Parameter-identification investigations on the hysteretic Preisach model improved by the fuzzy least square support vector machine based on adaptive variable chaos immune algorithm

Abstract

In order to solve the hysteretic character of the piezoelectric material for application, the initial weight factors of the hysteretic units are calculated by the Preisach theory and the first-order reversal curves test data, a hysteretic Preisach model based on the improved fuzzy least square support vector machine (improved FLS-SVM) is established. In the established model, the fuzzy least square support vector machine is introduced to calculate more weight factors of the hysteretic units and the adaptive variable chaos immune algorithm is introduced to optimize the penalty factor and the kernel parameter of the FLS-SVM (the penalty factor c = 35 and the kernel parameter σ = 1.35 are obtained). Moreover, the quadratic polynomial interpolation method is used to eliminate the sawtooth phenomenon. The validity of established model reveals that fuzzy least square support vector machine method based on adaptive variable chaos immune algorithm (FLS-SVMAVCIA) is more accurate than FLS-SVM method according to application results of the real actuators (the absolute mean error of the FLS-SVMAVCIA model is less than 1 µm and its maximum error is less than 2 µm). As a result, the hysteretic phenomenon can be effectively eliminated by the hysteretic Preisach model based on the FLS-SVMAVCIA method.

Keywords

Fuzzy least square support vector machine hysteretic Preisach model adaptive variable chaos immune algorithm interpolation method actuator

Introduction

There are many noise and vibration phenomena in the complicated mechanical and electrical systems.^1–5 Noise and vibration in the complicated mechanical and electrical systems and their control still represent serious challenges to researchers, engineers, and constructors. Piezoelectric material is one type of crystalline materials, which has the ability to generate voltage between the two ends when the materials are under pressure,^6,7 and the effect is called the positive piezoelectric effect. Relatively, when the piezoelectric materials are under the electric field, the materials can undergo deformations, which is to say the materials can convert the electrical energy into mechanical energy or mechanical movement, and the effect is called the inverse piezoelectric effect. The piezoelectric actuator^8–10 based on the inverse piezoelectric effect is designed as the ideal driving components. The advantages of piezoelectric actuator are included as follows: high resolution of the displacement, strong driving force, high efficiency of electromechanical coupling, swift response and noiseless.¹¹ Therefore, the piezoelectric actuators are widely adopted in the realm of active control and vibration and noise reduction. However, the piezoelectric materials are of the inherent characteristics of hysteresis, nonlinearity, creep deformation, etc.^12,13 These inherent characteristics lead to some undesirable results such as low control accuracy, non-repeatability in the engineering and the slow transient response of the materials.¹⁴ As a result, it is difficult to employ piezoelectric actuators in industrial applications.

At present, there were some investigations on the hysteresis problem of the piezoelectric materials and researchers had proposed some of the hysteretic models such as Preisach model,^15,16 multi-objective parameter optimization model,¹⁷ Dahl model,¹⁸ Bouc-Wen model,¹⁹ etc. And the most widely used models are the Preisach model and its improved models. The principle of the Preisach model is based on the double integral, its integral item is the weight factor of the hysteretic unit. But the Preisach model is of its own imperfections such as the sawtooth phenomenon in the output curve of the Preisach model with insufficient computational discrete data points, the large output error, and the long computational time as the number of computational discrete data points is increased.

Therefore, some research on solving the hysteresis problem of the piezoelectric materials had been carried out. A neural network model for the identification of Preisach-type hysteresis was proposed and a hysteretic operator was introduced to transform the multi-valued mapping of hysteresis into a one-to-one mapping in Zhao and Tan.²⁰ A model based on the Preisach operator and analytic weight function was used to simulate the hysteretic large-signal behaviours in ferroelectric materials in Wolf et al.²¹ Moreover, discrete Preisach model²² and support vector machine (SVM)^23,24 were applied to establish a hybrid model for piezoelectric actuators. In order to compensate a piezostage driven by piezoelectric stack actuators, the extended least squares support vector machines (LS-SVMs)^25–29 were used to establish the domain of nonlinear modelling and control other nonlinear systems.³⁰ An improved hysteresis model³¹ for magnetostrictive actuators was presented and the proposed four hybrid genetic algorithms (HGAs) were applied to identify parameters of the improved model.

