Response Error Prediction and Feedback Control Method for Electro-Hydraulic Actuators Based on LSTM

Abstract

The application of hydraulic systems in aerospace and engineering machines is becoming widespread. With the use of electro-hydraulic actuators, designing efficient and intelligent controllers can help the rapid expansion of electromechanical equipment in various scenarios. In response to the difficulty of slow response in the EHA control process, the paper designs an error prediction algorithm to predict the system response curve and replace the real-time error of PID input, achieving advanced correction of the controller. The experiment shows that the proposed method has a lower response time and smoother control curve while ensuring accuracy. It might have potential value in improving hydraulic system efficiency, reducing switching shock, and increasing system service life.

1. Introduction

Against the backdrop of the booming development of automated terminals, global container terminals have also entered a new era, and a new round of comprehensive upgrading of port machinery is gradually unfolding [1]. At present, automated terminal container equipment [2], especially transport vehicles, is developing towards electric drive direction, adopting volume servo integrated electro-hydraulic actuator systems, and combining the advantages of electrical and hydraulic systems [3]. Compared with traditional systems, the performance in various aspects has been significantly improved [4]. Volume servo integrated electro-hydraulic actuator(EHA) is a typical application of electro-hydraulic servo pump control technology [5], which can effectively solve the inherent defects of electro-hydraulic servo valve control technology [6], such as poor anti-pollution ability [7], high equipment installation cost [8], and inconvenient maintenance [1,9]. Compared with the electro-hydraulic servo valve control system equipment, the volume servo integrated electro-hydraulic actuator has the characteristics of small equipment volume [10], simple pipeline layout [11], no throttling overflow loss [12], high reliability [13], high safety [14], and high precision [15]. It occupies 60–80% less space [16], reduces energy consumption by 50–80% [17], increases power-to-weight ratio by 50–60% [6], reduces noise by 10–20%, saves energy and reduces noise, is easy to install, and is easy to maintain [18].

Scholars have made significant efforts in the control methods of EHA; Navatha et al. from the Madras Institute of Technology in India [19] established a proportional integral derivative (PID) controller to dynamically analyze, track, and control the position of an EHA system by changing the speed of the driving motor. The Ziegler Nichols (ZN) method was used for PID adjustment, and it was ultimately found that the PD controller had a better response. Tsuda et al. from Saitama University [20] proposed a dual inertia model to suppress vibrations that may occur during EHA operation and verified its dynamic characteristics. The US Air Force Technical College Pachte et al. [21] used quantitative feedback theory robust control algorithm to design a controller for EHA. In the design process, parameter changes, sensor noise, and flight condition changes were fully considered. The designed controller not only has robustness in terms of actuator parameter changes and flight conditions but also is insensitive to sensor noise, thereby improving the performance of the entire flight control system. Long Xianxue et al. [22] applied the Target Particle Swarm Segmentation algorithm in the control of EHA and found through simulation that this method is very useful for engineers, helping to determine the design parameters of EHA in the design phase. Fan Jizhong [23] established a nonlinear block diagram model based on SIMULINK and designed a nonlinear PID controller. Simulation results showed that the nonlinear PID controller can make the system have better dynamic characteristics than conventional PID controllers. Li Ruizhe et al. [24] found through simulation that fuzzy PID can better control EHA, with very low overshoot [25] and steady-state error [26]. This study will discuss a gear pump driven by a high-speed servo motor to control the hydraulic rod movement.

The classic control methods mentioned above can be roughly divided into two categories: model-based control and model-free control [27]. Model-free control methods such as PID are more widely used in industry [28], while model-based control has received more attention in academia [29]. Both methods have their own advantages and disadvantages, so some studies attempt to combine the strengths of both to obtain better control methods. Wu et al. [30] proposed a control framework based on error prediction, which no longer pursues more accurate dynamic models or characteristic curves, but actively introduces deep learning methods and uses various neural networks to achieve end-to-end control instruction generation. This type of method imitates the idea of PID control [19,24], taking the error between sensors and expected values as the input parameters of the neural network, attempting to teach the neural network human control strategies and habits through manual motion control data [31], which is called imitation learning [32,33]. There is currently no research team in the EHA field that systematically studies end-to-end control based on neural networks [34].

By optimizing the model structure and parameters continuously, as well as combining more data features and prior knowledge, LSTM’s prediction accuracy in various time series prediction tasks has been improved by researchers [35]. In comparative experiments with traditional prediction methods such as ARIMA and other deep learning models such as the RNN [36], LSTM often achieves lower evaluation metrics such as mean square error and mean absolute error, and more accurately predicts the future trends of time series. The robustness of LSTM gradually increases in the face of noisy data [37], missing data, abnormal data, and other situations. For example, by adding the reconstruction mechanism of the Variational Autoencoder [38], it can better handle incomplete or inaccurate data, maintain relatively stable predictive performance, enhance the practicality and reliability of the model, and promote its promotion and application in practical applications. As a powerful generative model, the Variational Autoencoder(VAE) has significant advantages in feature extraction [39]. It can not only compress the input signal into latent space but also model the distribution of latent features. Compared to traditional autoencoders, the variational autoencoder can better capture the complex structure inherent in motor-related signals [40].

