Adaptive Filtering and Feedforward Control for Suppression of Vibration and Jitter

Transcription

1 Adaptive Filtering and Feedforward Control for Suppression of Vibration and Jitter Eric H. Anderson Ross L. Blankinship Leslie P. Fowler Roger M. Glaese Paul C. Janzen CSA Engineering Mountain View, CA Albuquerque, NM SPIE Defense & Security Symposium Orlando, FL April, 2007 Copyright 2007 Society of Photo-Optical Instrumentation Engineers. This paper is made available as an electronic reprint (preprint) with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited.

2 Adaptive Filtering and Feedforward Control for Suppression of Vibration and Jitter Eric H. Anderson a, Ross L. Blankinship b, Leslie P. Fowler b, Roger M. Glaese a and Paul C. Janzen a a CSA Engineering Inc., Mountain View, CA, b CSA Engineering Inc., Albuquerque, NM, ABSTRACT This paper describes the use of adaptive filtering to control vibration and optical jitter. Adaptive filtering is a class of signal processing techniques developed over the last several decades and applied since to applications ranging from communications to image processing. Basic concepts in adaptive filtering and feedforward control are reviewed. A series of examples in vibration, motion and jitter control, including cryocoolers, ground-based active optics systems, flight motion simulators, wind turbines and airborne optical beam control systems, illustrates the effectiveness of the adaptive methods. These applications make use of information and signals that originate from system disturbances and minimize the correlations between disturbance information and error and performance measures. The examples incorporate a variety of disturbance types including periodic, multi-tonal, broadband stationary and non-stationary. Control effectiveness with slowly-varying narrowband disturbances originating from cryocoolers can be extraordinary, reaching 60 db of reduction or rejection. In other cases, performance improvements are only 30-50%, but such reductions effectively complement feedback servo performance in many applications. Keywords: adaptive filtering, signal processing, vibration control, jitter control, beam control, active optics, cryocoolers, stabilization INTRODUCTION Active vibration control has most often been realized via feedback control, and jitter control for optical systems generally uses feedback servo architectures. Such realizations make use of actuators, error sensors, control system hardware and software, and a variety of control architectures and algorithms. While feedback designs often incorporate information about system disturbance inputs, for example through the use of weighting filters in H2-optimal designs, they are generally not configured to process and adapt in real time to changing disturbances. Adaptive filtering methods, beginning with techniques such as the least mean squares (LMS) algorithm, have been successful and are used widely in signal processing applications. The extension to processing and control of measured vibration signals has been demonstrated over the last two decades. Recent advances in digital signal processors (DSPs) and other digital hardware have made real-time adaptive filtering feasible for a broader range of vibration control. New control implementation options and sensors have allowed applications including optical jitter control. Active noise cancellation or control, taken here to mean active control of sound, was a first major extension of adaptive filtering algorithms developed originally for signal processing. Long before adaptive filtering algorithms, cancellation of sound waves in a one-dimensional duct was conceived. 1 With the development and formalization of algorithms by Widrow 2 and others, active control of sound 3,4 and later vibration 5 have become practical. Further, in the last years, processing electronics and computational capability have advanced, enabling applications that were not technically or economically feasible earlier. The following section reviews basic concepts in adaptive filtering and feedforward control, including the relationship between and simultaneous implementation of feedback and feedforward algorithms. The majority of the paper describes examples in vibration, motion and jitter control that illustrate the effectiveness of the adaptive methods. Beginning with single and multiple-tone disturbances, and progressing from vibration control with rejection of jitter 1

