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Analysis of a Simple Neuromorphic Controller for Linear Systems: A Hybrid Systems Perspective

E. Petri, K. J. A. Scheres, E. Steur, W. P. M. H. Heemels

TL;DR

This paper analyzes a neuromorphic controller inspired by the leaky integrate-and-fire neuronal model in closed loop with a single-input single-output linear time-invariant system, and proves a practical stability property for the closed-loop system and ensures the existence of a strictly positive dwell-time between spikes.

Abstract

In this paper we analyze a neuromorphic controller, inspired by the leaky integrate-and-fire neuronal model, in closed-loop with a single-input single-output linear time-invariant system. The controller consists of two neuron-like variables and generates a spiking control input whenever one of these variables reaches a threshold. The control input is different from zero only at the spiking instants and, hence, between two spiking times the system evolves in open-loop. Exploiting the hybrid nature of the integrate-and-fire neuronal dynamics, we present a hybrid modeling framework to design and analyze this new controller. In the particular case of single-state linear time-invariant plants, we prove a practical stability property for the closed-loop system, we ensure the existence of a strictly positive dwell-time between spikes, and we relate these properties to the parameters in the neurons. The results are illustrated in a numerical example.

Analysis of a Simple Neuromorphic Controller for Linear Systems: A Hybrid Systems Perspective

TL;DR

This paper analyzes a neuromorphic controller inspired by the leaky integrate-and-fire neuronal model in closed loop with a single-input single-output linear time-invariant system, and proves a practical stability property for the closed-loop system and ensures the existence of a strictly positive dwell-time between spikes.

Abstract

In this paper we analyze a neuromorphic controller, inspired by the leaky integrate-and-fire neuronal model, in closed-loop with a single-input single-output linear time-invariant system. The controller consists of two neuron-like variables and generates a spiking control input whenever one of these variables reaches a threshold. The control input is different from zero only at the spiking instants and, hence, between two spiking times the system evolves in open-loop. Exploiting the hybrid nature of the integrate-and-fire neuronal dynamics, we present a hybrid modeling framework to design and analyze this new controller. In the particular case of single-state linear time-invariant plants, we prove a practical stability property for the closed-loop system, we ensure the existence of a strictly positive dwell-time between spikes, and we relate these properties to the parameters in the neurons. The results are illustrated in a numerical example.
Paper Structure (6 sections, 1 theorem, 57 equations, 2 figures)

This paper contains 6 sections, 1 theorem, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Neuromorphic controller and hybrid modeling framework
  4. Main analysis and design
  5. Numerical example
  6. Conclusions

Key Result

Theorem 1

Consider system eq:HybridSystem with $n_x = 1$ and $A = a \in \mathbb{R}_{>0}$, $B = C = 1$. Select $\rho \in (0,1)$. Define $\Psi:= \frac{\rho + 1}{(\rho + 1)^2 -1}\alpha$. Then for all $\Delta \in \left(0, \frac{\rho\alpha}{\mu +a}\right]$, and all $\sigma \in \left[ \frac{(\rho + 1)^2 -1}{(\rho

Figures (2)

  • Figure 1: Block diagram representing the system architecture
  • Figure 2: State $x$ and input $u$ when $\alpha = 0.5$, $\mu = 0.5$, $\Delta = 0.1$ (blue) and $\alpha_1 = 0.3$, $\alpha_2 = 0.5$, $\mu_1 = 0.2$, $\mu_2 = 0.5$, $\Delta_1 = 0.1$, $\Delta_2 = 0.2$ with noise and disturbances (red).

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4