Port-Hamiltonian Systems: from Geometric Network Modeling to Control

Transcription

1 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Port-Hamiltonian Systems: from Geometric Network Modeling to Control Arjan van der Schaft, University of Groningen Dimitri Jeltsema, Delft University of Technology In collaboration with Bernhard Maschke, Romeo Ortega, Jacquelien Scherpen, Stefano Stramigioli, Alessandro Macchelli, Peter Breedveld, Hans Zwart, Morten Dalsmo, Guido Blankenstein, Damien Eberard, Goran Golo, Ram Pasumarthy, Javier Villegas, Gerardo Escobar, Guido Blankenstein, Aneesh Venkatraman, Rostyslav Polyuga..

2 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Part II (Tuesday afternoon): From Network Modeling to Port-Hamiltonian Systems 1. From junction structures to Dirac structures 2. Port-Hamiltonian systems 3. Examples 4. Input-state-output port-hamiltonian systems 5. Multi-modal physical systems 6. Representations and transformations

3 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, In Part I we have seen how port-based network modeling of lumped-parameter physical systems leads to a representation of physical systems as generalized bond graphs, where each bond corresponds to a (vector) pair of flow variables f, and effort variables e. Energy-storing elements: Power-dissipating elements: ẋ = f S e S = H x (x) R(f R, e R ) = 0, e T Rf R 0 Power-conserving elements: transformers, gyrators, (ideal) constraints.

4 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, and 1-junctions: e 1 = e 2 = = e k, f 1 + f f k = 0 f 1 = f 2 = = f k, e 1 + e e k = 0 Note: 0- and 1-junctions are also power-conserving: e 1 f 1 + e 2 f e k f k = 0

5 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Transformers, gyrators, etc., are energy-routing devices, and may correspond to exchange between different types of energy. Ideal powerless constraints such as kinematic constraints. 0- and 1-junctions correspond to basic conservation laws such as Kirchhoff s laws. All power-conserving elements have the following properties in common. They are described by linear equations: Ff + Ee = 0, f, e R l satisfying e T f = e 1 f 1 + e 2 f e l f l = 0, [ ] rank F E = l

6 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Geometric definition: Definition 1 A (constant) Dirac structure on a finite-dimensional space V is a subspace D V V such that (i) e T f = 0 for all (f, e) D, (ii) dim D = dim V. For any skew-symmetric map J : V V its graph {(f, e) V V f = Je} is a Dirac structure!

7 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Alternative, more general, definition of Dirac structure Power is defined by P = e(f) =: < e f >= e T f, (f, e) V V. where the linear space V is called the space of flows f (e.g. currents), and V the space of efforts e (e.g. voltages). Symmetrized form of power is the indefinite bilinear form, on V V : (f a, e a ), (f b, e b ) := < e a f b > + < e b f a >, (f a, e a ), (f b, e b ) V V.

8 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Definition 2 A (constant) Dirac structure is a subspace D V V such that D = D, where denotes orthogonal complement with respect to the bilinear form,. Key element in the definition of port-hamiltonian systems

9 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, An k dimensional storage element is determined by a k-dimensional state vector x = (x 1,, x k ) and a Hamiltonian H(x 1,, x k ) (energy storage), defining the lossless system ẋ i = f Si, i = 1,, k e Si = H x i (x 1,, x k ) d dt H = k i=1 f Sie Si Such a k- dimensional storage component is written in vector notation as: ẋ = f S e S = H x (x) The elements of x are called energy variables, those of H x (x) co-energy variables.

10 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Geometric definition of a port-hamiltonian system without energy-dissipation f S = ẋ f x f P H D(x) e S = H x (x) e x e P Figure 1: Port-Hamiltonian system The dynamical system defined by the DAEs ( ẋ(t) = f x (t), H x (x(t)) = e x(t), f P (t), e P (t)) D(x(t)), is called a port-hamiltonian system. t R

11 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Basic property dh dt (x(t)) = H x (x(t))ẋ(t) = et x (t)f x (t) = e T P(t)f T P (t) Example: The ubiquitous mass-spring-damper system: Two storage elements: Spring Hamiltonian H s (q) = 1 2 kq2 (potential energy) q = f s = velocity e s = dh s dq (q) = kq = force Mass Hamiltonian H m (p) = 1 2m p2 (kinetic energy) ṗ = f m = force e m = dh m dp (p) = p m = velocity

12 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, interconnected by the Dirac structure f s = e m = y, f m = e s + u (power-conserving since f s e s + f m e m = uy) yields the port-hamiltonian system q = ṗ H q H p (q, p) + 0 u (q, p) 1 y = [ ] 0 1 H q H p (q, p) (q, p) with H(q, p) = H s (q) + H m (p)

13 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Power-dissipation is included by adding an extra port to the Dirac structure, terminated by power-conserving relations: R(f R, e R ) = 0, e T Rf T R 0 Example: For the mass-spring system, the addition of the damper e d = dr df d = cf d, R(f d ) = 1 2 cf2 d (Rayleigh function) via the extended interconnection (Dirac structure) f s = e m = f d = y, f m = e s e d + u leads to the mass-damper-spring system q = ( ) H q (q, p) + H ṗ c p (q, p) [ ] y = 0 1 H q (q, p) (q, p) H p 0 u 1

14 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example: Electro-mechanical systems q ṗ = ϕ R H q H p H ϕ (q, p, φ) 0 (q, p, φ) + 0 V, (q, p, φ) 1 I = H ϕ (q, p, φ) Coupling electrical/mechanical domain via Hamiltonian H(q, p, φ).

15 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example: LC circuits Two inductors with magnetic energies H 1 (ϕ 1 ), H 2 (ϕ 2 ) (ϕ 1 and ϕ 2 magnetic flux linkages), and capacitor with electric energy H 3 (Q) (Q charge). V denotes the voltage of the source. L 1 L 2 C ϕ 1 Q ϕ 2 V

16 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Hamiltonian equations for the components of the LC-circuit: Inductor 1 ϕ 1 = f 1 (voltage) (current) e 1 = H 1 ϕ 1 Inductor 2 ϕ 2 = f 2 (voltage) (current) e 2 = H 2 ϕ 2 Capacitor Q = f 3 (current) (voltage) e 3 = H 3 Q All are port-hamiltonian systems with J = 0 and g = 1. If the elements are linear then the Hamiltonians are quadratic, e.g. H 1 (ϕ 1 ) = 1 2L 1 ϕ 2 1, and H 1 ϕ 1 = ϕ 1 L 1 = current, etc.

17 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Kirchhoff s interconnection laws in f 1, f 2, f 3, e 1, e 2, e 3, f = V, e = I are f e 1 f e 2 = f e 3 e f Substitution of eqns. of components yields port-hamiltonian system ϕ 1 ϕ 2 = Q e = H ϕ H ϕ 1 H ϕ 2 H Q f 0 with H(ϕ 1, ϕ 2, Q) := H 1 (ϕ 1 ) + H 2 (ϕ 2 ) + H 3 (Q) total energy.

18 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, However, this class of port-hamiltonian systems is not closed under interconnection: Figure 2: Capacitors and inductors swapped Interconnection leads to algebraic constraints between the state variables Q 1 and Q 2.

19 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Network modeling is prevailing in modeling and simulation of lumped-parameter physical systems (multi-body systems, electrical circuits, electro-mechanical systems, hydraulic systems, robotic systems, etc.), with many advantages: Modularity and flexibility. Re-usability ( libraries ). Multi-physics approach. Suited to design/control. Disadvantage of network modeling: it generally leads to a large set of DAEs, seemingly without any structure. Port-based modeling and port-hamiltonian system theory identifies the underlying structure of network models of physical systems, to be used for analysis, simulation and control.

20 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, For many systems, especially those with 3-D mechanical components, the interconnection structure will be modulated by the energy or geometric variables. This leads to the notion of non-constant Dirac structures on manifolds. Definition 3 Consider a smooth manifold M. A Dirac structure on M is a vector subbundle D TM T M such that for every x M the vector space D(x) T x M TxM is a Dirac structure as before.

21 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example: Mechanical systems with kinematic constraints Constraints on the generalized velocities q: A T (q) q = 0. This leads to constrained Hamiltonian equations q = H p (q, p) ṗ = H q (q, p) + A(q)λ + B(q)f 0 = A T (q) H p (q, p) e = B T (q) H p (q, p) with H(q, p) total energy, and λ the constraint forces. Dirac structure is defined by the symplectic form on T Q together with constraints A T (q) q = 0 and force matrix B(q). Can be systematically extended to general multi-body systems.

