On the global stabilization of a class of nonlinear singularly perturbed systems using nonlinear H_∞ control approach

Abstract

In this paper, a stabilizing controller for a class of nonlinear singularly perturbed systems is designed. Firstly, some of the fast states are considered to construct an uncertainty block. Then, using the nonlinear ${\mathcal{H}}_{\mathrm{\infty }}$ approach, a controller is designed for the nominal system containing the slow modes and the remaining part of the fast modes. Through two new nonlinear forms of the small-gain theorem, the local and global stability of the controlled system are proven. Since the stability and ${\mathcal{L}}_2$-gain of the uncertainty block do not depend on the perturbation parameter, the designed controller can also be referred to as a robust controller. The proposed approach is used to design a stabilizing controller for the Lorenz system. The controller can globally stabilize the Lorenz system, while the previous methods can only stabilize it locally or even are unable to stabilize it. Material has been presented in ICCIA 2021.

KEYWORDS:

• Nonlinear singularly perturbed systems
• nonlinear H_∞ control
• small-gain theorem

Disclosure statement

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

Notes

1 7th International Conference on Control, Instrumentation and Automation 2021

2 Hamilton-Jacobi-Isaacs’s inequality (Aliyu, 2011)