LSTM is a special type of recurrent neural network that is particularly adept at processing time series data [40,41]. It can selectively remember and forget information through gating mechanisms [42], thereby effectively integrating and utilizing historical information in the signal. To sum up, this paper proposes an error prediction and control method based on LSTM. The framework includes an improved time series prediction network and an error prediction PID control strategy. Through the operation data of the device, the neural network can learn the true response curve of the system, predict the precise value of the error convergence process after inputting control instructions, and use the controller with an estimated correction format. The differential stage replaces the current error with the predicted error at the future time and generates the control instruction for the next time, thus achieving the combination of deep learning and traditional control methods. The main contributions of this article are as follows:

• An LSTM-based prediction structure is proposed, which extracts signals of different components separately as inputs and uses the variational autoencoder to mine the best potential predictable features, thereby improving the prediction accuracy of error sequences with noise.
• A modified PID method based on error prediction correction is proposed, which predicts the time series of errors in the differential stage and replaces the current potential error with future potential errors, thereby improving the adaptive control of the controller.
• By establishing a system simulation model of a real pump-controlled symmetrical hydraulic cylinder system, a control error sequence dataset was generated, and the robustness and speed of the designed controller were verified under different control modes.

The subsequent chapters of this article are arranged as follows: Section 2 introduces the proposed control method, Section 3 introduces various experiments and their results, Section 4 discusses the characteristics of the experimental results, and the fifth section summarizes all the work and proposes future improvement directions.

2. Materials and Methods

This section will introduce the basic principles of the EHA system and propose a new error prediction algorithm, which will ultimately form an improved PID algorithm. Including the study of the structure and physical examples of the hydraulic steering system for port AGVs, related transfer function models, and necessary physical derivation processes. In addition, this section also introduces the structure and working principle of a neural network, including necessary signal processing steps and equation expressions for related mathematical tools.

2.1. System Model of EHA

The volume servo-integrated electro-hydraulic actuator of this study is applied to the steering system. The steering system works in real-time during vehicle operation. In the straight holding state, the upper computer sets the steering signal as a constant value, and the steering system is in a position holding condition. The servo motor drives the hydraulic pump to overcome the bidirectional fluctuation of the steering cylinder force caused by hydraulic pump leakage and external load changes, maintaining the cylinder position unchanged. In the turning state, the upper computer provides a steering signal based on the target, and the steering system is in a loaded motion condition. The servo motor drives the hydraulic pump to provide power to the hydraulic cylinder. At the same time, the servo motor and the angle sensor closed-loop real-time control of the angle output and input command synchronization, ensuring real-time synchronization of the angles of different wheels. The steering system consists of a symmetrical hydraulic cylinder and a variable hydraulic pump, without any hydraulic valves involved in control, so it can be simplified into a linear system.

The AGV steering hydraulic system is shown in Figure 1, which is jointly driven by two hydraulic cylinders to rotate the steering frame of the AGV, thereby turning the direction of the wheels. The traditional steering system controls the displacement of the hydraulic cylinder through components such as balance valves, while the EHA system directly drives the piston displacement through a motor-driven gear pump. The model parameter identification process of the pump control system is simpler than that of the valve control system because the essence of EHA is a pipeline with a constant opening, a gear pump with a constant displacement, and a servo motor. The pressure loss and resistance along the way of the entire fluid system are relatively stable, and there is no need to consider the pressure and flow effects caused by the complex flow path of the valve core when modeling.

Establishing a mathematical model of the servo motor using the d-q axis control method.

The torque equation of the servo motor is as follows:

$$\begin{array}{c}{T}_m=1.5p_n{\psi }_{f}i=k_{t}i\end{array}$$

(1)

The motion equation of the servo motor is as follows:

$$\begin{array}{c}{T}_m=J_m{\displaystyle \frac{d{\omega }_m}{dt}}+D_m{\omega }_m+T_L\end{array}$$

(2)

The voltage equation of the servo motor is as follows:

$$\begin{array}{c}{U}_s=L_s{\displaystyle \frac{di}{dt}}+R_{s}i+k_e{\omega }_m\end{array}$$

(3)

The variables in the Equations (1)–(3) are motor related parameters, which describe the relationship between the output torque of the motor shaft and various parameters from the perspectives of circuit theory and dynamics. Variable $T_m$ is the motor torque, variable $p_n$ is the number of motor poles, variable ${\psi }_f$ is the motor magnetic flux, variable $i$ is the phase current, variable $k_t$ is the torque coefficient, variable $J_m$ is the rotor moment of inertia, and variable ${\omega }_m$ is the motor rotor angular velocity. Variable $D_m$ is the viscous friction coefficient of the motor, variable $T_L$ is the load torque, variable $U_s$ is the motor voltage, variable $L_s$ is the winding inductance, variable $R_s$ is the winding resistance, and variable $k_e$ is the back electromotive force coefficient. The rated frequency of the servo motor is 200 Hz, which is much higher than the natural frequency of the hydraulic system and actuator. In the modeling of the control system, the model of the servo motor can be simplified as an inertia link.

$$\begin{array}{c}{\displaystyle \frac{\omega \left(S\right)}{U\left(S\right)}}={\displaystyle \frac{k_v}{T_s+1}}\end{array}$$

(4)

Equations (1)–(4) introduce the important parameters and equivalent models of the motor. In the EHA system, the pump driven by the motor rotate directly, driving the hydraulic cylinder to move, avoiding the modeling complexity caused by valve systems.

