Strip thickness prediction method based on improved border collie optimizing LSTM

Border collie optimization

Inspired by the herding behavior of border collies in daily life, Tulika Dutta et al. proposed Border Collie Optimization (BCO) algorithm (Dutta et al., 2021). Three border collies are randomly initialized: the lead dog, the left guide dog, and the right guide dog. The fitness values are fit_f, fit_le and fit_ri respectively. The rest of the population consists of sheep, and the fitness value is denoted as fit_s. The updates of the velocity of the three guide dogs at time t +1 are shown in Eq. (1): (1)${\displaystyle V_{f,ri,le}\left(t+1\right)=\sqrt{V_{f,ri,le}{\left(t\right)}^2+2\times Ac{c}_{f,ri,le}\left(t\right)\times Po{p}_{f,ri,le}\left(t\right)}}$ where Acc (t) represents the acceleration of the three dogs at moment t, and Pop (t) represents their position at moment t. The update of the velocity of the aggregated sheep is shown in Eq. (2): (2)${\displaystyle V_{sg}\left(t+1\right)=\sqrt{V_f{\left(t+1\right)}^2+2\times Ac{c}_f\left(t\right)\times Po{p}_{sg}\left(t\right)}.}$

Three guide dogs control the global search of the entire algorithm. They move in different directions and are independent of each other. They can quickly find regions in the large search space where optimal solutions are likely to exist. The movement of the flocks are influenced by the three guide dogs. Also, the flocks can focus on local searches in the space, and strive to find a better position. The updates of positions of the three border collies and the flock are shown in Eqs. (3) and (4). ${\displaystyle Po{p}_{f,ri,le}\left(t+1\right)=V_{f,ri,le}\left(t+1\right)\times Tim{e}_{f,ri,le}\left(t+1\right)}$ (3)${\displaystyle +\frac{1}{2}Ac{c}_{f,ri,le}\left(t+1\right)\times Tim{e}_{f,ri,le}{\left(t+1\right)}^2}$ ${\displaystyle Po{p}_{sg}\left(t+1\right)=V_{sg}\left(t+1\right)\times Tim{e}_{sg}\left(t+1\right)}$ (4)${\displaystyle +\frac{1}{2}Ac{c}_{sg}\left(t+1\right)\times Tim{e}_{sg}{\left(t+1\right)}^2.}$

LSTM

LSTM optimizes the hidden layer structure based on memory information like the recurrent neural network, and introduces a “gate” structure into the hidden layer neurons, namely input gate, forget gate and output gate, which control the update of historical data (Zhai et al., 2021).

Forget gate: The forget gate is responsible for selectively forgetting the state information transmitted in the previous moment, namely, forgetting the redundant information. The calculation process of the forget gate is shown in Eq. (5): (5)${\displaystyle f_t=\sigma \left(W_f\cdot \left[h_{t-1},x_t\right]+b_f\right)}$ where f_t is the output of the forget gate. h_t−1 is the hidden state at moment t-1. x_t is the input at moment t. W_f represents the weight matrix of the forget gate. b_f denotes the bias of the forget gate. σ represents the sigmoid activation function.

Input gate: The input gate determines the extent to which the current input information x_t is stored in the long-term state C_t and it controls which new information is added to the unit state C_t. The calculation process of the input gate is shown in Eqs. (6) to (8): (6)${\displaystyle i_t=\sigma \left(W_i\cdot \left[h_{t-1},x_t\right]+b_i\right)}$ (7)${\displaystyle {\bar{C}}_t=tanh\left(W_c\cdot \left[h_{t-1},x_t\right]+b_c\right)}$ (8)${\displaystyle C_t=f_t\times C_{t-1}+i_t\times {\bar{C}}_t}$ where i_t represents the output of the sigmoid activation function in the input gate. ${\bar{C}}_t$ denotes the candidate input. C_t is the unit state at time t.

Output gate: The output gate is responsible for determining which values the memory unit outputs at the current moment, namely, calculating the output based on the unit state. The calculation process of the output gate is shown in Eqs. (9) and (10): (9)${\displaystyle o_t=\sigma \left(W_o\cdot \left[h_{t-1},x_t\right]+b_o\right)}$ (10)${\displaystyle h_t=o_t\times tanh\left(C_t\right)}$ where o_t is the output gate; h_t represents the hidden state of the memory unit at moment t.

The proposed method

LSTM optimized by IBCO

The important parameters of LSTM, namely the number of neurons of hidden layer and the learning rate, are difficult to determine and are often based on personal experience, which is random and can cause the prediction performance of the model to be very unstable. Therefore, IBCO-LSTM strip thickness prediction model is proposed, and the parameters of LSTM are optimized by using the improved BCO algorithm. The overall flow chart of IBCO-LSTM strip thickness prediction model is shown in Fig. 2.

