Utilization of noise for the control of a class of non-linear systems

TL;DR

The paper investigates stabilization of nonlinear systems using state-multiplicative noise, i.e., stochastic anti-resonance (SAR), and reviews Linear Matrix Inequality (LMI) based stability conditions for sector-bounded nonlinearities in the model $dx(t)=A x(t)dt + F f(y(t))dt + D x(t)d\beta(t)$ with $y(t)=C x(t)$. It then extends these results to more general nonlinearities by leveraging the universal approximation theorem to replace $f$ with a sector-bounded neural network, enabling the same LMI framework to certify stability. A Morris-Lecar neuron model serves as a case study where a shallow neural network approximates $f_i$, and the condition $\mathcal{N}(\nu,\Lambda,\mathcal{T})<0$ is checked via YALMIP to identify a stable SAR gain range, with simulations confirming stabilization near $\sigma \approx 0.85$ albeit with noisy outputs. The work highlights the potential of SAR in broader nonlinear control applications and points to future directions, including performance-based objectives such as the $H_2$ norm and mixed stochastic-deterministic controllers.

Abstract

Utilization of noise for the control of a class of non-linear systems is presented. The application of state-multiplicative noise as a mean of control is far more limited then the use of standard determinis?tic gains. Nevertheless, so called Stochastic Anti Resonance (SAR) with state-multiplicative noise based control, do arise in a variety of situations such as in engineering applications, physics modelling, bi?ology, and models of visuo-motor tasks. Linear Matrix Inequalities based conditions from recent publications are reviewed, that character?ize stochastic stability of such nonlinear systems applying SAR. While those results dealt with systems that are, apriori, modelled using sec?tor bounded nonlinearities, we demonstrate that more general systems that can be approximated as such, can be also controlled using SAR.

Figures (4)

• Figure 1: $x_1$-vs. $x_2$ Sector Bounded Nonlinearities
• Figure 2: $x_1(t)$ and $x_2(t)$ Sector Bounded Nonlinearities
• Figure 3: $x_1(t)$ and $x_2(t)$ Moris-Lecar Model - with multiplicative noise
• Figure 4: Sweep of $\sigma$ for $\mathcal{N}<0$ condition