Artificial Neural Network-Based Feed-Forward and Feedback Control Design and Convergence Analysis

Chen, Guoshao; Liu, Zhiping

doi:https://doi.org/10.1155/2022/1238020

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1238020 | https://doi.org/10.1155/2022/1238020

Artificial Neural Network-Based Feed-Forward and Feedback Control Design and Convergence Analysis

Guoshao Chen¹and Zhiping Liu¹

Academic Editor: Jose Vicente Salcedo

Received30 Jul 2021

Revised14 Apr 2022

Accepted15 Apr 2022

Published06 May 2022

Abstract

A feed-forward and feedback control scheme based on artificial neural network (ANN) and iterative learning control is proposed. Iterative learning control and ANN are combined as a feed-forward controller, which makes the output track the desired trajectory. Feedback control is introduced to reduce the effect of disturbances. To combine the feed-forward controller and the feedback controller, the ANN is employed to simulate the plant. Since the ANN can update the weights online, it is always consistent with the plant. The convergence and robustness of the system are analyzed, and the simulation shows the feasibility of the proposed control scheme.

1. Introduction

A Fourier transform spectrometer can get interferograms through modulations in the interference signal strength with a scanning mirror, which then provides a spectrogram through the Fourier transform. The mirror is the only moving part of a Fourier transform spectrometer. The interferogram data are impacted by nonuniform scanning. The accuracy of the scanning mirror system determines the accuracy of the Fourier transform spectrometer. Proportional integral derivative (PID) was adopted to control the scanning mirror. The fuzzy PID was adopted with a relative error in the output of less than 4% [1–3]. We found that the iterative learning control (ILC) can achieve satisfactory results when controlling the scanning mirror in the absence of disturbances. In the presence of disturbances, the ILC cannot immediately reduce the influence of disturbances; furthermore, the disturbance may make the error larger in the subsequent iteration. To reduce the influence of disturbance, feedback control is added to the ILC.

The ILC is an intelligent control approach for systems that perform a specific task repeatedly over a limited interval [4]. The ILC was first proposed in the late 1970s as a method to accurately track repetitive processes. After more than 40 years of development, it has become an important field of intelligent control theory and has achieved promising results in theoretical analyses and practical applications [5–7]. Several papers have discussed problems associated with ILC, including the initial state of the system, the repeatability of the input signal, and the plant; however, robustness and anti-disturbance have rarely been mentioned [4–8]. Chow proposed a neural network-based ILC but did not discuss convergence. Patan proposed a hybrid controller with an iterative learning neural network and feedback control to solve nonlinear systems. They implemented ILC using neural networks, but the disturbances affected the associated weights [9–11]. To avoid disturbances affecting the feed-forward controller, the feedback and feed-forward control are completely separated in this paper. In the part of feed-forward control, the plant is replaced with a mathematical model.

It is difficult to get an accurate mathematical model. Several papers proposed system identification with artificial neural networks (ANNs) over many years. Zamarreño demonstrated the usefulness of ANN in target recognition or predictions in linear systems [12, 13]. A physics-based recurrent neural network (RNN) was proposed for a general nonlinear dynamic system to improve the prediction accuracy by incorporating a priori process knowledge [14]. Adaptive neural network (NN) control was designed for robot systems and approximated the computer position model to address uncertain values of the system and improve its stability [15]. An adaptive neural network and proposed full state feedback and output feedback control were designed to ensure the uniform limit boundedness of the closed-loop system [16]. A radial basis function neural network was utilized to compensate for the actuator gain fault [17–19]. The neural network strategy was used to adapt to the unknown dynamics and disturbances [20]. Neural network systems have been widely used in various aspects, such as pattern recognition, calculations, and combinatorial optimization. As a powerful tool to model systems, neural networks have been widely used to solve control problems of unknown nonlinear systems [21–23]. A neural network-based observer was designed to identify the system dynamics where a piecewise update rule of neural network weights is adopted to handle the challenge of the complex time series [24, 25]. One neural network was applied to reconstruct the approximations of unknown nonlinearities [26, 27]. An adaptive neural network was designed, by combining its general online estimation ability with an adaptive backstepping design framework, and the probabilistic asymptotic tracking control was realized [28].

