The Application of Energy Operator Demodulation Approach Based on EMD in Mechanical System Identification

Transcription

1 0 9th International Conference on Mechatronics and Machine Vision in Practice (MVIP), 8-30th Nov 0, Auckland, New-Zealand The Application of Energy Operator Demodulation Approach Based on EMD in Mechanical System Identification Jie Guo,, Shaoqian Qin,3, Chang an Zhu,4 Department of Precision Machinery and Precision Instruments, University of Science and Technology of China, Hefei, China 3 4 Abstract The dynamic properties, such as the natural frequencies and damping ratios, of ultra-precise positioning systems are needed for accurately predicting the response to external excitations and achieving high precision. This paper proposes the energy operator demodulation approach based on the empirical mode decomposition (EMD) method for realizing parametric identification, which has high identification precision and low calculation. Firstly, the EMD method is used to decompose measured impulse response of a MDOF structure into a number of modal responses. Then the energy operator demodulation approach is applied to each modal response to calculate the instantaneous amplitudes and instantaneous frequencies. Finally, the linear least-square fit procedure is performed to accurately estimate the system parameters, such as the natural frequencies and damping ratios. The theoretical analysis and simulation results demonstrate that the energy operator demodulation approach based on EMD can be successfully applied to system identification. Keywords-system identification; empirical mode decomposition; Hilbert transform; energy operator I. INTRODUCTION The ultra-precise positioning system plays an important role[, ] in many industries, such as grating ruling machine, semiconductor manufacturing system and high-precision microscope (STM, AFM). Due to its high precision, accurately predicting the response to external excitations requires the information of dynamic properties including natural frequencies and damping ratios, while these can be obtained by means of the techniques of mechanical model analysis and system identification. Many methods of linear system identification have been developed in [3]. Generally, these methods can be classified into two classes: time-domain and frequency-domain[4], for example, time-domain data-driven methods and frequencydomain spectrum-driven methods. Other than these methods, techniques based on time-frequency analysis have also received considerable attention in recent years. Recently, a method of decomposing a signal in the time-frequency domain using the empirical mode decomposition (EMD) method and the Hilbert transform (HT) has been proposed by Huang et al.[5, 6]. With the help of the EMD method, a signal can be decomposed into intrinsic mode functions (IMFs) while each IMF admits well-behaved Hilbert transform. Then after applied the HT for demodulation, the instantaneous phase angle and amplitude of each IMF can be obtained, which form the decomposition of the signal in the time-frequency domain. Such an approach is referred to as the Hilbert Huang transform (HHT) and is applicable to non-stationary and nonlinear signals. Meanwhile the HHT method has also been successfully adopted to identify modal parameters of multiple-degrees-offreedom (MDOF) linear structures[7]. The simulation results demonstrated that the method had much remarkable accuracy in identifying modal parameters compared with other system identification methods and owned the ability to identify not only the natural frequencies and damping ratios, but also the mode shapes as well as the mass, stiffness, and damping matrices of structures. In the HHT applications, the HT plays an important role. However, owing to the inevitable window effect of HT, the results may present non-instantaneous response characteristic, that is, at the two ends of the modulated signal there would produce modulation again. This will produce demodulation errors and cause inconvenience to parametric identification. To remedy this problem, the Teager energy operator (TEO) demodulation approach is proposed and applied to the system identification. The approach of TEO demodulation also can extract the instantaneous amplitude and frequency of a modulated signal and the demodulation effect is superior to the method of HT while the computation time is decreased greatly [8, 9]. In [0], the TEO demodulation in conunction with the EMD method was proposed and successfully applied to the machinery fault diagnosis. In this paper, the TEO method for system identification is proposed. Through combining it with the EMD method, the theoretical analysis and simulation results confirm its effectiveness. The paper is organized as follows. The ultraprecise positioning system is introduced in Section. Then in Section 3 the fundamental theories and methods including impulse response formulae of MDOF system, EMD, HT and TEO are presented. The application of the proposed identification method and its simulation results are finally illustrated in Section 4. Copyright AUT University Auckland,New-Zealand 80

