Lyapunov approach on a homogeneous family of controllers for robotic manipulator

Jang, Wilson (2019) Lyapunov approach on a homogeneous family of controllers for robotic manipulator. PhD thesis, University of Nottingham.

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Second order sliding mode has been successfully implemented for solution of real problems for its inherent features such as finite-time convergence and robustness to disturbances. For the first order sliding modes, it is common to deal with the issues of stability, robustness, and convergence rate of the equilibrium by means of a Lyapunov approach. For higher order sliding modes, however, a similar treatment has not been developed until recently. The focus of this thesis is the construction of strong Lyapunov function, i.e. for which its time derivative can be upper bounded by negative-definite functions, for the design of control strategies for robotic manipulator, which is a nonlinear system, subject to combinations of parametric uncertainty, bounded disturbances, actuator saturation, and output feedback.

The first contribution of this work is the development of a strong Lyapunov function for a parameterized family of homogeneous sliding mode based controller comprising twisting algorithm, continuous finite time control, linear PD control law, and uniformly stable control law, all of which belongs to a general homogeneous family of control algorithms. The strict locally Lipschitz homogeneous Lyapunov function proposed permits the estimation of convergence time for the trajectories of the system to the origin, in finite-time, exponentially, or uniformly asymptotically, even in the case when it is affected by bounded non-vanishing or growth bounded vanishing external perturbations. Moreover, the relationship between the control gains and its convergence performance can be analyzed.

Leveraging on these results, a strong Lyapunov function is developed for a closely related second order sliding mode algorithm, the super-twisting algorithm based controller. In particular, the construction of these strong homogeneous Lyapunov function is able to show the relationship between the twisting and super-twisting algorithms and allows linear combination of two homogeneous control of different degree.

Extending the results for MIMO robot manipulator, a type of Euler-Lagrange dynamic systems, a family of integral sliding mode-based controller is introduced for trajectory tracking. In particular, the homogeneous dynamics is employed as the desired error dynamics for the controller. Additionally, the conventional PID control is shown to be a special case and the present formulation presents the relationship between the gains of the controller and the desired performance, which provides a systematic method for gain selection for a robust PID control. In addition, for the special problem of regulation, employing the results of homogeneous control, finite-time regulation of the robot manipulator is achieved.

Since actuator saturation is a phenomenon that affects the performance of dynamic systems under closed-loop control, a saturated version of the controller is also developed that achieved global stability while maintaining the features of the unbounded version of the controller in terms of trajectory tracking and finite time regulation. Extending the results for system with position measurements only, a saturated output feedback version of the controller is introduced that can achieved global stability as well. Each of the proposed controllers provides advantages over the previous literature in their ability to design desired error dynamics and the time derivative of the disturbance is not required in the stability analyses.

Throughout the work, Lyapunov-based stability, in particular the nonsmooth Lyapunov analysis techniques, and numerical experiments are provided to highlight the performance of each controller design.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Abbas, Haider
Kumar, Nandha
Keywords: Lyapunov stability, robotic manipulator, control, Nonlinear system
Subjects: Q Science > QA Mathematics
Faculties/Schools: University of Nottingham, Malaysia > Faculty of Science and Engineering — Engineering > Department of Electrical and Electronic Engineering
Item ID: 55404
Depositing User: JANG, WILSON
Date Deposited: 25 Feb 2019 07:24
Last Modified: 07 May 2020 13:00

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