Port-Hamiltonian Systems:

Transcription

1 Port-Hamiltonian Systems: From Geometric Network Modeling to Control Passivity-Based Control (PBC) RECAP Physical systems satisfy stored energy = supplied energy + dissipated energy This suggest the natural control objective desired stored energy = new supplied energy + desired dissipated energy Essence of PBC (Ortega/Spong, 1989) Module M1: HYCON-EECI Graduate School on Control Dimitri Jeltsema and Arjan van der Schaft April 6 9, 21 PBC = energy shaping + damping assignment Main objective: rendering the (closed-loop) system passive w.r.t. some desired storage function. EECI-M1: Port-Hamiltonian Systems 1 Delft University of Technology April 6 9, 21 2 EECI-M1: Port-Hamiltonian Systems Recall: Passivity of ph Systems Stabilization by Damping Injection The port-hamiltonian system April 6 9, 21 3 EECI-M1: Port-Hamiltonian Systems ẋ =[J(x) R(x)] H x (x)+g(x)u, y = g T (x) H x (x), with state x R n, and port-variables u,y R m, is passive if H[x(t)] H[x()] stored energy for some Hamiltonian H : R n R +. u T (τ)y(τ)dτ, ( ) supplied energy Use storage function (read: Hamiltonian) as yapunov function for the uncontrolled system. Passive systems can be asymptotically stabilized by adding damping via the control. In fact, for a passive port-hamiltonian system we have Ḣ(x) u T y. Hence letting u = K d y, with K d = Kd T, we obtain Ḣ(x) y T K d y, asymptotic stability, provided an observability condition is met (i.e., zero-state detectability of the output). April 6 9, 21 4 EECI-M1: Port-Hamiltonian Systems

2 Stabilization by Damping Injection If H(x) non-negative, total amount of energy that can be extracted from a passive system is bounded, i.e., u T (τ)y(τ)dτ H[x()] <. Energy-Balancing PBC Usually, the point where the open-loop energy is minimal is not of interest. Instead some nonzero eq. point, say x, is desired. Standard formulation of PBC: Find u = β(x)+v s.t. H d [x(t)] H d [x()] stored energy = v T (τ)z(τ)dτ d d (t), diss. energy supplied energy where the desired energy H d (x) has a minimum at x, and z is the new output (which may be equal to y). Hence control problem consist in finding u = β(x)+v s.t. energy supplied by the controller is a function of the state x. April 6 9, 21 5 EECI-M1: Port-Hamiltonian Systems April 6 9, 21 6 EECI-M1: Port-Hamiltonian Systems Energy-Balancing PBC Indeed, from the energy-balance inequality ( ) we see that if we can find a β(x) satisfying β T [x(τ)]y(τ)dτ = H a [x(t)] + κ, for some H a (x), then the control u = β(x)+v will ensure v y is passive w.r.t. modified energy H d (x)=h(x)+h a (x). Energy-Balancing PBC Proposition: Consider the port-hamiltonian system ẋ =[J(x) R(x)] H x (x)+g(x)u, y = g T (x) H x (x). If we can find a function β(x) and a vector function K(x) satisfying [J(x) R(x)]K(x) =g(x)β(x) such that K T i) x (x)= K (x) (integrability); x April 6 9, 21 7 EECI-M1: Port-Hamiltonian Systems April 6 9, 21 8 EECI-M1: Port-Hamiltonian Systems