2 II. ULTRA-PRECISION POSITIONING SYSTEM The ultra-precise positioning system is a macro/micro dualdrive system, composed of an outside stage and an inside stage as shown in Fig.. m is the th modal mass, and d = (-ζ ) / is the th damped modal frequency. So the displacement impulse response x p (t) of the structure at pth (p=,,n) DOF is given by n () = φ () = () (3) x t q t x t p p p = = where ζ () () t π xp t = φpq t = B ke cos dt+ + ϕ k n (4) B k F0 φp φk = (5) m d in which ϕ k is the phase difference between the pth element and the kth element in the th mode shape. With the existence of normal modes, all the mode shapes are real and hence ϕ k is either ±mπ or ±(m+) π where m is an integer. Figure. The ultra-precise positioning system The outside stage, with the purpose of realizing a long traveling range, is driven by a servo motor along the horizontal sliding guides. The transmission system includes a gear box, a turbine worm device and a feed screw nut pair. On the other hand, the inside stage, hung in the outside stage by four leaf springs, utilizes a piezoelectric actuator for nanometer-level positioning. The piezoelectric actuator is installed between the two stages and fixed on the outside stage as shown in Fig.. The other side of the piezoelectric actuator is tightly packed on the inside stage by two preloaded springs. The outside stage driven by servo motor can realize macro positioning while the piezoelectric-ceramic-driven inside stage achieves the micro. Finally the positioning precision of this system is less than 5nm. To grasp the dynamic properties of this MDOF system and realize high precision, mechanical system identification technique is required. III. THEORIES AND METHODS A. Modal Response of Multi-Degree-of-Freedom System The equation of motion of a general n-dof system can be expressed as MX t + CX t + KX t = F t () () () () () in which X(t)=[x,x,,x n ] T is a n-displacement vector, F(t) is a n-excitation vector, and M, C and K are (n n) mass, damping and stiffness matrices. With the assumption of the existence of normal modes and an impact loading applied to the kth DOF, i.e. f k (t)=f 0 δ(t) and f (t)=0 for all k, where f (t) is the th element of F(t), the displacement response of the th generalized modal co-ordinate is given by F0φ () k ζ q t t = e sindt () m in which φ k is the kth element of the th modal shape vector Φ, is the th modal frequency, ζ is the th modal damping ratio, d B. Empirical Mode Decomposition In (3),the impulse displacement response is a sum of n amplitude-modulated (AM) mono-component signals. With the help of the method of EMD proposed by Huang et al. [5, 6], one can decompose the response into a number of IMFs which include the prototypes of the n AM mono-component signals x p (t). The EMD method, a data-based, intuitive and adaptive technique, decomposes the signal into m IMFs as follows: m x () t = c () t + r () t (6) p p p = in which c p (t) is an IMF and r p (t) is the residue which represents the mean trend of x p (t). However, above IMF under the original EMD may contain more than one frequency, which is not a mono-component signal. So the intermittency criteria [5, 6] is imposed to improve the extraction of the modal response x p (t). Meanwhile, the EMD method also contributes to remove measurement noise inevitably introduced from external interference. Thus, n modal response functions and many other IMFs can be decomposed from the measure impulse response as follows: n m n () () + () + () (7) x t x t c t r t p p p p = = It should be noted that the intermittency criteria may bring large amount of calculation. For simplification, an alternative approach based on the band-pass filter and EMD proposed in [] is used here. C. Signal Demodulation and System Identification After extracting the modal response for each mode using the EMD above, the methods of demodulation analysis will be employed to process the modal response to obtain the instantaneous amplitudes and frequencies, from which the corresponding modal frequency and damping ratio can be easily identified. 8

