APPLICATION NOTE Esimaing Transfer Funcions wih SigLab Accurae ransfer funcion esimaion of linear, noise-free, dynamic sysems is an easy ask for DSPT SigLab. Ofen, however, he sysem being analyzed is noisy or no perfecly linear. All real-world sysems suffer from hese deficiencies o some degree, bu conrol sysems are usually he wors offenders. Obaining an accurae ransfer funcion esimaion from a noisy and non-linear sysem requires an undersanding of measuremen radeoffs. Overview Imperfecions in dynamic sysems SigLab is rouinely used o esimae ransfer funcions associaed wih dynamic sysems including conrol sysems. The firs half of his applicaion noe addresses he ask of making accurae ransfer funcion esimaes on dynamic sysems which are boh noisy and non-linear. The second half covers measuremen echniques focused specifically on conrol sysems. Elecro-mechanical conrol sysems ypically are noisier and less linear han he ypical elecrical or purely mechanical sysems. The non-linear behavior is ofen due o he elecro-mechanical componens involved in he sysems. The measuremen noise is ofen a resul of he sysems being characerized under acual operaing condiions. The head posiioning servo of a disk drive is a good example of such a conrol sysem. The dynamics of his sysem are usually measured under closed-loop condiions wih he disk media spinning. The mechanical imperfecions of he servo rack and plaer injec boh periodic and random signals o he conrol sysem. In he disk drive, he non-linear behavior is primarily due o he head posiion error signal. In order o characerize he servo dynamics, SigLab injecs a es signal ino he servo sysem. The head posiion error signal has a limied linear region. As he head is driven furher off is arge rack by his es signal, he posiion error signal response o he es signal becomes increasingly non-linear. The end resul of his combined noise and non-lineariy is a measuremen challenge. Balancing noise and non-lineariy To make a ransfer funcion measuremen on a dynamic sysem, an exciaion is supplied o he sysem and he sysem's response o his exciaion is measured. If he sysem is noisy bu linear, he exciaion level can be increased o improve he signal o noise raio by simply overpowering he noise. If he sysem non-linear a high exciaion ampliudes, bu free of noise, he exciaion can be lowered o a poin where he lineariy is accepable. When he sysem is boh non-linear and noisy, a radeoff mus be made balancing he poor signal-o-noise raio a low exciaion ampliudes wih he non-linear sysem behavior a high exciaion ampliudes. Two primary ools for ransfer funcion esimaion SigLab comes wih wo sofware applicaions for ransfer funcion esimaion: swep-sine and a broad-band FFT based nework analyzer. The swep-sine (or more accuraely seppedsine) esimaion echnique is he leas Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 1
affeced by noise and non-linear sysem behavior. The main radeoff wih he swepsine echnique is measuremen ime. The ransfer funcion is esimaed over a userdefined frequency range a single frequency poin a a ime. In measuremen siuaions where here is reasonable lineariy and good signal o noise raio, SigLab's broad-band FFT based nework analyzer provides he ransfer funcion esimae in a fracion of he ime ha is normally needed by he swep-sine approach. The exciaion is usually a bandlimied random or periodic chirp signal, bu, he user is no limied o hese exciaions. A signal selecion from he funcion generaor applicaion or an exernal source, may be used. The following five facors have he mos impac on he amoun of ime required for a ransfer funcion esimae: 1) sysem's noise 2) sysem's lineariy 3) number of frequencies poins in he esimae 5) required frequency resoluion Measuremen Configuraions The opimal measuremen configuraion The goal is o deermine he sysem ransfer funcion H( ω ) from measuremens on he sysem. The opimal measuremen configuraion for he Device Under Tes (DUT) and SigLab is shown in Figure 1. A his poin, no assumpions are being made abou he ype of sysem being measured. I could be purely elecrical, elecromechanical, mechanical, acousical, and so on. The measuremen configuraion in Figure 1 assumes ha he exciaion x() can be measured wih minimal error. If SigLab's oupu source is direcly conneced o boh he DUT inpu and a SigLab inpu, his is a perfecly valid assumpion. However, Figure 1 assumes ha he exciaion o he sysem is a volage, bu his SigLab OK Model 20-22 Power On xm() ym() x() h () H( ω) y() + Device Under Tes (sysem being analyzed) ny() Unwaned Noise Figure 1 - Opimal SigLab Measuremen Configuraion 4) required accuracy is ofen no he case in mechanical or 2 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
