Variational integrators for electric and nonsmooth systems

Transcription

1 Variational integrators for electric and nonsmooth systems Sina Ober-Blöbaum Control Group Department of Engineering Science University of Oxford Fellow of Harris Manchester College Summerschool Applied Mathematics and Mechanics: Geometric Methods in Dynamics 5 9 September 216 Sina Ober-Blöbaum p.1

2 Overview Lecture 1: Variational integrators for electric circuits Ober-Blöbaum, Tao, Cheng, Owhadi, Marsden: Variational integrators for electric circuits. Journal of Computational Physics, 213. Lecture 2: Variational integrators for nonsmooth systems Fetecau, Marsden, Ortiz, West: Nonsmooth Variational Mechanics and Variational Collision Integrators. SIAM Applied Dynamical Systems, 23. Lecture 3: Recent topics in Variational Integration Higher order variational integrators, error analysis, relation to Runge-Kutta methods,... Sina Ober-Blöbaum p.2

3 Overview Lecture 1: Variational integrators for electric circuits Ober-Blöbaum, Tao, Cheng, Owhadi, Marsden: Variational integrators for electric circuits. Journal of Computational Physics, 213. Sina Ober-Blöbaum p.3

4 Variational (symplectic) integration " ! Sina Ober-Blöbaum p.4

5 Variational (symplectic) integration " ! " " " ! ! ! explicit Euler symplectic Euler implicit Euler Sina Ober-Blöbaum p.5

6 Variational (symplectic) integration " ! preservation of symplecticity due to variational structure (generating function) momentum map preservation (e.g. linear and rotational momentum) due to discrete Noether theorem good long term energy behavior due to symplecticity and backward error analysis Sina Ober-Blöbaum p.6

7 Motivation electro-mechanical and mechatronic systems (mechanical and electrical parts) RailCab Neue Bahntechnik Paderborn Motor with circuit Goal: unified framework for simulation and optimization Sina Ober-Blöbaum p.7

8 +! Motivation Mechanical systems: variational integrators well established based on discrete variational principle in mechanics Electrical systems: L R C 1 U V C 2 variational principle required Lagrangian formulation with degenerate symplectic form forces and constraints discrete variational scheme Future goal: powerful unified variational scheme for the simulation and optimization of electromechanical systems Sina Ober-Blöbaum p.8

9 Outline Basics on circuit modeling Geometric setting Variational formulation: forces, constraints, degeneracy and reduced formulations Construction of integrators: discrete variational formulation Structure preserving properties Numerical examples and comparison to standard approaches Nonlinear circuits Literature overview Sina Ober-Blöbaum p.9

10 +! Basic notations C 1 L U V C 2 R device linear nonlinear resistor i R = Gu R i R = g(u R, t) capacitor i C = C d dt u C i C = d dt q C (u C, t) inductor u L = L d dt i L u L = d dt ϕ L(i L, t) device independent controlled voltage source u V = v(t) u V = v(u ctrl, i ctrl, t) current source i I = i(t) i I = i(u ctrl, i ctrl, t) characteristic equations for basic elements Sina Ober-Blöbaum p.1

11 Energies in electric circuits magnetic energy stored in inductor E mag (i L ) = il electrical energy stored in capacitor E el (q C ) = qc ϕ L (y) dy u C (y) dy linear circuit: ϕ L (i L ) = Li L and u C (q C ) = C 1 q C E mag (i L ) = 1 2 Li 2 L, E el (q C ) = 1 2 C 1 q 2 C Sina Ober-Blöbaum p.11

12 +! Graph representation graph consisting of n branches, m + 1 nodes on each branch there are inductor L i, capacitor C i, resistance R i, voltage source ɛ i L i C i R i " i Sina Ober-Blöbaum p.12

13 Kirchhoff Laws Kirchhoff Current Law (KCL) The sum of currents leading to and leaving from any node is equal to zero. i = Kirchhoff Voltage Law (KVL) The sum of voltages along each mesh of the network is equal to zero. u = Sina Ober-Blöbaum p.13

