimir function ifdt (x(t)) = ∂T C∂x(x(t))ẋ(t) = 0 for all ẋ(t) ∈Im ES T (x(t)). Hence C : X → R is a Casimir of the implicit port-Hamiltonian system (98) ifit satisfies the set of p.d.e.’s∂C∂x (x) ∈ Ker E S(x) (110)26

Remark 4.8. Note that C : X → R satisfying (110) is a Casimir function of (98) in a strongsense: it is a dynamical invariant ( dCdt (x(t)) = 0) for every port behavior and every resistiverelation (97).Example 4.9. [11, 51] The constrained Hamiltonian equations (22) can be viewed as animplicit port-Hamiltonian system, with respect to the Dirac structure D, given in constrainedinput-output representation (91) byD = {(f S , f P , e S , e P )|0 = A T (q)e S , e P = B T (q)e S ,−f S =[ 0 Ik−I k 0] [e S +0A(q)] [λ +0B(q)]f P , λ ∈ R r }(111)In this case, the algebraic constraints on the state variables (q, p) are given as0 = A T (q) ∂H (q, p) (112)∂pwhile the Casimir functions C are determined by the equations∂ T C∂q (q) ˙q = 0, for all ˙q satisfying AT (q) ˙q = 0. (113)Hence, finding Casimir functions amounts to integrating the kinematic constraints A T (q) ˙q =0. In particular, if the kinematic constraints are holonomic, and thus can be expressed as(19), then ¯q k−r+1 , · · · , ¯q k generate all the Casimir functions. ✷Remark 4.10. For a proper notion of integrability of non-constant Dirac structures, generalizingthe integrability conditions (12) of the structure matrix J(x), we refer e.g. to [11].In principle, the theory presented before for explicit port-Hamiltonian systems can be directlyextended, mutatis mutandis, to implicit port-Hamiltonian system. In particular, the standardfeedback interconnection of an implicit port-Hamiltonian system P with port variables f P , e P(the “plant”) with another implicit port-Hamiltonian system with port variables f C P , eC P (the“controller”) is readily seen to result in a closed-loop implicit port-Hamiltonian system withport variables. Furthermore, as in the explicit case, the Hamiltonian of this closed-loop systemis just the sum of the Hamiltonian of the plant port-Hamiltonian system and the Hamiltonianof the controller port-Hamiltonian system. Finally, the Casimir analysis for the closed-loopsystem can be performed along the same lines as before.5 Distributed-parameter port-Hamiltonian systemsFrom a modeling and control point of view it is very desirable to be able to include distributedparametercomponents into the Hamiltonian description of complex physical systems. However,in extending the Hamiltonian theory as for instance exposed in [37] to distributedparametercontrol systems a fundamental difficulty arises in the treatment of boundary conditions.Indeed, the treatment of infinite-dimensional Hamiltonian systems in the literatureseems mostly focussed on systems with infinite spatial domain, where the variables go to zerofor the spatial variables tending to infinity, or on systems with boundary conditions such thatthe energy exchange through the boundary is zero. On the other hand, from a control and27

interconnection point of view it is essential to be able to describe a distributed-parametersystem with varying boundary conditions inducing energy exchange through the boundary,since in many applications interaction with the environment (e.g. actuation or measurement)takes place through the boundary of the system. Clear examples are the telegraph equations(describing the dynamics of a transmission line), where the boundary of the system is describedby the behavior of the voltages and currents at both ends of the transmission line,or a vibrating string (or, more generally, a flexible beam), where