elecro-mechanical work. A ransducer mus be used o conver he sysem's exciaion ino a volage which can hen be measured wih SigLab. For insance, a force-o-volage ransducer can be used o measure he inpu exciaion force o a mechanical sysem. In hese cases, precauions mus be aken o insure ha minimal elecrical noise corrups he signal from he ransducer and ha he ransducer is operaed in is linear region. When hese precauions are observed, he assumpion ha he sysem's exciaion can be measured wih minimal error is sill valid. Therefore, as shown in Figure 1, he enire measuremen uncerainy is accouned for in he DUT's response signal ym() by he addiional (unwaned) noise erm ny(). When he exciaion o he sysem is zero, he response, y (), of he sysem is zero, and herefore ym() = ny(). There are no oher assumpions abou he characer of his noise. I is imporan o recognize ha he measuremen noise erm ny() is usually no provided by an acual exernal noise source. I simply represens he sysem's oupu wih no inpu. A firs i migh seem ha he assumpion of having a noiseless exciaion channel measuremen is unreasonable. In pracice, however, his requiremen is usually aainable. If an accurae measuremen of he exciaion o he sysem can be made, i is relaively easy o obain an unbiased esimae of he sysem's ransfer funcion. This is he primary reason for configuring he DUT and SigLab as shown in Figure 1. The ransfer funcion esimaion, H ( ω ), is compued from cross and auo power specra esimaes 1 as shown in (1): ) ) Pxy( ω) H( ω) = ) (1) P ( ω) xx where P ) xy ( ω ) is he cross power specrum beween he exciaion xm() and response ym() and P ) xx ( ω ) is he auo power specrum of he exciaion signal. These specral esimaes ( P ) xx ( ω ), P ) xy ( ω )) are compued inernally o SigLab using he FFT, windowing, and frequency-domain averaging. When more averaging is specified, more daa is acquired and processed o refine hese esimaes. SigLab's hardware and sofware is opimized o make hese calculaions in real-ime. As he amoun averaging used in he compuaions is increased, he esimae H ( ω) will converge o he acual ransfer funcion H( ω ). This is a key propery of an unbiased esimaor. The amoun of averaging required o aain a given accuracy for he ransfer funcion esimae is a funcion of he noise ny(): less noise, less averaging. The coherence is an auxiliary compuaion ofen made in conjuncion wih he ransfer funcion esimae. The coherence calculaion in (2) provides an indicaion of he porion of he sysem s oupu power ha is due o he inpu exciaion. P xy ( ω ) C ( ω ) = P ( ω ) P ( ω ) xx yy 2 (2) The coherence, C( ) ω ), has a range of 0.0 o 1.0, where 1.0 indicaes ha all of he measured oupu power is due o he inpu exciaion. This, of course, is he mos desirable siuaion and will only be rue a frequencies where he energy of he noise ny()is negligible. The coherence may be viewed as an indicaor of measuremen qualiy. When a significan porion of he Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 3
measured oupu is no relaed o he exciaion (e.g. he noise erm ny() is large), a low coherence will resul. For a given amoun of averaging, he variance of he ransfer funcion, a frequencies where he coherence is low, will be higher han he variance where he coherence is closer o 1.0. However, since he ransfer funcion esimae is unbiased, he esimae will evenually converge o he sysem's acual ransfer funcion given sufficien averaging. This is rue even if he coherence is low. To minimize measuremen ime, he above ransfer funcion and coherence esimaion calculaions are implemened inernally in SigLab. The broad-band FFT echnique The FFT based nework analyzer compues he ransfer funcion and coherence simulaneously over a band of frequencies using he mehod oulined in (1) and (2). The frequency range usually spans from dc o a user-defined