Port-Hamiltonian systems: an introductory survey

Transcription

1 Port-Hamiltonian systems: an introuctory survey Arjan van er Schaft Abstract. The theory of port-hamiltonian systems provies a framework for the geometric escription of network moels of physical systems. It turns out that port-base network moels of physical systems immeiately len themselves to a Hamiltonian escription. While the usual geometric approach to Hamiltonian systems is base on the canonical symplectic structure of the phase space or on a Poisson structure that is obtaine by (symmetry) reuction of the phase space, in the case of a port-hamiltonian system the geometric structure erives from the interconnection of its sub-systems. This motivates to consier Dirac structures instea of Poisson structures, since this notion enables one to efine Hamiltonian systems with algebraic constraints. As a result, any power-conserving interconnection of port-hamiltonian systems again efines a port-hamiltonian system. The port-hamiltonian escription offers a systematic framework for analysis, control an simulation of complex physical systems, for lumpe-parameter as well as for istribute-parameter moels. Mathematics Subject Classification (2000). Primary 93A30,70H05,70H45,70Q05, 70G45, 93B29, 37J60; Seconary 93C10, 93C15, 93C20, 37K05. Keywors. Interconnection, Dirac structures, constraine systems, Hamiltonian DAEs, stabilization, bounary control, conservation laws. 1. Introuction Historically, the Hamiltonian approach has its roots in analytical mechanics an starts from the principle of least action, an procees, via the Euler-Lagrange equations an the Legenre transform, towars the Hamiltonian equations of motion. On the other han, the network approach stems from electrical engineering, an constitutes a cornerstone of mathematical systems theory. While most of the analysis of physical systems has been performe within the Lagrangian an Hamiltonian framework, the network point of view is prevailing in moelling an simulation of (complex) physical engineering systems. The framework of port-hamiltonian systems combines both points of view, by associating with the interconnection structure of the network moel a geometric structure given by a (pseuo-) Poisson structure, or more generally a Dirac structure. The Hamiltonian ynamics is then efine with respect to this Dirac structure an the Hamiltonian given by the total store energy. Furthermore, port-hamiltonian systems are open ynamical systems, which interact with their environment through ports. Re- Proceeings of the International Congress of Mathematicians, Mari, Spain, European Mathematical Society

2 2 Arjan van er Schaft sistive effects are inclue by terminating some of these ports on energy-issipating elements. Dirac structures encompass the geometric structures which are classically being use in the geometrization of mechanics (that is, Poisson structures an pre-symplectic structures), an allow to escribe the geometric structure of ynamical systems with algebraic constraints. Furthermore, Dirac structures allow to exten the Hamiltonian escription of istribute-parameter systems to inclue variable bounary conitions, leaing to istribute-parameter port-hamiltonian systems with bounary ports. Acknowlegements. This survey is base on joint work with several co-authors. In particular I thank Bernhar Maschke an Romeo Ortega for fruitful collaborations. 2. Finite-imensional port-hamiltonian systems In this section we recapitulate the basics of finite-imensional port-hamiltonian systems. For more etails we refer e.g. to [19], [17], [20], [33], [34], [30], [36], [12], [5] From classical Hamiltonian equations to port-hamiltonian systems. The stanar Hamiltonian equations for a mechanical system are given as q = H (q, p), p ṗ = H (q, p) + F q (1) where the Hamiltonian H(q,p)is the total energy of the system, q = (q 1,...,q k ) T are generalize configuration coorinates for the mechanical system with k egrees of freeom, p = (p 1,...,p k ) T is the vector of generalize momenta, an the input F is the vector of external generalize forces. The state space of (1) with local coorinates (q, p) is calle the phase space. One immeiately erives the following energy balance: t H = T H q (q, p) q + T H p (q, p)ṗ = T H p (q, p)f = qt F, (2) expressing that the increase in energy of the system is equal to the supplie work (conservation of energy). This motivates to efine the output of the system as e = q (the vector of generalize velocities).

3 Port-Hamiltonian systems: an introuctory survey 3 System (1) is more generally 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)f, f Rm, (3) e = B T (q) H p (q, p) (= BT (q) q), e R m, with B(q)f enoting the generalize forces resulting from the input f R m. In case m<kwe speak of an uneractuate system. Similarly to (2) we obtain the energy balance H t (q(t), p(t)) = et (t)f (t). (4) A further generalization is to consier systems which are escribe in local coorinates as ẋ = J(x) H x (x) + g(x)f, x X, f Rm, e = g T (x) H (5) x (x), e Rm, where J(x) is an n n matrix with entries epening smoothly on x, which is assume to be skew-symmetric, that is J(x) = J T (x), an x = (x 1,...,x n ) are local coorinates for an n-imensional state space manifol X (not necessarily evenimensional as above). Because of skew-symmetry of J we easily recover the energybalance H t (x(t)) = et (t)f (t). We call (5) a port-hamiltonian system with structure matrix J(x), input matrix g(x), an Hamiltonian H ([17], [19], [18]). Remark 2.1. In many examples the structure matrix J will aitionally satisfy an integrability conition (the Jacobi-ientity) allowing us to fin by Darboux s theorem canonical coorinates. In this case J is the structure matrix of a Poisson structure on X. Example 2.2. An important class of systems that naturally can be written as port- Hamiltonian systems, is constitute by mechanical systems with kinematic constraints [22]. Consier a mechanical system locally escribe by k configuration variables q = (q 1,...,q k ). Suppose that there are constraints on the generalize velocities q, escribe as A T (q) q = 0, (6) with A(q) an r k matrix of rank r everywhere. The constraints (6) are calle holonomic if it is possible to fin new configuration coorinates q = ( q 1,..., q k ) such that the constraints are equivalently expresse as q k r+1 = q n r+2 = = q k = 0, in which case the kinematic constraints integrate to the geometric constraints q k r+1 = c k r+1,..., q k = c k (7)