3 broadband disturbances, the paper concludes with a discussion of ongoing applications and a summary of common characteristics in the use of adaptive feedforward control. ADAPTIVE FEEDFORWARD CONTROL Feedforward control involves sending information, usually information about a disturbance to the system, forward so a control system can anticipate and counters its effect. Adaptive feedforward implies that the information is processed in changing ways during operation of the control system. In this paper we consider feedback control to be fixed, i.e. compensation dynamics are not changed once they are set. The first step in adapting a feedback controller might be gain scheduling, and many more sophisticated techniques are available. Feedback and feedforward need not be considered completely separate approaches as Figure 1 may suggest. Rather, as some of the applications in this paper make clear, there are cases where the two can combine. For example, an optical beam control system can combine the lower left and upper right quadrants of the figure. Figure 1: Four classes of control are not exclusive, and fixed feedback combined with adaptive feedforward work well together in jitter control applications This paper describes examples based on the least mean squares (LMS) algorithm, a particular simple method for estimation and real-time minimization of signal errors. 2 In LMS applied to control, the control output at time step n is y ( n) = n 1 i= 0 w ( n) x( n i) where x is a reference signal and w is a set of weights computed and updated by w ( n + 1) = w ( n) µ x( n i) e( n) i i which uses the reference signal and an error signal, e, along with an adaptation parameter, µ. Tapped delay lines are typically used to implement all-zero finite impulse response (FIR) filters as the architecture around which error minimization works. There are many variations on this basic algorithm, but the one of interest here is the filtered-x LMS (FXLMS) algorithm in which the reference signal, x, is filtered through an estimate of the system dynamics, sometimes called the secondary path, before it is used in the standard LMS way. The estimate of the secondary path can be determined on-line or off-line, and it can be represented in several ways. Figure 2 diagrams the algorithm, considering a discrete time representation in which step n is the present sample. The reference signal x is assumed to be filtered by an unknown set of primary path dynamics represented by P(z). The result of this filtering is the signal d. This signal corresponds to the vibratory load produced by the disturbance in response to a drive signal. There is also a secondary path, S(z), corresponding in the present case to the transfer function between the control actuator and the error sensor. Although S(z) is not known perfectly, a good model, S ˆ( z ), can be found i 2

4 through system identification. That model is used to filter the reference signal, and feed the LMS adapter to adapt the weights that describe the filter W(z). The result is the minimization of the error, e. x(n) P(z) d(n) W(z) y(n) S(z) + Σ - y'(n) e(n) ^ S(z) x'(n) LMS Figure 2: General version of filtered-x least mean square (FXLMS) algorithm (ref. 5) The case of a multi-tonal input is a special one for which it is possible to streamline the FXLMS algorithm (Figure 3). The response, d, through the primary path is the sum of the responses d(1), d(2),, d(m) to m different tones. A reference or synch signal feeds the controller. A sinusoid at the fundamental drive frequency is extracted from the reference. This allows the creation of a cosine for that tone as well as sines and cosines for integer multiples of the fundamental. Only two weights, w0, and w1, are necessary to characterize the response at each frequency. When converged, this algorithm performs just as an equivalent feedback compensator would. 6 Ref./ Synch Sine wave generator (tone 1) 90 deg. phase shift x(1) sine(1) cosine(1) w1(1) w0(1) P(1) + Σ + y(1) S(1) d(1) + Σ - e(1) ^ S(1) x'(1) LMS + Σ e e(2) e(3) e(m) sine and cosine waves for m other tones error to m other tone controllers Figure 3: FXLMS for the multi-harmonic control algorithm For all the applications discussed, three quantities should be noted: the truth quantity is most important to the application, the error measurement is the best and/or most accessible surrogate for the truth, and the reference signal conveys information forward to be used with the error signal in the control system. 3

5 SINGLE-AXIS VIBRATION CONTROL FOR CRYOCOOLERS Cryocoolers are a relatively common element in sensing systems of many types. Unfortunately, these thermoelectric coolers generate vibration as a byproduct of their cooling function, with the nature of the vibration depending on the type of cryocooler. Vibration is a dominant feature of the common Gifford-McMahon (G-M) and Oxford-style Stirlingcycle coolers. The discussion returns briefly to more difficult vibration control for G-M coolers at the end of this section, but Stirling coolers are the main focus here. The majority of the Stirling cycle disturbance derives from mass imbalance as the compressor piston and expander components reciprocate. One practical way to reduce vibration output is to operate dual opposing coolers. This can be effective in countering the dominant vibration at a fundamental drive frequency, but cancellation at harmonic frequencies vibration that results from nonlinear gas springs and other effects is not feasible by this simple method. Fortunately, adaptive cancellation can counter vibration beyond the fundamental. Figure 4: Basic conceptual layout of one axis of an integral Stirling cooler vibration cancellation system Active vibration cancellation produces forces that nearly (in theory, exactly) counter the forces generated by the cryocooler. The cancellation system includes a vibration-generating actuator and an error sensor or sensors, typically an accelerometer or load cell. Through an adaptive control system, the actuator is driven to apply forces to the cryocooler assembly in order to reduce the amplitude of the error signal. In the physical system (Figure 4), an inertial actuator is connected to the cryocooler. This type of actuator figures prominently in many active cancellation schemes. Also called a proof mass actuator, its physical realization is a voice coil driving a mass suspended on springs within a housing. The accelerometer is mounted on the main cooler body. In principle, the accelerometer can be located anywhere in the assembly, though further separation from the actuator may lead to somewhat decreased performance. Drive waveform PWM drive amplifier Compressor (nonlinear gas spring,etc.) Drive freq., temperature set points, other commands Temp. control Temp. conditoning Temp. sensor(s) Vibration DSP CONTROLLER BOARD TRANSDUCER I/F AND DRIVER BOARD Adaptive filters PWM drive amplifier Counterbalance RS-232 or other programming and data interface Power input (18-36 V) Error sens. conditoning Accelerometer(s) or load cell(s) Figure 5: Architecture of a typical Stirling cryocooler drive and adaptive vibration compensation system 4

