Rate-Dependent Hysteresis Model Based on LS-SVM for Magnetic Shape Memory Alloy Actuator

Abstract

Magnetic shape memory alloy-based actuators (MSMA-BAs) have extensive applications in the field of micro-nano positioning technology. However, complex hysteresis seriously affects its performance. To describe the hysteresis of MSMA-BA, this study proposes integrating a hysteresis operator and the rate-of-change function of the input signal into the least squares support vector machine (LS-SVM) framework to construct a rate-dependent dynamic hysteresis model for MSMA-BAs. The hysteresis operator converts the multi-valued mapping of hysteresis into a one-to-one mapping, while the rate-of-change function of the input signal captures the rate dependence of the hysteresis, thereby enhancing the model’s ability to describe complex hysteresis. In addition, with the powerful nonlinear fitting capability and good generalization of LS-SVM, the dynamic performance of the proposed model is effectively improved. Experimental results show that the proposed model accurately describes the hysteresis of MSMA-BA.

1. Introduction

The development of science and technology has promoted the wide application of high-precision positioning and micro-nano manipulation technologies in many fields such as aerospace, biomedicine, and precision instruments. MSMA is an alloy material composed of Ni, Mn and Ga elements. Relying on its magnetostrictive characteristics, MSMA can generate a large strain in a magnetic field. As a new type of smart material-based actuator, the magnetic shape memory alloy-based actuator (MSMA-BA) has the advantages of a fast response speed, large output force, and high positioning accuracy, and has broad application prospects in the field of micro-nano positioning technology [1,2,3,4]. The two images in Figure 1 are photos of the MSMA and the MSMA-BA, and shows that the main components of the MSMA-BA include a push rod, a spring, an MSMA, and two coils. The energized coil generates a magnetic field, which drives the MSMA material to elongate and then pushes the actuator upward, resulting in the output displacement of the actuator. Figure 2 shows the input and output curves of the actuator. It can be seen from the figure that the hysteresis loop of MSMA-BA has characteristics of multi-value mapping, high asymmetry, and high saturation. The hysteresis characteristic of MSMA-BA is mainly affected by the magnetostriction effect, which is different from that of a traditional SMA-based actuator due to thermal phase transition. This makes MSMA-BA hysteresis sensitive to magnetic fields, with a large displacement and more complex hysteresis. At the same time, MSMA-BA is not as sensitive to temperature hysteresis as a conventional SMA-based actuator and has a long phase transition time.

Table 1. Specifications of the MSMA-BA.

Basic Properties	Values
Dimensions	50 × 50 × 50 ${\mathrm{mm}}^3$
Operating temperature range	0–45 °C
Maximum output displacement	450 $\mathrm{\mu }$m
Maximum input current	8 $\mathrm{A}$
Resistance	3 $\mathrm{\Omega }$

provides an extensive list of the MSMA-BA’s technical specifications and performance parameters. Despite the many advantages of the MSMA-BA, the complex hysteresis exhibited by the MSMA-BA seriously affects its performance. Therefore, establishing an accurate hysteresis model to describe its hysteresis is of great significance for improving the control accuracy and reliability of the MSMA-BA.

In recent years, many modeling methods have been proposed for the hysteresis in the MSMA-BA. Generally, the hysteresis modeling method can be divided into a physics-based modeling method and a phenomenon-based modeling method. Due to the extremely complex physical modeling process and the certain difficulty in deriving mathematical expressions, it is tough to achieve high-precision modeling of hysteresis by using a physics-based model. As a result, most of the research focuses on phenomena-based modeling approaches. The phenomenon-based model mainly includes the Krasnoselskii–Pokrovskii (KP) model [5,6,7,8], Prandtl–Ishlinskii model [9,10,11,12], Bouc–Wen model [13,14,15], Duhem model [16,17,18], etc. However, these phenomenon-based modeling methods face issues such as difficulty in accurately capturing complex characteristics, challenges in parameter identification, and a limited ability to describe highly asymmetric and highly saturated hysteresis behaviors when modeling complex hysteresis.