The above research revealed that the hysteresis problem of the piezoelectric materials had been focused on the establishment of hybrid model based on fusions of Preisach model and artificial intelligence techniques such as neural network, SVM, LS-SVM and HGAs. However, the hysteresis problem of the piezoelectric materials was not resolved effectively.

Compared with other artificial intelligence techniques such as Gaussian process(GP),^32,33 Bayesian regression,³⁴ neural networks (NNs),³⁵ Gaussian process mixed model (GPMM),³⁶ fuzzy least squares support vector machines (FLS-SVM) is of unique advantages such as requiring less modelling data, simple calculation and quick identification capacity in resolving the nonlinear and high dimension problems with small sample sets. Moreover, an adaptive variable chaos immune algorithm (AVCIA) is useful for global optimization of nonlinear and high dimension problems. Therefore, a novel idea is to use the FLS-SVM optimized by an AVCIA to effectively resolve the hysteresis problems of the piezoelectric materials, and new contributions in the paper are expressed as follows: (1) A hysteretic Preisach model based on the improved FLS-SVM was established; (2) The AVCIA was introduced to optimize the penalty factor and the kernel parameter of the FLS-SVM; (3) The hysteretic phenomenon was eliminated effectively by the hysteretic Preisach model based on the FLS-SVMAVCIA method. The research results reveal that the validity of the hysteretic Preisach model based on the improved FLS-SVM has been tested.

Identification of parameters of the Preisach hysteretic model based on the improved FLS-SVM

Theory of the generalized Preisach hysteretic model

The computational method of the generalized Preisach hysteretic model is through the double integral of the hysteretic factor, and the result of the method represents the character of the generalized Preisach hysteretic model. The mathematical relationship between the input and output can be expressed as

f (t) = R [u (t)] = \int \int_{α \leq β} μ (α, β) γ_{α, β} [u (t)] d α d β

(1)

where R[u(t)] is the computational operator of the Preisach hysteretic model; t is the non-dimensional time; α and β are the lower and upper threshold values respectively; μ(α, β) is the weight function of the hysteretic model based on the values of α and β; u(t) is the input voltage of the Preisach hysteretic model; γ_α_, _β [u(t)] is the hysteretic unit, and its output switches between 0 and 1.

The principle of the hysteretic unit is shown in Figure 1.

Figure 1.

Principle of hysteretic unit.

The output will not switch to 1 until the input of u(t) is greater than β₁ when the input is increased. Similarly, the output will not switch to 0 until the input of u(t) is smaller than α₁ when the input is decreased.

The hysteretic unit (α₁, β₁) presented in Figure 1 can also be expressed in the Preisach plane in which α and β are used as the coordinates and β ≥ α, and the schematic diagram is shown in Figure 2. The value ranges of α and β are between u_min and u_max, which are the minimum and maximum value of the input, respectively.

Figure 2.

Hysteretic unit shown in α-β plane.

In the triangle area surrounded by α and β, the value of the weight function μ(α, β) is not equal to 0 and the value of the weight function μ(α, β) is equal to 0 at out of the triangle area. When the input voltage u(t) is increased, the horizontal axis whose value is equal to the input is moving up and the output of the hysteretic unit below the horizontal axis switches to 1. Contrastively, when the input voltage u(t) is decreased, the vertical axis whose value is equal to the input is moving to the left and the output of the hysteretic unit on the right of this vertical axis switches to 0. As shown in Figure 2, the value of the hysteretic unit in area S⁺ which β ≤ β₁ and α ≤ α₁ is equal to 1, the other area is equal to 0.

For discrete hysteretic model, the triangle area shown in Figure 2 will be divided into infinite units and each unit represents different (α_i, β_j). For example, if the input voltage signal is divided into n segments, then the total number of the hysteretic model is n(n + 1)/2, and the hysteretic model consists of the n(n + 1)/2 units which are connected in parallel. The distance between two hysteretic unit is 1/(n + 1). So when the n is increased gradually, the output of the discrete hysteretic model will be more precise. But with the increase of the n, the computation of the weight factor is more difficult, and this will lead to consuming more computational time. The discrete hysteretic model can be expressed as equation (2).

f = \sum_{i, j}^{n} μ_{n} (i, j) \cdot γ_{i, j} (t)

(2)

where μ _n is a n × n matrix, the elements of the matrix are the weight factor of each hysteretic unit and the values of the weight factors are obtained from the FORCs, γ _ij (t) is the hysteretic computational unit matrix with time variation, and the elements of the matrix are 0 and 1.

