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IDENTIFICATION OF MODAL PARAMETERS USING KALMAN FILTER
Kim Kwang Ju ОПРЕДЕЛЕНИЕ МОДАЛЬНЫХ ПАРАМЕТРОВ С ИСПОЛЬЗОВАНИЕМ ФИЛЬТРА КАЛМАНА Ким Кван Чжу
Ким Кван Чжу / Kim Kwang Ju — преподаватель, Политехнический университет им. Ким Чака, г. Пхеньян, Корейская Народно-Демократическая Республика
Abstract: the application of system identification to vibrating structures consists of identifying the modal parameters (eigenfrequencies, damping ratios and mode shapes) from vibration data. For the dynamic characteristics, the control theory based on the transfer function representation is called the classical control theory, in contrast with, the methodology of the linear system theory based on the analysis of the time series by Kalman filter and the representation of the state space is called modern control theory. In this paper, we consider the methodology of identifying the mode parameters of the dynamic system of structures by using the Kalman filter, which is a powerful means of modern control theory. The effectiveness of this structure identification method is evaluated through simulated analysis of multi - degrees of freedom vibration syteam.
Аннотация: применение системы идентификации для вибрационных структур состоит в определении модальных параметров (собственных частот, демпфирующих коэффициентов и формы колебаний) из данных о вибрации. Для динамических характеристик, теория управления, основанная на передаточной функции представления, называется классической теорией управления, в отличие от методологии линейной теории систем на основе анализа временных рядов с помощью фильтра Калмана, и представление пространства состояний называется современной теорией управления. В этой статье мы рассмотрим методику идентификации модальных параметров динамической системы структур с помощью фильтра Калмана, который является мощным средством современной теории управления. Эффективность этого метода идентификации структуры оценивается через моделируемый анализ нескольких степеней свободы вибрации.
Keywords: Kalman filter, modal analysis, normal frequence, mode shap, damping ratio. Ключевые слова: фильтр Калмана, модальный анализ, собственная частота, формы колебаний, коэффициент демпфирования.
1. State-space model
The equations of motion for an nd degrees-of-freedom (DOF) linear, time invariant, viscously damped system subjected to external excitation are expressed as
Mz (t) + Ccz(t) + Kz(t) =/u(t) (1)
Where are the mass, damping and stiffness matrices, respectively;
is the excitation influence matrix that relates the -dimensional input vector to the -dimensional response vector; is the -dimensional displacement response vector; dot denotes taking derivatives with respect to time.
By defining the state vector , equation (1) can be converted into the
continuous state space form
x(t) = Ac x(t) + Bcu(t) (2)
Where
A„ =
Br =
0
м-1;.
(3)
0 I
-M~1K
In practice, only a limited number of measurements are available; therefore, the dimension of the measurement output is less than or equal to the total number of degrees of freedom. The -dimensional output vector can be expressed as
y ( t) = C d( t) z ( t) + Cv Z(t) + Ca Z(t) (4) Where are the measurement location matrices corresponding to the
displacement, velocity and acceleration responses of the structural system. We can rewrite the output vector into the continuous state space form,
y ( t) = Cc x(t) + Dc u(t) (5)
Where
c c = [ cd - CaM - ^ CF - CaM - 1 cc] , Dc = CaM - 1 / (6)
In practical application, accelerations are often used commonly, so in this work, only accelerations are considered. Therefore, the Cc of equations (6) is as simple as follows.
c c = c a [ -M - ^ -M - 1 Cf] (7)
Equations 2 and 5 define the state space equation in continuous time:
x ( t) = Ac x( t) + Bc u( t) (8a) y( t) = Q x (t) + Dc u (t) (8b)
Equation (8a) is known as the State Equation and equation (8b) is known as the Observation Equation. But measurements are taken in discrete time instants, so equations must be expressed in discrete time too.
Typical for the sampling of a continuous-time equation is a Zero-Order Hold assumption, which means that the input is piecewise constant over the sampling period, that is
Vt e [tk,tk+1) = [kAt, (fe + 1)A£) =>x(t) = x(tk) = u ( 0 = u (tfe) = ufc,y( t) = y (tfe) = yfc (9)
Under this assumption, the continuous time state-space model (8a) and (8b) is converted to the discrete time state-space model:
x 1=A xfc+B ufc (10a) yfc = C xfc + D ufc (10b)
Where xk is the discrete time state vector containing the sampled displacements and velocities; uk and yk are the sampled input and output; A is the discrete state matrix; B is the discrete input matrix; C is the discrete output matrix; D is the discrete direct transmission matrix. They are related to their continuous-time counterparts as ([2])
A = e A ^ A £, B = (A-/) A - 1 Bc (11) , (12)
In system identification, system response disturbance might be caused by different phenomena. The most obvious one is noise generated by the sensors, or noise arising from round off errors during A/D conversion.
