X - Semantic Scholar
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The Simulation and Application Research of an Improved SSI Algorithm Wang Xiao-Dong, Zhou Kai, Ye Qing-Wei and Wu Li-Gang
Information Science and Engineering College of Ningbo University, Ningbo, China Email: [email protected]
Abstract—As a linear system identification method developed in recent years, Random Subspace Algorithm (SSI) can effectively extract modal parameters from environmental incentive structure. However, this algorithm is not suitable for the embedded system due to its long calculation and low efficiency. For this consideration, improved from the data-based classical stochastic subspace algorithm, this article puts forward the partial projection SSI algorithm with higher efficiency. The basic idea of the improvement is to use the part of the output data rather than all data as a “past” signal, which greatly reduce the calculation and improve the efficiency of the algorithm as the result. Finally the simulation test and actual application of the improved algorithm show that the improved algorithm can achieve experimental results faster, which is still ideal even in strong noise environment. This algorithm improves the calculation efficiency with no loss of accuracy. Index Terms—random subspace algorithm, part of the projection, embedded, computational efficiency
I. INTRODUCTION The time-domain identification of experimental modal parameters refers to the identification of the structural modal parameters in time domain. As a technology developed with the application of computers in recent years, the research and application of time domain method go later than frequency domain method. It’s difficult to measure the work load of some large complex structure such as bridges, dams, high buildings, aircrafts and ships when affected by wind, wave or the pulse of the earth while the response signal is easy to detect. Therefore, the direct use of response signal for parameter identification is undoubtedly of great significance. When applying model analysis to some large building structures, it’s often difficult to identify the model parameters due to the high experimental cost when imposing the artificial incentive on these building structures so that the modal parameter identification method is developed based on environment excitation. Vibration modal parameter identification method is the core component of modal analysis theory for vibration signal [1-4]. At present, based on environment excitation, Manuscript received January 21, 2013; revised March 11, 2013; accepted April 3, 2013.
© 2013 ACADEMY PUBLISHER doi:10.4304/jsw.8.9.2246-2252
modal parameter identification methods are mainly frequency domain decomposition method [3-4], ERA [56] and the stochastic subspace method [7-9].Literature [7] proposed a stochastic subspace method algorithm based on eigenvalue decomposition, which removes the Hankel matrix QR decomposition and projection matrix SVD algorithm, so it greatly saves money and computing time and improves the computational efficiency as the result. Literature [8] proposed the stochastic subspace method to identify vibration system modal parameters based on reference point with no output covariance estimation. The basic idea improved in ref. [9] is to adopt a point of “the past” as a signal output instead of the total station. Databased random subspace is one of two algorithms commonly used in SSI algorithm. The essence of databased SSI algorithm is: to directly formed Hankel matrices with the output data, to calculate the output in the future to the past on the output of the projection with QR decomposition in linear algebra method, to do singular value decomposition (SVD)the projection and expand the observable matrix and system state kalman filter. Due to environmental stimulation it has nothing to do with the state of the system, the kalman filter and the output can be plugged into the system state space equation to resolve the correlation matrix of the system, and obtain the system's modal parameters. II. STATE SPACE DESCRIPTIONS OF SYSTEM VIBRATION EQUATION Suppose that one system which has n2 degrees of freedom is stimulated. The discrete mechanics system of second order expression is as follows [10-13]: ••
•
M U (t ) + C 2 U (t ) + KU (t ) = F (t )
Where M ∈ R
n2 *n2
F(t ) = B2u(t)
the mass matrix; C 2 ∈ R
the damping matrix K ∈ R
n2 *n2
n2 *n2
(1) is
is the stiffness matrix;
F (t ) ∈ R n 2 *1 is the exciting force, U (t ) ∈ R n2 *1 for n *m the displacement vector; B2 ∈ R 2 matrix to describe m*1 the input position; u (t ) ∈ R for describing the input
vector changes with time. By the state space theory in control theory, the equation (1) can be transformed and modals are reduced to get the continuous deterministic state model of the system
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⎧⎪ • (t ) = Ac x(t ) + Bc u (t ) ⎨x ⎪⎩ y (t ) = Cx(t ) + Du (t ) ⎛ U (t ) ⎞ ⎟, where, x(t ) = ⎜ • ⎜ (t )⎟ ⎝U ⎠
⎛ 0 Ac = ⎜⎜ -1 ⎝- M K
(2)
⎛ 0 ⎞ Bc = ⎜⎜ -1 ⎟⎟ ⎝ M B2 ⎠
In ⎞ ⎟, - M -1C 2 ⎟⎠
,
C ∈ R l *n as output matrix; D ∈ R l*m as direct transmission matrix , y (t ) ∈ R
l *l
Y0, 2i -1
for the output.
