Intelligent Compensation Control Using Fuzzy Inference and Recurrent Neural Network for Magnetic Levitation System

Abstract

To address the control uncertainty and improve the control performance of the magnetic levitation system, an intelligent feedforward compensation controller is proposed based on fuzzy inference (FI) and recurrent neural network (RNN). It consists of a PID-based baseline controller, an RNN-based inverse model identifier, an RNN-based feedforward compensator, and an FI-based automatic regulator. The PID-based baseline controller is applied to provide the initial control law. The RNN-based inverse model identifier is built to online learn the controlled object, while the RNN-based feedforward compensator is built to generate a real-time compensation control law based on the learned parameters. Moreover, the FI-based automatic regulator is designed to dynamically adjust the compensation quantity and adaptively restrain the control uncertainty according to the control error and its change. The effectiveness and advancement of the proposed intelligent controller are experimentally verified by the position-tracking control of the magnetic levitation system. Tracking results of step and square signals indicate that the proposed intelligent controller not only enhances the transient quality but also improves the overall control performance compared to other comparative controllers.

Keywords

intelligent compensation control feedforward compensation fuzzy inference recurrent neural network magnetic levitation system

1 Introduction

Magnetic levitation technology, with the merits of being noiseless and frictionless, has been widely used in many fields, such as maglev trains (Lv et al., 2023), magnetic couplers (Wu et al., 2024), magnetic bearings (Yao et al., 2015), magnetic levitation motors (Wang et al., 2023b), etc. Since the magnetic levitation control system has the characteristics of nonlinearity, hysteresis, and instability, it is hard to establish an exact mathematical model, which increases the real-time control difficulty of the time-varying system (Wang et al., 2023a). Therefore, it is necessary to design an intelligent controller that does not rely on a precise mathematical model of the control system and has strong robustness in real-time control.

To improve the control performance of the magnetic levitation system, many efforts have been made by researchers and various advanced control approaches have been successively designed, such as proportion integration differentiation (PID) control (Acharya et al., 2022; Chopade et al., 2018), model predictive control (Wang et al., 2022; Zhang et al., 2020), sliding mode control (Abdollahzadeh & Pourgholi, 2024; Kang et al., 2024), nonlinear adaptive control (Lin et al., 2027; Yaseen et al., 2022), robust control (Sun et al., 2019), etc. Among them, the control performance of most methods highly depends on precise mathematical models of the time-varying systems, and almost only the model-free PID control is frequently applied in practical control (Sahoo et al., 2023). However, it is hard to simultaneously obtain the optimum dynamic and steady-state performance by only tuning the PID parameters due to the nonlinearity and instability of the magnetic levitation system.

In practice, advanced feedforward compensation strategies can promote the improvement of the tracking performance of the time-varying control systems (Cao & Zhang, 2020; Nakashima et al., 2010). For instance, Sarkar et al. combined an order-separated feedforward and PID control to enhance the tracking performance of the time-varying control system with a good rejection ability of external disturbance (Sarkar et al., 2023). Morales and Sira-Ramirez developed an exact feedforward compensation control approach based on a model-based integral reconstructor and a generalized proportional-integral controller, which improved the trajectory tracking accuracy of the magnetic levitation ball (MLB) system (Morales & Sira-Ramirez, 2010). Zhao and Zhu proposed a PID-based feedforward decoupling control for the magnetic bearing system and enhanced the anti-interference ability of the control system (Zhao & Zhu, 2019). However, the high performance of these compensation strategies still relies on the accurate mathematical model of the control system, hindering the practical application of these advanced control methods in time-varying magnetic levitation systems.

Neural networks can accurately model the time-varying nonlinear system due to their learning and mapping abilities (Huang et al., 2024) and have been increasingly applied to improve the real-time tracking performance of the control systems (Chen et al., 2024; Zhao et al., 2023). Wahyudie et al. introduced neural networks into position tracking control for the magnetic levitation system, which enhanced robustness and fault tolerance compared with the traditional method (Wahyudie et al., 2024). Sun et al. proposed a neural network-based supervisor controller and improved the control accuracy of the time-varying maglev vehicles (Sun et al., 2022). Furthermore, by leveraging historical delay data through its internal feedback loops, the recurrent neural network (RNN) is capable of learning the nonlinear time-varying system more accurately (Huang et al., 2022), and consequently improving the control performance of the time-varying system. Fatemimoghadam et al. applied RNN-based adaptive backstepping control to lower the position control error of the magnetic levitation system (Fatemimoghadam et al., 2020). Ou et al. developed an intelligent feedforward compensation method based on gated recurrent units for precision motion control of maglev motors, which obtains an excellent tracking performance (Ou et al., 2021). However, when the training sample is extremely insufficient at the sudden leap moment of the tracking signal, the controlled object cannot be identified accurately by neural networks, thereby resulting in control uncertainty of the time-varying system.

