Adaptive Feedforward Control for Actively Isolated Spacecraft Platforms

ADAPTIVE FEEDFORWARD CONTROL FOR ACTIVELY ISOLATED SPACECRAFT PLATFORMS

Eric H. Anderson CSA Engineering, Inc. 2850 West Bayshore Road, Palo Alto A 94303-384C , A US 3 Jonathan P. How Dept. of Aeronautics and Astronautics Stanford University, Stanford 9430A A C , US 5

Abstract Vibration isolation systems separate vibration sources from vibration-sensitive components. There Active vibration isolation system beine ar s g consid- basio tw ce typear isolationf so , classifie come th y -db ered to improve the performance of spacecraft instru- ponent being isolated: 1) instrument or payload isola- ments and sensors. Because of uncertainties inherent bas) 2 tioed isolationnan . Both options have potential in on-orbit operation, adaptive control strategied an s use on-board spacecraft immediate Th . e motivation algorithms have relevanc theso et e systems thin I .s for this paper is payload isolation. paper, analysis of the algorithms, numerical simula- Most isolators in use are compliant passive systems tion, and laboratory test data are used to evaluate which support vibrating machines. Passive isolators adaptive feedforward control f particulaO . r interest effectively reduce transmissio f higno h frequenc- en y are performance characteristics and limitations of the ineffective ar ergy t bu , e belo suspensioe wth n frequen- filtered-x LMS (FXLMS) algorithm and its finite im- cies of the isolated system. These mounts are also pulse response (FIR) filter implementation. Combina- difficult to tailor to narrowband applications involving tion feedback/feedforward Augmentecontroe th d lan d transmissio singlf no multiplr eo e tones extensioe On . n Error algorith meano tw e sm ar investigate exteno dt d is adaptive passive isolators — inherently passive de- capabilitiee th FXLMf o s desensitiziny Sb e algogth - vices adjusted periodically by an external control sys- specifie rithth mo t c dynamic plante th f so . Several tem. Examples include stiffening a mount to counter experiments were conducte laboratora n o d y testbed maneuvering loads in a vehicle, and then softening the which serves as the prototype for a planned active vi- mount during cruise. Adaptation hi this case adds bration isolation flight demonstration. versatility, but the approach is still constrained by fundamental performance limitations associated with passive isolation. Adaptive Vibration Isolation Fully active isolation systems introduce fundamental advantages over their passive or adaptive passive jittee Th r requirement certair fo s n spacecraft sensors counterparts transmissibilite Th . isolatoe th n f yo ca r and instrument e stringentar s , resultin n submii g - tailoree b selectivelo dt y attenuate important inputs cron motion specifications. In the presence of reac- withou passive th t e constraint relating isolation cor- tion wheels, motors othed an , r articulating devices, ner frequency and static deformation. Overall perfor- the level of vibration on the satellite may exceed re- mance improvement significante b n sca . Active isola- quirements. Vibration isolation is one means of re- tio alreads ni y commo terrestrian i l applicationst bu , ducin e effecg th f motio o t t criticana l locationsA . this pape guides potentiare i spacecraftth n y o db e us l , well-designed isolation system would begin with pas- wher flighe eth t heritag numbee th shors e i d f an tro sive control and add active and finally adaptive control application increasings si casee on three-axi a , n I . - sac needes a countedo t r specific disturbance environments tive isolation system was demonstrated for a vibrating or meet particular instrument specifications. cryocooler [1]. Other active systems have been de- * [email protected]; 415-494-7351; www.csaengineering.com; velope r microgravitdfo y isolation within modulef so [email protected] manned vehicles e Shuttlth n e o ,e th Orbited an ] [2 r ©1997 by Eric H. Anderson and Jonathan P. How Space Station [3]. More recent systems [4] will isolate Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