it is natural to consider theevolution of the forces and velocities at the ends of the string. Furthermore, in both examplesit is obvious that in general the boundary exchange of power (voltage times current in thetransmission line example, and force times velocity for the vibrating string) will be non-zero,and that in fact one would like to consider the voltages and currents or forces and velocitiesas additional boundary variables of the system, which can be interconnected to other systems.Also for numerical integration and simulation of complex distibuted-parameter systems itis essential to be able to describe the complex system as the interconnection or coupling ofits subsystems via their boundary variables; for example in the case of coupled fluid-soliddynamics.From a mathematical point of view, it is not obvious how to incorporate non-zero energyflow through the boundary in the existing Hamiltonian framework for distributed-parametersystems. The problem is already illustrated by the Hamiltonian formulation of e.g. theKorteweg-de Vries equation (see e.g. [37]). Here for zero boundary conditions a Poissonbracket can be formulated with the use of the differential operator , since by integrationby parts this operator is obviously skew-symmetric. However, for boundary conditions correspondingto non-zero energy flow the differential operator is not skew-symmetric anymore(since after integrating by parts the remainders are not zero ).In [54, 28, 29] we have provided a framework to overcome this fundamental problem byusing the notion of an infinite-dimensional Dirac structure. The infinite-dimensional Diracstructure employed in these papers has a specific form by being defined on certain spaces ofdifferential forms on the spatial domain of the system and its boundary, and making use ofStokes’ theorem. Its construction emphasizes the geometrical content of the physical variablesinvolved, by identifying them as differential k-forms, for appropriate k. This framework hasbeen used [54] for a port-Hamiltonian representation of the telegrapher’s equations of an idealtransmission line, Maxwell’s equations on a bounded domain with non-zero Poynting vectorat its boundary, a vibrating string with traction forces at its ends, and planar beam models[17, 18]. Furthermore the framework has been extended to cover Euler’s equations for an idealfluid on a domain with permeable boundary, see also [53].Throughout, let Z be an n-dimensional smooth manifold with smooth (n−1)-dimensionalboundary ∂Z, representing the space of spatial variables.Denote by Ω k (Z), k = 0, 1, · · · , n, the space of exterior k-forms on Z, and by Ω k (∂Z), k =0, 1, · · · , n − 1, the space of k-forms on ∂Z. (Note that Ω 0 (Z), respectively Ω 0 (∂Z), is thespace of smooth functions on Z, respectively ∂Z.) Clearly, Ω k (Z) and Ω k (∂Z) are (infinitedimensional)linear spaces (over R). Furthermore, there is a natural pairing between Ω k (Z)and Ω n−k (Z) given by∫< β|α >:=Zβ ∧ α (∈ R) (114)with α ∈ Ω k (Z), β ∈ Ω n−k (Z), where ∧ is the usual wedge product of differential formsyielding the n-form β ∧ α. In fact, the pairing (114) is non-degenerate in the sense that if28ddx

