Sliding sector control using new equivalent sector control

ABSTRACT

This paper presents a variable structure control design method for linear and nonlinear continuous-time systems with a special type of sliding sector using a new equivalent sector control. The PR-sliding sector is the subset of state space in which the derivative of the square P-norm of the state, $\parallel x{\parallel }_P^2$, is less than or equal to $-x^{T}Rx$ with a positive semi-definite matrix $R=C^{T}C$ where $(A,C)$ is observable. The new equivalent sector control provides the derivative of $v=s\left(x\right)-\delta \left(x\right)$ being negative. The design method of $s\left(x\right)$ and $\delta \left(x\right)$ is proposed for the general linear single input system by using the Riccati equation. Under the proposed equivalent sector control, the state trajectory may jump at the particular uncontrollable states when $\mathrm{\partial }v/\mathrm{\partial }u=0$, and the control is set zero. If $v$ becomes positive, the state is outside of the sector, and then the state shall be transferred to the inside of the sector by the discontinuous nonlinear control input. Numerical simulations show that the proposed method gives satisfactory results.

KEYWORDS:

• Variable structure control
• sliding sector
• equivalent control
• Riccati equation

Acknowledgments

The authors appreciate the thoughtful comments and valuable suggestions made by the reviewers. The comments and suggestions have motivated the authors to rethink the essence of the proposal and to improve the manuscript. The authors also appreciate Dr Yaodong Pan and his contribution to variable structure control with sliding sector which is the standing point of this article, hence some related theorems and proofs are cited in the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.