Based on the introduction of Chua's periodic table, the position of the existing circuit elements(resistor, capacitor, inductor, memristor, memcapacitor, meminductor, fractor and fracmemristor) in the periodic table are given. In recent years, the concept of Fractional Order Memristor(FOMR) has been developed, but the use of this name is not consistent. In this paper, the existing FOMR include circuit are summarized into four types, and the locations of the elements in Chua's periodic table are determined. Fractional integration is a powerful tool to represent memory loss. This paper not only gives the unified constitutive relation expressions for circuit elements, but also gives the definitions for fractional integral control type memory elements―Current Fractional integral controlled Memcapacitor (CFMC), Voltage Fractional integral controlled Memcapacitor(VFMC), Current Fractional integral controlled Meminductor(CFMI), Voltage Fractional integral controlled Meminductor(VFMI), and Current Fractional integral controlled Fracmemristor(CFFM), and Voltage Fractional integral controlled Fracmemristor(VFFM), etc. In addition, the operation structure diagrams of some fractional integral control memory elements are given, and their characteristic curves are analyzed theoretically.