An exact active sensing strategy for a class of bio-inspired systems

TL;DR

The paper addresses stabilization challenges in bio-inspired translation-invariant systems with nonlinear output that mimics sensory adaptation, showing that traditional dynamic output feedback cannot stabilize to a point. It introduces an exact active-sensing strategy combining a low-amplitude periodic input with nonlinear output feedback to induce and stabilize an arbitrarily small limit cycle around the origin. The authors prove local exponential stability of the resulting time-periodic system via a Lyapunov function and corroborate the analysis with Floquet theory, revealing a conservative analytical bound and a larger numerically observed stability region. They also quantify the domain of attraction for the nonlinear system and discuss implications for understanding active sensing in animals and for broader control problems lacking observability. Overall, the work provides a provably stable, active-sensing-based feedback mechanism that can replicate observed biological behaviors while offering insights into control design for non-observable nonlinear systems.

Abstract

We consider a general class of translation-invariant systems with a specific category of output nonlinearities motivated by biological sensing. We show that no dynamic output feedback can stabilize this class of systems to an isolated equilibrium point. To overcome this fundamental limitation, we propose a simple control scheme that includes a low-amplitude periodic forcing function akin to so-called "active sensing" in biology, together with nonlinear output feedback. Our analysis shows that this approach leads to the emergence of an exponentially stable limit cycle. These findings offer a provably stable active sensing strategy and may thus help to rationalize the active sensing movements made by animals as they perform certain motor behaviors.

Figures (5)

• Figure 1: (A) Weakly electric fish control their position using active sensing to remain within a refuge; $x(t)$ is the fish's position relative to the refuge. (B) Simplified model.
• Figure 2: The system (\ref{['eq:simple_system']}) cannot be stabilized to an equilibrium point by the dynamic feedback in (\ref{['eq:outputfeedback']}).
• Figure 3: Evolution of the system states, $x(t)$ and $z(t)$ for $\delta = 1/2$ with $k =1, a= 1/\sqrt{2}$ from initial condition $(x_0,z_0)=(1,1)$. (A) Time traces. (B) State trajectories on $x$-$z$ plane. The black dashed line represents the steady-state circular orbit of radius $a$.
• Figure 4: Eigenvalues of the linearized system (\ref{['eq:LTV']}) for different values of $\delta = ka^2$. (A) Modulus of the eigenvalues. The gray region represents the stable area, where the modulus of both eigenvalues is less than one. The gray solid line denotes the square root of the product of eigenvalues $(\exp(-\pi/2))$ and the blue dotted line denotes the critical value $\delta^* \approx 3.2$ above which the system (\ref{['eq:LTV']}) becomes unstable. (B) The real (solid line) and imaginary part (dashed line) of the eigenvalues. The eigenvalues are real for low values of $\delta$, become complex conjugates between $(\delta_1,\delta_2) = (0.54,1.94)$, and return to being real for higher values of $\delta$. (C) Evolution of the eigenvalues on the complex plane with increase in $\delta$. At $\delta = 0$, the eigenvalues are $1$ and $\exp(-\pi)$, respectively. As $\delta \to \infty$ one eigenvalue approaches zero, while the other tends to infinity, with their product remaining constant $\exp(-\pi)$. The gray regions in (B, C) are the same as in (A) representing the stable area.
• Figure 5: Domain of attraction (DoA) for the nonlinear system (\ref{['eq:LTV']}) for $\delta = 1/2$ with $k =1, a= 1/\sqrt{2}$. (A) Trajectories from the same initial location $(x_0,z_0)=(2.5,2.5)$ (gray marker) initiated at $t=0$ (green) converges whereas the one initiated at $t=7\pi/8$ (red) diverges from the periodic solution, $\xi^*(t)$ (black dashed circle). (B) The green region is the conservative DoA, $D^*$, from within which all initial conditions converge to the periodic solution irrespective of the $t_0$. The red region denotes the set of all initial conditions that diverge irrespective of $t_0$. The convergence (or not) of trajectories whose initial conditions lie within the gray region depends on the initial time as illustrated in (A).

Theorems & Definitions (12)

• Proposition III.1
• proof
• Theorem 1
• proof
• Remark V.1
• Corollary V.2
• proof
• Remark V.3
• Remark VI.1
• Remark VI.2