For the hydraulic-related parts, it is necessary to select important parameters and establish a system model according to the steps of fluid mechanics. The flow continuity equation of a quantitative pump is as follows:

$$\begin{array}{c}{q}_p=D_p{\omega }_p-C_{ep}{p}_L\end{array}$$

(5)

The flow continuity equation of a hydraulic cylinder is as follows:

$$\begin{array}{c}{q}_L=A_p{\displaystyle \frac{d{x}_p}{dt}}+{\displaystyle \frac{V_t}{4{\beta }_e}}{\displaystyle \frac{d{p}_L}{dt}}+C_{ic}{P}_L\end{array}$$

(6)

The force balance equation of a hydraulic cylinder is as follows:

$$\begin{array}{c}{A}_p{p}_L=m_L{\displaystyle \frac{d^2{x}_p}{d{t}^2}}+B_m{\displaystyle \frac{d{x}_p}{dt}}+K{x}_p+F_L\end{array}$$

(7)

The feedforward compensation for leakage inside the hydraulic pump is fitted using the measured volumetric efficiency curve of the hydraulic pump, and the fitting polynomial is as follows:

$$\begin{array}{c}{\eta }_p=-7\times {10}^{-5}{p}_L^2-5\times {10}^{-4}{p}_L+0.997\end{array}$$

(8)

The variables in Equations (5)–(8) are the parameters of the hydraulic pump and hydraulic cylinder. The following variables are related parameters of the pump: variable $q_p$ is the output flow rate of the pump, variable $D_p$ is the displacement, variable ${\omega }_p$ is the angular velocity, variable $C_{ep}$ is the leakage coefficient, variable $p_L$ is the load pressure, and variable ${\eta }_p$ is the volumetric efficiency. The following variables are parameters of the hydraulic cylinder: variable $q_L$ is the flow rate entering the hydraulic cylinder, variable $A_p$ is the effective working area of the hydraulic piston, variable $x_p$ is the piston displacement, $V_t$ is the total volume of the hydraulic cylinder, variable ${\beta }_e$ is the elastic modulus of the oil, variable $C_{ic}$ is the internal leakage coefficient, variable $m_L$ is the load mass, variable $B_m$ is the viscous damping coefficient, variable $K$ is the elastic coefficient of the load, and variable $F_L$ is the external force acting on the hydraulic cylinder.

The system diagram in Figure 2 can be obtained through above relationships. According to the Equations (5)–(8), the block diagram of the pump controlled hydraulic system is obtained as follows.

The control objective of the integrated electro-hydraulic actuator is the steering angle, while also considering the system’s flexibility and low energy consumption characteristics. Using position feedback as the outer loop ensures the achievement of the final angle control target. Using speed feedback as the inner loop improves the static stiffness of the system and ensures that the influence of nonlinear factors in the system is minimized. Adopting hydraulic pump internal leakage feedforward compensation to ensure the improvement of system response speed and the relative stability of servo motor speed. Based on the above analysis, the block diagram of the volume servo-integrated electro-hydraulic actuator system is obtained in Figure 3.

The derivation details of the control system involve some trade secrets of product designers. This article does not discuss them in detail, but directly uses the system parameters provided by them to establish a digital simulation model to support subsequent discussions.

2.2. Error Prediction Framework

This section will provide a detailed introduction to the steps and overall process of error prediction. Through a neural network and necessary data cleaning steps, the network can accurately extract the long-term trends and short-term fluctuations of noisy position error sequences. In addition, it also includes necessary mathematical representations such as loss function, activation function, and neuron information transfer equation.

2.2.1. Architecture Fundamentals

This article uses an improved LSTM prediction framework, which includes two parts: a variational autoencoder and an LSTM. The encoder maps the filtered signal to a high-dimensional latent vector space, which is then passed to the LSTM for prediction. Finally, the predicted feature sequence is restored to the target space through a decoder and then passed through a delay filter to become an error prediction value that matches the system time. Before predicting time series, noise needs to be processed, and high-frequency and low-frequency components need to be predicted separately to avoid interference with each other. The signal preprocessing module separates the noise components and multi-band components in the signal through DFT and IDFT and sends them to the autoencoder separately.