In addition, artificial neural networks (ANNs) provide an effective method for hysteresis modeling [19,20,21,22]. However, there is no general approach for determining the optimal ANN structure, whether in terms of the number of hidden layers or the number of neurons per layer. The selection of the network architecture is often time-consuming and may fail to achieve optimal performance. Furthermore, ANNs are prone to overfitting, and the training process may get stuck in local optima, leading to suboptimal solutions.

Meanwhile, support vector machine (SVM) is a promising method for accurately modeling nonlinear systems [23,24]. SVM can transform regression problems into quadratic convex optimization problems, demonstrating good performance in hysteresis modeling [25,26,27]. However, the SVM training process requires solving a constrained quadratic programming problem, which can be time-consuming. To address this, Suykens proposed the least squares support vector machine (LS-SVM), which transforms the quadratic programming problem in support vector machines into solving linear equations, significantly improving training efficiency [28]. However, LS-SVM technology has certain limitations in describing the complex rate-dependent hysteresis of MSMA-BA. That is, the traditional LS-SVM usually assumes that there is a single-valued mapping between input and output and lacks the ability to capture dynamic memory effects, and thus it is difficult to describe the hysteresis with multi-valued mapping. Therefore, to improve the performance of the LS-SVM hysteresis model, an effective solution is to introduce exogenous functions to convert the hysteresis multi-valued mapping into a one-to-one mapping and effectively characterize the rate-dependent properties of hysteresis.

In this paper, we propose an improved LS-SVM hysteresis model by introducing a hysteresis operator and an input rate-related function, aimed at describing the complex dynamic rate-dependent hysteresis in MSMA-BA. Through these improvements, the performance of the LS-SVM model is effectively improved. The innovations are summarized as follows:

(1) The integration of the hysteresis operator converts the multi-valued mapping into a one-to-one mapping, improving the LS-SVM’s ability to describe hysteresis.

In the LS-SVM, the hysteresis operator, as a newly introduced feature, enriches the input space, enabling LS-SVM to depict hysteresis with multi-valued mapping features. Additionally, by combining the exponential decay form, it is possible to adjust the output amplitude of the hysteresis operator under different inputs, thus ensuring the model’s performance under various input conditions, and addressing the limitations of the traditional LS-SVM model in handling complex hysteresis.

(2) The introduction of the input rate-related function compensates for the shortcomings of LS-SVM in dealing with rate-related hysteresis.

The input rate-related function describes the rate of change of input signals over adjacent time intervals. By introducing this function as a new input feature into the LS-SVM model, it effectively captures the dynamic characteristics of input signals as they change over time, addressing the shortcomings of traditional LS-SVM when dealing with rate-related hysteresis. By incorporating the input rate feature, the model can more accurately describe the rate-dependent hysteresis behavior of the MSMA-BA.

The arrangement of this paper is as follows: Section 2 explains the modeling method based on LS-SVM. Section 3 verifies the model performance from an experimental perspective. Section 4 summarizes the study.

2. LS-SVM Hysteresis Model

The core principle of support vector machine (SVM) is to map the sample space to a high-dimensional feature space through the nonlinear transformation defined by the kernel function. In this high-dimensional space, the nonlinear relationship between input and output is explored. Least squares support vector machine (LS-SVM) replaces the inequality constraints in the standard SVM with equality constraints and uses the sum of squared errors as the loss function of the training set. In this way, the originally complex quadratic programming problem is transformed into a system of linear equations. This transformation brings significant advantages, effectively reducing the computational complexity and increasing the solving speed of the problem.

