Finite-time stabilization of a non-autonomous chaotic rotating mechanical system

Abstract

This paper presents robust adaptive controllers for finite-time stabilization of non-autonomous chaotic horizontal platform systems (HPSs). The effects of uncertainties, unknown parameters and input nonlinearities are fully taken into account. Appropriate adaptation laws are designed to undertake the unknown parameters. Using the adaptation laws and finite-time control theory, first a robust adaptive controller is derived to stabilize the non-autonomous uncertain HPS in finite time. Then, considering the effects of input nonlinearities, another finite-time controller is introduced to stabilize the uncertain HPS with nonlinear control inputs. The finite-time stability and convergence of the proposed schemes are analytically proved. Two illustrative examples are given to show the robustness and feasibility of the proposed finite-time controllers.

Introduction

Chaotic dynamical systems are very complex nonlinear systems that display unpredictable and irregular behavior. The main special attribute of chaotic systems is that a tiny change in the initial conditions and the system parameters leads to an enormous difference in the long-term behavior of the system. Since the pioneering work by Ott et al.[1], stabilization and synchronization of autonomous chaotic systems has received considerable attention in recent years [2], [3], [4], [5], [6]. On the other hand, with the discovery of more and more non-autonomous chaotic systems in engineering sciences and physics, stabilization and/or synchronization of non-autonomous chaotic systems has received significant interests of many researchers and various control techniques for controlling and synchronizing non-autonomous chaotic systems have been developed, such as adaptive control [7], [8], [9], [10], fractional-order control [11], [12], [13], sinusoidal state error feedback control [15], nonlinear control [16], fuzzy control [17], variable substitution control [18] and so on. Most of these methods are based on the Lyapunov direct method and prove the stability of the closed-loop system asymptotically. Moreover, most of the above-mentioned methods assume that either the system parameters are fully known in advance or there are no model uncertainties and external disturbances affecting the system dynamics.

In recent years, several mechanical systems with chaotic phenomena have been developed [14], [18], [19], [20]. One of the most interesting and attractive nonlinear dynamical systems is the horizontal platform system (HPS). It is a mechanical device that can freely rotate around the horizontal axis. The horizontal platform devices are widely used in offshore and earthquake engineering. It has been shown that these systems display a diverse range of dynamic behavior including both chaotic and regular motions [21].

In the past few years, complex dynamics, stabilization and synchronization of the HPS have been studied in the literature. Ge et al. [21] have numerically verified that two identical HPS coupled by linear, sinusoidal and exponential state error feedback controllers can be synchronized. The coupling strengths resulting in chaos synchronization have been detected according to negativity of all Lyapunov exponents of the slave system. However, the condition that all Lyapunov exponents are negative has been confirmed to be only necessary but not sufficient condition for chaos synchronization [22], [23]. Wu et al. [24] have used Lyapunov direct method to achieve a sufficient criterion for global chaos synchronization between two identical HPS coupled by linear state error feedback controller. By means of a linear state error feedback controller, the robust synchronization of the chaotic HPS with phase difference and parameter mismatches has been studied in [25]. Based on the Lyapunov stability theorem and Sylvester's criterion, some algebraic sufficient criteria for synchronization of two HPS coupled by sinusoidal state error feedback control have been derived in [26]. Recently, Pai and Yau [27] have designed an integral-type sliding mode controller for generalized projective synchronization of two HPS with uncertainties. Pai and Yau [28] have also designed an adaptive sliding mode control scheme for controlling the chaos in the state trajectories of the uncertain HPS.

However, most of the above-mentioned methods and control strategies have been proposed to stabilize the HPS asymptotically. In other words, these studies guarantee that the state trajectories of the HPS (or the resulted synchronization error system) can converge to zero with infinite settling time. Nevertheless, from a practical engineering point of view, it is more reasonable to stabilize the HPS in a finite time rather than merely asymptotically. This means that in practical situations we require to reach the steady state in a finite time. But previous works can reach to the control goal after infinite time horizon. Realization of a fast controller is based on the finite time control theory. On the other hand, finite-time controllers may realize physically similar to other nonlinear controllers by suitable circuits. To achieve faster convergence speed in control systems, the finite-time control method is an effective technique. Finite-time stabilization means the optimality in settling time. Moreover, the finite-time control techniques have demonstrated better robustness and disturbance rejection properties [29]. Furthermore, in real world practical applications, there are always some model uncertainties and external disturbances in the system dynamics. And, the system parameters are inevitably disturbed by external inartificial factors, such as environment temperature, voltage fluctuation, mutual interfere between components, etc, and cannot be exactly known in advance. The control goal can be destroyed with the effects of these uncertainties. In this paper, therefore, the effects of the model uncertainties, external disturbances and unknown parameters in stabilization of the HPS are fully taken into account. Moreover, to the best knowledge of the authors, the problem of finite-time stabilization of the non-autonomous uncertain chaotic HPS has received less attention and still remains as an open and challenging issue to be solved.

Motivated by the above discussion, in this paper, the problem of finite-time stabilization and chaos suppression of the non-autonomous chaotic HPS is investigated. It is assumed that the parameters of the HPS are fully unknown in advance. Model uncertainties and external disturbances are also imposed to the system dynamics. To tackle the unknown parameters of the system, suitable adaptation laws are introduced. On the basis of the adaptation laws and finite-time control technique, a robust adaptive controller is designed to stabilize the uncertain non-autonomous HPS in finite time. Subsequently, the effects of input nonlinearities are taken into account and another robust adaptive finite-time controller for stabilization of the HPS with input nonlinearities is proposed. The finite-time stability and convergence of the proposed schemes are analytically proved. Some numerical simulations are given to illustrate the robustness and applicability of the proposed techniques and to validate the theoretical results of the paper.