The discretization equation (2) consideration of the disturbance of noise:
taking
into
⎧ x k +1 = Axk + Bu k + wk (3) ⎨ ⎩ y k = Cx k + Du k + v k Here: xk = x(kΔt ) is the state vector. A = exp ( Ac Δ t ) ,
the discrete state matrix ;the dynamic performance of the system characterized by its characteristic value; l *n -1 B = [ A - I ]A c B c for the discrete input matrix; C ∈ R
for the output matrix, D ∈ R
y j-2 y j-1 ⎤ ⎡ y0 y1 y2 ⎢y y y3 … y j-1 y j ⎥⎥ 2 ⎢ 1 y j+1 ⎥ Y0，2i-1 = ⎢ y2 y3 y4 … y j ⎢ ⎥ ⎢ ⎥ ⎢ y2i-1 y2i y2i+1 ⎥ y y 2i+ j-3 2i+ j-2 ⎦ ⎣
l*m
for matrix directly,
u k ∈ R m*l and vector y k ∈ R l*L respectively for the system’s input signal and output signal. wk v k is zero mean white noise, which means : Ewk = 0 and vector
Ev = 0 .In equation (3),if u (t ) ≈ 0, equation (3) will k
change into a stochastic state space model.
⎧ x k +1 = Axk + wk ⎨ ⎩ y k = Cx k + v k III. CLASSICAL DATA-BASED SSI METHOD
Hankel matrix constructed by output data is defined. In equation (5), the first number of the suffix of Hankel Matrix is the time coefficient of the left upper corner elements while the second number is the time coefficient of the left lower corner elements. Since random system need sufficient statistical analysis, Hankel matrix is divided into two parts, which is called “the past" output and "the future" output respectively. If j → ∞ is assumed. In equation (6), the number of rows of the "past" and the "future" matrix are I, in which Y p and Y f are respectively the output of the "past" and "future".
y j -1 yj
(6)
The defined O i is the projection on the "future" output to the "past" output.
Oi ≡
Yf
(7)
Yp
Singular value decomposition (SVD) is carried out on the projection
⎛S 0⎞⎛V T ⎞ Oi =USV= (U1U2 )⎜⎜ 1 ⎟⎟⎜⎜ 1T ⎟⎟ =U S1V1T ⎝ 0 0⎠⎝V2 ⎠ 1 Among
them:
U
and
V
(8)
for
orthogonal
matrix, U U = UU = I , V V = VV = I ; S is composed from the singular value of the order of the diagonal matrix. Projection of Oi is the product of the T
T
observable
T
matrix ∧
X
i
Γi
and
T
Kalman
filter
.
∧
(9) O i = Γi X i Through the projection of the singular value decomposition, the observable matrix and Kalman filter estimation can be obtained. 1
Γi = U 1 S1 2 ∧
X
(10) (11)
1
i
= S 1 2 V 1T
According to equation (4), the system's state equation can get the characteristics of the system matrices: ⎡ X k +1 ⎤ ⎡A⎤ ⎥X ⎢C ⎥ = ⎢ Y ⎢⎣ k ⎦⎥ ⎣ ⎦
Yk = ( y 0
y1
(12)
+ k
⎡ w k ⎤ ⎡ X k +1 ⎤ ⎡ A ⎤ ⎢ v ⎥ = ⎢ Y ⎥ − ⎢C ⎥ ( X k ) ⎣ k⎦ ⎣ k ⎦ ⎣ ⎦ Among the X k = (x k x k +1 inverse of X k .
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⎤ ⎥ ⎥ ⎥ ⎥ y i + j - 2 ⎥ ⎡ Y0,i -1 ⎤ ⎡ Y p ⎤ ≡⎢ ⎥≡⎢ ⎥ y i + j -1 ⎥ ⎢⎣ Yi , 2 i -1 ⎥⎦ ⎣⎢ Y f ⎦⎥ ⎥ yi+ j ⎥ ⎥ ⎥ y 2 i + j -2 ⎥⎦
⎡ y 0 y1 ⎢ y ⎢ 1 y2 ⎢ ⎢ y i -1 y i ≡⎢ ⎢ yi y i +1 ⎢ y y i+2 ⎢ i +1 ⎢ ⎢ ⎢⎣ y 2 i -1 y 2 i
estimation (4)
(5)
(13)
x k +i −1 )
,
y k −1 ) , X k+ is the generalized
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IV. FUNDAMENTAL FREQUENCY EXTRACTION ALGORITHM BASED ON IMPROVED SSI ALGORITHM A. The Improved Random Subspace Algorithm - part of the Projection Method of SSI In this algorithm, the basic idea of the improved calculation efficiency of SSI method is to replace all the data with part of the output data as a "past" signal in equation (6). First of all, with the output data of Hankel matrices ⎡ y0r ⎢ r ⎢ y1 ⎢ y2r ⎢ ⎢ ⎢ yr i -1 Y0,2i-1 = ⎢ ⎢ yi ⎢y ⎢ i +1 ⎢ yi +2 ⎢ ⎢ ⎢ y2i-1 ⎣
y1r y2r y3r r i
y2r y3r y4r
yrj-2 yrj-1 yrj
r i +1
r i + j -3
y yi +1 yi +2 yi +3
y yi +2 yi +3 yi +4
y yi + j -2 yi + j -1 yi + j
y2i
y2i +1
y2i + j -3
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ yir+ j -2 ⎥ ⎡Ypim ⎤ ⎥ =⎢ ⎥ yi + j -1 ⎥ ⎢Yfim ⎥ ⎣ ⎦ yi + j ⎥ ⎥ yi + j +1 ⎥ ⎥ ⎥ y2i + j -2 ⎥ ⎦ yrj-1 yrj yrj+1
(14)
In equation (14), the cross line divides the matrix into two parts, the upper part is "the past" output Y pim composed of part of the output signal ,the lower part is im
the "future" output Y f
,composed of all the output
signals. Among them, y i matrix is constituted by a part r
of the output signal. While y i is matrix is constituted by all the output signals.
y = Ly i
Among them:L is the selection matrix. Suppose that there are m output data, in which the number of the data selected as "the past" is r. The first come-out output data totaled r is taken as the selected output, r=m/64 test.
