Port-Hamiltonian Based Modelling.

Transcription

1 Modelling and Control of Flexible Link Multi-Body Systems: Port-Hamiltonian Based Modelling. Denis MATIGNON & Flávio Luiz CARDOSO-RIBEIRO and July 8th, 2017 HAMECMOPSYS D. Matignon Port-Hamiltonian Based Modelling 1 / 38

2 Overview 1 Introduction 2 Finite-dimensional case: Ordinary Differential Equations (ODEs) Useful Tools Closed Systems Open Systems A short word on dissipation? 3 Infinite-dimensional case: Partial Differential Equations (PDEs) New Useful Tools Closed systems Open systems 4 Outlook

3 Introduction Introduction The port-hamiltonian formalism brings together: Port-based modeling approach (bond-graph); Geometric mechanics (coordinate-free); Systems and control theory. D. Matignon Port-Hamiltonian Based Modelling 2 / 38

4 Introduction Bond graphs provide a unified framework for the modeling of physical systems Henry Paynter (1959) Different domains (mechanical, electrical, hydraulic, thermal) Energy is the lingua franca; Complex systems are written as a composition of ideal components: energy-storage, energy-dissipation, energy-routing, etc. D. Matignon Port-Hamiltonian Based Modelling 3 / 38

5 Introduction Bond graph vs block diagram Block diagram Bond Graph Bonds are energy-conserving and acausual! D. Matignon Port-Hamiltonian Based Modelling 4 / 38

6 Introduction Bond graphs provides a unified language for systems of different physical domains Mass-spring-damper RLC Flow variables (f) speed current Effort variables (e) force voltage I storage (I) mass (Inertia) inductor C storage (C) spring capacitor Dissipation element (R) damper resistor D. Matignon Port-Hamiltonian Based Modelling 5 / 38

7 Introduction Bond graphs are used to model COMPLEX systems Many commercial software based on bond-graphs: AMEsim 20-sim (U. Twente) Simscape (Matlab/Simulink) Dymola (Catia / Dassault) D. Matignon Port-Hamiltonian Based Modelling 6 / 38

8 Introduction Port-Hamiltonian systems = bond graph elements + Hamiltonian Storage function is the Hamiltonian (energy) x H(x) Storage flow variable: f c := ẋ, Storage effort variable: e c := H x (x) = grad xh(x), Energy flow: Ḣ(t) = T H x ẋ = et c f c = e T p f p + e T r f r D. Matignon Port-Hamiltonian Based Modelling 7 / 38

9 Introduction Simple example of port-hamiltonian system From Newton 2nd Law: System energy: mẍ + kx = F ext, E = m 2 ẋ2 + k 2 x2 By choosing p = mẋ then H(p, x) = 1 2m p2 + k 2 x2 [ṗ ] ẋ [ ] [ 0 1 p ] = m kx ẋ = [ 1 0 ] [ ] p m kx [ ] 1 F 0 ext, D. Matignon Port-Hamiltonian Based Modelling 8 / 38

10 Introduction Simple example of port-hamiltonian system From Newton 2nd Law: System energy: mẍ + kx = F ext, E = m 2 ẋ2 + k 2 x2 By choosing p = mẋ then H(p, x) = 1 2m p2 + k 2 x2 [ṗ ] [ ] [ ] 0 1 ph = + ẋ 1 0 xh }{{}}{{}}{{} f c J e c [ ] [ ] ph 1 0 xh }{{} ẋ = f p [ ] 1 F 0 }{{} ext, e p D. Matignon Port-Hamiltonian Based Modelling 8 / 38

11 Introduction Simple example of port-hamiltonian system From Newton 2nd Law: System energy: mẍ + kx = F ext, E = m 2 ẋ2 + k 2 x2 By choosing p = mẋ then H(p, x) = 1 2m p2 + k 2 x2 [ṗ ] [ ] [ ] 0 1 ph = + ẋ 1 0 xh }{{}}{{}}{{} f c J e c [ ] [ ] ph 1 0 xh }{{} ẋ = f p [ ] 1 F 0 }{{} ext, e p = Ḣ = F extẋ F ext and ẋ are the interconnection ports. D. Matignon Port-Hamiltonian Based Modelling 8 / 38