6 The phase relationship relative to the input forcing is fixed when a passive absorber is used. In an active system, where residual error is measured, the counter-force can be adjusted adaptively in magnitude and phase to create a nearly-ideal anti-vibration. Unlike a passive absorber, the active system can compensate vibration at multiple frequencies simultaneously, updating the magnitude and phase of the vibration compensation in real-time using a digital signal processor (DSP). For most applications, the incremental benefit of controlling another (higher frequency) tone decreases, especially because displacement, not acceleration, is often the ultimate quantity of interest. Typical systems may control 8-10 tones in nominal operation, but only one tone during cool-down. This information feedforward allows the system to extract the frequency and phase of the drive. Thus, it has a reference signal to which it synchronizes its outputs at the fundamental frequency and all the harmonics. Figure 5 illustrates how the control ties in to the primary cryocooler drive and control system. Figure 6: Measured acceleration with and without axial active vibration control for a Stirling cryocooler assembly If the drive frequency shifts, the cancellation remains effective to a slew rate inversely related to adaptation time given a steady disturbance. If the model of the secondary path changes very significantly, for example if the phase of the plant changes by tens of degrees at a particular frequency, performance may suffer. Figure 6 and Figure 7 show vibration data acquired for one assembly showing dramatic reduction at the fundamental frequency (~60 Hz) and 9 harmonics. This was a controlled system and similar performance is certainly not possible in most cases. The rms vibration measured from Hz was reduced by 99.7% from g rms to g rms. Accelerations measured Hz, Hz and Hz were reduced by 99.8%, 99.8% and 99.9% respectively. This approach has been applied to several sensor systems that use cryocoolers, including one installation with two cryocoolers operating independently. Vibration reduction at the error sensor location is not difficult. However, vibration at the specific location of interest is not likely to be reduced as much. The real truth measurement is another process, one that may involve CCDs or other sensors. Adaptive feedforward control techniques that have been applied with great efficacy to Oxford Stirling coolers are not nearly as effective for G-M coolers. The disturbance mechanisms in the two cooler types are fundamentally different. In the Stirling case, the largest disturbance is cause by reciprocating compressor motion and there is significant harmonic content. In the G-M cooler there are low frequency impulsive inputs and substantial vibration is caused by gas flow. This flow is not the same on each cycle, and it is difficult for adaptive feedforward schemes to be effective. 5