0] ∈ R
r *l
(16) Again on this basis, the definition of the "future" output is the projection of the part of the "past" output: im i
O
≡
Y fim
∧
Y pim
= Γi X i
(17) im
In the same way, singular value decomposition of Oi
gets the observation matrix Γi and the kalman filter ∧
sequence X
i
. According to (9), it can be got:
∧
X i = Γi+ Oiim The same method can be obtained:
(18)
∧
(19)
O iim− 1 ≡
( Y fim ) −
( Y pim ) +
Among them Γi −1 =
= Γ i −1 * X
Γi , Γi _
i +1
is Γi line off the last
_
piece of the matrix. According to the equation (19): ∧
X
i +1
B. Cable Vibration Signal's Fundamental Frequency Extraction Algorithm The improved SSI algorithm described above shows that this algorithm can be used to extract the modal parameter of all orders from embedded system quickly and conveniently after obtaining the vibration response signal. The algorithm's main idea is shown in Figure 1, first, the output data directly form a block Hankel matrix, QR decomposition method is used to calculate the projection of the future output over the past output, the observable matrix expanded and system state kalman filter can be obtained by decomposing singular value of the projection; Then Kalman filter and output are substituted into the system’s state space equation to resolve the system function. Finally the fundamental frequency is obtained by system function. Free response signal output data yk The improved SSI algorithm The kalman filter state sequence The least square method
(15)
r i
L = [I r
According to the equation (12) , the system matrix A of the structure and least square solution of output matrix C can be got. The literature (7) shows that system modal parameters can be obtained after the system matrix A and C (natural frequency, damping ratio, modal shape).
= Γ i+−1 O i −1
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(20)
System matrix
The fundamental frequency
Cable force
Figure1. Schema of whole algorithm
V. PRECISION AND EFFICIENCY TEST OF THE IMPROVED ALGORITHM ON ARM DEVELOPMENT BOARD Software and hardware of the development board: S3C6410 ARM11 processor, ARM1176JZF -s kernel[14], dominant frequency 533 MHZ, 128 MB Mobile DDR memory, 256 MB of RAM, android version: 2.3.3 Now suppose equation (21) as the vibration the analog signal for simulation. Set t as uniform sampling point on the interval [0, 2], sampling points of each signal are 4096.
y (t ) = 2 * e −5*t * cos(2 * pi *11* t + 0.1) + 1.7 * e −3*t * cos(2 * pi * 21* t + 0.3) w(t ) = 0.5 * Math.random( );
(21)
(22) The signal-to-noise ratio of the vibration system is defined as:
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SNR = 10 * log 10
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(∑ y (t ) / ∑ Noise (t , D )) 2
2
(23)
Among them, Noise (t , D ) is the random even distribution noise with strength D. y (t ) is the response signal of the system, w(t) is the noise signal. Under the condition of different noise intensity (on the same piece of development board) results of improved SSI algorithm
were shown in the figure 2 below: The ordinate is the error between the system frequency and ideal frequency, the abscissa is the signal-to-noise ratio. Figure2 shows that this algorithm also can achieve ideal result under the strong noise.
Figure 2. Algorithm in the development board different SNR of the results
Figure 3. The improved algorithm running time contrast
The improved algorithm on the development board on the running time is shown in figure 3, the abscissa for the
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simulated signal points. Y coordinate for the corresponding points on the Android terminals running
2250
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time, the unit is in seconds. The improved algorithm efficiency increased about 4 times. VI. ALGORITHM APPLICATION Taking vibration signal measured by Ningbo cablestayed bridge as the test data; this paper compares the calculation results and the default settings of the improved algorithm with the data-based classic stochastic
Figure 4. Ningbo bridge vision diagram
subspace algorithm to verify the feasibility and accuracy of the algorithm. Data acquisition sensors are acceleration sensor and the sampling frequency Fs =128 Hz. The sampling points of each signal are 1024 points. Bridge and sensor installation diagram are shown in figure 4,5. The signal waveform graph acquired by cable vibration is shown in figure 5:
Figure 5. The bridge sensor installation method
High modes The fundamental frequency
Figure 6. Random vibration response signal waveform figure
Figure 5 is the actual acquisition signal waveform. The above is the time domain signal waveform diagram of the cable signal. The following is the corresponding signal spectrum of Fig. Spectrum diagram, the highest peak is high-order modes, about 0.8Hz peak is the required fundamental modal. Due to dense modal interference
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between, as well as the noise interference, and the error of the FFT spectrum itself leads to fundamental frequency peak recognition error is bigger, it is difficult to obtain precise stable fundamental frequency values directly from the FFT spectrum.