12 Introduction Typical mathematical representation of finite-dimensional port-hamiltonian systems ẋ = J H x + Bu, y = B T H x. where: H(x): system Hamiltonian; x R n : energy variables; u R m : inputs; y R m : outputs; J: interconnection matrix (skew-symmetric); = Ḣ = y T u D. Matignon Port-Hamiltonian Based Modelling 9 / 38

13 Introduction The interconnection of two (or N) phs is still a phs Individual systems H 1 ẋ 1 = J 1 + B 1u 1, x 1 y 1 = B T H 1 1, x 1 H 2 ẋ 2 = J 2 + B 2u 2, x 2 y 2 = B T H 2 2, x 2 InterconnectionH(x 1, x 2) = H 1(x 1) + H 2(x 2) u 1 = y 2 + u e, u 2 = y 1 = Coupled system dh dt ] [ [ẋ1 J1 B 1B T 2 = ẋ 2 = y T e u e y e = [ B T 1 ] [ H ] B 2B T x 1 H + 1 J 2 x 2 0 ] [ ] H x 1 H x 2 [ ] B1 u 0 e, D. Matignon Port-Hamiltonian Based Modelling 10 / 38

14 Introduction PHSs are convenient for (non-linear) control design Typical port-hamiltonian representation: We ve seen that: Ḣ = y T u ẋ = J H (x) + Bu, x y = B T H x (x). What happens if u = K(x)y, with K(x) > 0? Ḣ = y T K(x)y 0, If H(x) is lower bounded, the controlled system is stable! D. Matignon Port-Hamiltonian Based Modelling 11 / 38

15 Introduction General ideas on Port Hamiltonian Systems (phs) 1 strongly structured mathematical dynamical systems: both linear and non-linear, both finite-dimensional and infinite-dimensional, 2 based on physical grounds, allowing for different modelling levels, 3 all physics permitted: solid mechanics, structural mechanics, fluid mechanics, electromagnetism, electrical circuits,... 4 comes along with specific numerical methods, which do preserve, at the discrete level, the structure of the continuous equations, 5 allows for open dynamical systems, with interacting ports, 6 modularity: interconnection of sub-systems, e.g. Multi-Body Systems, and... easy multiphysics modelling, e.g. Fluid-Structure Interaction, 7 physically-based strategy for control and stabilization 1, 8 extensions to dissipative dynamical systems are available. 1 wait till... Part II, by Paul Kotyczka, this afternoon! D. Matignon Port-Hamiltonian Based Modelling 12 / 38

16 Introduction Outline 1 Introduction 2 Finite-dimensional case: Ordinary Differential Equations (ODEs) Useful Tools Closed Systems Open Systems A short word on dissipation? 3 Infinite-dimensional case: Partial Differential Equations (PDEs) New Useful Tools Closed systems Open systems 4 Outlook D. Matignon Port-Hamiltonian Based Modelling 13 / 38

17 Overview 1 Introduction 2 Finite-dimensional case: Ordinary Differential Equations (ODEs) Useful Tools Closed Systems Open Systems A short word on dissipation? 3 Infinite-dimensional case: Partial Differential Equations (PDEs) New Useful Tools Closed systems Open systems 4 Outlook

18 ODEs Useful Tools Useful Tools in Finite Dimension These 3 tools will be of major help in the study: the gradient: to be able to compute grad X H(X), when X R 2n, skew symmetric matrices: J T = J, in the special case of quadratic Hamiltonian, hence linear dynamical systems Ẋ = A X: matrix exponentials. Straightforward consequences on stability: 1 spec(a) C 0 = (exp(t A) 0, as t ), i.e. asymptotic stability. 2 λ spec(a), Re(λ) > 0 = ( exp(t A), as t ), i.e. (exponential) instability. 3 the case with eigenvalues on ir is more subtle, and requires geometric insight on the eigenspaces to be solved: either stability or (polynomial) instability can be found. D. Matignon Port-Hamiltonian Based Modelling 14 / 38

