Variational Integrators for Electrical Circuits

Variational Integrators for Electrical Circuits Sina Ober-Blöbaum California Institute of Technology Joint work with Jerrold E. Marsden, Houman Owhadi, Molei Tao, and Mulin Cheng Structured Integrators Workshop May 8, 29 Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.1

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

Outline basics on circuit modeling variational modeling, constraints, and degeneracy approaches to handle constraints and degeneracy construction of integrators standard approach: Modified Nodal Analysis (MNA) and Linear Multistep Methods numerical examples future work Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.3

Circuit modeling C 1 L 1 U V C 2 R 1 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 + Kirchoff constraint laws Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.4

Kirchoff constraint laws Network topology graph consisting of n branches, m + 1 nodes incidence matrix K R n,m describes brach-node connection 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 K T I ] T, K J R n J,m, J {L, C, R, V, I } Constraints current law (KCL) K T I (t) = I (t): branch currents voltage law (KVL) K u(t) = U(t) u(t): node voltages U(t): branch voltages Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.5

Variational formulation mechanical system electrical circuit (linear relations) configuration charge q Q R n velocity current v T q Q R n momentum flux p T q Q R n 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 R n with L = diag(l 1,..., L n ), C = diag( 1 C 1,..., 1 C n ), R = diag(r 1,..., R n ), E = (ɛ 1,..., ɛ n ) Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.6

Variational formulation Lagrangian L(q, v) = 1 2 v T Lv 1 2 qt Cq on TQ forces f = R v + E T q Q distribution Q (q) = {v T q Q K T v = } = N (K T ) annihilator Q (q) = {w T q Q w, v = v Q (q)} = R(K) Legendre transformation FL(q, v) = (q, L/ v) if FL is not invertible degenerate system Lagrange-d Alembert-Pontryagin Principle on TQ T Q gives implicit Euler-Lagrange equations δ T (L(q, v) + p, q v ) dt + T f δqdt =, δq Q (q) q = v charge conservation (n eq.) ṗ = L q + f + K λ KVL with node voltages λ Rm (n eq.) = L v p flux conservation (n eq.) = K T v KCL (m eq.) Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.7

Literature overview Lagrangian / Hamiltonian formulation of circuits MacFarlane (1967): tree structures (capacitor charges or inductor fluxes as generalized coordinates) Chua, McPherson (1973): mixed set of coordinates Chua, McPherson and Milic, Novak: enlarge applicability of Lagrangian methods to electrical networks 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 Leok, Ohsawa (29): discrete version Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.8

Reduction integrable distribution Q (q) = {v T q Q K T v = } = N (K T ) configuration submanifold C = {q Q K T (q q ) = } K 2 R n,n m with R(K) R(K 2 ): q = K 2 q + α q Q R n m α = for consistent q v = K 2 ṽ ṽ T q Q R n m constrained Lagrangian L c ( q, ṽ) := L(K 2 q, K 2 ṽ) constrained Legendre transformation FL c ( q, ṽ) = ( q, L c / ṽ) forces f = K T 2 f, momenta p = K T 2 p T q Q R n m reduced L-d A-Pontryagin Principle on T Q T Q δ T ( L c ( q, ṽ) + p, q ṽ ) T dt + f δ qdt = q = ṽ p = Lc q = Lc ṽ + f p Is Lc ṽ invertible? Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.9

Cancellation of degeneracy by constraints linear RCLV circuit FL c ( q, ṽ) = ( q, K T 2 LK 2ṽ) constrained Lagrangian L c is non-degenerate if R(K 2 ) N (L) = {} it holds N (L) = {v T q Q v L = } with R(K 2 ) R(K) and R(K) N (K T ) we have R(K 2 ) = N (K T ) = {v T q Q K T L v L +K T C v C +K T R v R +K T V v V = } and thus R(K 2 ) N (L) = {v T q Q [K T C K T R K T V ] [v C v R v V ] T =, v L = } it follows that for N ([K T C K T R K T V ]) = {} the constrained Lagrangian L c is non-degenerate Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.1

Examples L 1 L 1 C 1 C2 L 2 C1 C2 C 3 K T L = 1 1 «, K T C = 1 1 1 «K T L = 1 «, K T C = 1 1 1 1 N (KC T ) = {} N (K C T ) {} non-degenerate constrained Lagrangian degenerate constrained Lagrangian non-degeneracy iff number of capacitors = number of independent constraints involving capacitors equivalent statements, when resistors and voltage sources are included «Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.11