22 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 4 (Rolling coin) Let x, y be the Cartesian coordinates of the point of contact of the coin with the plane. Furthermore, ϕ denotes the heading angle, and θ the angle of Queen Beatrix head (on the Dutch version of the euro). The rolling constraints are ẋ = θ cosϕ, ẏ = θ sin ϕ (1) (rolling without slipping). The total energy is H = 1 2 p2 x p2 y p2 θ p2 ϕ, and the constraints can be rewritten as p x = p θ cosϕ, p y = p θ sinϕ. (2)

23 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Mathematical intermezzo: Jacobi identity and holonomic constraints There is an important notion of integrability of a Dirac structure on a manifold. Definition 5 A Dirac structure D on a manifold M is called integrable if < L X1 α 2 X 3 > + < L X2 α 3 X 1 > + < L X3 α 1 X 2 >= 0 for all (X 1, α 1 ), (X 2, α 2 ), (X 3, α 3 ) D. For constant Dirac structures the integrability condition is automatically satisfied. The Dirac structure D defined by the canonical symplectic structure and kinematic constraints A T (q) q = 0 satisfies the integrability condition if and only if the constraints are holonomic; that is, can be integrated to geometric constraints φ(q) = 0.

24 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Examples (a) Let J be a (pseudo-)poisson structure on M, defining a skew-symmetric mapping J : T M TM. Then graph J T M TM is a Dirac structure. Integrability is equivalent to the Jacobi-identity for the Poisson structure. (b) Let ω be a (pre-)symplectic structure on M, defining a skew-symmetric mapping ω : TM T M. Then graph ω TM T M is a Dirac structure. Integrability is equivalent to the closedness of the symplectic structure. (c) Let K be a constant-dimensional distribution on M, and let annk be its annihilating co-distribution. Then K annk TM T M is a Dirac structure. Integrability is equivalent to the involutivity of distribution K.

25 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Input-state-output port-hamiltonian systems: Particular case is a Dirac structure D(x) T x X T x X F F given as the graph of the skew-symmetric map f x = J(x) g(x) e x e P g T (x) 0 f P leading (f x = ẋ, e x = H x (x)) to a port-hamiltonian system as before ẋ = J(x) H x (x) + g(x)f P, x X, f P R m, e P = g T (x) H x (x), e P R m

26 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Power-dissipation is included by terminating some of the ports by static resistive elements f R = F(e R ), where e T RF(e R ) 0, for alle R. d dt H et Pf P This leads, e.g. for linear damping, to input-state-output port-hamiltonian systems in the form ẋ = [J(x) R(x)] H x (x) + g(x)f P e P = g T (x) H x (x) where J(x) = J T (x), R(x) = R T (x) 0 are the interconnection and damping matrices, respectively.

27 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Multi-modal physical systems Physical systems with switching constraints and/or switching network topology: locomotion behavior of robots and animals, power converters with switches and diodes, systems with inequality constraints. Many multi-modal physical systems can be formulated as port-hamiltonian systems with switching Dirac structure.

28 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 6 (Boost converter) The circuit consists of an inductor L with magnetic flux linkage φ L, a capacitor C with electric charge q C and a resistance load R, together with a diode and an ideal switch S, with switch positions s = 1 (switch closed) and s = 0 (switch open). The diode is modeled as an ideal diode: v D i D = 0, v D 0, i D 0. (3) we Port-Hamiltonian model (with H = 1 q C = 1 R 1 s H q C = q C C H φ L s 1 0 φ L = φ L L 2C q2 C + 1 2L φ2 L ): E + si D (s 1)v D I = φ L L (4)

29 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 7 (Bouncing pogo-stick) Consider a vertically bouncing pogo-stick consisting of a mass m and a massless foot, interconnected by a linear spring (stiffness k and rest-length x 0 ) and a linear damper d. g m spring/damper y x d k sum of forces in series foot fixed to ground spring/damper parallel zero on foot Figure 3: Model of a bouncing pogo-stick: definition of the variables (left), situation without ground contact (middle), and situation with ground contact (right).

30 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The mass can move vertically under the influence of gravity g until the foot touches the ground. The states of the system are x (length of the spring), y (height of the bottom of the mass), and p (momentum of the mass, defined as p := mẏ). Furthermore, the contact situation is described by a variable s with values s = 0 (no contact) and s = 1 (contact). The Hamiltonian of the system equals H(x, y, p) = 1 2 k(x x 0) 2 + mg(y + y 0 ) + 1 2m p2 (5) where y 0 is the distance from the bottom of the mass to its center of mass.

31 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, When the foot is not in contact with the ground total force on the foot is zero (since it is massless), which implies that the spring and damper force must be equal but opposite. When the foot is in contact with the ground, the variables x and y remain equal, and hence also ẋ = ẏ. For s = 0 (no contact) the system is described by the port-hamiltonian system d dt y = p 0 1 mg p 1 0 m (6) dẋ = k(x x 0 )

32 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, while for s = 1 the port-hamiltonian description is d dt x k(x x 0 ) y = mg (7) p 1 1 d p m

33 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The two situations can be taken together into one port-hamiltonian system with variable Dirac structure: d dt s 1 x d 0 s k(x x 0 ) y = mg (8) p s 1 sd The conditions for switching of the contact are functions of the states, namely as follows: contact is switched from off to on when y x crosses zero in the negative direction, and contact is switched from on to off when the velocity ẏ ẋ of the foot is positive in the no-contact situation, i.e. when p m + k d (x x 0) > 0. p m

34 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, In both examples above we obtain a switching port-hamiltonian system, specified by a Dirac structure D s depending on the switch position s {0, 1} n (here n denotes the number of independent switches), a Hamiltonian H : X R and a resistive structure R. Furthermore, every switching may be internally induced (like in the case of a diode in an electrical circuit or an impact in a mechanical system) or externally triggered (like an active switch in a circuit or mechanical system). Problems Well-posedness questions: e.g., systems with reverse Coulomb friction may have multiple solutions. Computation of the next mode may be difficult. Collision rules. Investigation of limit cycles/periodic orbits.

35 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, REPRESENTATIONS AND TRANSFORMATIONS Dirac structures, and therefore port-hamiltonian systems, admit different representations, with different properties for simulation and control. Let D V V, with dim V = n, be a Dirac structure. 1. Kernel and Image representation D = {(f, e) V V Ff + Ee = 0}, for n n matrices F and E (possibly depending on x) satisfying (i) EF T + FE T = 0, (ii) rank[f...e] = n. It follows that D can be also written in image representation as D = {(f, e) V V f = E T λ, e = F T λ, λ R n }.

36 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Constrained input-output representation Every Dirac structure D can be written as D = {(f, e) V V f = Je + Gλ, G T e = 0} for a skew-symmetric matrix J and a matrix G such that im G = {f (f, 0) D}. Furthermore, kerj = {e (0, e) D}.

37 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Hybrid input-output representation Let D be given by square matrices E and F as in 1. Suppose rank F = m( n). Select m independent columns of F, and group them. into a matrix F 1. Write (possibly after permutations) F = [F..F2 1 ], and correspondingly E = [E 1...E2 ], f = f 1 f 2, e = e 1 e 2. Then the matrix [F 1...E2 ] is invertible, and D = f 1 f 2, e 1 e 2 f 1 e 2 = J e 1 f 2 with J := [F 1...E2 ] 1 [F 2...E1 ] skew-symmetric.

38 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Canonical coordinates For simplicity take F F to be void (no external ports). If the Dirac structure on X is integrable then there exist coordinates (q, p, r, s) for X such that D(x) = {(f q, f p, f r, f s, e q, e p, e r, e s ) T x X Tx X } f q = e p, f p = e q f r = 0, 0 = e s Hence the port-hamiltonian system on X takes the form q = H p (q, p, r, s) ṗ = H q (q, p, r, s) ṙ = 0 0 = H s (q, p, r, s)

39 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, DAE representation of port-hamiltonian systems Represent the Dirac structure D in kernel representation as D = {(f x, e x, f, e) F x (x)f x + E x (x)e x + F(x)f + E(x)e = 0}, with (i) E x F T x + F x E T x + EF T + FE T = 0, (ii) rank [F x...ex...f...e] = dim(x F). Since the flows f x and efforts e x corresponding to the energy-storing elements are given respectively as f x = ẋ and e x = H x (x), it follows that the system is described by the set of differential-algebraic equations (DAEs) F x (x(t))ẋ(t) = E x (x(t)) H (x(t)) + F(x(t))f(t) + E(x(t))e(t) x with f, e the external port variables.

40 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Mixture of constrained and hybrid input-output representation By a hybrid input-output partition of the vector of port flows (f, e) F F as (u, y) we can represent any port-hamiltonian system in constrained form as ẋ = J(x) H x (x) + G(x)λ + g(x)u, 0 = G T (x) H x (x) + D(x)u, y = g T (x) H x (x), x X, u Rm y Rm where J(x) = J T (x), D(x) = D T (x) This is the form as encountered before in the case of kinematic constraints.

