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Logic-based switching finite-time stabilization with applications in mechanical systems

Shiqi Zheng, Shihao Wang, Xiang Chen, Yuanlong Xie

Abstract

This paper investigates the finite time stabilization problem for a class of nonlinear systems with unknown control directions and unstructured uncertainties. The unstructured uncertainties indicate that not only the parameters but also the structure of the system nonlinearities are uncertain. The contributions are mainly in the following two aspects: First, a new adaptive control method is proposed for the considered system. Logic-based switching rule is utilized to tune the controller parameters online to stabilize the system in finite time. Different from the existing adaptive controllers for structured/parametric uncertainties, a new switching barrier Lyapunov method and supervisory functions are introduced to overcome the obstacles caused by unstructured uncertainties and unknown control directions. Second, based on a time-varying backstepping scheme, an extension is made to the proposed method such that all the system states can be regulated to zero in prescribed finite time. Moreover, a new analysis tool is presented to show the boundedness of the control signals. Simulations are conducted to verify the effectiveness of the proposed methods.

Logic-based switching finite-time stabilization with applications in mechanical systems

Abstract

This paper investigates the finite time stabilization problem for a class of nonlinear systems with unknown control directions and unstructured uncertainties. The unstructured uncertainties indicate that not only the parameters but also the structure of the system nonlinearities are uncertain. The contributions are mainly in the following two aspects: First, a new adaptive control method is proposed for the considered system. Logic-based switching rule is utilized to tune the controller parameters online to stabilize the system in finite time. Different from the existing adaptive controllers for structured/parametric uncertainties, a new switching barrier Lyapunov method and supervisory functions are introduced to overcome the obstacles caused by unstructured uncertainties and unknown control directions. Second, based on a time-varying backstepping scheme, an extension is made to the proposed method such that all the system states can be regulated to zero in prescribed finite time. Moreover, a new analysis tool is presented to show the boundedness of the control signals. Simulations are conducted to verify the effectiveness of the proposed methods.

Paper Structure

This paper contains 15 sections, 8 theorems, 42 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Contributions
  4. Organizations
  5. Preliminaries and problem formulation
  6. Problem formulation
  7. Technical lemmas
  8. Finite time stabilization
  9. Controller design
  10. Logic-based switching rule
  11. Main result and stability analysis
  12. Examples
  13. Conclusions
  14. Proof of Lemma \ref{['lem:C1']}
  15. Proof of Claim 1a

Key Result

Lemma 1

(key-15-1) Consider the following Young's inequality where $x,y\in\mathbb{R}$, $a,b$ are positive constants, $\zeta(x,y)>0$ is any real valued function.

Figures (6)

  • Figure 1: One possible variation of $\hat{\chi}_{i},\sigma_{i},\eta_{i},V_{i}$. $\hat{\chi}_{i},\sigma_{i}$ will keep constant when $V_{i}\leq \eta_{i}$. As long as the event $V_{i}> \eta_{i}$ occurs, the switching will happen and the parameters $\hat{\chi}_{i},\sigma_{i}$ will be updated. Then, $\eta_{i}$ is reset to make $V_{i}< \eta_{i}$ and the previous procedures will be repeated. (For simplicity, $\hat{\Theta}_i$ and other variables are not shown in this figure.)
  • Figure 2: Control performance comparison for Case A.
  • Figure 3: Control performance comparison for Cases B and C.
  • Figure 4: Control performance comparison for Cases D and E.
  • Figure 5: Control performance for second order system.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Remark 3
  • Proposition 3
  • ...and 7 more