specific spacecraft instruments. Tabl : Applicatione1 adaptivr sfo e active isolation. e disturbanceTh s on-board operational spacecraft include reaction wheels, thrusters, solar array drives, motors, solar arrays, cryocoolers, and other instru- Physical Effect Ratf eo Examples ments. Thei large rb inpuy e enougma t disrupo ht t Variation operatio more th f eno sensitive instrument send san - Time-varying dist. low-high Reaction wheels, sors onboard satellite imagin d communicationan g s solar arrays systems. An active mount can isolate either the dis- Base/component modes low Flexible satellite turbance generato sensitive th r o r e equipment. Inertia changes low-med New instrument, Unfortunately e benefitth , f activo s e isolatioe nar moving stage partiall ye potentia offseth y b t l stabilit perford yan - Change reqdn si . perf low-high Launch/on-orbit pointing mance difficulties associated with implementine gth Subsystem failure low-high Transducer, spring controllers in a space environment that is challenging Nonlinearities low Aging, friction to simulate accurately on the ground. Furthermore, the disturbance environment is often poorly charac- problems [8]. terized. However, many of these issues can be over- This paper support separate sth e developmend an t come using adaptive control to modify an active con- testin plannea f go d passive/active/adaptive isolation trol structure, function, or form to improve perfor- space flight experiment. Whil e Satelliteth e Ultra- mance based on a new understanding of the system quiet Isolation Technology Experiment (SUITE- de ) dynamic r propertieso s (See Tabl . Adaptive1) e con- velopment continues seee addresw ,o k t s more broadly trol makes possible the full performance benefits of an applicable area adaptivf so e vibration isolatione Th . active isolation system. work here parallels some of the developments of Som- The application of adaptive control to vibration sup- merfeldt and Spanos, but we also explore both feedfor- pressio generan ni increasings i l , driveavaile th y -nb ward and feedback control, slewing disturbances, and cosabilitw tlo digitaf yo l signal processors (DSPs)- Vi . potentiae th l rolbenefit d adaptivw ean ne f so e feedfor- bration isolation is a more tractable problem than gen- ward control algorithms. Adaptive feedback and adap- eral vibration control because of the potentially lower tive feedforward are the two broadest classes of control order dynamics and inherent limitations on the num- with potential applicatio probleme th o nt . Adaptive physicaf o r be l path energr sfo y flow. feedbac e morth s ei k general formulation t typibu , - Sommerfeld approache] [5 t d adaptive isolation from cally depends accuratmorn a n eo e system modeld an , an active noise control perspective initias Hi . l labora- places greater computational demand e physth n -o s tory demonstration use off-linn da filteR modeeo FI t r l ical control system. The main advantages cited for secondare th y path dynamic on-linn a filted R esan FI r a feedforward e straightforwarsysteth e mar d physi- for control. Performance was demonstrated for single l implementatioca enhanced nan d stability properties. tones, two tones, and limited broadband frequency re- We discuss several connections between the feedfor- gions. This was later replaced with on-line estimation ward vibration isolation architectur adaptive th d ean e modelR FI MIMe f oso th f O system e simsth [5]-n I . signal processing framework. plest cases demonstratee h , o t pura df g o ela tha e us t The remainder of the paper focuses on adaptive mode secondare lth y pat adequates hwa primare Th . y feedforward control, beginning with a discussion of emphasis of his work was on the interaction between LMS-based algorithms e limitationTh . f finitso - im e the on-line estimation and FXLMS algorithms. pulse response (FIR) filter r controfo s f lightlo l y Spanos et al. [6] demonstrated isolation on a six- damped systems are reviewed. The hardware testbed strut laboratory hexapod system and are participat- used for experimental demonstration is described. Fi- ing in a flight demonstration system [4]. Beginning nally, results are presented and implications of these from a feedback background, and then moving toward result applicatior sfo spacecrafno t t vibration isolation FXLMS algorithms, their approach makes use of accu- are assessed. rate identified model e inversth f so e plant dynamics. Hayne . [7al ] t usee s n adaptivda feedforwarR eFI d controller to minimize the acceleration error while si- Adaptive Feedforward Algorithms multaneously runnin gsecona filteR modedo FI t r e lth plant dynamics. Relatively high orde filterR rFI s were Lease Th t Mean Square algorith lona s gm ha heritag e use achievo dt attenuatio B averagd n ea 5 1 f e- o nbe in the fields of adaptive signal processing and control, tween 50-250 Hz. Other promising recent work has and thus need not be developed in detail here. How- applied a neural framework to vibration suppression ever, there are several features of this algorithm that Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

are of particular interest because they strongly influ- in Fig. l(b) [13, 14]. This approach consists of a rela- ence the resulting compensator and the implementa- tively simple feedback compensator Gc, the coefficients tion, and these are discussed below. of adaptivele whicb n hca y tuned usin gfrequenca y fol- standare algorithTh S LM d e mforwrittes th i f mo n lowing circuit [13]. The feedback approach is relatively as a feedforward compensator H with finite impulse easy to implement for harmonic disturbances because response (FIR) filter coefficients hi (see Figure 1). The simple closed form expression available s. ar c G r efo objective is then to select N filter weights to generate a possibls i t I gaio et n much further insight into these control signal u to cancel (through 5) the disturbance adaptive algorithm analyziny b s e staa cas gth f -eo created by the input x through the primary path P. tionary harmonic disturbance at frequency U>Q. For e SISThusth r O fo ,case controe th , l signa t tima l ek this case, several authors have shown that the LMS generates i d fro filtee mth r and FXLMS controllers have the LTI form [14, 11]

•>JV - Z)i=o hi(k)xk-i= (1) (4) which includes the vector of weights hfc = T .-..cos fas + sin00^0

