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Dynamic System Stability Verification Using Numerical Simulator

Jongrae Kim

TL;DR

The paper tackles the challenge of providing stability guarantees for safety-critical systems controlled via high-fidelity numerical simulators, including RL-based designs, by extending the inverse Lyapunov framework to compute constants that certify exponential stability in numerically simulated dynamics. It derives state-propagation bounds for Euler and Runge-Kutta integrations with discontinuities, and introduces energy-based bounds together with a δ-sampling algorithm to yield deterministic stability guarantees. Under an exponential-stability assumption, long-horizon bounds simplify to a decaying bound $||φ_N(N Δt, t, x) - φ_N(N Δt, t, y)|| ≤ 2 k r_0 e^{-λ T}$ and a square-root bound that depends on $a$, $b$, and the sampling parameter $δ$. The approach enables formal stability certification for safety-critical, simulation-based controllers in robotics, and suggests avenues for improvement in sampling efficiency and parameter estimation to broaden applicability.

Abstract

There are recent shifts in demand for design controllers from simplified to complex model-based. Although simplification approaches are successful in many areas of engineering control systems, high-fidelity simulation-based control design, for example, reinforcement learning, has been rising in robotics areas. On the other hand, the lack of assurances about the stability and robustness of simulation-based control design restricts its applications to safety-critical systems. We develop computational methods to verify the stability and robustness of safety-critical systems. By extending the inverse Lyapunov theorem, we present a practical method to compute the constants required to check the exponential stability conditions of dynamic systems implemented in a numerical simulator. It is shown that the norm-bound of the propagated states is a function of the numerical integration steps, where the numerical simulator may include discontinuous jumps of states. The energy bounds for the transition states are obtained based on the exponential stability assumption of the inverse Lyapunov theorem. Finally, a finite sampling algorithm provides the deterministic stability guarantee for the continuous state space.

Dynamic System Stability Verification Using Numerical Simulator

TL;DR

The paper tackles the challenge of providing stability guarantees for safety-critical systems controlled via high-fidelity numerical simulators, including RL-based designs, by extending the inverse Lyapunov framework to compute constants that certify exponential stability in numerically simulated dynamics. It derives state-propagation bounds for Euler and Runge-Kutta integrations with discontinuities, and introduces energy-based bounds together with a δ-sampling algorithm to yield deterministic stability guarantees. Under an exponential-stability assumption, long-horizon bounds simplify to a decaying bound and a square-root bound that depends on , , and the sampling parameter . The approach enables formal stability certification for safety-critical, simulation-based controllers in robotics, and suggests avenues for improvement in sampling efficiency and parameter estimation to broaden applicability.

Abstract

There are recent shifts in demand for design controllers from simplified to complex model-based. Although simplification approaches are successful in many areas of engineering control systems, high-fidelity simulation-based control design, for example, reinforcement learning, has been rising in robotics areas. On the other hand, the lack of assurances about the stability and robustness of simulation-based control design restricts its applications to safety-critical systems. We develop computational methods to verify the stability and robustness of safety-critical systems. By extending the inverse Lyapunov theorem, we present a practical method to compute the constants required to check the exponential stability conditions of dynamic systems implemented in a numerical simulator. It is shown that the norm-bound of the propagated states is a function of the numerical integration steps, where the numerical simulator may include discontinuous jumps of states. The energy bounds for the transition states are obtained based on the exponential stability assumption of the inverse Lyapunov theorem. Finally, a finite sampling algorithm provides the deterministic stability guarantee for the continuous state space.
Paper Structure (4 sections, 9 theorems, 58 equations, 4 figures, 1 algorithm)

This paper contains 4 sections, 9 theorems, 58 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Norm-bound of simulator propagated states
  3. Stability Verification
  4. Conclusions & Future Works

Key Result

Theorem 1

For any ${\bf x}$ and ${\bf y}$ in ${\mathbb D}$, if ${\bf f}({\bf x})$ satisfies then and

Figures (4)

  • Figure 1: $\log(a)$ and $\log(b)$ with respect to $T$
  • Figure 2: The exponential bounds and the 1000 Monte-Carlo simulations
  • Figure 3: $\log_{10}(2 k r_0 e^{-\lambda T} a)$ and $\log_{10}(2 k r_0 e^{-\lambda T} b)$ with respect to $T$
  • Figure 4: The inequality condition values for the ranges of the number of $x_\delta$ samplings and the simulation time length, $T$

Theorems & Definitions (24)

  • Example 1: Numerical Solution for Non-Lipschitz Systems
  • Theorem 1: Lipschitz Condition for ${\boldsymbol\Phi}_\text{\rm Euler}$ and ${\boldsymbol\Phi}_\text{\rm RK}$
  • Remark 1
  • Theorem 2: Bound for ${\boldsymbol\Phi}_\text{\rm Euler}$ and ${\boldsymbol\Phi}_\text{\rm RK}$
  • Example 2
  • Definition 1: State-Transition by Numerical Simulator
  • Theorem 3: Bound for Longer Propagation
  • Example 3
  • Definition 2: Exponential Stability
  • Theorem 4: Exponential Bounds
  • ...and 14 more