Let the input series be variable $x=\left[x_1,x_2,\dots ,x_T\right]$, where variable $T$ represents the length of the series. After the Discrete Fourier Transform, a frequency domain series variable $X=\left[X_1,X_2,\dots ,X_T\right]$ of length variable $N$ is obtained:

$$\begin{array}{c}X\left(k\right)=DFT\left(x\left[n\right]\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=0}^{N-1}}x\left(n\right)e^{-j\frac{2\pi }{N}kn},n=0,1,2,\dots ,N-1\end{array}$$

(9)

And the Inverse Discrete Fourier Transform (IDFT) is as follows:

$$\begin{array}{c}x\left(n\right)=IDFT\left(X\left(k\right)\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=0}^{N-1}}X\left(k\right)e^{j\frac{2\pi }{N}kn},n=0,1,2,\dots ,N-1\end{array}$$

(10)

The filter converts the signal into a spectrum through DFT, divides the transformed spectrum into three sub bands F1, F2, F3 according to high, medium, and low frequencies, then restores them into a time series through IDFT. These sequences are handed over to LSTM for prediction.

2.2.2. Error Prediction Based on LSTM

Although LSTM can predict time series, its robustness is often limited by the form of the sequence itself. It is a common method to decompose the signal into various simple subspaces in the preprocessing stage and predict them separately. This article uses VAE as the feature extractor for error sequences, maps the error sequences to subspaces through adversarial learning of the encoder and decoder, and inputs them to subsequent stages for prediction.

The Variational Autoencoder’s encoder is to map the input frequency domain series $x$ to the latent space. The output of the encoder is the mean $\mu$ and the variance ${\sigma }^2$ of the latent variable $z$.

$$\begin{array}{c}\mu =E_{\theta }\left(x\right)\end{array}$$

(11)

$$\begin{array}{c}{\sigma }^2=L_{\theta }\left(x\right)\end{array}$$

(12)

where $E_{\theta }(·)$ and $L_{\theta }\left(X\right)$ represent the functions of the encoder networks for calculating the mean and variance. To sample the latent variable $z$ from the latent distribution, the reparameterization trick is used as Equation (13).

$$\begin{array}{c}z=\mu +\sigma ϵ\end{array}$$

(13)

where $ϵ~\mathcal{N}(0,I)$ is a random vector sampled from the standard-normal-distribution. The latent variable $z$ output by the encoder is segmented into M new subsequences $z^{\left(1\right)},z^{\left(2\right)},z^{\left(3\right)},\dots ,z^{\left(M\right)}$, and the length of each subsequence is variable $d=\frac{\mathrm{d}\mathrm{i}\mathrm{m} \left(z\right)}{M}$. For the Long Short Term Memory (LSTM) network (with parameters ${\theta }_{lstm}^i$), the input is variable $z^{\left(i\right)}$. The update formulas for the hidden state of the LSTM are as follows:

• Input gate.

  $$\begin{array}{c}{i}_t=\sigma \left(W_i·\left[h_{t-1},x_t\right]+b_i\right)\end{array}$$

  (14)
• Forget gate.

  $$\begin{array}{c}{f}_t=\sigma \left(W_f·\left[h_{t-1},x_t\right]+b_f\right)\end{array}$$

  (15)
• Cell state update.

  $$\begin{array}{c}{\widehat{C}}_t=tanh\left(W_C·\left[h_{t-1},x_t\right]+b_C\right)\end{array}$$

  (16)

  $$\begin{array}{c}{C}_t=f_{t-1}·C_{t-1}+i_t·{\widehat{C}}_t\end{array}$$

  (17)
• Output gate.

$$\begin{array}{c}{o}_t=\sigma \left(W_o·\left[h_{t-1},x_t\right]+b_o\right)\end{array}$$

(18)

$$\begin{array}{c}{h}_t=o_t\tanh ⨀C_t\end{array}$$

(19)

where variable $t$ represents the time step, variable $\sigma$ is the sigmoid function, $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$ is the hyperbolic tangent function, $⨀$ represents element—wise multiplication, variable $\mathrm{W}$ is the weight matrix, and variable $b$ is the bias vector.

The output sequence of each LSTM is denoted as variable $h=h_1^i,h_2^i,h_3^i,\dots ,h_T^i$, where variable $T$ is the number of predicted time steps. The subsequences $h=h_1^i,h_2^i,h_3^i,\dots ,h_T^i$ predicted by multiple LSTMs are merged into a new sequence variable $H$, which can be simply concatenated in order. Then the merged sequence $H$ is input into the decoder with parameters $\theta$, the output of the decoder is the predicted main sequence Y.

$$\begin{array}{c}\mathrm{Y}=D_{\theta }\left(H\right)\end{array}$$

(20)

where variable $D_{\theta }$ represents the function of the decoder network. Finally, the output of the decoder is used to calculate the reconstruction loss function. If the true value at the predicted time is y and the predicted value is variable ${\widehat{y}}_i$, then the loss function is as Equation (21).

$$\begin{array}{c}MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\left(y_i-{\widehat{y}}_i\right)}^2\end{array}$$

(21)

Figure 4 illustrates the entire process of this error prediction network. The three sets of signals separated by a filter are passed through an encoder, LSTM, and decoder, and finally processed by a post-processing module to become an analyzable error prediction sequence that matches the standard time.

