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Efficient Simulation of Singularly Perturbed Systems Using a Stabilized Multirate Explicit Scheme

Yibo Shi, Cristian R. Rojas

TL;DR

The paper addresses stiff initial-value problems arising from singularly perturbed systems by introducing a Stabilized Multirate Explicit Scheme (SMES) that stabilizes explicit integration without requiring implicit solves or tiny steps. By interleaving a short, fast-time-step phase with a longer slow-time-step phase, SMES drives fast dynamics toward the equilibrium manifold $M_0$ while the slow dynamics advance with larger steps. Stability and error analyses show SMES achieves comparable accuracy to reduced models with significantly fewer function evaluations, yielding substantial efficiency gains. The numerical illustration demonstrates practical impact in nonlinear adaptive control with parasitic dynamics, highlighting SMES as a scalable, implementable approach for efficiently simulating SPSs in engineering applications.

Abstract

Singularly perturbed systems (SPSs) are prevalent in engineering applications, where numerically solving their initial value problems (IVPs) is challenging due to stiffness arising from multiple time scales. Classical explicit methods require impractically small time steps for stability, while implicit methods developed for SPSs are computationally intensive and less efficient for strongly nonlinear systems. This paper introduces a Stabilized Multirate Explicit Scheme (SMES) that stabilizes classical explicit methods without the need for small time steps or implicit formulations. By employing a multirate approach with variable time steps, SMES allows the fast dynamics to rapidly converge to their equilibrium manifold while slow dynamics evolve with larger steps. Analysis shows that SMES achieves numerical stability with significantly reduced computational effort and controlled error. Its effectiveness is illustrated with a numerical example.

Efficient Simulation of Singularly Perturbed Systems Using a Stabilized Multirate Explicit Scheme

TL;DR

The paper addresses stiff initial-value problems arising from singularly perturbed systems by introducing a Stabilized Multirate Explicit Scheme (SMES) that stabilizes explicit integration without requiring implicit solves or tiny steps. By interleaving a short, fast-time-step phase with a longer slow-time-step phase, SMES drives fast dynamics toward the equilibrium manifold while the slow dynamics advance with larger steps. Stability and error analyses show SMES achieves comparable accuracy to reduced models with significantly fewer function evaluations, yielding substantial efficiency gains. The numerical illustration demonstrates practical impact in nonlinear adaptive control with parasitic dynamics, highlighting SMES as a scalable, implementable approach for efficiently simulating SPSs in engineering applications.

Abstract

Singularly perturbed systems (SPSs) are prevalent in engineering applications, where numerically solving their initial value problems (IVPs) is challenging due to stiffness arising from multiple time scales. Classical explicit methods require impractically small time steps for stability, while implicit methods developed for SPSs are computationally intensive and less efficient for strongly nonlinear systems. This paper introduces a Stabilized Multirate Explicit Scheme (SMES) that stabilizes classical explicit methods without the need for small time steps or implicit formulations. By employing a multirate approach with variable time steps, SMES allows the fast dynamics to rapidly converge to their equilibrium manifold while slow dynamics evolve with larger steps. Analysis shows that SMES achieves numerical stability with significantly reduced computational effort and controlled error. Its effectiveness is illustrated with a numerical example.

Paper Structure

This paper contains 10 sections, 34 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Forward Euler Method
  4. Stabilized Multirate Forward Euler Method
  5. Stability analysis
  6. Error analysis
  7. Numerical illustration
  8. Efficiency of the SMFE method
  9. Error convergence
  10. Conclusions

Figures (3)

  • Figure 1: Adaptive control system with small parasitic dynamics kokotovic1999singular.
  • Figure 2: State trajectories of $y(t)$ and $z(t)$, and simulation errors with different methods. SMFE$(\Delta, N)$ represents the SMFE method using a large step of length $\Delta$ and small steps of number $N$.
  • Figure 3: 2D state trajectories of $y(t) - z(t)$ under different simulation schemes. SMFE is used to simulate the system with parasitic dynamics in \ref{['eq:adaori']}, while FEM is used to simulate the reduced system in \ref{['eq:adaslow']}. The large step size of SMFE and the step size of FEM are both $\Delta$ in each subplot. The direction of the state trajectories is indicated using arrows in the left subplot.