14 Kirchhoff Laws Kirchhoff Constraints matrix (incidence matrix) K R n,m 1 branch i connected inward to node j K ij = +1 branch i connected outward to node j otherwise K = [K T L K T C K T R K T V ] T, K J R n J,m, J {L, C, R, V } Mesh matrix K 2 R n,n m 1 branch i is a backward branch in mesh j K 2,ij = +1 branch i is a forward branch in mesh j branch i does not belong to mesh j, K 2 = [K T 2,L K T 2,C K T 2,R K T 2,V ] T, K 2,J R n J,n m, j {L, C, R, V } Sina Ober-Blöbaum p.14

15 Kirchhoff Laws Kirchhoff Current Law (KCL) The sum of currents leading to and leaving from any node is equal to zero. i = K T i(t) = Kirchhoff Voltage Law (KVL) The sum of voltages along each mesh of the network is equal to zero. u = K T 2 u(t) = Tellegen s theorem: im(k 2 ) im(k) Sina Ober-Blöbaum p.15

16 Geometric setting configuration, tangent and cotangent space charge space Q, branch charge q R n current space TQ, branch currents (i =)v T q Q R n flux linkage space T Q, flux linkages p Tq Q R n constraint distribution and annihilator Sina Ober-Blöbaum p.16

17 Geometric setting configuration, tangent and cotangent space charge space Q, branch charge q R n current space TQ, branch currents (i =)v T q Q R n flux linkage space T Q, flux linkages p Tq Q R n constraint distribution and annihilator constraint distribution Q (q) = {v T q Q w a, v =, a = 1,..., m} T q Q annihilator Q(q) = {w Tq Q w, v = v Q (q)} Tq Q Sina Ober-Blöbaum p.17

18 Geometric setting configuration, tangent and cotangent space charge space Q, branch charge q R n current space TQ, branch currents (i =)v T q Q R n flux linkage space T Q, flux linkages p Tq Q R n constraint distribution and annihilator constraint KCL space Q (q) = {v T q Q K T v = } T q Q constraint KVL space Q(q) = {u Tq Q K2 T u = } Tq Q Sina Ober-Blöbaum p.18

19 Geometric setting space of branches B = Q Q, Q R n space of meshes M = TM T M, M R n m space of nodes N = TN T N, N R m T ˆq N K Q (q) K T 2 {} T q M N B M nodes branches meshes {} Tˆq N K T Q (q) K 2 T q M with the linear maps K : T ˆq N Q (q) and K 2 : T q M Q (q), and their adjoints K T : ( Q (q)) Tˆq N and K2 T : ( Q(q)) T q M KCL K T v = or v = K 2 ṽ KVL K2 T u = or u = K û Tellegen s theorem: im(k 2 ) im(k) Sina Ober-Blöbaum p.19

20 Variational formulation mechanical system electrical circuit (linear relations) kinetic energy magnetic energy 1 2 v T Lv R potential energy electrical energy 1 2 qt Cq R friction dissipation R v R n external force external sources E u R n (non)holonomic constraints KCL K T v = with L = diag(l 1,..., L n ), C = diag( 1 C 1,..., 1 C n ), R = diag(r 1,..., R n ), E = diag(ɛ 1,..., ɛ n ) Sina Ober-Blöbaum p.2

21 Variational formulation Lagrangian L(q, v) = 1 2 v T Lv 1 2 qt Cq on TQ forces f = R v + E u T q Q distribution Q (q) = {v T q Q K T v = } = null(k T ) 1. dissipative and external forces (d Alembert) 2. KCL constraints (constrained variations) 3. degenerate Lagrangian Legendre transformation FL(q, v) = (q, L/ v) if 2 L = L is not invertible degenerate system v 2 constraint flux linkage subspace P = FL( Q ) T Q, Sina Ober-Blöbaum p.21

22 Constrained variational formulation Constrained Lagrange-d Alembert-Pontryagin Principle on TQ T Q gives implicit Euler-Lagrange equations [Yoshimura, Marsden 26] T T δ (L(q, v) + p, q v ) dt + f δqdt =, δq Q (q) Sina Ober-Blöbaum p.22

