Traditional approach to vibration control uses passive techniques, which are relatively large, costly and ineffective at low frequencies. Active Vibration Control (AVC) is used to overcome these problems and in AVC additional sources (secondary) are used to cancel vibration from primary source based on the principle of superposition theorem Since the characteristics of the vibration source and environment are time varying, the AVC system must be adaptive. Adaptive systems have the ability to track time varying disturbances and provide optimal control over a much broader range of conditions than conventional fixed control systems. In multi channel AVC vibration fields in large dimensions are controlled and is more complicated. Therefore to actively control low frequency vibrations on large structures, multi channel AVC requires a control system that uses multiple secondary sources to control the vibration field simultaneously at multiple error sensor locations. The error criterion that can be directly measured is the sum of squares of outputs of number of sensors. The adaptive algorithm is designed to minimize this and the algorithm implemented is the quot;Multiple error LMS algorithmquot;. The best known applications of multiple channel FXLMS algorithm is in real time AVC and system identification. Additionalr applications are in the control of propeller induced noise in flight cabin interiors. In the present paper the results of simulation studies carried out in MATLAB as well as on TMS320C32 DSP processor will be brought out for a two-channel case

Balasubramaniam, R

Prakash, Shashikala

Praseetha, KK

English

NAL-IR

Simulation studies for multichannel active vibration control

National Aerospace Laboratories Institutional Repository

Proceedings of ISSS-SPIE 2002 International Conference on Smart Materials Structures and Systems July 17-19, 2002, Indian Institute of Science, Bangalore, India  ISSS2002/SA-434   Simulation Studies for Multichannel Active Vibration Control   Shashikala Prakash,  R.Balasubramaniam K.K. Praseetha Structures Division National Aerospace Laboratories, PB 1779, Kodihalli, Bangalore 560017 India E mail: shaship@css.cmmacs.ernet.in *  ABSTRACT   Traditional approach to vibration control uses passive techniques, which are relatively large, costly and ineffective at low frequencies. Active Vibration Control (AVC) is used to overcome these problems & in AVC additional sources (secondary) are used to cancel vibration from primary source based on the principle of superposition theorem Since the characteristics of the vibration source and environment are time varying, the AVC system must be adaptive. Adaptive systems have the ability to track time varying disturbances and provide optimal control over a much broader range of conditions than conventional fixed control systems.    In multi channel AVC vibration fields in large dimensions are controlled & is more complicated. Therefore to actively control low frequency vibrations on large structures, multi channel AVC requires a control system that uses multiple secondary sources to control the vibration field simultaneously at multiple error sensor locations. The error criterion that can be directly measured is the sum of squares of outputs of number of sensors. The adaptive algorithm is designed to minimize this & the algorithm implemented is the  “Multiple error LMS algorithm1,2”. The best known applications of multiple channel FXLMS algorithm is in real time AVC and system identification. More wider applications are in the control of propeller induced noise in flight cabin interiors. In the present paper the results of simulation studies carried out in MATLAB as well as on TMS320C32 DSP processor will be brought out for a two-channel case.  Key words: Active Vibration Control, Adaptive filters, Smart Actuators, Smart Sensors, Digital Signal Processing, Least Mean Square algorithm, MATLAB, Simulation, Code compose studio. ACRONYMS                                                              P   Physical plant IIR  Infinite Impulse Response S   Secondary actuator LMS  Least Mean Square W   Adaptive FIR filter PZT  Lead Zerconate Titanate  AVC  Active Vibration Control d  Desired signal CFRP  Carbon Fiber Reinforced Plastic e  Error signal FIR  Finite Impulse response  x  Primary noise/Input FXLMS  Filtered –X LMS y  Control filter output  1y’  Secondary actuator output   INTRODUCTION  In many applications it is desirable to cancel / suppress vibration at multiple location i.e.. The simultaneous control of several rigid body modes are required in a mechanical structure Active Vibration Control is performed by introducing secondary forces to cancel the effects of undesired or primary vibration sources.  