It can be concluded that there are two challenges. (1) The ILC can track the desired response, and the feedback controller can reduce the influence of disturbance. How to combine the two controllers so that the ILC works without disturbance is one challenge. (2) Neural network has no memory unit, so it cannot simulate the state space equation. How to improve the neural network is another challenge. Therefore, a feed-forward and feedback control scheme based on ILC and ANN is proposed. Compared with previous works, this paper has the following main contributions.(1)The ANN is introduced to simulate the plant to avoid the influence of disturbance on the ILC.(2)We improved the ANN to make it similar to the difference discrete system.(3)The convergence and robustness of the system are analyzed. The robustness analysis shows that the anti-disturbance capability of the control scheme proposed in this paper is stronger than ILC.

The proposed control scheme is only suitable for the system where the desired trajectory is periodic, and it is difficult to achieve satisfactory results just by using PID. This paper is organized as follows. Section 2 gives the general model of the plant. Section 3 designs the structure of the controller. Section 4 designs the ANN and gives the framework for the weights to train the ANN. Section 5 analyzes the convergence and robustness of the feed-forward and feedback control. Section 6 presents the simulation. The last section gives the conclusions.

2. Problem Formulation

We consider the plant as the following discrete-time system:

Suppose () satisfies the Lipschitz condition and is the Lipschitz constant, where denotes the iteration instance and is the time instance. , , and are the state, input, and output of the system in the p-th cycle, respectively, and () is the function.

3. Controller Design

The structure of the system is shown in Figure 1. The system consists of two subsystems: one is the feedback control system and the other is the feed-forward control system. We adopt the ANN to simulate the plant. The input is generated by the ILC. The feedback controller generates the input . We adopt the PID as feedback control.

The input of the plant is the sum of and , while the input of the ANN is only . The feed-forward controller consists of ILC and ANN, so disturbances will not affect it. The ILC is p-type as

This paper presents an ILC scheme for the Arimoto prototype [29–31] aswhere and is the diagonal learning gain matrix.

The feedback control is the proportional control as [32]

After the system works for a period of time, the parameters of the plant may change, resulting in the difference between the plant and ANN. The weights of the ANN can be updated online. The dashed line in Figure 1 indicates the weight update for the ANN.

Remark 1. To accelerate the learning speed, a variable speed learning gain matrix should be used. The time-varying p-type version of the ILC update rule is .

4. Artificial Neural Network Design

An accurate mathematical model for the plant is difficult to obtain in some cases. Here, an ANN is used as it can flexibly represent any linear and nonlinear function. Being a state-space model, the number of outside connections is minimal. The inputs drive the operation of the system, and the outputs are the effects observed on the system. A parallel input/output is established between the neural model and the physical system [12–14, 28]. The ANN is given as follows:

4.1. ANN Structure Design

The structure of the ANN is shown in Figure 2, where is the input, is the state, is the output, and is the state at the previous time. The function has the following form:where , , and are the weights and is the bias vector.

Substituting (7) into (6) giveswhere is the activation function. The derivative is ; thus,

Letso

4.2. ANN Training

In this paper, the gradient descent method is applied to update the weights of the ANN. We design the ANN structure using prior knowledge or reasonable assumptions. The parameter estimation is performed in a highly experimental manner involving several parameters without a clear understanding of the dynamic model [12, 13]. When the system initializes, the weights of the ANN are random numbers between 0 and 1. The system works for a period of time, the input data and output data are recorded, and at the same time, the data are used to train the ANN. The structure of ANN training is shown in Figure 3, where is the output of the ANN, is the output of the plant, and is the desired trajectory.

The ANN needs an optimization target to update the weights. The optimization function in this paper is

Then,where is the constant learning rate. In practice, using an adaptive learning rate can improve training efficiency [5]. Note thatwhere is the derivative of the activation function.

Remark 2. The plant is usually regarded as a differential discrete system, which means that the output is related not only to the current input but also to the previous input and previous output. The backpropagation (BP) neural network can express , but it cannot express . We add to ANN to make it similar to the state-space equation of the plant.

5. Convergence and Robustness Analysis

Lemma 1. This lemma refers to that in the paper [33]. Let us suppose that a sequence of positive real numbers is defined aswhere is given if satisfies . The positive real number sequence has the limit and uniform bound (for any ). Then,

5.1. Convergence Analysis

We assume that the system satisfies the following conditions.(A1)Let be the desired trajectory. can be realized for each initial condition , where there is a unique that satisfies the following state-space equation:(A2)For all trials, the initial condition is the same as(A3) satisfies the global Lipschitz condition ofwhere is the Lipschitz constant.