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TABLEI. MANY ROOT STAY-CABLES VIBRATION SIGNAL OF THE ALGORITHM AND IMPROVED MODAL PARAMETER VALUE, ERROR, THIS ALGORITHM RUNNING TIME COMPARISON CHART
cable no. C18 C19 C20
Algorithm SSI The improved SSI SSI The improved SSI SSI The improved SSI
frtv(HZ) 0.78 0.76 0.72
In the table 1: frtv means fundamental frequency theoretical value; frmv: fundamental frequency measured values; fre: fundamental frequency error. This article mainly focuses on how to obtain the system’s fundamental frequency quickly in the embedded system by using stochastic subspace algorithm to calculate the modal parameters by the vibration signal. According to Formula 1 of literature [12], obtaining accurate fundamental frequency (the first order natural frequency) is the key to calculate cable force. To compare the fundamental frequency, error and calculation time calculated based on three groups of vibration signal data collected from the three cables randomly selected with the data-based classic stochastic subspace algorithm, Table 1 shows that the efficiency of the new algorithm in embedded system obviously improves. VII. CONCLUSIONS This paper introduces the improvement to the method of stochastic total space, i.e., using part of the output data to replace all the data as "the past" signal. Stochastic subspace method procedure is to establish the relationship between the “past” output data and future status to further obtain the system matrix through the output data and state equation. Finally the improved random subspace algorithm is applied to the embedded terminal to obtain the fundamental frequency of the cable-stayed bridge. According to the test results, the calculation time of the improved algorithm is much shorter while the calculation accuracy is no lower than the algorithm before the improvement. This advantage is more obvious when the output signal is large. It’s worth promotion since it achieves convenience and high efficiency for the application of the embedded project in the bridge inspection system. ACKNOWLEDGMENT This work was supported in part by a grant from China natural science fund projects (61071198), and this work also has been financially supported by Welfare Scientific Research of Zhejiang Province (2010C31087). This work has been financially supported by Nature Science Funds of Ningbo city, China, grant No. 2012A610019. REFERENCES
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frmv(HZ) 0.752 0.741 0.745 0.733 0.714 0.730
running time (s) 216 43 215 44 216 43
fre (dimensionless) 4% 5% 2% 4% 1% 2%
[1] Chengbing He, Yujiong Gu, “Fault Statistics and Mode Analysis of Steam-Induced Vibration of Steam Turbine Based on FMEA Method,” Journal of Computers, Vol. 6, No 10 (2011), 2098-2103, Oct 2011. [2] Junfeng Hu, Dachang Zhu, "Vibration Control of Smart Structure Using Sliding Mode Control with Observer," Journal of Computers, Vol. 7, No 2 (2012), 411-418, Feb 2012. [3] Ling Xiang, Aijun Hu, "Comparison of Methods for Different Time-frequency Analysis of Vibration Signal," Journal of Software, Vol. 7, No 1 (2012), 68-74, Jan 2012. [4] Tongying Li, Zhenchao Zhang, "Vibration Data Processing Based on Petri Network in Wireless Sensor Networks," Journal of Networks, Vol. 7, No 2 (2012), 400-407, Feb 2012. [5] Lingmi.Zhang,Tong, Wang,Yukio.Tamura, “A frequency– spatial domain decomposition (FSDD) method for operational modal analysis,” Mechanical Systems and Signal Processing, Volume 24, Issue 5, July 2010, pp. 1227-1239 [6] Jorge. Larrey-Ruiz,” Frequency domain regularization of d-dimensional structure tensor-based directional fields,” Image and Vision Computing, Volume 29, Issue 9, August 2011, pp.620-630 [7] Qi .quan-quan,XIN.Ke-gui,CUI.Ding-yu, “The Application of the extended eigensystem realization algorithm for structural modal parameter identification [J],” Engineering Mechanics,2011,28（3）:29-34 [8] Jerome K.Vanclay, “Publication patterns of award-winning forest scientists and implications for the Australian ERA journal ranking,” Engineering Mechanics,Journal of Informetrics, Volume 6, Issue 1, January 2012, Pages 1926 [9] ZHANG. Guo-wen, TANG. Bao-ping.MENG, Li-bo, “improved stochastic subspace identification algorithm based on eigenvalue decomposition [J],” JOURNAL OF VIBRATION AND SHOCK,2012, 31(7):74-78 [10] Xu. Fuyou, Chen.Airong , Zhu. Shaofeng,“Identification of modal parameters for bridge wind tunnel test by using a stochastic subspace technique[J],”China Civil Engineering Journal, Vol.40 (10):67-73, 2007 [11] CHANG. Jun, SUN. Limin,ZHANG Qiwei, “Improvement in stochastic subspace identification[J],” Journal of Earthquake Engineering and Engineering Vibration, 2007, 27(3):88-94 [12] Peeters, B. and De Roeck, G, “Reference based stochastic subspace identification for output-only modal analysis,” Mechanical Systems and Signal Processing, Vol.13 (6), pp.855-878, 1999 [13] Ren. W.X. and Zong. Z.H, “Output-only modal parameter identification of civil engineering structures,” Structural engineering and Mechanics, Vol.17, pp. 420-444, 2004.
