Today, there is a great tendency toward using fractional calculus to solve engineering problems. The control is one of the fields in which fractional calculus has attracted a lot of attention. On the one hand, fractional order dynamic models simulate characteristics of real dynamic systems better than integer order models. On the other hand, Fractional Order (FO) controllers outperform
Today, linear controllers cannot satisfy requirements of high-tech industry due to fundamental limitations like the waterbed effect. This is one of the reasons why nonlinear controllers, such as reset elements, are receiving increased attention. To analyze reset elements in the frequency domain, researchers use the Describing Function (DF) method. However, it cannot accurately predict the closed-loop frequency responses of the system because it neglects high order harmonics. To overcome these barriers, this paper proposes a mathematical framework to model the closed-loop frequency responses of reset systems including the closed-loop high order harmonics. Furthermore, pseudo-sensitivities for reset systems are defined to make their analyses more straight-forward. In addition, a user-friendly toolbox is developed based on the proposed approach to facilitate frequency analyses of reset systems. To show the effectiveness of the method, multiple illustrative examples on a high-tech precision positioning stage are used to compare the results of the closed-loop frequency responses obtained using our proposed method with DF method. The results demonstrated that the proposed method is significantly more precise than the DF method. Indeed, this developed toolbox can enable reset controllers to be widely-used in industry and academia.
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