The secondary forces are driven such that the super position of the primary and secondary fields destructively interfere with each other.  Thus in AVC, vibration in the form of rigid body modes are suppressed or cancelled by the super position of areas in a mechanical structure.  In this paper the simulation studies carried out in MATLAB for a single input multiple output (SIMO) control of a structure is brought out. The block diagram of the two channel AVC System is shown in Figure. 1. The algorithm used is the feed forward filtered – x LMS algorithm3,4 to cancel broadband disturbances.  It consists of a matrix of finite impulse response (FIR) control filters followed by a recursive section of FIR filters connecting each out put to each input.            Figure 1: 1x2x2 Multicahannel Active Vibration Control syst  The algorithm performs system  identification where Slm is defined from thsource to the output of lth error sensors.  This is required before the adaptationIn this paper the results from 1 X 2 X 2 channel   is presented (one input channactuators). Also DSP implementation of the same has been completed succeboard.  The real time testing on actual structure has to be carried out.  2. MULTIPLE ERROR LMS ALGORITHM  2.1 Derivation  Let the sampled output of the lth error sensor be el(n), which is equal to the suthe sensor, dl(n), due to the primary source operating alone, and an output duethe sampled input to the mth actuator be obtained by filtering the reference signPower Amplifier For B&K shaker GCFRP beam Charge AmplifierPower AmplifierFor PZTCharge Amplifier Power Amplifier For PZT PZT Sensor PZT Actuator em setup. TMS320C32  DSP Board  In Pentium I Host PC   Random Noise Sine Signal eneratoe input of the of filter paramel, 2 error sensssfully on a TMm of the “des to each of thal x (n) using Analog/ Digital ConverterDigital / Analog ConverterAnalog/ Digital ConverterDigital / Analog Converter mth secondary eters can begin. or and 2 control S320C32 DSP ired” signal from e actuators.  Let an adaptive FIR 2controller whose ith coefficient at the nth sample is wmi (n).  Let the transfer function between this input and the output of the lth sensor be modeled as the Jth – order FIR filter, whose jth coefficient is clmj, so that   el(n) =  dl(n) +        (1) )(110jnsc mMmJjlmj −∑ ∑=−=where     Sm(n)  =  ∑         (2)  )(10inxwIiim −−= It is assumed that there are L sensors and M actuators, and that L ≥ M.  Let the total error J be defined as        J = E 2e          (3) ∑=)(1nLli Where E { . } denotes an expectation value. If the reference signal x (n) is at least partly correlated with each dl(n), it is possible to reduce the value of J due to the primary source alone, by driving the secondary sources with a filtered version of the reference signal, as indicated in eqn(1).    It is clear that the total error will be a quadratic function of each of these filter coefficients. The optimum set of filter coefficients required to minimize J may thus be evaluated adaptively using gradient descent methods.  The differential of the total error with respect to one coefficient is       ∂∂=∂∂ ∑=Li imllmi wneneEwJ1)()(2        (4)   Assuming for the moment that each wmi is time invariant differentiating (1) with respect to one of these coefficients gives      imlwne∂∂ )(  =        (5) )(10jinxcJjjml −−∑−= This sequence is the same as the one which would be obtained at the lth sensor if the reference signal, delayed by i samples, were applied to the mth actuator.  Let this be equal to rlm ( n – i), a filtered reference.  If each coefficient is now adjusted at every sample time by an amount proportional to the negative instantaneous value of the gradient, a modified form of the well-known LMS algorithm is produced.      wmi (n+1) = wmi (n) - µ       (6) ∑=−Limll inrne1)()(  Where  µ is the convergence coefficient.  This is a form of  “stochastic gradient” adaptive algorithm.   3For a single input, single output system (L = M = 1), this corresponds exactly to the “filtered x LMS” algorithm discussed by Widrow and Stearns. The assumption of time invariance in the filter coefficients is equivalent, in practice, to assuming that the filter coefficients wmi change only slowly compared to the timescale of the response of the system to be controlled.  This time scale is defined by the values of the coefficients clmj.   2.2 Time Domain analysis  In order to analytically demonstrate the shape of the error surface, and so determine the optimum, wiener set of filter coefficients5, it is convenient to consider the case in which the filter coefficients are exactly time invariant.  