β|α >= 0 for all α, respectively for all β, then β = 0, respectively α = 0.Similarly, there is a pairing between Ω k (∂Z) and Ω n−1−k (∂Z) given by∫< β|α >:= β ∧ α (115)∂Zwith α ∈ Ω k (∂Z), β ∈ Ω n−1−k (∂Z). Now let us define the linear spaceF p,q := Ω p (Z) × Ω q (Z) × Ω n−p (∂Z), (116)for any pair p, q of positive integers satisfyingp + q = n + 1, (117)and correspondingly let us defineE p,q := Ω n−p (Z) × Ω n−q (Z) × Ω n−q (∂Z). (118)Then the pairing (114) and (115) yields a (non-degenerate) pairing between F p,q and E p,q(note that by (117) (n − p) + (n − q) = n − 1). As before, symmetrization of this pairingyields the following bilinear form on F p,q × E p,q with values in R:≪ ( f 1 p , f 1 q , f 1 b , e1 p, e 1 q, e 1 b),(f2p , f 2 q , f 2 b , e2 p, e 2 q, e 2 b)≫:=∫Zwhere for i = 1, 2[e1p ∧ f 2 p + e 1 q ∧ f 2 q + e 2 p ∧ f 1 p + e 2 q ∧ f 1 q]+∫f i p ∈ Ωp (Z), f i q ∈ Ωq (Z)e i p ∈ Ω n−p (Z), e i p ∈ Ω n−q (Z)∂Z[e1b∧ fb 2 + e2 b ∧ f b1 ](119)(120)f i b ∈ Ωn−p (∂Z), e i b ∈ Ωn−q (∂Z)The spaces of differential forms Ω p (Z) and Ω q (Z) will represent the energy variables of twodifferent physical energy domains interacting with each other, while Ω n−p (∂Z) and Ω n−q (∂Z)will denote the boundary variables whose (wedge) product represents the boundary energyflow. For example, in Maxwell’s equations (Example 5.4) we will have n = 3 and p = q = 2;with Ω p (Z) = Ω 2 (Z), respectively Ω q (Z) = Ω 2 (Z), being the space of electric field inductions,respectively magnetic field inductions, and Ω n−p (∂Z) = Ω 1 (∂Z) denoting the electric andmagnetic field intensities at the boundary, with product the Poynting vector.Theorem 5.1. Consider F p,q and E p,q given in (116), (118) with p, q satisfying (117), andbilinear form ≪, ≫ given by (119). Define the following linear subspace D of F p,q × E p,qD = {(f p , f q , f b , e p , e q , e b ) ∈ F p,q × E p,q |[ ] [fp 0 (−1)=r ] [ ]d ep,f q d 0 e q[ ] [ ] [ ]fb 1 0 ep|∂Z=e b 0 −(−1) n−q } (121)e q|∂Zwhere | ∂Z denotes restriction to the boundary ∂Z, and r := pq + 1. Then D = D ⊥ , that is,D is a Dirac structure.29

The proof of Theorem 5.1 relies on Stokes’ theorem, and the infinite-dimensional Diracstructure defined in Theorem 5.1 is therefore called a Stokes-Dirac structure.The definition of a distributed-parameter Hamiltonian system with respect to a Stokes-Dirac structure can now be stated as follows. Let Z be an n-dimensional manifold withboundary ∂Z, and let D be a Stokes-Dirac structure. Consider furthermore a Hamiltoniandensity (energy per volume element)H : Ω p (Z) × Ω q (Z) × Z → Ω n (Z) (122)resulting in the total energy∫H := H ∈ R (123)ZRecall, see (114), that there exists a non-degenerate pairing between Ω p (Z) and Ω n−p (Z),respectively between Ω q (Z) and Ω n−q (Z). This means that Ω n−p (Z) and Ω n−q (Z) can beregarded as dual spaces to Ω p (Z), respectively Ω q (Z) (although strictly contained in theirfunctional analytic duals). Let now α p , ∂α p ∈ Ω p (Z), α q , ∂α q ∈ Ω q (Z). Then under weaksmoothness conditions on H∫H(α p + ∂α p , α q + ∂α q ) = H (α p + ∂α p , α q + ∂α q , z) =∫Z∫H (α p , α q , z) + [δ p H ∧ ∂α p + δ q H ∧ ∂α q ] + higher order terms in ∂α p , ∂α q (124)Zfor certain differential formsδ p H ∈ Ω n−p (Z)δ q H ∈ Ω n−q (Z)Z(125)Furthermore, from the non-degeneracity of the pairing between Ω p (Z) and Ω n−p (Z), respectivelybetween Ω q (Z) and Ω n−q (Z), it immediately follows that these differential forms areuniquely determined. This means that (δ p H, δ q H) ∈ Ω n−p (Z) × Ω n−q (Z) can be regardedas the (partial) variational derivatives (see e.g. [37]) of H at (α p , α q ) ∈ Ω p (Z) × Ω q (Z).Throughout this paper we shall assume that the Hamiltonian H admits variational derivativessatisfying (124).Now consider a time-function(α p (t), α q (t)) ∈ Ω p (Z) × Ω q (Z), t ∈ R, (126)and the Hamiltonian H(α p (t), α q (t)) evaluated along this trajectory. It follows that at anytime t[dHdt∫Z= δ p H ∧ ∂α p∂t + δ qH ∧ ∂α ]q(127)∂tThe differential forms ∂αp∂trepresent the generalized velocities of the energy variablesα p , α q . They are connected to the Stokes-Dirac structure D by setting∂t , ∂αqf p = − ∂αp∂tf q = − ∂αq∂t(128)30