When the system converges, the following condition must be satisfied, so the error approaches zero [4].

Theorem 1. Consider that the feed-forward controller for the plant from equation (1) and the ILC from equation (4) satisfy the assumptions mentioned above. If , then the error moves closer to zero.

Proof. LetSubtract both sides of (4) from to attainSubtracting (6) from (17), we getSubstituting (23) into (22) givesAccording to the trigonometric inequality,As , let . According to (6) and (11),We know that , so (26) can be rewritten asSubstituting (27) into (25),Apply λ-norm to both sides, multiply by , and let :The last part of (29) isLet and , and (29) can be written asLet α > 1. Then, there always exists λ that is sufficiently large so thatWe can rewrite (31) asWhen , we obtainAccording to (22),The proof is complete.

5.2. Robustness Analysis

The control scheme proposed in this paper is to add feedback control based on ILC. Whether the proposed scheme has a stronger anti-disturbance ability is analyzed in this section.

The influence of the measurement noise on the system is represented by . We assume that ; thus, the p-type ILC and feedback control are

We define

According to (2) and (37), we can obtain

From (38),

Then, (40) is substituted into (34), and it is assumed that is reversible. Thus,

Apply the norm to both sides of equation (44) to get

Multiply by , which is known as the Bellman–Gronwall lemma, when as

Then,where

According to Lemma 1, when , there is a sufficiently large and , . Then,

Therefore, when there is measurement noise, the output will converge to the neighborhood of the desired trajectory, where the neighborhood size is related to the upper bound of the disturbance. As indicated in (45), the neighborhood becomes smaller when is appropriately valued.

6. Illustrative Simulations

To verify the effectiveness of the proposed feed-forward and feedback control based on the ILC and ANN, the traditional PID control, ILC, and feed-forward and feedback control are simulated. The feed-forward and feedback control system design has the following three steps:(1)A PID controller is employed to generate the input and output data so that the weights of the ANN can be updated. The structure is given in Figure 3. The ANN has three layers: input, hidden, and output. The input and output layers have one neuron node, and the hidden layer has two state nodes, as shown in Figure 2.(2)Design the feed-forward control. The ILC and the ANN constitute the feed-forward controller.(3)Design the feed-forward and feedback controller. The composite control system is shown in Figure 1, which uses the proportional controller and ILC.

Considering the mirror control system of the Fourier transform infrared spectrometer, the plant iswhere

The desired trajectory is

The expected iteration length . To ensure the convergence of the ILC, we take . The desired trajectory and output are shown in Figure 4. The black line represents the desired trajectory, and the pink line represents the output of the 80th iteration. The output is closer to the desired trajectory, and the maximum error is less than 0.00002 in the 200th iteration. It shows that ILC is convergent.

To verify the anti-interference ability of the system, random disturbance and constant disturbance are added, respectively.

6.1. Random Disturbance

The random disturbance is uniform white noise , and . When PID is employed, the maximum error is 0.006, when ILC is adopted, the maximum error is 0.004, and when feed-forward and feedback control is used, the maximum error is 0.0004. Figures 5 to 7 show that the feed-forward and feedback control system proposed in this paper has stronger anti-interference ability in the presence of random disturbance.

6.2. Constant Disturbance

Another disturbance is the constant disturbance, whose value is 0.02, from 15 s to 50 s. Figure 8 shows the result with the PID control. The maximum error occurs at two points where the disturbance appears and disappears. PID can reduce the influence of disturbance. When the system adopts the ILC, as shown in Figure 9, disturbances cause significant errors in the output. With the control scheme proposed in this paper, the output is shown in Figure 10. The error is significantly smaller than that of the previous two schemes. Thus, the feed-forward and feedback controllers are better than traditional ILC and PID, especially in the case of disturbances.

By comparing the errors, we can get that whether the disturbance is constant or random, the errors of PID control and ILC are larger than those of the proposed feed-forward and feedback control.

7. Conclusions

This paper proposes a control scheme that improves the practicality of ILC. This scheme is suitable for systems with the desired response being repetitive and periodic, and the system may be disturbed. Simulations show that the control scheme proposed in this paper outperforms PID and ILC. Although ANN can simulate the plant, it is still uncertain what structure and size of ANN are optimal for a specific plant, which is a problem to be addressed in future research.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This study was supported in part by the key projects of the Shaanxi Provincial Department of Science and Technology under grant no. 2019GY-092.

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Copyright © 2022 Guoshao Chen and Zhiping Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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