eqn(1) may be written el(n) =  dl(n) + .       (7) ∑∑=−=MmIiimw110)(10jinxcJjlmi −−∑−=            =  dl(n) + . r∑∑=−=MmIiimw110lm(n-i)  Where the filtered reference signal rlm (n) is defined as above.  This equation may be written as  el(n) = dl(n) +          (8) wr TlTlr =       (9) +−−+−−+−−+−−)1(),1(),()1(),1(),()1(),1(),()1(),1(),(333222111InrnrnrInrnrnrInrnrnrInrnrnrlMlMlMlllllllllΚΚΚΚ wT =     (10) −−−110122120111110,,,,,,IMMMIIwwwwwwwwwΚΚΜΚΚΚΚeT  = [e1(n), e2(n), . . . . eL(n)]      (11) dT  = [d1(n), d2(n), . . . . dL(n)]      (12) e = d + Rw,     RT = [r1, r2, r3,. . . . rL], The error criterion can now be written as  J = E  = E{ e∑=)(12 neLliTe} = E{ dTd} + 2wTE{RTd} + wT E{RTR}w (13)   4Thus the error surface is quadratic in nature the surface has a unique global minimum. This will be known by examining the positive definiteness of E {RTR}.  By setting the differential of this expression with respect to w to zero, the optimum Wiener set of coefficients may be obtained as      wopt  =  -[E{RTR}]-1    E{RT}       (14)   This set of filter coefficients gives a minimum error criterion equal to  Jmin = Jo  - E{dTR}[E{RTR}]-1    E{RTd}     (15)  Where  Jo  = E{ dTd} is the value of the error criterion with no control applied.  3. ALGORITHM SUMMARY  3.1. Adaptive filtering:   ( Figure. 2)   x(n))(w)( T11 nny =  &   , x(n))(w)( T22 nny =  ) )   ))∑ ) e1(n)  S^11(z) 1 ))) S^21(z) ) x’21(nS^12(z) 2 ))) S^22(zP1(z)W1(z )LMSLMS) ∑ ) e2(n) W2(z) ))P2(z)S11(zS21(zS12(zS22(zy’12(ny’11(ny’21(nx’11(nx’12(n x’22(ny’22(nX(n) Figure.2 Block diagram of a 1x2x2 Multiple Channel AVC System d2(ny1(n)y2(nd1(nd1(n5               3.2.  Multiple-channel pre filtering:  x(n)x' ml *)(')( nsn ml≡     ∑−==10, )(Iijml ns i)-x(n      l (sensors)  =1,2 & m (Actuators)  =1,2 and                                              J is the order of the FIR filter  ).(' zs ml 3. Weight vector updates:  )]()()()([)()1( 2111 nennennwnw 1211 x'x' ++=+ µ  )]()(x')()(x'[)()1( 22212122 nennennwnw ++=+ µ  4. IMPLEMENTATION ASPECTS  4.1 MATLAB6 Implementation: The simulation of a single input 2 error sensors, 2 control actuator case ( 1 X 2 X 2) case was attempted in MATLAB on a Pentium II 350 MHz PC.  This needs 2 adaptive filters each having only 2 weights.  Although the algorithm is valid for general broadband signals, considerable simplifications occur in the implementation if the reference signal is sinusoidal.  The desired signals were generated by passing the input signal x (n) through a 2 point FIR filter.  In the multi channel case there appears to be no more interference between the convergence of the coefficients of the multiple filters than between the coefficients of a single filter, even though the update terms for each filter are coupled. Once we got good convergence for the case, two other cases were tried.  In one case we used random signal with a sinusoidal component as the reference signal and tried to filter out all the random components after convergence i.e., the desired signal was a sinusoid (the same that was added to the random reference signal).  In the second case two sinusoids were added and used as reference signal and good convergence was obtained.  These are clearly shown in figures 3, 4 & 5 respectively.  We have overlapped the filter output  onto the desired signal spectrum and zoomed one portion to show how good the match is.  As we go for random and complex reference signal inputs, we see that we need more number of tap weights to achieve good convergence in a reasonable time. If we use two weight vectors, then the number of samples required will be enormous and also the time taken will be too high.    4.2 DSPImplementation: The multiple error LMS algorithm has been programmed in assembly language on a Texas Instruments TMS320C327 DSP board mounted in a Pentium I 100 MHZ PC in assembly language.  Circular buffers and the parallel instructions have been used, which considerably speeds up the computations.  Initially trials were made on sinusoidal reference signal accessed as a look up table.  Once confidence was gained signals were acquired from a signal generator and simulation studies were carried out.  For the case of random signal added to sinusoid & for system identification, the random signal was generated using the DSP processor using the pseudo random noise generation algorithm and this was added to the acquired sinusoid.  Good results have been obtained.    The robustness of the algorithm as proved by the random and complex input cases show that the convergence is largely unaffected by the introduction of considerable uncorrelated observation noise Also the programs have been put on the Texas Instruments Code Composer Studio8 and have been meticulously tested.  