Fractional order iterative learning control for fractional order system with unknown initialization

Abstract

Abstract —This paper reveals a previously ignored problemfor fractional order iterative learning control (FOILC) thatthe fractional order system may have different behaviors whenit is initialized differently. To implement a novel scheme ofFOILC for this so-called initialized fractional order system, aD a type control law is applied, and the convergence conditionis derived by using the short memory principle and the systempreconditioning, which guarantees the repeatability of initial-ized fractional order system. Given a permitted error bound,the minimum preconditioning time horizon is calculated fromthe short memory principle. The relationships of memory andconvergent performance are highlighted to show the necessityof preconditioning. A fractional order capacitor model withconstant history function is illustrated to support the aboveconclusions. I. I NTRODUCTION The formal concept of iterative learning control (ILC)was published in 1978 by Uchiyama (in Japanese) andin 1984 by Arimoto et al (in English) [1]. Some earlierworks of ILC can be traced back to 1967 in the US patent3555252 “Learning control of actuators in control systems”and the multi-pass system analysis in 1974 by Edwards andOwens [2]. The ILC method is a batch process that operatesa given objective system repeatedly on a xed time intervalso that the reference signal can be perfectly tracked as theoperation repeats. The ILC schemes can be easily appliedto globally Lipschitz nonlinear system with less prior modelknowledge [1], [2], [3], [4], [5], [6], [7], [8]. In simple words,the ILC scheme improves the system control performance byusing the historical data even without a complete knowledgeof the controlled system [5], [9], [10], [11], [12], [13].The fractional order iterative learning control (FOILC)is a relatively new topic in ILC, which can be tracedback to 2001 [14]. In this pioneering work, a D