2.2.3. PID Control Based on LSTM

As shown in Figure 5, the PID control strategy based on time series includes several key steps: error signal generation, feedback mechanism, and adaptive control strategy. Through the Filter-LSTM framework, the control deviation of the hydraulic system can be sampled to obtain historical data before error prediction and control can begin. However, actual hydraulic systems have different allowable delay requirements. The hydraulic systems of aircraft and robotic arms allow for relatively long control delays, and more historical data points can be collected as inputs. Equipment such as automotive and port positioning AGVs that require positioning and navigation require faster response speeds and fewer historical data samples. Therefore, although the ideas proposed in this article are relatively direct, some strategies are still needed when applied to control objects in practice.

If the set value of input is $r\left(t\right)$ and the output value is $y\left(t\right)$. The error is as follows:

$$\begin{array}{c}e\left(t\right)=r\left(t\right)-y\left(t\right)\end{array}$$

(22)

For a hydraulic system, if the allowed delay time is $\Delta t$, the sampling frequency is variable $F$, then the number of sampling points that trigger time series prediction is as follows.

$$\begin{array}{c}N=\Delta tF\times m\%\end{array}$$

(23)

When predicting, the FFT module will perform N-point Fourier transform accordingly. Then the controller performs the following three steps.

(a) Error signal generation: Based on the predicted error value, combined with the control objectives and actual operating status of the motor, generate corresponding control signals. If the input N-point error sequence is E, then the expression for the predicted error sequence $\widehat{e}\left(t\right)$ is as follows.

$$\begin{array}{ccc}\hfill {\mathrm{U}}_1& =& \mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{F}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\left({\mathrm{E}}_N\right)\hfill \\ \hfill {\mathrm{U}}_2& =& \mathrm{E}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{r}\left({\mathrm{U}}_1\right)\hfill \\ \hfill {\mathrm{U}}_3& =& \mathrm{L}\mathrm{S}\mathrm{T}\mathrm{M}\left({\mathrm{U}}_2\right)\hfill \\ \hfill {\mathrm{U}}_4& =& \mathrm{D}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{r}\left({\mathrm{U}}_3\right)\hfill \\ \hfill \widehat{e}\left(t\right)& =& \mathrm{T}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{D}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}\left({\mathrm{U}}_4\right)\hfill \end{array}$$

(24)

(b) Feedback mechanism: Establish a feedback control mechanism to feedback the actual operating state information of the motor back to the VAE-LSTM model. Adjust and optimize the model by comparing the differences between the actual state, predicted state, and control objectives.

$$\begin{array}{c}u\left(k\right)=K_p\left[e\left(k\right)+{\displaystyle \frac{T}{T_i}}{\displaystyle \sum _{j=0}^k}e\left(j\right)+T_d{\displaystyle \frac{\widehat{e}\left(k+D\right)-\widehat{e}\left(k+D-1\right)}{T}}\right]\end{array}$$

(25)

Among them, variable $u\left(k\right)$ is the control variable at the $k$-th sampling time, variable $e\left(k\right)$ is the error at the $k$-th sampling time, and variable T is the sampling period, $D$ is the prediction step size.

(c) Adaptive strategy: The first two steps each have a sensitive parameter related to the controlled object. Based on the type of hydraulic system, the reasonable selection of the predicted starting parameter m and the predicted step size variable $D$ is an important indicator of the balance control effect and error prediction accuracy.

Our goal is to achieve optimal performance of the control system while considering the balance between prediction time overhead and control effectiveness, as well as the balance between prediction accuracy and remaining control time. A common performance metric is the Integral of Time Multiple Absolute Error (ITAE). Based on the predicted time cost C ($D$) (assuming it is a monotonically increasing function with respect to $D$), construct the following objective function:

$$\begin{array}{c}J\left(D,m\right)={\displaystyle {\int }_{m+1}^{m+D}}t\left|\widehat{e}\left(t\right)\right|dt+\alpha C\left(D\right)\end{array}$$

(26)

Among them, variable $\alpha$ is a weight coefficient used to balance control error and prediction time overhead. The predicted start time needs to meet the following requirements:

$$\begin{array}{c}m>m_{min}\end{array}$$

(27)

Variable $m_{min}$ is the minimum historical data length required for prediction, the prediction step size needs to meet the following requirements:

$$\begin{array}{c}D>1 and D+m<T\end{array}$$

(28)

Organized as follows:

$$\begin{array}{c}\underset{D,m}{\min }J\left(D,m\right)={\displaystyle {\int }_{m+1}^{m+D}}t\left|\widehat{e}\left(t\right)\right|dt+\alpha C\left(D\right)\end{array}$$

(29)

$$\begin{array}{c}\begin{array}{cc}s.t.& \begin{array}{c}m>m_{min}\\ D>1\\ D+m<T\end{array}\end{array}\end{array}$$

(30)

Considering that most controls require multiple time scales to complete, this paper adopts a relaxation iteration strategy as a dynamic control method for sensitive parameters.

