IdeI J Ramakrishnan", A. Hutand R. VanderVoort" J. Berg+ and Lt. D.F. Cosseyt Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 bstract Identification experiments on the Air Force Phillips Laboratory's (formerly the Astronautics Laboratory) Advanced Space Structure Technology Research Experiment facility (ASTREX) is described in this paper. A finite element Model (FEM) of ASTREX was constructed to provide modal vector information (necessary for control law development) and also to provide a comparison for the identified models derived from experimental data. The q-Markov COVER (short for Covariance Equivalent Realization) and Frequency Response Functions (FRFs) were used for the identification. The identified models show good correlation with the time history and frequency response data obtained in the laboratory. Figure 1: ASTREX Facility The ASTREX facility provides the hardware to test and validate emerging space structures technology. The large precision structures laboratory provides three axis large angle slew capability, realistic missionrepresentative test article, the ability to change key structural members, actuators and/or sensors and a fully programmable real time computer. The test article is supported by a 0.5 meter airbearing on top of a 5 meter pedestal. The pedestal comprises of pneumatic components, electronics, power supplies and a computer at its base. A gimbal system, located alongside the airbearing, is part of a cable follower system that minimizes the boundary effects of all the electronic lines running from the floor to the slewing test article. The ASTREX test article is a dynamically scaled model of a 5-meter class Space Based Laser (SBL), consisting of primary, secondary and tertiary components (Figure 1). For now, the mirrors are simulated by masses placed at the appropriate locations. Identification of ASTREX facility is crucial from both the controller design and the finite element "Dynacs Engineering Co., Palm Harbor, FL 34684 t Airforce Phillips Lab, Edwards AFB, CA OCopyright 01991 by the AIAA, Inc. All rights reserved 1219 model validation perspective. The purpose of the laboratory experiments described herein is to obtain a precise characterization of the ASTREX structure. Some of the parameters of interest were the frequency spectrum, the damping associated with the different modes and for control design the linear time invariant state space models. The q-Markov COVER identification method was used to generate the state space models of ASTREX. The q-Markov COVER approach can be used on data generated by an applied pulse or white noise input and seeks to generate covariance equivalent realizations matching the first q covariance and (q 1) Markov parameters of the data. Frequency response characterizations were found using a disturbance electromagnetic shaker subjected to either a sine dwell or periodic chirp (fast sine sweep) source input. + The paper is organized as follows: a brief formulation of the g-Markov and the FRF methods is presented in section 2 followed by a description of t!ie setup in section 3. Results are presente section followed by conclusions from the s orrnulat ion where The q-Markov COVER approach for discrete systems is based on the results from reference [I], [2]. A brief description of the approach is presented here for the sake of completeness. 2.1 (5) q-Markov COVER Theory Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 The Markov and autocorrelation parameters of the system to be identified are determined from the experimental data. There are two approaches to this problem: (i) Excite the system by a pulse input to obtain the Markov parameters and by a white noise input to obtain the autocorrelation factors. (ii) Obtain both the Markov and autocorrelation parameters from the pulse response. The second approach is used in this work since one experiment yields both the Markov and autocorrelation data. Let the Markov and autocorrelation parameters obtained from the pulse response data be {Hi}? and {Rl}?. Further, the finite data sequences { H i } $ and {Ri}$-l are consistent, in the sense that it can be generated by a linear system, if and only if D, 2 D9 2 0 It can be shown [l]that D, and D q satisfy the paranieter identities d D, Then the multivariable discrete time system 3(k + 1) = A q k ) 4-B u ( k ) d(k) = & ( k ) + B u ( k ) k >_ 0 with a state covariance 2 = limk,, = A 0,XO; A De = O,AXA*Oi (1) 2(k)i*(k) (9) where the set { A ,B , C, D ,X} is the multivariable disCrete time system generating the output data used for realization and 0, is the corresponding observability satisfying X=AXia+BB*, x>o is a q-Markov COVER of the system generating the data sequences {Hi}:, and {&}: if Hi & = H i , i = O , 1 , 2 , . . . ,Q = &, i = o , l , . . . , q - l (3) The realization problem states that given the finite find all minimal data sequences { H i } : and stable q-Markov COVERS. The realization problem is based on the factorization and