It is necessary to extend the state space model (10a) and (10b) including stochastic components, so stochastic state space model is obtained.
xfc+1=A xfc+B ufc+wfc (13a) yfc = C xfc + D ufc + vfe (13b)
Where is the process noise due to disturbances and modeling inaccuracies;
is the measurement noise due to sensor inaccuracy.
We assume they are both independent and identically distributed, zero-mean normal vectors. wfc ~ JV ( 0 ) ~ JV ( 0 ) (14)
2. The Kalman filter
Due to the noise present in the stochastic state space Equations (13), it is only possible to predict the response in term of probability. For state space systems, this prediction is accomplished by the construction of the associated Kalman filter.
For the state space model specified in (13) with initial conditions x{J = [i 0 and Pq = S0, for k = 1,2,...,N
x fc- ^A x£- 1 (15)
(16)
1 .
xfc - ^/ffc (17)
(18)
Where Kfe = Pfc - 1 CT S-1 (19)
(20)
= Var( efe) = Var [ C (xfc - xfc - ^ + vfe] = C Pfc - 1 CT + P (21) is called the Kalman gain and are the innovations. Under stationary conditions,
1 i mk - 1 = P > 0 (22)
(23)
(24)
3. System identification and modal analysis in a state-space model
The natural frequencies and modal damping ratios can be retrieved from the eigenvalues of A, and the mode shapes can be evaluated using the corresponding eigenvectors and the output matrix C. The eigenvalues of A come in complex conjugate pairs and each pair represents one physical vibration mode.
Assuming low and proportional damping, the second order modes are uncoupled and the jth eigenvalue of A has the form
Aj = exp ^ ( - cj ± i cj Ji-%] j At j (25)
Where Cj are the natural frequencies, S j are damping ratios, and At is the time step. Natural frequencies and the damping ratios are given by
c = MM, s = (26)
j At ' ^ aij At v '
The jth mode shape p j E Rn0 evaluated at sensor locations can be obtained using the following expression: p j = C p j (27) Where p j is the complex eigenvector of A corresponding to the eigenvalue Aj 4. Verification through numerical simulation
In order to verify the validity of the proposed method in this paper, a three degree of freedom vibration structure system as following (figure 1).
Fig. 1. 3 degrees of freedom vibration structure system
In figure 1, external excitation is applied through point m3 and is expressed as u(t). The physical parameters in the given structure vibration system are set as follows. m1=10kg, m2=15kg, m3=20kg k1=10kg, k2=15kg, k3=20kg cj= 3n/s, c2 = 5n/s, c3=10n/s
As the external excitation u(t), we used triangluar form signal as shown in figure 2.
Fig. 2. External excitation diagram according to time
Random noise with a covariance corresponding to 10% of nominal values was added in viscous coefficiances c1, c2, c3. At the same time,a random noise with a variance corresponding to 5% of the excitation maximum value was added in excitation. We added random noise corresponding with the measurement noise level of low cost acceleration sensors to measurement values.
From table 1, it can be seen that relative error between theorical values and identification results is less than 15% in damping ratio and less than 10% in the eigenfrequences and mode shape. That is, modal parameters were well identificated even in the presence of process noise and measurement noise.
From now on, the validity of the method proposed in this paper was proved.
Comparison of theorical value and identification result Table 1.
Table 1. Shows the results of the identification of the modal parameters obtained by using the Kalman filter algorithm
Normal frequence, Hz
theory identification relative error,%
1th 2.1014 1.9901 5.4785
2th 7.9821 8.5580 7.2147
3th 11.9635 11.2068 6.3254
Damping ratio,%
theory identification relative error,%
1th 6.5 5.89 9.5
2th 0.4 0.443 10.8
3th 0.4 0.459 14.8
Mode shape
theory identification relative error,%
(0.5878 (0.6063 (3.1457
1th 0.8875 0.8354 5.8741
1.000) 1.000) 0)
(1.000 (0 (0
2th 0.3070 0.2875 6.3651
-0.4982) -0.4796) 3.7415)
(-0.8693 (-0.9436 (8.5417
3th 1.000 0 0
-0.4102) -0.3818) 6.9214)
5. Conclusion
In this paper, we proposed the methodology to indentify modal parameters of a structure vibration system by using kalman filter algorithm, which becomes one of the powerful methods of modern control theory.
By using the kalman filter algorithm, it is possible to identify modal parameter optimally even in the presence of process noise and measurement noise exists.
The performance and validity of the proposed methodology was verificated through simulation application.
References
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2. Grant P. M., Cowan C. F. N., Mulgrew B. and Dripps J. H. "Analogue and Digital Signal Processing and Coding". Chartwell-Bratt Ltd, 1989.
3. Catlin. D. E. Estimation, Control and the Discrete Kalman Filter. In Applied Mathematical Sciences 71, page 84. Springer-Verlag, 1989.
4. Shumway R. H. andStoffer D. S. Time series analysis and its applications. Springer, 2006.
5. Verhaegen M., Verdult V. Filtering and System Identification. A least squares approach Cambridge University Press., 2007.