3 Energy-Balancing PBC Proposition (cont d): ii) K(x )= H x (x ) (equilibrium assignment); iii) K x (x ) 2 H x 2 (x ) (yapunov stability). Then the closed-loop system is a port-hamiltonian system of the form ẋ =[J(x) R(x)] H d x (x), with H d (x)=h(x)+h a (x), K(x)= H a x (x), and x (locally) stable. Energy-Balancing PBC Note that (R(x)=R T (x) ) Ḣ d (x)= T H d x (x)r(x) H d (x). x Also note that x is (locally) asymptotically stable if, in addition, the largest invariant set is contained in { x D T H d x (x)r(x) H } d x (x)=, where D R n. April 6 9, 21 9 EECI-M1: Port-Hamiltonian Systems April 6 9, 21 1 EECI-M1: Port-Hamiltonian Systems Mechanical Systems Consider a (fully actuated) mechanical systems with total energy April 6 9, EECI-M1: Port-Hamiltonian Systems H(q, p)= 1 2 pt M 1 (q)p + P(q), with generalized mass matrix M(q)=M T (q). Assume that the potential energy P(q) is bounded from below. PH structure: q = I k H q + u, H ṗ I k p B(q) J= J T g(q) y = [ B T (q) ] H q. H p Mechanical Systems Clearly, the system has as passive outputs the generalized velocities: Ḣ(q, p)=u T y = u T B T (q) H p (q, p)=ut M 1 (q)p = u T q. Now, the Energy-Balancing PBC design boils down to JK(q, p)=g(q)β(q, p), April 6 9, EECI-M1: Port-Hamiltonian Systems K(x)= H a x (x) which in the fully actuated (B(q)=I k ) case simplifies to K 2 (q, p)= K 1 (q, p)=β(q, p).

4 Mechanical Systems The simplest way to ensure that the closed-loop energy has a minimum at (q, p) = (q,) is to select β(q)= P q (q) K p(q q ), K p = K T p. This gives the controller energy H a (q)= P(q)+ 1 2 (q q ) T K p (q q )+κ, so that the closed-loop energy takes the form H d (q, p)= 1 2 pt M 1 (q)p (q q ) T K p (q q ). Mechanical Systems To ensure that the trajectories actually converge to (q,) we need to render the closed-loop asymptotically stable by adding some damping v = K d H p (q, p)= K d q, as shown before. Note that the energy-balance of the system is now H d [q(t), p(t)] H d [q(), p()] stored energy = v T (τ) q(τ)dτ supplied energy q T (τ)k p q(τ)dτ. diss. energy April 6 9, EECI-M1: Port-Hamiltonian Systems April 6 9, EECI-M1: Port-Hamiltonian Systems Mechanical Systems Exercise: inear RC circuit Observe that the controller obtained via energy-balancing is just the classical PD + gravity compensation controller. However, the design via Energy-Balancing PBC provides a new interpretation of the controller, namely, that the closed-loop u + - y r C energy is (up to a constant) equals to H d (q, p)=h(q, p) u T (τ)y(τ)dτ, i.e., the difference between the open-loop and controller energy. Write the dynamics in PH form. Determine the equilibrium point. Find the passive input and output. Design an Energy-Balancing PBC that stabilizes the admissible equilibrium. April 6 9, EECI-M1: Port-Hamiltonian Systems April 6 9, EECI-M1: Port-Hamiltonian Systems

5 Dissipation Obstacle Unfortunately, EnergyBalancing PBC is stymied by the existence of pervasive dissipation. A necessary condition to satisfy the Energy-Balancing PBC proposition is R(x)K(x)=R(x) H a (x)= dissipation obstacle. x This implies that no damping is present in the coordinates that need to be shaped. Appears in many engineering applications. imitations and the dissipation obstacle can be characterized via the control by interconnection perspective. April 6 9, EECI-M1: Port-Hamiltonian Systems