7 Figure 7: Time domain measurements of axial vibration in a Stirling cryocooler assembly; plot on the right is a zoom ROBUSTNESS AND LIMITATIONS ON A LARGE STRUCTRE WITH FAST SLEWING TONES Another application illustrates both the flexibility of the adaptive feedforward approach and some of its limitations. The application was control of vibration transmission in a large wind turbine structure. Vibration generated in one region of the structure propagated through the structure and caused re-radiation of energy as acoustic noise. This system was characterized by multiple tonal disturbances, with the primary one slewing over about 50 Hz in a few seconds. Analysis and past tests indicated that the fast slew can be accommodated. 7 The vibration control system tested installed 14 inertial actuators with collocated accelerometers at a station of the structure near the vibration source (Figure 8). Tachometer signal Power in (~12-24 VDC) Amplifier Controller (DSP) Vibrating structure Voice coil force generator Reaction mass Actuator assembly (SA-10) Accelerometer PC (command, monitor, debug) Figure 8: One of 14 actuator-sensor assemblies for adaptive vibration control of a wind turbine structure Figure 9 shows how the measured collocated frequency response at three locations on the structure in the Hz range over which the primary disturbance tone slews. The overall responses are similar, with many closely spaced modes in this frequency band. In this case the plant model used was extremely simple: a pair of numbers magnitude and phase were used to describe the secondary path at each location. Based on Figure 8, this implies phase errors of up to ±55 degrees (theoretically ±90 could be tolerated), and magnitude errors of up to ±10 db. Measurements were made with the disturbance controlled in 10-degree increments, from 100 to 180 Hz. The average reduction at the 14 6

8 locations ranged from 8 db at 140 Hz to 17 db at 160 Hz. However, data showed only modest improvement of 1-7 db at a remote location on the structure, with vibration levels increasing for the 160 Hz case. Further, acoustic reductions with the actual fast-slewing disturbance were less than 2 db on average. Among the lessons from this application are: There is remarkable tolerance of the control algorithm to errors in the secondary path model, It is possible to implement multiple independent adaptive feedforward controls on the same structure, and Local control effectiveness does not always imply global control effectiveness. Figure 9: Magnitude and phase of collocated transfer functions for 3 locations over frequency range of interest MULTI-AXIS VIBRATION CONTROL AND ISOLATION Returning to the earlier application, we note that many systems that incorporate cryocoolers are sensitive to multiple axes of vibration. To the extent that axial vibration can be cancelled or controlled, response in other axes becomes dominant. In one case, the single-axis control used for other cryocoolers was adapted to the multi-axis case by integrated actuation within support struts (Figure 10). The FXLMS algorithm was implemented in the same way, and coupling, at least for the multi-tone disturbance case, was ignored. Disturbance force Inf ormation f eedf orward Pay load/disturbance source Voice coil force generator Controller Activ e stage To spacecraft or base Passiv e isolator Reaction mass Passiv e isolator Actuator assembly (SA-1) Force sensor EXTERNAL LAYOUT FUNCTIONAL LAYOUT Figure 10: Diagrams showing basic layout and architecture for each of six struts in hexapod demonstration systems for cryocooler disturbance isolation and spacecraft applications and photo of cryocooler system 7

9 The multi-axis system described in by Flint et al. 8 cancelled vibration in six axes from a cryocooler expander. This hexapod system (Figure 10, right) included passive vibration isolation and inertial actuators in each of six legs. Because force transmission was of interest, load cells within each strut were used as error sensors. With the primary cooler operating frequency at Hz, a five tone adaptive feedforward control system achieved the reductions of 47, 35, 35, 28 and 28 db on the five tones when the assembly was mounted to a relatively rigid base. This approach is likely to have reduced effectiveness for a compliant base with modes that couple in the frequency range of interest. A slightly more refined system was built to demonstrate a more general capability as part of a program originally expected to fly as a Space Shuttle Getaway Special experiment (Figure 11). The hexapod actively isolates disturbance forces transmitted from a mounted payload, such as a cryocooler or reaction wheels, to a supporting structure in all six degrees of freedom. Passive vibration isolation is provided by metallic flexures to which a variety of damping treatments can be added. Force transducers are used to provide closed loop control. Electromagnetic actuators provide the required counterforce as determined by the control system. Figure 11: A hexapod is one means of implementing disturbance cancellation in multiple axes Under a series of representative disturbances, the rms strut loads were reduced from mn to 8 to 24 mn. A noise floor limited levels to about 8 mn, but these were db lower than the case with control off. The strut loads will vary across different installations, since the dynamic behavior of the mounting surface affects the system. CONTROL OF OPTICAL JITTER Jitter is defined in this case as rotational motion of an optical beam away from line of sight. It is generally an angular quantity that might be measured in microradians, but errors are manifested when the light beam strikes some twodimensional surface such as a CCD and we can speak of numbers of pixels of jitter. One convention is to describe any such beam angular motion above a certain frequency, for example above 10 or 20 Hz, as jitter. Figure 12: Beam control / jitter control system with disturbance paths relevant to feedforward control 8