projection of two data matrices. Using the pulse response data, the two data matrices D, and De are constructed. D, A = R,-H,H; D, A= R,-H,Hi (4) 1220 The solution of the realization problem depends on the key observation that D, and Dq can be factored in terms of the observability matrix 0 , , the system matrix A and the state covariance matrix X . The full factorization of the first data matrix yields D, = P A P " , A > O A T = P E rank (D,) = rank (P)= rank ( A ) V y q X r , Our goal is to force P and A to play the role of the observability matrix 0 , and the state covariance ,f of the identified state space realization (1). In order to force P and A to play the roles of Oq and we partition s, dimension ny x r and define two new where G is an ny x r matrix to be determined such that Dq = PAPe P will play the role of the matrix product and the second parameter identity in (9) will be satisfied. The matrix Da can be If this G can be determined, then Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 6,a Figure 2: Laplace Plane Representation leading to the relations Dq-l = FhF* Eq = FAG* Dqq = GAG" The relation (15) is satisfied by virtue of the construction of p . Solution of relations (16) and (17) yields G and hence p . The identified matrices are a - A - A Po, D=H,, X = A A Mq = [H;, H ; , . . . , H;]" The interested reader is directed to references [l],[2] for more details and proofs. 2.2 Frequency Response Function It is well known that a system's transfer function correiy in.put/output pair can be represf mted the Laplace plane by plotting the system's complex poles and zeros (Figure 2). Because the poles of a 2nd-order, viscously damped system are intimately related to the modal natural frequencies and modal damping ratios as described in Equation 19, knowledge of a system's poles is sufficient for the identification of those modal parameters. The modal parameters of frequency and damping are global properties of the system; consequently, all input/output pairs d Figure 3: Bode Plot consist of identical poles. However, the modal amplitudes depend upon the choice of input and output and may be too small for the identification of some of the poles. This means that one would investigate multiple sets of input/output pairs in order to flush out all of the poles of interest. Unfortunately, it is not possible to measure the traasfer function directly. The best we can do is to measure the semi-infinite slice of the Laplace domain corresponding to the positive imaginary axis. The resulting 2-dimensional curve, known as the Frequency Response Function (FRF), is simply the ratio of the output to the input of a system with respect to the frequency of excitation. There are several ways of plotting FRFs, the most popular being the Bode Plot (Figure 3). The Bode Plot is actually two curves with a common abscissa being the log of the frequency: one curve is the magnitude of the F R F (in decibels) and the other is the phase of the FRF. Figure 4 displays a linear, timeinvariant system subjected to noisy sensor readings at both input and output. The FRF, symbolized as H ( u ) , is the ratio of output, y, over input, 2. are generally plagued by noisy sensor r make the exact measurements of z and A method of minimizing the effe 1 2 21 = cxy + c , u mu Therefore, the F R F estimates H1 and H:!become: nu Hi M cy, + cup cx, + c,, - k+% W X e+% Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 Figure 4: System Subjected to Noisy Signals out auto- and cross-correlations which are then used for F R F estimates. There are two common F R F estimates, often symbolized by H1 and Hz. H1 is the ratio of with both the numerator and denominator multiplied by the complex conjugate of u. Similarly, Hz is the ratio of with both the numerator and denominator multiplied by the complex conjugate of w. Hi cwu. cuu Hz -. C W W cuw The subscripts of C are the frequency dependent sequences which are correlated. For example, Cab is the correlation between A ( w ) and the complex conjugate of E ( w ) , E* ( w ) . Expanding the correlations of Equations (20) and (21) in terms of the actual signals and noise, and assuming that all noise is uncorrelated with the input and output, we can see how these two estimates compare with the true FRF: cwu E W(w)U*(w) = (Y+ N)(X" + M") = cy, cy, c u , + cuu + + cup CY,+C",. 