41 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Intermezzo: Relation with classical Hamiltonian equations ẋ = J(x) H x (x) with constant or integrable J- matrix admits coordinates x = (q, p, r) in which 0 I 0 J = I 0 0, q = H p (q, p, r) ṗ = H q (q, p, r) ṙ = 0 For constant or integrable Dirac structure one gets Hamiltonian DAEs q = H p (q, p, r, s) ṗ = H q (q, p, r, s) ṙ = 0 0 = H s (q, p, r, s)

42 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Recall of Hamiltonian dynamical systems from analytical mechanics Historically, the Hamiltonian approach starts from the principle of least action, via the Euler-Lagrange equations and the Legendre transformation, towards the Hamiltonian equations of motion. The standard Euler-Lagrange equations are given as ( ) d L (q, q) L (q, q) = τ, dt q q where q = (q 1,...,q k ) T are generalized configuration coordinates for the system with k degrees of freedom, the Lagrangian L equals the difference T P between kinetic co-energy T and potential energy P, and τ = (τ 1,...,τ k ) T is the vector of generalized forces acting on the system.

43 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The vector of generalized momenta p = (p 1,...,p k ) T is defined as p = L q = T q By defining the state vector (q 1,...,q k, p 1,...,p k ) T the k second-order equations transform into 2k first-order equations q = H p (q, p) ṗ = H q where the Legendre transform is the total energy of the system. (q, p) + τ H(q, p) = K(q, p) + P(q)

44 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, In standard mechanical systems the kinetic co-energy T is of the form T(q, q) = 1 2 qt M(q) q where the k k inertia (generalized mass) matrix M(q) is symmetric and positive definite for all q. Hence p = M(q) q and because of the fact that the kinetic co-energy T is a quadratic function of the velocities q it equals the kinetic energy K(q, p) = 1 2 pt M 1 (q)p.

45 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The above equations are called the Hamiltonian equations of motion, and H is called the Hamiltonian. The state space with local coordinates (q, p) is called the phase space. The following energy balance immediately follows: d dt H = T H q (q, p) q + T H p (q, p)ṗ = T H p (q, p)τ = qt τ, expressing that the increase in energy of the system is equal to the supplied work (conservation of energy).

46 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, A Hamiltonian system with collocated inputs and outputs 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)u, u Rm, y = B T (q) H p (q, p) (= BT (q) q), y R m, Here B(q) is the input force matrix. In case m < k we speak of an underactuated system. By definition of the output y = B T (q) q we again obtain dh dt (q(t), p(t)) = ut (t)y(t)

47 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, A major generalization of the class of Hamiltonian systems consists in considering systems which are described in local coordinates as ẋ = J(x) H x (x) + g(x)u, x X, u Rm y = g T (x) H x (x), y Rm Here J(x) is an n n matrix which is skew-symmetric: J(x) = J T (x), and x = (x 1,...,x n ) are local coordinates for an n-dimensional state space manifold X. We recover the energy-balance dh dt (x(t)) = ut (t)y(t). In the previous case we had J = 0 I0 I.

48 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 8 Consider a rigid body spinning around its center of mass in the absence of gravity. The energy variables are the three components of the body angular momentum p along the three principal axes: p = (p x, p y, p z ), and the energy is the kinetic energy H(p) = 1 2 ( ) p 2 x + p2 y + p2 z I x I y I z where I x, I y, I z are the principal moments of inertia. Euler s equations are p x 0 p z p y p y = p z 0 p x p z p y p x 0 }{{} J(p) H p x H p y H p z + g(p)u,, y = gt (p) H p

49 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, J(p) is the canonical Lie-Poisson structure matrix on the dual of the Lie algebra so(3) corresponding to the configuration space SO(3) of the rigid body.) Equations arise from the standard (6-dimensional) Hamiltonian equations by reduction ( symmetry ).

50 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Large-scale port-hamiltonian systems, and composition of Dirac structures The composition of two Dirac structures with partially shared variables is again a Dirac structure: D 12 V 1 V1 V 2 V2 D 23 V 2 V2 V 3 V3 V 1 V 1 V 2 D 12 D 23 V 2 } {{ } D 12 D 23 Figure 4: Composed Dirac structure V 3 V 3

51 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, A B f 1 f 2 f 2 f 3 D A D B A B e 1 e 2 e 2 e 3 Figure 5: Standard interconnection f A = f B F 2 e A = e B F2 The gyrating (or feedback) interconnection f A = e B e B = f B can be easily transformed to this case.

52 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Thus D A D B := {(f 1, e 1, f 3, e 3 ) F 1 F 1 F 3 F 3 (f 2, e 2 ) F 2 F 2 s.t. (f 1, e 1, f 2, e 2 ) D A and ( f 2, e 2, f 3, e 3 ) D B } Theorem 9 Let D A, D B be Dirac structures (defined with respect to F 1 F1 F 2 F2, respectively F 2 F2 F 3 F3 and their bilinear forms). Then D A D B is a Dirac structure with respect to the bilinear form on F 1 F1 F 3 F3.

53 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Proof Consider D A, D B defined in matrix kernel representation by D A = {(f 1, e 1, f A, e A ) F 1 F1 F 2 F2 F 1 f 1 + E 1 e 1 + F 2A f A + E 2A e A = 0} D B = {(f B, e B, f 3, e 3 ) F 2 F2 F 3 F3 F 2B f B + E 2B e B + F 3 f 3 + E 3 e 3 = 0} Make use of the following basic fact from linear algebra: ( λ s.t. Aλ = b) [ α s.t. α T A = 0 α T b = 0]

54 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Note that D A, D B are alternatively given in matrix image representation as D A = im E T 1 F T 1 E T 2A F T 2A 0 0 D B = im 0 0 E T 2B F T 2B E T 3 F T 3 Hence, (f 1, e 1, f 3, e 3 ) D A D B λ A, λ B such that

55 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April,

56 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, f 1 e f 3 e 3 = E T 1 0 F T 1 0 E T 2A F T 2A E T 2B F T 2B 0 F T 3 0 E T 3 λ A λ B (β 1, α 1, β 2, α 2, β 3, α 3 ) s.t. E1 T 0 F1 T 0 (β1 T α1 T β2 T α2 T β3 T α3 T E2A T ) F2A T 0 F3 T E T 2B F T 2B = 0, 0 E T 3 β T 1 f 1 + α T 1 e 1 + β T 3 f 3 + α T 3 e 3 = 0

57 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, (α 1, β 1, α 2, β 2, α 3, β 3 ) s.t. F 1 E 1 F 2A E 2A F 2B E 2B F 3 E 3 α 1 β 1 α 2 β 2 α 3 = 0, β 3 β T 1 f 1 + α T 1 e 1 + β T 3 f 3 + α T 3 e 3 = 0 (α 1, β 1, α 3, β 3 ) D A D B, β T 1 f 1 + α T 1 e 1 + β T 3 f 3 + α T 3 e 3 = 0 (f 1, e 1, f 3, e 3 ) (D A D B ) Thus D A D B = (D A D B ), and so it is a Dirac structure.

58 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Explicit expressions for the composition of two Dirac structures Consider Dirac structures D A F 1 F1 F 2 F2, D B F 2 F2 F 3 F3, given by matrix kernel/image representations (F A, E A ) = ([F 1 F 2A ], [E 1 E 2A ]), respectively (F B, E B ) = ([F 2B F 3 ], [E 2B E 3 ]). Define M = F 2A and let L A, L B be matrices with F 2B E 2A E 2B (9) L = [L A L B ], kerl = imm F = [L A F 1 L B F 3 ] E = [L A E 1 L B E 3 ] is a relaxed matrix kernel/image representation of D A D B.

59 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, This relaxed kernel/image representation can be readily understood by premultiplying the equations characterizing the composition of D A with D B F 1 E 1 F 2A E 2A F 2B E 2B F 3 E 3 f 1 e 1 f 2 e 2 f 3 = 0, (10) e 3 by the matrix L := [L A L B ]. Since LM = 0 this results indeed in the relaxed kernel representation L A F 1 f 1 + L A E 1 e 1 + L B F 3 f 3 + L B E 3 e 3 = 0 (11)

60 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Consequence The interconnection of a number of port-hamiltonian systems (X i, D i, H i ), i = 1,, k through an interconnection Dirac structure D I is a port-hamiltonian system (X, D, H), with H = H H k, X = X 1 D k and D the composition of D 1,, D k, D I.