10 Every servo used for jitter control is limited in bandwidth due to actuator, sensor or control hardware or software limitations, or sometimes due to flexible mode response. Feedback requiring certain levels of stability most simply gain margin and phase margin is bandwidth limited. Adaptive feedforward control can augment servo performance by reducing jitter correlated with disturbances particularly in the frequency range around or above the servo crossover. Figure 12 depicts a general jitter control scenario that may occur in a laser weapon system, a reconnaissance camera or a semiconductor manufacturing or inspection machine. Of the many possible contributors to jitter, vibration of optical elements is often the direct contributor. These physical elements are in turn driven by local vibration and acoustic sources and by more remote sources that are transmitted into the local environment. The feedback control servo depicted on the right generally will be designed to exhibit certain properties including command following and disturbance rejection. It will not explicitly acknowledge or take advantage of known disturbance paths. This is where feedforward control enters. Given the transmission of Figure 12, we recognize that information can travel faster over wires and fibers than it travels through a physical system. Therefore, it becomes feasible to measure response at points upstream from the beam control system, feed that information into the BCS, and use an adaptive filter to remove the portion of the jitter that is correlated with the reference signal or signals. The feedback servo already does this in a broad sense, but at higher frequencies its disturbance rejection capacity rolls off. The feedforward control complements the feedback and can be implemented simultaneously with the servo control. Experiments on Laboratory Testbeds The possible effectiveness of adaptive feedforward algorithms was evaluated in a series of testbeds incorporating acoustic disturbances designed to capture some of the relevant elements of systems such as the Airborne Laser (ABL). 9 Position sensing detector Disturbance enclosure Ø 18" Composite tube Upstream microphones Disturbance speaker Fast steering mirror Mirror Accelerometers Control speaker Ø 8.00" flat mirror Mirror microphones Laser source not shown for clarity Optics bench Figure 13: ABL acoustic jitter suppression testbed and control architecture 9

11 A testbed (Figure 13) that captured the fundamental physics of the ABL acoustically induced optical jitter problem was developed and exercised. It consisted of a flexure-mounted mirror exposed to an acoustic field that is generated outside a beam tube and then propagates within the tube. Both feedback and adaptive feedforward control topologies were implemented using either of two actuators (a fast steering mirror, or FSM, and a secondary acoustic speaker located near the precision mirror), and a variety of sensors (microphones measuring the acoustic disturbance, accelerometers and microphones mounted on the precision optic, and an optical position sensing detector). The active jitter suppression techniques studied were broadly classified in two categories depending on the actuator used to control the jitter, active noise control using secondary acoustic sources and active optical control using a fast steering mirror. Results of testing are summarized in Table 1 which highlights that expected broadband performance is significantly less than narrowband performance, and also notes how performance was affected by choice of reference signal. Table 2 identifies performance using a speaker as the correcting actuator and the FXLMS algorithm with feedback neutralization. In this case, performance at the PSD truth sensor was satisfactory only when that sensor was used as the error measurement. Table 1: Performance for FSM feedforward control of position sensing detector x-axis PSD Narrowband PSD Broadband Reference Sensor Error Sensor Performance (db) Performance (db) 1. Signal generator PSD Microphone on mirror PSD Microphone in disturbance enclosure PSD Accelerometer on mirror PSD Table 2: Performance for error sensors and position sensing detector for narrowband FXLMS-FN using acoustic actuator Error Sensor Position Sensing Detector Reference Sensor Error Sensor Performance (db) Performance (db) Microphone upstream Mirror microphone Microphone upstream Mirror accelerometer Microphone upstream Position sensing detector single axis Broadband FXLMS Implementation on a Second Testbed Acoustically-induced jitter may also be important in sensors or other systems contained in pods underneath aircraft, or in aircraft weapons bays. Figure 14 shows a generic pod under an aircraft with several potential jitter mitigation elements. An under-wing location is likely to lead to extremely high sound pressure levels. An experimental program was executed to understand some of the possibilities of feedforward control. This scenario was represented generically by the testbed shown in Figure 15. It allowed for introduction of acoustic disturbances within a 12 cubic foot enclosure shown on the right. A broadband FXLMS algorithm was implemented with the reference signal fed directly from the disturbance signal generator to the control system including FSM and PSD, and the measured jitter was reduced by a factor of 3-4 over Hz (Figure 16). 10

