Abstract
In this paper, fractional calculus is applied to establish a novel fractional-order ferroresonance model with fractional-order magnetizing inductance and capacitance. Some basic dynamic behaviors of this fractional-order ferroresonance system are investigated. And then, considering noncommensurate orders of inductance and capacitance and unknown parameters in an actual ferroresonance system, this paper presents a novel fractional-order adaptive backstepping control strategy for a class of noncommensurate fractional-order systems with multiple unknown parameters. The virtual control laws and parameter update laws are designed in each step. Thereafter, a novel fractional-order adaptive controller is designed in terms of the fractional Lyapunov stability theorem. The proposed control strategy requires only one control input and can force the output of the chaotic system to track the reference signal asymptotically. Finally, the proposed method is applied to a noncommensurate fractional-order ferroresonance system with multiple unknown parameters. Numerical simulation confirms the effectiveness of the proposed method. In addition, the proposed control strategy also applies to commensurate fractional-order systems with unknown parameters.
Highlights
Fractional calculus is a generalization of ordinary integration and differentiation to arbitrary order [1]
Considering noncommensurate orders of inductance and capacitance and unknown parameters in an actual ferroresonance system, this paper presents a novel fractional-order adaptive backstepping control strategy for a class of noncommensurate fractional-order systems with multiple unknown parameters
We try to investigate some dynamic behaviors of the ferroresonance system with noncommensurate fractional-order magnetizing inductance and capacitance
Summary
Fractional calculus is a generalization of ordinary integration and differentiation to arbitrary order [1]. Though the fractional-order extension of the Lyapunov direct method proposed in [30] has been used in some present works, designing an excellent controller for noncommensurate fractional-order systems is still an open and challenging problem [31]. Considering the uncertainties of a real ferroresonance system in practice, investigation of fractionalorder adaptive backstepping control for noncommensurate fractional-order systems with unknown parameters is necessary. To suppress undesirable chaotic behaviors of the proposed system, a novel fractional-order adaptive controller is designed for the noncommensurate fractionalorder system with unknown parameters via the fractionalorder Lyapunov direct method. Chaotic behaviors of the noncommensurate and commensurate fractional-order ferroresonance systems with multiple unknown parameters are eliminated, which illustrates the effectiveness and feasibility of the proposed method.
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