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[14] Viberg. M. “Subspace-based system identification circuits systems,” Signal Processing, Vol.21, No.1, pp23-27, 2002. [15] DING. Feng, XIAO. De-yun, “ Hierarchical identification of state space models for multivariable systems [J],” Control and Decision,2005,20（8）:848-853. [16] Daode Zhang, Yurong Pan, Xinyu Hu, "Design of HighSpeed Parallel Data Interface Based on ARM & FPGA," Journal of Computers, Vol. 7, No 3 (2012), 804-809, Mar 2012.
Wang Xiao-Dong received BSc from Ningbo University in 1992 and MSc from Zhejiang University in 2007. He is an associate professor in Ningbo University. His main research interests include vibration signal processing and multimedia.
© 2013 ACADEMY PUBLISHER
© 2013 ACADEMY PUBLISHER
Zhou Kai received BSc from Xinhua College in 2010. He is currently pursuing MSc in Ningbo University. He is a graduate student in Ningbo University. His main research interests include vibration signal processing. Ye Qingwei received BSc from Ningbo University in 1992 and PhD from Chongqing University in 2010.He is an associate professor in Ningbo University. His main research interests include vibration signal processing and optimzation. Wu Li-Gang received Ph .D from Shanghai Institute of Technical Physics, Chinese Academy of Sciences. He is an associate professor .His main research interests include photoelectric detection technology.
© 2013 ACADEMY PUBLISHER
The Simulation and Application Research of an Improved SSI Algorithm Wang Xiao-Dong, Zhou Kai, Ye Qing-Wei and Wu Li-Gang
Information Science and Engineering College of Ningbo University, Ningbo, China Email: [email protected]
Abstract—As a linear system identification method developed in recent years, Random Subspace Algorithm (SSI) can effectively extract modal parameters from environmental incentive structure. However, this algorithm is not suitable for the embedded system due to its long calculation and low efficiency. For this consideration, improved from the data-based classical stochastic subspace algorithm, this article puts forward the partial projection SSI algorithm with higher efficiency. The basic idea of the improvement is to use the part of the output data rather than all data as a “past” signal, which greatly reduce the calculation and improve the efficiency of the algorithm as the result. Finally the simulation test and actual application of the improved algorithm show that the improved algorithm can achieve experimental results faster, which is still ideal even in strong noise environment. This algorithm improves the calculation efficiency with no loss of accuracy. Index Terms—random subspace algorithm, part of the projection, embedded, computational efficiency
I. INTRODUCTION The time-domain identification of experimental modal parameters refers to the identification of the structural modal parameters in time domain. As a technology developed with the application of computers in recent years, the research and application of time domain method go later than frequency domain method. It’s difficult to measure the work load of some large complex structure such as bridges, dams, high buildings, aircrafts and ships when affected by wind, wave or the pulse of the earth while the response signal is easy to detect. Therefore, the direct use of response signal for parameter identification is undoubtedly of great significance. When applying model analysis to some large building structures, it’s often difficult to identify the model parameters due to the high experimental cost when imposing the artificial incentive on these building structures so that the modal parameter identification method is developed based on environment excitation. Vibration modal parameter identification method is the core component of modal analysis theory for vibration signal [1-4]. At present, based on environment excitation, Manuscript received January 21, 2013; revised March 11, 2013; accepted April 3, 2013.
© 2013 ACADEMY PUBLISHER doi:10.4304/jsw.8.9.2246-2252
modal parameter identification methods are mainly frequency domain decomposition method [3-4], ERA [56] and the stochastic subspace method [7-9].Literature [7] proposed a stochastic subspace method algorithm based on eigenvalue decomposition, which removes the Hankel matrix QR decomposition and projection matrix SVD algorithm, so it greatly saves money and computing time and improves the computational efficiency as the result. Literature [8] proposed the stochastic subspace method to identify vibration system modal parameters based on reference point with no output covariance estimation. The basic idea improved in ref. [9] is to adopt a point of “the past” as a signal output instead of the total station. Databased random subspace is one of two algorithms commonly used in SSI algorithm. The essence of databased SSI algorithm is: to directly formed Hankel matrices with the output data, to calculate the output in the future to the past on the output of the projection with QR decomposition in linear algebra method, to do singular value decomposition (SVD)the projection and expand the observable matrix and system state kalman filter. Due to environmental stimulation it has nothing to do with the state of the system, the kalman filter and the output can be plugged into the system state space equation to resolve the correlation matrix of the system, and obtain the system's modal parameters. II. STATE SPACE DESCRIPTIONS OF SYSTEM VIBRATION EQUATION Suppose that one system which has n2 degrees of freedom is stimulated. The discrete mechanics system of second order expression is as follows [10-13]: ••
•
M U (t ) + C 2 U (t ) + KU (t ) = F (t )
Where M ∈ R
n2 *n2
F(t ) = B2u(t)
the mass matrix; C 2 ∈ R
the damping matrix K ∈ R
n2 *n2
n2 *n2
(1) is
is the stiffness matrix;
F (t ) ∈ R n 2 *1 is the exciting force, U (t ) ∈ R n2 *1 for n *m the displacement vector; B2 ∈ R 2 matrix to describe m*1 the input position; u (t ) ∈ R for describing the input
vector changes with time. By the state space theory in control theory, the equation (1) can be transformed and modals are reduced to get the continuous deterministic state model of the system
© 2013 ACADEMY PUBLISHER
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⎧⎪ • (t ) = Ac x(t ) + Bc u (t ) ⎨x ⎪⎩ y (t ) = Cx(t ) + Du (t ) ⎛ U (t ) ⎞ ⎟, where, x(t ) = ⎜ • ⎜ (t )⎟ ⎝U ⎠
⎛ 0 Ac = ⎜⎜ -1 ⎝- M K
(2)
⎛ 0 ⎞ Bc = ⎜⎜ -1 ⎟⎟ ⎝ M B2 ⎠
In ⎞ ⎟, - M -1C 2 ⎟⎠
,
C ∈ R l *n as output matrix; D ∈ R l*m as direct transmission matrix , y (t ) ∈ R
l *l
Y0, 2i -1
for the output.