The optimal individual position (the position of the leading dog) in the algorithm is taken as the number and learning rate of neurons in the hidden layers of LSTM to establish the optimal prediction model. The optimization process of LSTM parameters by IBCO algorithm is as follows:

1. Specify the basic parameters in the optimization process of parameters LSTM, including the size of the algorithm population n, the maximum number of iterations Max_iterations. Limit the neurons number of hidden layer and the search range of learning rate. And randomly initialize the velocity, acceleration and time of individual movements in the population.

2. Initialize the population position of BCO algorithm by Tent chaotic mapping, and the individual positions p ₁, p₂, p₃ are set as the number of neurons and learning rate of each hidden layer of the model.

3. Train the model by tenfold cross validation method, and the root mean square error between the actual thickness and the predicted thickness is used as the fitness function of IBCO.

4. Carry out the iteration to calculate the position of individuals in each iteration and calculate the fitness value. The optimal historical position of individuals is determined by comparison and then the optimal position of the population is determined. When the maximum number of iterations is reached, the optimal individual position (the position of leader dog) is mapped to the number of neurons in the hidden layer of LSTM and the learning rate, and end the optimization process.

Data collection and analysis

In order to validate the performance of the proposed method, diverse experimental studies will be carried out. The experimental data are from the hot continuous rolling and finishing mill of a steel plant of a domestic iron and steel group, which includes nine stands. When collecting data, a thickness gauge is installed in the finishing mill to measure the thickness of the strip outlet. At the same time, sensors are installed on the hydraulic lower devices of nine flat roller stands in the finishing mill to collect the factors of the strip thickness. The strip thickness of the final collected data is numerical data, while the factor data exists in the form of signals, mainly including mill rolling force, strip outlet temperature, mill current, SONY value, rolling speed, and roll gap. Since the signal is complex and cannot be directly utilized as the experimental data, for this reason ibaAnalyzer is used to analyze the data and convert the signal data into numerical type sequences. After the analysis is completed, the multi-dimensional factor sequence and thickness are combined to form the original numerical strip data. Some factors are shown in Figs. 3 to 8.

Rolling feature selection

The original strip data is actually a nonlinear time series, which contains many factors affecting the accuracy of strip thickness, but the influence degree of each factor is not the same, and there may be redundant factors with weak influence. The weak correlation redundancy factors should be eliminated, and the important factors with strong representativeness should be used as the input data characteristics of the model training, which can reduce the complexity of the model, shorten the training time and improve the generalization ability of the model.

Mutual information is usually used as an effective standard to measure the correlation between two random variables. This measurement is not only applicable to linear correlation variables, but also applicable to nonlinear correlation variables (Vergara Jorge & Estévez, 2014). It is a feature selection method widely used in the field of machine learning (Zhang & Wang, 2016). Mutual information can be used to quantify the mutual information value between each factor and thickness. According to the mutual information value, the importance of each factor can be compared, and then the importance of each factor can be ranked. The larger the mutual information value is, the closer the variable relationship is. The definition of mutual information between two random variables X and Y is shown in Eq. (15). (15)${\displaystyle I\left(X,Y\right)=\iint \rho \left(x,y\right)log\frac{\rho \left(x,y\right)}{\rho \left(x\right)\rho \left(y\right)}dxdy}$ where, ρ(x) is the edge probability density function of X, ρ(y) is the edge probability density function of Y. ρ(x,y) represents the joint probability density function between random variables X and Y.

Mutual information is used to realize the feature selection of the original factors. The important factors are selected as the features of the input data set of the model. The specific process of feature selection is as follows:

Step 1: Calculate the mutual information between the factors and strip thickness according to Eq. (15), as shown in Table 1.

Step 2: According to the principle of mutual information feature extraction (Wang et al., 2022; Yu et al., 2022). Take the threshold α = 0.2.

Step 3: The factors with mutual information value I greater than α are selected. The factors selected according to Table 1 include rolling speed, roll gap, rolling current and rolling force.

Performance analysis of IBCO algorithm

In order to verify the superior performance of the improved border collie optimization algorithm, IBCO algorithm, border collie optimization (BCO) algorithm, whale optimization algorithm (WOA), grey wolf optimization (GWO) algorithm and particle swarm optimization (PSO) algorithm are jointly used for independent repeated experiments on six test functions to compare the optimization performance of the algorithm (Yin et al., 2021). In the experiment, the size of population is set to 30, the maximum number of iterations is set to 250, and each function is tested for 30 times independently. The test function expressions are shown in Table 2.