[h0(k) • • • /ijv(fc)] and a vector of filter inputs x& = [xfc • • • Xk-N]-T Typically, Xfc is obtained from the input x through a tap delay line. where S(juo) — Me^° models the secondary path s discusseA detain di Widron i l Stearnd wan s [9], dynamic disturbance th t sa e frequency cone , th /i-s i the filter weights hfc are selected to minimize the in- c vergence paramete continuoue th r rfo s systemd A d an , stantaneous mean-square value of the measured error is the amplitude of the harmonic signal. Note that the signal efc. The optimal weight selection is then given e controller forth show4 f m. o Eq s than i s t \JL deter- by the update equation mine overale th s l compensator gain. Also note that these compensators have infinite gain (theoretically) hfc+1 = hfc - (2) at the disturbance frequency U>Q, independent of the It is well known that the convergence parameter fj, value of fj, > 0. This indicates that fi should not influ- significantly impacts the performance of this adaptive enc deptnotce e eth th f ho filter introduce canceo dt l system, as it determines the bandwidth of the adap- the harmonic disturbance. However, since n controls tation process. Note tha resultine th t g compensator the loop gain, it will influence the width (bandwidth) lineaa s i ) r2 functiod (Eqs erroe an th 1 f .r n o signal , of the notch, which is determined by the frequency but a nonlinear function of the disturbance x, which range about UQ for which the loop gain is greater than complicate stabilite sth performancd yan e robustness one. Thus the feedback interpretation of the role of fj, analysis [10, 11]. is consistent wit feedforware hth d one. A variation on this problem, called filtered-x The FXLMS formulation has proven to be very use- (FXLMS), use estimatn sa secondare th f eo y path5 acoustie th r fufo l c control problem sinc oftes i t ei n dif- to generate a filtered version of the reference signal ficul accuratelo t t y mode systee th l m gaiphased nan , x = Sx that is used to drive the filter update equa- frequence eveon t na y point. However, several issues tions [12]. Wit assumptioe hth n tha filtee tth r update arise when using these approaches to adaptively con- proces relativels si y slow e FXLMth , S algoriths mha trol isolation systems: the same control output given in Eq. 1, but the update equation for the filter weights is of the form f additionaI . 1 l modeling informatio s availablei n , how could thi includee sb formulatione th n di ? x= x =-. (3) fe k algorithe th n 2 Ca reformulate.me b avoio dt d some Note that these algorithms were originally developed of the approximations in the FXLMS approach? as feedforward systems, with the architecture shown adaptive th n 3Ca e. controller desig a case nb i h t in Fig. l(a). The interpretation in this case is that the more formal settin enablo gt e additional exten- measured referenc forwardd e fe signa s i ,x l through sions? adaptive th secondare th ed filtean Eqsn y3 i r , pat1 . h S to generate a signal that cancels the output of the followine Th g outlines recent development fiele th d i sh primary pat . HoweverhP , wit erroe hth r signa usele d that address these point providiny sb overvien ga f wo updato t e (FX)LMeth S weights, therimplicin a s ei t the Augmented Error algorithm and a comment on the feedback loop in the overall system, the loop gain of T^oo formulation of LMS. which is determined by p, in Eq. 3. This feedback loop To compar e augmenteth e d error algorithm with is made explicit in an equivalent formulation shown FXLMS and LMS, we first rewrite these algorithms Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

x X p y -f e /

« 0 / O

) Feedforwar(a d Interpretatio ) Feedbac(b n [10]k Interpretation [10] (c) Augmented Error [11]

Figur : Terminologe1 varioue th r yfo s interpretation adaptive th f so e controllers. consistena n i t form. Followin derivatioe gth - Ba f no wher measureds i ee defines i y , Wd T— X.above y d an , yar dproblee [11]th ,cance o t ms i disturbancla e signal las e cancel9 t. Th tercontributioe Eq smth n i e th f no y(t) tha assumes i t relatee measuree b th o d t o dt d sig- error e(t), leavin augmentee gforth a r mfo d error that nal x(t) through an optimal, fixed set of N weights w°, is similar to the FXLMS error in Eq. 7. However, i.e. w°y(t)— Tx(t), wher e regressiona e TZx(t)s i N Eqshow0 1 . s tha e augmenteth t dcome b erro -n ca r vector taken from the measured data. The objective puted without knowledg optimae th f eo l weightsd an , is to tune the weights of the approximate filter w(t) e determineb thun ca s real-timen di . Furthermore, to cancel the disturbance using the output from the with this augmented error stabla , e tuning algorithm T adaptive controller GAE given by y(t) — w(t} x(t). is (see Refs. [16, 11]) Define <$>(€) = w° — w(t), and write the error as T w = y = W X (11) Bayard [11] presents a very detailed analysis of this wher implie] e[• s tha filterine th t g operatio applies ni d algorith varieta r m fo potentiaf yo l disturbance envi- to each element of the vector. The block diagram for ronments stationara r Fo . y harmonic disturbanct ea this formulation is shown in Fig. l(c). Note that y(t) WQ, the equivalent LTI controller is enter inpun a s sa t disturbanc systee th o emt dynamics (which actually corresponds to the secondary path). GF S(S) Q fs\ _ By redefinin possiblweights e i gth t i ° rewrito esw t e AEV ' S(s)G- FXLMS this proble more th mn ei standard form wit outpun ha t disturbance. where as before, S(s) is a model of the secondary path. As discussed, the standard assumption with the choseexpressios e Th wa controllee 2 n1 th r . nfo Eq n ri FXLMS algorithm is that the weights change slowly emphasizo t close eS th e linkLM e sth betweed an E nA so that the order of the filtering and adaptation can and FXLMS algorithms, and in particular, the very interchangee b d yielding similar properties that they share. For example, AE compensators have the same robustness properties at S(s)e= (7) e FXLMth Ws a Q S design n tolerats ca (i.e. e phase where x = S(s)[x]. The associated stabilizing weight error largs sa ±90°s ea ) [11]. update algorith mthes i sam e nth befor s ea e As discussed, the compensator poles of the FXLMS controller disturbancS e th t LM a d e an sar e frequency, effectiveld an y resul infinitn i t e gai t thina s frequency. w = /j,cex ; y = w x. (8) show4 . Eq sinclude S thaLM t furtheo n s r informa- Monopoli [15] introduce dtechniqua e which includes tion abou secondare th t y path dynamics. However, extr resula s termerrore a tth d doe n neest i an , sno d the FXLMS algorithm also includes information about to invoke this assumption. The technique, called Aug- phassystee e th th f eo m (secondary free path-th t a ) mented Error, is well known in the design of adap- quency of the applied disturbance. The phase infor- tive feedback controllers [16], but, until recentlys ha , mation determine locatione sth reaa f sle o zerth n oi t beeno n applie adaptive th do t e feedforward case [11]. compensator GFXLMS, which will have a large impact namAe sth e suggests, this approach hinge addinn so g e compensatoonth lood an rp phase. n Thera e ear extra d e erroterman th S ro t s e(t)LM e useth n di infinite number of secondary paths that could have FXLMS algorithms to develop the augmented error same th e magnitud givee phasd th ean nt ea distur-