The discretization equation (2) consideration of the disturbance of noise:
taking
into
⎧ x k +1 = Axk + Bu k + wk (3) ⎨ ⎩ y k = Cx k + Du k + v k Here: xk = x(kΔt ) is the state vector. A = exp ( Ac Δ t ) ,
the discrete state matrix ;the dynamic performance of the system characterized by its characteristic value; l *n -1 B = [ A - I ]A c B c for the discrete input matrix; C ∈ R
for the output matrix, D ∈ R
y j-2 y j-1 ⎤ ⎡ y0 y1 y2 ⎢y y y3 … y j-1 y j ⎥⎥ 2 ⎢ 1 y j+1 ⎥ Y0，2i-1 = ⎢ y2 y3 y4 … y j ⎢ ⎥ ⎢ ⎥ ⎢ y2i-1 y2i y2i+1 ⎥ y y 2i+ j-3 2i+ j-2 ⎦ ⎣
l*m
for matrix directly,
u k ∈ R m*l and vector y k ∈ R l*L respectively for the system’s input signal and output signal. wk v k is zero mean white noise, which means : Ewk = 0 and vector
Ev = 0 .In equation (3),if u (t ) ≈ 0, equation (3) will k
change into a stochastic state space model.
⎧ x k +1 = Axk + wk ⎨ ⎩ y k = Cx k + v k III. CLASSICAL DATA-BASED SSI METHOD
Hankel matrix constructed by output data is defined. In equation (5), the first number of the suffix of Hankel Matrix is the time coefficient of the left upper corner elements while the second number is the time coefficient of the left lower corner elements. Since random system need sufficient statistical analysis, Hankel matrix is divided into two parts, which is called “the past" output and "the future" output respectively. If j → ∞ is assumed. In equation (6), the number of rows of the "past" and the "future" matrix are I, in which Y p and Y f are respectively the output of the "past" and "future".
y j -1 yj
(6)
The defined O i is the projection on the "future" output to the "past" output.
Oi ≡
Yf
(7)
Yp
Singular value decomposition (SVD) is carried out on the projection
⎛S 0⎞⎛V T ⎞ Oi =USV= (U1U2 )⎜⎜ 1 ⎟⎟⎜⎜ 1T ⎟⎟ =U S1V1T ⎝ 0 0⎠⎝V2 ⎠ 1 Among
them:
U
and
V
(8)
for
orthogonal
matrix, U U = UU = I , V V = VV = I ; S is composed from the singular value of the order of the diagonal matrix. Projection of Oi is the product of the T
T
observable
T
matrix ∧
X
i
Γi
and
T
Kalman
filter
.
∧
(9) O i = Γi X i Through the projection of the singular value decomposition, the observable matrix and Kalman filter estimation can be obtained. 1
Γi = U 1 S1 2 ∧
X
(10) (11)
1
i
= S 1 2 V 1T
According to equation (4), the system's state equation can get the characteristics of the system matrices: ⎡ X k +1 ⎤ ⎡A⎤ ⎥X ⎢C ⎥ = ⎢ Y ⎢⎣ k ⎦⎥ ⎣ ⎦
Yk = ( y 0
y1
(12)
+ k
⎡ w k ⎤ ⎡ X k +1 ⎤ ⎡ A ⎤ ⎢ v ⎥ = ⎢ Y ⎥ − ⎢C ⎥ ( X k ) ⎣ k⎦ ⎣ k ⎦ ⎣ ⎦ Among the X k = (x k x k +1 inverse of X k .
