Characterizing all locally exponentially stabilizing controllers as a linear feedback plus learnable nonlinear Youla dynamics

TL;DR

This paper characterizes all locally exponentially stabilizing controllers for nonlinear systems as a sum of a linear feedback and learnable nonlinear dynamics, enabling neural network implementation.

Contribution

It provides a state-space nonlinear Youla-type parametrization for local exponential stabilization, facilitating neural network-based controller design.

Findings

All stabilizing controllers can be decomposed into linear feedback plus stable nonlinear dynamics.

The residual dynamics can be implemented with recurrent neural networks.

The approach enables high-performance neural ODE controllers in nonlinear tasks.

Abstract

We derive a state-space characterization of all dynamic state-feedback controllers that make an equilibrium of a nonlinear input-affine continuous-time system locally exponentially stable. Specifically, any controller obtained as the sum of a linear state-feedback $u=Kx$, with $K$ stabilizing the linearized system, and the output of internal locally exponentially stable controller dynamics is itself locally exponentially stabilizing. Conversely, every dynamic state-feedback controller that locally exponentially stabilizes the equilibrium admits such a decomposition. The result can be viewed as a state-space nonlinear Youla-type parametrization specialized to local, rather than global, and exponential, rather than asymptotic, closed-loop stability. The residual locally exponentially stable controller dynamics can be implemented with stable recurrent neural networks and trained as neural ODEs to achieve high closed-loop performance in nonlinear control tasks.