T T bance frequency, and we would use the same FXLMS ea = e + S(s)[x] - S(s)[ x] = . (9) controller for each of them. Clearly the stability of S(s)(y]-y+ e = (10) e compensatoth thesn o r e system givea r fo ns value Copyright© 1997, American Institute f Aeronautico Astronauticsd san , Inc.

of n would be determined by the loop gain and phase equation r equivalentlyo , ratia , polynomialf oo . z n si at other frequencies as well, but these are not taken generae Th l expressio z-transfore p th ta r nN fo n a f mo into account in the design. The analysis of the AE filteR FI s i r algorithm is more complex than LMS and FXLMS, as l N illustrated by the fact that the location of the poles H(z} = - + ... + bNz- , (13) of AE compensator are not obvious from Eq. 12. In particular algorithE A e th , m include entire sth e plant and considering H(z) as the transfer function to the input-output relationship of two variables, modee controlleth n i l r term SGPXLMS- This parf o t the control is used to cancel 5GFXLMS terms that ap- Y(z) characteristie peath n i r c equatio e systeth r mn fo i H(z= } (14) X(z) Figure l(c), thereby phase stabilizin systee gth d man guaranteeing stable closed-loop performance indepen- n rewritwfilteR ca e t FI eaca e outpu r n eth ha f o t e valudenth f p,f eo o t [11] f courseO . e impacth , f o t samples a , n , modeling errors hi S is a key issue, and for systems

with resonant modes, this will require study beyond y (n b)= 0x (n)+bix . (15 ) - ! •-+b)• )N n + ( - Nn x( the multiplicative error model used in Ref. [11]. e developmenTh a forma f o t l framewore th r fo k In contrast, the general expression for an IIR filter is adaptive controller enablo st e further extensions will be addressed hi more detail in subsequent work. It is interestin notgo t e however that Hassib hav. al - t ere e i cently shown that the LMS algorithm is HOO optimal, whil filtee eth r outpu t eaca t h samples i , n , sense inth e tha t minimizei t e maximuth s m energy gain fro disturbancee mth predictee th o st d errors [17]. y (n) = box (n) + b\x (n — 1) + ...bffx (n — N) This resul importans i t t becaus t showei clossa e con- -aiy (n - 1) - ... - amy (n-M). nection between adaptive control and modern robust (17) estimation/control. The approach provides the oppor- primare Th y difference betweefilteo tw r e typenth s tunity for many further developments such as adaptive ifilteR s II tha rn a tinherentl y contains feedbace kdu approaches based on optimal Hi/T-Loo estimators and delae th yo t termdenominatoe th n si transfee th f ro r formulations of FXLMS-like estimators that do not re- function. Comparing equation 15 and equation 17, quire any assumptions about the adaptation rate. it is clear that an FIR filter is simply an IIR filter feedbace th f wito l khal term t equasse zeroo lt . This Limitations of FIR Filters for Isolation lack of feedback places contraints on the ability of the FIR filter to match the magnitude and phase of an This section review importann sa t limitatio LMSf no - arbitrary transfer function. type algorithms implemented witnitersR e hFI Th . In general, FIR filters are simple to implement since limitation involves light damping in the physical sys- they require only knowledg pase presend th an tf eo t undem te r control certair Fo . n physical isolation archi- inputs. Stabilit weighte th f yo achieves si d when there tecture possibls i t si lightlr efo y damped mode cono st - is no feedback between the filter output and the filter tributprimare th o secondaryr et y(o ) path response. input. In contrast, the lack of input-output feedback In many cases, the attractive aspects of FIR filters, in the filter structure makes it difficult to model the carefud an l selectio actuatorf no sensorsd san mitn ,ca - pole transfea f so r function generaln I . larg,a e number igat effecte eth s discussed here. of tap requires si modeo dt dynamia l c system with The general issue is illustrated well by considering lightly-damped poles. single-channea algorithmS LM l . Complete cancella- Usin weighfiltere R gth II S n si t updateLM e th r sfo tion occurs if the weights converge to the primary path algorithm introduces advantages and disadvantages. time inverse secondare sth th f eo y path, P(z)S(z)~l, Poles and zeros can be matched exactly with an IIR fil- whic singla s hi e transfer functio i thinh s casee Th . r becaus te input-outpue th f eo t feedback low-ordea ; r performanc e adaptivth f eo e feedforward algorithms i IIR filter can achieve the same performance as an FIR strongly relate abilite weighte th th f o dyt o mode o st l filter that contain largsa e numbe tapsf prie o r -Th . the transfer function P(z)S(z)~l, and this ability is af- mary disadvantage of an IIR implementation is that fected by the filter architecture chosen for the weights, t guarantee filtee no th s ri stablee b do t , which requires specifically whether finite impulse response (FIR) fil- that the poles of filter be monitored during the on-line ters or infinite impulse response (IIR) filters are used. adaptation. Several pole monitoring techniques exist, Each filter can be expressed as a finite difference but they require intensive computations and therefore Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Table 2: Comparison of FIR and IIR niters for adaptive feedfoward design Advantages Disadvantages FIR • simple implementation • require large numbe tapf o r s • stability guaranteed modeo t l lightly-damped poles UK • poles and zeros can be • stabilit guaranteet yno d matched exactly • adaptation may converge • can achieve performance to a local minimum desigR equaFI o nt l with lower • convergence of adaptation order filters algorithm could be slow compared to FIR design