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⎤ ⎥ ⎥ ⎥ ⎥ y i + j - 2 ⎥ ⎡ Y0,i -1 ⎤ ⎡ Y p ⎤ ≡⎢ ⎥≡⎢ ⎥ y i + j -1 ⎥ ⎢⎣ Yi , 2 i -1 ⎥⎦ ⎣⎢ Y f ⎦⎥ ⎥ yi+ j ⎥ ⎥ ⎥ y 2 i + j -2 ⎥⎦
⎡ y 0 y1 ⎢ y ⎢ 1 y2 ⎢ ⎢ y i -1 y i ≡⎢ ⎢ yi y i +1 ⎢ y y i+2 ⎢ i +1 ⎢ ⎢ ⎢⎣ y 2 i -1 y 2 i
estimation (4)
(5)
(13)
x k +i −1 )
,
y k −1 ) , X k+ is the generalized
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IV. FUNDAMENTAL FREQUENCY EXTRACTION ALGORITHM BASED ON IMPROVED SSI ALGORITHM A. The Improved Random Subspace Algorithm - part of the Projection Method of SSI In this algorithm, the basic idea of the improved calculation efficiency of SSI method is to replace all the data with part of the output data as a "past" signal in equation (6). First of all, with the output data of Hankel matrices ⎡ y0r ⎢ r ⎢ y1 ⎢ y2r ⎢ ⎢ ⎢ yr i -1 Y0,2i-1 = ⎢ ⎢ yi ⎢y ⎢ i +1 ⎢ yi +2 ⎢ ⎢ ⎢ y2i-1 ⎣
y1r y2r y3r r i
y2r y3r y4r
yrj-2 yrj-1 yrj
r i +1
r i + j -3
y yi +1 yi +2 yi +3
y yi +2 yi +3 yi +4
y yi + j -2 yi + j -1 yi + j
y2i
y2i +1
y2i + j -3
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ yir+ j -2 ⎥ ⎡Ypim ⎤ ⎥ =⎢ ⎥ yi + j -1 ⎥ ⎢Yfim ⎥ ⎣ ⎦ yi + j ⎥ ⎥ yi + j +1 ⎥ ⎥ ⎥ y2i + j -2 ⎥ ⎦ yrj-1 yrj yrj+1
(14)
In equation (14), the cross line divides the matrix into two parts, the upper part is "the past" output Y pim composed of part of the output signal ,the lower part is im
the "future" output Y f
,composed of all the output
signals. Among them, y i matrix is constituted by a part r
of the output signal. While y i is matrix is constituted by all the output signals.
y = Ly i
Among them:L is the selection matrix. Suppose that there are m output data, in which the number of the data selected as "the past" is r. The first come-out output data totaled r is taken as the selected output, r=m/64 test.
0] ∈ R
r *l
(16) Again on this basis, the definition of the "future" output is the projection of the part of the "past" output: im i
O
≡
Y fim
∧
Y pim
= Γi X i
(17) im
In the same way, singular value decomposition of Oi
gets the observation matrix Γi and the kalman filter ∧
sequence X
i
. According to (9), it can be got:
∧
X i = Γi+ Oiim The same method can be obtained:
(18)
∧
(19)
O iim− 1 ≡
( Y fim ) −
( Y pim ) +
Among them Γi −1 =
= Γ i −1 * X
Γi , Γi _
i +1
is Γi line off the last
_
piece of the matrix. According to the equation (19): ∧
X
i +1
B. Cable Vibration Signal's Fundamental Frequency Extraction Algorithm The improved SSI algorithm described above shows that this algorithm can be used to extract the modal parameter of all orders from embedded system quickly and conveniently after obtaining the vibration response signal. The algorithm's main idea is shown in Figure 1, first, the output data directly form a block Hankel matrix, QR decomposition method is used to calculate the projection of the future output over the past output, the observable matrix expanded and system state kalman filter can be obtained by decomposing singular value of the projection; Then Kalman filter and output are substituted into the system’s state space equation to resolve the system function. Finally the fundamental frequency is obtained by system function. Free response signal output data yk The improved SSI algorithm The kalman filter state sequence The least square method
(15)
r i
L = [I r
According to the equation (12) , the system matrix A of the structure and least square solution of output matrix C can be got. The literature (7) shows that system modal parameters can be obtained after the system matrix A and C (natural frequency, damping ratio, modal shape).
= Γ i+−1 O i −1
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(20)
System matrix
The fundamental frequency
Cable force
Figure1. Schema of whole algorithm
V. PRECISION AND EFFICIENCY TEST OF THE IMPROVED ALGORITHM ON ARM DEVELOPMENT BOARD Software and hardware of the development board: S3C6410 ARM11 processor, ARM1176JZF -s kernel[14], dominant frequency 533 MHZ, 128 MB Mobile DDR memory, 256 MB of RAM, android version: 2.3.3 Now suppose equation (21) as the vibration the analog signal for simulation. Set t as uniform sampling point on the interval [0, 2], sampling points of each signal are 4096.
y (t ) = 2 * e −5*t * cos(2 * pi *11* t + 0.1) + 1.7 * e −3*t * cos(2 * pi * 21* t + 0.3) w(t ) = 0.5 * Math.random( );
(21)
(22) The signal-to-noise ratio of the vibration system is defined as:
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SNR = 10 * log 10
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(∑ y (t ) / ∑ Noise (t , D )) 2
2
(23)
Among them, Noise (t , D ) is the random even distribution noise with strength D. y (t ) is the response signal of the system, w(t) is the noise signal. Under the condition of different noise intensity (on the same piece of development board) results of improved SSI algorithm
were shown in the figure 2 below: The ordinate is the error between the system frequency and ideal frequency, the abscissa is the signal-to-noise ratio. Figure2 shows that this algorithm also can achieve ideal result under the strong noise.