12 Vibration isolator Microphones Piezo film sensor Speaker Surface actuato Acoustic blankets Accelerometers Figure 14: Pods of any kind under aircraft experience a harsh sound and vibration environment speakers fast steering mirror laser turning mirror microphones Figure 15: Testbed used for further study of jitter suppression by adaptive feedforward control Figure 16: Measured response of optical jitter with and without feedforward control shows significant improvement 11

13 This testbed confirmed the viability of broadband adaptive feedforward control and emphasized further the benefits of using optical quantities as error measures for jitter control. Tests at an Outside Facility Research was continued outside of the laboratory at a remote facility that incorporated more realistic beam control systems including various fast steering mirrors, cameras, light sources, etc. An established servo tracking loop with a bandwidth between 25 and 100 Hz was closed during all the tests. An artificial acoustic disturbance was provided by four JBL EON-G15 powered speakers mounted about 1.5 m off the floor, two meters from the main optical elements. They were driven such that the total sound pressure level around the main optics was about 114 db. The noise source was white noise low-pass filtered with a cutoff frequency of 500 Hz. An instrumentation microphone was mounted about 0.3 meters in front of each speaker. Jitter measurements were determined from analog signals exported from facility sensors. Initial tests verified that that acoustic disturbances dominated above 50 Hz and were approximately 40 db greater in magnitude than the jitter due to atmospheric turbulence or other sources. Testing determined the dynamic characteristics of our control actuator, the fast steering mirror (FSM). We measured the transfer function between the input to the FSM and the output of the jitter sensor for both axes. Two main classes of control were tested: feedforward control and feedback control. For these tests, two reference signals were used: the signal sent to the speakers (signal generator reference) and the signal from the microphones in the turret (microphone reference.) We implemented the FXLMS algorithm on a single-board computer with a Texas Instruments C6701 digital signal processor. The sampling rate for the broadband control was 2000 Hz, and the controller used 256 taps for the secondary path (S(z) in the diagram) filters and 640 taps for the primary path (W (z) in the diagram) filters. The two axes were controlled independently. A prerequisite for using a FXLMS controller is that the controller must know the impulse response of the control actuator to error sensor. In this case, the secondary path model was formed adaptively using the LMS algorithm. The secondary path impulse response was about 20 ms long and very similar in the two axes, with a significant latency (around 3 ms) associated with the FSM and analog reconstruction process. This latency could limit the performance of an FXLMS control algorithm under some circumstances. Figure 17: Controller performance, signal generator reference The first control test was feedforward control using the output of the signal generator as the reference signal. Only a single noise source was used; the same signal was fed to all four speakers. Although a real acoustic control system 12

14 would not have a signal generator output to use as its reference signal, it is a good test case to verify proper operation of the controller. The controller demonstrated good performance, 12 db in the X axis and 11 db in the Y axis, and did a particularly good job at quieting the large mode around 85 Hz, and good reductions were seen across the band. Figure 18 shows that the controlled performance is significantly closer to the ambient case (with the acoustic disturbance turned off) than it is to the disturbed case Controller performance, signal generator reference 10 1 Controller performance, microphone reference Jitter forward sum, µrad RMS Jitter forward sum, µrad RMS X, Uncontrolled (4.5 µrad RMS) X, Controlled (1.1 µrad RMS) X, Ambient (0.73 µrad RMS) Y, Uncontrolled (3.3 µrad RMS) Y, Controlled (0.95 µrad RMS) Y, Ambient (0.86 µrad RMS) Frequency, Hz X, Uncontrolled (4.5 µrad RMS) X, Controlled (1.2 µrad RMS) Y, Uncontrolled (3.3 µrad RMS) Y, Controlled (0.88 µrad RMS) Frequency, Hz Figure 18: Controller performance, forward sum, for signal generator reference (left) and microphone reference (right) Figure 19: Controller performance, microphone reference The next step was to repeat the controller test using a microphone to provide the reference signal. This represents a more practical configuration than the signal generator reference case. The performance for this configuration is shown on the right of Figure 18, and in Figure 19. The controller had very similar performance when using the microphone as the reference signal, providing 11 db of broadband attenuation in both the X and Y directions. The normalized rms 13

