High-order Sliding Mode Tracking Differentiator with Neural Network based Adaptive Parameter Estimation

This paper presents a new high-order sliding mode tracking differentiator with backpropagation neural network based adaptive parameter estimation (SMF-BP) to obtain accurate filtering and differentiation estimates from signals that contain noise. It is improved based on the Levant’s high-order sliding mode tracking differentiator. First, SMF-BP incorporates a sigmoid function to reduce overshoot. Second, the BP neural network is introduced to adjust the parameters adaptively to balance response speed and filtering performance. Finally, the validity of SMF-BP has been demonstrated through numerical simulation examples.


Introduction
When a control system acquires signals by sensors, the acquired signals can be contaminated by noise from the surrounding environment and the system itself, which can lower the system's performance and stability and even damage its components.Therefore, it is particularly important to use filters to remove the noise component from the signal between the sensor and the controller.
In control systems, linear filters are widely used in the field of filtering due to their simple structure and convenient operation [1].However, when the noise amplitude increases, it will produce a larger phase lag and lower filtering performance.To enhance filtering performance and avoid the disadvantages of linear filters, researchers have conducted research on non-linear filters.For instance, Jin et al. [2] introduced a real-time quadratic sliding mode filter for noise reduction.Unlike linear filters, it offers reduced phase lag and overshooting tendencies.Incorporating the utilization of the backward Euler discretization algorithm prevents chattering from occurring.In addition, a parameter selection criterion is provided [3].
The proposal and research of the quadratic sliding mode tracking differentiator [4] has implemented filtering and state estimation in the model-free state, and further improved its performance in terms of response speed, filtering performance and other aspects.However, the quadratic sliding mode tracking system still cannot estimate high-order derivatives of the input signal.Therefore, based on the quadratic sliding mode tracking differentiator, the new problem in this research field is to propose a high-order sliding mode tracking differentiator to obtain high-order derivatives of the signal.
Numerous researchers have focused on studying high-order sliding mode filters for this issue.Levant et al. [5] suggested a high-order sliding mode tracking differentiator (SMF-L2018).It not only can estimate high-order derivatives of a signal and converge in finite time, but also has unique advantages in processing signals subject to large amplitude small mean value noise interference.However, with the increasing frequency and amplitude of the input signal, this system needs to use a larger gain L to ensure the tracking performance of the system, which also affects the filtering effectiveness.Therefore, how to adjust the system parameter in accordance with the different frequency and amplitude of the input signal requires immediate resolution.
In response to the shortcomings of SMF-L2018 above, this paper presents a high-order sliding mode tracking differentiator with adaptive parameter based on BP neural network (SMF-BP).It can adaptively adjust parameter L , and adds Sigmoid function which can adjust adaptively according to the size of system error to reduce overshooting.The effectiveness of SMF-BP is verified by Numerical simulation examples.
The paper's remaining structure is as follows: Section Ⅱ illustrates the problems of SMF-L2018.Section Ⅲ presents a high-order sliding mode tracking differentiator (SMF-BP), and introduces BP neural network to make the system SMF-BP parameter L adaptive adjustment.Section Ⅳ verifies the efficiency of SMF-BP by Numerical simulation.Section Ⅴ offers a summary of the entire paper.

Problem statement
Levant and Yu [5] suggested a high-order sliding mode tracking differentiator (SMF-L2018), the discrete-time expression using the explicit Euler discretization algorithm is as follows： ) where u is the input, 0 x is the estimated output of u , 1 x R   is an auxiliary variable, k x R  is the estimation of the kth-order derivative of u , 0 L  is system parameter, and 0 1 2 , , 0 is a constant that can be obtained from the recursive sequence, T and i are respectively the sampling interval and the discrete-time index.The expression for the function sgn( ) x is as follows: Figure 1 and Figure 2 shows the output results with different L values of the 1st-order SMF-L2018 under the input signal (3).It is shown that a smaller L value enhances the filtering effect but cannot track well while a larger L value leads to better tracking performance but poorer filtering effect, and it produces a significant overshoot.Based on the above issues, a new sliding mode tracking  differentiator is proposed in Section Ⅲ of this paper to reduce overshoot, and the parameter L is adaptive adjusted based on BP neural network to balance the tracking performance and filtering effect.