increas convergence eth ealgorithme timth f eo . Fur- feedbace thermore th filtere o th algoe t th n e ,ki -du , rith longeo n ms i r guarantee convergo dt globae th o et l minima durin adaptatione gth . A stud conductes ywa understano dt e converdth - gence propertie filteR FI rf so design plantr sfo s with dynamics simila thoso t r f vibratioeo n isolation sys- frequency (Hz) tems. Of specific interest are the ability of FIR filters modeo t resonane th l t dynamics associated wite hth Figur filteR : Optimae2 FI r p desigta 0.01£2 = r l3 n fo : suspension modes of the isolation system, and the ef- frequency respons P(z)S(z)~f eo l (dashed), frequency fec f dampino t e convergenc th filteR n FI go r e th f eo response of the optimal 32-tap filter (solid). weights. For the investigation, the magnitude and phase response secondare th f so primard yan y paths wer- eas and phase of P(z)S(z)~l below 30 Hz and above 95 sumelineae b o dt r combination f second-ordeso r fil- Hz, resulting in a poor representation of the dynamics ters with natural frequencies at 50 Hz and 80 Hz. The at those frequencies. In contrast to the previous result, damping ratioe filteth f rso were eacr variedfo h d an , though, the FIR filter accurately models the dynamics damping ratio constrainea , d optimizatio pers nwa - of P(z)S(z)~l within the target band. The increased formed to minimize the error between the magnitude damping ratio smooth magnitudee th t sou phas d an e

l phasd an P(z)S(z)~f eo H(z),d an wheree H(z)th s i response of P(z)S(z)~, 1 making it easier for the FIR transfe filterR e constrainer FI functioTh .e th f o nd filte represeno t r dynamice tth . Hz s5 9 betwee d an 0 n3 optimizatio filteR performes FI nwa r n wita r , h4 dfo Another indicato e beneficiath f o r l effect dampf o s - 2 tap 3 r dampin , fo s • • • , g 12 ratio , 8 f £= 0.01o s , in shows gi Figurn ni thin I s. e4 figure normale th , - 0.03, 0.05 0.10d e targean ,Th . t frequency rangr efo ized mean-square error of the single-channel control constrainee o th N . Hz 5 9 d optimizatioo t z H 0 3 s nwa syste mplottes i functioa s da filtef no r lengt sysr hfo - weigh placee s frequencth wa t n do y response outside tems with increasing modal damping normalizee Th . d this band. mean-square error is defined for this case as optimizatioe Th n resul 32-ta a r C 0.0= ,fo d t 1an p FIR filter is shown in Figure 2. The dashed line in

both e magnitudplotth s i s d phasan e e responsf o e

P(z)S(z}~, 1 and the rapid changes in magnitude and phase of P(z)S(z)~ 1 as a function of frequency are due to the light modal damping. As the plot indicates, whic equivalens hi rooe th t o t meat n square response eve n32-taa filteR pFI r doe t accuratelsno y modee lth for an input with a unity power spectra between 30 and magnitud0 6 phasd o t ean 0 f P(z)S(z)~e5 o e th n i 1 zerd an o z 9powe5H r spectra elsewhere meane Th . - Hz region. Outside the target band, the FIR filter is a square erro s parameterizei r e dampinth y db g ratio, very poor representatio dynamice th f no s because eth £, and the filter length, N. As Figure 4 illustrates, in- optimization did not attempt to shape the magnitude creased damping increases the maximum amount of and phase of H(z) below 30 Hz or above 95 Hz. error attenuation achieved by the FIR filter and de- The optimization result for a damping rato of £ = creases the filter length required to obtain an achiev- 32-taa 0.1d 0an p filte shows i r Figurn wits i A h. e3 able mean-square error. For example, a 32-tap filter previoue th s result frequence th , y R responsFI e th f eo only achieves a thirtyfold decrease in the open-loop filter doe t evesno n attemp modeo t t magnitude th l e mean-square error when the damping ratio is 0.01, Copyright© 1997, American Institute Aeronauticf o Astronauticsd san , Inc.