2 U(w)U*(w) = (X M)(X* + M") = cxx cxp c ,x 4- c,, (22) + + + = cx, + c,,. cw, = W(w)W*(w) + + - We know that as we approach resonance, the required load to drive the system decreases rapidly yet the system maintains its relatively large vibrations. Because the input signal falls off, the noise within the input sensor signal becomes influential. Likewise, the noise within the output sensor signal becomes influential. Away from resonance, the opposite occurs. Using this insight with Equations 25 and 26, we can construct guidelines on the use of the two FRF estimates. We see that H1 is generally a better estimate of the FRF, except at resonance where H2 is the better estimator. Additionally, H1 tends to behave as a lower bound for the true F R F while Hz behaves as the upper bound. In the experiments conducted here it is found that the difference between H1 and H2 are slight. The coherence, which is the ratio remains very near 1.0 throughout the frequencies 11 Hz and greater. When a periodic chirp is used for the input source, there are shallow dips in the coherence at the anti-resonances, which is to be expected. To estimate the F R F to an acceptable degree of resolution, the poles are first found using a periodic chirp (fast sine sweep). One does this by using different actuator and sensor locations in order to miss as few poles as possible. Once the pole frequencies have 2, (23) = (Y+ N)(Y*+ N") = cy, + cy, cuy The terms N i l N o , Neil and, N,, are power ratios which represent the degree of noise within the sensor readings. Ni is the ratio of the power of the input sensor noise m e r the power of the input sensor signal. Similarly, N o corresponds to the output sensor signal. N,, and N,, are the ratios of the noise cross-correlation (input sensor noise and output sensor noise) over the power of the input sensor signal and output sensor signal, respectively. cuu cy,+ c , u . cuw = c,zu 1222 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 Figure 5: W i n d o w of Curve Fit estimated, the SISO Sine Dwell procedure is used on each of the individual poles keeping a frequency resolution of approximately 5-mHz. Various methods have been tried to determine the modal damping ratios, but the one used herein is the steady-state method of curve fitting the peaks of H I across the frequency band which encloses a 3-dB spread in the FRF amplitude (Figure 5). This is a straightforward sequence of steps using an HP-3562A Dynamic Signal Analyzer. Figure 6: Selected S e n s o r S e t Experiment 3.1 q-Markov ID The identification experiment was conducted at the AFAL ASTREX facility. The control data acquisition computer (CDAC) has 32 analog input and output channels with a maximum sampling rate of 1000 Hz. From an admissible set of sensors (accelerometers) 31 tical sensor locations were selected using the sensor d actuator selection algorithm [3]. The 32"d channel used to record the input pulse. The selected sensor et is shown in Figure 6. The accelerometers used are of he piezoelectric type with an eighth-order Butterworth ost-filter. The actuator locations for the q-Markov ID vperiment are given in Figure 7 and Table 1. The six ctuators include four thrusters and two roll actuators; n the absence of these actuators impulse hammers were to excite the ASTREX structure. The actuator ions are all on the primary truss. The sensors, on ther hand, have been installed on the primary, secondary and tertiary mirrors and the connecting tripods. I t A sampling frequency of 500 Hz w a s used in identification. This sampling frequency choice w a s dictated by ( i ) Sampling of the 90 Hz mode Figure 7: A c t u a t o r Locations (ii) Using the 500 Hz data, data at 250 Hz and 100 Hz could be constructed by deleting the appropriate 1.22' 3 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 samples. The experiment was repeated 10 times for each input channel. The repetition was done from statistical considerations to obtain well averaged data. The averaging process helps in smoothing measurements corrupted by high and low frequency noise. The experiment is repeated for all the input channels. For each input, 10 sets of sensor output data were collected for approximately 10 seconds. The data was detrended, scaled and averaged to produce a processed data file. The processed data files have 8 seconds (4000 samples) of data for each of the 31 output channels due to each input pulse. The input intensities are determined from the input channel (Output 32) to be: W = I-, I 1 I 1 I Apply pulse L collect Output data diag ([0.2305 0.2695, 0.2776, 0.2337 0.3065, 0.31611) pulse strength for each experiment Table 1: Input D a t a Input # I Type Node 15 Direction Pulse Location Lower Right Pulse Upper Right 14 +z Pulse Upper Left 13 +z Pulse Lower Left 16 +z +z Scale all outputs associated with i=i+l b Pulse TOP 27 +Y Pulse Bottom 28 -Y The data collection and processing procedure is shown in Figure 8. For modal testing, an electromagnetic shaker is attached to node 14 of the structure and the response is measured at node 23. The electromagnetic shaker has a stinger unit that transmits the force to the structure. A force transducer on the stinger sends the signal to channel 1 of a HP 3562A dynamic signal analyzer. The input signal is controller from the analyzer. The accelerometer is connected to channel 2 of of the analyzer and the transfer function is obtained from the input-output relationship. 