23 Constrained variational formulation Constrained Lagrange-d Alembert-Pontryagin Principle on TQ T Q gives implicit Euler-Lagrange equations [Yoshimura, Marsden 26] T T δ (L(q, v) + p, q v ) dt + f δqdt =, δq Q (q) see blackboard Sina Ober-Blöbaum p.23

24 Constrained variational formulation Constrained Lagrange-d Alembert-Pontryagin Principle on TQ T Q gives implicit Euler-Lagrange equations [Yoshimura, Marsden 26] T T δ (L(q, v) + p, q v ) dt + f δqdt =, δq Q (q) q = v charge-current relation ṗ = L + f + K û q = L p v = K T v KCL KVL form u = K û flux relation ( primary constraints ) n + n + n + m equations differential-algebraic system differential variables q and p, algebraic variables v and û non-autonomous system due to external force f (t) no ode description for singular Lagrangian Sina Ober-Blöbaum p.24

25 Reduction 1. less redundant formulation 2. degeneracy of Lagrangian is cancelled for specific systems 3. physical meaning of reduced space: mesh space M with coordinates ( q, ṽ, p), q, ṽ, p R n m T ˆq N K Q (q) K T 2 {} T q M N B M nodes branches meshes {} Tˆq N K T Q (q) K 2 T q M Sina Ober-Blöbaum p.25

26 Reduction 1. less redundant formulation 2. degeneracy of Lagrangian is cancelled for specific systems 3. physical meaning of reduced space: mesh space M with coordinates ( q, ṽ, p), q, ṽ, p R n m T ˆq N K Q (q) K T 2 {} T q M N B M nodes branches meshes {} Tˆq N K T Q (q) K 2 T q M Sina Ober-Blöbaum p.26

27 Reduction steps T ˆq N K Q (q) K T 2 {} T q M N B M nodes branches meshes {} Tˆq N K T Q (q) K 2 T q M replace KCL formulation K T v = by K 2 ṽ = v Q (q) = {v T q Q K T v = } is integrable configuration submanifold C = {q Q K T q = } for consistent initial values q C q = K 2 q. Sina Ober-Blöbaum p.27

28 Reduction steps T ˆq N K Q (q) K T 2 {} T q M N B M nodes branches meshes {} Tˆq N K T Q (q) K 2 constrained Lagrangian L M := K 2 L: TM R T q M L M ( q, ṽ) = L(K 2 q, K 2 ṽ) = 1 2ṽT K T 2 LK 2 ṽ 1 2 qt K T 2 CK 2 q reduced Legendre transformation FL M : TM T M FL M ( q, ṽ) = ( q, L M / ṽ) = ( q, K2 T LK 2 ṽ). reduced force f M ( q, ṽ, t) = K2 T f (K 2 q, K 2 ṽ, t) in T M reduced flux p = K2 T p in T M Sina Ober-Blöbaum p.28

29 Reduced constrained variational formulation reduced L-d A-Pontryagin Principle on TM T M T ( δ L M ( q, ṽ) + p, q ṽ ) T dt + f M δ qdt = implicit Euler-Lagrange equations on mesh space M q = ṽ p = LM q + f KVL form K T 2 u = = LM LM p with = K2 T LK 2 ṽ ṽ ṽ 3 (n m) equations differential variables q and p, algebraic variable ṽ Is 2 L M ṽ 2 = K T 2 LK 2 invertible? Sina Ober-Blöbaum p.29

30 Cancellation of degeneracy by constraints Cancellation of degeneracy For linear LRCV circuits the system is non-degenerate, if the number of capacitors n C, resistors n R and voltage sources n V equals the number of independent constraints l X involving the currents through the capacitor, resistor and source branches, i.e., if n C + n R + n V = l X Proof uses basic linear algebra arguments showing 1. null([kc T K R T K V T ]) = {} 2. null(k2 T LK 2) = {} Sina Ober-Blöbaum p.3