4 4 Arjan van er Schaft for certain constants c k r+1,...,c k etermine by the initial conitions. Then the system reuces to an unconstraine system in the remaining configuration coorinates ( q 1,..., q k r ). Ifitisnot possible to integrate the kinematic constraints as above, then the constraints are calle nonholonomic. The equations of motion for the mechanical system with constraints (6) are given by the constraine Hamiltonian equations q = H (q, p), p ṗ = H (q, p) + A(q)λ + B(q)f, q e = B T (q) H (q, p), p 0 = A T (q) H (q, p). p (8) The constraine state space is therefore given as the following subset of the phase space: { X c = (q, p) A T (q) H } p (q, p) = 0. (9) One way of proceeing is to eliminate the constraint forces, an to reuce the equations of motion to the constraine state space, leaing (see [32] for etails) to a port- Hamiltonian system (5). The structure matrix of this reuce port-hamiltonian system satisfies the Jacobi ientity if an only if the constraints (6) are holonomic [32]. An alternative way of approaching the system (8) is to formalize it irectly as an implicit port-hamiltonian system (with respect to a Dirac structure), as will be the topic of Section From port-base network moelling to port-hamiltonian systems. In this subsection we take a ifferent point of view by emphasizing how port-hamiltonian systems irectly arise from port-base network moels of physical systems. L 1 L 2 C ϕ 1 Q ϕ 2 V Figure 1. Controlle LC-circuit.

5 Port-Hamiltonian systems: an introuctory survey 5 In network moels of complex physical systems the overall system is regare as the interconnection of energy-storing elements via basic interconnection (balance) laws such as Newton s thir law or Kirchhoff s laws, as well as power-conserving elements like transformers, kinematic pairs an ieal constraints, together with energyissipating elements [3], [14], [13]. The basic point of eparture for the theory of port-hamiltonian systems is to formalize the basic interconnection laws together with the power-conserving elements by a geometric structure, an to efine the Hamiltonian as the total energy store in the system. This is alreay illustrate by the following simple example. Example 2.3 (LCTG circuits). Consier a controlle LC-circuit (see Figure 1) consisting of two inuctors with magnetic energies H 1 (ϕ 1 ), H 2 (ϕ 2 ) (ϕ 1 an ϕ 2 being the magnetic flux linkages), an a capacitor with electric energy H 3 (Q) (Q being the charge). If the elements are linear then H 1 (ϕ 1 ) = 1 2L 1 ϕ 2 1, H 2(ϕ 2 ) = 1 2L 2 ϕ 2 2 an H 3 (Q) = 2C 1 Q2. Furthermore let V = u enote a voltage source. Using Kirchhoff s laws one obtains the ynamical equations Q ϕ 1 = ϕ }{{} J y = H ϕ 1 H Q H ϕ 1 H ϕ u, 0 (= current through voltage source) with H(Q,ϕ 1,ϕ 2 ) := H 1 (ϕ 1 ) + H 2 (ϕ 2 ) + H 3 (Q) the total energy. Clearly (by Tellegen s theorem) the matrix J is skew-symmetric. In this way every LC-circuit with inepenent elements can be moelle as a port- Hamiltonian system. Similarly any LCTG-circuit with inepenent elements can be moelle as a port-hamiltonian system, with J now being etermine by Kirchhoff s laws an the constitutive relations of the transformers T an gyrators G Dirac structures an implicit port-hamiltonian systems. From a general moeling point of view physical systems are, at least in first instance, often escribe as DAE s, that is, a mixe set of ifferential an algebraic equations. This stems from the fact that in network moeling the system uner consieration is regare as obtaine from interconnecting simpler sub-systems. These interconnections usually give rise to algebraic constraints between the state space variables of the sub-systems; thus leaing to implicit systems. Therefore it is important to exten the framework of port-hamiltonian systems to the context of implicit systems; that is, systems with algebraic constraints Dirac structures. In orer to give the efinition of an implicit port-hamiltonian system we introuce the notion of a Dirac structure, formalizing the concept of (10)

6 6 Arjan van er Schaft a power-conserving interconnection, an generalizing the notion of a structure matrix J(x)as encountere before. Let F be an l-imensional linear space, an enote its ual (the space of linear functions on F )byf. The prouct space F F is consiere to be the space of power variables, with power efine by P = f f, (f,f ) F F, (11) where f f enotes the uality prouct. Often we call F the space of flows f, an F the space of efforts e, with the power of an element (f, e) F F enote as e f. Example 2.4. Let F be the space of generalize velocities, an F be the space of generalize forces, then e f is mechanical power. Similarly, let F be the space of currents, an F be the space of voltages, then e f is electrical power. There exists on F F the canonically efine symmetric bilinear form for f i F, e i F, i = 1, 2. (f 1,e 1 ), (f 2,e 2 ) F F := e 1 f 2 + e 2 f 1 (12) Definition 2.5 ([6], [8], [7]). A constant Dirac structure on F is a linear subspace D F F such that D = D (13) where enotes the orthogonal complement with respect to the bilinear form, F F. It immeiately follows that the imension of any Dirac structure D on an l-imensional linear space is equal to l. Furthermore, let (f, e) D = D. Then by (12) 0 = (f, e), (f, e) F F = 2 e f. (14) Thus for all (f, e) D we obtain e f =0. Hence a Dirac structure D on F efines a power-conserving relation between the power variables (f, e) F F, which moreover has maximal imension. Remark 2.6. For many systems, especially those with 3-D mechanical components, the Dirac structure is actually moulate by the energy or geometric variables. Furthermore, the state space X is a manifol an the flows f S = ẋ corresponing to energy-storage are elements of the tangent space T x X at the state x X, while the efforts e S are elements of the co-tangent space T x X. Moulate Dirac structures often arise as a result of kinematic constraints. In many cases, these constraints will be configuration epenent, causing the Dirac structure to be moulate by the configuration variables, cf. Section 2.2.

