Abstract
The paper proposes a low-pass filter adaptive iterative learning control (LPF-AILC) strategy for unmatched, uncertain, time-varying, non-parameterized nonlinear systems (NPNL systems). To address the difficulty of nonlinear parameterization terms in system models, a new function approximator (FSE-RBFNN), which combines the radial basis function neural network (RBFNN) and Fourier series expansion (FSE), is introduced to model each time-varying nonlinear parameterized function. The adaptive backstepping method is used to design control laws and parameter adaptive laws. In the process of controller design, we may encounter the problem of too many derivatives, which can cause parameter explosions after derivatives. Therefore, we introduce a first-order low-pass filter to solve this problem and simplify the structure of the controller. As the number of iterations increases, the maximum tracking error gradually decreases until it converges to the nearby region, approaching zero within the entire given interval , according to the Lyapunov-like synthesis. To mitigate the impact of initial state errors, a dynamically changing boundary layer is introduced, along with a series to deal with the unknown error upper bounds. Finally, two simulation examples prove the correctness of the proposed control method.
1 Introduction
Adaptive iterative learning control (AILC) is a useful control strategy for solving repetitive tracking control task problems for uncertain nonlinear systems. It continuously adjusts its control algorithm through iterative learning to gradually approach the ideal trajectory of the unknown system. AILC has extensive application value and promising development prospects for practical applications. Repeat systems include uncertain robotic manipulators and uncertain hard disk drivers. The task requirements specify that it can quickly achieve exact tracking as the number of iterations increases [1–4].
A non-parameterized nonlinear (NPNL) system refers to a dynamic characteristic that exhibits a complex nonlinear relationship and unknown parameters, making it difficult to design effective control strategies. It is particularly challenging to achieve high-precision tracking and control within a limited time frame. Traditional control methods often require the establishment of a mathematical model for the system, but for the NPNL system, this step is usually very difficult or even impossible to complete. AILC technology has become an important method for solving these problems [5, 6].
There are many challenging problems in the research of AILC. This paper considers three difficult problems of AILC. The first problem is the processing problem of uncertain nonlinear parameterization terms with time-varying parameters. In the field of control, the control problem of nonlinear systems with uncertain time-varying parameters is very challenging. Adaptive control and robust control are common methods to deal with uncertain problems [7, 8]. Through learning, adaptive control can mitigate the impact of uncertainties. In order to handle uncertain nonlinear terms, adaptive control is often combined with some approximation methods, such as neural networks (NNs) and Fuzzy Logic Systems (FLSs). However, these adaptive controls only solve the uncertain linearly parameterized disturbances and ensure the stability of the system [7–20]. For the uncertain system, a fuzzy AILC was presented [21]. The composite energy function–adaptive iterative learning control (CEF–AILC) is an effective scheme for systems with time-varying disturbances [21–23]. Few AILC research results focus on uncertain, non-parameterized nonlinear systems [24–26]. Specifically, for systems with non-separable time-varying parameters, the tracking control problem on finite time intervals is still an open problem.
The second problem of AILC is ensuring complete tracking over a finite time interval when the initial state has errors. In these studies [27–31], the stability analysis section requires that initial state errors be strictly zero. Although the research on this problem is well done in traditional D-type or P-type ILC [32–41], it has not been well solved based on Lyapunov analysis for AILC. Specifically, in the presence of an initial state error, ensuring the system’s completion of accurate tracking tasks within a specified time frame presents a complex challenge. [39] solved the tracking control problem of the unmatched uncertain NPNL systems. [41] solved the tracking problem of a class of high-order nonlinear systems with random initial state shifts, which relaxes the requirement of initial positioning in ILC. So far, no relevant research results have been found for AILC applied to NPNL systems with uncertain time-varying parameters and initial state errors.
The last problem is parameter explosions after the derivative of the virtual controller. When designing a controller, we may encounter the problem of too many derivatives, which can cause parameter explosions after derivatives. Addressing this issue and streamlining the controller’s structure to ensure the effective tracking of the non-parametric, nonlinear, time-varying system is a challenging and crucial problem. [42–44] employed a first-order low-pass filter to address the challenge of parameter explosions and achieve satisfactory performance. Therefore, we introduce a first-order low-pass filter to solve this problem and simplify the structure of the controller.
