An Adaptive Optimization Approach to Active Cancellation of Repeated Transient Vibration Disturbances

Transcription

1 An aptive Optimization Approah to Ative Canellation of Repeated Transient Vibration Disturbanes David L. Bowen RH Lyon Corp / Aenteh, 33 Moulton St., Cambridge, MA 138, U.S.A., owen@lyonorp.om J. Gregory MDaniel Dept. of Aerospae and Meh. Engineering, Boston University, 11 Cummington St., Boston, MA 134, U.S.A., jgm@bu.edu Vibration ontrol of lightly damped strutures suh as spae-borne optial platforms an be diffiult when the disturbanes are broadband transients. Suh transients an be produed by the ation of lathes or positioning devies suh as thrusters, inertia wheels and ontrolled moment gyros. In some ases it may not be possible to use the system until the vibrations have deayed. Ative vibration ontrol, whih relies on knowledge of strutural transfer funtions, disturbane waveforms, and spatial loation, an be used to blok the disturbane. However, the broadband nature of the disturbane and the natural variations in strutures due to the environment and aging make knowledge of the transfer funtions unertain. But the fat that the transient disturbane is broadband, and is the result of a ommand, gives us an opportunity to bring signal proessing tehniques originally developed for mahinery diagnostis to bear on this problem. The signal proessing method of epstral windowing allows us to substitute more robust and stable estimates of the needed transfer funtions. Knowledge of the time and loation of the disturbane allows us to present a orretion signal to a strategially plaed atuator that will oppose the disturbane fore. Redution an be further enhaned by adaptive optimization of the orretion signal, whereupon the response of the struture to repeated disturbanes is analyzed. This paper disusses the use of soure waveform reovery tehniques in ombination with an optimization proedure to redue the amplitude and duration of repeated transient disturbanes. Example results of applying these methods to a truss struture are given. 1 Introdution Vibratory disturbanes to aerospae strutures, and to deployable spaeraft in partiular, arise from transient fores applied to the struture. The resultant long settling time may interfere with the timing and auray of positioning, as well as other funtions. A spaeraft is generally a lightweight, stiff, and poorly damped struture. The fores on the struture are often planned sine they are the result of atuators, thrusters, and other mehanisms used for alignment, positioning, and funtioning of the spaeraft. Sine they are planned and ontrolled, the timing and loation of the exitation is known. That allows the appliation of an opposing exitation at the proper time and plae if the waveform of exitation an be determined. Therefore, ative ontrol of the vibrations that result from these known disturbanes is a viable prospet. However, most established methods are effetive on only a handful of modes, and require preise knowledge of the transfer funtion (TF) between the aneling fore and the vibration observation point. The TF an hange as a funtion of temperature, loading and age, and it an also vary from one struture to another of the same design. These effets of environment and usage result in TFs that are variable and therefore unertain. Our methodology is based on reduing the sensitivity of the ative anellation to details of the TF, as well as on introduing an adaptive optimization tehnique to aount for long term hanges in either the TF or the disturbane fores. Cepstral proessing offers a potential way to redue the sensitivity to the details of the TF, as well as stabilize the strutural TF for inverse filtering [1]. Used extensively in speeh proessing and to eliminate the effet of eho from reeived signals in highly reverberant environments, epstral analysis has also been shown to be effetive in mahinery diagnostis, allowing us to reover impats or other fore transients from the strutural vibration that results [, 3]. To determine the effetiveness of this approah, we have evaluated it on a sale model of an atual telesope, using an atuator and a solenoid to provide a variety of exitation fores. Figure 1 shows a photograph of the truss struture serving as the sale model. Bakground and theory The general method we have employed for reduing the severity of repeated transient disturbanes on a struture is summarized by the following steps: 1. Measure or otherwise determine orretion atuator to observer Transfer Funtion(s) (e.g., the TF(s) from voltage sent to an amplifier powering the orretion atuator to aeleration measured at an observer loation(s) on the struture), then epstrally smooth these aeleration/voltage TF(s). 881

