On the stability of event-based control with neuronal dynamics

TL;DR

This work addresses stability for event-based impulsive control when neuronal (leaky integrate-and-fire) dynamics drive the controller. The main idea is to form a continuous auxiliary state $x_c(t)=x(t)+B\Theta^{-1}z(t)$, whose discontinuities cancel, enabling analysis via an analogue control system and standard Lyapunov arguments. The authors prove global practical stability for nonlinear dynamics under ISS of the analogue closed-loop and global practical exponential stability for linear systems, along with explicit bounds and minimum inter-event-time guarantees. They extend the framework to networks of connected neuronal units and validate the results with numerical simulations, illustrating the practical relevance for neuromorphic control design.

Abstract

Event-based control, unlike analogue control, poses significant analytical challenges due to its hybrid dynamics. This work investigates the stability and inter-event time properties of a control-affine system under event-based impulsive control. The controller consists of multiple neuronal units with leaky integrate-and-fire dynamics acting on a time-invariant, multiple-input multiple-output plant in closed loop. Both the plant state and the neuronal units exhibit discontinuities that cancel if combined linearly, enabling a direct correspondence between the event-based impulsive controller and a corresponding analogue controller. Leveraging this observation, we prove global practical stability of the event-based impulsive control system. In the general nonlinear case, we show that the event-based impulsive controller ensures global practical asymptotic stability if the analogue system is input-to-state stable (ISS) with respect to specific disturbances. In the linear case, we further show global practical exponential stability if the analogue system is stable. We illustrate our results with numerical simulations. The findings reveal a fundamental link between analogue and event-based impulsive control, providing new insights for the design of neuromorphic controllers.

Figures (4)

• Figure 1: The control loop between the controller, which receives the error and generates a control input, and the plant.
• Figure 2: A one-dimensional linear plant controlled by an event-based impulsive controller. The state $x(t)$ (solid) with discontinuities (shaded), the continuous variable $x_c(t)$ (dashed), the bound \ref{['eq:thm_stab_lin_0']} (dash-dotted), and the tighter bound \ref{['eq:cor_lin_2']} (dotted) are shown in the upper plots for $\lambda=3$ (left, cyan), $\lambda = 1.5$ (middle, blue), and $\lambda=0$ (right, purple). The two neuronal variables $z(t)$ (solid), and the rescaled spikes $\theta s(t)$ (dashed) are shown in the lower plots (units 1 and 2 in the top and bottom half, respectively). The control system is given by $f(x) = x$, $B=[-1,1]$, $\theta=1/2.5$, and $g(x) = [[x]_+, [-x]_+]^\intercal$.
• Figure 3: Heatmap of the stability of the event-based impulsive control system as a function of $a$ and $b$. We consider $f(x)=ax$, $B/\theta\cdot g(x) = -bx$, $\lambda = 0$ (left) and $\lambda = 3$ (right), and $B,g$ as in Fig. \ref{['fig:2_1dim']}. The stability for a pair $a,b\in[0,5]$ is measured by $C=\log(| x(T)/x(0)|)/T$ for $x(0)=50.5$ and $T=200$. The black squares on the diagonal indicate $a=b$.
• Figure 4: A two-dimensional linear plant controlled by an event-based impulsive controller. The state $x(t)$ (solid) with discontinuities (shaded), the continuous variable $x_c(t)$ (dashed), and a ball with radius given by the ultimate bound (dotted) are shown for $\omega = 0.5$ (left) and $\omega=3$ (right). The control system is given by $f(x) = \left(1\omega-\omega1\right)\,x$, $B=\left(-110000-11\right)$, $g(x) = [[x_0]_+, [-x_0]_+, [x_1]_+, [-x_1]_+]^\intercal$, $\lambda=0.2$ and $\theta=1/1.5$.

Theorems & Definitions (12)

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