15 error in x and y converged from 1.0 to 0.4 in about 8 seconds and to its final value of about 0.3 in 30 seconds. By increasing the value of the normalized adaptation, µ, the LMS controller could be made to converge faster (at the cost of slightly reduced overall performance.) The final test set repeated the control using independent noise sources to drive the four speakers. This test was intended to more accurately represent the sound field likely in many applications. For this test, we used the sum of the four microphones as the reference signal. The performance in this case was notably worse than in the case where all speakers are driven with the same signal - just 2 db of broadband reduction in the X axis (5.0 to 3.0 microradians) and 3 db (5.5 to 2.3 microradians) in the Y axis. We also tried using the sum of the 4 signal generators as the reference output. However, this configuration yielded control performance similar to the case where we used the sum of microphone signals as the reference signal. Finally, we used a single microphone mounted about 1 foot away from the mirror as the reference source. Again, minimal reductions were found. For this case in particular, the 3 ms of latency in the impulse response of the FSM and camera might have limited the overall performance. This is because LMS requires that the time delay between the reference signal and its effect on the system be greater than the time delay between a command to the control actuator and its effect on the system. Test time was limited, but future efforts will address control in the presence of multiple incoherent sources. EXTENDING THE CAPABILITY IN SERVO APPLICATIONS AND ELSEWHERE This section describes ongoing research and application in adaptive filtering applied to motion control, beam control, and identification of disturbances. One area of interest is flight motion simulation, a good candidate for feedforward control because the command profile is often known. In this case, the control uses techniques described by Widrow and Stearns 2 and Widrow and Walach. 3,11 It begins with a feedback compensator around the open loop plant. This closed loop system itself becomes the new plant. Because the desired response is known, the goal is to find a controller, or adaptive filter, that effectively inverts the plant dynamics. This filter can be parameterized as a finite impulse response (FIR) filter, whose weights are adjusted to minimize the error. The adaptation uses standard variations of LMS. In Figure 20, if the controller correctly inverts the plant, the error is driven to zero, and the achieved angles equal the requested angles. Plant Output (achieved angles) Command Input (requested angles) Controller + - Σ Plant Controller (copy) Feedback compensator Error + Σ - Figure 20: Flight motion simulator control system uses both feedback and feedforward methods A promising area of ongoing work is in beam control enhancement for a flight system. A research program has paralleled the flight system implementation. A recent paper 12 presented an adaptive control strategy using FXLMS to improve upon the performance of a traditional Type II servo loop in the presence of multiple broadband disturbances. Simulation and experimental test results were shown for a multiple beam path optical test bed with two optical beam trains, FSMs for disturbance and control inputs, and PSDs for use as reference and error measurements. A pseudoreference was constructed using measurements from both a PSD and the fast steering mirror used for control. A second PSD was used as a truth measurement. Real-time control experiments were carried out comparing the performance of 14

16 the controller using adaptive filtering to that of a traditional servo control system design. Over a factor of two reduction in rms jitter at the truth sensor was achieved as compared to that achieved with the traditional servo design. The results show the ability to respond to and mitigate the effects of a complex broadband disturbance environment offers a more efficient and higher performance approach than conventional control system designs. It is not always easy to identify a reference quantity. One goal of ongoing research 13 is identification of disturbance paths through the system of interest, and parallel identification of secondary paths from actuators to error quantities. The adaptive nature of the algorithms allows there to be substantial inaccuracies in the latter. Real-time identification and updating of secondary path models can enable higher performance in some cases. Another area of interest is in adaptive feedforward control in the presence of more complex disturbances. Recall that the testbed experiment described above was much less effective when the four disturbance sources were incoherent. The response of a linear system can be decomposed into a number of independent disturbances that propagate through distinct paths. In many cases, we ignore all disturbances but one because it by far dominates the others. But in some cases, there may be multiple independent sources. It can be useful to decouple these sources and treat each independently in the control. The challenge is then in determining how many independent sources there are, and how many are important. To the extent the disturbance-to-performance paths are known or can b estimated, Hankel singular values form a useful tool in ranking importance of paths. Another area of interest is the application of adaptive filtering beyond jitter control to wavefront control. 14 Whereas jitter control does not convey spatial information, spatial information is at the heart of wavefront control. Thus, the adaptive filtering algorithms have been applied to both the response of individual channels and the processing of information from adjacent cells. SUMMARY AND CONCLUSIONS The paper illustrated the effectiveness of adaptive feedforward control for vibration and jitter suppression through a series of examples. Results of the various examples are summarized by Tables 3 and 4. Table 3: Summary of reference, error and truth quantities for each application Application / System Reference Error Truth Cryocooled sensors Cryocooler drive signal Accelerometer(s) Displacement at focal plane array or detector (cm distance for accel.) Isolated platform Disturbance signal Load cells in mount Load transmitted Wind turbine Ratio of gear tooth rates Local acceleration Global sound attenuation Laboratory testeds Remote facility Disturbance signal or microphones Disturbance signal or microphones Jitter (2 axes) position sending detector Jitter (2 axes) position sending detector Jitter (2 axes) Jitter (2 axes) Motion simulator Command (2 axes) Measured angles Achieved angles Servo control airborne application Wavefront control Acceleration plus command input Pseudo-reference with command Jitter (2 axes) PSD and other sensors Slopes from wavefront sensor Jitter (2 axes) Wavefront error 15