10"

j 10'

frequency Hz)

10 15 20 25 Number of FIR taps Figure 4: Normalized mean-squared error (Equa- a functio s a tio) f filteo n18 n r lengt d dampinhan g frequency (Hz) ° ratio is reduced for greater damping, £. Figure 3: Optimal 32 tap FIR filter design for C = 0.10: frequency response of P(z)S(z)~l (dashed), frequency respons optimae th f eo l 32-tap filter (solid).

whereas the same order filter attenuates the open-loop mean-squar dampina r fo e 0 errofactoa g 25 y f b ro r rati 0.10f oo . Similarly 24-taa , p filte requires i r o dt achieve almos fiftyfolta d reductio mean-square th n i e error for a damping ratio of 0.03, whereas the same error attenuation can be achieved with a 4-tap filter if dampine th g rati increases oi 0.10o dt . Primary result numericae th f so l investigation are: • Accurately modeling sharp change mage th -n i s nitud phasd ean P(z)S(z)~ f eo requireR FI n sa filter wit hlarga e numberl tapsf o . • Increasing the modal damping in the primary and secondary path transfer functions increases the amount of achievable attenuation using FIR fil- weighte adaptatione terth th s sa n si . Similarly, increased damping enables equivalent attenuation with a lower number of filter taps. Combination of feedback with feedforward is therefore expecte e beneficialb o dt , wit e presenchth f eo low-order feedback enabling use of substantially fewer weights in the FIR filters, and largest benefit expected for broadband performance objectives. •250 10 Frequency (Hz) Adaptive Feedforward Implementation Figure 5: Measured P(z)S(z)~1 (solid) and two recon- e adaptivTh e feedforward control approachee b n sca structe disturbancez dH FXLM5 9 d San sfilter9 7 r sfo implemented using both analog and digital methods. (dashed lines) on three-axis testbed Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

80 90 100 110 120 130 140 150 Frequency (Hz)

Figure 6: Measured phase difference between the fourth order quadrature niters used for AE algorithm Figur : 7 Steware t platform microprecision testbed implementation use vibratior dfo n isolation experiments

Collins [18] provides a detailed analysis of the advan- fourth order quadrature filters provides an accurate tage issued san s associated wit analon ha g implemen- 90° phase difference (less than 0.5 degree error over tation. However developmena r fo , t system (an- dpo 1.5 decades in frequency), which is useful for slewing tential space flight experiment), there are significant disturbances (Figure 6). advantages, such as increased flexibility and greater broadband performance, in a digital implementation. The ability of FXLMS to model P(z)S(z)~l was Experimental Results considered in hardware for a tonal disturbance. This propert verifies ywa d experimentall simpla n yo e three- This section describe e laboratorth s y experimental axis isolation testbe r whicdfo h feedforward control system and then summarizes the results of several ex- was implemented in the frequency range of the sus- periments wit adaptivhthe e feedforward algorithms. pension modes. A 16-tap FXLMS filter executed at 3500 Hz was allowed to converge for the same value Six-axis Payload Isolation Testbed o differentw o fr p.fo t disturbances, sinusoidal inputs at 95 and 79 Hz. The filter was then reconstructed Figur showe7 primare sth y test platform which serves and compared in each case to the previously measured as a prototype for a planned space flight system called P(z)S(z)~ . Figur plote5 measuree sth d phase data, SUITE (Satellite Ultraquiet Isolation Technology Ex- l showing the modes of S(z) at 82 and 105 Hz, and the periment). Each member in the six-strut Stewart plat- two different reconstructed filters botn I . h casese th , form uses a combined passive-active isolator arranged filter matches the magnitude and phase of P(z)S(z)~l mechanically in series. The stiffness of the damped appropriate th t a e disturbance frequency. Outside eth passiv evariee stagb n produco det ca desiree eth d sus- respective narrow frequency ranges reconstructee th , d pension frequencie particulaa r fo s r payload masd san filter is dissimilar to the measured P(z)S(z)~l. inertia. The suspension mode frequencies ranged from Numerous authors have investigated implementa- inertid 25-8an ) witz amase 0 H Kg usedh th e 6 s( Th . tion using various length FIR or IIR filters. It is active stage in each laboratory strut uses a piezoelec- well known that onlweighto ytw requiree ar s a r dfo tric actuator (50 fj,m stroke) and a velocity sensor that harmonic disturbance, but more weights are needed . ca motioHz f n o 0 resolv1 m n t n a 0 e5 for broadband disturbance rejection. FIR filters were The isolation syste mcontrolles i d wit dedicateha d used in this work. The FXLMS was implemented here electronic support system including a Texas Instru- using 4 weights in a standard tap delay filter. How- ments TMS320C31 processor which receives down- ever, the AE algorithm was implemented using an al- loaded control code from a PC. The Stewart platform ternate structure employing onlweightso ytw , witha was mounted on a generic satellite model (mass 40 quadrature filter developed usin e networkgth s from Kg). This structure was in turn connected to a rigid Bedrosian [19]. A digital implementation of the two plate through soft rubber mounts. Durin testse gth a , Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

10' 10' 10'

10"

. s: ' 10"

10' 10- 10'

-47.7 OB -40.5 dB FXLMS Performance AE Performance 10' 10" 10' 0 20 0 18 0 16 0 14 0 12 0 10 0 8 0 6 0 4 0 2 50 100 150 50 100 150 Frequency (Hz) Frequency (Hz) Frequency (Hz)