Projection for degenerate systems identify Dirac constraints: denote by FL c M : T Q M the restriction of FL c : T Q T Q to its image primary submanifold M = {( q, p) T Q φ( q, p) = R l } l: degree of degeneracy consistency with dynamics leads to more constraints (motion constrained to lie in M) Dirac-Bergmann: M M 2 M k T Q Gotay-Nester: N N 2 N k T Q knowledge about existence of solutions if constrained algorithm converges consistent equations of motion on N k useful for circuit modeling? Can we find a regular Lagrangian on the final constraint manifold N k? Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.12

Example: linear RLCV circuit q = ṽ K T 2 LK 2 ṽ = K T 2 C q K T 2 Rṽ + Ẽ = case 1: no resistors and no external sources φ( q, ṽ, Ẽ) =, linear case: P T q + U T ṽ + V T Ẽ = holonomic constraint defines constraint manifolds C 2 = { q Q P T q = }, T q C 2 = {ṽ T q Q PT ṽ = } reparametrizations q = P 2ˆq, ˆq ˆQ R n m l and ṽ = P 2ˆv, ˆv Tˆq ˆQ R n m l constrained Lagrangian L c2 (ˆq, ˆv) := L(K 2 P 2ˆq, K 2 P 2ˆv) case 2: with resistors and external sources affine non-holonomic constraint defines constraint manifold P 2 = {ṽ T q Q P T q + U T ṽ + V T Ẽ = } reparametrization ṽ = U 2ˆv + γ( q, Ẽ), ˆv R n m l, q R n m constrained Lagrangian L c2 ( q, ˆv) := L(K 2 q, K 2 U 2ˆv + K 2 γ( q, Ẽ)) Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.13

The constrained Variational Integrator same reduction steps can be performed on the discrete level to construct a discrete constrained Lagrangian discrete variational principle for discrete constrained Lagrangian case 1: N 1 ( δ L c2 (ˆq k, ˆv k ) + ˆp k, ˆq ) N 1 k ˆq k 1 ˆv k + ˆf k δˆq k dt = h k= with variations δˆq k, δˆv k, δˆp k case 2: N 1 δ k= ( L c2 ( q k, ˆv k ) + with variations δ q k P 2, δˆv k, δ p k k= p k, q ) k q k 1 (U 2ˆv k + γ( q k, Ẽ k ) h gives well-defined scheme without any constraints N 1 + f k δ q k dt = k= Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.14

The constrained Variational Integrator straight forward for linear systems loss of physical meaning of generalized coordinates for nonlinear systems the additional constraints are hard to identify no global reparametrization are there other ways to deal with the degeneracy? Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.15

Keep degeneracy Variational Principle results into DAE system q = v ṗ = L q + f + K λ = L v p = K T v I I q v p λ = F (q, v, p, λ, f ) for a linear circuit (no external sources) we have I I } {{ } E q v p λ } {{ } ẋ = I C R K L I K T } {{ } A q v p λ } {{ } x Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.16

Existence and uniqueness for linear DAE Definition: The pair (E, A) of square matrices is regular, if µ R s.t. det(µe A). Theorem: Let the pair (E, A) of square matrices be regular. Then it holds 1. The DAE system E ẋ(t) = A x(t) is solvable. 2. The set of consistent initial conditions x is nonempty. 3. Every initial value problem with consistent initial conditions is uniquely solvable. (Kunkel, Mehrmann: Differential-Algebraic equations) Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.17

Existence and uniqueness for linear DAE Lagrangian system: for µ = 1 we have µe A = E A = I I C R I K L I K T det(e A) N (K T ) N (L + C + R) = {} a linear RLC circuit has a unique solution (it holds N (L + C + R) = {}) a linear RLCV circuit has a unique solution if N (K T V ) = {} Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.18

Variational Integrator I discrete Lagrange-d Alembert-Pontryagin Principle N 1 ( δ L(q k, v k ) + p k, q ) N 1 k q k 1 v k + f k δq k dt = h k= k= δq k Q (q k ) q k+1 = q k + hv k+1 p k+1 = p k hcq k hrv k + K λ k = Lv k+1 p k+1 I hi I K L I K T } {{ } E d q k+1 v k+1 p k+1 λ k = K T v k+1 } {{ } x k+1 = I hc hr I } {{ } A d unique discrete solution exists if N (K T ) N (L) = {} integrator applicable if KCL cancels degeneracy q k v k p k λ k 1 } {{ } x k Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.19