31 Examples 1 L 1 C L 1 C1 L 2 C 1 C 2 C K T L = ( 1 1 ), K T C = ( ) K T L = ( 1 ), K T C = ( null(kc T ) = {} null(k C T ) {} non-degenerate constrained Lagrangian degenerate constrained Lagrangian interpretation: Each fundamental loop has to contain at least one inductor ) Sina Ober-Blöbaum p.31

32 Next steps Alternative 1: Construction of non-degenerate Lagrangian on submanifold projection onto image of Legendre transformation identification of more constraints ((non)holonomic) e.g. Gotay-Nester: C C 2 C k T Q loss of physical meaning for reduced variables, difficult to identify additional contraints Alternative 2: Keep degeneracy derive integrator for degenerate Lagrangian (DAE integrator) derive discrete version of implicit EL equations check for applicability Sina Ober-Blöbaum p.32

33 Discrete variational principle discrete time grid t = {t k = kh k =,..., N} discrete paths that approximate continuous paths q(t), v(t) and p(t) q d : {t k } N k= Q, v d : {t k } N k= T q Q, p d : {t k } N k= T q Q, q k := q d (t k ) q(kh) v k := v d (t k ) v(kh) p k := p d (t k ) p(kh) discrete reduced L.-d Alembert-Pontryagin principle { N 1 ( δ h L M ( q k, ṽ k ) + p k, q ) } N 1 k+1 q k ṽ k +h f M ( q k, ṽ k, t k )δ q k = h k= k= time derivative q(t) is approximated by the forward difference operator force evaluated at the left point Forward Euler (EFD) Sina Ober-Blöbaum p.33

34 Discrete variational principle discrete time grid t = {t k = kh k =,..., N} discrete paths that approximate continuous paths q(t), v(t) and p(t) q d : {t k } N k= Q, v d : {t k } N k= T q Q, p d : {t k } N k= T q Q, q k := q d (t k ) q(kh) v k := v d (t k ) v(kh) p k := p d (t k ) p(kh) discrete reduced L.-d Alembert-Pontryagin principle { N ( δ h L M ( q k, ṽ k ) + p k, q ) } k q k 1 N ṽ k +h f M ( q k, ṽ k, t k )δ q k = h k=1 time derivative q(t) is approximated by the backward difference operator force evaluated at the right point Backward Euler (EBD) k=1 Sina Ober-Blöbaum p.34

35 Discrete variation discrete variations δ q k with δ q = δ q N = discrete variations δṽ k and δ p k L M N 1 v ( q [ L M, ṽ ) p, δṽ + v ( q k, ṽ k ) p k, δṽ k k=1 L M + q ( q k, ṽ k ) 1 h ( p k p k 1 ) + f M ( q k, v k, t k ), δ q k δ p k 1, q ] k q k 1 q N q N 1 ṽ k 1 + δ p N 1 ṽ N 1 = h h Sina Ober-Blöbaum p.35

36 Discrete reduced Euler-Lagrange equations L M v ( q, ṽ ) = p, q N q N 1 h L M q ( q k, ṽ k ) 1 h ( p k p k 1 ) + f M ( q k, ṽ k, t k ) = q k q k 1 Forward Euler h L M v ( q k, ṽ k ) = ṽ N 1 = ṽ k 1 = p k k = 1,..., N 1 q 1 q L M = ṽ 1, h v ( q 1, ṽ 1 ) = p 1 L M q ( q k 1, ṽ k 1 ) 1 h ( p k p k 1 ) + f M ( q k 1, ṽ k 1, t k 1 ) = q k q k 1 = ṽ k k = 2,..., N h L M Backward Euler v ( q k, ṽ k ) = p k Sina Ober-Blöbaum p.36

37 Applicability apply e.g. Newton s scheme to determine x k+1 = ( q k+1, ṽ k+1, p k+1 ) by solving = F (x k, x k+1 ) for given x k = ( q k, ṽ k, p k ) unique solutions for regular Jacobian of F w.r.t. x k+1 applicability dependent on update rule Example: EBD for linear circuit leads to the iteration scheme I hi q k I q k 1 K2 T LK 2 I ṽ k = ṽ k 1 + u s(t k 1 ) } {{ I } p k hk2 T CK 2 hk2 T diag(r)k 2 I p k 1 hk2 T =A A is invertible iff K T 2 LK 2 is regular Sina Ober-Blöbaum p.37