Figur : 8 eClosed-loo p feedforward control perfor- Figure 9: Feedforward control with FXLMS and AE manc strutSteware 2 th n ef o so t platform (TITO)

proache r intermediatsfo e value f fj,so (approximately capacitf 50Ib shakeS yVT r drivin greactioa n mass swa one halvaluee th f s that resulte closed-loon di p insta- mounted verticall corne e satellite on th f n yro o e model FXLMS)e bilitth r yfo . These graphs ploratie th tf oo to generate a disturbance input to the base satellite. closed-looe th open-looo pt p system response th n ei The signal input to the shaker drive amplifier was used presence of background noise plus a strong stationary as the reference quantity in the control. harmonic disturbance at 100 Hz. The cancellation of the disturbance is clearly shown by the sharp notch MIMO Feedforward in the plots. As indicated, both approaches achieve The particular strut design of the hardware configura- greate rperformanc B thad 0 n4 e improvement. How- tion show Figurn ni allowe7 simplsa e extension from ever, much further wor s requireki o demonstratdt e SIS MIMOo t O control becaus f minimaeo l interac- the theoretical advantages of AE over FXLMS that tion betwee controe nth l loop eacn so h leg. Figure8 were discussed previously thin I . s implementatione th , shows typical results with feedforward (FF) contron lo added plant information in the AE controller did not two legs tha attachee tar dt adjacena p to t e portth n so allow increased gain compared to FXLMS. plate. The FF control was independently designed and implemented for each leg. The plot shows excellent re- Feedback Control duction (> 50 dB) in the transmission of a stationary Both feedforwar d feedbacan d k control have been harmonic disturbanc froe z base th H m th o 0 et 10 t ea tested on the isolation mount in Figure 7. Typical top plate through these two legs. These results sug- feedback results shown in Figure 10 which demonstrate ges controF t F tha e th tl should easily extene th o dt broadbana d 20-3 reductioB 0d e disturbancth n ni e full 6 leg MIMO control without requiring the addi- transmission singla (10-10r fo e) strut0Hz coursef O . , tional complication of successive loop closure. the feedforward and feedback can be combined to pro- FXLMS and Augmented Error vide both broadban tunabld dan e narrowband control. This was tested using a stationary harmonic distur- As discussed, various algorithms exist to design adap- , wher Hz feedbace e0 th banc 10 t ea k provides essen- tive feedforward controllers, and both FXLMS and tiall attenuationo yn t doebu , s modif secondare yth y the augmented error methods have been tested on path transfer function to an extent that the FXLMS the isolation mount. The FXLMS approach is well must incorporate a modified plant estimate. These ex- known and widely used within the active noise con- perimental results also demonstrate very good reduc- trol significan fielda thu d s an s,ha t experimental her- tion of the narrowband disturbance (« 40 dB), but othee itageth n approacrE O hand.A e th ,mucs hi h there was some additional amplification in the system less well known therd limitee an , ear d experimenta- lre respons whic, eHz neah0 indicate10 r dpotentiaa - in l sults within this field to support the available analysis. teraction betwee feedforware nth feedbacd an d k con- Figure 9 compares the performance achieved for initial trollers. Further investigatio requires ni identifo dt y SISO experiments using both the FXLMS and AE ap- the cause of this phenomenon. Copyright© 1997, American Institute f Aeronautico Astronauticsd san , Inc.

10'

10' 10* 65 70 75 80 85 90 95 100 105 110 115 Frequency (Hz) Frequency (Hz) Figure 10: Single strut response with feedback control Figure 11: Response to slews in the harmonic distur- Slewed Harmonic Disturbances bance fromz H 80— 0 >10 Figur compare1 e1 opee closed-lood sth nan p response of the system to a harmonic disturbance that is slewed Summary/ Conclusions at various rates from 80 to 100 Hz. This test is repre- sentative of the disturbance types that are expected on This pape s motivatewa r plannea y db d space flight spacecrafa resula imbalancn a s a f t o t reactioa n ei n experiment which will benefit from adaptive control. wheel that responsspinn i p su commandea eo t d input. Adaptive control will be implemented on-orbit to com- The curves show the square root of the power spectra plement a planned set of feedback algorithms. The computed fro second0 m2 measuref so d time data, with paper considers singl multipld ean e tone SISr sfo r Oo open-looe th p respons comparisonr efo . Cas showeA s two-input two-output control. MIM broadband Oan d the FXLMS response to a slow slew (80->100 Hz in 10 control problems wil consideree b l i futurdh e work. sec) with a relatively small value of n, and illustrates The need for adaptive isolation has been motivated tha controllee th t r track change frequence sth th n ei y adaptivy ke d an e feedforward algorithms have been inpue inth t signal, thereby providing some attenua- reviewed e clas f On limitationso . filterR FI f so rep - tion from the OL response. However, the tracking is resentations was discussed as a means of motivat- not fast enough to achieve the same levels of attenua- desigg in f systemno s fre f lightlo e y damped modes. tion achieved foinitialle rth y stationary disturbanct ea Basic FXLMS. contro extremels wa l y effectiv- re t a e comparisonn I . 80Hz , cas showeB s tha FXLMe tth S jecting single fixed tone and rapidly slewing inputs. response with a larger fj, (increased by a factor of 10) Augmentee Th d Error (AE) algorith s potentiamha l can adapt to the change in the signal frequency fast expano t e utilitdth f FXLMyo S controller syso t s - enough tha t providei t s much better attenuations A . tems with non-trivial dynamics presene th n I . t study, shown, the attenuation is particularly good in this case AE was implemented and reasonable performance was firse fo th rslew te halth f investigat.fD o Cased an sC e achieved. Combining feedback and feedforward isola- the system performance for even faster slews of 5 sec tion, another extensio standaro nt d FXLMS, demon- and 1 sec, respectively. Since the slews are faster, the strateperformancB d 0 d4 e improvemen casee on .n i t power at a given frequency is lower, which should lower There remain open issuee controth n si l interaction sensoe th r response. However givea r fo n, valun f eo investigatee whicb y hma d furthe consideriny b r g feed- we would expec trackine tth g performanc degradeo et , forward controllers from the feedback perspective. which should increase the system response. In case vere th y , higD h slew rate cause secone sth d effeco t t Acknowledgements dominate. Note that when the value of (j, was increased by another facto fourf o rclosed-looe th , p systes mwa The authors thank Donald Leo for his contributions destabilized. These results, which sho excellene wth t to the analysis and discussion of FIR filters, and Mark dynamic performanc FXLMe th f eo S controllers, com- Holcomb, Scott Pendleton, and Michael Evert for de- plement the stationary results shown previously. velopin testbede gth .

