Nearest Neighbor Control For Practical Stabilization of Passive Nonlinear Systems

TL;DR

The paper studies static output-feedback stabilization of continuous-time nonlinear passive systems when the actuation is restricted to a finite set. It introduces a nearest-neighbor mapping from the measured output to a finite input set, leveraging passivity and large-time observability to ensure practical stabilization to a ball $\mathbb{B}_\epsilon$. A key result shows that for an $m$-input system, at most $m+1$ nonzero input points (plus zero or a target input) suffice, with a constructive design procedure; the framework extends to incrementally passive systems with constant references and provides explicit Voronoi-cell bounds and minimal-action constructions. The paper also presents two concrete minimal input constructions, derives closed-form bounds for the Voronoi radius $\delta$ in terms of a scale parameter, and illustrates the approach with examples and simulations. Overall, the work enables stabilization under actuator constraints with a small, well-structured control alphabet and offers practical guidance for quantized control of nonlinear passive systems.

Abstract

This paper studies static output feedback stabilization of continuous-time (incrementally) passive nonlinear systems where the control actions can only be chosen from a discrete (and possibly finite) set of points. For this purpose, we are working under the assumption that the system under consideration is large-time norm observable and the convex hull of the realizable control actions contains the target constant input (which corresponds to the equilibrium point) in its interior. We propose a nearest-neighbor based static feedback mapping from the output space to the finite set of control actions, that is able to practically stabilize the closed-loop systems. Consequently, we show that for such systems with $m$-dimensional input space, it is sufficient to have $m+1$ discrete input points (other than zero for general passive systems or the target constant input for incrementally passive systems). Furthermore, we present a constructive algorithm to design such $m+1$ nonzero input points that satisfy the conditions for practical stability using our proposed nearest-neighbor control.

Figures (3)

• Figure 1: Illustration of the nearest neighbor regions, or Voronoi cells, for discrete control actions set $\mathcal{U}_{\rm ex}=\left\lbrace 0,u_{\rm ex,1},u_{\rm ex, 2},u_{\rm ex,3} \right\rbrace$ given in Example \ref{['example:exU']}. The triangular region around the origin is $V_{\mathcal{U}}(0)$ and it is contained in $\mathbb{B}_\delta$, for some $\delta=\alpha > 0$.
• Figure 2: Simulation results of $\Sigma_{ex}$ using the control approach proposed in the Proposition \ref{['prop_nearest_control']} with discrete input set $\mathcal{U}_{ex}$ as in \ref{['ex:U']} and fixed parameters $\theta_{\rm ex}=0$ and $\alpha=0.1$. It can be seen that once both the state $x$ and the output $y$ enters their respective convergence ball, the control input is zero.
• Figure 3: Simulation results of $\Sigma_{ex}$ using the control approach proposed in the Propostion \ref{['prop:phi_incrementalpassivity']} with discrete input set $\overline{\mathcal{U}}_{ex} := \mathcal{U}_{ex} + u^*$ with $\mathcal{U}_{ex}$ be as the input set used in Example \ref{['ex:1']}. Here we have that $u^*\in\overline{\mathcal{U}}_{ex}$. Similar to before, in this simulation, once both the state $x$ and the output $y$ enter their respective convergence ball, the control action is switched to $u^*$ for the rest of the simulation.

Theorems & Definitions (30)

• Definition 1
• Remark 1
• Remark 2
• Definition 2: $m$-simplex
• Example 1
• Definition 3
• Remark 3
• Lemma 1
• proof
• Example 2