Feed-Forward Control for Magnetic Shape Memory Alloy Actuators Based on the Radial Basis Function Neural Network Model

RBFNN model

The RBFNN is used to establish the functional relationship model (20–21–22). But the input-output of MSMA actuators is multi-mapping (23). To solve this problem, hysteresis nonlinearity in two dimensions is transformed into one-to-one mapping linear relationship in three dimensions. There are two neurons in the input layer of RBFNN. One is the current value and another is the previous value of actuators. There is one neuron in the output layer. The multi-input multi-output relationship can be transformed into one-to-one mapping (24). The structure diagram of RBFNN is shown in Figure 1.

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Fig. 1

Structure diagram of radial basis function neural network (RBFNN).

As shown in Figure 1 there are two neurons in input layer. Where v(t) is input value of actuators at time t, u(t) is output value of actuators at time t, u(t-1) is output value of actuators at time t-1. The radial basis function is used as transfer function and the weight between input and hidden layer is 1. It has radial symmetry for the center-point in n-dimensional space. In this paper, Gaussian function Rt is function in hidden layer which transforms hysteresis nonlinearity of MSMA actuators to linear relationship.

R t = exp ( − h t )

[Eq. 1]

where h_t represents the independent variable of the activation function. In the RBFNN, h_t is normally Euclidean norm, which represents the separation from input vector to weight vector. The two-dimensional Gaussian function can be expressed as:

h t = ( x p 1 i − c 1 i ) 2 σ 1 i 2 + ( x p 2 i − c 2 i ) 2 σ 2 i 2

[Eq. 2]

where p is the number of input samples; x_p-c represents the Euclidean norm of independent variable in two-dimensional Gaussian function; c represents the center for nodes in hidden layer; s represents the variance for Gaussian function. The output of RBFNN is:

y = ∑ i = 1 n w i exp ( − h i )

[Eq. 3]

where y is the output for output layer; n represents the number of hidden layer nodes, which automatically increases with different design requirements; w_i represents the weight from hidden layer's ith neuron to output neuron. The basic idea of RBFNN is used in this paper (22): input singles are transferred to hidden layer by using the two neurons. The RBFNN is nonlinear transformation, which is an activation function between input layer and hidden layer. The sum of output in hidden layer is linear weighted as input value for output layer, where the weight is adjustable. There are three parameters need to be solved in RBFNN: the weight w from hidden layer to output layer, the center c of basis function and the variance s. Besides, the RBFNN model is established by using Neural Network Toolbox in Matlab to increase hidden layer nodes to achieve high precision of neural network structure.

Inverse model

MSMA inverse model is established by using RBFNN structure. The structure diagram for inverse hysteresis model is the same as Figure 1. While the inputs of input layer are the input v(t-1) of MSMA actuators at time t-1 and the output u(t) of MSMA actuators at time t. The output of output layer is the input v(t) of MSMA actuators at time t. By training experiment data, the availability of the inverse model is proved (25).

Feed-forward controller design

In contrast with feedback control, the method of feed-forward control can adjust the disturbance before the actual output departing from the desired output. Feed-forward control has the predictive compensation capacity with the behavior of disturbance. Satisfied with requirements such as high precision of model, measurable disturbance and high precision of device, feed-forward controller can be established (26, 27).

A feed-forward control system is established based on inverse RBFNN model. Schematic diagram is shown as Figure 2. Feed-forward controller can be designed to eliminate hysteresis nonlinearity for MSMA actuators.

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Fig. 2

Inverse model feed-forward control scheme.

Model simulation

The RBFNN hysteresis model can be proposed by using self-including neurons method in this paper. Modeling precision of MSMA can be improved with enough sample data. The curves of actual input and actual output are shown in Figure 3 which is used to establish the MSMA actuators model.

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Fig. 3

Actual input and actual output of the magnetic shape memory alloy (MSMA) actuators.

Both modeling speed and precision are improved by adjusting scatter coefficient sc and target accuracy coefficient eg. When scatter coefficient is 1, input ranges are covered and the control precision cannot be influenced. Figure 4 shows contrast diagrams of RBFNN model with different parameters.

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Fig. 4

Contrast diagrams of the radial basis function neural network (RBFNN) model with different parameters. (A) Hysteresis curves of the RBFNN model at eg = 0.02. (B) Hysteresis curves of the RBFNN model at eg = 4.1 × 10⁻⁶. (C) Modeling error of RBFNN model at eg = 0.02. (D) Modeling error of RBFNN model at eg = 4.1 × 10⁻⁶.

In Figure 4 (A) and (C) the structure of neural network is simple and running speed is fast at eg = 0.02. The maximum modeling error is 0.04 mm and the maximum modeling error rate is 4.42%. So that the neural network model is not consistent with MSMA actuators at eg = 0.02. While at eg = 4.1 × 10⁻⁶, neurons of hidden layer will be increased to 25. In Figure 4 (B) and (D) running speed is ensured and maximum modeling error based on RBFNN is decreased to 0.007 mm and maximum error rate is 0.79%. The effectiveness of hysteresis model based on the RBFNN is proved.

Inverse model simulation

In order to prove effectiveness of inverse RBFNN model for MSAM actuators, it is compared with the simulation of inverse KP model. Simulation results are as shown Figure 5.

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Fig. 5

Contrast diagrams of inverse model based on different methods. (A) Hysteresis curves of inverse KP model (11). (B) Hysteresis curves of inverse radial basis function neural network (RBFNN) model. (C) Modeling error of inverse Krasnosel'skii-Pokrovskii (KP) model (11). (D) Modeling error of inverse RBFNN model.

In Figure 5 the inverse model based on RBFNN has better effectiveness in comparison with the inverse KP model in Zhou et al (11). In Figure 5 (B) and (D) maximum error rate for inverse model based on RBFNN is 0.24%. Compared with the inverse KP model established by traditional cut-and-try method (11), the maximum error has reduced 0.017T. The inverse RBFNN model has higher modeling precision of MSMA actuators.

Feed-forward control simulation

An inverse hysteresis model with high accuracy is established by using RBFNN, which is used to build the feed-forward controller as shown in Figure 2. Compared with the simulation of PID feedback hybrid control (11), the effectiveness for feed-forward control method can be proved. The simulation results are shown in Figure 6.

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Fig. 6

Contrast diagrams of results of feed-forward control based on different inverse models. (A) Tracking displacement based on inverse Krasnosel'skii-Pokrovskii (KP) model (11). (B) Tracking displacement based on inverse radial basis function neural network (RBFNN). (C) Tracking error based on inverse KP model (11). (D) Tracking error based on inverse RBFNN model. (E) Input-output function based on inverse KP model (11). (F) Input-output function based on inverse RBFNN model.

As shown in Figure 6 compared with the feed-forward control based on inverse KP model in (11), feed-forward control based on inverse RBFNN model has higher control precision. Figure 6 (B) is the curve between input and output of feed-forward control system. The tracking effect of the system established by inverse RBFNN model is better than method in Zhou et al (11). In Figure 6 (D) the controlling error for feed-forward control system is 0.00334 mm and the maximum error rate is 0.38%. Compared with the method in Zhou et al (11), it has reduced by 0.72%. Figure 6 (F) shows the hysteresis nonlinearity can be transformed to a linear relationship. The simulation results show the feed-forward controller can suppress the hysteresis nonlinearity of the MSMA actuators effectively.