Thus, the output of the hysteretic model expressed as equation (2) is the weight sum of the hysteretic unit which is not equal to 0.

Improved FLS-SVM based on AVCIA

Theory of the FLS-SVM

SVM is a mechanical learning method which was developed by Vapnik³⁷ for solving problems of pattern recognition and regression based on the statistics theory. The FLS-SVM theory is based on the SVM, and the basic principle is depicted as follows^38–40: The fuzzy theory is introduced into the SVM for constructing the corresponding objective function, and different fuzzy punishment weight coefficients are introduced into the samples so that the noise sample data can be eliminated.

The assumed sample data of the FLS-SVM model is expressed as follows

[x_{1}, y_{1}, λ (x_{1})], [x_{2}, y_{2}, λ (x_{2})], \dots, [x_{i}, y_{i}, λ (x_{i})], i = 1, 2, \dots, l

(3)

where x_i is the i-th input parameter of the FLS-SVM model, y_i is the i-th output parameter of the FLS-SVM model, λ(x_i) is the subjection degree whose range is 0 < λ(x_i) ≤ 1.

The λ(x_i) represents the fuzzy rules of the characteristic parameters of sample data after fuzzification. And in the process of training the FLS-SVM, the λ(x_i) represents the weight effect, which is different between the training data. There is no general guideline to follow to construct the fuzzy subjection function. According to Abad et al.,⁴¹ the method called the linear distance fuzzy subjection function construction is applied. The principle is expressed as follows: using the distance between the sample data and the class-center as the value of the subjection degree, so this is a linear function and the larger distance is, the bigger subjection degree is.

For the sample set {x₁, x₂, … , x_n}, x₀ is treated as the class-centre, r is treated as class-radius, and then

x_{0} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}

(4)

r = {max}_{x_{i}} ‖ x_{0} - x_{i} ‖

(5)

The function of subjection degree λ(x_i) of the sample data can be expressed as equation (6).

λ (x_{i}) = 1 - \frac{‖ x_{i} - x_{0} ‖}{r + δ}

(6)

where δ is a very small constant which can avoid the subjection function being equal to 0.

The objective function of the FLS-SVM is modified after introducing the fuzzy subjection degree.

min J (w, ξ) = \frac{1}{2} ‖ w ‖^{2} + \frac{C}{2} \sum_{i = 1}^{n} λ (x_{i}) \cdot ɛ_{i}^{2}

(7)

s . t . y_{i} = w^{T} \cdot ϕ (x_{i}) + b + ɛ_{i}, ɛ_{i} > 0, i = 1, 2, \dots, l

where w is the weight vector of the hyperplane, ɛ_i is the slack variable, C is the penalty factor and b is the threshold value.

To solve the optimal problem which shows in equation (7), the Lagrangian factor a_i (i = 1, 2, … , n) is introduced, and the corresponding Lagrangian function is expressed as equation (8).

L (w, ɛ_{i}, b, a_{i}) = \frac{1}{2} ‖ w ‖^{2} + \frac{C}{2} \sum_{i = 1}^{n} λ (x_{i}) \cdot ɛ_{i}^{2} - \sum_{i = 1}^{n} a_{i} [w_{i} \cdot ϕ (x_{i}) + b + ɛ_{i} - y_{i}]

(8)

In order to obtain the minimum of the objective function, making the partial derivative of the parameters w, ɛ_i, b, a_i in equation (8) equal to 0, thus the equation (9) can be expressed as follows.

{\frac{\partial L}{\partial w} = w - \sum_{i = 1}^{n} a_{i} \cdot ϕ (x_{i}) = 0 \frac{\partial L}{\partial ɛ_{i}} = C \cdot λ (x_{i}) \cdot ɛ_{i} - a_{i} = 0 \frac{\partial L}{\partial b} = \sum_{i = 1}^{n} a_{i} = 0 \frac{\partial L}{\partial a_{i}} = w \cdot ϕ (x_{i}) + b + ɛ_{i} - y_{i} = 0

(9)

Eliminating the parameters w and ɛ_i, then the optimal problem is transferred into the problem of solving the linear system of equation (10).

[0 E^{T} E Ω + C \cdot λ (x_{i})^{- 1} l] [b a] = [0 y]

(10)

where y = [y₁, y₂, … , y_l]^T; E = [1, 1, … , l]^T; a = [a₁, a₂, … , a_l]^T and Ω_ij = ϕ(x_i)·ϕ(x_j) = K(x_i, x_j).