38 Applicability apply e.g. Newton s scheme to determine x k+1 = ( q k+1, ṽ k+1, p k+1 ) by solving = F (x k, x k+1 ) for given x k = ( q k, ṽ k, p k ) unique solutions for regular Jacobian of F w.r.t. x k+1 applicability dependent on update rule Example: EFD for linear circuit leads to the iteration scheme I q k I hi q k 1 K2 T LK 2 I ṽ k = ṽ k 1 + u s(t k ) hk2 T CK 2 } hk2 T diag(r)k 2 {{ I } p k I p k 1 hk2 T =A A is invertible iff K T 2 (L+hR)K 2 is regular Sina Ober-Blöbaum p.38

39 Applicability For linear RLCV circuits we have EBD applicable if K T 2 LK 2 is regular (i.e., if reduced system is non-degenerate) EFD applicable if K T 2 (L + R)K 2 is regular VI based on implicit midpoint rule applicable if K T 2 (L + R + C)K 2 is regular extension to higher order (symprk) methods possible [Bou-Rabee et al. 26] trade off: the more implicit the better but the more expensive Sina Ober-Blöbaum p.39

40 Structure preservation symplecticity symlecticity: conservation of symplectic form Ω = dq i dp i (F t ) Ω = Ω t conformal symlecticity (in presence of uniform dissipative forces f L = c p, c R): (F t ) Ω = exp (ct)ω t variational integrator preserves the symplectic form, or the rate of decay of the symplectic form, respectively leads to good long-term energy behavior of variational integrators e.g. Hairer, Lubich 24 Sina Ober-Blöbaum p.4

41 Structure preservation momentum maps Noether: exact preservation of momentum maps in presence of symmetries Lie group G with Lie group action ψ : G Q Q (and ψ g = ψ(g, ) with g G), tangent lift ψg TQ : TQ TQ infinitesimal generator ξ Q (q) := d dt t= ψ(exp (tξ), q) with ξ g and the exponential function exp : g G invariance of Lagrangian under Lie group action L ψg TQ = L g G invariance of holonomic constraints h(q) = under Lie group action h ψ g = g G force f M L orthogonal to group action, i.e., f L (q, v), ξ Q (q) = for all (q, v) TQ and all ξ g = J F t = J with flow F t and L J(q, v), ξ = v, ξ Q(q) = p, ξ Q (q) Sina Ober-Blöbaum p.41

42 Structure preservation momentum maps Examples from mechanics G = R temporal energy G = R 3 translational linear momentum G = SO(3) rotational angular momentum variational integrator preserves the invariance property if the discrete Lagrangian has the same symmetry What are momentum maps for electric circuits? Which symmetries can we expect? Sina Ober-Blöbaum p.42

43 Momentum maps for electric circuits Lagrangian L(q, v) = 1 2 v T Lv 1 2 qt Cq force f = Rv + Eu, KCL K T v = Invariance of Lagrangian The Lagrangian of the unreduced system is invariant under the translation of q L. Proof: Let G = R n L, Φ TQ (g, (q, v)) = (q L + g, q C, q R, q V, v L, v C, v R, v V ) L Φ TQ g (q, v) = 1 2 v L v C v R v V T L v L v C v R v V 1 2 q L + g q C q R q V T C q L + g q C q R q V ( ) = 1 2 v T Lv 1 2 qt 1 Cq = L(q, v) since C = diag C 1,..., 1 C n with the first n L diagonal elements being zero. Sina Ober-Blöbaum p.43

44 Momentum maps for electric circuits Lagrangian L(q, v) = 1 2 v T Lv 1 2 qt Cq force f = Rv + Eu, KCL K T v = Orthogonality of external force The external force f is orthogonal to the action of the group G = R n L being translations of ql. Proof: Let ξ g = R n L. For Φ g (q) = (q L + g, q C, q R, q V ) we compute ξ Q (q) = d dt Φ exp tξ (q) = d t= dt (q L +exp tξ, q C, q R, q V ) = (ξ,,, ). t= Thus, we have f L, ξ Q (q) = Rv + Eu, ξ Q (q) = since R and E have zero entries in the first n L lines and columns. Sina Ober-Blöbaum p.44