Variational Integrator II discrete Lagrange-d Alembert-Pontryagin Principle N 1 ( δ L(q k, v k ) + p k, q ) N 1 k+1 q k v k + f k δq k dt = h k= k= δq k Q (q) q k+1 = q k + hv k p k+1 = p k hcq k+1 hrv k+1 + K λ k+1 = Lv k+1 p k+1 I hc hr I K L I K T } {{ } E d = K T v k+1 q k+1 v k+1 p k+1 = λ k+1 } {{ } x k+1 I hi I } {{ } A d unique discrete solution exists if N (K T ) N (L + hr) = {} q k v k p k λ k } {{ } x k Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.2

Variational Integrator III higher order discrete Lagrange-d Alembert-Pontryagin Principle of second order resulting into implicit midpoint rule 1 q k+1 = q k + hv k+ 1 2 p k+1 = p k hc q k +q k+1 2 hr v k +v k+1 2 + K λ k+ 1 2 = Lv k+ 1 p k +p k+1 2 2 = K T v k+ 1 2 I hi q k+1 h 2 C hr I K I L 2 v k+ 1 2 p k+1 = K T λ k+ 1 2 }{{}}{{} E d x k+1 I h 2 C I I 2 } {{ } A d unique solution exists if N (K T ) N (L + hc + hr) = {} integrator applicable if continuous system has unique solution 1 cf. Nawaf s thesis: Variational Partitioned RK method q k v k 1 2 p k λ k 1 2 }{{} x k Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.21

Nonlinear case e.g. Variational Integrator II q k+1 + q k + hv k p = k+1 + p k + hd 1 L(q k+1, v k+1 ) + f k+1 + K λ k+1 D 2 L(q k+1, v k+1 ) p k+1 K T v k+1 = F (x k, x k+1 ) find x k+1 such that F (x k, x k+1 ) = for given x k Jacobian of F w.r.t. x k+1 has to be regular (e.g. Newton scheme) applicability dependent on update rule trade off: the more implicit the better but the more expensive what is state of the art (e.g. simulation software SPICE)? Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.22

Modified Nodal Analysis (MNA) (charge-flux) apply KCL to every node except ground insert representation for the branch current of resistors, capacitors and current sources add representation for inductors and voltage sources explicitely to the system KC T C q(k C u, t) + KR T g(k Ru, t) + KL T v L + KV T v V + KI T v I (Ku, q, v L, v V, t) = φ(v L, t) K L u = linear relations v V (Ku, q C, v L, v V, t) K V u = K T C CK C u + K T R GK Ru + K T L v L + K T V v V + K T I v I (Ku, q, v L, v V, t) = L v L K L u = v V (Ku, q C, v L, v V, t) K V u = DAE for node voltages u, inductor currents v L Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.23

Numerical integration schemes DAE system A[d(x(t), t)] + b(x(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 1 1 1 d(x n+1 ) d(x n ) = hd n+1 (implicit Euler) 2 2 3 4 1 3 3 d(x n+1 ) 4 3 d(x n) + 1 3 d(x n 1) = h 2 3 d n+1 advantage: low computational cost (1 function evaluation per step) compared to RK methods disadvantage: bad stability properties (Voigt: General linear methods for integrated circuit design, PhD thesis, 26) i= Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.24

Example I C 1 C2 L 1 L = B @ L 1 L 2 L C = B 1 2 @ C 1 1 C 2 1 C A, K T L = 1 1 1 C A, K T C = 1 1 1 ««KCL: [KL T KC T ] [v L v C ] T = N (KC T ) = {} non-degenerate constrained Lagrangian N (KC T ) = {} all variational integrators are applicable Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.25

Example I results 1.5 1.5 q C1 q C2 f L1 f L2 1 1.5.5 q [C] f [A].5.5 1 1 1.5 5 1 15 2 25 3 t [sec] 1.5 5 1 15 2 25 3 t [sec] charges on capacitors currents on inductors 2 1.5 u 1 u 2 1 x 1 16 8 KCL charge conservation flux conservation 1 6 u [V].5.5 1 conservation properties 4 2 2 4 1.5 6 2 5 1 15 2 25 3 t [sec] 8 5 1 15 2 25 3 t [sec] voltages on nodes conserved quantities Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.26