17 Table 4: Summary of temporal / frequency content and variation for each application Application / System Content Time variation Cryocooled sensors Multi-tonal Very slow Isolated platform Multi-tonal Slow-moderate Wind turbine Multi-tonal (one dominant) Rapid Laboratory testbeds Tonal and broadband Slow Remote facility Broadband Slow Motion simulator Multi-tonal, some broadband Moderate Servo control airborne application Broadband Slow and moderate Wavefront control Broadband plus spatial variation Slow and moderate As it is applied in more places, adaptive feedforward control continues to be an area of research. Progress on methods for developing secondary path models, identification and control with multiple disturbance sources, use of IIR filters, and overlap with adaptive feedback methods will only expand the applicability of the techniques illustrated here. REFERENCES 1. Lueg, P., Process of Silencing Sound Oscillations, US Patent 2,043,416 (1936). 2. Widrow, Bernard and Stearns, Samuel D., Adaptive Signal Processing, Prentice-Hall (1985). 3. Elliott, S.J. and Nelson, P.A., Active Control of Sound, Academic Press (1992). 4. Kuo, S. and Morgan, D., Active Noise Control Systems, John Wiley and Sons (1996). 5. Fuller, C.R., Elliott, S.J. and Nelson, P.A., Active Control of Vibration, Academic Press (1996). 6. Sievers, L. and von Flotow, A., A Feedback Perspective on the LMS Disturbance Feedforward Algorithm, American Control Conf., Vol. 2, pp (1994). 7. How, Jonathan P. and Anderson, Eric H., Active Vibration Isolation Using Adaptive Feedforward Control, American Control Conference (1997). 8. Flint, E.M., Flannery, P.S., Evert, M.E. and Anderson, E.H., Cryocooler Disturbance Reduction with Single and Multiple Axis Active/Passive Control Systems, SPIE Smart Structures Symp., paper (2000). 9. Glaese, R.M., Anderson, E.H. and Janzen, P.C., Active Suppression of Acoustically Induced Jitter for the Airborne Laser, SPIE Aerospace Sensing Conference (2000). 10. Widrow, B. and Walach, E., Adaptive Inverse Control, prentice Hall (1995). 11. Anderson, E.H., Evert, M.E., Flannery, P.F., Fowler, B.F., Glaese, R.M., Image Stabilization Testbed (ISTAT), SPIE Aerospace Sensing Conf., (2001). 12. Fowler, L.P. and Blankinship, R.L., Experimental Adaptive Beam Train Control with Multiple Beams and Disturbance Paths, Directed Energy Symposium, Monterey, CA (2007). 13. Moon, S-M, Fowler, L.P, Clark, R.L. and Anderson, E.H., Real-time Optimal Sensing Strategy for Adaptive Control of Optical Systems, SPIE Defense and Security Symp., (2007). 14. Quintana, S. and Blankinship, R., Adaptive Feedforward Control for Adaptive Optics Systems, Ninth Directed Energy Symposium, Albuquerque, NM (2006). 16