When modeling a hysteresis system using LS-SVM, the goal is to find a function $f\left(x\right)$ that can predict the most accurate output y for any given input x, based on a training dataset ${\{x_i,y_i\}}_{i=1}^N$, where $x_i$ is the input vector, $y_i$ is the output value, and N is the number of samples. However, hysteretic systems often have complex multi-valued mapping relationships, while LS-SVM itself can only perform one-to-one mapping. These shortcomings limit its accurate modeling of hysteretic systems. We introduce the hysteresis operator to address the limitation of the LS-SVM, which is restricted to one-to-one mapping. The expression of the hysteresis operator is as follows:

$$\begin{array}{c}\hfill \mathrm{\Gamma }\left(x\right)=\left(1-e^{-\left|x-x_p\right|}\right)\left(x-x_p\right)+\mathrm{\Gamma }\left(x_p\right)\end{array}$$

(1)

where $\mathrm{\Gamma }\left(x\right)$ represents the current output of the hysteresis operator, x is the input of the operator, $x_p$ is the previous maximum input, and $\mathrm{\Gamma }\left(x_p\right)$ is the output of the hysteresis operator corresponding to $x_p$. In addition, we also consider the rate-dependent effect of hysteresis, that is, the rate-of-change function of the input signal is used to further improve the ability of the model to describe hysteresis. The expression is as follows:

$$\begin{array}{c}\hfill \dot{x}\left(k\right)={\displaystyle \frac{x\left(k\right)-x(k-1)}{T}}\end{array}$$

(2)

where T denotes the sampling time of the system.

To further improve the modeling effect of LS-SVM, it is very important to select the appropriate kernel function. Common kernel functions include the linear kernel, polynomial kernel, and radial basis function (RBF) kernel. The RBF kernel has good generalization ability and flexibility, and its expression is as follows:

$$\begin{array}{c}\hfill K(x_i,x_j)=exp\left(-{\displaystyle \frac{\parallel x_i-x_j{\parallel }^2}{2{\sigma }^2}}\right)\end{array}$$

(3)

where $\sigma$ is the kernel parameter.

For LS-SVM, the optimization problem can be expressed as follows:

$$\begin{array}{c}\hfill \underset{w,b,e}{min}{\displaystyle \frac{1}{2}}{w}^{T}w+{\displaystyle \frac{\gamma }{2}}\sum _{i=1}^N{e}_i^2\end{array}$$

(4)

subject to:

$$\begin{array}{c}\hfill y_i=w^T\varphi \left(x_i\right)+b+e_i,i=1,2,\cdots ,N\end{array}$$

(5)

where w is the weight vector, b is the bias term, $e_i$ is the error term, $\varphi \left(x_i\right)$ is the mapping function that maps the input vector $x_i$ to a high-dimensional feature space, and $\gamma$ is the penalty factor.

To solve the optimization problem, the Lagrangian function is introduced:

$$\begin{array}{cc}\hfill L(w,b,e,\alpha )=& {\displaystyle \frac{1}{2}}{w}^{T}w+{\displaystyle \frac{\gamma }{2}}\sum _{i=1}^N{e}_i^2-\hfill \\ \hfill & \sum _{i=1}^N{\alpha }_i\left[w^T\varphi \left(x_i\right)+b+e_i-y_i\right]\hfill \end{array}$$

(6)

where ${\alpha }_i$ represent the Lagrange multipliers. The term ${\displaystyle \frac{1}{2}}{w}^{T}w$ indicates a regularization term, which serves to control the model complexity and avert overfitting. The expression ${\displaystyle \frac{\gamma }{2}}{\sum }_{i=1}^N{e}_i^2$ stands for the measured squared error. The last term is the constraint term, which enforces the equality constraint and balances the impact of the constraint and the error term.

Take the partial derivatives of the Lagrangian function with respect to w, b, ${\alpha }_i$, and $e_i$, and set them to zero:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\partial L}{\partial \mathit{w}}}=\mathit{w}-\sum _{i=1}^N{\alpha }_i\phi \left({\mathit{x}}_i\right)=0\hfill \\ {\displaystyle \frac{\partial L}{\partial b}}=-\sum _{i=1}^N{\alpha }_i=0\hfill \\ {\displaystyle \frac{\partial L}{\partial {\alpha }_i}}=w_{\phi }\left({\mathit{x}}_i\right)+b+{\epsilon }_i-y_i=0\hfill \\ {\displaystyle \frac{\partial L}{\partial e_i}}=\gamma e_i-{\alpha }_i=0\hfill \end{array}\right.\end{array}$$

