Finite-time stabilization of a non-autonomous chaotic rotating mechanical system
Highlights
► An adaptive robust finite-time controller for chaos suppression of HPS is proposed. ► The effects of nonlinear control inputs are taken into account. ► Two main mathematical theorems are proposed and proved. ► Some numerical simulations are provided to justify the theoretical results.
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.
Section snippets
Description of HPS dynamics
The HPS is a mechanical device composed of a platform and an accelerometer located on the platform (see Fig. 1). The platform can freely rotate about the horizontal axis, which penetrates its mass center. The accelerometer produces an output signal to the actuator, subsequently generating a torque to inverse the rotation of the platform to balance the HPS, when the platform deviates from horizon. The motion equations of the HPS are given by [21]where A=0.3,
Design of a robust adaptive finite-time controller without input nonlinearities
Lemma 1 [31]. Assume that a continuous, positive-definite function V(t) satisfies the following differential inequality:where c>0, 0<ξ<1 are two constants. Then, for any given t0, V(t) satisfies the following inequality:and with t1 given by
Lemma 2 For , the following inequality holds: In order to guarantee the finite-time stabilization of the uncertain chaotic non-autonomous
Numerical simulations
It is known that if the HPS operate in a state of chaotic motion, the subsequently large broad-band vibration may increase the likelihood of fatigue failure and shorten the system lifetime [28]. Therefore, designing a controller to suppress the chaotic behavior in the HPS is a very important problem and of practical value. Therefore, we validate the performance of the proposed controllers in suppressing the chaotic behavior of the HPS by numerical simulations.
Conclusions
The problem of finite-time stabilization of a non-autonomous horizontal platform system (HPS) is studied in this paper. It is assumed that the parameters of the HPS are completely unknown in advance. Besides, some model uncertainties and external disturbances are added to the system dynamics. To tackle the system unknown parameters, proper adaptation laws are designed. Then, a finite-time robust adaptive controller is proposed to stabilize the HPS. Subsequently, it is supposed that
References (32)
- et al.
Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller
Communications in Nonlinear Science and Numerical Simulation
(2011) - et al.
Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique
Applied Mathematical Modelling
(2011) - et al.
Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities
Applied Mathematical Modelling
(2012) - et al.
A chattering-free robust adaptive sliding mode controller for synchronization of two different chaotic systems with unknown uncertainties and external disturbances
Applied Mathematics and Computation
(2012) - et al.
Adaptive finite-time stabilization of uncertain non-autonomous chaotic electromechanical gyrostat systems with unknown parameters
Mechanics Research Communications
(2011) Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller
Communications in Nonlinear Science and Numerical Simulation
(2012)- et al.
Synchronization of two different non-autonomous chaotic systems using fuzzy disturbance observer
Physics Letters A
(2010) - et al.
Global chaos synchronization of electro-mechanical gyrostat systems via variable substitution control
Chaos, Solitons & Fractals
(2009) - et al.
Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor
Chaos, Solitons & Fractals
(2007) - et al.
Chaos and chaos synchronization for a non-autonomous rotational machine systems
Nonlinear Analysis: Real World Applications
(2008)
Chaos synchronization of a horizontal platform system
Journal of Sound and Vibration
The synchronization of chaotic systems
Physics Reports
Master–slave chaos synchronization criteria for the horizontal platform systems via linear state error feedback control
Journal of Sound and Vibration
Robust synchronization of chaotic horizontal platform systems with phase difference
Journal of Sound and Vibration
Suppression of chaotic behavior in horizontal platform systems based on an adaptive sliding mode control scheme
Communications in Nonlinear Science and Numerical Simulation
An equation for hyper chaos
Physics Letters A
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