Recently, fuzzy inference (FI) has been gradually used to address uncertain problems of time-varying systems due to its clear interpretation, simple computation, and reliable reasoning (Liu et al., 2023; Tan et al., 2023; Yang et al., 2024). Kamal and Ghorbani proposed a fuzzy decentralized constrained controller based on robust fuzzy anti-windup, which addressed parameter uncertainties and disturbances and improved the overall robustness and tracking capability of nonlinear time-varying systems (Kamal & Ghorbani, 2024). Castillo et al. presented a type-2 fuzzy logic-based intelligent controller for complex nonlinear plants and obtained a lower simulation tracking error by modeling uncertainty more closely (Castillo et al., 2016). Sun et al. established a fuzzy logic-based inference rule to estimate uncertainties of the control system without sufficient prior knowledge and achieved a smaller prediction error and better control performance than other control schemes (Sun et al., 2023). Tang et al. introduced FI into the hybrid controller to address the control uncertainty of the neural network, thereby enhancing the dynamical performance of position control in the magnetic levitation system (Tang et al., 2022). Tong et al. developed a fuzzy inference-based decentralized state observer to obtain uncertain states, which achieved satisfactory control performance of the switching signals (Tong et al., 2016). These studies demonstrate that FI-based intelligent controllers can achieve better control performance by suppressing uncertainty more effectively. However, the above methods applied the traditional feedforward neural networks to identify the control system, but these neural networks do not take into account the historical information of an internal feedback loop during the process of parameter updating. This restricts the network's ability to accurately approximate the nonlinear dynamics of the controlled object. In addition, in the FI module of these methods, membership degree decreases rapidly with distance and ignores important edge information of the control interval, thereby hindering further improvement in the tracking performance of the position control.

Therefore, motivated by the aforementioned advantages of the RNN in the modeling of the nonlinear system and the capability of the FI in addressing uncertainties of the time-varying systems, an intelligent feedforward compensation controller based on fuzzy inference and recurrent neural network (FRNN) is proposed to address the control uncertainty and further improve the control performance of the MLB system in this research. It consists of four modules, namely, a PID-based baseline controller, an RNN-based inverse model identifier, an RNN-based feedforward compensator, and an FI-based automatic regulator. The main novelty and contributions of this research are summarized as follows:

1)
A novel FRNN-based intelligent feedforward compensation controller is proposed. Unlike traditional model-based feedforward controllers that require precise mathematical models, the proposed controller integrates the RNN's modeling capabilities with FI's decision-making potential, enabling it to effectively learn the time-varying dynamic of the control system without prior system identification.
2)
The RNN-based data-driven inverse model identifier and feedforward compensator are designed. The former online learns the inverse model of the controlled object without relying on prior knowledge, whereas the latter utilizes the learned parameters to generate a dynamical compensation, enhancing system adaptability and responsiveness.
3)
An FI-based automatic regulator is developed to address uncertainties caused by an undertrained RNN model. This component dynamically adjusts the output of the feedforward compensator based on real-time tracking error and its rate of change, thereby maintaining high control accuracy even when the RNN identifier is not fully trained.
4)
The proposed FRNN-based intelligent controller is rigorously evaluated through position-tracking experiments on a nonlinear and time-varying MLB system. Experimental results demonstrate superior transient and steady-state performance compared to existing control methods, validating the effectiveness and advancement of the proposed approach.

The remainder of this study is arranged as follows. Section 2 reviews the related works. Then, an intelligent compensation controller is explicated in detail in Section 3. Next, Section 4 presents the experimental platform and discusses the experimental results. Finally, the main works are concluded in Section 5.
2 Intelligent Compensation Controller

In this study, an intelligent compensation controller based on fuzzy inference and recurrent neural network (FRNN) is designed to implement reliable and precise position control of the MLB system. As illustrated in Figure 1, the proposed intelligent compensation controller consists of four modules: a PID-based baseline controller, an RNN-based inverse model identifier, an RNN-based feedforward compensator, and an FI-based automatic regulator.

Figure 1.

Overall Control Structure of the FRNN-Based Intelligent Compensation Controller.

In the proposed FRNN-based intelligent controller, the PID-based baseline controller is introduced to provide the basic control quantity and generate online training samples. The RNN-based inverse model identifier is employed to learn a precise inverse model of the MLB system online. The learned weight parameters are passed in real-time to the RNN-based feedforward compensator with the same network architecture, which is applied for adaptive position compensation of the MLB system. To suppress the control uncertainty of the RNN-based feedforward compensator caused by the undertrained RNN identifier, the FI-based automatic regulator is designed to dynamically adjust the compensation control quantity. In this way, the designed intelligent compensation controller ensures control reliability at the transient phase and high precision at the steady state phase.

Concretely, within each control cycle, the online input and output of the MLB system serve as training samples for the RNN-based inverse model identifier. By comparing the identifier's output $\hat{u} (t)$ with the control variable $u (t)$ , the resulting error is used to refine the network parameters through backpropagation, ultimately forming a dynamic inverse model of the MLB system. The parameters of the identifier are synchronously replicated in the RNN-based feedforward compensator at each control cycle, enabling it to adapt promptly to system variations. Using the error $e (t)$ and its change rate $e c (t)$ of the controlled system as inputs, the FI-based regulator infers a dynamic adjustment factor $c (t)$ that enables the control system to dynamically fine-tune the output of the compensator. Finally, after being dynamically tuned by the adjustment factor $c (t)$ , the compensator's output u(t) acts together with the PID controller's output u(t) on the controlled MLB system. Therefore, the total control quantity $u (t)$ of the intelligent compensation controller is calculated as follows

\begin{aligned} u (t) = u_{o} (t) + c (t) \cdot u_{c} (t) \end{aligned}

(1)

where

c (t)

is an adjustment factor decided by the FI-based automatic regular at the t-th moment.

2.1 PID-Based Baseline Controller

Since its advantages of clear principle and simple structure, the PID algorithm has been widely applied to solve various control problems in industrial applications. Therefore, the PID-based baseline controller is introduced into the proposed intelligent controller to provide the basic control quantity of the MLB system and generate online training samples for the RNN-based inverse model identifier. Furthermore, the PID-based baseline controller is used to maintain control stability of the MLB system when there is control uncertainty of the RNN-based feedforward compensator caused by the undertrained RNN-based inverse model identifier.