6 Brayton-Moser Equations Port-Hamiltonian Systems: From Geometric Network Modeling to Control Dimitri Jeltsema and Arjan van der Schaft Module M1 Special Topics HYCON-EECI Graduate School on Control April 6 9, 21 From Port-Hamiltonian Systems to the Brayton-Moser Equations Main Motivation: Stability Theory State-of-the-Art Passivity and Power-Shaping Control Generalization Final Remarks Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 1 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 2 From PH Systems to the Brayton-Moser Equations Consider a port-hamiltonian system without dissipation and external ports ẋ = J(x) H x (x), with J(x)= J T (x). Suppose that the mapping from the energy variables x to the co-energy variables e is invertible, such that ( ) x = ˆx(e)= H e (e), with H (e) the egendre transformation of H(x) given by H (e)=e T x H(x). From PH Systems to the Brayton-Moser Equations Assume that we can find coordinates x =(x q,x p ) T, with dim x q = k and dim x p = n k, such that [ ] B(x) J(x)= B T, (x) with B(x) a k (n k) matrix. Furthermore, assume that H(x q,x p )=H q (x q )+H p (x p ). In that case, the co-hamiltonian can be written as H (e q,e p )=H q (e q )+H p(e p ), Then the dynamics ( ) can also be expressed in terms of e as (e)ė = J(x)e. e2 and thus e 2 q e 2 p [ėq ė p ] [ = B(x) B T (x) ][ eq e p ]. Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 3 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 4

7 From PH Systems to the Brayton-Moser Equations Defining the (mixed-potential) function P(e q,e p,x)=e T q B(x)e p, From PH Systems to the Brayton-Moser Equations Note that the mixed-potential can be written as P(e q,e p,x)=e T q B(x)e p = e T q ẋ q = ẋ T p e p, which represent the instantaneous power flow between Σ q and Σ p. it follows that P e 2 ] q [ėq 2 H = e q ė p P. e 2 p e p Q(e q,e p ) These equations, called the Brayton-Moser equations, can be interpreted as a gradient system with respect to P and the indefinite pseudo-riemannian metric Q. Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 5 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 6 From PH Systems to the Brayton-Moser Equations Recall that dissipation can be included in the PH framework via with Hence we can write e 2 q 2 H e 2 p [ėq ė p ẋ = J(x) H x (x)+g R(x) f R, e R = g T R(x) H x (x), ] [ = f R = D e (e). B(x) B T (x) e R = g T R(x)e. ][ eq e p ] g R (x) D e (e), From PH Systems to the Brayton-Moser Equations For simplicity we assume that so that e 2 q e 2 p [ ] Ik g R =, I n k ] [ [ėq = ė p e R = B(x) B T (x) [ eq e p ]. ][ eq e p ] + D e q D e p Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 7 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 8

8 From PH Systems to the Brayton-Moser Equations Brayton-Moser Equations Finally, defining P(e q,e p,x)=e T q B(x)e p + D (e q,e p ), we (again) obtain the Brayton-Moser equations e 2 q e 2 p et us zoom in to electrical circuits... P ] [ėq = e q ė p P. e p Consider an electrical network Σ with n inductors, n C capacitors, and n R resistors. et i R n and v R n C, then [Brayton and Moser 1964] Σ = Σ p Σ q : (i) di dt = ip(i,v) C(v) dv dt = vp(i,v), ( ) =, with mixed-potential P : R n R n C R. For topologically complete networks P(i,v)=v T Bi + R(i) G(v), D (i,v) Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 9 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 1 Brayton-Moser Equations Main Motivation: Stability Theory Example: Tunnel-diode circuit [Moser 196] Rewrite BM equations as Q(x)ẋ = x P(x), R(i)= 1 2 Ri2 U in i G(v)= v ig (v )dv v T Bi = vi, (B = 1) Mixed-potential function: P(i,v)=iv + 1 v 2 Ri2 U in i ig (v )dv di dt = ip = U in + Ri + v (Σ p ) Σ : C dv dt = vp = i i g (v). (Σ q ) with Observation: Special cases: x = Ṗ = ẋ Q(x)ẋ ( ) ( ) i (i), Q(x)=. v C(v)? P(x) candidate yapunov function. R networks (x = i, Q(x)= (i)): Ṗ = Ṙ ; RC networks (x = v, Q(x)=C(v)): Ṗ = Ġ. However, in general Ṗ = ẋ Q(x)ẋ, or equivalently, Q(x)+Q (x). Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 11 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 12