19 ODEs Closed Systems Ex 0: a 1-d.o.f.linear oscillator (1) Original dynamics: mẍ + kx = 0, with (x 0, ẋ 0 ) initial data, (1) usually rewritten in state-space form, as: [ ] [ ] [ ] d x 0 1 = m x, with dt ẋ k 0 ẋ [ ] x0 ẋ 0 initial data. (2) = Choose the Hamiltonian formalism! Energy variables: position q := x, momentum p := m ẋ, D. Matignon Port-Hamiltonian Based Modelling 15 / 38

20 ODEs Closed Systems Ex 0: a 1-d.o.f.linear oscillator (1) Original dynamics: mẍ + kx = 0, with (x 0, ẋ 0 ) initial data, (1) usually rewritten in state-space form, as: [ ] [ ] [ ] d x 0 1 = m x, with dt ẋ k 0 ẋ [ ] x0 ẋ 0 initial data. (2) = Choose the Hamiltonian formalism! Energy variables: position q := x, momentum p := m ẋ, Hamiltonian function: H(X) := 1 2 m p k q2, with X := (q, p), D. Matignon Port-Hamiltonian Based Modelling 15 / 38

21 ODEs Closed Systems Ex 0: a 1-d.o.f.linear oscillator (1) Original dynamics: mẍ + kx = 0, with (x 0, ẋ 0 ) initial data, (1) usually rewritten in state-space form, as: [ ] [ ] [ ] d x 0 1 = m x, with dt ẋ k 0 ẋ [ ] x0 ẋ 0 initial data. (2) = Choose the Hamiltonian formalism! Energy variables: position q := x, momentum p := m ẋ, Hamiltonian function: H(X) := 1 2 m p k q2, with X := (q, p), Co-energy variables: q H = k q force, p H = 1 m p := ẋ velocity, D. Matignon Port-Hamiltonian Based Modelling 15 / 38

22 ODEs Closed Systems Ex 0: a 1-d.o.f.linear oscillator (1) Original dynamics: mẍ + kx = 0, with (x 0, ẋ 0 ) initial data, (1) usually rewritten in state-space form, as: [ ] [ ] [ ] d x 0 1 = m x, with dt ẋ k 0 ẋ [ ] x0 ẋ 0 initial data. (2) = Choose the Hamiltonian formalism! Energy variables: position q := x, momentum p := m ẋ, Hamiltonian function: H(X) := 1 2 m p k q2, with X := (q, p), Co-energy variables: q H = k q force, p H = 1 m p := ẋ velocity, Dynamical system: d dt X = [ with skew-symmetric matrix J := ] [ ] k q = J grad X H(X). 1 m p [ ], i.e. J T = J. D. Matignon Port-Hamiltonian Based Modelling 15 / 38

23 ODEs Closed Systems Ex 0: a 1-d.o.f.linear oscillator (2) Theorem J skew-symmetric = d dth(x(t)) = 0. Hence, the dynamical system is conservative, i.e. H(X(t)) = H(X 0 ). dh dt (t) = (grad d XH(X), dt X) R 2 = (grad X H(X), J grad X H(X)) R 2 = 0, since J is skew symmetric! D. Matignon Port-Hamiltonian Based Modelling 16 / 38

24 ODEs Closed Systems Ex 1: the n-d.o.f. linear oscillator Original dynamics: Mẍ + Kx = 0, with (x 0, ẋ 0 ) initial data, (3) with mechanical parameters M = M T > 0, K = K T 0. Energy variables: q := x R n, p := M ẋ R n, set X = (q, p) R 2n, Hamiltonian function: H(X) := 1 2 pt M 1 p qt K q, Co-energy variables: grad q H = K q, and grad p H = M 1 p := ẋ, Dynamical system in standard form: ] [ ] K q with J := d dt X = [ 0 ] I I 0 [ 0 I I 0 M 1 p is skew-symmetric in R 2n. = J grad X H(X), D. Matignon Port-Hamiltonian Based Modelling 17 / 38