61 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 10 (1-D mechanical system) Consider a spring with elongation q and energy function H s (q) = 1 2 kq2. Let (v s, F s ) represent the external port through which energy can be exchanged with the spring, where v s is equal to the rate of elongation (velocity) and F s is equal to the elastic force. This port-hamiltonian system can be written in kernel representation as 1 1 q kq = 0 (12) 0 0 v s 1 1 F s

62 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Similarly we model a moving mass m with scalar momentum p and kinetic energy H m (p) = 1 2 p2 as the port-hamiltonian system 1 1 ṗ F m p m v m = 0 (13) where (F m, v m ) are respectively the external force exerted on the mass and the velocity of the mass.

63 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The mass and the spring can be interconnected to each other using the symplectic gyrator v s F m = 0 1 F s (14) 1 0 Collecting all equations we have obtained a port-hamiltonian system with energy variables x = (q, p), total energy H(q, p) = H s (q) + H m (p) and with interconnection port variables (v s, F s, F m, v m ). After elimination of the interconnection variables (v s, F s, F m, v m ) one obtains the port-hamiltonian system v m 1 0 q kq = 0 (15) p 0 1 ṗ 1 0 m which is the ubiquitous mass-spring system.

64 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example: Coupled masses Consider two point masses m 1 and m 2 which are rigidly linked to each other. When decoupled the masses are described by the port-hamiltonian systems ṗ i = F i, v i = p i m i i = 1, 2 (16) with F i the force exerted on mass m i. Rigid coupling amounts to F 1 = F 2, v 1 = v 2 (17) This leads to the port-hamiltonian system ṗ1 = 1 λ 1 ṗ 2 0 = [ ] 1 1 p 1 m 1 p 2 m 2 where λ = F 1 = F 2 now denotes the internal constraint force. (18)

65 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The resulting interconnnected system does not have external ports anymore. On the other hand, external ports for the interconnected system can be included by either extending (16) to ṗ i = F i + F ext i v i = p i m i vi ext = p i m i i = 1, 2 (19) with Fi ext and vi ext denoting the external forces and velocities, or by modifying the interconnection constraints (17) to e.g. F 1 + F 2 + F ext = 0, v 1 = v 2 = v ext, (20) with F ext and v ext denoting the external force exerted on the coupled masses, respectively the velocity of the coupled masses.

66 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Consider the port-hamiltonian system ẋ = J(x) H x (x) + g(x)u + b(x)λ Σ : y = g T (x) H x (x) x X 0 = b T (x) H x (x) The Lagrange multipliers λ can be eliminated by constructing a matrix b (x) of maximal rank such that b (x)b(x) = 0

67 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, By premultiplication with b (x) one obtains b (x)ẋ = b (x)j(x) H x (x) + b (x)g(x)u Σ : y = g T (x) H x (x) x X (21) 0 = b T (x) H x (x) This is a kernel representation of the port-hamiltonian system.

68 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 11 (Coupled masses continued) Consider the system of two coupled masses. [ ] Premultiplication of the dynamic equations by the row vector 1 1 yields the equations ṗ 1 + ṗ 2 = 0, p 1 m 1 p 2 m 2 = 0, (22) which constitutes a kernel representation of the port-hamiltonian system.

69 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, A more difficult question concerns the possibility to solve for the algebraic constraints of a port-hamiltonian system: 0 = b T (x) H x (x) (23) Under constant rank assumptions the set X c := {x X b T (x) H x (x) = 0} defines a submanifold of the total state space X; the constrained state space. In order that this constrained state space qualifies as the state space for a port-hamiltonian system without further algebraic constraints one needs to be able to restrict the dynamics of the port-hamiltonian system to the constrained state space. This is always possible under the condition that the matrix b T (x) 2 H (x)b(x) (24) x2

70 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, has full rank, since in this case the differentiated constraint equations 0 = d dt (bt (x) H x (x)) = + bt (x) 2 H (x)b(x)λ (25) x2 can be always uniquely solved for λ.

71 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 12 (Coupled masses continued) Differentiating the constraint equation p 1 m 1 p 2 m 2 = 0 and using ṗ 1 = λ and ṗ 2 = λ one obtains ( 1 m m 2 )λ = 0 (26) which determines the constraint force λ (to be equal to 0). Defining the total momentum p = p 1 + p 2 one obtains the reduced system ṗ = p 1 + ṗ 2 = 0.

72 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, On the other hand, suppose that the mass m 1 is connected to a linear spring with spring constant k 1 and elongation q 1 and that the mass m 2 is connected to a linear spring with spring constant k 2 and elongation q 2. Then the dynamical equations change into ṗ 1 = k 1 q 1 + λ and ṗ 2 = k 2 q 2 λ, and differentiation of the constraint p 1 m 1 p 2 m 2 = 0 leads to k 1 m 1 q 1 + k 2 m 2 q 2 + ( 1 m m 2 )λ = 0 (27) which determines the constraint force λ as λ = m 1m 2 m 1 +m 2 ( k 1 m 1 q 1 k 2 m 2 q 2 ), and results in the dynamical equation for the total momentum p given by ṗ = k 1 q 1 k 2 q 2 (28)

73 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Consider the equations of a general mechanical system subject to kinematic constraints. The constrained Hamiltonian equations define a port-hamiltonian system, with respect to the Dirac structure D (in constrained input-output representation) D = {(f S, e S, f C, e C ) 0 = [ ] 0 A T (q) e S, e C = [ ] 0 B T (q) e S, f S = 0 I n I n 0 e S + 0 A(q) λ + 0 B(q) f c, λ R k }

74 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The algebraic constraints on the state variables (q, p) are 0 = A T (q) H p (q, p) and the constrained state space is X c = {(q, p) A T (q) H p (q, p) = 0}. We may solve for the algebraic constraints and at the same time eliminate the constraint forces A(q)λ in the following way. Since rank A(q) = k, there exists locally an n (n k) matrix S(q) of rank n k such that A T (q)s(q) = 0 Now define p = ( p 1, p 2 ) = ( p 1,..., p n k, p n k+1,..., p n ) as p 1 := S T (q)p, p 2 := A T (q)p, p 1 R n k p 2 R k The map (q, p) (q, p 1, p 2 ) is a coordinate transformation. Indeed, the rows of S T (q) are orthogonal to the rows of A T (q).

75 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, In the new coordinates the constrained Hamiltonian system becomes q p 1 = p 2 0 n S(q) ( S T (q) p T [S i, S j ](q) ) i,j λ + B c (q) u A T (q)a(q) B(q) A T (q) H p = A T (q)a(q) H p 2 = 0 H q H p 1 H p 2 + with H(q, p) the Hamiltonian H expressed in the new coordinates q, p.

76 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Here S i denotes the i-th column of S(q), i = 1,...,n k, and [S i, S j ] is the Lie bracket of S i and S j, in local coordinates given as: [S i, S j ](q) = S j q (q)s i(q) S i q S j(q)

77 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Since λ only influences the p 2 -dynamics, and the constraints A T (q) H H p (q, p) = 0 are equivalently given by p (q, p) = 0, the 2 constrained dynamics is determined by the dynamics of q and p 1 (coordinates for the constrained state space X c ) q p = J c (q, p 1 ) + 0 u, 1 B c (q) H c q(q, p 1 ) H c p 1 (q, p 1 ) where H c (q, p 1 ) equals H(q, p) with p 2 satisfying H p 2 = 0, and where the skew-symmetric matrix J c (q, p 1 ) is given as the left-upper part of the structure matrix, that is J c (q, p 1 ) = O n S(q) ( S T (q) p T [S i, S j ](q) ), i,j where p is expressed as function of q, p, with p 2 eliminated from H p 2 = 0.

78 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Furthermore, in the coordinates q, p, the output map is given in the form y = [ B T c (q) ] B T (q) H p 1 H p 2 which reduces on the constrained state space X c to y = B T c (q) H p 1(q, p1 ) These equations define a port-hamiltonian system on X c, with Hamiltonian H c given by the constrained total energy, and with structure matrix J c.

79 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 13 (Example 4 continued: The rolling euro) Define the new p-coordinates p 1 = p ϕ p 2 = p θ + p x cosϕ + p y sinϕ p 3 = p x p θ cosϕ p 4 = p y p θ sinϕ The constrained state space X c is given by p 3 = p 4 = 0, and the

80 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, dynamics on X c is computed as ẋ 0 cos ϕ ẏ 0 sinϕ θ O = ϕ 1 0 p p 2 cosϕ sinϕ y 1 = y p 2 p 1 H c x H c y H c θ H c ϕ H c p 1 H c p u u where H c (x, y, θ, ϕ, p 1, p 2 ) = 1 2 p p2 2.