9 A Family of BM Descriptions { The key observation is to generate a new pair, say Q, P }, such that Q(x)+ Q (x), Q =. For any M = M and λ R, new pairs can be found from Q(x) := [ 2 xp(x)m + λi ] Q(x) P(x) := λp(x)+ 1 2 [ xp(x)] M x P(x). Note that the system behavior is preserved since ẋ = Q 1 x P(x) ẋ = Q 1 x P(x). Example: Tunnel Diode Circuit Obviously, for the tunnel diode circuit Q + Q. However, selecting yields and M = Q(v)= ( 1 ) 1 R, λ = 1, RC ( ) R R C R, vi g (v) P(i, v)= v Hence, it is easily seen that i g (v )dv + 2RC (i i g(v)) R (v U in) 2. min v v i g (v) > RC Q(v)+ Q (v). Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 13 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 14 State-of-the-Art Constructive procedures to obtain stability criteria using the mixed-potential function are given in [Brayton and Moser 1964] Three theorems, each depending on the type of nonlinearities in R or C part. Generalizations can be found in e.g., [Chua and Wang 1978] (for cases that 2 i R = or 2 vg = ); [Jeltsema and Scherpen 25] (RC simultaneously nonlinear). Over the past four decades several notable generalizations of the BM equations itself have been developed, e.g., [Chua 1973] (non-complete networks: Pseudo-Hybrid Content); [Marten et al. 1992] (on the geometrical meaning); [Weiss et al. 1998] (on the largest class of RC networks). PBC of power converters [Jeltsema and Scherpen 24]; Extension to other domains, e.g., [Jeltsema and Scherpen 27]; Passivity and control: Power-Shaping stabilization... Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 15 Energy-Balancing Control [Ortega et al.] To put our ideas into perspective let us briefly recall the principle of Energy-Balancing (EB) Control. Consider general system representation ẋ = f (x)+g(x)u, y = h(x), x R n, u,y R m. ( ) Assume that ( ) satisfies H[x(t)] H[x()] Open-loop stored energy u (τ)y(τ)dτ, Supplied energy with storage function H : R n R. If H, then ( ) is passive wrt (u,y). Usually desired operating point, say x, not minimum of H. Idea is to look for û : R n R m st û [x(τ)]h[x(τ)]dτ = H a [x(t)] H a [x()], for some H a : R n R. Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 16

10 Energy-Balancing Control [Ortega et al.] Energy-Balancing Control [Ortega et al.] Hence, the control u = û(x)+v will ensure that the closed-loop system satisfies H d [x(t)] H d [x()] Closed-loop stored energy v (τ)y(τ)dτ, Supplied energy where H d = H + H a is the closed-loop energy storage. If, furthermore, x = arg min H d (x), then x will be stable equilibrium of the closed-loop system (with yapunov function the difference between the stored and the control energies.) However, applicability of EB severely stymied by the existence of pervasive dissipation. Indeed, since solving for û is equivalent to solving the PDE [ f (x)+g(x)û(x)] x H a (x)= û (x)h(x), where the left-hand side is equal to zero at x, it is clear that the method is only applicable to systems verifying û (x )h(x )=. This is known as the dissipation obstacle. Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 17 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 18 Passivity: Power-Balance Inequality [Jeltsema et al. 23] Extract sources (controls) and rewrite BM equations as Q(x)ẋ = x P(x)+G(x)u. We then observe that if Q(x)+Q (x) the network satisfies the power-balance inequality P[x(t)] P[x()] u (τ)y(τ)dτ, with outputs y = h(x,u)= G (x)q 1 (x)[ x P(x)+G(x)u]. Power as storage function instead of energy. Trivially satisfied by all R and RC networks with passive elements. Notice that y = G (x)ẋ natural derivatives in the output! Control: Power-Shaping Stabilization [Ortega et al. 23] The open-loop mixed-potential is shaped with the control u = û(x), where G(x)û(x)= x P a (x), for some P a : R n R. This yields the closed-loop system Q(x)ẋ = x P d (x), with total power function P d (x)=p(x)+p a (x). The equilibrium x will be stable if x = arg min P d (x). Power-Balancing: closed-loop power function equals difference between open-loop power function and power supplied by controller, i.e., Ṗ a = û (x)h(x,û(x)) = û (x)g (x)ẋ. No dissipation obstacle since û (x )G (x )ẋ =! Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 19 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 2