The control quantity of the PID-based baseline controller at the t-th moment is calculated by

\begin{aligned} u_{o} (t) = k_{p} e (t) + k_{i} \sum_{i = 0}^{i} e (i) Δ t + k_{d} (e (t) - e (t - 1)) / Δ t \end{aligned}

(2)

where

k_{p}, k_{i}

, and

k_{d}

are proportion gain, integration gain, and differentiation gain, respectively.

2.2 RNN-Based Inverse Model Identifier

The RNN-based inverse model identifier is designed to learn a precise inverse model of the MLB system online. As shown in Figure 2, the RNN-based inverse model identifier includes one input layer, one hidden layer, and one output layer. It takes the system output $y (t)$ as the network input and the estimated control quantity $\hat{u} (t)$ as the network output.

Figure 2.

Network Architecture of the RNN-Based Inverse Model Identifier.

At the t-th moment in the control cycle, the network output $\hat{u} (t)$ of the RNN-based inverse model identifier is calculated by forward propagation. Its calculated expression is shown as follows

\begin{aligned} \hat{u} (t) = \sum_{i = 1}^{n} v_{i} h_{t}^{i} = \sum_{i = 1}^{n} v_{i} σ (w_{i} y (t) + a_{i} h_{t - 1}^{i} + b_{i}) \end{aligned}

(3)

where

v_{i}

represents the i-th weight between the hidden layer and the output layer,

h_{t}^{i}

is the i-th neuron output of the hidden layer,

σ (\cdot)

is the tanh activation function of the hidden layer,

w_{i}

denotes the i-th weight between the input layer and the hidden layer,

a_{i}

is the i-th internal weight of the hidden layer between the adjacent moments, b_i is the i-th bias of the hidden layer.

During the process of backward propagation, the Euclidean distance between the actual control quantity $u (t)$ and the estimated control quantity $\hat{u} (t)$ is taken as the training loss function. Its expression is defined as follows

\begin{aligned} L = \frac{1}{2} e_{u} {(t)}^{2} = \frac{1}{2} (u (t) - \hat{u} (t))^{2} \end{aligned}

(4)

By minimizing the training loss function online, the RNN-based inverse model identifier is expected to identify a real-time inverse model of the MLB system as precisely as possible. In contrast to previous feedforward neural networks-based approaches that employed stochastic gradient descent for network updates, the proposed FRNN-based controller adds a gradient differentiation term to the original Adagrad algorithm. Gradient differentiation serves to forecast the evolution trend of neural network parameters, aiming to enhance the convergence accuracy and rate of the RNN-based inverse model identifier. In addition, to simplify the expression, let $θ$ denote $v_{i}, w_{i}, a_{i}$ , and $b_{i}$ , then the parameters of the RNN-based inverse model identifier are updated by

\begin{aligned} {\begin{cases} s (t) \leftarrow α \cdot s (t - 1) + (g_{θ} (t))^{2} \\ θ (t) \leftarrow θ (t - 1) - \frac{η}{\sqrt{s (t) + δ}} g_{θ} (t) + β (g_{θ} (t) - g_{θ} (t - 1)) \end{cases} \end{aligned}

(5)

where

s (t)

is a quadratic sum of the gradient

g_{θ} (t)

at the t-th moment, the decay factor

α

is set to the default value of 0.95,

η

is the learning rate,

β

is the penalty coefficient, smoothing factor

δ

is equal to the default value of

1 e - 8

2.3 RNN-Based Feedforward Compensator

While identifying the inverse model of the MLB system using the RNN-based inverse model identifier, the RNN-based feedforward compensator with the same network architecture is built for real-time precise position compensation in the MLB system. As shown in Figure 1, the RNN-based inverse model identifier takes the reference signal $r (t)$ as the network input and the compensation control quantity $u_{c} (t)$ as the network output. Particularly, it only propagates forward rather than backward, and its weight parameters are real-time passed by the RNN-based inverse model identifier.

According to the above control principle, the compensation control quantity $u_{c} (t)$ of the RNN controller at the k-th moment can be calculated by

\begin{aligned} u_{c} (t) = \sum_{i = 1}^{n} v_{i} h_{t}^{i} = \sum_{i = 1}^{n} v_{i} σ (w_{i} r (t) + a_{i} h_{t - 1}^{i} + b_{i}) \end{aligned}

(6)

where parameters

v_{i}, w_{i}, a_{i}

, and

b_{i}

are the same as the RNN-based inverse model identifier shown in Equation (3),

σ (\cdot)

is also the tanh activation function of the hidden layer.

It can be found that the compensation precision of the RNN-based feedforward compensator highly depends on the identification precision of the inverse model of the MLB system. When at the early phase of the control cycle or the tracking signal mutates instantaneously, the training samples are insufficient for training the excellent RNN-based inverse model identifier, thereby causing the compensation uncertainty of the RNN-based feedforward compensator. At this moment, the compensation output needs to be suppressed to ensure control stability.

2.4 FI-Based Automatic Regulator

The fuzzy inference technique is frequently applied to address various uncertain problems of time-varying systems due to its clear interpretability and efficient reasoning. Therefore, in the proposed intelligent controller, the FI-based automatic regulator is designed to suppress the control uncertainty of the RNN-based feedforward compensator caused by the untrained RNN. As shown in Figure 3, the FI-based automatic regulator consists of a fuzzifier, a rule knowledge base, a fuzzy inference engine, a defuzzifier, a gain, and a saturation operation.

Figure 3.

Structure Illustration of the FI-Based Automatic Regulator.