7 Port-Hamiltonian systems: an introuctory survey 7 In general, a port-hamiltonian system can be represente as in Figure 2. The port variables entering the Dirac structure D have been split in ifferent parts. First, there are two internal ports. One, enote by S, is corresponing to energy-storage an the other one, enote by R, is corresponing to internal energy-issipation (resistive elements). Secon, two external ports are istinguishe. The external port enote by C is the port that is accessible for controller action. Also the presence of sources may be inclue in this port. Finally, the external port enote by is the interaction port, efining the interaction of the system with (the rest of) its environment. R S D C Figure 2. Port-Hamiltonian system Energy storage port. The port variables associate with the internal storage port will be enote by (f S,e S ). They are interconnecte to the energy storage of the system which is efine by a finite-imensional state space manifol X with coorinates x, together with a Hamiltonian function H : X R enoting the energy. The flow variables of the energy storage are given by the rate ẋ of the energy variables x. Furthermore, the effort variables of the energy storage are given by the co-energy variables H x (x), resulting in the energy balance H t H = x (x) ẋ = T H (x)ẋ. (15) x (Here we aopt the convention that H x (x) enotes the column vector of partial erivatives of H.) The interconnection of the energy storing elements to the storage port of the Dirac structure is accomplishe by setting f S = ẋ, e S = H x (x). (16)

8 8 Arjan van er Schaft Hence the energy balance (15) can be also written as t H = T H x (x)ẋ = et S f S. (17) Resistive port. The secon internal port correspons to internal energy issipation (ue to friction, resistance, etc.), an its port variables are enote by (f R,e R ). These port variables are terminate on a static resistive relation R. In general, a static resistive relation will be of the form with the property that for all (f R,e R ) satisfying (18) R(f R,e R ) = 0, (18) e R f R 0. (19) In many cases we may restrict ourselves to linear resistive relations. This means that the resistive port variables (f R,e R ) satisfy linear relations of the form R f f R + R e e R = 0. (20) The inequality (19) correspons to the square matrices R f properties of symmetry an semi-positive efiniteness an R e satisfying the R f R T e = R er T f 0, (21) together with the imensionality conition rank[r f R e ]=im f R. Without the presence of aitional external ports, the Dirac structure of the port- Hamiltonian system satisfies the power-balance e T S f S + e T R f R = 0 which leas to t H = et S f S = er T f R 0. (22) An important special case of resistive relations between f R an e R occurs when the resistive relations can be expresse as an input-output mapping f R = F(e R ), where the resistive characteristic F : R m r R m r satisfies e T R F(e R) 0, e R R m r. (23) For linear resistive elements this specializes to f R = Re R, for some positive semiefinite symmetric matric R = R T External ports. Now, let us consier in more etail the external ports to the system. We istinguish between two types of external ports. One is the control port C, with port variables (f C,e C ), which are the port variables which are accessible for controller action. Other type of external port is the interaction port, which enotes the interaction of the port-hamiltonian system with its environment. The

9 Port-Hamiltonian systems: an introuctory survey 9 port variables corresponing to the interaction port are enote by (f I,e I ). By taking both the external ports into account the power-balance extens to e T S f S + e T R f R + e T C f C + e T I f I = 0 (24) whereby (22) extens to t H = et R f R + ec T f C + ei T f I. (25) Port-Hamiltonian ynamics. The port-hamiltonian system with state space X, Hamiltonian H corresponing to the energy storage port S, resistive port R, control port C, interconnection port, an total Dirac structure D will be succinctly enote by = (X,H,R, C,, D). The ynamics of the port-hamiltonian system is specifie by consiering the constraints on the various port variables impose by the Dirac structure, that is (f S,e S,f R,e R,f C,e C,f I,e I ) D, an to substitute in these relations the equalities f S = ẋ, e S = H x (x). This leas to the implicitly efine ynamics ( ẋ(t), H ) x (x(t)), f R(t), e R (t), f C,(t),e C (t), f I (t), e I (t) D (26) with f R (t), e R (t) satisfying for all t the resistive relation (18): R(f R,e R ) = 0. (27) In many cases of interest the ynamics (26) will constrain the allowe states x, epening on the values of the external port variables (f C,e C ) an (f I,e I ). Thus in an equational representation port-hamiltonian systems generally will consist of a mixe set of ifferential an algebraic equations (DAEs). Example 2.7 (General LC- circuits). Consier an LC-circuit with general network topology. Kirchhoff s current an voltage laws take the general form A L T I L + A C T I C + A P T I P = 0, V L = A L λ, V C = A C λ, V P = A P λ for some matrices A L, A C, A S. Here I L, I C, I P enote the currents, respectively through the inuctors, capacitors an external ports. Likewise, V L, V C, V P enote the voltages over the inuctors, capacitors an external ports. Kirchhoff s current an voltage laws efine a Dirac structure between the flows an efforts: f = (I C,V L,I P ) = ( Q, φ,i P ), ( H e = (V C,I L,V P ) = Q, H ) φ,v P

10 10 Arjan van er Schaft with Hamiltonian H(φ,Q)the total energy. This leas the to port-hamiltonian system in implicit form φ = A L λ, H Q = A Cλ, V P = A P λ, T 0 = A H L φ A C T Q + A T P I P. Example 2.8 (Electro-mechanical system). Consier the ynamics of an iron ball in the magnetic fiel of a controlle inuctor: The port-hamiltonian escription of this V R I g φ q m Figure 3. Magnetically levitate ball. system (with q the height of the ball, p the vertical momentum, an ϕ the magnetic flux of the inuctor) is given as H q q 0 ṗ = H ϕ 0 0 R 1 p + 0 V, 1 (28) I = H ϕ. This is a typical example of a system where the coupling between two ifferent physical omains (mechanical an magnetic) takes place via the Hamiltonian H ϕ H(q,p,ϕ) = mgq + p2 2m + ϕ 2 2k 1 (1 q k 2 )

