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This research delves into the efficient simulation of physical system models through the use of inlined implicit Runge-Kutta algorithms. The study explores techniques for simulation, results, and an application introduction, focusing on handling stiffness and widely varying eigenvalues. The text covers both explicit and implicit algorithms, highlighting the simplicity of implementing explicit algorithms and the computational load associated with implicit ones. Inline-integration is discussed as a method to merge the integration algorithm with the model to eliminate differential equations, resulting in easily implemented implicit algorithms. Step-size control techniques are examined to improve computational efficiency while maintaining accuracy, and various embedding methods are explored. Numerical experiments are conducted using benchmark ODEs to validate the proposed methods. The application section showcases real-time simulations for embedded control systems, specifically in model predictive control and missile dynamics. This innovative approach aims to enhance simulation speed and accuracy while minimizing computational costs.
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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 85721 U.S.A
Topics • Introduction • Techniques for Simulation • Results • An Application
Introduction • Stiffness • Widely varying eigenvalues • Explicit algorithms • Straightforward to implement • Step size limited by numerical stability • Implicit algorithms • More difficult to implement • Additional computational load • Needed to simulate stiff systems • May use larger step sizes
Inline-Integration • Merges the integration algorithm with the model • Eliminates differential equations • Results in difference equations (∆Es) • Easily implement implicit algorithms • Circuit example inlining Rad3
Inlined with Rad3 Evaluate at Rad3 time instants Eliminate derivatives Integrator equations
Sorting • 10 equations immediately causalized • Need to perform tearing • Make assumptions about variables being ‘known’
Tearing Tearing variable Residual Eq.
Tearing Residual Eq. #2 Tearing variable #2
Tearing • Completely causalized equations • 2 iteration variables, vc and i1 • Could use this set of equations for simulation • Want step-size control
Step-Size Control • Want larger step sizes • Reduce the overall computational cost • Maintain desired accuracy • Compute error estimate • Embedding method • Shares computations with original method
Step-Size Control • Explicit RKs • Embedding methods have been found • Implicit RKs • Difficult problem • Algorithms are compact • Can find embedding methods using two steps • Linear polynomial approximation
HW-SDIRK Embedding • 3rd-order accurate • Behaves like an explicit method • May unnecessarily restrict step size for stiff systems • Search for an alternate embedding method
Alt. HW-SDIRK Embedding • 3rd-order accurate • Implicit method
Alt. HW-SDIRK Embedding Stability Domain Damping Plots
Lobatto IIIC(6) • No embedding method exists • Expensive to perform step size control • Can search for an embedding method
Lobatto IIIC(6) Embedding Method • 5th-order accurate • A-Stable • Large asymptotic region
Lobatto IIIC(6) Embedding Method Stability Domain Damping Plots
Numerical Experiments • Tested various algorithms with selected benchmark ODEs • Implemented in Dymola/Modelica
ODE Set B Inlined with HWSDIRK and alternate error method ode15s
ODE Set B Error estimate stays near 10-3 Step size grows and shrinks appropriately
ODE Set D Inlined with Lobatto IIIC(6) ode15s
An Application • Real-Time, Limited Resources • Embedded control systems • Model Predictive • Add additional system dynamics • Simulate missile dynamics in flight for trajectory shaping • First solution is faster computer • Model may still be too complex • Try inlining