Equation (29) is a typical two-dimensional optimization problem. Although the objective function is a complex expression, its calculation results are only related to $m$ and $D$. Considering $m$ and $D$ as the coordinates of a two-dimensional plane, $J\left(D,m\right)$ is a two-dimensional distribution, the parameter update formula based on relaxation iteration strategy can be written in the following form:

$$\begin{array}{cc}\hfill g_m& ={\displaystyle \frac{J\left(D,m+ϵ\right)-J\left(D,m\right)}{ϵ}}\hfill \\ \hfill g_D& ={\displaystyle \frac{J\left(D+ϵ,m\right)-J\left(D,m\right)}{ϵ}}\hfill \\ \hfill m_t& =m_{t-1}-{\beta }_1·g_{m,t}+{\beta }_2·\left(m_{t-1}-m_{t-2}\right)\hfill \\ \hfill D_t& =D_{t-1}-{\beta }_1·g_{D,t}+{\beta }_2·\left(D_{t-1}-D_{t-2}\right)\hfill \end{array}$$

(31)

Among them, ${\beta }_1$ is the learning rate, and ${\beta }_2$ is the relaxation factor between 0 and 1. Using a simulation system, after testing the corresponding $J$ under different parameter combinations, the optimal parameters for the next step can be inferred through Formula (31). Repeat this iterative process until the objective function remains unchanged, and then stop the loop. We will discuss in detail the impact of these two parameters on the control effect in the future.

3. Experiment

As shown in Figure 6, port AGVs are large heavy-duty equipment that operate in outdoor scenarios. They are usually used to transport large cargo such as containers.

Due to the influence of ground and outdoor weather, AGV computers can only use highly reliable industrial computer systems, and existing equipment usually does not have Nvidia’s GPU computing conditions. When the steering system starts working, the real-time displacement of the hydraulic cylinder is sent to the upper computer through the readable register of the PLC, and the difference between it and the commanded displacement is the error sequence to be predicted in this article. At the end of the experiment, two sets of position error curves corresponding to sine input were collected from a real AGV platform.

Figure 7 records the position error curve of the hydraulic steering system during testing, and under the condition of a sine control signal, the position error curve also exhibits a regular sine pattern. More real data also confirms that there is a regular pattern of changes in the response error of the system, which will vary with the input instructions.

There is one thing that needs to be declared, due to the commercial secrets involved in the operation data of port AGVs, this article did not conduct further testing on the AGV engineering platform. In order to further test the performance data of the proposed algorithm in detail, simulation data will be used for verification in subsequent stages. Through a variable parameter simulation system, the performance of the algorithm can be comprehensively and flexibly tested.

3.1. Experiment Setup

This article tested the proposed control method through a series of simulation experiments. Referring to an AGV steering hydraulic system, we built a system simulation environment. Different control objectives and parameters were set up in the experiment to analyze in detail the advantages and disadvantages of the proposed method.

The simulator, as shown in Figure 8, reads the input and output waveform data through two oscilloscopes and uses them for training the error prediction model. Considering that the control process of AGV on-site may involve different durations, the experiment randomly inputted standard waveform models containing different frequencies of random noise into the simulator, including sine wave signals, square wave signals, triangular wave signals, and polynomial signals of different frequencies. By analyzing the real sample data of AGV, the frequency of the position error sequence returned by PLC is usually below 100 Hz. A total of 1600 sets of data with different waveforms and durations were generated in the experiment. The shortest running time of the simulation was 1 s, and the longest was 25 s, of which three-quarters were used as the training set and the rest as the testing set.

The neural network used for the experiment consists of three LSTM modules and one VAE module. The VAE module consists of six fully connected layers and one probability distribution layer. The number of neurons in the fully connected layers is 128, 64, 32, 32, 64, and 128, respectively. The activation function is ReLU. The probability distribution layer contains 64 neurons, but the activation function is ReLU multiplied by a Gaussian distribution. After trying different combinations, the experiment found that the optimal learning rate during training is 0.001, drop out is 0.1, batch size is set to 64, and other training hyperparameters have little impact on the training process. The default values of PyTorch-1.4.1 can be used.

The signal waveform is observed through a simulation oscilloscope, and the input of control instructions is simulated using various given excitation sources. The

Table 1. Parameters of EHA and simulation.

	Variable	Value	Unit
Physical parameters of hydraulic steering system	hydraulic cylinder	80/45–264	mm
	Max speed	90	mm/s
	Cylinder number	2	-
	pull	14	ton
	Servo power	10	kW
	pump displacement	16	cm3/r
	Working presure	18	MPa
	rated speed	2500	r/min
	Hydraulic oil tank capacity	10	L
State feedback parameters of EHA Simulation	Dp	0.6	-
	K	0.364	-
	Kff	2.169	-
	kp	1.038	-

shows the important parameters’ values. Through the simulator shown in Figure 5, we collected the controlled curves under different excitations and input them into the neural network to train the response sequence prediction function, which can directly output the predicted values based on the starting mode of the sequence.