11 Port-Hamiltonian systems: an introuctory survey 11 where the last term epens both on a magnetic variable (in this case ϕ) anamechanical variable (in this case the height q) Input-state-output port-hamiltonian systems. An important special case of port-hamiltonian systems is the class of input-state-output port-hamiltonian systems, where there are no algebraic constraints on the state space variables, an the flow an effort variables of the resistive, control an interaction port are split into conjugate input output pairs. Input state output port-hamiltonian systems without interaction port are of the form ẋ =[J(x) R(x)] H (x) + g(x)u, x y = g T (x) H x (x) (29) where u, y are the input output pairs corresponing to the control port C. Here the matrix J(x) is skew-symmetric, while the matrix R(x) = R T (x) 0 specifies the resistive structure, an is given as R(x) = gr T (x) Rg R (x) for some linear resistive relation f R = Re R, R = R T 0, with g R representing the input matrix corresponing to the resistive port. The unerlying Dirac structure of the system is then given by the graph of the skew-symmetric linear map J(x) g R (x) g(x) gr T (x) 0 0. (30) g T (x) Control by interconnection of port-hamiltonian systems The basic property of port-hamiltonian systems is that the power-conserving interconnection of any number of port-hamiltonian systems is again a port-hamiltonian system. To be explicit, consier two port-hamiltonian systems A an B with Dirac structures D A an D B an Hamiltonians H A an H B, efine on state spaces X A, respectively X B. For convenience, split the ports of the Dirac structures D A an D B into the internal energy storage ports an all remaining external ports whose port-variables are enote respectively by f A, e A an f B, e B. Now, consier any interconnection Dirac structure D I involving the port-variables f A, e A, f B,e B possibly together with aitional port-variables f I,e I. Then the interconnection of the systems A an B via D I is again a port-hamiltonian system with respect to the compose Dirac structure D A D I D B, involving as port-variables the internal storage port-variables of D A an D B together with the aitional port-variables F I, e I. For etails we refer to [5], [34], [30].

12 12 Arjan van er Schaft Furthermore, the state space of the interconnecte port-hamiltonian system is the prouct of the two state spaces X A X B, while its Hamiltonian is simply the sum H A + H B of the two Hamiltonians. This basic statement naturally extens to the interconnection of any number of port-hamiltonian systems via an interconnection Dirac structure. Control by port-interconnection is base on esigning a controller system which is interconnecte to the control port with port-variables (f C,e C ). In principle this implies that we only consier collocate control, where the controller will only use the information about the plant port-hamiltonian system that is containe in the conjugate pairs (f C,e C ) of port variables of the control port, without using aitional information about the plant (e.g. corresponing to observation on other parts of the plant system). In the secon place, we will restrict attention to controller systems which are themselves also port-hamiltonian systems. There are two main reasons for this. One is that by oing so the close-loop system is again a port-hamiltonian system, allowing to easily ensure some esire properties. Furthermore, it will turn out that the port-hamiltonian framework suggests useful ways to construct port-hamiltonian controller systems. Secon reason is that port-hamiltonian controller systems allow in principle for a physical system realization (thus linking to passive control an systems esign) an physical interpretation of the controller action. Since we o not know the environment (or only have very limite information about it), but on the other han, the system will interact with this unknown environment, the task of the controller is often two-fol: 1) to achieve a esire control goal (e.g. setpoint regulation or tracking) if the interaction with the environment is marginal or can be compensate, 2) to make sure that the controlle system has a esire interaction behavior with its environment. It is fair to say that up to now the evelopment of the theory of control of port-hamiltonian systems has mostly concentrate on the secon aspect (which at the same time, is often unerevelope in other control theories). Most successful approaches to eal with the secon aspect of the control goal are those base on the concept of passivity, such as issipativity theory [38], impeance control [13] an Intrinsically Passive Control (IPC) [36]. In fact, the port-hamiltonian control theory can be regare as an enhancement to the theory of passivity, making a much closer link with complex physical systems moeling at one han an with the theory of ynamical systems (in particular, Hamiltonian ynamics) at the other han. As sai above, we will throughout consier controller systems which are again port-hamiltonian systems. We will use the same symbols as above for the internal an external ports an port-variables of the controller port-hamiltonian system, with an ae overbar or a superscript c in orer to istinguish it from the plant system. (The interaction port of the controller system may be thought of as an extra possibility for aitional controller action (outer-loop control).) In orer to further istinguish the plant system an the controller we enote the state space of the plant system by X p with coorinates x p, the Dirac structure by D p an its Hamiltonian by H p, while we will enote the state space manifol of the controller system by X c with coorinates x c, its Dirac structure by D c an its Hamiltonian by H c : X c R. The interconnection

13 Port-Hamiltonian systems: an introuctory survey 13 of the plant port-hamiltonian system with the controller port-hamiltonian system is obtaine by equalizing the port variables at the control port by f C = f C, e C =ē C (31) where f C, ē C enote the control port variables of the controller system. Here, the minus sign is inserte to have a uniform notion of irection of power flow. Clearly, this synchronizing interconnection is power-conserving, that is ec T f C +ēc T f C = 0. Remark 3.1. A sometimes useful alternative is the gyrating power-conserving interconnection f C = ē C, e C = f C. (32) In fact, the stanar feeback interconnection can be regare to be of this type. For both interconnection constraints it irectly follows from the theory of composition of Dirac structures that the interconnecte (close-loop) system is again a port-hamiltonian system with Dirac structure etermine by the Dirac structures of the plant PH system an the controller PH system. The resulting interconnecte PH system has state space X p X c, Hamiltonian H p + H c, resistive ports (f R,e R, f R, ē R ) an interaction ports (f I,e I, f I, ē I ), satisfying the power-balance t (H p + H c ) = e T R f R +ē T R f R + e T I f I +ē T I f I e T I f I +ē T I f I (33) since both er T f R 0 an ēr T f R 0. Hence we immeiately recover the state space formulation of the passivity theorem, see e.g. [31], if H p an H c are both nonnegative, implying that the plant an the controller system are passive (with respect to their controller an interaction ports an storage functions H p an H c ), then also the close -loop system is passive (with respect to the interaction ports an storage function H p + H c.) Furthermore, ue to the Hamiltonian structure, we can go beyon the passivity theorem, an we can erive conitions which ensure that we can passify an/or stabilize plant port-hamiltonian systems for which the Hamiltonian H p oes not have a minimum at the esire equilibrium Stabilization by Casimir generation. What oes the power-balance (33)mean for the stability properties of the close-loop system, an how can we esign the controller port-hamiltonian system in such a way that the close-loop system has esire stability properties? Let us first consier the stability of an arbitrary port- Hamiltonian system = (X,H,R, C,, D) without control or interaction ports,

