Abstract
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 norm. Firstly, according to the Laplace transform and M-L function, the concept of mild solutions of the system is derived. Secondly, we give the sufficient conditions of robustness analysis of the -type ILC algorithm with uncertain disturbances and then study the robust analysis of the second-order -type ILC algorithm. At last, two fractional examples are given to demonstrate the results.
1. Introduction
The aim of the paper is to analyze the robustness of a type of iterative algorithm in the sense of norm of the following R-L fractional system:where denotes the R-L derivative of order , , ,, is a control vector, and .
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 [1–8], and many fractional nonlinear systems were investigated [9–17]. In recent years, the fractional ILC algorithm has played a great role in multiagent control information transmission, and for more information, one can see the references [13–16].
In Li et al.’s study [17], the authors discussed a P-type ILC scheme for a class of fractional-order nonlinear systems with delay by using the -norm and Gronwall inequality and obtained the sufficient condition for the robust convergence of the tracking errors.
In view of that the -norm often causes tracking errors that exceed the actual engineering range and cause inaccurate data, the authors Lan and Lin [18] used the norm to discuss the convergence of iterative learning algorithms, and it objectively quantifies the essential characteristics of the tracking error and comprehensively reflects the behavior of the system. Zhang and Peng [19] used the generalized Young inequality of convolution and discussed the robustness of the PD-type fractional-order iteration and learning control algorithm in the sense of norm, and the conditions of its robust convergence are obtained.
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 2, according to the Laplace transform and M-L function, the concept of mild solutions of the system is derived. In Section 3, we give the sufficient conditions of robustness analysis of the -type ILC algorithm with uncertain disturbances and then study the robust analysis of the second-order -type ILC algorithm. In Section 4, two fractional examples are given to demonstrate the results.
2. Some Preliminaries for Fractional Systems
In this section, we show some definitions and preliminaries of the norm and Mittag–Leffler function. From [20–23], one can see the definitions of the R-L fractional integral and derivative.
Definition 1. The norm for the -dimensional vector is defined as , and the norm is defined as , where .
Definition 2. The definition of the two-parameter function of the Mittag–Leffler type is described byIf , one has the Mittag–Leffler function of one parameter as follows:Now, according to the results of the papers [17, 24–27], we will give the following lemma.
Lemma 1. (Lemma 3, see [25]). The general solution of equation (1) is given bywhere
Lemma 2. (Definition 2.4, see [27]). The operators are exponentially bounded, and there is a constant , , , and .
Lemma 3. (Hlder inequality). Set , and ; if , and , then .
3. Robustness Analysis of the -Type ILC Algorithm with Uncertain Disturbances
In this section, we consider the following fractional equation:where and are uncertain disturbances.
For system (6), we apply the following open- and closed-loop -type ILC algorithm:where , and are the parameters which will be determined, is the given function, , and . For convenience, one can see Figure 1. The initial state of each iterative learning is as follows:
We denote that
Theorem 1. Assume that each iteration state meets algorithm (7) and the initial state is ; then, there exists such that , and then, the sufficient condition for being uniformly bounded on is .
Proof. DefineFor , one has and .
According to system (6), we haveand thus, using the ILC algorithms (7) and (8), we derivesoHence,By taking the norm, we obtaindenotingConsequently, . So, there exists a positive , such that and , and then , which implies is uniformly bounded on .
4. Robust Analysis of the Second-Order -Type ILC Algorithm
In this section, we consider the following second-order -type ILC algorithm:where .
The initial state of the system is as follows:
For convenience, one can see Figure 2.
Assume that the initial state of each iterative learning meets (18), where are the parameters which will be determined.
Note
Theorem 2. Suppose system (6) satisfies the second-order -type ILC algorithm and the initial state of each iteration satisfies (18), then there exists positive such that . Since , is uniformly bounded, which guarantees that .
Proof. According to Lemma 1, we yieldand then,By taking the norm, it yieldsFor brevity, note thatand one can deduce .
There exists a constant , which satisfies . Since , is uniformly bounded. The proof is completed.
5. Simulations
In this section, we will give two simulation examples to demonstrate the validity of the algorithms.
5.1. -Type ILC with Initial State Error
Consider the following one-dimensional systems as follows:
with the iterative learning control and initial state error
where . Now, we can choose , , , , , . For the system, we use the -type ILC algorithm and set the initial control , , and . One can calculate , and then, all conditions of Theorem 1 are satisfied.
The state trajectories of system (24) with initial conditions are given in Figure 3 and Table 1, and with the increase of the number of iterations, it can track the desired trajectory gradually. Consistent with the theoretical analysis in the previous section, the algorithm has a faster convergence speed. At the end of the fourth iteration, the algorithm has converged. From Figures 3 and 4, the curve is basically completely fitted, showing that the system algorithm is well robust.
5.2. -Type ILC with Random Disturbance
Consider a two-dimensional ILC system; we set , , and construct the second-order -type ILC algorithm as follows:
We also select other parameters and initial values of the algorithm as follows: , , . It is easy to show that , and all conditions of Theorem 2 are satisfied. In the simulation, denotes the desired trajectory of state 1, denotes the desired trajectory of state 2, and solid lines (—–) in different colors denote the output of the system. In Figure 5, we use k1 to represent the iteration of state 1 and use k2 to represent the iteration of state 2, and the tracking error is shown in Figure 6, which implies the number of iterations and tracking error.
From Figure 6 and Table 2, one can find that the tracking error tends to zero quickly, so the output of the system can track the desired trajectory almost perfectly.
6. Conclusion
In this paper, we show the concept of mild solutions of the R-L fractional system and considered two cases of the -type ILC algorithm. The sufficient conditions of robustness analysis of the -type ILC algorithm with uncertain disturbances were given by the corresponding theorems and proved. At last, two R-L fractional examples are given to demonstrate the results.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
The authors contributed equally to this work, and all authors read and approved the final manuscript.
Acknowledgments
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).