11 Power-Shaping Stabilization However, as mentioned before, in general Q(x)+Q (x), which requires first the generation of a new pair { Q, P }, such that Q(x)+ Q (x), Q =. Proposition. For any M(x) = M (x) and λ R, new pairs can be found from [ 1 Q(x) := 2 2 xp(x)m(x) ( x M(x) x P(x) ) ] + λi Q(x) Hence, P(x) := λp(x)+ 1 2 [ xp(x)] M(x) x P(x). ẋ = Q 1 x P(x)+G(x)u ẋ = Q 1 x P(x)+ G(x)u, with G(x)= Q(x)G(x). Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 21 Example: Tunnel Diode Circuit Again, selecting M = diag ( 1 R, and Assumption. Q(v)= P(i, v)= ) 1 RC and λ = 1 yields ) (, G = ( R R C R vi g (v) v i g (v )dv + R ), 2RC (i i g(v)) R v2. min v i g (v) > RC v. This assumption can be relaxed applying a preliminary feedback Ra i, with [ ] R a > C min v v i g (v)+r. Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 22 Example: Tunnel Diode Circuit Poincare s emma Using the Power-Shaping procedure we obtain: Proposition. The control with control parameter K > satisfying u = K(v v )+u, (= U in ) K > [1 + R v i g (v )], globally asymptotically stabilizes x = col(i,v ) with yapunov function P(i, v)= v i g (v )dv + 2RC (i i g(v)) 2 + K 2R (v v ) R (v u ) 2. Existence of P follows from Poincare s emma. Indeed, suppose the network is described by ẋ = f (x)+g(x)u, with f : R n R n and f C 1, then there exists a P : R n R s.t. x P = Q(x) f (x) iff x (Q(x) f (x)) = [ x (Q(x) f (x))]. Power-Shaping can be applied to general nonlinear systems! Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 23 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 24

12 Power-Shaping of Nonlinear Syst. [Garcia-Canseco et al. 6] Remark: Assumption. (A.1) There exists Q : R n R n n, Q =, satisfying x (Q(x) f (x)) = [ x (Q(x) f (x))], and Q(x)+Q (x). (A.2) There exists P a : R n R verifying g (x)q 1 (x) x P a =, with g (x)g(x)=, rank{g (x)} = n m; x = arg min P d (x), where P d (x) := x[ x (Q(x ) f (x ))] dx + P a (x). Proposition. Under A.1 and A.2, the control law [ 1 u = g (x)q (x)q(x)g(x)] g (x)q (x) x P a (x) ensures x is (locally) stable with yapunov function P d (x). Observe that Assumption A.1 includes the class of port-hamiltonian systems with invertible interconnection and damping matrices. Indeed, recall ẋ =[J(x) R(x)] x H(x)+g(x)u y = g (x) x H(x). Now, if J(x) R(x) a trivial solution for the PDE x (Q(x) f (x)) = [ x (Q(x) f (x))], is obtained by setting Q(x) = [J(x) R(x)] 1, and f (x)= x H(x). Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 25 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 26 Final Remarks Power-Shaping applicable to general nonlinear systems. Similar to Energy-Balancing. However, no dissipation obstacle involved. Tunnel diode example shows that Power-Shaping yields a simple linear (partial) state-feedback controller that ensures robust global asymptotic stability of the desired equilibrium point. Current research includes: Solvability of the PDE x (Q(x) f (x)) = [ x (Q(x) f (x))] subject to Q(x)+Q (x) for different kind of systems (e.g., mechanical, electromechanical, hydraulic, etc.). Connections with IDA-PBC. Distributed-parameter systems... Control of chemical reactors (see pub list)... The real voyage of discovery consists not in seeking new landscapes but in having new eyes. Marcel Proust Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 27 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 28