25 ODEs Closed Systems Ex 2: a 1-d.o.f. nonlinear oscillator: the pendulum Original dynamics: J θ + g sin(θ) = 0, with (θ 0, θ 0 ) initial data. (4) = Choose the Hamiltonian formalism! Energy variables: position q := θ, momentum p := J θ, Hamiltonian function: H(X) := 1 2 J p2 + g (1 cos(θ)), Co-energy variables: q H = g sin(θ) torque, p H = 1 J p := θ angular velocity, Dynamical system with X = (q, p): [ ] [ ] d 0 1 g sin(θ) dt X = = J grad 1 0 θ X H(X). Theorem J skew-symmetric = d dth(x(t)) = 0. Hence, the non-linear system is conservative, i.e. H(X(t)) = H(X 0 ), with non-quadratic Hamiltonian. D. Matignon Port-Hamiltonian Based Modelling 18 / 38

26 ODEs Open Systems Open Systems, Ports, and Energy Balance (1) Suppose we have interaction with the environment, by means of: actuators, with control u R m, sensors, with co-localized measurements or observations y R m, then, the port-hamiltonian system (phs) is defined by: Theorem Ẋ = J grad X H(X) + g(x) u(t), (5) y(t) = g(x) T grad X H(X). (6) J skew-symmetric = the system is lossless. Indeed, d dt H(X(t)) = (y(t), u(t)) R m, or H(X(t)) = H(X 0) + t 0 (y(τ), u(τ)) R m dτ. D. Matignon Port-Hamiltonian Based Modelling 19 / 38

27 ODEs Open Systems What about the linear / quadratic case? Suppose the Hamiltonian function is quadratic H(X) := 1 2 XT Q X, with Q = Q T > 0, we then easily compute grad X H(X) = QX, and we can define the closed linear dynamical system: Ẋ = J Q X, that is, the matrix of dynamics reads A := J Q. Let B := g be the control matrix of size n m, then the open dynamical system is given by: Ẋ = J Q X + B u(t), (7) y(t) = B T Q X. (8) that is, the m n observation matrix reads C := B T Q. D. Matignon Port-Hamiltonian Based Modelling 20 / 38

28 ODEs Ex 1: n-d.o.f. damped oscillator A short word on dissipation? Original dynamics: Mẍ+(C+G)ẋ+Kx = 0, with (x 0, ẋ 0 ) initial data, (9) with M = M T > 0, K = K T 0 and C = C T, G = G T. Energy variables: q := x R n, p := M ẋ R n, Hamiltonian function: H(X) = 1 2 pt M 1 p qt K q, Dynamical system in standard form: [ ] d 0 I dt X = grad I (G + C) X H(X) = (J R) grad X H(X), [ ] [ ] 0 I 0 0 with J :=, and R :=. I G 0 C = Examine the two terms separately : Role of the skew-symmetric G matrix, the gyroscopic term. Damping effect is ensured, provided that C = C T 0. D. Matignon Port-Hamiltonian Based Modelling 21 / 38

29 ODEs Ex 1: about the G matrix A short word on dissipation? 1 This matrix is often not considered in modelling processes of damping! Why? When C = 0, whatever G = G T, the system proves conservative: d dt H 0(X(t)) = 0! 2 Is it a naive generalizations due to mathematicians? No! Classical mechanical example: Coriolis force f = ω ẋ, with rotational speed ω = (p, q, r) T, f = G ω ẋ where 0 r q G ω := r 0 p is skew symmetric. q p 0 3 Also in electromagnetics: Lorentz force in a uniform magnetic field. D. Matignon Port-Hamiltonian Based Modelling 22 / 38

