| With the extensive application of fractional calculus in various circuits and systems,fractional-order circuits and systems have become a new subject of research.Fractors and fractional-order memristors are the critical components required for hardware implementation in fractional-order circuits and systems.Fractor is divided into constant-order fractor and variable-order fractor.Among them,the latter is applicable to construct or model the new phenomena and problems encountered in fractional calculus that depend on time,space,or other variables.Fracmemristors,as a novel component,have demonstrated their potential of applications in intelligent prediction models,neural weighted circuits,chaotic circuits,and other fields.Currently,there is still a lack of commercially available variable-order fractors and fractional-order memristors.Approximation circuits,which make use of the existing components such as resistors,capacitors,and operational amplifiers,play an important role in approximating variable-order fractors and fractional-order memristors,showing a certain level of precision.The significance of conducting research on approximation circuit implementation for variable-order fractors and fractional-order memristors is as follows:(i)To enrich the fundamental theory of variable-order fractors and fractionalorder memristor.(ii)To verify the fractional electrical characteristics of variable-order fractors and fractional-order memristor.(iii)It helps to bridge the gap between theoretical research and applications.However,there are some difficulties posed in the implementation of approximation circuits for variable-order fractor and fractional-order memristor.Therefore,the key to overcoming these difficulties is to develop effective circuit configurations.The difficulties in implementing variable-order fractor approximation circuits are as follows:(i)To develop circuit configurations.(ii)To improve approximation performance.(iii)To design the high-resolution wide-range programmable resistance and programmable capacitance emulators.(iv)To design the high-resolution wide-range programmable resistancecapacitance series circuit emulators.The difficulties in implementing fractional-order memristor approximation circuits are as follows:(i)To determine the fracmemristance.(ii)To represent the time-domain electrical characteristics of corresponding fractionalorder memristors.(iii)To maintain the time constants for each sub-circuit.(iv)To improve the approximation performance of string scaling fractional-order memristor approximation circuits.(v)Considering the existing methods,it is challenging to replace floating memristor emulators with the memristors directly in the configuration of string scaling fractional-order memristor approximation circuit.To resolve these difficulties,the present study encompasses the following aspects:(1)A high-resolution wide-range variable-order ladder scaling fractor approximation circuit is proposed.According to the results of hardware circuit experiment,the proposed variable-order ladder scaling fractor approximation circuit can alter the operational order from-0.7 to-0.3 and approximate the frequency range from 7.72 Hz to 4.82 k Hz,showing an input signal peak value of up to 10 V.This is conducive to solving the difficulties faced by the existing variable-order fractor approximation circuits,such as the small number of operating order variations,narrow range,and the difficulty in making adjustment.Besides,high-resolution wide-range variable-order ladder scaling fractor approximation circuits can also be implemented in a way that satisfies the specific requirements.The main work achieved in this aspect are as follows.Firstly,a variable scaling factor expansion method is proposed and then the configuration of the variable-order ladder scaling fractor approximation circuit is introduced.Secondly,with high-resolution multiplication digital-to-analog converters as the core,programmable resistance-capacitance series circuit emulators and general component parameter programmable circuit emulators are designed,which can be used to programmatically adjust the resistance and capacitance in the variable-order ladder scaling fractor approximation circuit configuration.Lastly,a circuit theory-based method is proposed for variable-order fractional calculus operations,which improves the understanding of variable-order fractional calculus operations and promotes further research for wider applications.(2)A method of frequency domain characteristic analysis is proposed for fractionalorder memristors,which can be used to numerically calculate the performance of fractionalorder memristor approximation circuits in operating bandwidth and frequency domain approximation.Also,the problem of how to determine the fracmemristance in scaling fractional-order memristor approximation circuits is solved.The main work achieved in this aspect are as follows.Firstly,the order-frequency characteristic function and Ffrequency characteristic function in the frequency domain are applied to determine the approximation bandwidth,operational order,and fractional-order memristance of scaling fractional-order memristor approximation circuits.Secondly,an ideal approximation-type half-order chain fractional-order memristor approximation circuit is proposed to validate the method of frequency domain characteristic analysis for fractional-order memristors from the perspective of circuit.Lastly,the proposed method of frequency domain characteristic analysis for fractional-order memristors can be widely applied to the analysis of frequency domain characteristics for memristors,memcapacitors,and meminductors.This is beneficial in exploring the characteristics of memristors,memcapacitors,and meminductors from the perspective of fractional calculus.(3)The configuration of a constant approximation frequency range scaling fractionalorder memristor approximation circuit is proposed to recalculate the effective range of approximation frequency for different input signals in the existing scaling fractional-order memristor approximation circuits.The main work achieved in this aspect are as follows.Firstly,a scaling expansion method is proposed for variable reference resistance and reference capacitance,so as to maintain the time constant of each sub-circuit in the scaling fractional-order memristor approximation circuit configuration.Then,the configuration of constant approximation frequency range string and chain scaling fractional-order memristor approximation circuits are proposed to meet the needs for different applications.Secondly,the approximation performance of the string scaling fractional-order memristor approximation circuit configuration is optimized.Lastly,the new configuration inspires the idea of replacing the resistance and capacitance components in the impedance function of arbitrary-order fractors with the memristance and memcapacitance components controlled by state variables.Thus,a small-signal impedance function is proposed for the natural implementation form of arbitrary-order fractional-order memristors.(4)A schematic design approach is proposed for scaling fractional-order memristor approximation circuits,including scaling fractor approximation circuits,which removes the need for a large number of memristor emulators in the existing scaling fractional-order memristor approximation circuits.The main work achieved in this aspect are as follows.Firstly,the schematics of string and chain scaling fractional-order memristor approximation circuits are proposed,with string and chain scaling fractional-order memristors implemented to meet the requirements of different applications.Secondly,the expression of time-domain electrical characteristics is determined for string and chain scaling fractional-order memristor approximation circuits.On this basis,the results of electrical characteristic theoretical analysis are obtained,and they are verified through hardware circuit experiments.Lastly,the proposed schematics inspire the fractance of impedance function of arbitrary-order fractors by a variable coefficient controlled by state variables.Thus,another small-signal impedance function is proposed for the natural implementation form of arbitrary-order fractional-order memristors.The above works facilitate the application of variable-order fractors and fractionalorder memristors in such sectors as electronic information,computer science,and technology.For example,in the research on the application in artificial neural network circuits,future research can be focused on the following subjects.The first one is to construct variable-order fractional-order neuron models and fractional-order memristor neuron models.The second one is to develop the hardware circuit architecture intended for fractional-order neural networks suited to variable-order fractional-order neuron models or fractional-order memristor neuron models.The third one is to implement the hardware of variable-order fractional-order neural networks and fractional-order memristor neural networks.The fourth one is to conduct research on training algorithms and learning rules for variable-order fractional-order neural networks and fractional-order memristor neural networks.The last one is to explore the applications of variable-order fractional-order neural networks and fractional-order memristor neural networks. |
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