(7)

By eliminating w and $e_i$, the following system of linear equations can be obtained:

$$\left[\begin{array}{cc}0& {\mathbf{1}}^T\\ \mathbf{1}& \mathrm{\Omega }+{\gamma }^{-1}\mathit{I}\end{array}\right]\left[\begin{array}{c}b\hfill \\ \mathbf{\alpha }\hfill \end{array}\right]=\left[\begin{array}{c}0\hfill \\ \mathit{y}\hfill \end{array}\right]$$

(8)

where ${\mathrm{\Omega }}_{ij}=K(x_i,x_j)$, $\mathbf{1}={\{1,1,\cdots ,1\}}^T$, $\mathit{y}={\left\{y_1,y_2,\cdots ,y_N\right\}}^T$, $\alpha ={\left\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_N\right\}}^T$, and I is the identity matrix.

By solving the above system of linear equations, the values of b and $\alpha$ can be obtained. Thus, the model for the hysteresis nonlinear system is as follows:

$$\begin{array}{c}\hfill f\left(x\right)=\sum _{i=1}^N{\alpha }_{i}K(x,x_i)+b\end{array}$$

(9)

By selecting an appropriate kernel function, constructing the optimization problem, introducing the Lagrangian function, and solving it, the final model expression for hysteresis nonlinearity using the least squares support vector machine is obtained. In practical applications, adjusting the kernel parameter ${\sigma }^2$ and other parameters can improve the model’s performance.

In this paper, the particle swarm optimization (PSO) algorithm is used to optimize the parameters of the LS-SVM. The parameter setting of the PSO algorithm is taken from [26]. Finally, the penalty factor and kernel parameters of the LS-SVM are selected as 79,000 and 0.0147, respectively.

3. Experimental Verification

To validate the effectiveness of the proposed modeling method, experiments were conducted on the platform shown in Figure 3. Figure 3a illustrates the schematic diagram of the experimental setup, while Figure 3b presents a photograph of the experimental equipment. The experimental platform consisted of six main components: the MSMA-BA, a high-precision programmable DC power, a high-precision linear variable differential transformer (LVDT), a host computer, a temperature sensor (TS), and a PCI-1716 data-acquisition card.

The programmable DC power served as the driving source for the MSMA-BA, providing a controllable and stable driving current. The high-precision LVDT acted as a crucial displacement measurement tool, accurately capturing the displacement of the MSMA-BA. The PCI-1716 data-acquisition card was responsible for collecting sensor data and transmitting the data to the host computer. The host computer, connected to the PCI-1716 data-acquisition card, performed data analysis and processing, while also controlling the programmable DC power to adjust the driving current. The basic sampling time for the experiment was set to 0.001 s to ensure high temporal accuracy and precision of the data. The TS was used to monitor the operating temperature of the actuator to ensure that the experiment is carried out within the operating temperature range specified by the MSMA-BA.

To validate the effectiveness of the proposed rate-dependent dynamic hysteresis model based on LS-SVM, a series of experiments were conducted to examine the model’s ability to describe the hysteresis behavior of the MSMA-BA under different input frequencies and loads, as well as a complex harmonic signal and a triangular wave signal. Additionally, the KP model is a phenomenological model based on the hysteresis operator and is frequently employed to depict the hysteretic characteristics in smart material actuators. Thus, the experimental results of this study (denoted as M2) are compared with the results obtained from the KP model proposed in [29] (denoted as M1).

The experiment first verified the performance of the model under input signals of different frequencies, ranging from 1 Hz to 10 Hz, and the experimental results are shown in Figure 4. The results demonstrate that the proposed model can accurately capture the hysteresis characteristics of the MSMA-BA under various input frequencies. This improvement is attributed to the introduction of the input rate-related function, which enables the model to more accurately describe complex hysteresis behavior.

In the variable load experiment, the model was tested under different load conditions ranging from 100 g to 300 g. As shown in Figure 5, as the load increased, the accuracy of M1 significantly decreased, failing to effectively handle complex hysteresis. By contrast, the proposed model effectively handled the impact of load changes on hysteresis characteristics and maintained high accuracy under different load conditions, demonstrating the robustness of the model to load variations.