References D-13556 t PropulsioJe , n Laboratory, Pasadena, CA. [I] Glase , GarbObad rR. . STRV-1an , lM a J. B Cry- ocooler Vibration Suppression (1995 AIAAn I ) [12] Morgan, D. (1980) An Analysis of Multiple Structures, Structural Dynamics, Materialsand Correlation Cancellation Loop with a filter in Conference, April 1995 AIAA 95-1122. e Auxiliarth y Path. IEEE Trans. Acoustics,n o Speech, and Signal Proc. ASSP-28(4), 454-467. [2] Edber , BouchegD. , SchencR. r , NurrkD. , G. e d Alhoran , . ResultnD . Y Whortom sKi , nM. [13] Scribne . ActivrK e narrow band Vibration Iso- STABLe oth f E Microgravity Vibration Isolation latio f Machinerno y Noise From Resonant Sub- Flight Experiment (1996 19thn I ) American- As structures (1990) Master's thesis, Departmenf to tronautical Soc. Guidance and Control Confer- Aeronautic Astronauticsd san , MIT, Cambridge, ence, February 96-071199S 6AA . MA. [3] Bushnel , Jaco G. , lAnderso D. tBecraf d an , nT. t [14] Sievers L. and von Flotow A. (1992) Comparison M. Active Rack Isolation System Developmenr tfo Extensiond an f Controso l Method Narrowr sfo - the International Space Station (1997) In AIAA Band Disturbance Rejection. IEEE Trans. on Sig- Structures, Structural Dynamics, Materialsand nal Proc. 40, 2377-2391. Conf., April 1997 AIAA 97-1203. [15] Monopoli } (1974R. , ) Model Reference Adaptive [4] Sullivan J., Cobb R., Rahman Z., and Spanos J. Control wit Augmenten ha d Error. IEEE Trans. Closed Loop Performanc Vibratioa f o e n Isola- on Auto. Control AC-19(5), 474-484. tion and Suppression System (1997) In American Control Conference, June 1997 I-97115D. [16] Narendr d Annaswaman . aK . y(1989A ) Sta- ble Adaptive Systems Prentice Hall, Englewood [5] Sommerfeld . (1989S t ) Adaptive Vibration Con- Cliffs, NJ. trol Vibrationof Isolation Mounts Using LMS-an based Control Algorithm PhD thesis, The Penn- [17] Hassibi B. , Sayed A. , and Kailath T. (1995) sylvania State University. WooOptimality of the LMS Algorithm. IEEE Trans. on Acoustics, Speech, and Signal Proc. [6] Spanos J. and Rahman Z. Narrow-band Control AC-44(2), 267-280. Experiment Activn si e Vibration Isolation (1994) In SPIE Proc., Vibration Monitoring and Con- [18] Collin . S s(1994 ) Multi-Axis Analog Adaptive trol, July 1994 SPIE 2264-01. Feedforward Cancellation Cryocoolerof Vibration PhD thesis, Dept.of Aero.and Astro. Mass. Inst. [7] HayneGend an . g Z . (1994sL Degrex )Si Freef eo - of Tech., Cambridge, MA MIT SERC Report #8- Activm do e Vibration Control Usin Steware gth t 94. Platform. IEEE Transactions Controlon Systems Technology, pages 45-53. [19] Bedrosian . S ,(1960 ) Normalized Desig° 90 f no Phase-Difference Networks. IRE Trans. on Cir- [8] d Multiple-InputHylanHarriA an . . L sD d , cuit Theory, pages 128-136. Multiple-Output Neural Architecture for Sup- pressio a Disturbanc f o n f Multipleo e Tones (1996 19thn )I American Astronautical Soc. Guid- ance and Control Conference, February 1996 AAS 96-067. [9] Widrow B. and Stearns S. (1985) Adaptive Signal Processing Prentice-Hall, Englewood Cliffs, New Jersey. [10] MacMartin D. A Feedback Perspective of the LMS Feedforward Algorithm (1994) In Ameri- can Control Conference, pages 1632-1636, 1994. [II] Bayar . (1996dD GeneraA ) l Theor Lineaf yo r Tune Invariant Adaptive Feedforward Systems with Harmonic Regressor InternaL JP s l Report