upper limi. Analysis of a band of frequencies cenered abou a specified cener frequency is also suppored. In order o carry ou he ransfer funcion measuremen, he exciaion o he sysem mus conain frequency componens covering he seleced frequency range (ergo, no a sine wave). Selecing eiher he chirp or random exciaion from he conrol panel of he nework analyzer applicaion is he simples way o mee his objecive. However, if cusomized exciaions are desired, an exernal source or he funcion generaor applicaion can be used. racking band-pass filers. The cener frequency of hese filers is se o mach he frequency of he sine exciaion. These bandpass filers can drasically reduce measuremen noise. A he expense of measuremen speed, a lower filer bandwidh may be seleced, providing greaer noise immuniy if needed. The ransfer funcion is sill obained by aking he raio of he cross and auo specra, bu, now i is compued using he band-pass filered ime hisories. When he ransfer funcion esimae has been compued, he oupu source frequency is advanced o he nex frequency desired for he ransfer funcion esimaion, and he measuremen is repeaed. The swep-sine uses: single frequency exciaion racking digial band-pass filers unbiased cross-auo ransfer funcion esimaor hereby providing an accurae ransfer funcion esimae under he mos demanding measuremen condiions. Some Measuremen Examples The linear/noise-free measuremen Figure 2 shows a ransfer funcion esimae, made by he broad-band FFT based nework analyzer, on a linear, noise free, dynamic sysem. SigLab is conneced o he DUT as shown in Figure 1. The swep-sine echnique Swep-sine analysis differs from he broadband echnique in ha a single frequency sine signal is used as he exciaion o he sysem. SigLab's inpu daa acquisiion subsysem can be configured as digial 4 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
Figure 2 - Transfer Funcion of a linear, noise-free sysem. The ransfer funcion is measured simulaneously a 401 discree frequency poins over he dc-10 khz range herefore providing a 25 Hz frequency resoluion. The magniude of he ransfer funcion, in db, is ploed on he y-axis wih a logarihmic frequency x-axis. The oal measuremen ime was well under 1 second. The coherence is ploed on he axis above he ransfer funcion esimae. Only en averages were used o make he measuremen since he sysem is virually noise-free. In fac, lile or no averaging was acually required o obain an excellen ransfer funcion esimae, however, he coherence calculaion is no meaningful unless here is some amoun of averaging. Figure 3 - Noise ime hisory ny() and is power specrum. Fify frequency-domain averages were used o esimae he noise specrum. I can readily be seen ha he resuling specrum in he lower plo is no whie, i.e. no fla. The noise is a combinaion of boh random and periodic componens. This noise specrum is similar o wha migh be found in disk drive head posiioning servo sysem. Figure 4 shows he effec of his noise on he ransfer funcion esimae. I is clear ha he magniude curve is no longer smooh, especially a he lower frequencies, in spie of doing 50 measuremen averages. This is five imes he amoun of averaging done in he previous measuremen example. Measuremens on a noisy, bu linear, sysem The nex example demonsraes a ransfer funcion measuremen made on he same sysem under more realisic condiions: noise is presen. To characerize he measuremen noise, he sysem's exciaion was se o zero. Figure 3 shows a snapsho of a ime hisory (upper plo), and specrum (lower plo) of he measured sysem response ym(). Since he sysem's inpu is zero, his is he noise ny()in Figure 1. Figure 4 - Transfer funcion esimae of a noisy sysem. The coherence is no longer uniy across he measuremen band. The coherence is closes o uniy where he noise power is minimal and he sysem response o he exciaion is Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 5