2 Forum Austium 5 Budapest Bowen, MDaniel.1 Cepstral smoothing # 9 # 1 # 1 # # # 1 # 8 (atuator loation) # 1 (atuator loation) # 8 Figure 1: Test struture (1:9 sale model of telesope). Operate the disturbane mehanism by itself, measure and reord vibration response at observer loation(s), then epstrally window the vibration signal(s). 3. Calulate a orretion voltage waveform to send out to the orretion atuator power amplifier, via inverse filtering of the epstrally windowed vibration pulse(s) obtained from Step with the epstrally smoothed TF(s) obtained from Step 1. The result will in effet be a (epstrally windowed) soure voltage pulse, designed to produe an equal but opposite vibration response to that observed in Step above. 4. Use triggering or other means to operate orretion atuator the next time a disturbane ours, observe and determine redution in vibration. 5. Employ aptive Optimization to revise the orretion voltage for subsequent disturbane events, based on the residual vibration observed in Step 4. This method has appliation in situations where the disturbanes are transient (e.g., where the duration of eah disturbane fore pulse is small ompared to the periodiity of their ourrene, and muh less than the deay time of the strutural response), the loation of the disturbane is known, and the time at whih the disturbane operates is known (e.g., it is periodi, a loal sensor detets onset, and/or the disturbane is the result of a ommand). The next setions provide more detail on the epstral smoothing, orretion voltage alulation and adaptive optimization aspets of the method. The epstrum is found by first taking a Fourier transform of a signal. We take the natural logarithm of this funtion - yielding log magnitude for the real part and (unwrapped) phase for the imaginary part - and then take the inverse Fourier transform of that. The resulting time domain signal is the omplex epstrum. We now multiply the epstrum with a short time window, so as to keep the low time portion of the epstrum while zeroing out the high-time portion, resulting in smoother frequeny domain log magnitude and phase funtions. TFs smoothed in this way will generally be less sensitive to small variations in modal resonane frequenies and shapes [1]. The degree of epstral smoothing is ontrolled by the type and length of the time window applied to the omplex epstrum. We have used a Hanning window for this, the length of whih is governed by the overall duration of the disturbane fore pulse. Using a Hanning funtion for the epstrum window, the length of the window needs to be at least about twie the duration of the disturbane in order to aurately estimate the voltage waveform to be sent out to the orretion atuator. The epstrum window length an be made longer than this, but eventually we start to lose the robustness advantage that the smoothing gives us. Figure shows the effet that epstral smoothing has on the magnitude and phase values of a typial TF measured on the truss struture (using a Hanning window with an effetive length of about 5 ms). (a), re 1 m/se²/volt (b) radians Freq. in Hz original after pre-treatment epstrally smoothed original after pre-treatment epstrally smoothed Freq. in Hz Figure : Original and epstrally smoothed TF from loation 1 to 8 - (a) magnitude, (b) unwrapped phase 88

3 Forum Austium 5 Budapest Bowen, MDaniel. Estimation of orretion voltage The transient disturbanes that this approah is designed to deal with may have a simple waveform of fore or motion that has a frequeny spetrum that will exite a large number of the strutural modes, leading to a ompliated response. It is this strutural response that the system has to use to determine the signal to be sent to a orretion atuator. The fore or motion that the orretion atuator must provide will mimi the transient disturbane generally in time duration and waveform. Estimation of the orretion waveform is fundamentally a soure reovery problem, in whih the system transfer funtion(s) and response(s) are known, but the soure is unknown. In this ase, though, the response measured is due not to the orretion soure we wish to reover, but to a separate, disturbane soure. We wish to determine the best soure waveform that will produe the observed response, so that it an be sent out to the orretion atuator the next time this (same) disturbane ours. By using the TF from voltage supplied to the orretion atuator amplifier, to aeleration at an observation point on the struture, we are trying to determine the voltage waveform to supply to the orretion atuator that will reate this same measured vibration at the observation point. If the orretion atuator and disturbane soure are o-loated or lose enough together, then supplying the negative of the reovered orretion voltage to the orretion atuator should result in a fore that is equal and opposite to the fore supplied by the disturbane soure, thus leading to a global redution in the vibration response. But in general, the role of the orretion atuator is to anel the vibration at the observer. In some appliations, depending on the onfiguration of disturbane and atuator, that will not be the same as aneling the disturbane fore. Estimation of the orretion voltage soure waveform is ahieved by first epstrally windowing the measured aeleration response(s) to the disturbane. In priniple, the epstrally windowed response is