30 ODEs A short word on dissipation? Open Systems with Damping, Energy Balance (2) The port-hamiltonian system (phs) is defined by: Theorem Ẋ = (J R) grad X H(X) + g(x) u(t), (10) y(t) = g(x) T grad X H(X). (11) J skew-symmetric and R positive = the system is passive. Indeed, d dt H(X(t)) = (grad XH(X), R grad X H(X)) R n + (y(t), u(t)) R m, (y(t), u(t)) R m. Hence, H(X(t)) H(X 0 ) + t 0 (y(τ), u(τ)) R m dτ. D. Matignon Port-Hamiltonian Based Modelling 23 / 38

31 Overview 1 Introduction 2 Finite-dimensional case: Ordinary Differential Equations (ODEs) Useful Tools Closed Systems Open Systems A short word on dissipation? 3 Infinite-dimensional case: Partial Differential Equations (PDEs) New Useful Tools Closed systems Open systems 4 Outlook

32 PDEs New Useful Tools Useful Notions in Infinite Dimension These notions will be of major help in the following: functions u, instead of vectors X, an infinite-dimensional Hilbert functional space H for functions, instead of a finite-dimensional Euclian vector space R 2n, a Hamiltonian functional H, defined on functions u, instead of a Hamiltonian function defined on vectors, e.g.: H : H R u H(u) := 1 2 L 0 u(z) 2 dz D. Matignon Port-Hamiltonian Based Modelling 24 / 38

33 PDEs New Useful Tools Useful Tools in Infinite Dimension These tools will be of major help in the following: the variational derivative of a functional: δ u H, in place of the gradient of the function, defined by H(u + ε w) = H(u) + ε (δ u H, w) H + O(ε 2 ) N.B. in the above easy example, δ u H = u. formally skew symmetric operators: J T = J, w.r.t the scalar product in the Hilbert space H, i.e. (u, J v) H = (J u, v) H in the special case of quadratic Hamiltonian, hence linear dynamical systems Ẋ = A X: semigroups. D. Matignon Port-Hamiltonian Based Modelling 25 / 38

34 PDEs Definitions for (linear) phs (1) Consider the dynamical system Closed systems d dt X(z, t) = J δ XH(X) (12) with (quadratic) Hamiltonian functional: H(X) = 1 2 L its (linear) variational derivative reads: 0 X(z, t) T LX(z, t) dz, δ X H(X) = LX(z, t). We suppose that operator J is formally skew-symmetric. Indeed in the linear case, we get back to semigroups, with A = J L: d X(z, t) = (J L) X(z, t). dt D. Matignon Port-Hamiltonian Based Modelling 26 / 38

35 Definitions for phs (2) PDEs Closed systems The energy balance associated to this system is: dh L dt (t) = = 0 L 0 δ X H(X). dx dt dz, δ X H(X).J.δ X H(X) dz, = 0, since J is skew-adjoint? Almost! Be careful, J is skew-adjoint, but only... formally. = Energy flows through the boundary, only! There is no internal dissipation of any kind in this case. = Let us compute things more concretely on two examples: 1 Webster horn equation (linear, space varying coefficients) 2 Euler-Bernoulli beam equation (linear, constant coefficients) D. Matignon Port-Hamiltonian Based Modelling 27 / 38

36 PDEs Closed systems Ex 3: Webster horn equations (1) Consider an axi-symmetric horn with varying cross-section z S(z). energy variables: density ρ, and particle velocity v, with pressure p := c 2 0 ρ, and energy density U(ρ) := c2 0 Hamiltonian H(ρ, v) := L 0 2 ρ 2 0 ρ 2, ( 1 2 ρ 0v 2 + ρ 0 U(ρ)) S(z) dz, co-energy variables: δ ρ H = ρ 0 U (ρ) = 1 ρ 0 p and δ v H = ρ 0 v, the dynamical system reads: [ ] [ d ρ 0 1 = S ( )] [ ] z S δρ H. (13) dt v z 0 δ v H Proposition Operator J is formally skew-symmetric, w.r.t the scalar product (e, f) H := L (e 1 f 1 + e 2 f 2 ) S(z) dz. 0 D. Matignon Port-Hamiltonian Based Modelling 28 / 38

