Backstepping Design for Incremental Stability of Stochastic Hamiltonian Systems with Jumps
This work provides a theoretical framework and constructive method for ensuring incremental stability in stochastic systems, which is relevant for control engineers dealing with noisy environments.
The paper introduces a notion of incremental stability for stochastic control systems and provides a backstepping controller design scheme that renders stochastic Hamiltonian systems with jumps incrementally stable, demonstrated on a spring pendulum system in a noisy environment.
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.