Motivated by the above discussion, we will use a low-pass filter AILC (LPF-AILC) method for uncertain time-varying NPNL systems. The AILC is given by the adaptive backstepping technique and Lyapunov-like theorem. In response to the difficult issues discussed above, the main contributions of this article are as follows:
1) An LPF-AILC strategy is proposed for a class of strongly time-varying, non-parameterized, nonlinear systems combined with a new approximation method.
2) The processing problem of uncertain time-varying nonlinear parameterization terms was solved. This is a very important and difficult problem. Specifically, in the field of AILC, no relevant research results have been found.
3) The difficulty problem of AILC is ensuring complete tracking on a given interval when the initial state has errors.
4) The problem of parameter explosions was solved by applying a derivative to the virtual controller and simplifying its structure.
In this paper, a combination of Fourier series expansion and radial basis function neural network (RBFNN) (FSE-RBFNNs) is used to model the uncertain, time-varying nonlinear dynamics by using their uniform approximation [24, 38]. An updating time-varying boundary layer is used to design the error function to deal with the initial state error. A common convergence series sequence is employed to mitigate the impact of approximation errors on the control performance of the system. A low-pass filter was introduced to solve the problem of parameter explosions resulting from the derivative of the virtual controller and simplify the structure of the controller. Theoretical analysis can demonstrate the bounded nature of all signals within the closed-loop system. The maximum value of errors will gradually converge to a narrow range close to zero as the boundary layer width satisfies the convergence condition with the number of iterations. Finally, two simulation examples are given to prove the effectiveness and correctness of the control method.
2 Problem description and mathematical foundations
2.1 Problem description
Uncertain time-varying NPNL systems are considered:where and represents measurable state vectors. is the control input. is the system output. , , and are uncertain time-varying functions, and represents unknown time-varying parameters. denotes the iteration time.
The design objective of this article is to find for system (1) to ensure that follows the ideal trajectory on .
2.2 Mathematical foundations
The mathematical knowledge used in this article is provided with relevant references, and the specific definitions and principles will not be elaborated. Here, we only provide the conclusions that need to be used in this article.
In system (1), the processing of unknown time-varying, nonlinear, parameterized function terms is a challenge. Since the function is not known, is expanded using Fourier series as ; based on this, uncertain time-varying nonlinear functions can be approximated asA new FSE-RBFNN approximator is built:representing aswhere
: In the compact domain , the weights and are constrained, and and with being unknown positive numbers.
Lemma 1[38]: For , in (5) is bound, andwhere represents the supremum of .
Because and are unknown, we estimate them with and , respectively. and are estimation errors.
Lemma 2[38]: In the surrogate model (4), the following conclusion holds:where with and , and the remainder is bounded by
For the processing of the supremum of each error term, this article introduces the following typical series sequence:
Lemma 2[39] For a sequence , where and , the following result exists:
Assumption 2: The initial error value at the beginning of each iteration should meet with being a convergence series sequence, where .
Considering the initial errors, a new function is accepted:where is the saturation function given aswith being an updating time-varying boundary layer. When and considering assumption 2 again, we have .
In order to prevent the problem of gradient explosion, we introduce the first-order low-pass filter , which is given as follows:where results from filtering an instruction with as its input, with being the virtual controller, , and . Because part of cannot pass through the filter, an error compensation mechanism is introduced to overcome the influence of the instruction filter. Therefore, a new function is introduced as follows:
3 AILC design
Based on the above mathematical foundations, we present the specific controller design process.
3.1 Designing the AILC controller
Step 1: Denote , which will be defined later. and , where is the virtual controller. Because the initial state values of the system have errors and gradient explosion, the new error functions and are given asWe recall thatGiven the derivative of ,
Therefore, the derivative of with respect to time is as follows:
The error compensation mechanism is considered as follows:Using Equation 18, we can find the time derivative of the error function as follows:
The unknown time-varying, nonlinear functions and may be approximated by FSE-RBFNN and RBFNN, respectively.where and are the truncation errors after approximation and and are weight vectors.
