Exponential Stability and Stabilization of Extended Linearizations via Continuous Updates of Riccati Based Feedback

TL;DR

This paper develops a method for stabilizing nonlinear systems by continuously updating Riccati-based feedback, enabling convergence to nonattractive steady states from well-developed initial states.

Contribution

It introduces a novel update scheme for Riccati feedback that maintains stability during state evolution, extending stabilization capabilities beyond local linearization methods.

Findings

The proposed method ensures stability if the coefficient matrix remains uniformly stable nearby.

Numerical examples demonstrate the effectiveness of the approach for setpoint stabilization.

The scheme is computationally efficient and applicable to various nonlinear systems.

Abstract

Many recent works on stabilization of nonlinear systems target the case of locally stabilizing an unstable steady state solutions against small perturbation. In this work we explicitly address the goal of driving a system into a nonattractive steady state starting from a well developed state for which the linearization based local approaches will not work. Considering extended linearizations or state-dependent coefficient representations of nonlinear systems, we develop sufficient conditions for stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. Based on these analytical results we propose an update scheme that is designed to maintain the stabilization property of Riccati based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in two examples.