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Rhythmic neuromorphic control of a pendulum: A hybrid systems analysis

E. Petri, R. Postoyan, W. P. M. H. Heemels

TL;DR

The paper develops a hybrid-systems framework to analyze rhythmic neuromorphic control of a pendulum driven by Dirac impulse spikes. It proves the existence, uniqueness, and uniform exponential stability of a hybrid limit cycle in the closed-loop, linearized pendulum model and demonstrates numerical convergence to the cycle. It also proposes a spike-amplitude adaptation mechanism to set the oscillation amplitude and sketches a learning-augmented model with preliminary simulations showing amplitude control via adaptation. The work highlights the potential of hybrid dynamical tools to provide formal design and robustness guarantees for neuromorphic control systems, bridging neuroscience-inspired control with rigorous analysis.

Abstract

Neuromorphic engineering is an emerging research domain that aims to realize important implementation advantages that brain-inspired technologies can offer over classical digital technologies, including energy efficiency, adaptability, and robustness. For the field of systems and control, neuromorphic controllers could potentially bring many benefits, but their advancement is hampered by lack of systematic analysis and design tools. In this paper, the objective is to show that hybrid systems methods can aid in filling this gap. We do this by formally analyzing rhythmic neuromorphic control of a pendulum system, which was recently proposed as a prototypical setup. The neuromorphic controller generates spikes, which we model as a Dirac delta pulse, whenever the pendulum angular position crosses its resting position, with the goal of inducing a stable limit cycle. This leads to modeling the closed-loop system as a hybrid dynamical system, which in between spikes evolves in open loop and where the jumps correspond to the spiking control actions. Exploiting the hybrid system model, we formally prove the existence, uniqueness, and a stability property of the hybrid limit cycle for the closed-loop system. Numerical simulations illustrate our approach. We finally elaborate on a possible spiking adaptation mechanism on the pulse amplitude to generate a hybrid limit cycle of a desired maximal angular amplitude.

Rhythmic neuromorphic control of a pendulum: A hybrid systems analysis

TL;DR

The paper develops a hybrid-systems framework to analyze rhythmic neuromorphic control of a pendulum driven by Dirac impulse spikes. It proves the existence, uniqueness, and uniform exponential stability of a hybrid limit cycle in the closed-loop, linearized pendulum model and demonstrates numerical convergence to the cycle. It also proposes a spike-amplitude adaptation mechanism to set the oscillation amplitude and sketches a learning-augmented model with preliminary simulations showing amplitude control via adaptation. The work highlights the potential of hybrid dynamical tools to provide formal design and robustness guarantees for neuromorphic control systems, bridging neuroscience-inspired control with rigorous analysis.

Abstract

Neuromorphic engineering is an emerging research domain that aims to realize important implementation advantages that brain-inspired technologies can offer over classical digital technologies, including energy efficiency, adaptability, and robustness. For the field of systems and control, neuromorphic controllers could potentially bring many benefits, but their advancement is hampered by lack of systematic analysis and design tools. In this paper, the objective is to show that hybrid systems methods can aid in filling this gap. We do this by formally analyzing rhythmic neuromorphic control of a pendulum system, which was recently proposed as a prototypical setup. The neuromorphic controller generates spikes, which we model as a Dirac delta pulse, whenever the pendulum angular position crosses its resting position, with the goal of inducing a stable limit cycle. This leads to modeling the closed-loop system as a hybrid dynamical system, which in between spikes evolves in open loop and where the jumps correspond to the spiking control actions. Exploiting the hybrid system model, we formally prove the existence, uniqueness, and a stability property of the hybrid limit cycle for the closed-loop system. Numerical simulations illustrate our approach. We finally elaborate on a possible spiking adaptation mechanism on the pulse amplitude to generate a hybrid limit cycle of a desired maximal angular amplitude.

Paper Structure

This paper contains 18 sections, 9 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem statement
  4. Hybrid model of closed-loop system
  5. Variables
  6. Hybrid model
  7. Hybrid limit cycles
  8. Existence and stability of a hybrid limit cycle
  9. Forward invariant set $\mathbb{X}_0$
  10. Solution expression
  11. Existence and uniqueness of hybrid limit cycle
  12. Stability properties of the hybrid limit cycle
  13. Numerical illustration
  14. Discussion on pulse amplitude adaptation
  15. Adaptation rule
  16. ...and 3 more sections

Key Result

Proposition 1

Given any $I\in\mathbb{R}_{>0}$, the following holds for system $\mathcal{H}_I$ in (eq:hybrid-model-compact).

Figures (4)

  • Figure 1: Illustration of the pendulum.
  • Figure 2: Angular position (red), velocity (blue) and $\sigma$ (yellow) with nonlinear dynamics and initial condition $(\frac{\pi}{3},2,-1)$.
  • Figure 3: Hybrid limit cycle with nonlinear dynamics with different initial conditions.
  • Figure 4: Angular position (blue) and velocity (red) with nonlinear dynamics and spiking adaptation loop to adapt the amplitude of the spikes $I$. Angular position (black) without adaptation (constant $I = 0.1$).

Theorems & Definitions (13)

  • Remark 1
  • Proposition 1
  • Definition 1
  • Definition 2
  • Proposition 2
  • Lemma 1
  • Proposition 3
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • ...and 3 more