In the control system, during the early phase of control and at moments of tracking signal transitions, the RNN does not have adequate training samples. Due to insufficient training, the RNN-based compensator's output carries strong uncertainty, which negatively affects the system's transient performance. Consequently, the FI-based regulator is incorporated to counteract the effects of the undertrained RNN during the initial training phase. This adaptive adjustment helps diminish the instability of the RNN-based compensator, enhancing the transient performance of the control system. Once the control system attains a steady state, there is a gradual increase in effective training samples, resulting in a more accurate dynamic inverse model for the MLB system. This improvement reduces the control error and slows down the rate of error change. At this stage, to ensure the RNN-based compensator's dominant role in the control system's steady state, the adaptive adjustment effect of the FI-based regulator is gradually reduced, thereby maintaining the steady-state performance of the control system. Therefore, under the action of the regulator illustrated in Figure 3, the compensation controller can balance the transient performance of the control system and superior steady-state accuracy.

Since the control error $e (t)$ and the change of the control error $e c (t)$ can directly reflect the control performance, they are used as the input variables of the FI-based automatic regulator. In particular, considering the reasoning sensitivity and rule compatibility, each input variable is divided into seven fuzzy subsets: $- B, - M, - S, Z, + S, + M, + B$ , corresponding to negative big, negative middle, negative small, zero, positive small, positive middle, and positive big.

In addition, considering the computational efficiency, their membership functions on both sides are set to trapezoid functions, and the other membership functions are set to triangle functions. To simplify the expressions, let x represent the input variables $e (t)$ and $e c (t)$ , let $[- U, U]$ denote their universes of discourse $[- U_{e}, U_{e}]$ and $[- U_{e c}, U_{e c}]$ , then their membership functions are expressed as follows

\begin{aligned} λ_{- B} (x) & = {\begin{cases} 1, & - U \leq x < - 3 U / 5 \\ (- 2 U - 5 x) / U, & - 3 U \leq x < - 2 U / 5 \end{cases} \end{aligned}

(7)

\begin{aligned} λ_{- M} (x) & = {\begin{cases} (5 x + 3 U) / U, & - 3 U / 5 \leq x < - 2 U / 5 \\ (- U - 5 x) / U, & - 2 U / 5 \leq x < - U / 5 \end{cases} \end{aligned}

(8)

\begin{aligned} λ_{- S} (x) & = {\begin{cases} (5 x + 2 U) / U, & - 2 U / 5 \leq x < - U / 5 \\ - 5 x / U, & - U / 5 \leq x < 0 \end{cases} \end{aligned}

(9)

\begin{aligned} λ_{Z} (x) & = {\begin{cases} (5 x + U) / U, & - U / 5 \leq x < 0 \\ (U - 5 x) / U, & 0 \leq x < U / 5 \end{cases} \end{aligned}

(10)

\begin{aligned} λ_{+ S} (x) & = {\begin{cases} 5 x / U, & 0 \leq x < U / 5 \\ (2 U - 5 x) / U, & U / 5 \leq x < 2 U / 5 \end{cases} \end{aligned}

(11)

\begin{aligned} λ_{+ M} (x) & = {\begin{cases} (5 x - U) / U, & U / 5 \leq x < 2 U / 5 \\ (3 U - 5 x) / U, & 2 U / 5 \leq x < 3 U / 5 \end{cases} \end{aligned}

(12)

\begin{aligned} λ_{+ B} (x) & = {\begin{cases} (5 x - 2 U) / U, & 2 U / 5 \leq x < 3 U / 5 \\ 1, & 3 U / 5 \leq x \leq U \end{cases} \end{aligned}

(13)

The reasoning output $c (t)$ of the FI-based automatic regulator is employed to adjust the control quantity of the RNN-based feedforward compensator. The reasoning output is divided into five fuzzy subsets: ${Z, S, M, B, G}$ , corresponding to zero, small, middle, big, and great. Its membership functions on both sides are also set to trapezoid functions, and the other membership functions are also set to triangle functions. Let $[- U_{c}, U_{c}]$ represent its universe of discourse, then its membership functions are expressed as follows

\begin{aligned} μ_{Z} (c) = {\begin{cases} 1, 0 \leq c < 0.5 U_{c} \\ (0.6 U_{c} - c) / U_{c}, 0.5 U_{c} \leq c < 0.6 U_{c} \end{cases} \end{aligned}

(14)

\begin{aligned} μ_{S} (c) = {\begin{cases} (c - 0.5 U_{c}) / U_{c}, 0.5 U_{c} \leq c < 0.6 U_{c} \\ (0.7 U_{c} - c) / U_{c}, 0.6 U_{c} \leq c < 0.7 U_{c} \end{cases} \end{aligned}

(15)

\begin{aligned} μ_{M} (c) = {\begin{cases} (c - 0.6 U_{c}) / U_{c}, 0.6 U_{c} \leq c < 0.7 U_{c} \\ (0.8 U_{c} - c) / U_{c}, 0.7 U_{c} \leq c < 0.8 U_{c} \end{cases} \end{aligned}

(16)

\begin{aligned} μ_{B} (c) = {\begin{cases} (c - 0.7 U_{c}) / U_{c}, 0.7 U_{c} \leq c < 0.8 U_{c} \\ (0.9 U_{c} - c) / U_{c}, 0.8 U_{c} \leq c < 0.9 U_{c} \end{cases} \end{aligned}

(17)

\begin{aligned} μ_{G} (c) = {\begin{cases} (c - 0.8 U_{c}) / U_{c}, 0.8 U_{c} \leq c < 0.9 U_{c} \\ 1, 0.9 U_{c} \leq c \leq U_{c} \end{cases} \end{aligned}

(18)

After setting the above fuzzy variables and membership functions, the fuzzy rule knowledge base is built to decide the adjustment factor. In particular, the setting of fuzzy rules follows the following principles:

When the available training samples are insufficient at the initial control phase or the moment of a sudden change of the tracking signal, it is difficult for the untrained RNN-based inverse model identifier to build an accurate inverse model of the MLB system online. It leads to the compensation uncertainty of the RNN-based feedforward compensator, resulting in a large control error and error change ratio in the MLB system. At this time, the FI-based automatic regulator is required to restrict the compensated output of the RNN-based feedforward compensator. In other words, the larger the control error or error change ratio in the MLB system, the smaller the reasoning output of the FI-based automatic regulator.