45 Momentum maps for electric circuits Topology assumption: For every node j = 1,..., m in the circuit (except ground) the same amount of inductor branches connect inward and outward to node j. the sum of each row of KL T i = 1,..., m. Invariance of distribution is zero, i.e. n L j=1 (K T L ) ij = for Under the topology assumption the KCL on configuration level are invariant under equal translation of q L. Proof: Let g = a1 G with a R and 1 being a vector in R n L with each component 1. It follows that q L + g K T Φ g (q) = K T q C q R = K T q + KL T g = K T q + KL T 1a = K T q q V since the sum of each row of K T L is zero. Sina Ober-Blöbaum p.45

46 Momentum maps for electric circuits Theorem (Preservation of flux) Under the topology assumption the sum of all inductor fluxes in the electrical circuit described by the Lagrangian L, the external forces f, and the KCL, is preserved. Proof Invariance of Lagrangian L and KCL f orthogonal to group action induced momentum map J(q, v), ξ = L v, ξ Q(q) J(q, v) = n L i=1 p n Li is preserved by Noether = L v i ξ i Q(q) = L v Li ξ i more general: for any circuit η T p L = const for η null(kl T ). Sina Ober-Blöbaum p.46

47 Example: LC transmission line L 1 1 L 2 2 L 3 flux C 1 C 2 p L1 K = K L = ( ) 1 K C = 1 p L2 p L3.5 p L1 +p L2 +p L time n L j=1 (K T L ) ij =, i = 1, 2 p L1 + p L2 + p L3 = const Sina Ober-Blöbaum p.47

48 Structure preservation what else? Preservation of frequency spectrum consider the example of the 1D harmonic oscillator define one-step update scheme as (q k+1, p k+1 ) T = A (q k, p k ) T = QVQ 1 (q k, p k ) T with A having linearly independent eigenvectors λ 1, λ 2 and being diagonalizable with V = diag(λ 1, λ 2 ) with the coordinate transformation (x k, y k ) T = Q 1 (q k, p k ) T we obtain (x k+1, y k+1 ) T = V (x k, y k ) T i.e., x k+1 = λ 1 x k and y k+1 = λ 2 y k Sina Ober-Blöbaum p.48

49 Structure preservation frequency spectrum We show 1. A has two eigenvalues of norm 1 iff scheme is symplectic 2. methods defined by matrices with norm 1 eigenvalues preserve frequeny spectrum defined on different time spans Proof of 2. discrete inverse Fourier transformation x k = 1 N N n=1 x n exp ( 2πi N kn), k = 1,..., N Consider a sequence of discrete points {X k } N k=1 that is shifted by one time step such that X k = x k+1 = λ 1 x k, k = 1,..., N discrete inverse Fourier transformation X k = 1 N N n=1 λ 1 x n exp ( 2πi N kn), k = 1,..., N, i.e., X n = λ 1 x n by definition of the frequency spectrum, we have X n X n = x n λ 1 λ 1 x n = x n λ 1 2 x n = x n x n preservation of the frequency spectrum Sina Ober-Blöbaum p.49

50 Standard methods in circuit theory Circuit simulator SPICE (Simulation Program with Integrated Circuit Emphasis) Electronics Research Laboratory of the University of California, Berkeley Modeling: Modified Node Analysis (MNA) Simulation: Backward Differentiation Formulas (BDF) methods Sina Ober-Blöbaum p.5

51 Modified Nodal Analysis (MNA) (charge-flux) 1. Apply KCL to every node except ground 2. Insert representation for the branch current of resistors, capacitors and current sources 3. Add representation for inductors and voltage sources explicitely to the system (KCL) K T C q C (K C û, t) + K T R g(k Rû, t) + K T L v L + K T V v V +K T I v I (Kû, q C (K C û, t), v L, v V, t) = (inductors) ṗ L (v L, t) K L û = (voltage sources) u V (Ku, q C (K C u, t), v L, v V, t) K V û = inductor current v L, capacitor charge q C, inductor flux p L node voltage û, voltage source current v V, conductance g controlled current and voltage source v I and u V Sina Ober-Blöbaum p.51

