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Approximate Analytical Solution of the
Bouc-Wen Hysteresis Model by the Fourier Transform
Denis V. Kozlov
Abstract — an analytical method for solving nonlinear differential equation that describes the Bouc-Wen model of dynamic hysteresis is proposed. Solution is sought in the form of successive approximations based on the Fourier transforms. The results confirm the efficiency of this method.
IndexTerms — Bouc-Wen model, Fourier transform, Hysteresis, Method of Successive Approximations
I. INTRODUCTION
B
ouc-Wen model is often used to describe the dynamic hysteresis loop. It has an analytical form, and well describes such effects as the degrading of stiffness and strength, the pinching effect, etc. [22]. However, analysis of qualitative and quantitative properties of systems with hysteresis causes serious difficulties. This is related to the fact that the model is described by nonlinear differential equations and contains the unknown coefficients which must be identified.
To use a well-developed apparatus of the analysis of linear systems, as a rule, many authors pass from nonlinear model to linear model. The most effective and commonly used methods of transition are equivalent linearization under the assumption of Gaussian [2, 3, 24] and non-Gaussian [12, 19, 20] nature of the coefficients of the linearized model, the direct linearization technique [8]. Also known to use various types of polynomial approximations [3, 11, 14], Volterra series [5], and Volterra/Wiener neural networks [16, 17].
In this paper proposed find an approximate analytic solution of nonlinear equation of the Bouc-Wen model, without resorting to the procedures of linearization and approximation.
II. Determination of approximate solutions
Generalized Bouc-Wen model has the form [22]
(1)
where and denote the input and output model of hysteresis respectively, and are constants, which control the scale and shape of the hysteresis curve.
Function in the general case is
(2)
where is constants.
A. Proposed approach
Equation (1) can be written as
(3)
Integral Laplace transform is often used for solving ordinary differential equations [6, 18]. This method is so effective that known attempts to use it in the case of nonlinear problems. For example, in [13, 15] have been obtained approximate solutions of nonlinear second order systems at a specified time interval on application of right-sided Laplace transform. Solution was considered to be zero for.
As opposed to [13, 15], equation (3) depends on the variable, that can take not only positive but also negative values, and for at that. Therefore necessary to use the bilateral Laplace transform instead of right-sided Laplace transform. However, in this case may arise significant difficulties in calculating the images of some functions, associated with definition the path of integration and strip of convergence [23]. For this reason, will use the Fourier transform, which is similar to the bilateral Laplace transform and defined as
(4)
where is the operator of the direct Fourier transform, is the operator of the inverse Fourier transform.
For simplicity, all further calculations will be made with .
B. First-order approximation
Perform a direct Fourier transform on the equations (3)
(5)
where is Dirac delta-function.
From (5) have
(6)
or as a first-order approximation, following [6],
(7)
Passing from the image to the original, obtain
(8)
C. Second-order approximation
To find the second-order approximation, substitute (8) into (6) and perform the inverse Fourier transform
(9)
As is well known [4, 21]
(10)
where is the gamma-function and
Then (9) takes the form
(11)
Also from [4] follows that
(12)
for
Expression (11) after simplifications, based on (12), transformed to
(13)
The subsequent approximation of can be defined similarly
(14)
for and .
D. Third-order approximation
To find the third-order approximation at first compute the
(15)
where
(16)
The multiplier can expand into Newton’s binomial series
(17)
The module in (17) is omitted, because the series is convergent if .
Further represent (17) as
(18)
where
(19)
After calculating the product (19), write (17) as
(20)
Substituting (20) into (15) and again using (10), obtain
(21)
Define the inverse Fourier transform
(22)
Finally, in accordance with (13) the third-order approximation write in the following form:
(23)
where
(24)
If in (23), (24) restrict to the first term in the infinite sum, the expression is completely coincide with the second-order approximation (13). If assume that than in the limit can obtain a first-order approximation (8). Thus, high-order approximations contain all low-order approximations.
E. Fourth-order approximation
Compute the following expression:
(25)
where
(26)
Similarly (20), possible to write
(27)
Consider separately the raising to the power of infinite series (26). Introduce the notations for convenience
(28)
From (26) and (28) after some transformations, obtain
(29)
where
(30)
From [1, 9, 10] is known following recurrent relation:
(31)
for and .
It is allows to define the first coefficients of series by known coefficients of series (30). If , then obtain the coefficients of the series .
Recording (31) can be generalized to determine all coefficients of the series. Omitting the cumbersome computations, write in the final form
(32)
where , , and for .
From (28) find, that
(33)
Substituting (33) and (32) into (27), have
(34)
By the additivity of the Fourier transform, write
(34)
Define the inverse Fourier transform
(35)
Substituting (35) into (14), find the fourth-order approximation
(36)
The third-order approximation can be obtained from (36) if omit the second term in brackets and change variable.
Calculation of the fifth and subsequent approximations is based on using the formulas (10), (12), (20), (31), (32) and does not cause serious difficulties, except the transformation cumbersome expressions.
III. Analysis of the solutions
Graphs of the numerical solution of equation (1) and approximate analytical solutions of different orders when the input initial condition and coefficients are shown in figures1-3. In this case, the value satisfies the condition
Accuracy of the solution of equation (3) is increasing slightly with increasing of approximation order. For these reasons, as the function will take the fourth approach, ie
Graphs under the same conditions, but the other input are shown in figures4, 5. In the region observed large deviations of all approximations from the actual behavior of z (t) because the value
IV. Conclusion
Proposed in this paper a method of successive approximations for solution of the nonlinear Bouc-Wen model of hysteresis is very effective. It allows to obtain an analytical solution without resorting to the procedure of linearization. In subsequent works will be constructed approximate solutions when and , based on provision method.
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Denis Kozlov received the B.S. degree in electrical engineering, electromechanics and electrotechnics from Tula State University, Tula, Russia, in 2004, and the M.S. degree in electrical equipment of enterprise in 2005. He is currently working toward the Ph.D. degree in system analysis, management and information processing at Tula State University, Tula, Russia.
His research interests include analysis of dynamic systems, optimal control, sliding control, high-precision motion control, and position control of elastic mechanical systems.
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Manuscript received May 30, 2011.
D.V.Kozlov is with the Department of Automatic Control System, Tula State University, Russia (e-mail: ).