14 14 Arjan van er Schaft that is, an autonomous port-hamiltonian system = (X,H,R, D). Clearly, the power-balance (33) reuces to t H = et R f R 0. (34) Hence we immeiately infer by stanar Lyapunov theory that if x is a minimum of the Hamiltonian H then it will be a stable equilibrium of the autonomous port- Hamiltonian system = (X,H,R, D), which is actually asymptotically stable if the issipation term er T f R is negative efinite outsie x, or alternatively if some sort of etectability conition is satisfie, guaranteeing asymptotic stability by the use of LaSalle s Invariance principle (see for etails e.g. [31]). However, what can we say if x is not a minimum of H, an thus we cannot irectly use H as a Lyapunov function? A well-known metho in Hamiltonian systems, sometimes calle the Energy- Casimir metho, is to use in the Lyapunov analysis next to the Hamiltonian other conserve quantities (ynamical invariants) which may be present in the system. Inee, if we may fin other conserve quantities then caniate Lyapunov functions can be sought within the class of combinations of the Hamiltonian H an those conserve quantities. In particular, if we can fin a conserve quantity C : X R such that V := H + C has a minimum at the esire equilibrium x then we can still infer stability or asymptotic stability by replacing (34) by t V = et R f R 0 (35) an thus using V as a Lyapunov function. For the application of the Energy-Casimir metho one may istinguish between two main cases. First situation occurs if the esire equilibrium x is not a stationary point of H, an one looks for a conserve quantity C such that H +C has a minimum at x. This for example happens in the case that the esire set-point x is not an equilibrium of the uncontrolle system, but only a controlle equilibrium of the system. Secon situation occurs when x is a stationary point of H, but not a minimum. Functions that are conserve quantities of the system for every Hamiltonian are calle Casimir functions or simply Casimirs. Casimirs are completely characterize by the Dirac structure of the port-hamiltonian system. Inee, a function C : X R is a Casimir function of the autonomous port-hamiltonian system (without energy issipation) = (X,H,D) if an only if the graient vector e = T C x satisfies e T f S = 0 for all f S for which there exists e S such that (f S,e S ) D. (36) Inee, (36) is equivalent to t C = T C x (x(t))ẋ(t) = T C x (x(t))f S = e T f S = 0 (37)

15 Port-Hamiltonian systems: an introuctory survey 15 for every port-hamiltonian system (X,H,D) with the same Dirac structure D. By the generalize skew-symmetry of the Dirac structure (36) is equivalent to the requirement that e = T C x satisfies (0,e) D. Similarly, we efine a Casimir function for a port-hamiltonian system with issipation = (X,H,R, D) to be any function C : X R satisfying Inee, this will imply that (0,e,0, 0) D. (38) t C = T C x (x(t))ẋ(t) = T C x (x(t))f p = e T f p = 0 (39) for every port-hamiltonian system (X, H, R, D) with the same Dirac structure D. (In fact by efiniteness of the resistive structures the satisfaction of (39) for a particular resistive structure R implies the satisfaction for all resistive structures R.) Now let us come back to the esign of a controller port-hamiltonian system such that the close-loop system has esire stability properties. Suppose we want to stabilize the plant port-hamiltonian system (X p,h p, R, C, D p ) aroun a esire equilibrium xp. We know that for every controller port-hamiltonian system the close-loop system satisfies t (H p + H c ) = er T f R +ēr T f R 0. (40) What if x is not a minimum for H p? A possible strategy is to generate Casimir functions C(x p,x c ) for the close-loop system by choosing the controller port-hamiltonian system in an appropriate way. Thereby we generate caniate Lyapunov functions for the close-loop system of the form V(x p,x c ) := H p (x p ) + H c (x c ) + C(x p,x c ) where the controller Hamiltonian function H c : X c R still has to be esigne. The goal is thus to construct a function V as above in such a way that V has a minimum at (xp,x c ) where x c still remains to be chosen. This strategy thus is base on fining all the achievable close-loop Casimirs. Furthermore, since the close-loop Casimirs are base on the close-loop Dirac structures, this reuces to fining all the achievable close-loop Dirac structures D D. Another way to interpret the generation of Casimirs for the close-loop system is to look at the level sets of the Casimirs as invariant submanifols of the combine plant an controller state space X p X c. Restricte to every such invariant submanifol (part of) the controller state can be expresse as a function of the plant state, whence the close-loop Hamiltonian restricte to such an invariant manifol can be seen as a