In Table 2, f ₁, f₂ andf₃ are single-peaked test functions, and f₄ is multi-peaked test function. The number of local minimum values of multi-peaked function increases exponentially with the increase of problem dimension, which is the most difficult test function for algorithm optimization. The f₅ and f₆ are the fixed-dimensional multi-peaked test functions, which have only a few local minimums. The optimal values of f₁ to f₅ are 0, and the optimal value of f₆ is 0.0003. The related parameters of each algorithm are initialized: the logarithmic spiral coefficient b of the whale algorithm is set to 1, and the search coefficient a decreases from 2 to 0. Grey Wolf algorithm collaborative coefficient vector c = [0, 2], and convergence factor a = [0, 2]. The maximum speed of the particle swarm algorithm is set to 6, the inertia weight w = (0.2, 0.9), and the learning factor C1 = C2 = 2. The convergence curves of the five algorithms on six benchmark test functions are shown in Figs. 9 to 14.

Figures 9 to 11 are single-peak function convergence curves. It can be seen that the convergence accuracy of IBCO on function f₁ and f₂ is better than that of BCO, and it is also better than other algorithms. Besides, its convergence effect is obvious. IBCO has the fastest speed of convergence and the highest accuracy on function f₃, and it can still perform well when other algorithms fall into stagnation. Therefore, IBCO has the best ability to find the best when solving for the single-peak function. Figures 12 to 14 show the convergence curves of the multi-peak and fixed multi-peak functions. IBCO can jump out of the local optimum and converge to the optimal value on both the multi-peak function f₄ and fixed multi-peak function f₆ with the highest accuracy of the optimization, and the convergence speed on the function f₄ is very fast.

The algorithm can’t converge to the optimal value of the function f₅, but IBCO still has the best performance and the highest convergence accuracy among the algorithms. The optimization effect of each algorithm on function f₅ is not good and does not converge to the optimal value, but IBCO is still the best in many algorithms and has the highest convergence accuracy. Although the convergence rate of IBCO on function f₅ and f₆ is not the fastest, it converges to a better value at the expense of a little convergence rate and it is generally acceptable. In short, IBCO can jump out of local optimum in multi-peak and fixed multi-peak function. In addition, the optimization accuracy is better than other algorithms and the speed is faster. In order to further analyze the performance of IBCO, the optimal value, average value and standard deviation of each algorithm are compared in Table 3. The optimal value and average value measure the optimization accuracy of the algorithm, and the standard deviation is used to measure the robustness and stability of the algorithm. It can be seen from the table that the optimal value, average value and standard deviation of IBCO in functions f₁, f₂ andf₄ are the lowest, indicating that IBCO has the highest optimization accuracy, the strongest stability and robustness. In functions f₃ and f₆, the optimal value of IBCO is the lowest and the optimization accuracy is the highest standard deviation are at the medium level but better than BCO.

Performance analysis of IBCO-LSTM

The prediction performance of IBCO-LSTM, BCO-LSTM, LSTM, BP, SVM and LSSVM applied by scholars in the field of strip thickness prediction are compared and analyzed through experiments. First, the parameters involved in the algorithm are initialized, the population size n = 50, the dimension d = 3, and the maximum number of iterations Max_iterations = 5.

In the experiments, the training data set contains 800 training samples, and the test data set contains 100 test samples. The root mean square error (RMSE) is adopted as an evaluation index to evaluate the prediction accuracy of the six models (Sun et al., 2022). Each model is repeated five times to obtain the average RMSE. The comparison results of different models are shown in Table 4.

According to Table 4, it is obvious that the prediction accuracy of IBCO-LSTM is higher than those of BCO-LSTM, LSTM, BP, SVM and LSSVM in terms of RMSE and IBCO-LSTM possesses the highest prediction accuracy. In order to visualize the superiority of the prediction performance of the proposed model, the error comparison curves between the prediction results of the six models and the actual values are plotted, as shown in Figs. 15 to 17:

According to the standard of GB709-88, the strip with high rolling accuracy between 1.60 mm and 2.00 mm in strip thickness has the thickness allowable deviation of ±0.13 mm. In Fig. 15, it can be directly seen that the error between the predicted thickness and the actual thickness of IBCO-LSTM is within ±0.02 mm, which is much lower than 0.13 mm. Therefore, the proposed IBCO-LSTM prediction model can meet the actual demand in the rolling field, and the prediction effect is good. If this algorithm is applied to the control system, the rolled strip of the control system can reach the qualified standard.

Figure 18 is the comparison diagram of the convergence curve of IBCO-LSTM, BCO-LSTM and WOA-LSTM after 30 iterations. From the diagram, it can be seen that IBCO-LSTM has the highest convergence accuracy, namely the lowest prediction error, which indicates that its prediction effect is the best.