Defining K(x_i, x_j) = ϕ(x_i)ϕ(x_j) and K(x_i, x_j) can be called the kernel function, thus equation (7) can be expressed as equation (11) which is the FLS-SVM model.

f (x) = \sum_{i = 1}^{n} a_{i} K (x, x_{i}) + b

(11)

The commonly used kernel function is as follows: radial basis kernel function, polynomial kernel function, linear kernel function, sigmoid kernel function, etc. The widely used kernel function is the radial basis kernel function; due to that, the error of the function is smaller in application and the mathematical expression is written as equation (12).

K (x, x_{i}) = \exp (- | x - x_{i} |^{2} / σ^{2})

(12)

where σ is the kernel parameter.

Optimization of the radial basis kernel function by AVCIA

In order to solve the problems of the immune algorithm⁴² which is easy to fall into local optimum in the whole iterative process and is of a low convergence rate in the late iterative process, an AVCIA is introduced to optimize the penalty factor C and kernel parameter σ.

Considering that the key to AVCIA is to determine the fitness function, the fitness function is selected as follow

F (C, σ) = \frac{1}{\sqrt{\sum_{i = 1}^{n} [y_{i} - f (x_{i})]^{2}} + e} s . t . ɛ_{i} \leq x_{i} \leq η_{i} i = 1, 2, \dots, n

(13)

where y_k is the desired output, f(x_k) is the actual output, e is a small real number, which is used to prevent a situation that the denominator is zero, here e is 10⁻³.

The error function mean squared error (MSE) MSE is defined as the evaluation index of generalization performance of FLS-SVM:

MSE = \frac{1}{n} \sqrt{\sum_{i = 1}^{n} [f (x_{i}) - y_{i}]^{2}}

(14)

Figure 3 shows flow chart of optimizing parameters of FLS-SVM by AVCIA.

Figure 3.

Flow chart of optimizing parameters of fuzzy least squares support vector machine by adaptive variable chaos immune algorithm.

The concrete steps of AVCIA to optimize parameters of FLS-SVM are expressed as follows:

Step 1: The standardization process is done for the input antigens {A_g₁, A_g₂, … , A_g_j, … , A_g_m}.

Step 2: The logistic model x_r₊₁ = sin(2/x_r)⁴³ is selected as chaotic variables of initialization antibodies {A_b₁, A_b₂, … , A_b_j, … , A_b_m} in interval (0, 1), which are randomly generated by chaotic model.

Step 3: For each antigen A_g_j(j = 1, 2, … , m), it is operated as follows:

Step 3.1: The affinity β_r of each antibody A_br and antigen A_g are calculated as

β_{r} = \sqrt{\sum_{j = 1}^{m} (A_{r} - A_{j})^{2}}

(15)

Step 3.2: Antibodies with highest affinity are treated as network cells and they are cloned simultaneously, then the corresponding number of clonal antibody cells S₀ will be obtained.

Step 3.3: The following is used for mutation of the clonal cells

S_{1} = S_{i} (1 - α) + α T_{0} (i = 1, 2, \dots, R_{0})

(16)

where S_i is the number of clonal antibody cells, T₀ is the number of clonal antigen cells and α is the mutation rate.

Step 3.4: The affinity β_r of each antibody A_br and antigen A_g_j after mutation is recalculated using equation (15).

Step 3.5: 30% of the cells with highest affinity are treated as the memory cell data set T_p.

Step 3.6: The similarity degree λ_r₍ _R _- _r ₎(r ≠ R−r) between each antibody A_br and A_b₍ _R ₋ _r ₎ is calculated based on

λ_{r (R - r)} = \sqrt{\sum_{r = 1}^{m} [A_{r} - A_{(R - r)}]^{2}}

(17)

and the individuals whose similarity degrees λ_r₍ _R ₋ _r ₎ are greater than the threshold value σ_s (in this paper it is set to 0.95) in the data set, T_p will be eliminated.

Step 4: The memory cell data set T_p is added into the memory dataset T.

Step 5: The better antibody and antigen individuals are selected for chaotic fine search.

Fifteen per cent of the individuals whose fitness values are relatively large will be selected for chaotic fine search, and optimum individual is set as Y = (Y₁, Y₂, … , Y_k), and the search interval of chaotic variables is narrowed as⁴³

{ɛ_{i}^{/} = Y_{i} - μ (η_{i} - ɛ_{i}) η_{i}^{/} = Y_{i} + μ (η_{i} - ɛ_{i})

(18)

where μ is the shrinkage factor, μ∈(0, 0.55).