Consider , , and . The virtual control law is designed as
By substituting Equations 20, 21 into Equation 19, we obtain
where , , , and are estimations of , , , and , respectively. , , , and are the estimation errors. It can be proved that the following result is correct.
Using Equations 7, 23, Equation 22 can be rewritten as
Let , where is the remaining term of the estimation error after FSE-RBFNN expansion, and is also the same; then, Equation 24 becomes
The remainder is bounded with and .
Remark 1: This assumption is easily satisfied because 1) , , and are bounded and 2) when is large enough, is sufficiently small.
The Lyapunov-like function is chosen as follows:
where , , , and are adjustable matrices, each being positive, definite, and symmetric. Consider the derivative of by system (25), we obtain
where for any and , .
We chooseso Equation 27 becomes
Step 2: Denote , which will be defined later. Due to initial state errors and gradient explosion, we introduce the following error function as
The derivative of is shown as follows:
Let the error compensation mechanism be defined as follows:
Using Equation 32, we can find the time derivative of error function as
The uncertain time-varying, nonlinear functions and are approximated by FSE-RBFNN and RBFNN, respectively.where and are reconstructed errors and and are optimal weight vectors.
Let the virtual control be defined as follows:
Substituting Equations 34, 35 into Equation 33, we obtain
where , , , and are the estimators of , , , and , respectively. , , , and are estimation errors. It can be proved that the following results are correct.
Using Equations 7, 37, Equation 36 can be written as
Let , then Equation 38 becomes
The Lyapunov-like function was chosen as follows:
where , , , and are adjustable, positive, definite, and symmetric matrices. According to Equation 39, Assumption 3, and Remark 1, can be expressed as
We chooseThen, Equation 41 can be changed as
Step i: . Denote , which will be defined later. Because there exist initial state errors and gradient explosion, the error functions and are defined as
Therefore, can be deduced as follows:
Let the error compensation mechanism be defined as
Using Equation 47, we can find the time derivative of the error function as
The uncertain time-varying, nonlinear functions and are approximated by FSE-RBFNN and RBFNN, respectively, and reconstruction errors and are as given follows:where and are the approximation errors and and are ideal weight vectors.
Define , where is any arbitrary number with ; meanwhile, . Let the virtual control be defined as
By substituting Equations 49, 50 into Equation 48, we obtain
where , , , and are the estimations of , , , and , respectively. , , , and are estimation errors. We can rephrase the final three components on the right side of Equation 51 as
Using Equations 7, 52, Equation 51 can be reformulated as
Let , then Equation 53 becomes
Consider the following nonnegative function:
where , , , and are adjustable, positive, definite, and symmetric matrices. According to Equation 54, Assumption 3, and Remark 1, can be expressed as
We chooseThen, Equation 56 can be written as
Step n: Denote , which will be defined later. Because there exist initial state errors and gradient explosion, the function , denoting the error, is defined as
The derivative of with respect to time is expressed as
Let the error compensation mechanism be defined as
Using Equation 61, we can obtain the time derivative of the error function as
The overall approximation capability of the RBFNN asserts that the unknown nonlinear functions and are capable of approximation within a defined scope by FSE-RBFNN and RBFNN, respectively, and reconstruction errors and are as follows:where and are the approximation errors and and are ideal weight vectors.
Define ,where is any arbitrary number such that ; meanwhile, . Let the virtual control be defined as
By substituting Equations 63, 64 into Equation 62, we can conclude that
where , , , and are the estimations of , , , and , respectively. , , , and are estimation errors. We can rephrase the final three components on the right side of Equation 65 as
Using Equations 7, 66, Equation 65 can be reformulated as
Let , then Equation 67 becomes
Assumption 4: The remainder is bounded with and .
Remark 2: This assumption is reasonable because 1) , , and are constrained within the specified area by Equations 6, 8.
