Iterative learning control for impulsive fractional order time‐delay systems with nonpermutable constant coefficient matrices

Abstract

We introduce an impulsive fractional order time-delay systems with nonpermutable constant coefficient matrices whose solution is given by delayed perturbation of Mittag-Leffler type matrix function. The existence and uniqueness of the solution of the system is proved by using Banach contraction principle. The Ulam-Hyers stability of the given system below are demonstrated. Construct the iterative learning control (ILC) problem obtaining from the mentioned system.The conditions of convergence of ILC problem of each of P, D, and D 𝛼 -types are presented and proved. A comprehensive example with three different original reference trajectories is given to illustrate some theoretical results. This article provides novel outcomes.

ψ$$ \psi $$‐Caputo type time‐delay Langevin equations with two general fractional orders

“…𝛼 , the integral operator 𝔊 is a contraction. By using contraction mapping principle, there is a unique fixed point of 𝔊, which is the unique global continuous solution of system (7). This completes the proof.…”

Joint two‐stage multi‐innovation recursive least squares parameter and fractional‐order estimation algorithm for the fractional‐order input nonlinear output‐error autoregressive model

“…The proposed parameter estimation algorithms in this paper are based on the identification model in (13). Some identification methods are derived based on the identification models of the systems [54][55][56][57][58][59] and these methods can be used to estimate the parameters of other linear systems and nonlinear systems [60][61][62][63][64][65] and can be applied to other fields [66][67][68][69][70][71] such as chemical process control systems.…”

“…In this section, we separate the identification model in (13) into the two sub-models by using the hierarchical identification principle, 72 one is the IN-OE model with white noise containing the parameter vector 𝜽 l , the other is the AR noise model containing the parameter vector 𝜽 h . The two-stage multi-innovation recursive least squares (2S-MIRLS) algorithm can estimate the parameters of the IN-OE model white noise and the AR noise model.…”

“…12 It is more reliable to describe a very large class of nonlinear systems than integer case. [13][14][15] As opposed to the integer-order models, the identification difficulty of the fractional-order models contains not only the parameters of the models but also the fractional-order. Therefore, many researchers have been attracted to develop effective identification algorithms for the fractional-order Hammerstein models with focus on the fractional-order estimation.…”

“…The fractional‐order models have found widespread applications in many fields such as rock creep characteristics 11 and electric vehicle Li‐ion batteries 12 . It is more reliable to describe a very large class of nonlinear systems than integer case 13‐15 …”