16 16 Arjan van er Schaft shape version of the plant Hamiltonian. To be explicit (see e.g. [31], [24], [25] for etails) suppose that we have foun Casimirs of the form x ci F i (x p ), i = 1,...,n p where n p is the imension of the controller state space, then on every invariant manifol x ci F i (x p ) = α i, i = 1,...,n p, where α = (α 1,...,α np ) is a vector of constants epening on the initial plant an controller state, the close-loop Hamiltonian can be written as H s (x p ) := H p (x p ) + H c (F (x p ) + α), where, as before, the controller Hamiltonian H c still can be assigne. This can be regare as shaping the original plant Hamiltonian H p to a new Hamiltonian H s Port Control. In broa terms, the Port Control problem is to esign, given the plant port-hamiltonian system, a controller port-hamiltonian system such that the behavior at the interaction port of the plant port-hamiltonian system is a esire one, or close to a esire one. This means that by aing the controller system we seek to shape the external behavior at the interaction port of the plant system. If the esire external behavior at this interaction port is given in input output form as a esire (ynamic) impeance, then this amounts to the Impeance Control problem as introuce an stuie by Hogan an co-workers [13]; see also [36] for subsequent evelopments. The Port Control problem, as state in this generality, immeiately leas to two funamental questions: 1). Given the plant PH system, an the controller PH system to be arbitrarily esigne, what are the achievable behaviors of the close-loop system at the interaction port of the plant? 2). If the esire behavior at the interaction port of the plant is not achievable, then what is the closest achievable behavior? Of course, the secon question leaves much room for interpretation, since there is no obvious interpretation of what we mean by closest behavior. Also the first question in its full generality is not easy to answer, an we shall only aress an important subproblem. An obvious observation is that the esire behavior, in orer to be achievable, nees to be the port behavior of a PH system. This leas alreay to the problem of characterizing those external behaviors which are port behaviors of port-hamiltonian systems. Seconly, the Port Control problem can be split into a number of subproblems. Inee, we know that the close-loop system arising from interconnection of the plant PH system with the controller PH system is specifie by a Hamiltonian which is just the sum of the plant Hamiltonian an the controller Hamiltonian, an a resistive structure which is the prouct of the resistive structure of the plant an of the controller system, together with a Dirac structure which is the composition of the plant Dirac structure an the controller Dirac structure. Therefore an important subproblem is again to characterize the achievable close-loop Dirac structures. On the other han, a funamental problem in aressing the Port Control problem in general theoretical terms is the lack of a systematic way to specify esire behavior.

17 Port-Hamiltonian systems: an introuctory survey 17 The problem of Port Control is to etermine the controller system in such a way that the port behavior in the port variables f I,e I is a esire one. In this particular (simple an linear) example the esire behavior can be quantifie e.g. in terms of a esire stiffness an amping of the close-loop system, which is easily expresse in terms of the close-loop transfer function from f I to e I. Of course, on top of the requirements on the close-loop transfer function we woul also require internal stability of the close-loop system. For an appealing example of port control of port-hamiltonian systems within a context of hyraulic systems we refer to [15] Energy Control. Consier two port-hamiltonian systems i (without internal issipation) in input state output form ẋ i = J i (x i ) H i + g i (x i )u i, x i y i = gi T (x i) H i, i = 1, 2, x i both satisfying the power-balance t H i = yi T u i. Suppose now that we want to transfer the energy from the port-hamiltonian system 1 to the port-hamiltonian system 2, while keeping the total energy H 1 + H 2 constant. This can be one by using the following output feeback [ u1 ] = u 2 [ 0 y1 y T 2 y 2 y T 1 0 ][ y1 (41) y 2 ]. (42) Since the matrix in (42) is skew-symmetric it immeiately follows that the close-loop system compose of systems 1 an 2 linke by the power-conserving feeback is energy-preserving, that is t (H 1 + H 2 ) = 0. However, if we consier the iniviual energies then we notice that t H 1 = y1 T y 1y2 T y 2 = y 1 2 y (43) implying that H 1 is ecreasing as long as y 1 an y 2 are ifferent from 0. Conversely, as expecte since the total energy is constant, t H 2 = y2 T y 2y1 T y 1 = y 2 2 y (44) implying that H 2 is increasing at the same rate. In particular, if H 1 has a minimum at the zero equilibrium, an 1 is zero-state observable, then all the energy H 1 of 1 will be transferre to 2, provie that y 2 is not ientically zero (which again can be guarantee by assuming that H 2 has a minimum at the zero equilibrium, an that 2 is zero-state observable). If there is internal energy issipation, then this energy transfer mechanism still works. However, the fact that H 2 grows or not will epen on the balance between the energy elivere by 1 to 2 an the internal loss of energy in 2 ue to issipation.

18 18 Arjan van er Schaft We conclue that this particular scheme of power-conserving energy transfer is accomplishe by a skew-symmetric output feeback, which is moulate by the values of the output vectors of both systems. Of course this raises, among others, the question of the efficiency of the propose energy-transfer scheme, an the nee for a systematic quest of similar power-conserving energy-transfer schemes. We refer to [9] for a similar but ifferent energy-transfer scheme irectly motivate by the structure of the example (control of a snakeboar) Achievable close-loop Dirac structures. In all the control problems iscusse above the basic question comes up what are the achievable close-loop Dirac structures base on a given plant Dirac structure an a controller Dirac structure, which still is to be etermine. Theorem 3.2 ([5]). Given any plant Dirac structure D p, a certain interconnecte D = D p D c can be achieve by a proper choice of the controller Dirac structure D c if an only if the following two equivalent conitions are satisfie: where Dp 0 D0, D π Dp π Dp 0 := {f 1,e 1 ) (f 1,e 1, 0, 0) D p }, Dp π := {(f 1,e 1 ) there exists (f2 P,eP 2 ) with (f 1,e 1,f2 P,eP 2 ) D p}, D 0 := {(f 1,e 1 ) (f 1,e 1, 0, 0) D}, D π := {(f 1,e 1 ) there exists (f 3,e 3 ) with (f 1,e 1,f 3,e 3 ) D}. An important application of the above theorem concerns the characterization of Casimir functions which can be achieve by interconnecting a given plant port- Hamiltonian system with a controller port-hamiltonian system. 4. Distribute-parameter port-hamiltonian systems The treatment of infinite-imensional Hamiltonian systems in the literature is mostly confine to systems with bounary conitions such that the energy exchange through the bounary is zero. On the other han, in many applications the interaction with the environment (e.g. actuation or measurement) will actually take place through the bounary of the system. In [35] a framework has been evelope to represent classes of physical istribute-parameter systems with bounary energy flow as infinite-imensional port-hamiltonian systems. It turns out that in orer to allow the inclusion of bounary variables in istribute-parameter systems the concept of (an