Let the following non-negative function be defined as
where , , , and are adjustable, positive, definite, and symmetric matrices. The derivative of is considered as follows (Equation 68):
We chooseThen, Equation 70 can be written as
For the initial state, we rely on the following set of assumed conditions:
Assumption 2: When , , , , and holds true for all values of .
3.2 Stability and convergence analysis
Theorem 1: For nonlinear system (1) with assumptions 2, 3, and 4, if we design virtual controllers (21), (35), (50), controller (64), and parameter updating laws (28), (42), (57), (71),then all signals in the closed-loop system are bounded within the interval [0, T]. We obtainIn other words, , and then , where is the boundary of the difference between and . Let be chosen sufficiently large, ensuring that and can be minimized as much as possible throughout the entire time interval [0, T].
Proof: In accordance with Assumption 2, we find that . Consider that . Using Equation 69, we obtain , ,, , and. Using Equation 72,
Let , then Equation 74 can be rewritten asUsing Equation 9, we obtain and is bounded. , soBased on Equation 69, for any given value of ; substituting Equation 72 obtainBased on Equation 76, is bounded. According to definition 1, is bounded and , so is also bounded. In addition, , , , and ; based on Equation 77, for any given value of , is bounded. So, is also bounded; from above all, for any given value of , if is bounded, then we can deduce that , , , , and are bounded. According to Equation 64, is bounded. According to Equation 53, is bounded, so is continuous uniformly. Thus, we can deduce Equation 73.
Then, we need to prove that will converge to a neighborhood that approaches 0. Initially, let be a signal satisfying and for all . The compensation error within the compensation system is defined asWith specified initial conditions, , i.e., . From (11), we obtainAs shown in Equation 79, choosing an appropriate value for confines the error within a narrow range, approximately equating to . In addition, based on the compensation system, the Lyapunov function is defined on the interval as follows:The derivative of along systems (78) with respect to time is expressed aswhere. To ensure the stability of the compensation system, it is sufficient to satisfyEquation 82 leads to the conclusion that is bounded. Hence, is also bounded. Moreover, we can choose a parameter to arbitrarily reduce , thereby causing the compensation of the system to approach 0. In this way, by ensuring that the error approaches 0, will converge to the neighborhood approaching 0. Thus, we conclude Theorem 1.
4 Illustrative examples
4.1 Number simulation
This section includes an example illustrating the effectiveness of the proposed adaptive iterative learning controller.
The second-order pure-feedback nonlinear system described is considered as follows:where , , and are state variables and is the input variable. Utilizing the widely recognized van der Pol oscillator as the reference model, we obtainwhere and are state variables. The primary control objective is to synchronize the output of systems (82) with the reference trajectory generated by system (84) over the interval [0,5] under the condition .
In accordance with Theorem 1, the adaptive iterative learning controller is chosen as
The error compensation mechanism iswhere .
The parameter adaptive iterative learning laws are provided by (57):where , , , , , , , , and .
Figures 1–3 show the tracking performance of the system output and expected output without iteration and at 50th and 100th iterations, respectively. Figures 4, 5 show that as the number of iterations increases, the system error may converge to a small region near the zero point. Furthermore, observations shown in Figures 6–10 confirm that both control signals and and estimated parameters, , , , , ,and exhibit bounded behavior within the [0,5] range. The validity of the control strategy presented in this research is reaffirmed by the simulation results shown in Figures 11–20 over the interval .
FIGURE 1
FIGURE 2
FIGURE 3
FIGURE 4
FIGURE 5
FIGURE 6
FIGURE 7
FIGURE 8
FIGURE 9
FIGURE 10
FIGURE 11
FIGURE 12
FIGURE 13
FIGURE 14
FIGURE 15
FIGURE 16
FIGURE 17
FIGURE 18
FIGURE 19
FIGURE 20
4.2 Simulation of a single-joint robotic arm
In this section, we conducted simulation verification on a single degree-of-freedom robotic arm system to assess the performance of the proposed control method. The dynamic equation of a single degree-of-freedom robotic arm iswhere is the angle between the robotic arm and the reference frame. is the input of the DC motor.where is the output of the reference model. is the reference input signal. According to Equations 88, 89, the state equation of the system is derived asand its reference model is derived aswhere equals to can be defined as the angle between the robotic arm and the reference frame. is the time derivative of , i.e., . The primary control objective is to synchronize the output of systems (88) with the reference trajectory generated by system (89) over the interval [0,5] under the condition .