(again the minus sign is included to have a consistent energy flow description). Since theright-hand side of (127) is the rate of increase of the stored energy H, we sete p = δ p He q = δ q H(129)Definition 5.2. The distributed-parameter port-Hamiltonian system with n-dimensionalmanifold of spatial variables Z, state space Ω p (Z) × Ω q (Z) (with p + q = n + 1), Stokes-Dirac structure D given by (121), and Hamiltonian H, is given as (with r = pq + 1)[− ∂αp∂t− ∂αq∂t][fbe b]==[ 0 (−1) r dd 0] [δp Hδ q H[ ] [ ]1 0 δp H| ∂Z0 −(−1) n−q δ q H| ∂Z](130)By the power-conserving property of any Dirac structure it immediately follows that forany (f p , f q , f b , e p , e q , e b ) in the Stokes-Dirac structure D∫∫[e p ∧ f p + e q ∧ f q ] + e b ∧ f b = 0 (131)Z∂ZHence by substitution of (128), (129) and using (127) we obtainProposition 5.3. Consider the distributed parameter port-Hamiltonian system (130). ThendHdt∫∂Z= e b ∧ f b , (132)expressing that the increase in energy on the domain Z is equal to the power supplied to thesystem through the boundary ∂Z.The system (130) can be called a (nonlinear) boundary control system in the sense of e.g.[14]. Indeed, we could interpret f b as the boundary control inputs to the system, and e b asthe measured outputs (or the other way around).Example 5.4 (Maxwell’s equations). Let Z ⊂ R 3 be a 3-dimensional manifold withboundary ∂Z, defining the spatial domain, and consider the electromagnetic field in Z. Theenergy variables are the electric field induction 2-form α p = D ∈ Ω 2 (Z):D = 1 2 D ij(t, z)dz i ∧ dz j (133)and the magnetic field induction 2-form α q = B ∈ Ω 2 (Z) :B = 1 2 B ij(t, z)dz i ∧ dz j (134)The corresponding Stokes-Dirac structure (n = 3, p = 2, q = 2) is given as (cf. (121))[ ] [ ] [ ] [ ] [ ] [ ]fp 0 −d ep fb 1 0 ep|∂Z=, =f q d 0 e q e b 0 1 e q|∂Z(135)31

Usually in this case one does not start with the definition of the total energy (Hamiltonian)H, but instead with the co-energy variables δ p H, δ q H, given, respectively, as the electric fieldintensity E ∈ Ω 1 (Z) :E = E i (t, z)dz i (136)and the magnetic field intensity H ∈ Ω 1 (Z) :H = H i (t, z)dz i (137)They are related to the energy variables through the constitutive relations of the medium (ormaterial equations)∗D = ɛE∗B = µH(138)with the scalar functions ɛ(t, z) and µ(t, z) denoting the electric permittivity, respectivelymagnetic permeability, and ∗ denoting the Hodge star operator (corresponding to a Riemannianmetric on Z), converting 2-forms into 1-forms. Then one defines the Hamiltonian Has∫1H = (E ∧ D + H ∧ B), (139)2Zand one immediately verifies that δ p H = E, δ q H = H.Nevertheless there are other cases (corresponding to a nonlinear theory of the electromagneticfield, such as the Born-Infeld theory, see e.g. [21]) where one starts with a more generalHamiltonian H = ∫ Zh, with the energy density h(D, B) being a more general expressionthan 1 2 (ɛ−1 ∗ D ∧ D + µ −1 ∗ B ∧ B).Assuming that there is no current in the medium Maxwell’s equations can now be writtenas (see [21])∂D∂t= dH∂B∂t= −dE(140)Explicitly taking into account the behavior at the boundary, Maxwell’s equations on a domainZ ⊂ R 3 are then represented as the port-Hamiltonian system with respect to the Stokes-Diracstructure given by (135), as[ −∂D∂t− ∂B∂t][fbe b]==[ 0 −dd 0[ ]δD H| ∂Zδ B H| ∂Z] [δD Hδ B H](141)Note that the first line of (140) is nothing else than (the differential version of) Ampère’s law,while the second line of (140) is Faraday’s law. Hence the Stokes-Dirac structure in (140),(141) expresses the basic physical laws connecting D, B, H and E.The energy-balance (132) in the case of Maxwell’s equations takes the formdHdt∫∂Z= δ B H ∧ δ D H =∫∂Z∫H ∧ E = −with E ∧ H a 2-form corresponding to the Poynting vector (see [21]).32∂ZE ∧ H (142)