4 seconds of scaled data for each Average the data from the 10 Processed data esults The results from the q-Markov COVER identification are compared with the results from modal testing in Table 2. The frequencies from modal testing, finite element modeling and ID experiments are presented. Table 2 shows that the frequencies of the identified model are close to the frequencies obtained from modal testing. The outputs sampled a t 500 Hz were used to identify the ASTREX facility. The q-Markov procedure w a s executed with different values for q ( q = 10, 20 and 30). 1224 Figure 8: Data Collection and Processing Procedure am io Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 405 Figure 9: Singular Values of D, (Ny = 6, q = 30, Hz 1 . . . .- . For the identification, the first six sensors alone were used. One of the decisions made by the user is the selection of the rank of the data matrix D,. This rank determines the order of the state space realization. In this work, the singular values of D, were plotted and rank determination was made based on the relative nitudes of the singular values. The results from the ces of q of 10 and 20 were not very good. In order to ve the model, and to obtain better low frequency teristics a q parameter value 30 was used. The ot of the singular values of the data matrix is shown Figure 10: Time response comparison y1 due to = 30, 500 Hz) ~1 (9 A Based on the singular values, an identified model of der 100 was obtained. The time responses of output annels 1 and 2 due to the first input and the correonding lab response are plotted in figures 10 and 11. The correlation between the experiment and the identified response is very good. Next, the Fast Fourier I (FFT) of the two data sets were compared. e FFT of outputs 1 and 2 are shown in figures 12 and . The figures show poor agreement below 50 Hz and ement in the higher frequency ranges. The r this may be the use of accelerometers as senhe high sampling rate used. With acceleromeeven substantial low-frequency vibration displacet may result only in small accelerations, since acration is proportional to the square of the natural 1225 Figure 11: Time response comparison y2 due to ( q = 30, 500 Hz) 211 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 L M D Diu W 1 Figure 14: Time response comparison yl due t o ( q = 40, 250 Figure 12: FFT of y1 due to u1 (q Hz) = 30, 500 Hz) Increasing q will result in better low frequency characfkistics. However, this also results in higher computational burden and introduces more noise modes in the model. The q-Markov procedure was repeated at 250 Hz to alleviate the problem. The processed lab data file is converted to a 250 Hz sampling rate by retaining every other data point. Using a smaller sample rate increase the response time window for the same number of samples. The first 500 samples of the data at 250 Hz (T = 2s) were used. Using a q of 40, a 120th order identified model was obtained. T h e time history response of outputs 1 and 2 in figures 14 and 15 show good correlation with the lab data. Further the FFT of these outputs (figures 16 and 17) show good results for the low frequency spectrum. The identified model's frequencies match the results of other ASTREX modal identification experiments. Figure 13: FFT of y2 due to u1 (q = 30, 500 Hz) 1226 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 Figure 17: FFT of yz due to rb u1 ( q = 40, 250 H z ) Table 2 Frequency Comparison (10-20 H z ) : Experiment, FE Model, ID Model Sine Dwell Spectral Curve Fit Periodic Chirp Curve Fit (0-20 Hz) I Curve Fit (0-100 Hz) Models ID Hz F E Hz I 10.5 onclusion Identification experiments on ASTREX are described in this paper. The q-Markov COVER approach is used to identify models at 500 Hz and 250 Hz. The identified model at 500 Hz has poor low frequency characteristics. This is caused by the choice of accelerometers as sensors and the high sampling rate used. The identified model at 250 Hz shows a significantly better response at the lower frequencies. The comparison between the tion. The modal characteristics of finite element models are compared modal data well. The results of q-Markov COVER metho 122 1 6 rences 1. A.M. King, U.B. Desai and R.E. Skelton, A Generalized Approach to q-Markov Covariance Equivalent Realizations for Discrete Systems, 1987 American Control Conference, Minneapolis, MN. 2. U.B. Desai and R.E. Skelton, Partially Nested q-Markov Covariance Equivalent Realizations, CTAT, Vol. 3, No. 4, pp. 323-342, December 1987. Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737 3. A. Hu,R.E. Skelton, G.A. Norris and D.F. Cossey, Selection of Sensors and Actuators with Applications to the A S T R E X Facility, Proceedings of the 41h NASA/DoD CSI Conference, Orlando, FL, Nov. 1990. cknowledgements Prof. Robert Skelton was a consultant for this work. His help and suggestions a t different stages of development are gratefully acknowledged. Thanks are also due to Dr. A Das for his helpful suggestions. Ms. Linda Large’s dedication to typesetting and preparing this document is appreciated. 1228

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