In order to ensure that the new range is not out of bound, some measurements are dealt with it as follows: if ɛ_i^/<ɛ_i, ɛ_i^/= ɛ_i; if η_i^/>η_i, η_i^/= η_i.

Therefore, vector T_i of Y_i after reduction in the new interval [ɛ^/ _i , η^/ _i ] can be determined using

T_{i} = \frac{Y_{i} - ɛ_{i}^{/}}{η_{i}^{/} - ɛ_{i}^{/}}

(19)

The linear combination of T_i and Y_i_, _n ₊₁ will be expressed as a new chaotic variable and it is applied to chaotic fine search.

Y_{i_{, n + 1}}^{/} = Y_{i_{, n + 1}} + β_{i} (T_{i} - 0.55)

(20)

where β_i is the adaptive control coefficient, 0 < β_i < 1.

The adaptive control coefficient δ_i can be determined adaptively as follow

β_{i} = 1 - (\frac{K - i}{K})^{θ}

(21)

where θ is a positive integer that is determined according to the objective function (in this paper it is set to 2.5) and K is the evolutional generation.

And then the individuals whose similarity is greater than σ_s in 10% of individuals with larger fitness value will be eliminated in the memory base.

Step 6: x_r₊₁ = sin(2/x_r) is selected for generating N^/ individuals in (0, 1) to replace the individuals with poor affinity, then they are set as the antibodies for next immune computing together with the memory dataset T_p which is obtained from last immune computing, return to Step 3 until the network reaches to convergence.

Step 7: Use the fitness function f(C_rj, σ_rj) to evaluate C_rj* and σ_rj*, calculate the corresponding f(C_rj*, σ_rj*), if f(C_rj*, σ_rj*) > f(C_rj, σ_rj), then f(C_rj, σ_rj) = f(C_rj*,σ_rj*), or else, abandon C_rj* and σ_rj*.

Step 8: If the fitness function is the maximum which is greater than 1.0, then stop searching and output the optimal solution C_rj* and σ_rj*, otherwise return to Step 1.

Simulation verification of identification effect of FLS-SVM optimized by AVCIA

In order to verify the identification effect of AVCIA to optimize fuzzy least squares support vector machine(FLS-SVMAVCIA), three commonly used standard test (UCI) datasets are used in the experiments, and its identification effect is compared with the identification effect of the FLS-SVM and FLS-SVM optimized by particle swarm optimization algorithm (PSO) (FLS-SVMPSO).

Ripley dataset: The second Ripley dataset is adopted as the training set and the test set. The training set contains 800 samples (the number of positive class and negative class is 400, respectively), and the test set contains 1000 samples (the number of positive kind and negative kind is 500, respectively).

Monk dataset: The third Monk dataset in which noise points are added randomly is adopted, and the training set contains 800 samples (including 400 positive class and 400 negative class), the test set contains 1000 samples (including 500 positive class and 500 negative class).

Pima dataset: The total sample number of Pima dataset is 1800 (including 900 positive class and 900 negative class); 800 samples are randomly selected for training set and the remaining 1000 samples are set as test set in accordance with the dataset documents.

When the optimal precision of the identification and classification is satisfied, the optimal test precision and the corresponding parameters are shown in Tables 1 and 2, respectively, through these three methods, FLS-SVM method, the FLS-SVMPSO method and the FLS-SVMAVCIA method.

Table 1.

Optimal identification precision/%.

Datasets	FLS-SVM	FLS-SVMPSO	FLS-SVMAVCIA
Ripley	92.5	94.4	96.8
Monk	95.8	97.2	98.6
Pima	84.7	86.6	88.8

FLS-SVMPSO: fuzzy least square-support vector machine particle swarm optimization algorithm; FLS-SVM: fuzzy least square support vector machine; FLS-SVMAVCIA: fuzzy least square-support vector machine adaptive variable chaos immune algorithm.

Table 2.

Corresponding parameters correspond to identification precision.

Datasets	Parameters	FLS-SVM	FLS-SVMPSO	FLS-SVMAVCIA
Ripley	C	1.28	1.39	1.55
Ripley	σ	0.87	0.73	0.66
Monk	C	15.1	15.4	15.8
Monk	σ	0.82	0.76	0.74
Pima	C	17.8	18.1	18.6
Pima	σ	2.35	2.46	2.54

As shown in Table 1, the FLS-SVMAVCIA proposed in this paper can effectively improve the identification precision of data in the dataset with noisy points and anomalous points.