81 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Part III: Analysis of port-hamiltonian systems Port-Hamiltonian systems and passivity A square nonlinear system ẋ = f(x) + g(x)u, u R m Σ : y = h(x), y R m where x R n are coordinates for an n-dimensional state space X, is passive if there exists a storage function V : X R with V (x) 0 for every x, such that V (x(t 2 )) V (x(t 1 )) t2 t 1 y T (t)u(t)dt for all solutions (u( ), x( ), y( )) and times t 1 t 2. The system is lossless if is replaced by =.

82 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, If H is differentiable then passive is equivalent to d dt V yt u which reduces to (Willems, Hill-Moylan) T V x (x)f(x) 0 h(x) = g T (x) V x (x) while in the lossless case is replaced by =. In the linear case ẋ = y = Cx Ax + Bu is passive if there exists a quadratic storage function V (x) = 1 2 xt Qx, with Q = Q T 0 satisfying the LMIs A T Q + QA 0, C = B T Q

83 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Clearly, any port-hamiltonian system with Hamiltonian H 0 is passive, since d dt H = et Rf R + e T Pf P e T Pf P and thus H is a storage function. Furthermore, if there are no power-dissipating elements R, then a port-hamiltonian system with H 0 is lossless.

84 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Every linear passive system with storage function V (x) = 1 2 xt Qx, satisfying kerq kera can be rewritten as a linear port-hamiltonian system ẋ = (J R)Qx + Bu, J = J T, R = R T 0 y = B T Qx, in which case the storage function V (x) = 1 2 xt Qx is called the Hamiltonian H. Passive linear systems are thus port-hamiltonian with non-negative Hamiltonian.

85 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Mutatis mutandis most nonlinear lossless systems can be written as a port-hamiltonian system ẋ = J(x) H x (x) + g(x)u with J(x) = J T (x) and H x derivatives. Note that y = g T (x) H x (x) (x) the column vector of partial ẋ = J(x) H x (x) is the internal Hamiltonian dynamics known from physics, which in classical mechanics can be written as q = H p (q, p) ṗ = H q (q, p) with the Hamiltonian H the total (kinetic + potential) energy.

86 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Similarly, most nonlinear passive systems can be written as a port-hamiltonian system (with dissipation) ẋ = [J(x) R(x)] H x (x) + g(x)u y = g T (x) H x (x) with R(x) = R T (x) 0 specifying the energy dissipation d dt H = T H x (x)r(x) H x (x) + ut y u T y

87 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Further analysis of port-hamiltonian systems Port-Hamiltonian systems modeling encodes more information than energy-balance. The Dirac structure determines all the Casimir functions (conserved quantities which are independent of H). Example: In the first LC circuit the total flux φ 1 + φ 2 is a conserved quantity that is solely determined by the circuit topology. (In Part IV this will be used for set-point control.) Furthermore, the Dirac structure directly determines the algebraic constraints. Example: In the second LC-circuit the state variables Q 1 and Q 2 are related by Q 1 C 1 = Q 2 C 2

88 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Port-Hamiltonian systems are more than just energy-conserving. For any Dirac structure D define G 1 := {f x e x, f, e s.t. (f x, e x, f, e) D} T x X P 1 := {e x f x, f, e s.t. (f x, e x, f, e) D} T x X The space G 1 expresses the set of admissible flows, and therefore the conserved quantities or Casimir functions: C is a Casimir function iff C x (f x) = 0 for all f x G 1. Indeed, then (x(t))ẋ(t) = 0 for all solutions. dc dt = C x P 1 determines the set of admissible efforts, and therefore the algebraic constraints: x should satisfy the equations dh(x) P 1 (x). Extension: symmetries and reduction of port-hamiltonian systems.

89 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Pole- and zero-dynamics of port-hamiltonian systems Start with a general port-hamiltonian system in kernel representation F x ẋ = E x H x (x) F RF(e R ) + E R e R + F P f P + E P e P Various pole/zero-dynamics, which inherit the port-hamiltonian structure, can be defined. Simplest two possibilities: f P = 0, or e P = 0 For e P = 0 (while leaving f P free) we obtain the port-hamiltonian system LF x ẋ = LE x H x (x) LF RF(e R ) + LE R e R (29) where L is any matrix of maximal rank satisfying LF P = 0.

90 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Indeed, the equations LF x f x + LE x e x + LF R f R + LE R e R = 0 define the reduced Dirac structure D red F x E x F R E R, which results from interconnection of the original Dirac structure D with the Dirac structure on the space of external port variables F P E P defined by e P = 0. The choice f P = 0 is similar, the difference being that L should now satisfy LE P = 0. For a hybrid partitioning of the port-variables f P, e P, we may define for every subset K {1,, m} the reduced Dirac structure corresponding to setting the variables e Pi, i K, f Pi, i / K, equal to zero (while leaving the complementary part free).

91 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Model reduction of port-hamiltonian systems Network modeling of complex lumped-parameter systems (circuits, multi-body systems) often leads to high-dimensional models. Structure-preserving spatial discretization of distributed-parameter port-hamiltonian systems yields high-dimensional port-hamiltonian models. Lumped-parameter modeling of systems like MEMS gives high-dimensional port-hamiltonian systems. Controller systems may be in first instance distributed-parameter, and need to be discretized to low-order controllers.

92 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, In many cases we want the reduced-order system to be again port-hamiltonian: Port-Hamiltonian model reduction preserves passivity. Port-Hamiltonian model reduction may (approximately) preserve other balance laws /conservation laws. Physical interpretation of reduced-order model. Reduced-order system can replace the high-order port-hamiltonian system in a larger context. Thus there is a need for structure-preserving model reduction of high-dimensional port-hamiltonian systems.

93 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Controllability analysis Consider a linear port-hamiltonian system, written as ẋ = FQx + Bu, F := J R, J = J T, R = R T 0 y = B T Qx, Q = Q T 0 Take linear coordinates x = (x 1, x 2 ) such that the upper part of ẋ1 ẋ 2 = F 11 F 12 F 21 F 22 Q 11 Q 12 Q 21 Q 22 x 1 x 2 + B 1 B 2 u y = [ B T 1 B T 2 ] Q 11 Q 12 Q 21 Q 22 x 1 x 2 is the reachability subspace R.

94 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, By invariance of R this implies F 21 Q 11 + F 22 Q 21 = 0 B 2 = 0 If follows that the dynamics restricted to R is given as ẋ 1 = (F 11 Q 11 + F 12 Q 21 )x 1 + B 1 u y = B T 1 Q 11 x 1 Now solve for Q 21 as Q 21 = F 1 22 F 21Q 11. This yields ẋ 1 = (F 11 F 12 F 1 22 F 21)Q 11 x 1 + B 1 u y = B T 1 Q 11 x 1 which is again a port-hamiltonian system.

95 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Observability analysis Suppose the system is not observable. Then there exist coordinates x = (x 1, x 2 ) such that the lower part is the unobservability subspace N. By invariance of N it follows that F 11 Q 12 + F 12 Q 22 = 0 B1 T Q 12 + B2 T Q 22 = 0 Then the dynamics on the quotient space X/N is ẋ 1 = (F 11 Q 11 + F 12 Q 21 )x 1 + B 1 u y = B T 1 Q 11 x 1 + B T 2 Q 21 x 1

96 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, It follow from that F 12 = F 11 Q 12 Q 1 22 and BT 2 = B T 1 Q 12 Q Substitution yields ẋ 1 = F 11 (Q 11 Q 12 Q 1 22 Q 21)x 1 + B 1 u y = B T 1 (Q 11 Q 12 Q 1 22 Q 21)x 1 which is again a port-hamiltonian system with Hamiltonian H = 1 2 xt 1 (Q 11 Q 12 Q 1 22 Q 21)x 1. Remark Note that the Schur complement (Q 11 Q 12 Q 1 22 Q 21) 0 if Q 0.

97 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, This suggests two canonical ways for structure-preserving model reduction. Effort-constraint method Define e = Qx = e 1 and set e2 = 0: e 2 ẋ 1 = F 11 (Q 11 Q 12 Q 1 22 Q 21)x 1 + B 1 u y = B T 1 (Q 11 Q 12 Q 1 22 Q 21)x 1 Flow-constraint method (if B 2 = 0) Set x 2 = ẋ 2 = 0: ẋ 1 = (F 11 F 12 F 1 22 F 21)Q 11 x 1 + B 1 u y = B T 1 Q 11 x 1

98 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, General structure-preserving model reduction Let us assume that we have been able to find a splitting of the state space variables x = (x 1, x 2 ) having the property that the x 2 coordinates hardly contribute to the external port behavior of the system, and thus could be omitted from the state space description. In which way is it possible to retain the port-hamiltonian structure in model reduction?