Example 5.5 (Vibrating string). Consider an elastic string subject to traction forces atits ends. The spatial variable z belongs to the interval Z = [0, 1] ⊂ R. Let us denote byu(t, z) the displacement of the string. The elastic potential energy is a function of the straingiven by the 1-formα q (t) = ɛ(t, z)dz = ∂u (t, z)dz (143)∂zThe associated co-energy variable is the stress given by the 0-formσ = T ∗ α q (144)with T the elasticity modulus and ∗ the Hodge star operator. Hence the potential energy isthe quadratic functionU(α q ) =∫ 10σα q =∫ 10T ∗ α q ∧ α q =∫ 10T( ) ∂u 2dz (145)∂zand σ = δ q U.The kinetic energy K is a function of the kinetic momentum defined as the 1-formα p (t) = p(t, z)dz (146)given by the quadratic functionK(α p ) =∫ 10p 2dz (147)µThe associated co-energy variable is the velocity given by the 0-formv = 1 µ ∗ α p = δ p K (148)In this case the Dirac structure is the Stokes-Dirac structure for n = p = q = 1, with anopposite sign convention leading to the equations (with H := U + K)[ ] [ ] [ ]− ∂αp∂t0 −d δp H=− ∂αq −d 0 δ∂tq H[fbe b]=[ 1 00 1or, in more down-to-earth notation∂p∂t= ∂σ∂z = ∂ ∂z(T ɛ)( )∂ɛ∂t= ∂v∂z = ∂ 1∂z µ p] [ ]δp H| ∂Zδ q H| ∂Z(149)(150)f b = v| {0,1}33

with boundary variables the velocity and stress at the ends of the ( string. ) Of course, bysubstituting ɛ = ∂u∂zinto the 2nd equation of (150) one obtains ∂ ∂u∂z ∂t − p µ= 0, implyingthatp = µ ∂u∂t+ µf(t) (151)for some function f, which may be set to zero. Substitution of (151) into the first equationof (150) then yields the wave equationµ ∂2 u∂t 2 = ∂ (T ∂u )∂z ∂zThis framework can be extended to general beam models, see [17, 18].(152)6 Conclusions and future researchThe theory presented in this paper opens up the way for many analysis, simulation andcontrol problems. Its potential for set-point regulation has already received some attention(see [33, 34, 39, 40, 47]), while the extension to tracking problems is wide open. In this contextwe also like to refer to some recent work concerned with the shaping of the Lagrangian, see e.g.[5]. Also, the control of mechanical systems with nonholonomic kinematic constraints can befruitfully approached from this point of view, see e.g. [15], as well as the modelling and controlof multi-body systems, see [32, 27, 58]. The framework of port-Hamiltonian systems seemsperfectly suited to theoretical investigations on the topic of impedance control; see already[56] for some initial results in this direction. Furthermore, the connection with multi-modal(hybrid) systems, corresponding to port-Hamiltonian systems with varying interconnectionstructure, needs further investigations. Some applications of the framework of distributedparameterport-Hamiltonian systems have been already reported in [53, 42]; see also [55] forrelated work on smart structures. Finally, current research is concerned with the spatialdiscretization of port-Hamiltonian distributed-parameter systems to finite-dimensional port-Hamiltonian systems [16], with direct applications towards simulation and control of suchsystems.References[1] R.A. Abraham & J.E. Marsden, Foundations of Mechanics (2nd edition), Reading, MA:Benjamin/Cummings, 1978.[2] V. I. Arnold, B.A. Khesin, Topological Methods in Hydrodynamics, Springer Verlag, AppliedMathematical Sciences 125, New York, 1998.[3] G. Blankenstein, Implicit Hamiltonian Systems; Symmetry and Interconnection, PhDthesis, University of Twente, The Netherlands, November 2000.[4] G. Blankenstein, A.J. van der Schaft, “Symmetry and reduction in implicit generalizedHamiltonian systems”, Rep. Math. Phys., 47, pp. 57–100, 2001.34

