Design of feedback stabilisers using Wiener processes for nonlinear systems

Abstract

A Lyapunov-type theorem is developed to design feedback-stabilizers using Wiener processes for dynamical systems that can contain non-vanishing disturbances and arbitrarily nonlinear growths in their system functions. The proposed stabilizers guarantee that the resulting closed-loop system is globally well-posed, and is globally practically ${\mathcal{K}}_{\mathrm{\infty }}$-exponentially $p$-stable, almost surely globally practically ${\mathcal{K}}_{\mathrm{\infty }}$-exponentially stable, and globally practically ${\mathcal{K}}_{\mathrm{\infty }}$-exponentially stable in probability. Examples on a highly nonlinear system and mobile robots are included to illustrate the fact that although the theory is complicated, its application is straightforward. It is shown that the developed stabilizers can be applied to stabilize dynamical systems that cannot be stabilized by existing deterministic control laws.

Keywords:

• Lyapunov-type theorem
• stabilisation by noise
• almost sure stability
• stability in moment
• stability in probability
• mobile robots

Disclosure statement

No potential conflict of interest was reported by the author(s).