99 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Recall that the vector of flow and effort variables is required to be in the Dirac structure (fx, 1 fx, 2 e 1 x, e 2 x, f R, e R, f P, e P ) D, while the flow and effort variables f x, e x are linked to the constitutive relations of the energy-storage by ẋ 1 = f 1 x, ẋ 2 = f 2 x, H x 1 (x 1, x 2 ) = e 1 x H x 2 (x 1, x 2 ) = e 2 x, The basic idea of structure-preserving model reduction is to cut the interconnection ẋ 2 = f 2 x, H x 2 (x1, x 2 ) = e 2 x between the energy storage corresponding to x 2 and the Dirac structure, in such a way that no power is transferred.

100 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, This is done by making both power products ( H x 2 ) T ẋ 2 and (e 2 x) T f 2 x equal to zero. ẋ 1 H x 1 e 1 x f 1 x f R R H D e R e P ẋ 2 H x 2 e 2 x f 2 x f P Figure 6: Model Reduction Scheme

101 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The following main scenario s arise: 1 Set H x 2 (x1, x 2 ) = 0, e 2 x = 0 The first equation imposes an algebraic constraint on the space variables x = (x 1, x 2 ). Under general conditions this constraint allows one to solve x 2 as a function x 2 (x 1 ) of x 1, leading to the reduced Hamiltonian H ec red(x 1 ) := H(x 1, x 2 (x 1 )) Furthermore, the second equation defines the reduced Dirac structure D ec red := {(f 1 x, e 1 x, f R, e R, f P, e P ) f 2 x (f 1 x, e 1 x, f 2 x, 0, f R, e R, f P, e P ) D} such that

102 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, leading to the reduced port-hamiltonian system ( ẋ 1, Hec red x 1 (x1 ), F(e R ), e R, f P, e P ) D ec red This reduction method is the Effort-constraint reduction method.

103 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Set ẋ 2 = 0, f 2 x = 0 The first equation imposes the constraint x 2 = c (constant) and thus defines the reduced Hamiltonian H fc red(x 1 ) := H(x 1, c), while the second equation leads to the reduced Dirac structure D fc red := {(f 1 x, e 1 x, f R, e R, f P, e P ) e 2 x such that (f 1 x, e 1 x, 0, e 2 x, f R, e R, f P, e P ) D} and the corresponding reduced port-hamiltonian system ( ẋ 1, Hfc red x 1 (x1 ), F(e R ), e R, f P, e P ) D fc This is the Flow-constraint reduction method. red

104 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Set ẋ 2 = 0, e 2 x = 0 This leads to the reduced-order port-hamiltonian system with reduced Hamiltonian Hred fc (x1 ) and reduced Dirac structure Dred ec. 4 Set H x 2 (x1, x 2 ) = 0, fx 2 = 0 This leads to the port-hamiltonian system with reduced Hamiltonian Hred ec (x1 ) and reduced Dirac structure Dred fc.

105 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The above reduction schemes have different physical interpretations and consequences. Consider an electrical circuit where x 2 corresponds to the charge Q of a single (linear) capacitor. Application of the Effort-constraint method corresponds to removing the capacitor (and setting its charge equal to zero) and short-circuiting the circuit at the location of the capacitor. The Flow-constraint method corresponds to open-circuiting the circuit at the location of the capacitor, and keeping the charge of the capacitor constant. Method 3 is in this case very similar to the Effort-constraint method, and corresponds to short-circuiting, with the minor difference of setting the charge of the capacitor equal to a constant. Method 4 corresponds to open-circuiting while setting the charge of the capacitor equal to zero (and thus is similar to the

106 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Flow-constraint method).

107 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Explicit equational representations of the four methods starting from the full-order model: F x ẋ = E x H x (x) F RF(e R ) + E R e R + F P f P + E P e P Corresponding to the splitting of the state vector x into x = (x 1, x 2 ) and the splitting of the flow and effort vectors f x, e x into f 1 x, f 2 x and e 1 x, e 2 x we write F x = [ F 1 x F 2 x ], E x = [ E 1 x The reduced Dirac structure Dred ec corresponding to the effort-constraint e 2 x = 0 is given by the explicit equations L ec F 1 xf 1 x + L ec E 1 xe 1 x + L ec F R f R + L ec E R e R + L ec F P f P + L ec E P e P = 0 where L ec is any matrix of maximal rank satisfying L ec F 2 x = 0 E 2 x ]

108 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Similarly, the reduced Dirac structure Dred fc corresponding to the flow-constraint fx 2 = 0 is given by the equations L fc F 1 xf 1 x + L fc E 1 xe 1 x + L fc F R f R + L fc E R e R + L fc F P f P + L fc E P e P = 0 where L fc is any matrix of maximal rank satisfying L fc E 2 x = 0

109 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, It follows that the reduced-order model resulting from applying the Effort-constraint method is given by L ec F 1 xẋ 1 = L ec E 1 x H ec red x 1 (x1 ) L ec F R F(e R )+L ec E R e R +L ec F P f P +L ec E P e P, whereas the reduced-order model resulting from applying the Flow-constraint method is given by L fc F 1 xẋ 1 = L fc E 1 x H fc red x 1 (x1 ) L fc F R F(e R )+L fc E R e R +L fc F P f P +L fc E P e P Similar expressions follow for the reduced-order models arising from applying Methods 3 and 4.

110 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Part V (Thursday morning): Control of port-hamiltonian systems; continued Contents Use of passivity for control Control by interconnection: set-point stabilization The dissipation obstacle A state feedback perspective; shaping the Hamiltonian New control paradigms

111 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Use of passivity for control and beyond The storage function can be used as Lyapunov function, implying some sort of stability for the uncontrolled system. The standard feedback interconnection of two passive systems is again passive, with storage function being the sum of the individual storage functions. Passive systems can be asymptotically stabilized by adding artificial damping. In fact, d dt H ut y together with the additional damping u = y yields d dt H y 2 proving asymptotic stability provided an observability condition is met.

112 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example The Euler equations for the motion of a rigid body revolving about its center of gravity with one input are I 1 ω 1 = [I 2 I 3 ]ω 2 ω 3 + g 1 u I 2 ω 2 = [I 3 I 1 ]ω 1 ω 3 + g 2 u I 3 ω 3 = [I 1 I 2 ]ω 1 ω 2 + g 3 u, Here ω := (ω 1, ω 2, ω 3 ) T are the angular velocities around the principal axes of the rigid body, and I 1, I 2, I 3 > 0 are the principal moments of inertia. The system for u = 0 has the origin as an equilibrium point. Linearization yields the linear system A = B = I 1 1 g 1 I 1 2 g 2 I 1 3 g 3 Hence the linearization does not say anything about stabilizability..

113 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Stability and asymptotic stabilization by damping injection Rewrite the system in port-hamiltonian form by defining the angular momenta p 1 = I 1 ω 1, p 2 = I 2 ω 2, p 3 = I 3 ω 3 and defining the Hamiltonian H(p) as the total kinetic energy H(p) = 1 2 (p2 1 I 1 + p2 2 I 2 + p2 3 I 3 ) Then the system can be rewritten as ṗ 1 ṗ 2 ṗ 3 0 p 3 p 2 = p 3 0 p 1 p 2 p 1 0 H p 1 H p 2 H p 3 + g 1 g 2 g 3 [ ] u, y = g 1 g 2 g 3 H p 1 H p 2 H p 3

114 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Since Ḣ = 0 and H has a minimum at p = 0 the origin is stable. Damping injection amounts to the negative output feedback u = y = g 1 p 1 I 1 g 2 p 2 I 2 g 3 p 3 I 3 = g 1 ω 1 g 2 ω 2 g 3 ω 3, yielding convergence to the largest invariant set contained in S := {p R 3 Ḣ(p) = 0} = {p R 3 g 1 p 1 I 1 + g 2 p 2 I 2 + g 3 p 3 I 3 = 0} It can be shown that the largest invariant set contained in S is the origin p = 0 if and only if g 1 0, g 2 0, g 3 0, in which case the origin is rendered asymptotically stable (even, globally).

115 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Beyond control via passivity: What can we do if the desired set-point is not a minimum of the storage function?? Recall the proof of stability of an equilibrium (ω 1, 0, 0) (0, 0, 0) of the Euler equations. + 2I 2 p I 3 p 2 3 = 1 2 I 1ω I 2ω I 3ω 2 3 has a The total energy H = 2I 1 p 2 1 minimum at (0, 0, 0). Stability of (ω1, 0, 0) is shown by taking as Lyapunov function a combination of the total energy K and another conserved quantity, namely the total angular momentum This follows from C = p p p 2 3 = I 2 1ω I 2 2ω I 2 3ω 2 3 [p 1 p 2 p 3 ] 0 p 3 p 2 p 3 0 p 1 = 0 p 2 p 1 0

116 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, In general, for any Hamiltonian dynamics ẋ = J(x) H x (x) one may search for conserved quantities C, called Casimirs, as being solutions of T C x (x)j(x) = 0 Then d dtc = 0 for every H, and thus also H + C is a candidate Lyapunov function. Note that the minimum of H + C may now be different from the minimum of H.