52 Numerical integration schemes DAE system A[d(x(t), t)] + b(x(t), t) = with x = [û, v L, v v ] T and d(x, t) = [q C (K C û, t), p L (v L, t)] T conventional approach: Implicit linear multi-step formulas k α i d(x n+1 i, t n+1 i ) = h i= k β i d n+1 i BDF method: β 1 = = β k =, α = 1 k β α 1 α 2 scheme d(x n+1 ) d(x n ) = hd n+1 (implicit Euler) i= 1 3 d(x n+1 ) 4 3 d(x n) d(x n 1) = h 2 3 d n+1 low computational cost (1 function evaluation per step) compared to RK methods BUT non-symplectic (Tang 1993) Sina Ober-Blöbaum p.52

53 Example computations Comparison of different solutions exact solution (exact) variational integrator of second order (VI) variational integrator of first order (EBD) variational integrator of first order (EFD) Runge Kutta of order 4 (RK) BDF method of order 2 applied to MNA system (MNA BDF) (fixed time-stepping for all methods) Sina Ober-Blöbaum p.53

54 Example: RLC circuit on edge 1 5 pair of capacitor and inductor (C i = L i = 1) on edge 6 only capacitor (C 6 = 1) n = 6 branches, m + 1 = 4 nodes, l = 3 meshes Sina Ober-Blöbaum p.54

55 LC circuit (no resistors) with step size h = current on branch 1 current on branch time time current branch 1 current branch exact VI VI EBD VI EFD RK4 MNA BDF current on branch energy time current branch time energy behavior Sina Ober-Blöbaum p.55

56 LC circuit (no resistors) with step size h = exact VI VI EBD VI EFD RK4 MNA BDF exact VI VI EBD VI EFD RK4 MNA BDF.5.5 current on branch 1.5 current on branch time time current branch 1 current branch exact VI VI EBD VI EFD RK4 MNA BDF current on branch energy exact VI VI EBD VI EFD RK4 MNA BDF time current branch time energy behavior Sina Ober-Blöbaum p.56

57 Frequency spectrum of first branch current frequency of branch current exact VI VI EBD VI EFD RK4 MNA BDF frequency of branch current exact VI VI EBD VI EFD RK4 MNA BDF ! ! h =.1 h =.2 frequency of branch current exact VI VI EBD VI EFD RK4 MNA BDF frequency of branch current exact VI VI EBD VI EFD RK4 MNA BDF ! ! h =.4 h =.6 Sina Ober-Blöbaum p.57

58 Current of first branch of LC circuit (no resistors) 1 exact VI RK4 MNA BDF 1 exact VI RK4 MNA BDF.5.5 current on branch 1 current on branch time time h =.1 h =.2 1 exact VI RK4 MNA BDF 1 exact VI RK4 MNA BDF.5.5 current on branch 1 current on branch time time h =.4 h =.6 Sina Ober-Blöbaum p.58

59 LCR circuit (with resistors) with step size h = exact VI VI EBD VI EFD RK4 MNA BDF exact VI VI EBD VI EFD RK4 MNA BDF.5.5 current on branch 1.5 current on branch time time current branch 1 current branch exact VI VI EBD VI EFD RK4 MNA BDF exact VI VI EBD VI EFD RK4 MNA BDF current on branch energy time current branch time energy behavior Sina Ober-Blöbaum p.59

60 Example: Oscillating LC circuit two capacitors C 1 = 1, C 2 = 1 two inductors L 1 = 1, L 2 = 1 n = 4 branches, m + 1 = 3 nodes, l = 2 meshes C L 1 K = C1 2 L 2 K 2 = Sina Ober-Blöbaum p.6