When the control system gradually enters a stable state, the increasing number of samples enables the RNN-based inverse model identifier to learn the inverse model of the MLB system more and more accurately. At this time, the control error and error change ratio are small and the FI-based automatic regulator is used to promote the compensated output of the RNN-based feedforward compensator. Therefore, the smaller the control error or error change ratio in the MLB system, the larger the reasoning output of the FI-based automatic regulator.

Concretely, based on the fuzzy subset settings of two input variables (i.e., each input has seven fuzzy subsets), a total of forty-nine fuzzy rules can be obtained. Finally, combining the above expert experience, the fuzzy rule knowledge is shown in Table 1.

Table 1.

Rule Knowledge of the FI-Based Automatic Regulator.

		E
C		−B	−M	−S	Z	+S	+M	+B
EC	−B	Z	Z	Z	S	Z	Z	Z
	−M	Z	Z	S	M	S	Z	Z
	−S	Z	S	M	B	M	S	Z
	Z	Z	M	B	G	B	M	Z
	+S	Z	S	M	B	M	S	Z
	+M	Z	Z	S	M	S	Z	Z
	+B	Z	Z	Z	S	Z	Z	Z

The weighted average method, as a frequently used defuzzification method in industrial control, is used for the defuzzification approach in the FI-based automatic regulator. After the defuzzification, the initial adjustment factor $c_{o} (t)$ is obtained.

To avoid drastic changes in the reasoning output, the saturation operation is adopted in the RNN-based feedforward compensator. Its calculation expression is as follows

\begin{aligned} c (t) = {\begin{cases} c (t - 1) - ε, c (t) - c (t - 1) < - ε \\ K_{c} \cdot c_{o} (t), - ε \leq c (t) - c (t - 1) \leq ε \\ c (t - 1) + ε, c (t) - c (t - 1) > ε \end{cases} \end{aligned}

(19)

where

ε

represents the saturation coefficient of the change ratio of the adjustment factor

c (t)

Finally, after the saturation operation, the final adjustment factor $c (t)$ is obtained. It is used to adjust the compensation quantity of the RNN-based compensator dynamically.

3 Experimental Verification

3.1 Experimental Platform

To evaluate the effectiveness and advancement of the proposed intelligent compensation controller, comparative experiments are conducted in the MLB system. The experimental platform is shown in Figure 4. It consists of an LED light source, an electromagnet, a steel ball, a photoelectric sensor, and a data acquisition and control card installed in a computer (Tang et al., 2022).

Figure 4.

Illustration of the Experimental Platform of the Magnetic Levitation Ball System (Tang et al., 2022).

In the control experiment, the photoelectric sensor converts the displacement signal of the steel ball into a voltage signal. After signal modulation and analog-digital conversion, it enters the computer and compares with the command position of the maglev ball. The control quantity is determined by the intelligent controller that runs on the MATLAB/Simulink real-time workshop.

After digital-analog conversion and power amplification, it controls the current of the electromagnet coil. This produces an electromagnetic attraction to the magnetic levitation ball and then controls the position of the steel ball. Therefore, the control precision of the maglev ball position depends on the performance of the intelligent controller. In particular, the initial position of the maglev ball was set near the equilibrium point to evaluate the transient response characteristics of the MLB control system.

3.2 Performance Metrics

In this study, to verify the effectiveness and advancement of the proposed FRNN-based intelligent compensation controller, it is compared with the baseline PID controller and the RNN-based feedforward compensation controller. These control methods are implemented in the tracking experiments of step and square signals.

First, overshoot and settling time are employed in the tracking experiments to compare the transient performance of these approaches quantitatively. Furthermore, four metrics named mean absolute error (MAE), root-mean-square error (RMSE), integral time absolute error (ITAE), and integral time square error (ITSE) are applied to evaluate the overall control performance of these control approaches. Their calculation expressions are shown as follows

\begin{aligned} {\begin{cases} M A E = \frac{1}{T} \sum_{t = 0}^{T} | r (t) - y (t) | \\ R M S E = \sqrt{\frac{1}{T} \sum_{t = 0}^{T} {(r (t) - y (t))}^{2}} \\ I T A E = \int_{0}^{T} t | r (t) - y (t) | d t \\ I T S E = \int_{0}^{T} t {(r (t) - y (t))}^{2} d t \end{cases} \end{aligned}

(20)

where

r (t)

and

y (t)

represent the desired input and the actual output of the control system, respectively.

