Iterative learning control for fractional order nonlinear system with initial shift

In this study, a closed-loop Dα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D^\alpha }$$\end{document}-type iterative learning control (ILC) with a proportional D-type iterative learning updating law for the initial shift is applied to nonlinear conformable fractional order system. First, the system with the initial shift is introduced. Then, fractional order ILC (FOILC) frameworks that experience the initial shift problem for the path tracking of nonlinear conformable fractional order systems are addressed. Moreover, the sufficient condition for the convergence of tracking errors is obtained in the time domain by introducing λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-norm and Hölder’s inequality. Lastly, numerical examples are provided to illustrate the effectiveness of the proposed methods.

described with the use of fractional derivatives and integrals [3]. Besides, numerous fractional order controllers have been applied in engineering areas [4][5][6]. Iterative learning control (ILC), which is used in control systems that repeat the same task via an unknown model during a finite duration, was first introduced by Uchiyama in Japanese [7]. Arimoto et al. [8] further developed this method in English. Thereafter, ILC has achieved considerable development [9] and has been found to be a good alternative in practical application [10]. The combination of fractional differential calculus and ILC was first reported by Chen and Moore [11] who proposed a D α -type control law analyzed in the frequency domain. FOILC has currently become a new topic and has received increasing attention [12][13][14] because it can enhance tracking performance in control systems.
The basic idea of Fo-ILC is illustrated in Fig. 1, where u k (t) and y k (t) are, respectively, the system input and output in the kth iteration, u k+1 (t) is the system input of the (k + 1)th trial, and y d (t) is the given desired trajectory. The goal of ILC is that lim k→∞ y k (t) = y d (t) for all t ∈ [0, T ], where T is a fixed constant.
To the best of our knowledge, Li et al. [15] described a fractional order linear system in state space form, and the convergent conditions for a D α -type law were provided. In [16], the asymptotic stability of P D α -type ILC for a fractional order linear time invariant (LTI) system was studied. The convergence condition for open-loop P-type ILC for fractional order nonlinear system was investigated in [17]. High-order fractional order P I D-type ILC strategies for a class of Caputotype fractional order LTI system was discussed in [18]. FOILC for both linear and nonlinear fractional order multi-agent systems was applied to solve the consensus tracking problem in [19]; for nonlinear and linear conformable fractional differential equations, Wang et al. [20][21][22][23] provided standard analysis technique for standard open-loop P-type, D α -type, and conformable P I α D α -type ILC; unfortunately, it is supposed that the initial state is coincident with the desired initial state.
However, the aforementioned existing literature, FOILC laws generally assume that the initial state of a system must be strictly in accordance with the desired states. Therefore, the initial shift problem should be involved to extend the application of FOILC [24][25][26]. For example, Zhao et al. [27] introduced an initial state learning scheme coupled with D α -type ILC updating law to eliminate initial shift states in fractional order linear invariant systems. The possibility of application to the motion control of robot manipulators was discussed under specific conditions. Moreover, the D α -type FOILC scheme was applied to the fractional order linear system with different historical functions under the Riemann-Liouville definition [28], and the relationship between memory and convergent performance was highlighted. Lan [29] presented a Ptype ILC scheme with initial state learning for a single-inputCsingle-output fractional order nonlinear system, and the asymptotic stability of P D α -type ILC was studied by Li et al. [30]. Li and Zhou [31] discussed fractional order nonlinear systems with delay by P-type ILC scheme, initial state delay was considered. Li et al. [35] discussed the consensus problem of fractionalorder multiagent systems with nonzero initial states, both open-and closed-loop P D α -type fractional-order iterative learning control are presented. For the consensus problem of nonzero initial states of fractionalorder multiagent systems, Li et al. [32], Luo et al. [33] and Zhou et al. [34] discussed relevant fractional-order iterative learning control,respectively. In addition, an adaptive generalized FOILC, which expands the practical applications of FOILC, was illustrated in fractional order nonlinear systems [35].
Motivated by the above-mentioned research, a comfortable closed-loop D α -type ILC with an initial state learning law was proposed in the current study to eliminate the influence of the initial shift in nonlinear comfortable fractional systems. It can expand the application scope of ILC method and eliminate the influence of initial value change on tracking effect in practical engineering. Notably, sufficient conditions in the time domain were derived by introducing λ-norm and using Hölder's inequality. Based on this convergence condition, the learning gain of the initial learning and input learning updating law can be determined. Unlike the existing methods, the ILC scheme will not fix the initial value on the expected condition at the beginning of each iteration. And the availability of this contribution is examined using two numerical examples.
The rest of the paper is organized in the following manner. Preliminary knowledge is briefly reviewed in Sect. 2. The nonlinear conformable fractional differential equations and the design of learning control are presented in detail in Sect. 3. Learning convergence is analyzed in Sect. 4. A number of simulations demonstrate the effectiveness of the theory in Sect. 5. Conclusions are drawn and suggestions for future work are provided in Sect. 6.

