A novel condition for fixed-time stability and its application in controller design for robust fixed-time chaos stabiliz
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METHODOLOGIES AND APPLICATION
A novel condition for fixed-time stability and its application in controller design for robust fixed-time chaos stabilization against Ho¨lder continuous uncertainties Alireza Khanzadeh1 • Mahdi Pourgholi1
•
Elham Amini Boroujeni2
Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents a new nonlinear state-feedback controller design for robust fixed-time chaos stabilization of chaotic systems in the presence of a relatively large class of uncertainties known as Ho¨lder continuous uncertainties. Based on Lyapunov’s second method, a novel sufficient condition for fixed-time stability is derived. The spectacular property of the proposed controller is that the upper bound of the convergence time exists as an explicit parameter in the control’s law, thus the true fixed stabilization time can be set in advance. To show the effectiveness of the proposed controller, two scenarios are provided, and the simulation results are reported. Keywords Fixed-time stability Fixed stabilization time Chaos stabilization Ho¨lder continuous uncertainties
1 Introduction In many of practical applications, some strict time response requirements must be met. For instance, chaos appearing in a permanent magnet synchronous motor (PMSM) must rapidly be controlled because it ravages the stability of the motor and even can leads to the collapse of driven system. In micro-electro-mechanical systems (MEMS), chaotic behaviour resulting from nonlinear dynamic characteristic can seriously impair the performance of MEMS and should be suppressed very soon. To prevent voltage collapse in power system grids, it is required to damp chaotic oscillations within a limited time. Hence, in such applications, being able to adjust the stabilization time in accordance with the performance requirements is of great importance. Finite-time control is a well-established technique widely employed in the above-mentioned applications. This type of the controllers, which are designed on the Communicated by V. Loia. & Mahdi Pourgholi [email protected] 1
Faculty of Electrical Engineering, Shahid Beheshti University, P.O. Box 1658953571, Tehran, Iran
2
Department of Electrical and Computer Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran
basis of the finite-time stability condition, has been applied to the different applications with hard time response constraints such as attitude control of spacecraft (Du et al. 2011), consensus control of a group of uncertain mechanical systems (Huang et al. 2015), longitudinal control of an air-breathing hypersonic vehicle (Sun et al. 2013), control of switched nonlinear systems (Cai and Xiang 2015), stabilization of chaos in PMSMs (Wang et al. 2014), finitetime stabilization of uncertain non-autonomous chaotic gyroscopes (Aghababa and Aghababa 2012), chaos synchronization of PMSMs (Chen et al. 2015), chaos control of MEMS systems (Zhankui and Sun 2013), stabilization of chaotic rotating m
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