3.3 Position Tracking Results

3.3.1 Step Signal Tracking Results

In the tracking experiment of continuous step signal, to make the comparison among all control methods as fair as possible, the proportion, integration, and differentiation parameters for all comparative controllers are set to the same optimal value using the trial-and-error method. Concretely, these parameters are set to $k_{p} = 1.005, k_{i} = 0.003$ and $k_{d} = 16$ , respectively. In addition, balancing the computational time and learning performance, the number of neurons in the hidden layer is set to 5. The learning rate $η$ is 0.12, the penalty coefficient $β$ is set to 0.05, and the saturation coefficient $ε$ is set to 0.007, respectively. Correspondingly, experimental results of different control methods in tracking continuous step signals are drawn in Figure 5.

Figure 5.

Comparative Results of Tracking Step Signal Under Different Control Methods.

As shown in Figure 5, after introducing the feedforward compensation, the RNN-based compensation controller obtains a lower overshoot and settling time in tracking continuous step signals compared with the baseline PID controller. Furthermore, by establishing the FI-based automatic regulator to adaptively adjust the compensation quantity, the FRNN-based intelligent controller significantly reduces overshoot and settling time compared to the baseline PID controller and the RNN-based compensation controller. It can be found from Figure 6 that, when the continuous step signals suddenly jump at positions I, II, and III, the proposed FRNN-based intelligent controller provides dynamic compensation quantity more quickly than the RNN-based compensation controller. Meanwhile, the FI-based automatic regulator effectively restrains the compensation output of the RNN-based feedforward compensator at these jump moments. Therefore, the proposed FRNN-based intelligent controller makes the control system reach a steady state faster than the RNN-based control method.

Figure 6.

Control Quantities of Different Control Methods Under Tracking Step Signal.

In addition, to facilitate a more comprehensive and quantitative comparison of the control performance of these methods, both the transient response metrics (overshoot and settling time) and the integrated error metrics (MAE, RMSE, ITAM, ITSE) are computed and listed in Table 2, using the experimental data derived from Figure 5. It can be found that, at the beginning phase of the control experiment, the overshoot and settling time of the traditional PID method are 0.7119 mm and 1.5332 s. After using the RNN-based compensation, the overshoot and setting time are lower to 0.2236 mm and 0.2765 s. Especially, combining the RNN's modeling capability with FI's decision-making ability, the FRNN-PID method further reduces the overshoot and settling time to 0.0039 mm and 0.10753 s. Compared to the RNN-PID method, the proposed FRNN-based intelligent controller lowers the overshoot by 98.26% and reduces the settling time by 36.60%.

Table 2.

Quantitative Comparisons of Different Control Methods Under Tracking Step Signal.

	PID	RNN-PID	FRNN-PID	Improvement
Overshoot/mm	0.7119	0.2236	0 . 0039	98.26%
Settling time/s	1.5332	0.2765	0.1753	36.60%
MAE/mm	0.0594	0.0159	0.0108	32.08%
RMSE/mm	0.1656	0.0503	0.0476	5.37%
ITAE/mm	6.6769	1.9690	1.4476	26.48%
ITSE/mm	3.1951	0.2640	0.2298	12.95%

For the entire control cycle, the control errors (i.e., MAE, RMSE, ITAE, and ITSE) of the traditional PID method are 0.0594 mm, 0.1656 mm, 6.6769 mm, and 3.1951 mm, respectively. The intelligent RNN-PID method reduces the errors to 0.0159 mm, 0.0503 mm, 1.9690 mm, and 0.2640 mm, correspondingly. By contrast, the control errors of the FRNN-PID method are lower, reaching 0.0108 mm, 0.0476 mm, 1.4476 mm, and 0.2298 mm. That is, the FRNN-based intelligent controller lowers the MAE, RMSE, ITAE, and ITSE by 32.08%, 5.37%, 26.48%, and 12.95% than the RNN-based compensation controller, respectively. Consequently, combined Figure 5 and Table 2, it can be found that the proposed FRNN-based intelligent controller can not only enhance transient response but also improve control precision in the step signal tracking experiment.

3.3.2 Square Signal Tracking Results

To make the comparison as fair as possible in the tracking experiment of square signal, the proportion, integration, and differentiation parameters of all the comparative control methods are set to the same optimal value by using the trial-and-error method, that is, $k_{p} = 1.005, k_{i} = 0.003$ , and $k_{d} = 16$ , respectively. For the RNN-based and FRNN-based intelligent controllers, the number of neurons in the hidden layer is set to 5. The learning rate $η$ is set to 0.16, the penalty coefficient $β$ is set to 0.04, and the saturation coefficient $ε$ is set to 0.007, respectively. Correspondingly, square signal tracking results of magnetic levitation ball position using different control methods are drawn in Figure 7.

Figure 7.

Experimental Results of Tracking Square Signal Under the Different Controllers.

It can be seen from Figure 7 that, compared to the baseline PID controller in the tracking experiment of square signal, the RNN-based compensation controller obviously reduces overshoot and settling time by introducing the feedforward compensation. Moreover, after developing the FI-based automatic regulator to adaptively adjust the control output of the RNN-based feedforward compensator, the proposed FRNN-based intelligent controller achieves the lowest overshoot and settling time compared with the PID and RNN-based controllers. It shows that the proposed FRNN-based intelligent controller has excellent dynamic performance in the square signal tracking process. As displayed in Figure 8, the proposed FRNN-based intelligent controller with the FI-based automatic regulator can quickly compensate for the control quantity compared with the RNN-based compensation controller at the signal jump moment. Since the FI-based automatic regulator can effectively restrain the control quantity of the RNN-based feedforward compensator at the jump moments of the tracking signal, the proposed FRNN-based intelligent controller promotes the control system into a steady state faster than the RNN-based controller.

Figure 8.

Control Quantities of Different Control Methods Under Tracking Step Signal.