high. The sharp dips in he coherence occur where here is significan power from he periodic noise componens. For a linear sysem, here are wo ways o improve he measuremen: increase he sysem exciaion level or increase he amoun of averaging. Figure 5 is he ransfer funcion esimae under idenical operaing condiions bu wih 1000 averages. The acquisiion, processing, and averaging ook abou 40 seconds o complee. There is a clear improvemen in he ransfer funcion esimae over he esimae in Figure 4. Noice however ha aside from being a smooher curve, he coherence has no changed significanly. The noise has no been lowered nor he exciaion increased, herefore he coherence has no improved. The imporan poin is, ha in spie of he low coherence, he ransfer funcion measuremen has converged quie nicely o he esimae made in Figure 2. funcion esimae. The affecs of nonlineariy will now be considered. Firs, i is beneficial o undersand and quanify he non-linear behavior of he DUT. An assessmen of lineariy is easy o do if he sysem is noise-free. Injecing sine waves, possibly a muliple frequencies, and using specrum analysis o measure harmonic or inermodulaion erms is a common approach. Wih he addiion of sysem noise, non-linear behavior becomes more difficul o quanify. For example, Figure 6 shows a ime hisory snapsho of he sysem noise (no averaging) along wih he power specrum of he noise compued from 100 frequency domain averages. This daa is similar o ha shown in Figure 3, bu he analysis bandwidh is 20 khz and a linear x-axis is used for he specrum plo. Figure 5 - Transfer funcion esimae wih 1000 averages. Due o he consrucion of his paricular sysem, increasing he exciaion level is no an opion. The sysem becomes non-linear wih larger inpu signals and his behavior will be now be discussed. A non-linear measuremen example. The previous examples provides an idea of how noise in a sysem affecs he ransfer Figure 6 - Response ime hisory and noise specrum wih no exciaion. If a periodic funcion (such as a sine wave) is used as he sysem's exciaion in he aemp o measure lineariy, i is difficul o ell which harmonics are due o he exciaion given he many (and possibly large) periodic componens in he sysem noise. One soluion o his problem is o use a riggered mode of daa acquisiion and average he ime hisories. For his procedure o work, he rigger source mus be synchronized o he fundamenal period of 6 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
he exciaion. SigLab has he abiliy o generae a variey of periodic exciaions. An inernal digial signal is produced wih a period idenical o he chosen exciaion period. This signal provides virually perfec rigger synchronizaion. The ime averaging hen reduces, o an arbirarily small level, he porion of he measured response signal due o he sysem noise ny() if (and only if) his noise is no correlaed wih he exciaion provided by SigLab. I is herefore imporan o choose an exciaion period ha is unrelaed o any of he periodic componens in he noise. specrum plo. The square wave has a perfec 50% duy cycle. Therefore, only he odd harmonics of he fundamenal should be presen in he specrum plo. Under he curren operaing condiions, his proves o be he case. Figure 7 - Time averaging reduces he sysem noise and enhances he sysem response o he squarewave exciaion specified by he funcion generaor applicaion o he Figure 7 shows he effec of ime averaging he sysem's response. The exciaion is a square wave of 0.3 vols peak ampliude (0.6 vols peak-peak). The ime hisory on he upper display is he response of he sysem o he 1422.22 Hz square wave. Noice ha he noise has been reduced o below -60 db Vrms, excep for he componen a approximaely 500 Hz. Also noice ha he DUT is acing like a low pass filer and only he firs hree componens of he square wave exciaion are visible in he Figure 8 - Increasing he ampliude of he squarewave gives rise o even harmonics of he fundamenal frequency, a sign of midly nonlinear behavior. The resuls shown in Figure 8 are due o increasing he ampliude of he square wave o 0.6 vols peak (1.2 vols peak o peak). The ime hisory has increased in ampliude, and appears o have he same general shape as ha of Figure 7. However, he specrum now clearly shows even harmonics of he fundamenal frequency. This is an indicaion of he sysem becoming non-linear. The onse of non linear behavior is usually a gradual process. The increases in he even harmonic conen could be acually be seen a levels on he order of 0.4 vols peak, bu hese harmonics are hard o inerpre and he qualiy of he ransfer funcion esimae is no affeced significanly by mildly nonlinear behavior. To provide a beer idea of he advanage of he synchronous ime averaging, Figure 9 shows he same analysis bu wih frequency domain averaging. The square wave fundamenal and he firs wo odd harmonics can be easily idenified, bu he random and periodic sysem noise obscures he even harmonic informaion. Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 7
