SATELLITE VIBRATION CONTROL USING FREQUENCY SELECTIVE FEEDBACK A C H Tan, T Meurers, S M Veres, G Aglietti and E Rogers School of Engineering Sciences, Department of Electronics and Computer Science, University of Southampton, SO17 1BJ, UK Email: sandy@mech.soton.ac.u, Fax:44-2380-593230 Abstract This paper addresses the problem of actively attenuating the vibration of plates on satellites. A pure feedbac controller is demonstrated which operates at a set of dominant frequencies in a disturbance spectrum, where the control path model is estimated online. A new feature of the frequency selective feedbac is the use of the inverse Hessian to improve adaptation speed. The control scheme also incorporates a frequency estimation technique to determine the relevant disturbance frequencies with higher precision than the standard fast Fourier transform (FFT). The controller is implemented on a test rig to demonstrate the practical feasibility of the method. A disturbance with three rational dominant frequencies is introduced. If FFT were used instead of the frequency estimation method proposed, then a large number of samples would be required to accurately estimate the disturbance frequencies, and, most importantly, FFT-estimated frequencies could lead to an unstable control system due to their granularity. Using the proposed frequency estimation method, the total achieved attenuation is 26dB on the experimental rig. 1 Introduction Active vibration control is an effective means of attenuating unwanted vibrations on-board satellites as an alternative or complement to passive absorbers. The most commonly used scheme is the so called filtered reference least mean squares algorithm. Here the reference (or detection) signal is normally passed through an FIR filter whose coefficients are constantly updated. Two main disadvantages of this method are firstly the need for a second sensor to pic up the reference signal; and secondly a model of the plant dynamics from the actuator to the error sensor is needed for the adaptation process. Microvibrations (taen as frequencies below 1Hz) 1, 6 create various problems for sensitive equipment mounted on satellites. Active vibration control in this frequency band plays a vital role for satellites both during the launch of a satellite 2 and during orbital cruising 3, 1. Due to trajectory course corrections or movements of mechanical parts, vibrations are created on the satellite which then influence the reliability of equipment used on-board it. Major sources of such unwanted vibrations include motorized pumps and reaction wheels. The induced vibrations also pose severe problems in the case when optical measurement devices are part of the payload, leading in the absence of effective compensation to blurring of images etc. In this paper, we determine the performance of a newly developed controller 4, 5 with increased convergence when applied to microvibration control. The proposed control strategy uses an enhanced frequency estimation method and only a limited set of data samples. Initially, the disturbance spectrum is estimated by performing a standard FFT, and this rough estimate is then used to design a second estimation nowing that the true frequencies has to be in the vicinity of the FFT-estimated frequencies. A least mean squares optimisation is then performed, resulting in a more accurate frequency estimation. This paper also contains the analysis of the performance of this new control scheme when applied to a laboratory test rig especially constructed to emulate the production of microvibrations. This paper is structured as follows. In the next section, the basic control algorithm is presented, followed in the next section by a detailed description of the improved convergence time of the algorithm. The following section deals with frequency estimation on a limited set of data samples and the next gives the experimental results and discussion of them. 2 Control algorithm development Figure 1 shows a bloc diagram representation of the system to be controlled. The controller aims to minimise the output signal e and the signal ε, which is a measure of the error in the plant model. For perfect output attenuation the control signal u is given by u(jω n ) = d(jω n), n = 1,, N (1) G(jω n )