117 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Control by interconnection: set-point stabilization: Consider first a lossless Hamiltonian plant system P ẋ = J(x) H x (x) + g(x)u y = g T (x) H x (x) where the desired set-point x is not a minimum of the Hamiltonian H, while the Hamiltonian dynamics ẋ = J(x) H x (x) does not possess useful Casimirs. How to (asymptotically) stabilize x?

118 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Control by interconnection: Consider a controller port-hamiltonian system C : ξ = J c (ξ) H c ξ (ξ) + g c(ξ)u c, y c = g T (ξ) H c ξ (ξ) ξ X c via the standard feedback interconnection u = y c, u c = y u P y y c C u c

119 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Then the closed-loop system is the port-hamiltonian system ẋ J(x) g(x)g = c T (ξ) H x (x) ξ g c (ξ)g T H (x) J c (ξ) c ξ (ξ) with state space X X c, and total Hamiltonian H(x) + H c (ξ). Main idea: design the controller system in such a manner that the closed-loop system has useful Casimirs C(x, ξ)! This may lead to a suitable candidate Lyapunov function with H c to-be-determined. V (x, ξ) := H(x) + H c (ξ) + C(x, ξ)

120 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Thus we look for functions C(x, ξ) satisfying [ T C x (x, ξ) ] T C ξ (x, ξ) J(x) g c (ξ)g T (x) g(x)g T c (ξ) J c (ξ) = 0 such that the candidate Lyapunov function V (x, ξ) := H(x) + H c (ξ) + C(x, ξ) has a minimum at (x, ξ ) for some (or a set of) ξ stability. Remark: The set of such achievable closed-loop Casimirs C(x, ξ) can be fully characterized. Subsequently, one may add extra damping (directly or in the dynamics of the controller) to achieve asymptotic stability.

121 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example: the ubiquitous pendulum Consider the mathematical pendulum with Hamiltonian H(q, p) = 1 2 p2 + (1 cos q) actuated by a torque u, with output y = p (angular velocity). Suppose we wish to stabilize the pendulum at a non-zero angle q and p = 0. Apply the nonlinear integral control ξ = u c = y u = y c = H c ξ (ξ) which is a port-hamiltonian controller system with J c = 0.

122 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Casimirs C(q, p, ξ) are found by solving [ C q C p C ξ ] = leading to Casimirs C(q, p, ξ) = K(q ξ), and candidate Lyapunov functions V (q, p, ξ) = 1 2 p2 + (1 cosq) + H c (ξ) + K(q ξ) with the functions H c and K to be determined.

123 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, For a local minimum, determine K and H c such that Equilibrium assignment sinq + K z (q ξ ) = 0 K z (q ξ ) + H c ξ (ξ ) = 0 Minimum condition cosq + 2 K z (q ξ ) 0 2 K 2 z (q ξ ) > 0 2 K z (q ξ ) 0 2 K 2 z (q ξ ) + 2 H c 2 ξ (ξ ) 2 Many possible solutions.

124 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The dissipation obstacle Surprisingly, the presence of dissipation R 0 may pose a problem! C(x) is a Casimir for the Hamiltonian dynamics with dissipation iff ẋ = [J(x) R(x)] H x (x), J = JT, R = R T 0 T C x [J R] = 0 T C [J R] C x x = 0 T C x R C x = 0 T C x R = 0 and thus C is a Casimir iff T C x (x)j(x) = 0, T C x (x)r(x) = 0 The physical reason for the dissipation obstacle is that by using a passive controller only equilibria where no energy-dissipation takes place may be stabilized.

125 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Similarly, if C(x, ξ) is a Casimir for the closed-loop port-hamiltonian system then it must satisfy [ T C x (x, ξ) T C ξ (x, ξ) ] R(x) 0 = 0 0 R c (ξ) implying by semi-positivity of R(x) and R c (x) T C x (x, ξ)r(x) = 0 T C ξ (x, ξ)r c(ξ) = 0 This is the dissipation obstacle, which implies that one cannot shape the Lyapunov function in the coordinates that are directly affected by energy dissipation. Remark: For shaping the potential energy in mechanical systems this is not a problem since dissipation enters in the differential equations for the momenta.

126 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, To overcome the dissipation obstacle Suppose one can find a mapping C : X R m, with its (transposed) Jacobian matrix K T (x) := C x (x) satisfying [J(x) R(x)]K(x) + g(x) = 0 Construct now the interconnection and dissipation matrix of an augmented system as J aug := J K T J By construction JK, R aug := R K T JK K T R [K T (x) I]J aug = [K T (x) I]R aug = 0 RK K T RK implying that the components of C are Casimirs for the Hamiltonian dynamics

127 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, ẋ = [J aug R aug ] ξ H x (x) H c ξ (ξ) Furthermore, since [J(x) R(x)]K(x) + g(x) = 0 J aug R aug = J R [J R]K K T [J R] K T JK K T RK = J R g [g 2RK] T K T JK K T RK Thus the augmented system is a closed-loop system for a different output!

128 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Port-Hamiltonian systems with feedthrough term take the form ẋ = [J(x) R(x)] H x (x) + g(x)u y = (g(x) + 2P(x)) T H x (x) + [M(x) + S(x)]u, with M skew-symmetric and S symmetric, while R(x) P(x) 0 P T (x) S(x)

129 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, The augmented system is thus the feedback interconnection of the nonlinear integral controller ξ = u c y c = H c ξ (ξ) with the plant port-hamiltonian system with modified output with feedthrough term ẋ = [J(x) R(x)] H x (x) + g(x)u y mod = [g(x) 2R(x)K(x)] T H x (x) + [ KT (x)j(x)k(x) + K T (x)r(x)k(x)]u

130 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Generalization to feedback interconnection with state-modulation. Recall that K T (x) := C x (x) is a solution to [J(x) R(x)]K(x) + g(x) = 0. This can be generalized to [J(x) R(x)]K(x) + g(x)β(x) = 0 with β(x) an m m design matrix. The same scheme as above works if we extend the standard feedback interconnection u = y c, u c = y to the state-modulated feedback u = β(x)y c, u c = β T (x)y Note that K(x) is a solution for some β(x) iff g (x)[j(x) R(x)]K(x) = 0 (In fact, β(x) := (g T (x)g(x)) 1 g T (x)[j(x) R(x)]K(x) does the job.)

131 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Further possibilities to generate Lyapunov functions Recall that the set of storage functions H of a passive system generally has aminimal and maximal element (Willems, 1972): S a (x) H(x) S r (x), for all x. where the available storage S a (x) at x is given as T S a (x) = sup u,t 0 0 u T (t)y(t)dt while the required supply S r (x) to reach x at t = 0 starting from x 0 equals S r (x) = 0 inf u,t 0,x( T)=x 0 T u T (t)y(t)dt In the lossless case S a = S r, and thus H is unique.

132 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Let H be a different storage function, then there exist J(x) and R(x) such that [J(x) R(x)] H x (x) = [ J(x) R(x)] H x (x) Hence, the same story as before can be repeated for the new data. Remark: An effective characterization of the class of possible storage functions H, together with a characterization of the achievable Casimirs corresponding to J(x) and R(x) seems to be lacking currently

133 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, A state feedback perspective: shaping the Hamiltonian Restrict (without much loss of generality) to Casimirs of the form C(x, ξ) = ξ j G j (x) It follows that for all time instants ξ j = G j (x) + c j, c j R Suppose that in this way all control state components ξ i can be expressed as function ξ = G(x) of the plant state x. Then the dynamic feedback reduces to a state feedback, and the Lyapunov function H(x) + H c (ξ) + C(x, ξ) reduces to the shaped Hamiltonian H(x) + H c (G(x))

134 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, A direct state feedback perspective: Interconnection-Damping Assignment (IDA)-PBC control A direct way to generate candidate Lyapunov functions H d is to look for state feedbacks u = û IDA (x) such that [J(x) R(x)] H x (x) + g(x)û IDA(x) = [J d (x) R d (x)] H d x (x) where J d and R d are newly assigned interconnection and damping structures. Remark: For mechanical systems IDA-PBC control is equivalent to the theory of Controlled Lagrangians (Bloch, Leonard, Marsden,.).