The computational complexity measured by CPU time is also compared among the FLS-SVM method, the FLS-SVMPSO method and the FLS-SVMAVCIA method. As shown in Table 3, compared with the CPU time of the FLS-SVM method and the FLS-SVMPSO method, the CPU time of the FLS-SVMAVCIA method is the shortest. Obviously, it is easy to simply conclude that the FLS-SVMAVCIA method is less computational expensive than the FLS-SVM method or the FLS-SVMPSO.

Table 3.

CPU Time corresponds to identification precision datasets.

Datasets	CPU time/s
Datasets	FLS-SVM	FLS-SVMPSO	FLS-SVMAVCIA
Ripley	0.967514	0.961424	0.958264
Monk	0.870963	0.861245	0.851268
Pima	0.795675	0.786256	0.778654

Establishing of the Preisach hysteretic model based on the improved FLS-SVM

Establishing of the primary hysteretic piezoelectric model and its parameters optimization

The primary attributes of the piezoelectric actuator in paper are expressed as follows: the range voltage of driver is from 0 to120 V and the amplitude is 300 µm.

To establish the hysteretic piezoelectric model, the main task is to determine the weight coefficients of the hysteretic units. So the FORCs method is introduced to obtain the test data and compute the weight coefficients with these data. The input voltage is divided into six segments, so the value of n in μ _n is equal to 6, and then the initial values of the weight coefficients are calculated. Figure 4 shows the FORCs curves through the test.

Figure 4.

FORCs curves.

In Figure 4, the outer loop represents the voltage–displacement diagram, while input increases from 0 V to 120 V and then decreases to 0 V. The inner loop represents the voltage–displacement diagram while input increases from 0 V to 20 V, 40 V, 6 0 V, 80 V and 100 V, respectively, and then decreases to 0 V. From Figure 4, it can be concluded that the piezoelectric actuator in paper has a large hysteresis phenomenon. If the actuator is used directly, there are problems that the accuracy cannot be controlled and some large errors will appear.

In order to calculate the initial weight factors based on the test data, the simplified model is introduced. According to equation (1), the integrand is the weight factor. So a suitable weight factor value can be treated as a mean value in the integration domain that is divided into some units. Product values of mean weight factor values and the corresponding areas are treated as the total value of output and the mean weight factor value is expressed as the weight matrix elements in equation (2). Thus, equation (2) can be rewritten as

f (t) = R [u (t)] = \int \int_{α \leq β} μ (α, β) γ_{α, β} [u (t)] d α d β = \bar{μ} (α, β) \cdot A (α, β)

(22)

where

\bar{μ} (α, β)

is the mean value matrix of each integral unit, and A(α, β) is the area matrix of the integral unit.

Before applying the FLS-SVMAVCIA method, assuming the coordinate of each unit’s centroid according to the element in matrix $\bar{μ} (α, β)$ as the input of the FLS-SVMAVCIA method and using the weight value as the output of FLS-SVMAVCIA method to train the FLS-SVMAVCIA model.

The initial mean weight factor value is obtained based on the FORCs curves in Figure 5 and the value matrix is shown as

{\bar{μ}}_{6} = [12.5 00000 2.5 11.68 0000 1.75 3.16 120002 2.25 4.75 7.9 00 2.5 2.5 0.5 4.8 12.8 0 3.25 2.95 3.18 3.43 1.93 10.25] \times 10^{- 5}

(23)

Figure 5.

Curve of the fitness function.

Finding the centroid of initial weight value of each unit in α-β plane, the coordinate of each centroid was calculated. Using the coordinate of each centroid as the input of the FLS-SVMAVCIA method and using the mean weight value of each unit as the output of the FLS-SVMAVCIA method, the model is trained and established. Applying the radial basis function as the kernel function, but the penalty factor c and kernel parameter σ must be optimized in the process of training due to the effect on the accuracy of the training model.

Figure 5 shows the curve of the fitness function in the process of parameters optimizing. From Figure 5, the value of fitness function is getting smaller when the iteration is increased, and finally the value of fitness function attains the required precision.

