Identification of System with Bouc-Wen Hysteresis

A method of adaptive identification of parameters for a system with Bouc-Wen hysteresis has been developed. The method is based on the use of adaptive observers and resolves the problem of stability of the identification system. Adaptive algorithms of identification were obtained using the Lyapunov second method. The stability proof of the adaptive system is based on the application of Lyapunov vector functions. Adaptive algorithms for adjusting model parameters were developed obtained and finiteness of trajectories in the adaptive system was pointed out.


Introduction
The Bouc-Wen (BWM) model is widely used to describe a hysteresis [1][2][3][4][5][6][7][8][9][10][11]. System with BWM has the form where 0 m > is mass, 0 c > is damping, ( , , ) F x z t is the recovering force, 0 d > , 0 n > , 0 k > , (0,1) α ∈ , ( ) f t is exciting force, , , a β g are some numbers. Equation (3) is the BWM. Many modifications of BWM [3] have been proposed. Each proposed model considers features of the considered object. The degree of success of the BWM application strongly depends on the identification of its parameters. The solution of the nonlinear equation (3) is the main problem of the BWM identification. A three-level algorithm [4], based on regression analysis, least squares or Gauss-Newton methods, and the extended Kalman filter, is applied to Bouc-Wen model identification. The relevant approach has been applied in [5,6]. Adaptive algorithms were proposed in [8,9] for the BWM parameters estimation with the data forgetting [7]. Paper [10] presents an adaptive on-line identification methodology with a variable trace method to adjust the adaptation gain matrix.
Examples [11] are known when BWM parameters estimations do not agree with results obtained for other inputs. Such examples confirm the identification ambiguity which causes instability of the model and emphasize that the Bouc-Wen model should be stable in order to provide an adequate description of a physical process [12].
The analysis of literature shows that accounting for features of the object being examined is critically important for the development of algorithms and procedures for the identification of Bouc-Wen model parameters and that. the main difficulties in the estimation of the BWM parameters are related to the model stability and choice of the input . As a rule, the range, in which BWM parameters are varying, should be specified and all the derivatives of the object are should be measured. However, this is not always happenning that makes the application of the proposed algorithms ineffecient.
In this work, the adaptive identification method based on adaptive observer application [13] is used for the solution of the model (3) stability problem. The systemspecified by the equations (1)-(3) is considered. It is assummedthat the input ( ) f t and the output ( ) x t are measured.

Problem statement
Let us consider the system BW S (1)-(3). Let y be the output of the system. The set of the experimental data has the form Problem: to design the adaptive observer for vector estimation A of the system BW S that satisfy the condition where ŷ R ∈ is the output of the adaptive observer, 0 y π ≥ .

About identifiability of BW S -system
The identification effectiveness of the system BW S depends on the features of the input ( ) f t . The requirements to ( ) f t in identification problems are known. The force ( ) f t satisfies the condition of constant excitation (CE). This condition is necessary, but not sufficient [14]. The input having the CE property cannot ensure the identifiability of the hysteresis structure. The structural identifiability of the hysteresis is guaranteed with the input ( ) f t having the Sstabilisation property of the system BW S [14]. Conditions of this property verification are given in [14]. x a x a p a p bp

Adaptive system identification
, , , (5) and (6) contain only measurable variables, except for z . This complicates the identification process of the system BW S parameters. Now we apply the model to the parameters estimation of (5), where 0 x k > is the specified number; ˆ( ) i a t , 1, 2, 3 i = , and ˆ( ) b t are adjusted parameters. Let ê x x = − . Obtain the equation for the identification error from (5), (7) The (8) is not solvable as the variable z in (6). is unknown Now we shall obtain the current estimation for ( ) z t . Consider model After that we etermine by the misalignment ẑ z x x ε = − and use it for the variable z estimation.
Let r z ε be the current estimation z . After applying the model to the estimation of z , we get where ( ) Let present (7) as Then, (8) can be rewriten as and adaptive algorithms is described , , where 0, 1, 2,3 Tuning algorithms for β ∆ and g ∆ into (10) have the following form where 0 1 , υ υ are given positive numbers, 0 n g > .
Remark. Stability of the identification procedure is the main problem the solution of the system with BWM. We proposed the method based on the application of adaptive observers. Another solution to the stability problem is to change the structure of the equation (3). We proposed the equation ω > . to describe hysteresis.

Properties of adaptive system
Let consider the subsystem X AS described by (14) and (15).
where ( ) Assumption 1. The input ( ) f t is constantly exciting and limited.
The theorem 1 shows the limitation of adaptive system trajectories. The asymptotical stability ensuring the system demands to impose additional conditions.
where 0 ν > is some number.
Then the system X AS is exponentially stable with the estimation Theorem 2 shows that the adaptive system X AS gives the true estimates for parameters of the system (1). This is fair at the fulfilment of conditions (24). We assume that the variable ẑ p restricted. The boundedness of the variable ˆz x follows from the boundedness of the system X AS trajectories. Let us consider subsystem Z AS described by equations (12), (16) and introduce Lyapunov functions exists, where Ω is the definition range of the subsystem Hence, the boundedness of trajectories in the adaptive system is proved. The analysis showed that the subsystem X AS is asymptotically stable. The prove of trajectories boundedness for the subsystem Z AS is a more complex problem in parametrical and output spaces. This problem is solvable if the condition (28) is satisfied. The estimation (27) shows that the quality of processes in the Z AS -system depends on the output derivative of the BW S -system. The following result given more exact estimations for processes in the Z AS -system.  Let us construct the framework ey S  (Fig. 2), using the method [14], and evaluate the structural identifiability of the system BW S . The variable e R ∈  is x  is the estimation of the steady state (process) in the BW S -system for 9.85 t ∀ ≥ s, and e  is the hysteresis output estimation. Fig. 1 and 2 show that definition ranges z and e  coincides. The analysis of ey S  shows that the system BW S is structurally identifiable, and the input ( ) f t is S-stabilizing.
Let usonsider the identification of the system BW S parameters. After that we determine by the parameter µ of the system Eq. (13) using the transient process analysis for e  and 9.85 t < s. Now we alculate Lyapunov exponents (LE) [16]. The estimation for the maximum LE is -0.9. Therefore, we set 0.8 µ = . Initial conditions in (6) are equal to zero.
The results of the work of the adaptive system are presented in Figs. 3-7. Parameters x k , z k are equal to 2.5 and 0.75. The tuning process of X AS -systems (the model (7)) parameters are shown in Fig. 3, Fig. 4 shows the tuning parameters of the model (10). The modification of identification errors , e ε are shown in Fig. 5. We can see that the accuracy of obtained estimations depends on numbers of tuned parameters and the x  level, and properties ( ) f t .
Obtained results confirm statements of theorems 3, 4.
The results of Z AS -system work influence the tuning processes in the X AS -system. Gain coefficients in (15), (16) and (17)  The hysteresis output estimation process is shown in Fig. 6.

Conclusion
The adaptive parameter identification method for the system with Bouc-Wen hysteresis has been developed. The method is based on the application of adaptive observers and solves the stability problem. Adaptive algorithms of model parameter tuning have been designed, and the boundedness of trajectories in the adaptive system is shown. Estimates of uncertainty, which are used for the tuning of the hysteresis model parameters, have been obtained.
We show that the boundary of the system exponential dissipativity area is determined by the system output derivative level.