[5] A. Bloch, N. Leonard & J.E. Marsden, “Matching and stabilization by the method ofcontrolled Lagrangians”, in Proc. 37th IEEE Conf. on Decision and Control, Tampa, FL,pp. 1446-1451, 1998.[6] A.M. Bloch & P.E. Crouch, “Representations of Dirac structures on vector spaces andnonlinear LC circuits”, Proc. Symposia in Pure Mathematics, Differential Geometry andControl Theory, G. Ferreyra, R. Gardner, H. Hermes, H. Sussmann, eds., Vol. 64, pp.103-117, AMS, 1999.[7] P.C. Breedveld, Physical systems theory in terms of bond graphs, PhD thesis, Universityof Twente, Faculty of Electrical Engineering, 1984[8] R.W. Brockett, “Control theory and analytical mechanics”, in Geometric Control Theory,(eds. C. Martin, R. Hermann), Vol. VII of Lie Groups: History, Frontiers andApplications, Math. Sci. Press, Brookline, pp. 1-46, 1977.[9] T.J. Courant, “Dirac manifolds”, Trans. American Math. Soc., 319, pp. 631-661, 1990.[10] P.E. Crouch & A.J. van der Schaft, Variational and Hamiltonian Control Systems, Lect.Notes in Control and Inf. Sciences 101, Springer-Verlag, Berlin, 1987.[11] M. Dalsmo & A.J. van der Schaft, “On representations and integrability of mathematicalstructures in energy-conserving physical systems”, SIAM J. Control and Optimization,37, pp. 54-91, 1999.[12] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, JohnWiley, Chichester, 1993.[13] G. Escobar, A.J. van der Schaft & R. Ortega, “A Hamiltonian viewpoint in the modellingof switching power converters”, Automatica, Special Issue on Hybrid Systems, 35, pp.445-452, 1999.[14] H.O. Fattorini, ”Boundary control systems”, SIAM J. Control, 6, pp. 349-385, 1968.[15] K. Fujimoto, T. Sugie, “Stabilization of a class of Hamiltonian systems with nonholonomicconstraints via canonical transformations”, Proc. European Control Conference’99, Karlsruhe, 31 August - 3 September 1999.[16] G. Golo, V. Talasila, A.J. van der Schaft, “Approximation of the telegrapher’s equations”,Proc. 41st IEEE Conf. Decision and Control, Las Vegas, Nevada, December 2002.[17] G. Golo, V. Talasila, A.J. van der Schaft, “A Hamiltonian formulation of the Timoshenkobeam”, Mechatronics 2002, pp. 838–847, Enschede, 24-26 June 2002.[18] G. Golo, A.J. van der Schaft, S.Stramigioli, “Hamiltonian formulation of planar beams”,Proceedings 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for NonlinearControl, Eds. A. Astolfi, F. Gordillo, A.J. van der Schaft, pp. 169–174, Sevilla, 2003.[19] G. Golo, A. van der Schaft, P.C. Breedveld, B.M. Maschke, “Hamiltonian formulation ofbond graphs”, Nonlinear and Hybrid Systems in Automotive Control Eds. R. Johansson,A. Rantzer, pp. 351–372, Springer London, 2003.35

[20] D.J. Hill & P.J. Moylan, “Stability of nonlinear dissipative systems,” IEEE Trans. Aut.Contr., AC-21, pp. 708-711, 1976.[21] R.S. Ingarden, A. Jamiolkowski, Classical Electrodynamics, PWN-Polish Sc. Publ.,Warszawa, Elsevier, 1985.[22] A. Isidori, Nonlinear Control Systems (2nd Edition), Communications and Control EngineeringSeries, Springer-Verlag, London, 1989, 3rd Edition, 1995.[23] J.E. Marsden & T.S. Ratiu, Introduction to Mechanics and Symmetry, Texts in AppliedMathematics 17, Springer-Verlag, New York, 1994.[24] B.M. Maschke & A.J. van der Schaft, “Port-controlled Hamiltonian systems: Modellingorigins and system-theoretic properties”, in Proc. 2nd IFAC NOLCOS, Bordeaux, pp.282-288, 1992.