The penalty factor c and kernel parameter σ are obtained after the AVCIA optimization, c = 35, σ = 1.35. Thus, the parameters of the FLS-SVMAVCIA method are determined and the calculated model of the hysteretic model is established.

In order to solve more mean weight value, taking n = 12 which divides the α-coordinate and the β-coordinate into 12 segments, the interval is 10 V. Then centroid coordinates of each unit are calculated. The 78 calculated coordinates are used as the input of the FLS-SVMAVCIA method, then the 78 mean weight values is calculated. The result of the calculation is shown in Figure 6.

Figure 6.

Calculated mean weight values.

For obtaining the weight value which can be used in the discrete hysteretic model, it is important to multiply the mean weight value with the corresponding area. Thus, the initial piezoelectric hysteretic model is established.

Improvement of the initial piezoelectric hysteretic model

There is a disadvantage according to the theory of the discrete Preisach hysteretic model. The output of the discrete Preisach hysteretic model is toothed and the phenomenon is much more severe especially when the number of hysteretic units is small. In order to eliminate the inherent disadvantages of the toothed phenomenon, the interpolation method is introduced, thus the output curve of the model could be smooth and precise.

In order to compute the output of the hysteretic unit whose input is not on the specific voltage that was divided by 12 segments, the quadratic polynomial interpolation method is introduced to calculate the output of the model. Assume at one time, the input voltage is increased but the voltage value is not equal to the specific voltage, the quadratic polynomial interpolation can be expressed as

y = A_{0} + A_{1} \cdot β + A_{2} \cdot β^{2}

(24)

where A₀, A₁ and A₂ are the undetermined coefficients of the polynomial, respectively.

To obtain the values of the undetermined coefficients A₀, A₁, A₂, three values of β are needed. Assuming that the input voltage u(t) is between value β_q and value β_q+₁, then the β equals to β_q-₁, β_q, β_q+₁ and the corresponding output f(β_q-₁), f(β_q), f(β_q+₁) are obtained from the discrete Preisach hysteretic model, respectively; this leads to

{f (β_{q - 1}) = A_{0} + A_{1} \cdot β_{q - 1} + A_{2} \cdot β_{q - 1}^{2} f (β_{q}) = A_{0} + A_{1} \cdot β_{q} + A_{2} \cdot β_{q}^{2} f (β_{q + 1}) = A_{0} + A_{1} \cdot β_{q + 1} + A_{2} \cdot β_{q + 1}^{2}

(25)

The above equation reveals that q must be greater than or equal to 1. When the input voltage is less than β₁, the quadratic polynomial interpolation method is not used to compute the output of the hysteretic unit because two values of the input voltage are just obtained. And the linear interpolation method is introduced to compute intermediate value between two values of the input voltage.

The undetermined coefficients A₀, A₁ and A₂ are determined through equation (25) and the determined A₀, A₁ and A₂ can be put into equation (24) to establish the polynomial interpolation model. So the output of the hysteretic model is known when the input is u(t) to equation. (24). Similarly, when the input voltage is decreased, the following interpolation equation can be obtained

y = B_{0} + B_{1} \cdot α + B_{2} \cdot α^{2}

(26)

where B₀, B₁ and B₂ are the undetermined coefficients of the polynomial.

The determination of B₀, B₁ and B₂ is the same as the determination of A₀, A₁ and A₂.

The final hysteretic piezoelectric model is obtained after combining with the polynomial interpolation. The toothed phenomenon is eliminated and the accuracy of the model is better. Figure 7 shows the comparison of the outer loops between the unimproved discrete Preisach hysteretic model and the improved one.

Figure 7.

Comparison between unimproved and improved model. 1- the output of the hysteretic piezoelectric model which combines with the polynomial interpolation; 2- the output of the discrete Preisach hysteretic model.

In Figure 7, the output of the hysteretic piezoelectric model combined with the polynomial interpolation is improved, and the output of the discrete Preisach hysteretic model is unimproved. The comparison of the figure suggests that the curve of the output is smoother after combining with the polynomial interpolation than that without the polynomial interpolation, so the method of the polynomial interpolation has a good effect on the discrete hysteretic piezoelectric model.

