Ultimate uniform bounded-stability of inertial coupling electromechanical system via nonlinear time-varying feedback

ABSTRACT

This paper concerns with the problem of stabilisation of inertial coupling electromechanical system. The concept of inertial coupling electromechanical system is an interesting topic from the control theory point of view and its universe of applications. This is because, for stabilisation proposes, the complete model of electromechanical system significantly reduces higher frequency functions in their dynamics caused by high gain controllers. Here, the system stabilisation is achieved by applying nonlinear time-varying controllers. The reason to use a time-varying controller is to give some learning or adaptation in the control process. Moreover, to get a good result for the closed-loop system, the structure of a storage function is based on the classical quadratic function in addition to quadratic function of system total energy. So, the main goal of this paper is obtain a controller that guarantees that the system state error arrives near the required equilibrium in exponential form. Hence, the Ultimate Uniform Bounded-Stability of the system trajectories of desired equilibrium is concluded. Finally, experimental results of the control algorithm developed in this paper are presented briefly.

KEYWORDS:

• Inertial coupling electromechanical system
• nonlinear time-varying feedback
• dynamic matrix inequality
• ultimate uniform bounded-stability

Acknowledgments

The authors would like to express their appreciation to the Automatic-Control Laboratory team from CINVESTAV México, for help in the realisation of the presented experimental example. Thanks to the reviewers for their thoughtful review which helps improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. As is the case of external perturbation, mechanical pulsation, variations given from data acquisition, among others

2. By definition, a matrix $\mathit{A}={\mathit{A}}^{⊺}\in {\mathbb{R}}^{n\times n}$ be a positive definite matrix if and only if $0<x^{⊺}\mathit{A}x$ for all $x\in {\mathbb{R}}^n$ and $0=x^{⊺}\mathit{A}x$ for $x={\bar{0}}_n$, this means the matrix ${\mathit{A}}^{-1}$ exists and it is positive definite (for instance see Boyd & Lieven, 2004; Poznyak, 2009).

3. The proofs of this statement and propositions are provided on appendix.

4. See the proof of Proposition 3.1 in the appendix A.2.