135 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, For J d = J and R d = R (Basic IDA-PBC) this reduces to [J(x) R(x)] (H d H) x (x) = g(x)û BIDA (x) and thus in this case, there exists an û BIDA (x) if and only if g (x)[j(x) R(x)] (H d H) (x) = 0 x which is the same equation as obtained for stabilization by Casimir generation with a state-modulated nonlinear integral controller! Conclusion: Basic IDA-PBC State-modulated Control by Interconnection.

136 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Shifted passivity w.r.t. a controlled equilibrium (see Jayawardhana, Ortega). Consider a port-hamiltonian system ẋ = Fz + gu, z = H x (x) y = g T z where F = J R, g are constant, and a controlled equilibrium x 0 : Fz 0 + gu 0 = 0, Define the shifted storage function z 0 = H x (x 0) V (x) := H p (x) (x x 0 ) T H p x (x 0) H p (x 0 ) Note that V x = z z 0. It follows that d dt V = (z z 0) T ẋ = (z z 0 ) T (Fz + gu) = (z z 0 ) T F(z z 0 ) + (z z 0 ) T g(u u 0 ) + (z z 0 ) T (Fz 0 + gu 0 ) (y y 0 ) T (u u 0 ) implying passivity w.r.t. the shifted inputs u u 0 and outputs y y 0.

137 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Application to switching control Consider the port-hamiltonian model of a power-converter ẋ = F(ρ)z + g(ρ)e + g l u, z = H p x (x), F(ρ) := J(ρ) R(ρ) with vector of Boolean variables ρ {0, 1} k, H p (x) the total stored electromagnetic energy, and output vector y = g T l z. Let x 0 be an equilibrium of the averaged model, that is F(ρ 0 )z 0 + g(ρ 0 )E + g l u 0 = 0, for some ρ 0 [0, 1] k and u 0. Then ẋ = F(ρ)(z z 0 ) + F(ρ)z 0 + g(ρ)e + g l u z 0 = H x (x 0) = F(ρ)(z z 0 ) + [F(ρ) F(ρ 0 )]z 0 + [g(ρ) g(ρ 0 )]E + g l (u u 0 ) +F(ρ 0 )z 0 + g(ρ 0 )E + g l u 0 = F(ρ)(z z 0 ) + [F(ρ) F(ρ 0 )]z 0 + [g(ρ) g(ρ 0 )]E + g l (u u 0 )

138 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, For many power converters we know that and thus ẋ = F(ρ)(z z 0 ) + F(ρ) F(ρ 0 ) = p i=1 F i(ρ i ρ 0i ) g(ρ) g(ρ 0 ) = p i=1 g i(ρ i ρ 0i ) p [F i z 0 + g i E](ρ i ρ 0i ) + g l (u u 0 ) i=1 Take as Lyapunov/storage function Then V (x) := H p (x) (x x 0 ) T H p x (x 0) H p (x 0 ) d dt V (x) = [ H p x (x) H p x (x 0)] T ẋ = (z z 0 ) T ẋ = (z z 0 ) T F(ρ)(z z 0 ) + p i=1 (z z 0) T [F i z 0 + g i E](ρ i ρ 0i ) + (z z 0 ) T g l (u u 0 ) with (z z 0 ) T F(ρ)(z z 0 ) 0.

139 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Thus at any time we can choose ρ i {0, 1} such that d dt V (x) (z z 0) T g l (u u 0 ) implying passivity of the switched system with respect to the input vector u u 0 and output vector y y 0 = g T l (z z 0). As a consequence, if the converter is terminated on a static resistive load then the switched converter is (asymptotically) stable around x 0. Thus the voltage over the resistive load can be stabilized around any set-point. This can be immediately generalized to converters connected to a load via a transmission line (see Part VII). Note that for linear capacitors and inductors we have H p (x) = 1 2 xt Qx, V (x) = 1 2 (x x 0) T Q(x x 0 )

140 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, New control paradigms Example: Energy transfer control Consider two port-hamiltonian systems Σ i ẋ i = J i (x i ) H i x i (x i ) + g i (x i )u i y i = g T i (x i) H i x i (x i ), i = 1, 2 Suppose we want to transfer the energy from the port-hamiltonian system Σ 1 to the port-hamiltonian system Σ 2, while keeping the total energy H 1 + H 2 constant.

141 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, This can be done by using the output feedback u 1 = u 2 0 y 1y2 T y 2 y1 T 0 y 1 It follows that the closed-loop system is energy-preserving. However, for the individual energies y 2 d dt H 1 = y T 1 y 1 y T 2 y 2 = y 1 2 y implying that H 1 is decreasing as long as y 1 and y 2 are different from 0. On the other hand, d dt H 2 = y T 2 y 2 y T 1 y 1 = y 2 2 y implying that H 2 is increasing at the same rate. Has been successfully applied to energy-efficient path-following control of mechanical systems (cf. Duindam & Stramigioli).

142 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Impedance control Consider a system with two (not necessarily distinct) ports ẋ = [J(x) R(x)] H x y = g T (x) H x (x) e = k T (x) H x (x) (x) + g(x)u + k(x)f, u, y Rm f, e Rm x X, u Rm The relation between the f and e variables is called the (30) impedance of the (f, e)-port. In Impedance Control (Hogan) one tries to shape this impedance by using the control port corresponding to u, y. Typical application: the (f, e)-port corresponds to the end-point of a robotic manipulator, while the (u, y)-port corresponds to actuation. Basic question: what are achievable impedances of the (f, e)-port?

143 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Conclusions of Part V Beyond passivity by port-hamiltonian systems theory. Control by interconnection and Casimir generation, IDA-PBC control. Allows for physical interpretation of control strategies. Suggests new control paradigms for nonlinear systems. Use of passivity generally yields good robustness, but performance theory is yet lacking. See arjan for further info. See forthcoming book: Modeling and Control of Complex Physical Systems; the Port-Hamiltonian Approach, Geoplex consortium, Springer, 2009.

144 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, VII Part Distributed-parameter port-hamiltonian systems f a e a a b f b e b Figure 7: Simplest example: Transmission line Telegrapher s equations define the boundary control system Q t (z, t) = z I(z, t) = z φ t (z, t) = z V (z, t) = z f a (t) = V (a, t), e 1 (t) = I(a, t) f b (t) = V (b, t), e 2 (t) = I(b, t) φ(z,t) L(z) Q(z,t) C(z)

145 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Transmission line as port-hamiltonian system Define internal flows f x = (f E, f M ) and efforts e x = (e E, e M ): electric flow magnetic flow electric effort magnetic effort f E : [a, b] R f M : [a, b] R e E : [a, b] R e M : [a, b] R together with external boundary flows f = (f a, f b ) and boundary efforts e = (e a, e b ). Define the infinite-dimensional Dirac structure f E = 0 z e E f M e M z 0 f a,b e a,b = e E a,b e M a,b

146 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, This defines a Dirac structure on the space of internal flows and efforts and boundary flows and efforts. Substituting (as in the lumped-parameter case) f E = Q t f M = ϕ f x = ẋ t e E = Q C = H Q e M = ϕ L = H ϕ e x = H x with, for example, quadratic energy density H(Q, ϕ) = 1 2 Q 2 C ϕ 2 L we recover the telegrapher s equations.

147 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Of course, the telegrapher s equations can be rewritten as the linear wave equation 2 Q t 2 = z I t = z t φ L = z 1 L φ t = z 1 L z Q C = 1 LC 2 Q z 2 (provided L(z), C(z) do not depend on z), or similar expressions in φ, I or V. The same equations hold for a vibrating string, or for a compressible gas/fluid in a one-dimensional pipe. Basic question: Which of the boundary variables f a, f b, e a, e b can be considered to be inputs, and which outputs?

148 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Example 2: Shallow water equations; distributed-parameter port-hamiltonian system with non-quadratic Hamiltonian The dynamics of the water in an open-channel canal can be described by t h + v h z h = 0 v g v v with h(z, t) the height of the water at position z, and v(z, t) the velocity (and g gravitational constant). This can be written as a port-hamiltonian system by recognizing the total energy H(h, v) = 1 2 b a [hv 2 + gh 2 ]dz

149 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, yielding the co-energy functions a e h = H h = 1 2 v2 + gh Bernoulli function e v = H v = hv mass flow It follows that the shallow water equations can be written, similarly to the telegraphers equations, as h t (z, t) = z H v v t (z, t) = z H h with boundary variables hv a,b and ( 1 2 v2 + gh) a,b. a Daniel Bernoulli, born in 1700 in Groningen as son of Johann Bernoulli, professor in mathematics at the University of Groningen and forerunner of the Calculus of Variations (the Brachistochrone problem).

150 Port-Hamiltonian Systems: from Geometric Network Modeling to Control, EECI, April, Paying tribute to history: Figure 8: Johann Bernoulli, professor in Groningen Figure 9: Daniel Bernoulli, born in Groningen in 1700.