Figure 2. Algorithm in the development board different SNR of the results
Figure 3. The improved algorithm running time contrast
The improved algorithm on the development board on the running time is shown in figure 3, the abscissa for the
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simulated signal points. Y coordinate for the corresponding points on the Android terminals running
2250
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time, the unit is in seconds. The improved algorithm efficiency increased about 4 times. VI. ALGORITHM APPLICATION Taking vibration signal measured by Ningbo cablestayed bridge as the test data; this paper compares the calculation results and the default settings of the improved algorithm with the data-based classic stochastic
Figure 4. Ningbo bridge vision diagram
subspace algorithm to verify the feasibility and accuracy of the algorithm. Data acquisition sensors are acceleration sensor and the sampling frequency Fs =128 Hz. The sampling points of each signal are 1024 points. Bridge and sensor installation diagram are shown in figure 4,5. The signal waveform graph acquired by cable vibration is shown in figure 5:
Figure 5. The bridge sensor installation method
High modes The fundamental frequency
Figure 6. Random vibration response signal waveform figure
Figure 5 is the actual acquisition signal waveform. The above is the time domain signal waveform diagram of the cable signal. The following is the corresponding signal spectrum of Fig. Spectrum diagram, the highest peak is high-order modes, about 0.8Hz peak is the required fundamental modal. Due to dense modal interference
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between, as well as the noise interference, and the error of the FFT spectrum itself leads to fundamental frequency peak recognition error is bigger, it is difficult to obtain precise stable fundamental frequency values directly from the FFT spectrum.
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TABLEI. MANY ROOT STAY-CABLES VIBRATION SIGNAL OF THE ALGORITHM AND IMPROVED MODAL PARAMETER VALUE, ERROR, THIS ALGORITHM RUNNING TIME COMPARISON CHART
cable no. C18 C19 C20
Algorithm SSI The improved SSI SSI The improved SSI SSI The improved SSI
frtv(HZ) 0.78 0.76 0.72
In the table 1: frtv means fundamental frequency theoretical value; frmv: fundamental frequency measured values; fre: fundamental frequency error. This article mainly focuses on how to obtain the system’s fundamental frequency quickly in the embedded system by using stochastic subspace algorithm to calculate the modal parameters by the vibration signal. According to Formula 1 of literature [12], obtaining accurate fundamental frequency (the first order natural frequency) is the key to calculate cable force. To compare the fundamental frequency, error and calculation time calculated based on three groups of vibration signal data collected from the three cables randomly selected with the data-based classic stochastic subspace algorithm, Table 1 shows that the efficiency of the new algorithm in embedded system obviously improves. VII. CONCLUSIONS This paper introduces the improvement to the method of stochastic total space, i.e., using part of the output data to replace all the data as "the past" signal. Stochastic subspace method procedure is to establish the relationship between the “past” output data and future status to further obtain the system matrix through the output data and state equation. Finally the improved random subspace algorithm is applied to the embedded terminal to obtain the fundamental frequency of the cable-stayed bridge. According to the test results, the calculation time of the improved algorithm is much shorter while the calculation accuracy is no lower than the algorithm before the improvement. This advantage is more obvious when the output signal is large. It’s worth promotion since it achieves convenience and high efficiency for the application of the embedded project in the bridge inspection system. ACKNOWLEDGMENT This work was supported in part by a grant from China natural science fund projects (61071198), and this work also has been financially supported by Welfare Scientific Research of Zhejiang Province (2010C31087). This work has been financially supported by Nature Science Funds of Ningbo city, China, grant No. 2012A610019. REFERENCES
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frmv(HZ) 0.752 0.741 0.745 0.733 0.714 0.730
running time (s) 216 43 215 44 216 43
fre (dimensionless) 4% 5% 2% 4% 1% 2%
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[14] Viberg. M. “Subspace-based system identification circuits systems,” Signal Processing, Vol.21, No.1, pp23-27, 2002. [15] DING. Feng, XIAO. De-yun, “ Hierarchical identification of state space models for multivariable systems [J],” Control and Decision,2005,20（8）:848-853. [16] Daode Zhang, Yurong Pan, Xinyu Hu, "Design of HighSpeed Parallel Data Interface Based on ARM & FPGA," Journal of Computers, Vol. 7, No 3 (2012), 804-809, Mar 2012.
Wang Xiao-Dong received BSc from Ningbo University in 1992 and MSc from Zhejiang University in 2007. He is an associate professor in Ningbo University. His main research interests include vibration signal processing and multimedia.
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Zhou Kai received BSc from Xinhua College in 2010. He is currently pursuing MSc in Ningbo University. He is a graduate student in Ningbo University. His main research interests include vibration signal processing. Ye Qingwei received BSc from Ningbo University in 1992 and PhD from Chongqing University in 2010.He is an associate professor in Ningbo University. His main research interests include vibration signal processing and optimzation. Wu Li-Gang received Ph .D from Shanghai Institute of Technical Physics, Chinese Academy of Sciences. He is an associate professor .His main research interests include photoelectric detection technology.