[25] B.M. Maschke & A.J. van der Schaft, “System-theoretic properties of port-controlledHamiltonian systems”, in Systems and Networks: Mathematical Theory and Applications,Vol. II, Akademie-Verlag, Berlin, pp. 349-352, 1994.[26] B.M. Maschke & A.J. van der Schaft, “Interconnection of systems: the networkparadigm”, in Proc. 35th IEEE Conf. on Decision and Control, Kobe, Japan, pp. 207-212,1996.[27] B.M. Maschke & A.J. van der Schaft, “Interconnected Mechanical Systems, Part II:The Dynamics of Spatial Mechanical Networks”, in Modelling and Control of MechanicalSystems, (eds. A. Astolfi, D.J.N. Limebeer, C. Melchiorri, A. Tornambe, R.B. Vinter),pp. 17-30, Imperial College Press, London, 1997.[28] B.M. Maschke, A.J. van der Schaft, “Port controlled Hamiltonian representation of distributedparameter systems”, Proc. IFAC Workshop on Lagrangian and Hamiltonianmethods for nonlinear control, Princeton University, Editors N.E. Leonard, R. Ortega,pp.28-38, 2000.[29] B.M. Maschke, A.J. van der Schaft, “Hamiltonian representation of distributed parametersystems with boundary energy flow”, Nonlinear Control in the Year 2000. Eds. A. Isidori,F. Lamnabhi-Lagarrigue, W. Respondek, Lect. Notes Control and Inf. Sciences, vol. 258,Springer-Verlag, pp. 137-142, 2000.[30] B.M. Maschke, A.J. van der Schaft & P.C. Breedveld, “An intrinsic Hamiltonian formulationof network dynamics: non-standard Poisson structures and gyrators”, J. FranklinInstitute, vol. 329, no.5, pp. 923-966, 1992.[31] B.M. Maschke, A.J. van der Schaft & P.C. Breedveld, “An intrinsic Hamiltonian formulationof the dynamics of LC-circuits, IEEE Trans. Circ. and Syst., CAS-42, pp. 73-82,1995.[32] B.M. Maschke, C. Bidard & A.J. van der Schaft, “Screw-vector bond graphs for thekinestatic and dynamic modeling of multibody systems”, in Proc. ASME Int. Mech.Engg. Congress, 55-2, Chicago, U.S.A., pp. 637-644, 1994.36

[33] B.M. Maschke, R. Ortega & A.J. van der Schaft, “Energy-based Lyapunov functionsfor forced Hamiltonian systems with dissipation”, in Proc. 37th IEEE Conference onDecision and Control, Tampa, FL, pp. 3599-3604, 1998.[34] B.M. Maschke, R. Ortega, A.J. van der Schaft & G. Escobar, “An energy-based derivationof Lyapunov functions for forced systems with application to stabilizing control”, in Proc.14th IFAC World Congress, Beijing, Vol. E, pp. 409-414, 1999.[35] J.I. Neimark & N.A. Fufaev, Dynamics of Nonholonomic Systems, Vol. 33 of Translationsof Mathematical Monographs, American Mathematical Society, Providence, RhodeIsland, 1972.[36] H. Nijmeijer & A.J. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990.[37] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, secondedition, 1993.[38] R. Ortega, A. Loria, P.J. Nicklasson & H. Sira-Ramirez, Passivity-based Control of Euler-Lagrange Systems, Springer-Verlag, London, 1998.[39] R. Ortega, A.J. van der Schaft, B.M. Maschke & G. Escobar, “Interconnection anddamping assignment passivity-based control of port-controlled Hamiltonian systems”,Automatica, vol. 38, pp. 585–596, 2002.[40] R. Ortega, A.J. van der Schaft, I. Mareels, & B.M. Maschke, “Putting energy back incontrol”, Control Systems Magazine, 21, pp. 18–33, 2001.[41] H. M. Paynter, Analysis and design of engineering systems, M.I.T. Press, MA, 1960.[42] H. Rodriguez, A.J. van der Schaft, R. Ortega, “On stabilization of nonlinear distributedparameter port-controlled Hamiltonian systems via energy-shaping”, Proc. 40th IEEEConf. on Decision and Control, Orlando, Florida, pp.131–136, 2001.