then inverse filtered (deonvolved) with the epstrally windowed TF impulse response. In pratie, we use fast deonvolution, in whih the inverse filtering takes plae in the frequeny domain. For the single-input, single-output (SISO) ase, this essentially amounts to dividing the epstrally smoothed aeleration spetrum by a modified form of the epstrally smoothed TF, then inverse Fourier transforming the result in order to obtain the estimated soure waveform. This is a valid approah as long as the inverse impulse response (i.e., that orresponding to the inverse of the modified epstrally smoothed TF) is not time-aliased. For the ase where we wish to make use of information from multiple response loations, we an use a singleinput, multiple-output (SIMO) approah to mitigate the effet of unertain TF zero values. In this approah, we seek a least-squared-error solution for the orretion voltage, V(ω), to the over-determined system of equations desribed by A(ω) = V(ω)H(ω), (1) at eah frequeny sample, ω. Here, A(ω) denotes a olumn vetor holding the (epstrally smoothed) aeleration spetra for the two response loations, and H(ω) is a olumn vetor ontaining the (epstrally smoothed) transfer funtions from the voltage soure to the aeleration responses at the two loations. An optimal solution to (1) is given by V est (ω) = H + (ω)a(ω), () where V est (ω) is the estimated orretion voltage, H + (ω) denotes the full rank pseudoinverse of H(ω), and where rank is defined as the number of singular values larger than some tolerane value governed by the omputing platform. The omputation of H + is based on the singular value deomposition of H, where any singular values less than the tolerane value are ignored by treating them as zero. The use of H + means that V est an only be determined in suh a way as to minimize the error between V est H and A, the true lefthand side of (1). The error minimization that we are using is based on the least-squares (Eulidean) norm, whih minimizes the mean square error between the true A and the value omputed by V est H..3 aptive optimization The final step in the proess involves updating the urrent estimate of the orretion voltage, based on the residual vibration observed during an attempted ontrol event. The resulting updated orretion voltage is then sent out the next time the disturbane event ours. We refer to this step as aptive Optimization (AO). Using this approah, the TF is measured (or otherwise estimated) only one, followed by a single measurement of the unontrolled disturbane. All subsequent measurements onsist only of the vibration with ontrol attempted. While adaptive optimization of the orretion voltage offers a hane to improve the vibration redution performane the next time the disturbane ours, the primary motivation for developing and inluding AO into the method arises from an expetation that the disturbane as well as the dynami properties of the struture and orretion atuator will drift over time. As these drifts our, the ontrol voltage must be orreted to preserve ontrol performane. The time sale over whih an improved orretion signal an be alulated will be greater than the duration of a single disturbane sine no adaptation an take plae as the orretion sig- 883

4 Forum Austium 5 Budapest Bowen, MDaniel nal is generated and sent out. But an improved version of the orretion voltage an be omputed off-line sine the vibration produed by the disturbane and the orretion atuator to observer is known. An adaptive algorithm has been designed that updates the orretion voltage based on error measurements at an observation point. The resulting routine is intended to aommodate slowly varying hanges in the disturbane or the TF, in that it relies on these quantities being onstant over at least two onseutive events in order for it to adapt to a new disturbane or TF. For illustration purposes, the algorithm is developed below for the SISO ase, in whih one TF and one observation loation are onsidered. Also, epstral smoothing of the TF and the vibration responses is not expliitly taken into aount in this analysis. In order to develop the analysis on whih the algorithm is based, two loations are identified: The ontrol loation, denoted by a subsript, at whih ontrol fores are applied. The observation loation, denoted by a subsript o, at whih the total aeleration is desired to be zero. The point at whih the disturbane is applied is irrelevant to this analysis, as our interest lies only in the aeleration produed by the disturbane at the observation point. The analysis begins by applying Fourier transforms to the transient quantities, resulting in the following frequeny-domain quantities = ontrol voltage applied to the atuator at (in V this analysis, the term ontrol voltage is used instead of orretion voltage to avoid onfusion with updates or orretions of ertain quantities involved in the analysis). = aeleration at o due to the ontrol voltage A applied to the atuator at A = aeleration at o due to the disturbane d G = A V = transfer funtion relating ontrol aeleration at o