3 ) Hilbert Transform Demodulation Analysis As one of the most common demodulation analysis method, the HT has been widely employed to extract the instantaneous amplitude and instantaneous phase angle of a vibration signal with its quick algorithm. Let x () t denote the HT of x p (t), i.e. p p ( τ ) ( t τ) + x x p () t = HT xp () t = dτ (8) π Then the analytical signal Y p (t) of x p (t) can be expressed as () = () + () = () () t i p p p p p (9) Y t x t i x t A t e θ in which A p (t) is the instantaneous amplitude and θ p (t) is the instantaneous phase angle. When ζ is very small and is large, after some approximate process [], the amplitude and phase angle are given by (), A t p t Bp ke ζ (0) π θp () t dt+ + ϕ k () After applying logarithm operation to (0), one obtains () ζ ln, ln A t t+ B () p p k So based on () and (), one can estimate d = (-ζ ) / and -ζ from the slope of the phase angle θ p (t) and decaying amplitude lna p (t) versus time t plot by the use of linear leastsquare fit procedure. Then the natural frequency and the damping ratio ζ can be easily calculated. However, the slope is obtained conveniently only for small ζ. When ζ is not small, a frequency modulation is introduced into the amplitude variation lna p (t), which arouses a frequency fluctuation around the mean value but not a change of it. Although the mean value can be estimated also through the linear least-squares fit procedure, the range of time interval for fit procedure need to be exactly selected, or estimation error will be brought in. Furthermore, the inevitable window effect of HT also causes inconvenience to parametric identification. ) Teager Energy Operator Compared with HT, the TEO demodulation approach is a very simple demodulating technique which has higher demodulation precision and less calculation and yields small errors for AM-FM demodulation especially. The energy operator Ψ( ) is defined for continuous-time signal x(t) as: ( x) x() t x() t x() t = (3) where x () t and x() t are the first and second time derivatives of x(t) respectively. For an AM-FM signal, the general expression is as follows () () cos φ () x t = a t t (4) By using the TEO demodulation method, the instantaneous amplitude and frequency of the AM-FM signal can be obtained approximately[] () at () t φ() t ( xt ()) ( xt ()) ( xt ()) ( xt ()) (5) = (6) Although the instantaneous amplitude and frequency can be only calculated approximately under the circumstance of the general expression, there will be no approximation for the modal response x p (t), proved as follows. For the modal response x p (t), substitute (4) into (3) ( ) k d t x = B e ζ (7) Similarly, one can calculate the energy operator of x () t ( ) k d d ζ t kd e ζ t x = B ( + ζ ) e = B From (5) and (6), one obtains ( x) ( x ) From (9), we obtain d = B e = B e ( x) ( x ) ζ t ζ t k ζ k ( x ) ( x) = ln = ζ t+ ln B + ln ( ζ ) k (8) (9) (0) () Finally from (0) and (), one can exactly estimate and - ζ without the weakness of HT and any error of approximation compared with (5) and (6). Consequently the natural frequency and the damping ratio ζ can be more precisely estimated. To sum up, after measuring the displacement impulse response x p (t) of the MDOF structure, the EMD method can be implemented to decompose the measurement into n modal response x p (t) for =,,,n, as the (7) shows. Then, the more effective demodulation method, TEO, is applied to each modal response to calculate the instantaneous amplitude and instantaneous frequency as shown in (0) and (). Finally, the modal parameters, such as the natural frequencies and damping ratios, can be accurately estimated by the use of linear leastsquare fit procedure. IV. SIMULATION RESULTS A. Identification of Single-Degree-of-Freedom system The procedure of linear least-square fits for θ p (t) and lna p (t) versus time t is fundamental to the success of parametric identification. To validate this procedure, the impulse response of a single mode used for simulation is given by 8