37 PDEs Closed systems Ex 3: Webster horn equations (2) Indeed, for the above horn equation, we can compute: dh L dt (t) = (e 1 f 1 + e 2 f 2 ) S(z) dz, with f = J e, = = 0 L 0 L 0 ( e 1 z (Se 2 ) Se 2 z e 1 ) dz, z (e 1 Se 2 ) dz, = S(0)e 1 (0) e 2 (0) S(L)e 1 (L) e 2 (L), = p(0).s(0)v(0) p(l).s(l)v(l). = p(0).u(0) p(l).u(l). = Energy flows through the boundary, only! Here u := Sv is volume velocity, and p.u is also meaningful. D. Matignon Port-Hamiltonian Based Modelling 29 / 38

38 PDEs Open systems The piezoelectric bending beam equations w(z) z L Classical Euler-Bernoulli equations: ( ) ρs 2 2 w(z, t) = EI 2 t2 z 2 z w. 2 D. Matignon Port-Hamiltonian Based Modelling 30 / 38

39 PDEs Open systems The piezoelectric bending beam equations a b L w(z) z Including the piezoelectric patch rigidity and mass: Π ab (z) (ρs + Π ab (z)ρ ps p) w(z, t) = }{{} t2 (EI + Π z 2 ab (z)e pi p }{{} µ(z) κ(z) ) 2 a b z w. 2 z D. Matignon Port-Hamiltonian Based Modelling 30 / 38

40 PDEs Open systems The piezoelectric bending beam equations a V b L Including the moment due to voltage: w(z) z 2 2 (ρs + Π ab (z)ρ ps p) w(z, t) = }{{} t2 (EI + Π z 2 ab E pi p) }{{} µ(z) κ(z) ) 2 Π ab (z) 1 a b z w + Π ab(z)k pv(z, t). 2 z D. Matignon Port-Hamiltonian Based Modelling 30 / 38

41 PDEs Open systems The piezoelectric bending beam equations a V b L Including the moment due to voltage: w(z) z 2 2 (ρs + Π ab (z)ρ ps p) w(z, t) = }{{} t2 (EI + Π z 2 ab E pi p) }{{} µ(z) κ(z) ( ) µ(z) 2 2 w(z, t) = κ(z) 2 t2 z 2 z w + 2 ) 2 2 z 2 (Π ab(z) }{{} Π ab (z) 1 a b z w + Π ab(z)k pv(z, t). 2 k pv(z, t)). unbounded input operator. Modeling with piezo, see BANKS, SMITH and WANG 1996 Smart Material Structures (Chapter 4), Wiley z D. Matignon Port-Hamiltonian Based Modelling 30 / 38

42 PDEs Open systems Ex 4: Bending eqn as port-hamiltonian system (1) Let us define the energy variables x 1 (z, t) and x 2 (z, t) as: x 1 (z, t) := µ(z)ẇ(z, t) = linear momentum x 2 (z, t) := 2 z2w(z, t) = strain. The system Hamiltonian is given by: H[x 1, x 2 ] = 1 L ( x1 (z, t) 2 ) + κ(z)x 2 (z, t) 2 dz, 2 µ(z) z=0 D. Matignon Port-Hamiltonian Based Modelling 31 / 38