to ontrol voltage The ontrol objetive is to produe zero aeleration at the observation point, so the ontrol error is defined as the total aeleration at the observation point; E = A + A = GV + A (3) d The ontrol voltage that produes zero aeleration (i.e., error) is V A d d =. (4) G This alulation relies, of ourse, on an aurate knowledge of A d and G. As mentioned above, these quantities are expeted to drift over time. To remedy this, we seek to orret A d and G based on the error measured during previous ontrol events. Two events are initially onsidered, denoted by subsripts 1 and, in whih ontrol was attempted and A d and G are assumed to be onstant over these two events. Control errors are then defined as E = G ( V ) + A 1 1 d E = G ( V ) + A d (5) Algebrai solution of these equations gives expressions for A d and G E ( ) ( ) 1 E E V E 1 1 V G = = ( V ) ( V ) ( V ) ( V ) 1 1 (6) For numerial reasons, it is advantageous to reast this result in terms of orretions to A d and G. Denoting the previous values by an overbar and orretions by a, we define G = G + G = + and solve for the orretions where (7) E1 E G = ( V ) 1 ( V ) ( V ) 1 E ( V ) E1 =, (8) ( V ) 1 ( V ) En = En En En = G( V ) n + (9) These orretions are used to alulate a modifiation ( orretion ) to the ontrol voltage, V, by writing V = V + V = (1) G so that the modifiation is + V = V (11) G + G In terms of the measured errors, this modifiation is [( V ) 1 ( V ) ] + ( V ) 1 E ( V ) E1 (1) V = V G [( V ) ( V ) ] + E E 1 If the normalized orretion G / G is small, then the binomial expansion may be used to derive the simplified approximation G V, (13) G G G in whih all produts of orretions have been negleted. In terms of the measured errors, this approximate modifiation is 1 884

5 Forum Austium 5 Budapest Bowen, MDaniel ( V ) 1 E ( V ) E1 V = G 1 E1 E [( V ) ( V ) ] G [( V ) ( V ) ] 1 (14) In general, given two onseutive ontrol attempts denoted by n and n-1, the governing equation an be expressed in matrix form as: ( V ) 1 ( G ) n n En = ( ) 1 ( ). (15) V A n 1 d n En 1 Although the above equation is written for the SISO ase, it is relatively straightforward to extend this form of the equation to the SIMO ase, where multiple observation points are inluded. The resulting algorithm onsists of the following steps: 1. Begin with alibrated values of A d and G (first event, zero error and ontrol voltage).. Attempt ontrol and measure error (seond event). 3. Reompute the ontrol voltage using above analysis and data from the previous two events. 4. Attempt ontrol and measure error. 5. Repeat steps 3 and 4 for eah new event, updating first and seond events to the previous two events. If the system is re-alibrated during operation by measuring A d and G diretly, the algorithm may be restarted by returning to Step 1. Although we have not done so, the algorithm ould be extended to inlude more than two previous events by solving for the updated A d and G in a least-squares sense. If the funtions G and A d do not atually hange between attempts, then the ontrol voltage obtained by (1) will not hange. This will produe two idential rows in the matrix of (15), and attempts to numerially solve for G and A d will be ill-onditioned. We enountered this in numerial simulations where G and A d were held fixed, but did not enounter it when the algorithm was applied to the atual truss struture. (a) (b) () (d) (e) at 1st loation (1) at nd loation (8) at 1st loation (1) at nd loation (8) at 1st loation (1) at nd loation (8) at 1st loation (1) at nd loation (8) Performane of the AO algorithm This setion presents examples of typial redutions obtained using a PC-based development system to anel various types of disturbanes applied to the truss struture of Figure 1. With this system, the disturbane output signal is first sent out through one of the two D/A outputs, while at the same time the response data from the truss struture is aquired on the A/D inputs. After alulating the orretion voltage, the system again sends out the disturbane signal, but this time the orretion signal is also sent out on the other D/A hannel. Vibration response data from aelerometers is also olleted during this time, onurrent with the two outputs. Figure 3(a) shows some example results obtained by letting the adaptive optimization algorithm run for 1 at 1st loation (1) at nd loation (8) "Event Number" Figure 3: Redutions in overall rms aeleration levels ahieved with adaptive optimization - (a) hirp as disturbane, (b) triangular wave as disturbane, () solenoid devie as disturbane, (d) hange in N-wave disturbane at event 6, (e) starting out with TFs estimated from FEA model of truss. ontrolled events (i.e., iterations ), with the disturbane being provided by sending out a 5 ms long hirp pulse from 4-4 Hz to the disturbane atuator attahed to the truss struture at loation #1. The orretion atuator is also mounted at loation 1, but on 885