19 Port-Hamiltonian systems: an introuctory survey 19 infinite-imensional) Dirac structure provies again the right type of generalization with respect to the existing framework [23] using Poisson structures. As we will iscuss in the next three examples, the port-hamiltonian formulation of istribute-parameter systems is closely relate to the general framework for escribing basic istribute-parameter systems as systems of conservation laws, see e.g. [11], [37]. Example 4.1 (Invisci Burger s equation). The viscous Burger s equation is a scalar parabolic equation efine on a one-imensional spatial omain (interval) Z = [a,b] R, with the state variable α(t, z) R, z Z, t I, where I is an interval of R, satisfying the partial ifferential equation α t + α α z ν 2 α = 0. (45) z2 The invisci (ν = 0) Burger s equations may be alternatively expresse as α t + z β = 0 (46) where the state variable α(t, z) is calle the conserve quantity an the function β := α2 2 the flux variable. Eq. (46) is calle a conservation law, since by integration one obtains the balance equation b αz= β(a) β(b). (47) t a Furthermore, accoring to the framework of Irreversible Thermoynamics [27], one may express the flux β as a function of the generating force which is the variational erivative of some functional H(α) of the state variable. The variational erivative δh δα of a functional H(α)is uniquely efine by the requirement b δh H(α + εη) = H(α)+ ε a δα ηz+ O(ε2 ) (48) for any ε R an any smooth function η(z, t) such that α + εη satisfies the same bounary conitions as α [23]. For the invisci Burger s equation one has β = δh δα, where b α 3 H(α) = z. (49) a 6 Hence the invisci Burger s equation may be also expresse as α t = δh z δα. (50) This efines an infinite-imensional Hamiltonian system in the sense of [23] with respect to the skew-symmetric operator z that is efine on the functions with support containe in the interior of the interval Z.

20 20 Arjan van er Schaft From this formulation one erives that the Hamiltonian H(α)is another conserve quantity. Inee, by integration by parts b t H = δh a δα δh z δα z = 1 ( β 2 (a) β 2 (b) ). (51) 2 We note that the right-han sie is a function of the flux variables evaluate at the bounary of the spatial omain Z. The secon example consists of a system of two conservations laws, corresponing to the case of two physical omains in interaction. Example 4.2 (The p-system, cf. [11], [37]). The p-system is a moel for e.g. a one-imensional gas ynamics. Again, the spatial variable z belongs to an interval Z R, while the epenent variables are the specific volume v(t, z) R +, the velocity u(t, z) an the pressure functional p(v) (which for instance in the case of an ieal gas with constant entropy is given by p(v) = Av γ where γ 1). The p-system is then efine by the following system of partial ifferential equations v t u z = 0, u t + p(v) = 0 z representing respectively conservation of mass an of momentum. By efining the state vector as α(t, z) = (v, u) T, an the vector-value flux β(t,z) = ( u, p(v)) T the p-system is rewritten as α t + β = 0. (53) z Again, accoring to the framework of Irreversible Thermoynamics, the flux vector may be written as function of the variational erivatives of some functional. Inee, consier the energy functional H(α) = b a H(v, u)z where the energy ensity H(v, u) is given as the sum of the internal energy an the kinetic energy ensities H(v, u) = U(v) + u2 (54) 2 with U(v) a primitive function of the pressure. (Note that for simplicity the mass ensity has been set equal to 1, an hence no ifference is mae between the velocity an the momentum.) The flux vector β may be expresse in terms of the variational erivatives of H as β = ( (52) ) ( ) δh. (55) The anti-iagonal matrix represents the canonical coupling between two physical omains: the kinetic an the potential (internal) omain. Thus the variational erivative δv δh δu

21 Port-Hamiltonian systems: an introuctory survey 21 of the total energy with respect to the state variable of one omain generates the flux variable for the other omain. Combining eqns. (53) an (55), the p-system may thus be written as the Hamiltonian system ( α1 t α 2 t ) = ( 0 z z 0 )( δh ) δα1 δh. (56) δα 2 Using again integration by parts, one may erive the following energy balance equation: t H = β 1(a)β 2 (a) β 1 (b)β 2 (b). (57) Notice again that the right-han sie of this power-balance equation is a quaratic function of the fluxes at the bounary of the spatial omain. The last example is the vibrating string. It is again a system of two conservation laws representing the canonical interomain coupling between the kinetic energy an the elastic potential energy. However in this example the classical choice of the state variables leas to express the total energy as a function of some of the spatial erivatives of the state variables. Example 4.3 (Vibrating string). Consier an elastic string subject to traction forces at its ens, with spatial variable z Z =[a,b] R. Denote by u(t, z) the isplacement of the string an the velocity by v(t, z) = u t. Using the vector of state variables x(t,z) = (u, v) T, the ynamics of the vibrating string is escribe by the system of partial ifferential equations ( ) x ( v ) = (58) t 1 μ z T u z where the first equation is simply the efinition of the velocity an the secon one is Newton s secon law. Here T enotes the elasticity moulus, an μ the mass ensity. The total energy is H(x) = U(u) + K(v), where the elastic potential energy U is a function of the strain u z (t, z) U(u) = b a 1 2 T an the kinetic energy K epens on the velocity v(t, z) = u K(v) = b Thus the total system (58) may be expresse as x t a = ( 0 1 μ 1 μ 0 ( ) u 2 z (59) z t as 1 2 μv(t, z)2 z. (60) )( δh δu δh δv ) (61)