Modification of SMF-L2018
Dealing with the overshoot issue of SMF-L2018, the discrete-time expression of a modification of SMF-L2018 is as follows (SMF-BP):  is transition variable, 0 m  is a parameter.
Figure 3 shows the response results of 1st-order SMF-BP ( 1 L  ) with different h values under noisy step input signal (6).It is shown that when the gain L remains constant, the convergence speed and overshoot of the system decrease with decreasing h value, and enhanced filtering effect.Therefore, h is designed to be a decreasing function that adapts to changes in the error.When the error increases, the system is in the overshoot phase, the value of h decreases to reduce the overshoot.When the error decreases, the value of h increases to speed up the convergence speed.
In conclusion, the overshoot of SMF-BP has significantly decreased, but the response speed slows down.If the value of L is increased, it will sacrifice filtering performance.Therefore, the trade-off between filtering performance and tracking speed needs to be resolved urgently.

Introducing backpropagation neural network
In response to the above question, this section introduces a backpropagation (BP) neural network to enable adaptive adjustment of the parameter L in SMF-BP.BP neural network is a popular type of feedforward network extensively employed in various applications.The BP algorithm is utilized to fine-tune the network's weights during training, aiming to minimize the discrepancy between the achieved output and the intended output.Due to its ability to model intricate nonlinear functions, the BP neural network finds utility across diverse domains, including image manipulation and control systems.
The algorithm and step of SMF-BP with adaptive parameter based on BP neural network, as shown in Figure 4, is presented as follow: (1) Choosing the BP neural network structure, determining the node number and giving the value of the inertial coefficient and learning speed rate, provide the initial values of the weight coefficients for each layer; (2) Sampling input variety ( ) u i , and obtained 0 ( ) x i through operation SMF-BP, calculating ( ) e i ; (3) Calculate neuron input and output for each layer in the network, with the output layer's output as the adjustable parameter L ; (4) Perform online adjustment of weight coefficients and neural network learning; (5) Return to (1).

Numerical simulation
In this section, two input signals assess SMF-BP 's validity: where ( ) ~(0,1) t N  represents zero-mean unit white Gaussian noise.The sampling period of all numerical simulation examples are 0.001s in this paper.
Figure 5 and Figure 6 show the output of SMF-L2018 and SMF-BP when the input (6), (7), respectively.As shown in the figure, SMF-BP has the best filtering effect, and it can track well when the input signal undergoes mutation.However, SMF-L2018 cannot track well due to L being a fixed value.
Figure 7 and Figure 9 show the trend of changes in ( ) L i and ( ) e i .As shown in the figure, the error approached zero, and ( ) L i can be adaptively adjusted.Figure 8 and Figure 10 show the average error (AE) and the reach time (RT) of SMF-L2018 and SMF-BP under the input (6), (7), respectively.AE and RT should both small, the performance will be better.As shown in the figure, SMF-BP is closest to the origin, it can balance the filtering performance and the responding speed.

Conclusion
This paper has introduced an innovative high-order sliding mode tracking differentiator with adaptive parameter based on neural network (SMF-BP) to get reliable signals from noisy input signals, which is improvement of SMF-L2018.First, SMF-BP applies a sigmoid function to reduce the overshoot.Then, use BP neural network to adaptively adjust parameters to balance tracking speed and filtering performance.Finally, numerical simulations indicate SMF-BP's enhanced filtering compared to SMF-L2018.

Figure 7 .
Figure 7.The trend of changes in ( ) L i and ( ) e i under input (6).

Figure 8 .
Figure 8.The average error (AE) and the reach time (RT) of SMF-L2018 and SMF-BP under input (6).

Figure 5 .
Figure 5. Responses of the third-order SMF-L2018 with 25 L  and 150 L ,and SMF-BP with 1 m  under input (6).

Figure 9 .
Figure 9.The trend of changes in ( ) L i and ( ) e i under input (7).

Figure 10 .
Figure 10.The average error (AE) and the reach time (RT) of SMF-L2018 and SMF-BP under input (7).