Fractional derivative and preliminary
In this section, some related mathematical definitions and properties are introduced, which will be applied in the following sections.

The λ-norm
Suppose C(J, R n ) is the space of vector-value continuous functions from J → R n . Consider C(J, R n ) endued with λ-norm, and 2.2 Hölder's inequality [36] Suppose that Ω is a measured space, p, q ∈ [1, ∞) and
where D α f (t) exists and is finite, Besides, the conformable fractional integral is defined as where I α presents fractional order integral operator in [0, t].

Solution of conformable fractional order function
According to the lemma 4 in [40], if the conformable fractional order function D α x(t) = f (t) is continuous and given, the solution

ILC design for fractional order nonlinear systems
In order to solve the above problem, a class of fractional order nonlinear system is described. The repetitive nonlinear conformable fractional differential system is considered as where There exists a L F > 0 such that satisfies where ∀t ∈ J, ∀x 1 , x 2 ∈ R n . According to Sect. 2.4, the solution of x k (t) in Eq. (5) can be derived as follows Let y d (t) be a continuous differentiable desired function on J and u d (t) be a expectation control variable. If the desired initial value is x d (0) = x d0 , and The ILC updating law with initial state learning is defined as follows, where the subscript k is iterative index, L ∈ R n×r and Λ ∈ R r ×r are the learning gains to be designed based on prior-knowledge about the system under investigation.

Numerical examples
In this section, numerical examples are presented to test the effectiveness of the designed methods. The following simulations are performed for the fractional order nonlinear system. Example 1 Consider the first fractional order nonlinear system The iterative learning control laws are chosen where the system state is x(t) , and the desired trajectory is y d (t) = 12t 2 (1 − t), the initial control is u 0 (t) = 0 and with initial condition x k (0) = 0.5 . In this case, it can be calculated that  It can be seen that the system output is capable of approaching the desired trajectory accurately within a few iterations. Next, let the iterative learning control law be and the initial control as well as the reference be the same. Figure 4 shows  It concludes that closed-loop D α type ILC updating law performs better in convergence rate during the learning process.
Example 2 Consider the second fractional order nonlinear system The iterative learning control laws are chosen and u k+1 (t) = u k (t) + 2D 0.6 e k+1 (t) (39) where the system state is x(t) , and the desired trajectory is y d (t) = 6sin(t), the initial control is u 0 (t) = 0 and with initial condition x k (0) = 0.6 . In this case, it can be calculated that The simulation results are demonstrated in Figs. 5, 6, and 7.
Example 3 Consider the Third fractional order nonlinear system  The iterative learning control laws are chosen and where the system state is x(t) , and the desired trajectory is y d (t) = cos(2π t)sin(4π t), the initial control is u 0 (t) = 0 and with initial condition x k (0) = −2 .
In this case, it can be calculated that  Therefore, from the aforementioned simulations, it concluded that the proposed laws with initial state laws perform well. Moreover, it can be seen that after iteration, they all arrive at the reference trajectory under the desired precision.

Conclusions and future work
This study presents D α -type ILC with D-type initial learning strategy for a class of nonlinear conformable fractional order systems with the initial shift. Its major feature is that disturbance in the initial state at each iteration is eliminated by introducing an initial state learning scheme. Furthermore, the robust convergent analysis of tracking errors with respect to initial errors is derived by introducing Hölder's inequality. Lastly, numerical simulations are provided to validate the obtained theoretical results.
In the future, P I λ D α -type ILC for general nonlinear fractional order systems with repetitive properties will be researched. Moreover, when P I λ D α -type ILC is applied to track the nonlinear fractional order system, nonrepetitive uncertainties (such as time delay, input saturation, and nonrepetitive desired trajectory) should be considered.
anonymous reviewers who contributed their valuable comments to this paper.
Data availability statements All data included in this study are available upon request by contact with the corresponding author.