6 Forum Austium 5 Budapest Bowen, MDaniel the opposite side of the juntion box from the disturbane atuator. The figure shows the redutions in the overall rms aeleration levels obtained for eah event when loations 1 and 8 are used as the response loations for the proedure operating in its SIMO mode (with a epstral window width of 5 samples, at a sampling rate of 5 khz). Here, event number 1 orresponds to no optimization, where the ontrol voltage has been alulated based on a previous disturbaneonly event and on previously measured TFs. Representative results for a triangular type of pulse ating as the disturbane are shown in Figure 3(b). As with the hirp pulse, response loations 1 and 8 were used with the proedure operating in its SIMO mode, with a epstral window width of 5 samples. With this broader band type of disturbane, the additional redutions ahieved at loation 1 during adaptive optimization were slightly less than those seen for the hirp pulse. The sudden, but temporary, derease at event 9, is likely due to a partiular hardware issue that allowed an oasional sample to be skipped in the A/D or D/A subsystems when both A/D and D/A were being used onurrently. The redutions ahieved with AO when a solenoid mehanism was used as the disturbane devie are shown in Figure 3(). The values ahieved are somewhat less than those obtained when an atuator is used as the disturbane soure, and there is also more of a spread between the redutions obtained at the two loations. Both of these effets may be related to the potential of the solenoid impat to generate fore omponents in diretions that are not aligned with the solenoid armature motion. These additional fore omponents, if transferred to the truss struture at the mounting point, an not be effetively anelled by the orretion atuator due to its orientation, and ould thus be fator in limiting the redutions ahieved with the solenoid as a disturbane devie. In the results presented above, the same nominal disturbane was used for eah event. If the disturbane should hange in a slowly varying way, however, then we expet the AO proedure to be able to update the orretion voltage aordingly, so that we have effetive anellation in subsequent events. Figure 3(d) shows what happens when the triiangular wave disturbane is delayed by.6 ms (3 samples), starting at event 6. In this ase, the proedure adapts to the hange in the theoretial minimum time of iterations. We an also examine the behavior of the AO proedure when it is provided with an initial inorret transfer funtion(s). To do this, we let the algorithm start off with TFs derived from an FEA model of the truss, instead of letting it start off with the diretly measured TFs. Figure 3(e) shows the result of doing this, with the disturbane being provided by the 4-4 Hz hirp pulse. Here, the redutions for the first ontrol attempt (i.e., before AO starts) are about 7 or so less than what is typially ahieved when diretly measured TFs are used. However, by the 4th iteration the algorithm is able to ahieve redutions similar to what they were when starting out with the measured TFs. The algorithm an also be started with both TFs simply set to unity. In this ase, the initial redutions are near zero, but then by the 5th iteration the redutions have onverged to roughly the same final values as we normally obtain. There may be some advantage to starting out with unity transfer funtions over starting out with badly estimated TFs, as the unity TFs provide signal energy at all frequenies and therefore better signal-tonoise ratio to allow for orretion. The proedure was also found to perform well if the disturbane soure onsisted of multiple pulses spaed lose together in time, as long as the width of the epstral window was inreased beyond that used for a single pulse in order to aommodate the longer length of the total disturbane. 4 Summary The method desribed here for ative vibration redution has appliation in situations where the disturbanes are transient, the loation of the disturbane soure is known, and the time at whih the disturbane operates is known (either through a ommand signal or a vibration-based trigger). Redutions of 1 to 15 were ahieved on a truss struture, and this ould be improved by another 5-1 via an adaptive optimization proedure. The AO proedure also allows for adaptation to long-term hanges to the struture or in the disturbane fore. We have implemented the neessary hardware and algorithms into a dediated embedded system whih uses one of the aelerometers to initiate sending out the orretion voltage, and the redutions obtained using this system are omparable to those desribed above in Setion 3. Referenes [1] J. T. Kim and R. H. Lyon, "Reduing Transfer Funtion Variability and Complexity by Cepstrum Windowing", Pro. Noise Con 88, Purdue University, West Lafayette, Indiana (1988). [] D. J. MCarthy and R. H. Lyon, "Reovery of Impat Signatures in Mahine Strutures", Mehanial Systems and Signal Proessing, 9(5), Sept., [3] D. L. Bowen, "Reovery of Combustion Pressure from Diesel Engine Vibration," Proeedings, 45th Meeting of the Mehanial Failures Prevention Group, Vibration Institute, Willowbrook, Ill., pp (1991). 886