Figure 8 shows the performance of the target EHA system under normal PID, the Figure 8a shows the control effect of a conventional PID controller on sine input. It can be observed that there is always a fixed phase difference in the system output, which is manifested as a response delay during AGV steering. Under normal circumstances, the response delay of using ordinary PID control is within 500 ms, and there are certain differences among different input commands

As the stability and controllability of the system have been validated, even if there are slight differences between the system parameters and the actual system, convergence can still be achieved within an appropriate time step. Subsequent experiments will be conducted under different control conditions to evaluate the proposed method from three perspectives: stability, speed, and accuracy.

3.2. Simple Control Experiment

Table 2. Adjustment performance after introducing error prediction under different control instructions.

Source Mode	Parameter	Overshoot-(%)	Rise Time (ms)	Steady-State Error (%)
Step	Without prediction	5.6	95.6	<0.3
	$m$ = $50, D$ = 1	2.37	55.7
	$\mathrm{Best} m, D$	2.69	49.6
Sin	Without prediction	5.19	53.8	<0.3
	$m$ = $50, D$ = 1	1.99	29.4
	$\mathrm{Best} m, D$	2.31	21.1
triangle	Without prediction	5.5	44.9	<0.3
	$m$ = $50, D$ = 1	1.88	26.5
	$\mathrm{Best} m, D$	1.91	17.2
normal distribution	Without prediction	4.31	199	<0.3
	$m$ = $50, D$ = 1	1.33	59.3
	$\mathrm{Best} m, D$	1.6	57.4

shows the differences in using the proposed error prediction method under several common control modes. In simple control tasks, the improved method of introducing deep learning to predict response curves has a significant positive impact on PID.

The first row of each column in the table represents the control indicators without error prediction, the second row represents the control indicators corresponding to the default values of 50 and 1 for parameters $m$ and $D$, and the third row represents the control indicators corresponding to the optimal parameters $m$ and $D$. The third line represents the optimal experimental scenario with default parameter settings. The final steady-state errors are all less than 0.3%, so the proposed method can be considered stable. From the experimental results, it can be seen that the LSTM overshoot and rise time are the highest in each group, indicating that the control effect without error prediction is relatively poor. The advantage of the optimal ($m$, $D$) over the default selection ($m$ = 50, $D$ = 1) is mainly reflected in the rise time. The improvement effect brought by parameter optimization varies under different inputs. The improvement effect of step signals, sine signals, and triangular pulse signals is more obvious, while the improvement effect of normal distribution signals is smaller.

3.3. Error Series Prediction Experiment

Figure 9 shows the error prediction performance under different parameters m, where the red represents the prediction error and the blue represents the actual error. Since the later the start time, the more comprehensive the historical information is, resulting in a significant improvement in the error prediction accuracy from subgraph a to d. The PID controller distinguishes whether the input error comes from the current value or the predicted value through a reducer during operation, ensuring a smooth and effective control process.

3.4. Real Condition Control Experiment

At the end of the experiment, the working data of the real system was used for verification. With the help of this data, the simulation model and the actual system are very close. By inputting the same given amount, the same response curve can be obtained. Based on this, the implementation form of the PID module was modified and tested, and the results shown in Figure 10 were obtained. Changing the starting point of the prediction will significantly affect the effectiveness of the response curve. The larger the prediction step size, the greater the difference between the response curve and the original prediction curve, and the smaller the jitter, indicating that the error prediction step reduces the number of system adjustments.

Figure 11 compares the controlled response curves corresponding to different parameters $D$. For the input standard step signal of the simulation system, the prediction strategy is more aggressive when $D$ = 20, so the convergence speed is faster. When $D$ = 1, the controller almost does not introduce an error prediction link, relying only on the controller’s own adjustment performance, thus requiring more time to adjust.

Table 3. Efficiency improvement brought by introducing error prediction process under different task durations.

Task Time Length/ms	Response Time Without EP/ms	Response Time With EP/ms
1000	1540	1260
2000	2694	2406
3000	3718	3560
5000	5956	5465
7000	8109	7506
9000	10,069	9206
12,000	13,697	12,474
15,000	16,995	15,819
18,000	20,374	18,771
22,000	24,695	22,695
26,000	28,794	26,469
30,000	33,116	30,348

EP is the error prediction module.

shows the relationship between response time and input instruction time under different conditions. In traditional PID control, the absence of an error prediction loop increases the control delay with the increase of input instruction time. When the task time reaches 30 s, the response delay reaches 3116 milliseconds, while the response delay after introducing the error prediction loop is only 348 milliseconds.

Moreover, as the input instruction becomes longer, the response delay time of the proposed method shows a trend of first increasing and then decreasing, proving that error prediction has outstanding performance advantages in long-period control tasks.

3.5. More Detailed Performance Testing

The experiment further validated the designed method on a system containing directional valves. Figure 12 shows the valve-controlled dual hydraulic cylinder and EHA dual hydraulic cylinder, respectively.