In accordance with Theorem 1, the adaptive iterative learning controller is chosen as
The error compensation mechanism iswhere .
The parameter adaptive iterative learning laws are provided by (57).
where , , , , , , .
Figures 11–13 show the tracking performance of the system output and expected output without iteration and at 15th and 30th iterations, respectively. Figures 14, 15 show that as the number of iterations increases, the system error may converge to a small region near the zero point. Furthermore, observations from Figures 16–20 confirm that both control signals and and estimated parameters, , , , and , exhibit bounded behavior within the [0,5] range. The validity of the control strategy presented in this research is reaffirmed by the simulation results shown in Figures 11–20 over the interval .
5 Conclusion
This article presents a solution to the complete trajectory, following challenges within a finite time frame for a category of nonlinearly parameterized systems characterized by time-varying disturbed functions and initial state errors. A new FSE neural network is used to learn the time-varying, nonlinearly parameterized term. Based on this and Lyapunov theory, we proposed the new LPF-AILC method. A low-pass filter is used to solve the problem of parameter explosion after obtaining the derivative of the virtual controller. The unmatched uncertainties, nonlinear parameterization, and initial state errors are well considered. Two simulation examples have proven the feasibility of the control approach. This article does not mention time-delay issues, but they often exist in practical systems. Our future work should consider solving the complete tracking problem on a finite time interval for these complex systems with time delays. This is a more interesting issue. In addition, there are two deficiencies in the controller design process: the assumption of time-varying parameters being periodic and the jitter issues caused by the low-pass filter. These challenges will be carefully considered and addressed in our future work.
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
CZ: conceptualization, funding acquisition, investigation, methodology, and writing–review and editing. LY: formal analysis, software, writing–original draft, and writing–review and editing. YG: investigation, validation, and writing–original draft. JY: funding acquisition, supervision, and writing–review and editing. FQ: funding acquisition, supervision, and writing–review and editing.
Funding
The authors declare that financial support was received for the research, authorship, and/or publication of this article. This work is supported by the National Natural Science Foundation (NNSF) of China (grants 62073259 and 61973094), the Natural Science Basis Research Plan in Shaanxi Province of China (2023-JC-QN-0752), the Science and Technology Plan Project of Xi’an City (No. 23GXFW0062), and the Shaanxi Provincial Key R& D Program General Project (No. 2024GX-YBXM-106).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
adaptive iterative learning control, time-varying non-parameterized nonlinear systems, backstepping method, Fourier series expansion-radial basis function neural network, initial state errors, low-pass filter
Citation
Zhang C, Yan L, Gao Y, Yao J and Qian F (2024) FSE-RBFNN-based LPF-AILC of finite time complete tracking for a class of time-varying NPNL systems with initial state errors. Front. Phys. 12:1442486. doi: 10.3389/fphy.2024.1442486
Received
02 June 2024
Accepted
02 August 2024
Published
21 August 2024
Volume
12 - 2024
Edited by
Fei Yu, Changsha University of Science and Technology, China
Reviewed by
Jinping Jia, Tianshui Normal University, China
Yichao Yan, University of Electronic Science and Technology of China, China
Njitacke Tabekoueng Zeric, University of Buea, Cameroon
Updates
Copyright
© 2024 Zhang, Yan, Gao, Yao and Qian.
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*Correspondence: Junliang Yao, yaojunliang@xaut.edu.cn
ORCID: Lei Yan, orcid.org/0000-0001-5894-7588; Yangjie Gao, orcid.org/0009-0009-9651-5757; Junliang Yao, orcid.org/0000-0001-6041-9813; Fucai Qian, orcid.org/0000-0001-8461-1420
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