43 PDEs Open systems Ex 4: Bending eqn as port-hamiltonian system (1) Let us define the energy variables x 1 (z, t) and x 2 (z, t) as: x 1 (z, t) := µ(z)ẇ(z, t) = linear momentum x 2 (z, t) := 2 z2w(z, t) = strain. The system Hamiltonian is given by: H[x 1, x 2 ] = 1 L ( x1 (z, t) 2 ) + κ(z)x 2 (z, t) 2 dz, 2 µ(z) z=0 The co-energy variables are the variational derivatives of the Hamiltonian with respect to x 1 and x 2 : e 1 (z, t) := δh = x 1(z, t) = ẇ(z, t) = vertical speed δx 1 µ e 2 (z, t) := δh = κ(z)x 2 (z, t) = κ(z) 2 z2w = restoring torque. δx 2 D. Matignon Port-Hamiltonian Based Modelling 31 / 38

44 PDEs Open systems Ex 4: Bending eqn as port-hamiltonian system (2) The piezoelectric beam equation can thus be rewritten as: [ ] [ ] [ ] [ ] x1 0 2 = z 2 e1 2 x z 2 Π z 0 e } ab (z)k p v(z, t), {{} J where J is a formally skew-symmetric operator on L 2 (0, L). D. Matignon Port-Hamiltonian Based Modelling 32 / 38

45 PDEs Open systems Ex 4: Bending eqn as port-hamiltonian system (2) The piezoelectric beam equation can thus be rewritten as: [ ] [ ] [ ] [ ] x1 0 2 = z 2 e1 2 x z 2 Π z 0 e } ab (z)k p v(z, t), {{} J where J is a formally skew-symmetric operator on L 2 (0, L). The time-derivative of the Hamiltonian is computed as: Ḣ = L z=0 L (e 1 x 1 + e 2 x 2 ) dz, ( ( = e1 2 z 2e z 2Π ab(z)k p v(z, t) ) + e 2 2 z 2e 1) dz, z=0 L b = e }{{} 1 z (e 2 ) + z (e 1 ) e 2 }{{}}{{}}{{} + k p 2 z 2e 1 v(z, t) dz. z=0 z=a }{{} speed force angular vel. torque y v(z,t) D. Matignon Port-Hamiltonian Based Modelling 32 / 38

46 PDEs Open systems Ex 4: Bending eqn as port-hamiltonian system (3) Force, speed v(z, t), y v (z, t) torque, angular speed With distributed input port v(z, t) and output port (conjugated to v(z, t)): And boundary ports*: ze 2(0) e 2(0) y := e = 1(L) ze 1(L) y v(z, t) := k p 2 z 2e1, a < z < b. force torque vert. speed angular speed Ḣ = y T u +, u = b e 1(0) vert. speed ze 1(0) angular speed = ze 2(L) force. e 2(L) torque v(z, t)y v(z, t) dz. z=a * other possible choices of boundary ports: LE GORREC, ZWART, MASCHKE 2005 D. Matignon Port-Hamiltonian Based Modelling 33 / 38

47 Overview 1 Introduction 2 Finite-dimensional case: Ordinary Differential Equations (ODEs) Useful Tools Closed Systems Open Systems A short word on dissipation? 3 Infinite-dimensional case: Partial Differential Equations (PDEs) New Useful Tools Closed systems Open systems 4 Outlook

48 Outlook Summary on port-hamiltonian Systems (phs) In finite-dimension: ẋx = J x H(x) + Bu, u J, B, D y y = B T x H(x) + Du, H(x) where J and D are skew-symmetric matrices. We easily verify the power-balance: Ḣ = y T u. D. Matignon Port-Hamiltonian Based Modelling 34 / 38

49 Outlook Summary on port-hamiltonian Systems (phs) In finite-dimension: ẋx = J x H(x) + Bu, u J, B, D y y = B T x H(x) + Du, H(x) where J and D are skew-symmetric matrices. We easily verify the power-balance: Ḣ = y T u. The infinite-dimensional case (in 1D): ẋx(z, t) = J δ xh(x) + B u(z, t), y(z, t) = B δ xh(x), with boundary control and observation: u = Bx, y = Cx Power-balance: Ḣ = y T u + u(z, t)y(z, t)dz. D. Matignon Port-Hamiltonian Based Modelling 34 / 38