In addition, to further evaluate the performance of the model under complex input conditions, the experiment used a complex harmonic signal and a triangular wave signal as the excitation signals of the MSMA-BA to verify the performance of the proposed model. The experimental results are shown in Figure 6, indicating that the proposed model can describe not only the main hysteresis loop of the MSMA-BA, but also the small hysteresis loop of the MSMA-BA. Specifically, when processing the triangular wave signal, M2 achieved significantly higher fitting accuracy than M1. This is primarily due to the strong nonlinear fitting ability of LS-SVM, as well as the exogenous functions (the hysteresis operator and input rate-related function) that enhance the model’s capacity to handle rate-dependent hysteresis. The hysteresis operator converts complex multi-valued mapping into one-to-one mapping, enabling LS-SVM to more effectively represent the hysteresis behavior under different input frequencies. Meanwhile, the input rate-related function provides additional input information for the model to capture the dynamic changes of the input signal, allowing for a more accurate description of complex hysteresis.

In addition, the root mean square error (RMSE), the maximum absolute error (MAE), and the maximum error ratio (MER) were used to evaluate the performance of the model. The experimental data are shown in

Table 2. Modeling performance comparison between the KP model and LS-SVM hysteresis model.

Frequency	Model	RMSE	MAE	MER	Im. (RMSE / MAE)
		($\mathrm{\mu }$m)	($\mathrm{\mu }$m)	(%)	(%)
1 Hz	KP model	1.0898	6.13	1.75	32.4/69.5
	LS-SVM model	0.7368	1.87	0.53
10 Hz	KP model	6.0826	11.36	3.79	35.7/57.5
	LS-SVM model	3.9087	4.83	1.61
2 Hz, 100 g	KP model	1.5162	7.13	2.85	26.1/71.1
	LS-SVM model	1.1208	2.06	0.82
2 Hz, 300 g	KP model	1.6017	4.87	3.83	10.8/56.7
	LS-SVM model	1.4295	2.11	1.66
Mixed Signal	KP model	2.2899	9.10	2.17	25.7/55.3
	LS-SVM model	1.7025	4.07	0.95
Triangular Signal	KP model	1.6959	21.62	5.54	22.1/28.7
	LS-SVM model	1.3204	15.41	3.95

Im.: M1 precision improvement relative to M2.

. The error index and Im. data confirm the ability of the LS-SVM model to accurately characterize the hysteresis behavior in the MSMA-BA.

In summary, in the proposed LS-SVM hysteresis model, although the introduction of hysteresis operators and input rate-related function increases the time and computational costs, the introduction of exogenous functions provides LS-SVM with an effective means of characterizing complex hysteresis behavior. Especially under the conditions of load variation and complex signal inputs, it can transform multi-valued mapping into one-to-one mapping and better capture the rate-dependent features, thus improving the accuracy and generalization ability of the model.

4. Conclusions

Hysteresis significantly impacts the performance of MSMA-BA, making it essential to develop an accurate hysteresis model to effectively characterize the actuator’s behavior. In this study, a dynamic hysteresis model of MSMA-BA is constructed by introducing an exogenous function into LS-SVM. This integration effectively addresses the challenges posed by traditional LS-SVM when dealing with multi-valued mapping and rate-dependent hysteresis. The improvement in model performance is attributed to the exogenous function, which simplifies the hysteresis mapping and enhances the model’s ability to address complex input information. The modeling error data show that the proposed modeling method not only enhances the LS-SVM’s capacity to describe hysteresis but also effectively characterizes the complex hysteresis behavior of smart material-based actuators. In addition to having a high modeling accuracy, this model can also meet the requirements of real-time control. Although this algorithm requires a certain amount of computation time during the training phase, this process is completed before the model is applied, so it will not directly affect the actual control process. In subsequent research, we will further explore the relationship between the model and the control real-time performance, weigh the computation time, model accuracy, and control effect, and seek the optimal balance.