Additionally, based on the experimental data displayed in Figure 5 and the expressions shown in Equation (20), the transient response metrics (overshoot and settling time) and the integrated error metrics (MAE, RMSE, ITAM, ITSE) are quantitatively computed and presented in Table 3.

Table 3.

Quantitative Comparisons of Different Control Methods Under Tracking Square Signal.

	PID	RNN-PID	FRNN-PID	Improvement
Overshoot/mm	1.3291	0.8506	0.0025	99 . 71%
Settling time/s	1.7523	0.0249	0.0156	37.35%
MAE/mm	0.1449	0.0377	0.0223	40.85%
RMSE/mm	0.3609	0.1416	0.1240	12.43%
ITAE/mm	17.7314	5.0009	3.1158	37.70%
ITSE/mm	15.7928	2.5417	2.0305	20.11%

As can be seen from Table 3, after introducing the RNN-based intelligent compensation, the RNN-PID method reduces the overshoot from 1.3291 mm to 0.8506 mm and shortens the settling time from 1.7523 s to 0.0249 s when compared with the traditional PID-based control method. Moreover, benefiting from the dynamic adjustment of the RNN compensation by the FI-based automatic regulator at the beginning phase of the control experiment, the overshoot and settling time of the FRNN-PID method are lowered to 0.0025 mm and 0.0156 s. In particular, compared to the RNN-based controller, the FRNN-based controller reduces overshoot by 99.71% and settling time by 37.335%.

Meanwhile, in comparison with the traditional PID-based control method during the entire control cycle, the RNN-PID method reduces the MAE from 0.1449 mm to 0.0377 mm, the RMS from 0.3609 mm to 0.1416 mm, the ITAE from 17.7314 mm to 5.0009 mm, and the ITSE from 15.7928 mm to 2.5417 mm, respectively. Particularly, after the dynamic adjustment of network compensation by the FI-based regulator at the moment of signal jump change, these errors (i.e., MAE, RMSE, ITAM, and ITSE) of the FRNN-based method are further reduced to 0.0223 mm, 0.1240 mm, 3.1158 mm, and 2.0305 mm, respectively. Significantly, compared to the RNN-based controller in terms of these errors, the proposed FRNN-based intelligent controller lowers MAE by 40.85%, RMSE by 12.43%, ITAE by 37.70%, and ITSE by 20.11%, respectively. As a result, through the comprehensive analysis of signal tracking results, it can be obviously found that the proposed FRNN-based intelligent controller can improve dynamic quality and tracking performance.

3.4 Robust Performance

To evaluate the disturbance rejection capacity of the proposed FRNN-based intelligent controller, the tracking experiment of the step signal with stochastic disturbance is implemented in this subsection. The experimental results are displayed in Figure 9, where the above subgraph shows the overall tracking results, and the below subgraph is a partially enlarged detail.

Figure 9.

Robust Result of the FRNN-Based Intelligent Compensation Controller.

It can be found from Figure 9 that the FRNN-based intelligent controller can quickly make the controlled object reach the desired position when the tracking signal is suddenly disturbed. Although there are not sufficient samples to train the RNN-based inverse model identifier well at the disturbed moment, the RNN-based automatic regulator can retain the uncertain output of the RNN-based feedforward compensator effectively. Therefore, the FRNN-based intelligent controller exhibits strong anti-interference ability and robustness in the position control system.

4 Discussion

4.1 Sensitivity Analysis of Hyperparameters

To evaluate the impact of key parameters on control performance, a hyperparameter sensitivity analysis is conducted in this section. Firstly, as shown in Equation (5), the learning rate $η$ and the penalty coefficient $β$ play a crucial role in enabling the RNN-based identifier to accurately model the inverse dynamics of the MLB system during the control process. In the context of step signal tracking, Figure 10 displays the sensitivity results for different learning rates and penalty coefficients. It can be seen that when the learning rate is below 0.12, a lower learning rate results in a larger MAE and RMSE, leading to degraded tracking performance of the control system. When the learning rate exceeds 0.12, the MAE and RMSE increase with the increase in the learning rate. Therefore, the optimal learning rate is selected as 0.12 for the step signal tracking experiment. Additionally, the lowest MAE and RMSE are observed when the penalty coefficient is set to 0.05, which makes it the preferred value in the experiment.

Figure 10.

Experimental Results of Tracking Step Signals Under Different Hyperparameters.

In the square signal tracking, Figure 11 presents the sensitivity results for different learning rates and penalty coefficients. It can be observed that when the learning rate is below 0.16, a lower value leads to increased MAE and RMSE, resulting in deteriorated tracking performance. Conversely, when the learning rate exceeds 0.16, both error metrics rise with increasing learning rate. Therefore, 0.16 is selected as the optimal learning rate. Additionally, the lowest MAE and RMSE occur at a penalty coefficient of 0.04, making it the preferred choice.

Figure 11.

Experimental Results of Tracking Square Signals Under Different Hyperparameters.

In addition, as shown in Equation (19), the saturation coefficient directly affects the change ratio of the adjustment factor in the FI-based automatic regulator. It can be seen from Figure 12 that, whether in step signals or square signals tracking experiments, the control errors (MAE and RMSE) are lowest when the saturation coefficient is equal to 0.007. Therefore, the value of 0.007 is selected as the final and optimal setting for the saturation coefficient in this study.

Figure 12.

Experimental Results Under Different Saturation Coefficients.