61 Oscillating LC circuit with step size h = current on branch current on branch time time Current on inductor 1 Current on inductor charge on branch charge on branch time time Charge on capacitor 1 Charge on capacitor 2 Sina Ober-Blöbaum p.61

62 Energy of oscillating LC circuit energy exact energy exact VI VI EBD.492 VI VI EBD.485 VI EFD RK4.49 VI EFD RK4 MNA BDF MNA BDF time time h =.1 h = energy.46 energy exact VI VI EBD VI EFD RK4 MNA BDF exact VI VI EBD VI EFD RK4 MNA BDF time time h =.4 h =.6 Sina Ober-Blöbaum p.62

63 Charge on capacitor 1 of LC circuit (h =.6) charge on branch exact VI VI EBD VI EFD RK4 MNA BDF time charge on branch exact VI VI EBD VI EFD RK4 MNA BDF time t [, 3] t [, 5] exact VI VI EBD VI EFD RK4 MNA BDF charge on branch charge on branch exact VI VI EBD VI EFD RK4 MNA BDF time time t [3, 35] t [25, 255] Sina Ober-Blöbaum p.63

64 Frequency spectrum of charge on capacitor exact VI VI EBD VI EFD RK4 MNA BDF.3.25 exact VI VI EBD VI EFD RK4 MNA BDF frequency of branch charge frequency of branch charge frequency of branch charge ! ! h =.1 h =.2 exact VI VI EBD VI EFD RK4 MNA BDF frequency of branch charge exact VI VI EBD VI EFD RK4 MNA BDF ! ! h =.4 h =.6 Sina Ober-Blöbaum p.64

65 Nonlinear circuits Lagrangian L(q, v) = n L forces f L (q, v, t) = k=1 v Lk ϕ Lk (y) dy n C j=1 (linear circuit: L(q, v) = 1v T Lv qt C 1 q) L C f R (q, v, t) = f S (q, v, t) q Cj u Cj (y) dy L C f R (v R ) f S (v S, t) distribution Q (q) = {v T q Q K T v = } Legendre transform ( FL(q, v) = q, L ) ( ( )) ϕl (v L ) (q, v) = q, = (q, ϕ) v C,R,S results on cancellation of degeneracy, preservation properties etc. extendable to nonlinear case Sina Ober-Blöbaum p.65

66 Nonlinear LCR circuit Lagrangian: L(q C, v L ) = c 2 v 2 L 1 + d 2 log((tan(v L 2 /d)) 2 +1) sgn(q C1 ) a 3 q3 C 1 1 2b q2 C 2 force: f R (v R ) = ev 3 R (a =.2, b = 1, c = 3, d = 1, e = 5) energy energy VI EFD VI EBD VI MPR RK4 ODE45 exact time.1 VI EFD VI EBD VI MPR RK4 ODE time energy, no resistor (h =.8) energy with resistor (h =.8) Sina Ober-Blöbaum p.66

67 Conclusions variational approach for circuit modeling (degenerate Lagrangian, dissipative and external forces, KCL constraints) reduced variational approach on mesh space (cancels degeneracy for some cases, reduced system) discrete variational approach provides structure-preserving integrator preservation of momentum maps preservation of frequency spectrum good energy behavior variational integrators are convenient tools for the simulation of electric and electromechanical systems Sina Ober-Blöbaum p.67

68 Literature overview Variational inegrators for electric circuits Ober-Blöbaum, Tao, Cheng, Owhadi, Marsden (213): linear circuits Ober-Blöbaum, Lindhorst (214): nonlinear circuits Lagrangian / Hamiltonian modeling of circuits MacFarlane (1967): tree structures (capacitor charges or inductor fluxes as generalized coordinates) Chua, McPherson (1973): mixed set of coordinates Maisser et al. (1995): Lagrangian modeling of electromechanical systems Degenerate Lagrangian / Hamiltonian Dirac, Bergmann (1956): construction of submanifolds Gotay, Nester (1979): Lagrangian viewpoint van der Schaft (1995): implicit Hamiltonian systems Yoshimura, Marsden (26): implicit Lagrangian systems Sina Ober-Blöbaum p.68