The difference between the two mainly lies in the different flow control media. The valve control system controls the opening degree of the servo valve by adjusting the valve core position, which in turn affects the flow rate. When hydraulic oil passes through a small hole, there is a pressure loss, which affects the flow rate. Therefore, the flow rate and opening degree of the valve have a nonlinear relationship. When modeling, the mapping relationship between the opening degree and flow rate at the rated state can only be obtained through the characteristic curve provided by the manufacturer. Once the pressure and temperature deviate from the rated working area, it will lead to the inability to determine the appropriate valve core displacement degree based on the flow rate. The experiment established the “spool displacement flow” characteristic curve of a virtual valve in a simulation environment and set the characteristic curve to deviate linearly with pressure to simulate the coupling effect of the working conditions of a real valve control system.

The experiment inputted first-order, second-order, and third-order signals into the valve control system and EHA system respectively, and tested their response using LSTM-PID and ordinary PID. Since all control results eventually reached stability, the table mainly records the percentage deviation of the time parameters of LSTM-PID and common PID during the control process, which is used to measure the improvement effect of the proposed method on different systems. The relevant results are summarized in

Table 4. Control improvement effects on valve system and pump system for polynomial input.

	Valve System			Pump System
	First Order	Second Order	Third Order	First Order	Second Order	Third Order
Rise Time Improvement	14.36%	1.69%	1.02%	13.95%	16.68%	19.61%
Settling Time Improvement	16.78%	0.05%	0.66%	14.18%	16.01%	18.73%

, the more the response time decreases, the larger the percentage indicator will be.

From the results, it can be seen that the valve control system only achieved control effects comparable to the EHA system under first-order input. In addition, the more the response time is shortened, the higher the percentage of experimental indicators. The time improvement indicators in the second and third columns are only about 1%, indicating that the response time of LSTM-PID is only slightly shorter than PID.

The experiment further compared the advanced control methods commonly used in LSTM-PID and EHA, including model predictive control, sliding mode control, and adaptive control, and applied them to the designed simulation model for testing. The relevant results are recorded in

Table 5. Performance of different control methods, input signal’s time length is 1000 ms.

	LSTM-PID	Model Predictive Control [43]	Sliding Mode Control [11]	Adaptive Control [28]
Settling Time/ms	1261	1755	1613	1309
steady-state error	0.376%	0.422%	0.632%	0.299%
overshoot	16%	29%	43%	21%

.

The results show that the proposed method has advantages in improving response speed, with the shortest steady-state time among all methods. The steady-state errors of the four methods have no significant difference. However, the overshoot of the adaptive control is smaller than that of the proposed method.

Finally, in order to analyze the impact of VAE and filters on model performance, we designed ablation experiments and removed these two components for control variable experiments. The experiment generated 400 sets of error sequences using signals randomly input to the simulator and used prediction models under different conditions for prediction. The average prediction error of each test condition was statistically analyzed, as shown in

Table 6. Ablation experiment.

Position Prediction Error	Without VAE	Without Filter	Without Both	Without None
RMSE/mm	11.5	8.75	21.53	4.30
MAE/mm	10.23	7.91	19.06	4.22
MAPE/%	4.79	3.07	2.65	2.16

, including root mean square error (RMSE), mean absolute error (MAE), and average absolute percentage error (MAPE). The relevant results are recorded in Table 6.

The experimental results show that removing VAE has the greatest impact while removing the frequency filter has a relatively small impact. If these two modules are not used to process the input sequence and error prediction is performed directly, the results may be affected by noise interference, resulting in the model being unable to accurately predict the error trend.

4. Discussion

From the above experimental results, it can be seen that adding the error prediction module significantly improves the regulation efficiency of the PID controller, exhibiting better overshoot and rise time in the regular excitation source experiment, and the final convergence accuracy remains consistent with the original PID effect. The results in Table 2 indicate that the control strategy based on the error prediction framework developed in this paper achieves faster lead correction without increasing additional control costs.

Figure 10 and Figure 12 demonstrate the impact of the two sensitive parameters introduced earlier on the control effect. From the experimental results, it can be seen that the starting time of prediction significantly affects the accuracy of error prediction, as it not only fixes the length of historical data read for the first prediction but also limits the historical length of each subsequent prediction. In addition, the prediction step size will significantly affect the control effect. Through actual working condition data, the PID introduced with error prediction shows a smoother response curve, and the larger the prediction step size, the smoother the corresponding response curve. This indicates that error prediction indeed improves the control efficiency of the controller and reduces response time.

5. Conclusions

This article discusses a new method for error prediction and feedforward control of hydraulic systems. Based on the LSTM prediction architecture, filters, and variational autoencoders are added to extract features by different frequency, improving the modeling ability of LSTM and achieving better error prediction results ultimately.

5.1. Advantages and Limitations

The experiment shows that the proposed method improves the controlled response speed and ensures that the final convergence accuracy is consistent with the original controller. The limitation of this article is that it mainly focuses on the research of an AGV steering system composed of EHA and lacks research on a wider range of hydraulic circuits, especially valve-controlled nonlinear hydraulic systems. The related error prediction and feedforward control methods have not been confirmed in high-order nonlinear systems.

5.2. Future Improvement Directions

Therefore, future research in this article will validate and improve the designed predictive control method in more hydraulic systems and design an intelligent controller that combines data-driven and knowledge-driven approaches.