Fractional-Order Memory Elements and Electrical Circuits for Future Electronics and Industrial Automation: A Survey

This paper reviews a novel class of elements for electrical circuits and the novel types of electrical circuits built from them. The common characteristic of the considered elements (fractional-order memristor; fractional-order (mem)capacitor; fractional-order (mem)inductor) is that they exhibit fractional-order behavior and include a memory effect. The survey first clarifies the distinction between constant-phase elements, linear fractional-order elements with hereditary memory, and nonlinear state-dependent memory elements (memristors, memcapacitors, meminductors), and summarizes the constraints of causality, passivity, stability, and finite bandwidth that every physical realization must satisfy. The main challenge in this field is designing and implementing such memory elements with adjustable fractional order. The developed memory components and circuits can be used in signal and image processing filters, controllers for process control and industrial automation, oscillators, and signal and image encoding/decoding. This survey compares analog and digital implementations of these devices and summarizes their reported characteristics, order adjustability, and technological maturity in a comparative table. Analog implementations reported in the literature use polymer–metal composites, supercapacitors, and SMD components, while digital implementations rely on computers, PLCs, FPGAs, microcontrollers, and embedded platforms such as Arduino or Raspberry Pi. With new fractional-order elements, researchers have investigated novel electrical circuits, including fractal-structure circuits, embedded ladder networks, and their combinations. The relevance of this field is underlined by the rapid growth in the use of electrical batteries practically everywhere, especially in vehicles, drones, and communication devices. Because batteries exhibit fractional-order behavior and memory, fractional-order modeling and fractional-order controllers can optimize charge/discharge processes, motivating the need for new fractional-order tools in control theory and the theory of chaotic systems. The paper also discusses a new type of non-uniform sampling, based on fractional-order noise, aimed at producing better fractional-order models from measurements, as well as the emerging use of fractional-order memory in neuromorphic and physical reservoir computing.