Plant with disturbance G n yn dn + + en where e r and e i denote the imaginary parts of e respectively. The control law will be designed to update the complex control gain u +1 in the negative gradient direction of Jc, where the gradient of this function is given by ~ U n J c u r = e r u e r + e i r u e i = G re r + G ie i (5) r ^ G n ^ yn - ~ dn - + ε n Jc u i = e r u e r + e i i u e i = G ie r + G re i (6) i Controller u n G ^ T n en or J c = GT e. (7) Figure 1: Bloc diagram of the system at frequency n where N is the number of disturbance frequencies to be controlled. The single multi-tone control problem is separated into a set of single tone control problems by the use of frequency selective filters (FSF)s. In the time domain the algorithm developed here based on iterative updating of gain-estimates: Iteration Iteration +1.. M n Samples M n Samples.. For each disturbance frequency an iterative step consists of M n samples. The length of an iterative step depends on the disturbance frequency but must be a multiple of the sampling to disturbance frequency ratio. During iterative step, an LMS estimation produces the complex output gain e (jω n ). (For ease of notation, the argument (jω n ) is neglected in the rest of this paper.) For the computation of the gradient only an estimate of G is available. With a design parameter µ the update of the complex gain of the control signal is given by u +1 = u µĝt e. (8) The same adjustment is made to the control path model and the error in the model can be represented by ε = y ŷ = (G Ĝ ) u (9) where Ĝ is an estimate of G in (4). As for controller tuning, the plant cost function is quadratic J p def = 1 2 ε (jω n ) 2 = 1 2 (ε r) 2 + 1 2 (ε i ) 2 (10) and the derivations with respect to the plant model parameters are Jp ĝr = ε ŷ r r ε ŷ i i = ε r u r ε i u i (11) g r g r The control criterion is to minimize the quadratic cost function J p ĝ i = ε ŷ r r g i ε ŷ i i g i = ε ru i ε i u r (12) J c def = 1 2 et e. (2) and the output can be written in the matrix form with e def = e r e i e = Gu + d (3) def u, u = r u i, g def gr =, g i Hence the gradient can be calculated as Jp = ur u i u i u r εr ε i = Ũ ε (13) and with a design parameter ν the plant model can be updated using ĝ +1 = ĝ + νũ ε (14) def d d = r d i, G def gr g = i g i g r (4) Note also that the update of the plant model does not have to be synchronized with the update of the controller coefficients.
3 Improved convergence time It can be shown that the control algorithm presented in the previous section is stable and converges 5, but this can tae quite some time. It also has to be decided how to select the adaptive gains µ and ν for optimal convergence. There is an easy and efficient way to increase the convergence speed using the inverse Hessian matrix 7. This also avoids the problem of choosing suitable adaptation gains for convergence. a resolution which is given by the ratio of the sampling frequency f s to the number of samples S. Therefore, true frequencies will be approximated by their nearest representative frequency depending on the values of f s and S. This information on the rough frequencies present will be used to determine N bandpass filters for the FSF scheme where the following frequency estimation algorithm is applied. First, the inverse Hessian matrix for the control parameter update is calculated using H c = ĝt ĝ 0 0 ĝ T (15) ĝ and its inverse using H 1 c = 1 ĝ T ĝ 1 0 0 1. (16) Hence the control signal in equation (8) can be updated using u +1 = u 1 ĝ T Ĝ T ĝ e. (17) Lumped mass Piezoelectric patches The Hessian matrix with respect to the plant model parameters is H u T p = u 0 0 u T u (18) and its inverse is H 1 p = 1 u T u 1 0 0 1. (19) Hence, the plant model (14) can be updated as ĝ +1 = ĝ + 1 u T u Ũ ε (20) The two update equations (17) and (19) are easy to implement and no extra signals need to be calculated for this purpose. 4 Frequency estimation Most analysis in this general area assumes a prior nowledge of the disturbance signal d, or that measurements can be taen separately. Here, however, a frequency estimator is proposed for use before the complex gain estimation. This is termed optimized frequency estimation (OFE). An initial estimation of the disturbance peas can be obtained by a standard application of FFT. For a fixed number of samples S, FFT can, however, only produce Figure 2: Photo of the vibrating plate The recorded output signal, without applying control is given by e(t) = d(t). (21) Basically, the frequency estimate is calculated using a nonlinear least mean squares method. To describe this method, assume that a sum of periodic N sinusoids is present in the disturbance signal and measured at the output e(t) = N a n sin (ω n t) + b n cos (ω n t) + ρ (22) n=1 where a n and b n denote the true coefficients and ρ some minor level noise. The rough location of the disturbance frequencies is nown. The output signal is filtered, using FSFs for each relevant frequency, to obtain a set of signals consisting of only one frequency and some minor level noise ρ n. The filtered output signal at frequency ω n can be written as e n (t) = sin(ω n t) cos(ω n t) a n b n + ρ n. where the real ω n is unnown and is to be estimated. Introduce the vector z(t, ˆω n ) = sin(ˆω n t) cos(ˆω n t). (23)