22 22 Arjan van er Schaft where δh δu = δu δu = z ( ) T u z is the elastic force an δh δv = δk δv = μv is the momentum. In this formulation, the system is not anymore expresse as a system of conservation laws since the time-erivative of the state variables is a function of the variational erivatives of the energy irectly, an not the spatial erivative of a function of the variational erivatives as before. Instea of being a simplification, this reveals a rawback for the case of non-zero energy flow through the bounary of the spatial omain. Inee, in this case the variational erivative has to be complete by a bounary term since the Hamiltonian functional epens on the spatial erivatives of the state. For example, in the computation of the variational erivative of the elastic potential energy U one obtains by integration by parts that U(u+ εη) U(u) equals b ( ε T u ) [ ( ηz+ ε η T u )] b + O(ε 2 ) (62) a z z z a an the secon term in this expression constitutes an extra bounary term. Alternatively we now formulate the vibrating string as a system of two conservation laws. Take as alternative vector of state variables α(t, z) = (ε, p) T, where ε enotes the strain α 1 = ε = u z an p enotes the momentum α 2 = p = μv. Recall that in these variables the total energy is written as H 0 = b a 1 2 ( Tα μ α2 2 ) z (63) an irectly epens on the state variables an not on their spatial erivatives. Furthermore, one efines the flux variables to be the stress β 1 = δh 0 δα 1 = Tα 1 an the velocity β 2 = δh 0 δα 1 = α 2 μ. In matrix notation, the flux vector β is thus expresse as a function of the variational erivatives δh 0 δα by ( ) 0 1 δh0 β = 1 0 δα. (64) Hence the vibrating string may be alternatively expresse by the system of two conservation laws α t = ( 0 z z 0 satisfying the power balance equation (57). ) δh0 δα 4.1. Systems of two conservation laws in interaction. Let us now consier the general class of istribute-parameter systems consisting of two conservation laws with the canonical coupling as in the above examples of the p-system an the vibrating string. Let the spatial omain Z R n be an n-imensional smooth manifol with smooth (n 1)-imensional bounary Z. Denote by k (Z) the vector space of (ifferential) k-forms on Z (respectively by k ( Z) the vector space of k-forms on Z). (65)

23 Port-Hamiltonian systems: an introuctory survey 23 Denote furthermore by = k 0 k (Z) the algebra of ifferential forms over Z an recall that it is enowe with an exterior prouct an an exterior erivation. Definition 4.4. A system of conservation laws is efine by a set of conserve quantities α i k i (Z), i {1,...,N} where N N,k i N, efining the state space X = i=1,...,n k i (Z), an satisfying a set of conservation laws α i + β i = g i t (66) where β i ki 1 (Z) enote the set of fluxes an g i k i (Z) enote the set of istribute interaction forms. In general, the fluxes β i are efine by so-calle closure equations β i = J (α i,z), i = 1,...,N (67) leaing to a close form for the ynamics of the conserve quantities α i. The integral form of the conservation laws yiels the following balance equations α i + β i = g i t S S S (68) for any surface S Z of imension equal to the egree of α i. Remark 4.5. A common case is that Z = R 3 an that the conserve quantities are 3-forms, that is, the balance equation is evaluate on volumes of the 3-imensional space. In this case () takes in vector calculus notation the familiar form α i t (z, t) + iv zβ i = g i, i = 1,...,n. (69) However, systems of conservation laws may correspon to ifferential forms of any egree. Maxwell s equations are an example where the conserve quantities are ifferential forms of egree 2. In the sequel, as in our examples sofar, we consier a particular class of systems of conservation laws where the closure equations are such that fluxes are linear functions of the variational erivatives of the Hamiltonian functional. First recall the general efinition of the variational erivative of a functional H(α) with respect to the ifferential form α p (Z) (generalizing the efinition given before). Definition 4.6. Consier a ensity function H : p (Z) Z n (Z) where p {1,...,n}, an enote by H := Z H R the associate functional. Then the uniquely efine ifferential form δh δα n p (Z) which satisfies for all α p (Z) an ε R [ ] δh H(α + ε α) = H (α) + ε Z Z δα α + O ( ε 2) is calle the variational erivative of H with respect to α p (Z).

24 24 Arjan van er Schaft Definition 4.7. Systems of two conservation laws with canonical interomain coupling are systems involving a pair of conserve quantities α p p (Z) an α q q (Z), ifferential forms on the n-imensional spatial omain Z of egree p an q respectively, satisfying p+q = n+1 ( complementarity of egrees ). The closure equations generate by a Hamiltonian ensity function H : p (Z) q (Z) Z n (Z) resulting in the Hamiltonian H := Z H R are given by ( ) ( ) ( ) βp 0 ( 1) r δh δαp = ε β q 1 0 δh (70) δα q where r = pq +1,ε { 1, +1}, epening on the sign convention of the consiere physical omain. Define the vector of flow variables to be the time-variation of the state, an the effort variables to be the variational erivatives ( ) ) ( ) fp ep =, = f q e q ( αp t α q t ( δh ) δαp δh δα q. (71) Their prouct equals again the time-variation of the Hamiltonian H = (e p f p + e q f q ). (72) t Z Using the conservation laws (4.5) for g i = 0, the closure relations (70) an the properties of the exterior erivative an Stokes theorem, one obtains H = εβ q ( β p ) + ( 1) r β p ε( β q ) t Z (73) = ε β q β p + ( 1) q β q β p = ε β q β p. Z Finally, as before we efine the power-conjugate pair of flow an effort variables on the bounary as the restriction of the flux variables to the bounary Z of the omain Z: ( ) ( ) f βq = Z. (74) e β p Z On the total space of power-conjugate variables, that is, the ifferential forms (f p,e p ) an (f q,e q ) on the omain Z an the ifferential forms (f,e ) efine on the bounary Z, one efines an interconnection structure by Eqn. (74) together with ( ) ( fq 0 ( 1) = ε r )( ) eq. (75) f p 0 e p This interconnection can be formalize as a special type of Dirac structure, calle Stokes Dirac structure, leaing to the efinition of istribute-parameter port-hamiltonian systems [35]. Z