We need to conduct experiments on an actual structure and we need to try to suppress the vibration levels at two modes using PZT sensors and smart PZT actuators.    6 0 5 0 0 1 0 0 0- 3- 2- 10123P lo t  s h o w in g  D e s i r e d  S ig n a l , F i l t e r  O u t p u tN u m b e r  o f s a m p le sAmplitude6 2 0 6 3 0 6 4 0 6 5 0-2-1012P lo t  s h o w in g  D e s i re d  S ig n a l , F i l t e r  O u t p u tN u m b e r  o f s a m p le sAmplitude0 5 0 0 1 0 0 0- 2- 1012P lo t  s h o w in g  O ve r l a p  o f D e s i re d  S ig n a l  &  E r ro r  S ig n a lN u m b e r  o f s a m p le sAmplitude0 5 0 0 1 0 0 0-2-1012E r ro r  S ig n a l  fr o m  c h a n n e l  1N u m b e r  o f s a m p le sAmplitude  Figure. 3  Matlab Simulation results – Sine Input/Sine Desired  0 2 0 0 0 4 0 0 0 6 0 0 0- 2- 1012O v e r l a p  o f  D e s i r e d  s i g n a l  &  F i l t e r  o u t p u t ( 1 + 2 )- - >   N u m b e r  o f  S a m p l e s-->  Amplitude3 6 0 3 8 0 4 0 0- 1 . 5- 1- 0 . 500 . 511 . 5O v e r l a p  o f  D e s i r e d  s i g n a l  &  F i l t e r  o u t p u t ( 1 + 2 )- - >   N u m b e r  o f  S a m p l e s-->  Amplitude0 2 0 0 0 4 0 0 0 6 0 0 0- 2- 1012- - >   N u m b e r  o f  S a m p l e s-->  AmplitudeD e s i r e d  S i g n a lE r r o r  S i g n a l0 2 0 0 0 4 0 0 0 6 0 0 0- 2- 1012- - >   N u m b e r  o f  S a m p l e s-->  Amplitude E r r o r  S i g n a l  ( M a g e n t a )E r r o r  s i g n a l  F i l t e r  O / P  ( r e d )Desired Signal /Error  Figure 4. MATLAB Simulation outputs– Complex I/P Complex Desire    7 CONCLUSIONS  This paper describes the implementation of Multi Channel LMS algorithm for the Active Vibration Control (AVC). This implementation has been carried out by using an adaptive filter that is a digital filter composed of a tapped delay line and adjustable weights, having a finite impulse response, which is controlled by the LMS algorithm. For stationary stochastic inputs the mean square error, which is the difference between the filter outputs and an externally supplied input called desired response is a quadratic function of the weights. This multi channel LMS control algorithm has been implemented on a Pentium II 350MHz PC using the MATLAB utility. The program has been validated for cases like sine I/P sine Desired, random input/sine desired and complex input/complex desired signals. This multi channel LMS algorithm has been implemented for the case of two adaptive filters, two error sensors and two control actuators. 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0- 2- 1 . 5- 1- 0 . 500 . 511 . 5O v e r l a p  o f  D e s i r e d  s i g n a l  &  F i l t e r  o u t p u t ( 1 + 2 )- - >   N u m b e r  o f  S a m p l e s-->  Amplitude3 6 5 0 3 7 0 0 3 7 5 0- 1- 0 . 500 . 51O v e r l a p  o f  D e s i r e d  s i g n a l  &  F i l t e r  o u t p u t ( 1 + 2 )- - >   N u m b e r  o f  S a m p l e s-->  Amplitude0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0- 1- 0 . 500 . 51- - >   N u m b e r  o f  S a m p l e s-->  AmplitudeFD e s i r e d  S i g n a lE r r o r  S i g n a l0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0- 1- 0 . 500 . 51- - >   N u m b e r  o f  S a m p l e s-->  Amplitude E r r o r  S i g n a l  ( M a g e n t a )E r r o r  s i g n a l i l t e r  O / P  ( r e d )Desired Signal /Error  d  This work is supported by ththank AR&DB & Director, NAleast, our thanks are due to M  1. S. J. Elliott, P. A. NelsSound & Vibration”, IE2. S. J. Elliott, P. A. Nels1985”, 10th October 13. Bernard Widrow, Sam4. Sen M. Kuo, Dennis RJohn Wiley And Sons5. John R. Treichler, C.RJohn Wiley & Sons 196. Matlab Reference Ma7. TMS320C32 users gu8. Code Composer Stud Figure 5. MATLAB Simulation outputs – RandomI/P & SineDesireACKNOWLEDGMENTS e Aeronautical Research and Development Board.  The authors would like to L for giving us the opportunity to carry out the work at NAL. Last, but not the rs. B.C.Shashikala for her meticulous help in preparing the manuscript REFERENCES  on “A Multiple Error LMS Algorithm & its application to the Active Control of EE – ASSP 35, No.10. (Oct 1987).  on “Algorithm For Multi Channel LMS Adaptive Filtering”, Electronics Letters 985, Volume-21, No. 21, Pp 979-981. uel D. Stearns-“Adaptive Signal Processing”, Prentice Hall, Inc., 1985. . Morgan, Active Noise Control Systems, Algorithms And DSP Implementation, -Inc, 1996. ichard Johnson, Michael G. Larimore-“Theory And Design Of Adaptive Filters”, 87. nual. ide io users Manual 8

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Simulation studies for multichannel active vibration control

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