4.2 Compared with State-of-the-Art Methods

To improve control accuracy and tracking capabilities of the nonlinear MLB system, this study presents an intelligent compensation controller. It comprises a PID-based controller, an RNN-based inverse model identifier, an RNN-based feedforward compensator, and an FI-based automatic regulator. Among them, the PID-based controller is employed to establish the initial control law. The RNN-based inverse model identifier is constructed to perform online learning of the dynamic inverse model of the MLB system, whereas the RNN-based feedforward compensator generates a real-time compensation control law based on the learned parameters. Additionally, the FI-based automatic regulator is designed to dynamically adjust the compensation amount and adaptively suppress control uncertainties.

To further verify the advancement of the proposed FRNN-PID control, it was compared with several advanced controllers on the same MLB system. Among them, the intelligent FI-BP-PID controller (Tang et al., 2022) applied a BP neural network to learn the inverse model of the MLB system and utilized a Mamdani-based FI block to adjust the compensation amount. It can be found from Table 4 that, whether tracking step or square signals, the overshoot achieved with the FI-BP-PID controller is significantly higher compared to the proposed FRNN-PID controller. This advantage stems from the ability of the RNN to exploit past data through internal recurrent connections in each control cycle, whereas traditional BP neural networks lack such memory mechanisms.

Table 4.
Experimental Comparisons of Different Advanced Controllers.

Signals Methods Overshoot Settling time ITAE

Step FI-BP-PID (Tang et al., 2022) 0.0540 0.0650

LSTM-ARX-PFC (Peng et al., 2024) 0.0032 0.1800 -

TR-NMPC (Peng et al., 2025) 0.0025 0.2900 -

FRNN-PID 0.0039 0.1753 1.4476

Square IEIDO-MPC (Wang et al., 2022) – – 15.8940

FI-BP-PID (Tang et al., 2022) 0.0980 0.0620 -

LSTM-ARX-PFC (Peng et al., 2024) 0.0086 0.2000 -

TR-NMPC (Peng et al., 2025) 0.0093 0.1600 -

FRNN-PID 0.0025 0.0156 3.1158

Signals	Methods	Overshoot	Settling time	ITAE
Step	FI-BP-PID (Tang et al., 2022)	0.0540	0.0650
LSTM-ARX-PFC (Peng et al., 2024)	0.0032	0.1800	-
TR-NMPC (Peng et al., 2025)	0.0025	0.2900	-
FRNN-PID	0.0039	0.1753	1.4476
Square	IEIDO-MPC (Wang et al., 2022)	–	–	15.8940
FI-BP-PID (Tang et al., 2022)	0.0980	0.0620	-
LSTM-ARX-PFC (Peng et al., 2024)	0.0086	0.2000	-
TR-NMPC (Peng et al., 2025)	0.0093	0.1600	-
FRNN-PID	0.0025	0.0156	3.1158

The LSTM-ARX-PFC method (Peng et al., 2024) used an LSTM-based auto-regressive model with exogenous input variables to learn the dynamic model of the MLB system and then applied a model predictive controller to improve real-time control performance. As shown in Table 4, although the overshoot and settling time of the LSTM-ARX-PFC method and the proposed FRNN-PID method are similar when tracking step signals, the control performance of the proposed method is significantly better than that of this compared method when tracking square signals.

The TR-NMPC approach (Peng et al., 2025) employed a radial basis function-based autoregressive model with exogenous variables to describe the dynamics of the MLB system and then utilized trust region nonlinear model predictive control to achieve superior transient performance. However, the overshoot and settling time of the TR-NMPC control method are still inferior to the proposed FRNN-PID method, especially when tracking square wave signals. Additionally, the proposed FRNN-PID method is superior to the IEIDO-MPC method (Wang et al., 2023a) in terms of the time integral evaluation metric. In summary, the results demonstrate that the proposed FRNN-PID control approach not only improves transient performance and lowers steady-state error but also exhibits exceptional robustness when contrasted with other advanced controllers.

For future studies, it is worthwhile to explore a feedforward compensation control method that combines artificial neural networks with variable universe fuzzy inference, particularly for improving the position control accuracy of MLB systems. Moreover, conducting a theoretical investigation into the interaction between RNN and FI components in the control framework would contribute to strengthening the methodological foundation and practical applicability in real-world control system.

5. Conclusion

This study focuses on high-performance position control of the magnetic levitation system and presents the FRNN-based intelligent compensation controller. Therefore, the RNN-based inverse model identifier online learns the time-varying controlled object while the RNN-based feedforward compensator generates compensated control quantity based on learned parameters in real-time. Moreover, the FI-based automatic regulator adjusts the compensated quantity and restrains the control uncertainty according to the control error and its change adaptively.

Compared with the RNN-based feedforward compensation controller, the FRNN-based intelligent controller lowers overshoot by 98.26% and 99.71%, and reduces settling time by 36.60% and 35.70% in the tracking experiments of step and square signals. Furthermore, the FRNN-based intelligent controller lowers ITAE by 26.48% and 37.70%, and lowers ITSE by 12.95% and 20.11%, respectively. It demonstrates the proposed FRNN-based intelligent controller can enhance the transient quality and improve the overall control performance. Therefore, the proposed FRNN-based intelligent feedforward compensation controller has been effectively applied in the magnetic levitation system and is expected to be used in other time-varying control systems with uncertain characteristics.

Footnotes

Acknowledgment

This work was financially supported by grants from the National Key Research and Development Program of China (No. 2022YFE0114300) and the Research Program of Shanghai Key Laboratory of Online Testing and Control Technology (No. Z2022304013).

ORCID iD

Jianmin Zhu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Research Program of Shanghai Key Laboratory of Online Testing and Control Technology, National Key Research and Development Program of China, (grant number Z2022304013, 2022YFE0114300).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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