Because hese errors are due o non-linear DUT behavior (no noise), more averaging will no lead o a beer measuremen. Figure 9 - Frequency domain averaging does no reduce he sysem noise which obscures he non-linear response of he sysem. How does non linear sysem behavior affec he ransfer funcion esimae? The concep of he ransfer funcion, as well as is esimaion echniques are boh based on he assumpion ha he sysem being analyzed is linear and ime-invarian. When his assumpion is violaed, i should no be a surprise ha errors in he esimaion can, and do, arise. Ripples are presen on low frequency porion of he Figure 10 - Transfer funcion esimae when sysem is driven ino non-linear region. As shown in Figure 8, he sysem exhibied mildly non-linear behavior when he peak exciaion ampliude reached 0.6 vols. The resuls of measuring he ransfer funcion wih an increased exciaion level (0.43 vols rms which is abou 0.6 vols peak for he chirp) are shown in Figure 10. Noice he prominen ripples in he low frequency porion of his measuremen. Also noe ha he coherence has improved over he resuls shown in Figure 5. This is due o he increase in exciaion ampliude. However, he coherence esimae is also based on linear sysem assumpions, so even hough he coherence is higher, he measuremen error is higher han ha of Figure 5. Clear evidence ha he coherence is no a rusworhy indicaor of measuremen qualiy when he sysem is non-linear. Measuremen resuls will change wih he ype of exciaion When he sysem is non-linear, differen ypes of exciaion will ypically produce differen ransfer funcion esimaes. I has jus been shown ha he ransfer funcion esimae can change wih he ampliude of he exciaion (he low frequency errors were higher a increased exciaion levels). Inconsisen ransfer funcion esimaes also provides anoher clue ha he sysem is nonlinear. If he DUT were perfecly linear, all ypes of exciaions and exciaion levels would produce consisen ransfer funcion measuremens. The chirp has hree nice properies when used as an exciaion for FFT based echniques. Firs, i is easy o consruc he chirp so is specral energy lies exacly on he analysis lines of he FFT. This removes he requiremen of using a window wih he FFT, herefore beer frequency resoluion can be obained for a given record lengh. Second, he cres facor (raio of peak o rms. volage) is 2 which is relaively low. This should allow he DUT o be driven a a higher rms level han random noise before clipping occurs. Third, is ime derivaives are coninuous so i is ofen a more genle 8 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
well behaved exciaion for mechanical sysems. level. Noe ha Figure 12 shows peaks over 1.2 vols in ampliude for he 0.43 vol rms oupu. Figure 11 - Time hisory and specrum of he chirp exciaion. Figure 11 shows a ime hisory of he chirp along wih is specrum. The chip repeas every inpu acquisiion frame, which is se a 1024 samples. When he chirp is consruced o repea every N samples, is energy mus lie a discree frequencies. This is a resul of basic Fourier analysis. The period of he chirp repeiion is hen N F where F S is he S sampling frequency. SigLab always samples a a rae equal o 2. 56 Bandwidh, herefore, for an analysis bandwidh of 10 khz, and a 1024 poin inpu frame, he underlying period is 40 ms. The reciprocal of 40 ms is 25 Hz hus exciaion energy is provided a 25, 50, 75, 100,... 