50 Outlook Summary on port-hamiltonian Systems (phs) In finite-dimension: ẋx = J x H(x) + Bu, u J, B, D y y = B T x H(x) + Du, H(x) where J and D are skew-symmetric matrices. We easily verify the power-balance: Ḣ = y T u. The infinite-dimensional case (in 1D): ẋx(z, t) = J δ xh(x) + B u(z, t), y(z, t) = B δ xh(x), with boundary control and observation: u = Bx, y = Cx Power-balance: Ḣ = y T u + u(z, t)y(z, t)dz. Our interest for phs: Interconnection and modularity; Obvious physical properties; Multiphysics; Energy-based control. D. Matignon Port-Hamiltonian Based Modelling 34 / 38

51 Outlook Summary Port-Hamiltonian formulation provides a: modular, physically motivated (energy-based), multi-domain framework for analyzing, simulating and controlling (complex) systems. Also available (not presented today): Damping models, including thermodynamical consistency, Energy preserving simulation in the discrete-time domain, Geometric discretization of PDEs into ODEs in phs form, Worked-out model and simulation a fluid-structure interaction (FSI) problem, like sloshing, see e.g. Realistic PDEs in 2D or 3D domain, with boundary control and observation. D. Matignon Port-Hamiltonian Based Modelling 35 / 38

52 Bibliography Introductory bibliography - Bond graph and phs Gawthrop, Peter J and Bevan, Geraint P. Bond-graph modeling. Control Systems, IEEE, Duindam, Vincent and Macchelli, Alessandro and Stramigioli, Stefano and Bruyninckx, Herman. Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach. Springer Science, van der Schaft, Arjan and Jeltsema, Dimitri. Port-Hamiltonian Systems Theory: An Introductory Overview. Now Publishers Incorporated, van der Schaft, A. and Maschke, B. Hamiltonian formulation of distributed parameter systems with boundary energy flow. J. of Geometry and Physics, 42, (2002). D. Matignon Port-Hamiltonian Based Modelling 36 / 38

53 Bibliography Some specific references on FSI and phs Cardoso-Ribeiro, F.L., Matignon, D. and Pommier-Budinger, V., A port-hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system, Journal of Fluids and Structures, vol. 69 (2017), pp Cardoso-Ribeiro, F.L., Matignon, D. and Pommier-Budinger, V. (2016). Piezoelectric beam with distributed control ports: a power-preserving discretization using weak formulation, in 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE. Cardoso-Ribeiro, F.L., Matignon, D. and Pommier-Budinger, V. (2015). Modeling of a fluid-structure coupled system using port-hamiltonian formulation. In IFAC Workshop on Lagrangian and Hamiltonian Methods in Non Linear Control (LHMNLC 15). Cardoso-Ribeiro, F.L., Pommier-Budinger, V., Schotte, J.S., and Arzelier, D. (2014). Modeling of a coupled fluid-structure system excited by piezoelectric actuators. In IEEE/ASME International Conference on Advanced Intelligent Mechatronics, D. Matignon Port-Hamiltonian Based Modelling 37 / 38

54 Bibliography Some specific references on Damping and phs MATIGNON, D. AND HÉLIE, T. (2013), A class of damping models preserving eigenspaces for linear conservative port-hamiltonian systems. European Journal of Control, 19(6), LE GORREC, Y. AND MATIGNON, D. (2013), Coupling between hyperbolic and diffusive systems: A port-hamiltonian formulation. European Journal of Control, 19(6), HÉLIE, T. AND MATIGNON, D., Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: a port-hamiltonian formulation, in Proc. 5th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Lyon, FR, July 2015, pp LE GORREC, Y. AND MATIGNON, D., Matrix-valued impedances with fractional derivatives and integrals in boundary feedback control: a port-hamiltonian approach, in Proc. 5th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Lyon, FR, July 2015, pp D. Matignon Port-Hamiltonian Based Modelling 38 / 38