The paper is concerned with the robustness analysis of a type of iterative algorithm for R-L fractional nonlinear control systems in the sense of

The aim of the paper is to analyze the robustness of a type of iterative algorithm in the sense of

Iterative learning control (ILC) was shown by Uchiyama in 1978 (in Japanese), and in recent years, more and more scholars have paid attention to the problems, among which are experts who study fractional calculus. The work of the fractional-order system in iterative learning control appeared in 2001. In the following decade, extensive attention has been paid to this field, great progress has been made [

In Li et al.’s study [

In view of that the

The above references have analyzed the robustness of the algorithm of the Caputo-type fractional system, and we find the Caputo fractional derivative is often used to solve general diffusion problems. The R-L type fractional derivative has a wider application in viscoelastic problems because it does not require the function to be differentiable at the origin. As far as we all know, analyzing robustness with interference of the R-L type fractional system is an extremely interesting and challenging work.

The rest of this paper is organized as follows. In Section

In this section, we show some definitions and preliminaries of the

The norm for the

The definition of the two-parameter function of the Mittag–Leffler type is described by

If

Now, according to the results of the papers [

(Lemma

(Definition 2.4, see [

(H

In this section, we consider the following fractional equation:

For system (

Block diagram of the open- and closed-loop

We denote that

Assume that each iteration state meets algorithm (

Define

For

According to system (

Hence,

By taking the

Consequently,

In this section, we consider the following second-order

The initial state of the system is as follows:

For convenience, one can see Figure

Block diagram of the second-order

Assume that the initial state of each iterative learning meets (

Note

Suppose system (

According to Lemma

By taking the

For brevity, note that

There exists a constant

In this section, we will give two simulation examples to demonstrate the validity of the algorithms.

Consider the following one-dimensional systems as follows:

with the iterative learning control and initial state error

where

The state trajectories of system (

Simulation results of output

.Numerical simulation of the output of the system in Section

1 | 2.0818 | 2.0539 | 18 | −4.9051 | −4.9774 | 35 | 5.0183 | 4.9945 |

2 | 1.2580 | 1.2454 | 19 | −0.9237 | −0.9645 | 36 | −4.0133 | −4.0915 |

3 | 0.3294 | 0.3088 | 20 | 4.7684 | 4.7910 | 37 | −4.6198 | −4.7037 |

4 | −0.7118 | −0.7414 | 21 | 0.8193 | 0.7912 | 38 | −3.3214 | −3.3956 |

5 | −1.8284 | −1.8677 | 22 | −4.7823 | −4.8564 | 39 | 3.2206 | 3.1983 |

6 | −2.9501 | −2.9989 | 23 | 4.1016 | 4.0857 | 40 | −4.7914 | −4.8791 |

7 | −3.9613 | −4.0190 | 24 | −2.2443 | −2.2993 | 41 | 4.9318 | 4.9216 |

8 | −4.6935 | −4.7576 | 25 | 2.1803 | 2.1603 | 42 | 2.1176 | 2.0840 |

9 | −4.9269 | −4.9936 | 26 | −3.9964 | −4.0669 | 43 | −4.8627 | −4.9535 |

10 | −4.4206 | −4.4837 | 27 | 4.3542 | 4.3339 | 44 | 4.5943 | 4.5790 |

11 | −2.9888 | −3.0410 | 28 | 3.8435 | 3.7982 | 45 | 4.2592 | 4.2403 |

12 | −0.6484 | −0.6824 | 29 | 2.6705 | 2.6515 | 46 | −2.3747 | −2.4479 |

13 | 2.1738 | 2.1620 | 30 | 4.9469 | 4.9048 | 47 | −4.3241 | −4.4139 |

14 | 4.4456 | 4.4755 | 31 | −1.9316 | −1.9891 | 48 | −2.2423 | −2.3162 |

15 | 4.7533 | 4.7927 | 32 | 4.4846 | 4.4505 | 49 | 3.3387 | 3.3090 |

16 | 2.0844 | 2.0698 | 33 | 3.1067 | 3.0883 | 50 | 2.8945 | 2.8603 |

17 | −2.5569 | −2.6095 | 34 | 2.1099 | 2.0826 |

The tracking error of the systems.

Consider a two-dimensional ILC system; we set

We also select other parameters and initial values of the algorithm as follows:

Simulation results of output

The tracking error of the system.

From Figure

Numerical simulation of the output of the system in Section

1 | 0 | 0 | 1 | 0 | 0 |

2 | 2.3808 | 2.3971 | 2 | 0.0181 | 0.0040 |

3 | 4.1782 | 4.2073 | 3 | 0.0534 | 0.0320 |

4 | 4.9538 | 4.9874 | 4 | 0.1349 | 0.1080 |

5 | 4.5182 | 4.5464 | 5 | 0.2860 | 0.2560 |

6 | 2.9784 | 2.9923 | 6 | 0.5299 | 0.5000 |

7 | 0.7378 | 0.7056 | 7 | 0.8898 | 0.8640 |

8 | −1.7256 | −1.7539 | 8 | 1.3887 | 1.3720 |

9 | −3.7370 | −3.7840 | 9 | 2.0900 | 2.0480 |

10 | −4.8293 | −4.8876 | 10 | 2.8958 | 2.9160 |

11 | −4.7343 | −4.7946 | 11 | 3.9500 | 4.0000 |

12 | −3.4746 | −3.5277 | 12 | 5.3063 | 5.3240 |

13 | −1.3580 | −1.3970 | 13 | 6.8846 | 6.9120 |

14 | 1.0981 | 1.0755 | 14 | 8.7488 | 8.7880 |

15 | 3.3259 | 3.2849 | 15 | 10.9227 | 10.9760 |

16 | 4.7019 | 4.6899 | 16 | 13.4300 | 13.5000 |

17 | 4.9573 | 4.9467 | 17 | 16.2946 | 16.3840 |

18 | 4.0060 | 3.9924 | 18 | 19.5403 | 19.6520 |

19 | 2.0936 | 2.0605 | 19 | 23.1909 | 23.3280 |

20 | −0.3191 | −0.3757 | 20 | 27.2702 | 27.4360 |

In this paper, we show the concept of mild solutions of the R-L fractional system and considered two cases of the

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

The authors contributed equally to this work, and all authors read and approved the final manuscript.

This work was supported by the NSF of China (no. 11661084) and Guizhou Province Department of Education Fund ([2016]046, Qian Jiao He KY[2020]093, and Qian Ke He Ping Tai Ren Cai [2018]5784-08).

^{03b1}-type fractional order iterative learning control in the sense of

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