and define a matrix Z n as z(t 0, ˆω n ) Z n =. z(t f, ˆω n ) (24) where t 0 to t f is the range of sampling points chosen to avoid the filter transient effects. Therefore, the output signal vector is given by e n = e n (t 0 )... e n (t f ) T. Now the complex gain coefficients can be calculated by a least mean squares estimation as â(ˆωn ) ˆb(ˆωn ) = ( Z T n Z n ) 1Z T n e n. (25) The evaluation of the estimated frequency, ˆω is undertaen by minimizing the mean-square error between the true and estimated frequency within the range of sampled points, and is given by ( e n Z n â(ˆωn ) ˆb(ˆωn ) r(t, ˆω) = arg min ˆω ) T ( e n Z n â(ˆωn ) ˆb(ˆωn ) ). (26) Repeating the procedure defined (23) to (26) for each of the frequencies, ω n, the residual signal r(t, ˆω n ) can be calculated by taing the difference between the disturbance signal and the corresponding estimated signal. Each of the new estimated frequencies ˆω derived from r(t, ˆω n ) can be used to calculate a complex output gain of measured data for control use. 5 Experimental setup An aluminum panel, shown in Figure 2, excited by a point disturbance vibration and a controller force was constructed to investigate satellite vibration control using the above algorithm with frequency estimation. The experimental investigation sees to minimize the plant error function so as to cancel out the disturbance vibration. Inaccurate frequency estimates tend to lead to unstable system even with a robust adaptive control system. A schematic single-input-single-output (SISO) vibration control system is shown in Figure 3 together with its electrical connections. The aluminum panel is glued perpendicularly on all of its four edges to thin metal sheets to ensure a firm support. The thin metal sheets are held by supports attached to a metal base bloc and the entire structure is suspended on springs. A small lumped mass is glued on the panel to act as the payload. The piezoelectric output sensor is positioned at the location where attenuation is intended and the control piezoelectric is directly beneath it on the opposite face of the aluminum panel. The disturbance signal is generated by a Bluewave PCI/C44S1-4-60 digital signal processing (DSP) board and a Bluewave PC/16IO8-477 is used for the A/D conversion, sampling at 4Hz. Both disturbance and controller signals are amplified by an electronic amplifier separately (LM1875 at gain=4.9 and LM1876 at gain=2 respectively) before each exciting the piezoelectric actuators. The piezoelectric sensor is effectively the same transducer as the actuator, converting mechanical vibrations to electrical signals, before feeding it bac to the worstation for control purposes. All channel signals are displayed on oscilloscopes to monitor the signals status, and the signals are filtered before transmission or after vibration reception. The bandpass filter is used to filter out the low frequency from the power mains and higher noise frequencies. Lowpass filters are used by the actuators to eliminate the effect of zero order hold components at the output of the D/A channel. A sum of three dominant disturbance frequencies are generated at 190.4762Hz and (correct to 4 decimal places). The disturbance spectrum received by the sensor is estimated by the OFE method. Estimated frequencies are used by the controller algorithm which will produce an anti-phase vibration cancellation at the controller patch. Figure 4(a) shows the disturbance time series signals for 0.1 second. For comparison, FFT and OFE methods are used to show the stability of the control system and these results are shown in Figure 4(b) and Figure 4(c) respectively. The disturbance frequencies estimated by FFT for SNR=40 are quite granular with values 190Hz and. The OFE method estimated the disturbance frequencies as 133.3319Hz, 190.4732Hz, 285.7071Hz for SNR=40. The FFT estimated frequencies here differ so much from the true frequencies that the controller cannot converge to minimize the cost function, leading to an unstable system. The corresponding frequency domain plots of the disturbance signal, unstable FFT and attenuated signal by OFE are given in Figures 5(a) to 5(c) respectively. Note the difference also the different y-axis scales. From these plots, it is clear that the three dominant tones are attenuated until near the level of the system measurement noise. The total power attenuation using OFE method was calculated to be 26dB. It is common to find that most real signals are corrupted by a certain level of noise, which can be caused by various sources, such as the quantization, anti-aliasing filtering, bacground noise and imperfect instrument responses, e.g.