13 Selected References Brayton, R.K. and Moser, J.K., A theory of nonlinear networks - I, Quart. of App. Math., vol. 22, pp 1 33, April, Moser, J.K., Bistable systems of differential equations with aplications to tunnel diode circuits, IBM Journal of Res. Develop., Vol. 5, pp , 196. Chua,.O., Wang, N.N.; Complete stability of autonomous reciprocal nonlinear networks, Int. Journal of Circ. Th. and Appl., July 1978, vol. 6, (no. 3): Jeltsema, D. and Scherpen, J.M.A., On Brayton and Moser s Missing Stability Theorem, IEEE Transactions on Circuits and Systems-II, Vol. 52, No. 9, 25, pp Chua,.O., Stationary principles and potential functions for nonlinear. networks,. J. Franklin Inst.,. vol. 296, no. 2, pp , Aug Marten, W., Chua,.O., and Mathis, W., On the geometrical meaning of pseudo hybrid content and mixed-potential, Arch. f. Electron. u. Übertr., Vol. 46, No. 4, Weiss,., Mathis, W. and Trajkovic,., A generalization of Brayton-Moser s mixed potential function, IEEE Trans. Circuits and Systems - I, April Jeltsema, D. and Scherpen, J.M.A., Tuning of Passivity-Preserving Controllers for Switched-Mode Power Converters, IEEE Trans. Automatic Control, Vol. 49, No. 8, August 24, pp Selected References Jeltsema, D., Ortega, R. and Scherpen, J.M.A., On Passivity and Power-Balance Inequalities of Nonlinear RC Circuits, IEEE Trans. on Circ. and Sys.-I: Fund. Th. and Appl., Vol. 5, No. 9, 23, pp Ortega, R., Jeltsema, D. and Scherpen, J.M.A., Power shaping: A new paradigm for stabilization of nonlinear RC circuits, IEEE Trans. Aut. Contr., Vol. 48, No. 1, 23. Garcia-Canseco, E., Ortega, R., Scherpen, J.M.A. and Jeltsema, D., Power shaping control of nonlinear systems: a benchmark example, In Proc. IFAC Workshop on agrangian and Hamiltonian Methods for Nonlinear Control, Nagoya, Japan (July 26). Brayton, R.K. and Miranker, W.., A Stability Theory for Nonlinear Mixed Initial Boundary Value Problems, Arch. Ratl. Mech. and Anal. 17, 5, , December Justh, E.W., and Krishnaprasad, P.S., Pattern-Forming Systems for Control of arge Arrays of Actuators, Journal of Nonlinear Science, Vol. 11, No. 4, January, 21. Jeltsema, D., and Van der Schaft, A.J., Pseudo-Gradient and agrangian Boundary Control Formulation of Electromagnetic Fields, J. Phys. A: Math. Theor., vol. 4, pp , 27. Garcia-Canseco, E., Jeltsema, D., Scherpen, J.M.A., and Ortega, R., Power-based control of physical systems: two case studies, in proc. 17th IFAC World Congress, Seoul, Korea, July 28. Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 29 Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 3 Some Recent Developments Favache Audrey and Dochain Denis, Analysis and control of the exothermic control stirred tank reactor: the power-shaping approach, in proc. 48th IEEE Conference on Decision and Control (CDC 9), 29, pp Favache Audrey and Dochain Denis, Power-shaping control of reaction systems: the CSTR case study, to appear in Automatica, 21. R. Ortega, A.J. van der Schaft, F. Castanos and A. Astolfi, Control by interconnection and standard passivity-based control of port-hamiltonian system, IEEE Transactions on Automatic Control, vol.53, pp , 28. E. Garcia-Canseco, D. Jeltsema, R. Ortega and J.M.A. Scherpen, Power-based control of physical systems, Automatica, Volume 46, Issue 1, January 21, Pages Module M1, Special topics EECI-M1: Port-Hamiltonian Systems 31