4 I0 ζ x () t p t = e sin ( dt) () d The impulse responses above with I 0 =, =3Hz and ζ =, 5, 0, 0 and 50% are shown in Fig.. By the demodulation of HT, the plots of θ p (t) and lna p (t) versus time t are displayed in Fig. 3 as blue dotted curves while the linear least-square fits of the selected intervals are shown as red solid lines. The plots using the demodulation of TEO are shown in Fig. 4. The identifying parameters are shown in Table. Note that the values in parentheses are the errors between the identification and the theoretical values. Figure 4. Plots of f p(t) and lna p(t) by the demodulation of Teager energy operator: ζ=% for (a) and (b), ζ=5% for (c) and (d), ζ=0% for (e) and (f), ζ=0% for (g) and (h) and ζ=50% for (i) and () TABLE I. IDENTIFICATION OF SINGLE-DEGREE-OF-FREEDOM SYSTEM Figure. Impulse response of SDOF system: (a) ζ=%, (b) ζ=5%, (c) ζ=0%, (d) ζ=0% and (e) ζ=50% Figure 3. Plots of θ p(t) and lna p(t) by the demodulation of Hilbert transform: ζ=% for (a) and (b), ζ=5% for (c) and (d), ζ=0% for (e) and (f), ζ=0% for (g) and (h) and ζ=50% for (i) and () ratio (%) Frequency=3Hz Hilbert Transform Teager Energy Operator Identification Identification Frequency(Hz) Frequency(Hz) ratio (%) ratio (%) ( %) (0.579%) (0.0000%) (0.0000%) (-0.05%) (0.536%) (0.0000%) (0.0000%) (0.490%) (.8388%) (0.000%) (-0.000%) (-0.47%) (5.333%) (0.0005%) ( %) ( (.886%) %) (0.0030%) (0.009%) From Fig. 3 and 4 it can be seen that the results obtained by HT have serious end effects, which need artificially select the range of time interval to perform the linear least-square fit procedure. Meanwhile, the end effects will increase along with the damping ratio, especially for the decaying amplitude, while the slope estimation is sensitive to the range of time interval used for fitting. So when the damping ratio is large, the selection of the range of time interval is difficult to determine and consequently large identifying errors will produce as Table shows. Oppositely, the TEO demodulation approach is obviously superior to HT. From the Fig. 4 and Table, it is obvious to see that the demodulation results of TEO are perfect with very small or even negligible error. So owing to no approximation being introduced in (7) to (), the energy operator is particularly suitable for the demodulation of the impulse response functions. Though the energy operator has the disadvantage of a moderate sensitivity to noise, with the help of preliminary of EMD method the mono-component impulse response can be purely extracted. As a result, the EMD 83

5 method in conunction with the energy operator method is a successful method for system identification. B. Identification of Multi-Degree-of-Freedom system To grasp the dynamic properties of the macro/micro dualdrive system and realize high positioning precision, a model simulation on a spring mass system for simplification is considered in Fig. 5. It consists of two blocks, with masses m and m, connected by two springs with spring constant k and k while between the masses damping c and c exist. To simplify the discussion, there is no friction between the blocks and the floor. The mass, stiffness and viscous damping of the positioning system are identical with m =00kg, k =980kN/m, c =.5kN s/m, respectively, for =,. Suppose an impulse loading is applied to the first mass, then the simulated time series of the first mass is shown in Fig. 6(a), from which all natural frequencies and damping ratios can be identified using the identification method above. In Fig. 6(b), a noise process is shown while its level is expressed by R=σ/max x(t) in which σ is the rms value of the noise associated with measurement. The noise level in Fig. 6(b) is R=5%. The noise-added measurement, that is the sum of Fig. 6(a) and 6(b), is shown in Fig. 6(c). The Fourier spectrum of Fig. 6(a) and 6(c) are displayed in Fig. 7, which shows two dominant frequencies around 7.4 and 3.8Hz. With the proposed identification synthesizing the method of EMD and TEO, the noise-added response is identified as the red solid curves in Fig. 8 while the corresponding theoretical responses are shown as blue dotted curves. It is observed that they coincide very well. Similarly, the identification results of the response without noise are shown in Fig. 9. The recognized modal parameters are listed in Table, where the theoretical values are also presented. Note that the values in parentheses are the errors between the identification and the theoretical values. From the list, it is indicated that both the HT and TEO methods can obtain nice results with very small errors. When the damping ratios are large, the TEO method behaves better. Furthermore, the addition of noise has little effect on the identification results because of the filter effect of the EMD method. Figure 6. Displacement impulse response of the spring-mass system: (a) the response of the first mass, (b) noises and (c) noise-added response Figure 7. Fourier transforms: (a) Fourier transform of the signal without noises, and (b) Fourier transform of the signal with 5% noises to peak response ratio Figure 5. The spring-mass system for simplification Figure 8. Impulse modal responses with noises: (a) second mode and (b) fiest mode 84

