Efficient Simulation of Physical System Models Using Inlined Implicit Runge-Kutta Algorithms Vicha Treeaporn Department of Electrical & Computer Engineering The University of Arizona Tucson, Arizona U.S.A

Topics Introduction Introduction Techniques for Simulation Techniques for Simulation Results Results An Application An Application

Introduction Stiffness Stiffness Widely varying eigenvalues Widely varying eigenvalues Explicit algorithms Explicit algorithms Straightforward to implement Straightforward to implement Step size limited by numerical stability Step size limited by numerical stability Implicit algorithms Implicit algorithms More difficult to implement More difficult to implement Additional computational load Additional computational load Needed to simulate stiff systems Needed to simulate stiff systems May use larger step sizes May use larger step sizes

Inline-Integration Merges the integration algorithm with the model Merges the integration algorithm with the model Eliminates differential equations Eliminates differential equations Results in difference equations (∆Es) Results in difference equations (∆Es) Easily implement implicit algorithms Easily implement implicit algorithms Circuit example inlining Rad3 Circuit example inlining Rad3

Simple Circuit

Circuit Equations

Inlined with Rad3 Integrator equations Eliminate derivatives Evaluate at Rad3 time instants

Sorting

Sorting

Sorting

Sorting 10 equations immediately causalized 10 equations immediately causalized Need to perform tearing Need to perform tearing Make assumptions about variables being ‘known’ Make assumptions about variables being ‘known’

Tearing Residual Eq. Tearing variable

Tearing Residual Eq. #2 Tearing variable #2

Tearing Completely causalized equations Completely causalized equations 2 iteration variables, v c and i 1 2 iteration variables, v c and i 1 Could use this set of equations for simulation Could use this set of equations for simulation Want step-size control Want step-size control

Step-Size Control Want larger step sizes Want larger step sizes Reduce the overall computational cost Reduce the overall computational cost Maintain desired accuracy Maintain desired accuracy Compute error estimate Compute error estimate Embedding method Embedding method Shares computations with original method Shares computations with original method

Step-Size Control Explicit RKs Explicit RKs Embedding methods have been found Embedding methods have been found Implicit RKs Implicit RKs Difficult problem Difficult problem Algorithms are compact Algorithms are compact Can find embedding methods using two steps Can find embedding methods using two steps Linear polynomial approximation Linear polynomial approximation

HW-SDIRK Embedding 3 rd -order accurate 3 rd -order accurate Behaves like an explicit method Behaves like an explicit method May unnecessarily restrict step size for stiff systems May unnecessarily restrict step size for stiff systems Search for an alternate embedding method Search for an alternate embedding method

Alt. HW-SDIRK Embedding 3 rd -order accurate 3 rd -order accurate Implicit method Implicit method

Alt. HW-SDIRK Embedding Stability Domain Damping Plots

Lobatto IIIC(6) No embedding method exists No embedding method exists Expensive to perform step size control Expensive to perform step size control Can search for an embedding method Can search for an embedding method

Lobatto IIIC(6) Embedding Method 5 th -order accurate 5 th -order accurate A-Stable A-Stable Large asymptotic region Large asymptotic region

Lobatto IIIC(6) Embedding Method Stability Domain Damping Plots

Numerical Experiments

Tested various algorithms with selected benchmark ODEs Tested various algorithms with selected benchmark ODEs Implemented in Dymola/Modelica Implemented in Dymola/Modelica

ODE Set B ode15s Inlined with HWSDIRK and alternate error method

ODE Set B Error estimate stays near Step size grows and shrinks appropriately

ODE Set D Inlined with Lobatto IIIC(6) ode15s

ODE Set D

An Application

Real-Time, Limited Resources Real-Time, Limited Resources Embedded control systems Embedded control systems Model Predictive Model Predictive Add additional system dynamics Add additional system dynamics Simulate missile dynamics in flight for trajectory shaping Simulate missile dynamics in flight for trajectory shaping First solution is faster computer First solution is faster computer Model may still be too complex Model may still be too complex Try inlining Try inlining

Questions?