[43] A.J. van der Schaft, System theoretic properties of physical systems, CWI Tract 3, CWI,Amsterdam, 1984.[44] A.J. van der Schaft, “Stabilization of Hamiltonian systems”, Nonl. An. Th. Math. Appl.,10, pp. 1021-1035, 1986.[45] A.J. van der Schaft, “Implicit Hamiltonian systems with symmetry”,Rep. Math. Phys.,41, pp. 203–221, 1998.[46] A.J. van der Schaft, “Interconnection and geometry”, in The Mathematics of Systemsand Control, From Intelligent Control to Behavioral Systems (eds. J.W. Polderman, H.L.Trentelman), Groningen, 1999.[47] A.J. van der Schaft, L 2 -Gain and Passivity Techniques in Nonlinear Control, 2nd revisedand enlarged edition, Springer-Verlag, Springer Communications and Control Engineeringseries, p. xvi+249, London, 2000 (first edition Lect. Notes in Control and Inf.Sciences, vol. 218, Springer-Verlag, Berlin, 1996).37

[48] A.J. van der Schaft, M. Dalsmo & B.M. Maschke, “Mathematical structures in the networkrepresentation of energy-conserving physical systems”, in Proc. 35th IEEE Conf.on Decision and Control, Kobe, Japan, pp. 201-206, 1996.[49] A.J. van der Schaft & B.M. Maschke, “On the Hamiltonian formulation of nonholonomicmechanical systems”, Rep. Math. Phys., 34, pp. 225-233, 1994.[50] A.J. van der Schaft & B.M. Maschke, “The Hamiltonian formulation of energy conservingphysical systems with external ports”, Archiv für Elektronik und Übertragungstechnik, 49,pp. 362-371, 1995.[51] A.J. van der Schaft & B.M. Maschke, “Mathematical modeling of constrained Hamiltoniansystems”, in Proc. 3rd IFAC NOLCOS ’95, Tahoe City, CA, pp. 678-683, 1995.[52] A.J. van der Schaft & B.M. Maschke, “Interconnected Mechanica5a Systems, Part I: Geometryof Interconnection and implicit Hamiltonian Systems”, in Modelling and Controlof Mechanical Systems, (eds. A. Astolfi, D.J.N. Limebeer, C. Melchiorri, A. Tornambe,R.B. Vinter), pp. 1-15, Imperial College Press, London, 1997.[53] A.J. van der Schaft, B.M. Maschke, “Fluid dynamical systems as Hamiltonian boundarycontrol systems”, Proc. 40th IEEE Conf. on Decision and Control, Orlando, Florida,pp.4497–4502, 2001.[54] A.J. van der Schaft, B.M. Maschke “Hamiltonian representation of distributed parametersystems with boundary energy flow”, Journal of Geometry and Physics, vol.42, pp.166-194, 2002.[55] K. Schlacher, A. Kugi, “Control of mechanical structures by piezoelectric actuatorsand sensors”. In Stability and Stabilization of Nonlinear Systems, eds. D. Aeyels, F.Lamnabhi-Lagarrigue, A.J. van der Schaft, Lecture Notes in Control and InformationSciences, vol. 246, pp. 275-292, Springer-Verlag, London, 1999.[56] S. Stramigioli, From Differentiable Manifolds to Interactive Robot Control, PhD Dissertation,University of Delft, Dec. 1998.[57] S. Stramigioli, B.M. Maschke & A.J. van der Schaft, “Passive output feedback and portinterconnection”, in Proc. 4th IFAC NOLCOS, Enschede, pp. 613-618, 1998.[58] S. Stramigioli, B.M. Maschke, C. Bidard, “A Hamiltonian formulation of the dynamicsof spatial mechanism using Lie groups and screw theory”, to appear in Proc. SymposiumCommemorating the Legacy, Work and Life of Sir R.S. Ball, J. Duffy and H. Lipkinorganizers, July 9-11, 2000, University of Cambridge, Trinity College, Cambridge, U.K..[59] M. Takegaki & S. Arimoto, “A new feedback method for dynamic control of manipulators”,Trans. ASME, J. Dyn. Systems, Meas. Control, 103, pp. 119-125, 1981.[60] A. Weinstein, “The local structure of Poisson manifolds”, J. Differential Geometry, 18,pp. 523-557, 1983.[61] J.C. Willems, “Dissipative dynamical systems - Part I: General Theory”, Archive forRational Mechanics and Analysis, 45, pp. 321-351, 1972.38