E E Example I results.75.7.65.716.714.712 VI/FD VI/BD VI/MP MNA/BDF2.6.71.55.78.5.76.45.4.35 VI/FD VI/BD VI/MP MNA/BDF1.74.72.7 5 1 15 2 25 3 t [sec].698 5 1 15 2 25 3 t [sec] Energy plots: Comparison with BDF1 (left) and BDF2 (right) (h =.1) Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.27

E E Example I results.75.7.65.716.714.712 VI/FD VI/BD VI/MP MNA/BDF2.6.71.55.78.5.76.45.4.35 VI/FD VI/BD VI/MP MNA/BDF1.74.72.7 5 1 15 2 25 3 t [sec].698 5 1 15 2 25 3 t [sec] Energy plots: Comparison with BDF1 (left) and BDF2 (right) (h =.1) Hairer, Lubich: multi-step methods are in general not symplectic (Tang (1993): The underlying one-step method of a linear multistep method cannot be symplectic.) same discrete solutions for full, reduced and projected system, BUT condition numbers full system reduced system projected system O(h 2 ) O(h 1 ) a Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.28

Example II L 1 C 1 R 1 L = @ L 1 1 «A 1, KL T = 1 «C = @ 1 A 1 C 1, KC T = 1 R = @ 1 «A, KR T = 1 R 1 N ([K T C K T R )] = {} non-degenerate constrained Lagrangian constrained Lagrangian L = 1 2 L 1v 2 1 2 C 1, dissipation R 1 v harmonic oscillator q 2 Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.29

E Example II results 1.8 q C v L q [C].8.6.4.2.2.4.6.8 v [A].6.4.2.2.4.6.8 1 2 4 6 8 1 12 14 16 18 2 t [sec] 1 2 4 6 8 1 12 14 16 18 2 t [sec] charge on capacitor current on inductor 1.8.6 u 1 u 2.45.4.35 VI/FD VI/BD VI/MP MNA/BDF1 MNA/BDF2.4.3.2.25 u [V].2.2.4.15.6.1.8.5 1 2 4 6 8 1 12 14 16 18 2 t [sec] 2 4 6 8 1 12 14 16 18 2 t [sec] voltages on nodes energy decay (h=.2) Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.3

Example III L 1 C 1 R 1 U V C 2 voltage source U V (t) = sin t resistor R 1 =.1 KL T = @ 1 1 A, KC T = @ 1 1 1 1 A KR T = @ 1 A, KV T = @ 1 1 A 1 1 N ([KC T K R T K V T )] {} degenerate constrained Lagrangian and VI I not applicable N ([K T C K T V )] {} VI II not applicable N (KV T ) = {} VI III applicable Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.31

Example III results 1 1 q C v L E 8 8 6 6 4 4 2 2 q [C] v [A] 2 2 4 4 6 6 8 8 1 1 2 3 4 5 6 7 8 9 1 t [sec] 1 1 2 3 4 5 6 7 8 9 1 t [sec] charges on capacitors currents on inductor 1 u 1 5 8 6 u 2 u 3 45 4 4 35 2 3 u [V] 25 2 2 4 15 6 8 1 1 2 3 4 5 6 7 8 9 1 t [sec] 1 5 VI/MP and MNA/BDF (h=.1) VI/MP (h=.1) MNA/BDF2 (h=.1) MNA/BDF1 (h=.1) 1 2 3 4 5 6 7 8 9 1 t [sec] voltages on nodes energy Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.32

Conclusions discrete variational approach provides symplectic integrator with good energy behavior different approaches to treat degeneracy: reduction to final constraint manifold + minimal number of variables (reduces cpu time for large circuits) + condition number of scheme independent on step size - difficult for nonlinear systems - no real physical meaning of generalized coordinates keep degeneracy + applicable to nonlinear systems + easily scalable + full information about charges, currents, node voltages - huge number of variables - condition becomes worse for small step sizes Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.33

Future work Open questions: determine degree of degeneracy dependent on circuit topology relation between degree of degeneracy and index of DAE systems Construction of integrators: can we construct multistep variational integrators that are symplectic to be computational more efficient? better way to find generalized coordinates, e.g. exploit graph structure (Todd) identify generate and degenerate sub-circuits and apply different integrators parallelize simulation of sub-circuits Future applications: application to noisy circuits (work in progress with Houman and Molei) application to electromechanical systems Sina Ober-Blöbaum Variational Integrators for Electrical Circuits p.34