Efficient Simulation of Physical Efficient Simulation of Physical System Models Using Inlined System Models Using Inlined

Implicit Runge-Kutta AlgorithmsImplicit Runge-Kutta Algorithms

Vicha TreeapornVicha Treeaporn

Department of Electrical & Computer EngineeringDepartment of Electrical & Computer Engineering

The University of ArizonaThe University of Arizona

Tucson, Arizona 85721 U.S.ATucson, Arizona 85721 U.S.A

TopicsTopics

IntroductionIntroduction Techniques for SimulationTechniques for Simulation ResultsResults An ApplicationAn Application

IntroductionIntroduction

StiffnessStiffness Widely varying eigenvaluesWidely varying eigenvalues

Explicit algorithmsExplicit algorithms Straightforward to implementStraightforward to implement Step size limited by numerical stabilityStep size limited by numerical stability

Implicit algorithmsImplicit algorithms More difficult to implementMore difficult to implement Additional computational loadAdditional computational load Needed to simulate stiff systemsNeeded to simulate stiff systems

May use larger step sizesMay use larger step sizes

Inline-IntegrationInline-Integration

Merges the integration algorithm Merges the integration algorithm with the modelwith the model Eliminates differential equationsEliminates differential equations Results in difference equations (∆Es)Results in difference equations (∆Es) Easily implement implicit algorithmsEasily implement implicit algorithms

Circuit example inlining Rad3Circuit example inlining Rad3

Simple CircuitSimple Circuit

Circuit EquationsCircuit Equations

Inlined with Rad3Inlined with Rad3

Integrator equations

Eliminatederivatives

Evaluate atRad3 time instants

SortingSorting

SortingSorting

SortingSorting

SortingSorting

10 equations immediately causalized10 equations immediately causalized Need to perform tearingNeed to perform tearing

Make assumptions about variables Make assumptions about variables being ‘known’being ‘known’

TearingTearing

Residual Eq.

Tearingvariable

TearingTearing

Residual Eq. #2

Tearing variable #2

TearingTearing

Completely causalized equationsCompletely causalized equations 2 iteration variables, v2 iteration variables, vcc and i and i11

Could use this set of equations for Could use this set of equations for simulationsimulation Want step-size controlWant step-size control

Step-Size ControlStep-Size Control

Want larger step sizes Want larger step sizes Reduce the overall computational costReduce the overall computational cost Maintain desired accuracyMaintain desired accuracy

Compute error estimateCompute error estimate Embedding methodEmbedding method

Shares computations with original methodShares computations with original method

Step-Size ControlStep-Size Control

Explicit RKs Explicit RKs Embedding methods have been foundEmbedding methods have been found

Implicit RKsImplicit RKs Difficult problemDifficult problem

Algorithms are compactAlgorithms are compact Can find embedding methods using two Can find embedding methods using two

stepssteps Linear polynomial approximationLinear polynomial approximation

HW-SDIRK EmbeddingHW-SDIRK Embedding

33rdrd-order accurate-order accurate Behaves like an explicit methodBehaves like an explicit method

May unnecessarily restrict step size for May unnecessarily restrict step size for stiff systemsstiff systems

Search for an alternate embedding Search for an alternate embedding methodmethod

Alt. HW-SDIRK Alt. HW-SDIRK EmbeddingEmbedding

33rdrd-order accurate-order accurate Implicit methodImplicit method

Alt. HW-SDIRK Alt. HW-SDIRK EmbeddingEmbedding

Stability Domain

Damping Plots

Lobatto IIIC(6)Lobatto IIIC(6)

No embedding method existsNo embedding method exists Expensive to perform step size controlExpensive to perform step size control

Can search for an embedding Can search for an embedding methodmethod

Lobatto IIIC(6) Embedding Lobatto IIIC(6) Embedding MethodMethod

55thth-order accurate-order accurate A-StableA-Stable Large asymptotic regionLarge asymptotic region

Lobatto IIIC(6) Embedding Lobatto IIIC(6) Embedding MethodMethod

Stability Domain

Damping Plots

Numerical Numerical ExperimentsExperiments

Numerical ExperimentsNumerical Experiments

Tested various algorithms with Tested various algorithms with selected benchmark ODEsselected benchmark ODEs

Implemented in Dymola/ModelicaImplemented in Dymola/Modelica

ODE Set BODE Set B

ode15s

Inlined with HWSDIRK and alternate error method

ODE Set BODE Set B

Error estimate stays near 10-3

Step size grows andshrinks appropriately

ODE Set DODE Set D

Inlined with Lobatto IIIC(6)

ode15s

ODE Set DODE Set D

An ApplicationAn Application

An ApplicationAn Application

Real-Time, Limited ResourcesReal-Time, Limited Resources Embedded control systemsEmbedded control systems

Model PredictiveModel Predictive Add additional system dynamicsAdd additional system dynamics Simulate missile dynamics in flight for trajectory Simulate missile dynamics in flight for trajectory

shapingshaping

First solution is faster computerFirst solution is faster computer Model may still be too complexModel may still be too complex

Try inliningTry inlining

Questions?Questions?