Figure 3: Schematic diagram of satellite panel experimental setup. (a) Disturbance signal. (a) Disturbance frequency domain signal. (b) Unstable signal of FFT-estimated frequencies. (b) Unstable signal in frequency domain after using FFTestimated frequencies. (c) Attenuated signal of OFE-estimated frequencies. Figure 4: Time series signal at SNR=40. (c) Attenuated disturbance signal in frequency domain after using OFE-estimated frequencies. Figure 5: Signals in frequency domain at SNR=40. actuators. Figure 6 shows the total power attenuation across different noise levels. Generally, OFE estimated frequencies attenuate the disturbance signal with increasing signal to noise ratio (SNR) (measured in dbs). This is as expected since accurate frequency estimates lead to better attenuation. Conversely,, FFT estimated frequencies lead to an unstable system where the total signal power is even higher than the disturbance signal itself. For SNR< 0, (which are considered noisy systems, the OFE method gave a very similar signal power to the disturbance signal and hence no discernible attenuation. The attenuation is more significant for SNR> 0 where the level is higher as this quantity increases. increasing the SNR gave higher attenuation. Table 1 gives the estimated frequencies of the disturbance signal for the two estimators for increasing noise levels and their stable/unstable property. The table clearly demonstrates that an FFT estimation is not good enough for control. In particular, it is clearly evident that the estimated frequencies using FFT are very much less accurate than those estimated by the OFE method. 6 Conclusions A scheme has been designed to estimated multisinusoidal disturbance frequencies and the control the output signal of a system subject to such disturbances.
Figure 6: Total signal power of 3 dominant frequencies for disturbance, Q&F and OFE methods. A practical example derived from the satellite industry was used a motivation for this wor and an experimental test rig built to verify overall performance. The experimental results obtained confirmed the superiority of the method proposed here over alternatives. Future wor could include the development of a supervisory system to react on changes in the disturbance signal. Also the experiments rig could be extended to multi-input-multi-output (MIMO) plants. References 1 Aglietti, G. and Langley, R.. and Gabriel, S. B. and Rogers, E.: An efficient model of an equipment loaded panel for active control design studies, 2000, The Journal of The Acoustical Society of America, 108, pp. 1673-1683. 2 van den Dool, T. and Doelman, S. and Häusler, S. and Baier, H.: Broadband MIMO ANC in an airframe fairing model, 1997, Proceedings of ACTIVE 97, pp. 861 872. 3 Nurre, G. S. and Ryan, R. S. and Scofield, H. N. and Sims, J. L.: Dynamcis and control of large space structures, 1984, AIAA Journal of Guidance, Control and Dynamics, 7(4), pp. 514 526. 4 Meurers, T. and Veres, S. M.: FSF based feedbac control with secondary path estimation, 2001, Porceedings of Internoise 01, 2, pp. 815 820. 5 Meurers, T. and Veres, S. M.: Stability analysis of secondary path estimation during FSF-based feedbac control, 2002, Proceedings IEEE Control Applications Conference, CD Rom Proceedings. 6 Tan, A. C. H. and Choudhury, A. and Ong, Y. S. and Veres, S. M.: Ultra low frequency estimation: a neglected apportion, 2002, Proceedings 9th Inter- Table 1: Estimated frequencies of the disturbance signal with various estimation methods and SNRs. FFT OFE True Freq SNR=-20 SNR=-10 SNR=0 SNR=10 SNR=20 SNR=30 SNR=40 133.4323Hz, 190.1451Hz, 285.9376Hz 133.2701Hz, 190.3691Hz, 285.8783Hz 133.4048Hz, 190.4548Hz, 285.7045Hz 133.3380Hz, 190.4735Hz, 285.7261Hz 133.3314Hz, 190.4756Hz, 285.7209Hz 133.3364Hz, 190.4752Hz, 285.7220Hz 133.3419Hz, 190.4732Hz, 285.7071Hz national Conference on Neural Information Processing (ICONIP), pp. 514 526. 7 Unbehauen, H. Regelungstechni II, Vieweg, Braunschweig, 1985.