Backstepping Design for Incremental Stability of Stochastic Hamiltonian Systems with Jumps

TL;DR

This paper introduces a backstepping control method to ensure incremental stability of stochastic Hamiltonian systems with jumps, using incremental Lyapunov functions, demonstrated on a noisy spring pendulum.

Contribution

It develops a novel backstepping controller design scheme for stochastic Hamiltonian systems with jumps, ensuring incremental stability via Lyapunov functions.

Findings

Controller successfully stabilizes a noisy spring pendulum system.

The approach guarantees uniform asymptotic stability of system trajectories.

Method extends incremental stability concepts to stochastic systems with jumps.

Abstract

Incremental stability is a property of dynamical systems ensuring the uniform asymptotic stability of each trajectory rather than a fixed equilibrium point or trajectory. Here, we introduce a notion of incremental stability for stochastic control systems and provide its description in terms of existence of a notion of so-called incremental Lyapunov functions. Moreover, we provide a backstepping controller design scheme providing controllers along with corresponding incremental Lyapunov functions rendering a class of stochastic control systems, namely, stochastic Hamiltonian systems with jumps, incrementally stable. To illustrate the effectiveness of the proposed approach, we design a controller making a spring pendulum system in a noisy environment incrementally stable.