6 procedure is performed to accurately estimate the modal parameters, such as the natural frequencies and damping ratios. The theoretical analysis and simulation results show that the TEO approach based on EMD can be successfully applied to system identification. REFERENCES Figure 9. Impulse modal responses without noises: (a) second mode and (b) first mode TABLE II. Theoretical values Identified values (R=0%) Identified values (R=5%) IDENTIFICATION OF MULTI-DEGREE-OF-FREEDOM SYSTEM Hilbert TEO Hilbert TEO Frequency (Hz) Mode Mode Frequency Ratio (%) (Hz) V. CONCLUSION Ratio (%) ( %) (0.095%) (0.049%) (.963%) ( (-0.004%) 0.84%) (0.060%) (.356%) ( (- ( %).493%) (-0.707%) 4.478%) ( (-0.348%).09%) (-.397%) (-.786%) The Hilbert-Huang transform analysis has been used in the field of system identification. After the decomposition of EMD method and demodulation of the Hilbert transform, the instantaneous phase angle and amplitude of each IMF are obtained, which form Hilbert-Huang spectrum of the signal in the time-frequency domain. From the spectrum, the modal parameters of MDOF system can be accurately identified. However, owing to the inevitable window effect of HT, the demodulation results may present obvious end effects, which will produce demodulation error and cause inconvenience to parametric identification. To remedy this problem, the TEO demodulation approach with higher demodulation precision and less calculation is proposed and applied to system identification. Firstly, the EMD method is used to decompose the measured impulse response of a MDOF structure into a number of modal responses. Then the TEO is applied to each modal response to calculate instantaneous amplitudes and instantaneous frequencies. Finally, the linear least-square fit [] L. H. Lu, Y. C. Liang, Y. F. Guo et al., Design and Testing of a Nanometer Positioning System, Journal of Dynamic Systems Measurement and Control-Transactions of the Asme, vol. 3, no., Mar, 00. [] J. L. Ha, R. F. Fung, C. F. Han et al., Effects of frictional models on the dynamic response of the impact drive mechanism, Journal of Vibration and Acoustics-Transactions of the Asme, vol. 8, no., pp , Feb, 006. [3] G. W. Housner, L. A. Bergman, T. K. Caughey et al., Structural control: Past, present, and future, Journal of Engineering Mechanics-Asce, vol. 3, no. 9, pp , Sep, 997. [4] B. Peeters, and G. De Roeck, Stochastic system identification for operational modal analysis: A review, Journal of Dynamic Systems Measurement and Control-Transactions of the Asme, vol. 3, no. 4, pp , Dec, 00. [5] N. E. Huang, Z. Shen, S. R. Long et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society of London Series a- Mathematical Physical and Engineering Sciences, vol. 454, no. 97, pp , Mar 8, 998. [6] N. E. Huang, Z. Shen, and S. R. Long, A new view of nonlinear water waves: The Hilbert spectrum, Annual Review of Fluid Mechanics, vol. 3, pp , 999, 999. [7] J. N. Yang, Y. Lei, and Sem, "System identification of linear structures using Hilbert transform and empirical mode decomposition," Imac-Xviii: A Conference on Structural Dynamics, Vols and, Proceedings, Proceedings of the Society of Photo-Optical Instrumentation Engineers (Spie), pp. 3-9, 000. [8] P. Maragos, J. F. Kaiser, and T. F. Quatieri, ON AMPLITUDE AND FREQUENCY DEMODULATION USING ENERGY OPERATORS, Ieee Transactions on Signal Processing, vol. 4, no. 4, pp , Apr, 993. [9] A. Potamianos, and P. Maragos, A Comparison of the Energy Operator and the Hilbert Transform Approach to Signal and Speech Demodulation, Signal Processing, vol. 37, no., pp. 95-0, May, 994. [0] J. S. Cheng, D. J. Yu, and Y. Yu, The application of energy operator demodulation approach based on EMD in machinery fault diagnosis, Mechanical Systems and Signal Processing, vol., no., pp , Feb, 007. [] J. N. Yang, Y. Lei, S. W. Pan et al., System identification of linear structures based on Hilbert-Huang spectral analysis. Part : normal modes, Earthquake Engineering & Structural Dynamics, vol. 3, no. 9, pp , Jul 5, 003. [] P. Maragos, J. F. Kaiser, and T. F. Quatieri, On separating amplitude from frequency modulations using energy operators, ICASSP-9: 99 IEEE International Conference on Acoustics, Speech and Signal Processing (Cat. No.9CH303-9), pp. -4 vol.4 vol., 99,