Validating of the hysteretic piezoelectric model

In order to validate the piezoelectric hysteretic model, the simulated model and the test platform are established. The MATLAB/Simulink software is used to establish the simulation model of the piezoelectric hysteretic model. The simulation model consists of two main modules, which are the discrete Preisach hysteretic model and the polynomial interpolation model. Moreover, 78 mean-weighted values are calculated after the α-coordinate and the β-coordinate are divided into 12 segments with the interval of 10 V and then 78 calculated coordinates are used as the inputs of the FLS-SVM. Therefore, the discrete Preisach hysteretic model consists of 78 Relay models which are connected with each other in parallel, the parameters of the Relay models are the values of weight factors calculated in “Establishing of the primary hysteretic piezoelectric model and its parameters optimization”. The polynomial interpolation model consists of equations in “Improvement of the initial iezoelectric hysteretic model” and the changing rate of the input. Figure 8 shows the connection of the computational model, where [A] represents the calculated median which can be transferred from the discrete hysteretic Preisach model to the polynomial interpolation model through “go to” module.

Figure 8.

Simulink model of the piezoelectric hysteretic model.

In order to meet the demands of contrast, the Preisach hysteretic model based on the FLS-SVM is established and the corresponding Simulink model is constructed, the principle and the steps of the constructing are similar to that of the FLS-SVMAVCIA. And Figure 9 shows the comparison diagram of the outer loops with different methods of FLS-SVM, FLS-SVMAVCIA and the test data.

Figure 9.

Testing data and the simulation data. 1-The testing data; 2-the simulated data of the FLS-SVMAVCIA model; 3-the simulated data of the FLS-SVM model.

In Figure 9, it can be concluded that the error of the FLS-SVM model is large most of the time and the error of the FLS-SVMAVCIA model is smaller than that of the FLS-SVM model. Therefore, the capability of the FLS-SVMAVCIA model to fit the test data is very well.

As shown in Figure 10, the errors of the fitted curve by using of the FLS-SVMAVCIA method are very small and the maximum error of the curve is about 2.2 µm, so it is a good method to solve the hysteretic problem.

Figure 10.

Error of the curve fitting. (a) Error of the bottom curve. (b) Error of the upper curve.

In order to validate the error of inner loop of the hysteretic loop, different reverse voltages are set. The input voltage is increased from 0 V to the set reversed voltage and then decreased to the input voltage, the errors of the simulation model and the test data in inner loop are shown in Table 4.

Table 4.

Comparison between simulated model and the test data in inner loop.

Reverse voltage/V	Model	Absolute mean error/µm	Maximum error/µm
20	FLS-SVM	1.63	2.52
20	FLS-SVMAVCIA	0.35	0.82
40	FLS-SVM	1.88	2.87
40	FLS-SVMAVCIA	0.41	1.08
60	FLS-SVM	2.15	3.11
60	FLS-SVMAVCIA	0.56	1.28
80	FLS-SVM	2.52	3.45
80	FLS-SVMAVCIA	0.66	1.56
100	FLS-SVM	2.84	3.58
100	FLS-SVMAVCIA	0.81	1.78

FLS-SVM: fuzzy least square support vector machine; FLS-SVMAVCIA: fuzzy least square-support vector machine adaptive variable chaos immune algorithm.

From Table 4, the absolute mean error and the maximum error of the FLS-SVM model are large, but the absolute mean error of the FLS-SVMAVCIA model is less than 1 µm and the maximum error of the FLS-SVMAVCIA model is less than 2 µm. So the hysteretic model based on the FLS-SVMAVCIA has a good fitting under the inner loop condition.

Through the comparison, the output of hysteretic model based on the FLS-SVMAVCIA has the same effect to the output of the real piezoelectric actuator, so the parameters identification of the hysteretic Preisach model based on the improved FLS-SVM in the paper is an effective method.

Conclusions

Based on the FORCs, the obtained 78 weight factors are used as the input of the FLS-SVM in the training model, the AVCIA is used to optimize the FLS-SVM, a method of FLS-SVMAVCIA is established to calculate the weight factors of the piezoelectric hysteretic model.

Simulink is applied to establish the piezoelectric hysteretic model with FLS-SVMAVCIA method, the comparison results show that the hysteretic Preisach model based on the FLS-SVMAVCIA method is an effective method to eliminate the hysteretic phenomenon.

The potentiality of the paper is the benefit for the development and usage of the piezoelectric actuators and the development of the precise control engineering. In further work, the frequency factor should be considered, especially at the working frequency range of the actuators.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge Project (9140A2011QT4801) supported by weapons and equipment pre-research fund and the National Studying Abroad Foundation Project (No.201208430262) supported by the China Scholarship Council.

Appendix

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