Design and Control of Fractional-Order Systems Based on Fractal Operators

In recent years, we have abstracted physical fractal space from biological structures and movements within living organisms, revealing the profound intrinsic connections between fractional order time and fractional-dimensional space, and providing partial explanations for the sources and orders of fractional order. We have confirmed that the topological invariants of fractal cells, the order of physical components, and the mismatch of spatiotemporal order are important factors determining the fractional order of operators. This paper is a continuation of the previous work. Inspired by bone fractal operators, this article attempts to identify other factors that affect the order of operators. Specifically, the following contents are included: (1) originating from the bone fractal operators, we present the construction process of the “apparent half-order” system; (2) using the Schiessel–Blumen model as the comparative object, we analyze the origin and characteristics of the “γ-order” system; (3) using the continued fraction theory and operatorization thought as the link, we establish the design and control method for general fractional-order systems, and discuss the factors affecting the order of fractional-order operators.