9975,10000 Hz. Noe ha he chirp's peak level is slighly less han 0.6 vols. Random noise exciaion is also a popular broad-band simulus. Of course, he random exciaion is acually a very long pseudorandom sequence. When SigLab is se o he 10 khz bandwidh, he sequence repeas every 46 hours. I is herefore, for all pracical purposes, random. Unlike he repeiive chirp, he random sequence has a virually coninuous energy vs. frequency disribuion, herefore, o minimize leakage effecs, a windowed FFT is ypically required. Random also has a higher cres facor han he chirp, so he peak excursions are higher han he chirp for he same rms Figure 12 - A random exciaion produces a higher peak-o-peak volage over ha of he chirp for he same rms level. Since non-linear behavior has been exhibied by his sysem a high exciaion ampliudes, i is reasonable o expec ha he random exciaion will provide poorer measuremens (wih respec o he chirp) due o is higher peak excursions. Figure 13 - A random exciaion produces a beer measuremen han he chirp for his paricular sysem. Acually, he random exciaion provided a decidedly beer measuremen han he chirp! Figure 13 shows a ransfer funcion esimae made wih random noise as he exciaion. The low frequency ripples in Figure 10 are no presen. Measuremens a various nodes wihin he sysem revealed ha he peak response Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 9
levels occurring wih random exciaion were lower in ampliude wih he random when compared o he he chirp exciaion. So in spie of he insananeous peak levels of he random exciaion being abou 2:1 greaer han he chirp, he inernal signals in he sysem sayed a lower peak levels and wihin a linear range. The chirp managed o ge significan energy ino he sysem's high- Q resonances and non-lineariy became an issue. The chirp can be viewed as a sweeping sine one. If he one frequency is a or near a resonance of he sysem for a significan period of ime, he response of his resonance will build in ampliude. The random sequence is naurally highly uncorrelaed and herefore he ampliudes of he response were less han wih he chirp. in he FFT based measuremens. In fac, due o he previously menioned sysem responses o he chirp a resonances, a higher ampliude could lead o a decrease in he measuremen qualiy. The frequency spacing of he measuremens was se o be logarihmic. The oal measuremen ime for 85 differen frequency poins was 6 minues and 20 seconds. The measuremen was made using 3 spans each wih differen racking filer bandwidh, averaging, and logarihmic frequency sep size. Differen exciaion levels for each span could have also been specified, bu were no. Swep-sine: when he going ges ough Up o his poin, good accuracy has been obained wih he broad-band FFT based ransfer funcion measuremens. When he noise and non-lineariy are exreme, swepsine is he ool of choice for hree reasons. Firs, a sinusoidal exciaion feeds he mos energy possible a a given measuremen frequency ino he sysem. Second, he use of racking filers reduces he unwaned affecs of noise on he measuremen. Third, he swep-sine echnique gives he mos flexibiliy o ailor he measuremen o he DUT. The swep-sine applicaion allows he user o decompose he overall analysis range ino from one o five sub-ranges called spans. In each of hese spans he acquisiion, analysis and simulus parameers can be opimized o radeoff measuremen speed, frequency resoluion, and/or accuracy. Unlike he FFT approach, he swep-sine applicaion can make frequency response measuremens a logarihmically spaced frequency poins. Figure 14 shows he swep-sine applicaion and he measuremen resuls. Noe ha he rms level of he exciaion was se o one-half he value used Figure 14 - The swep-sine echnique measures he ransfer funcion wih a lower exciaion ampliude a he expense of measuremen ime. Therefore, in spie of he noise and nonlineariy, he swep-sine produced an excellen measuremen using only half he exciaion drive level of he previous measuremens. Measuremens of Conrol Sysems The measuremen moivaion Transfer funcion esimaion is virually mandaory in conrol sysems engineering. The following are he hree mos common ransfer funcion measuremens made on conrol sysems: 1. Overall open-loop response (sabiliy analysis) 10 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
2. Plan dynamics (plan modeling) 3. Closed-loop response (sysem performance) Ofen hese measuremens mus be made on he conrol sysem under acual operaing condiions. For example, many plans conain an inrinsic inegraor making heir operaing poin difficul o sabilize. During he early sages of developmen, i is a common pracice o consruc a simple conroller o sabilize he operaing poin of he plan. This allows he dynamics of he plan o be sudied in greaer deail. Subsequenly, when he conroller design is refined, measuremens again need o be aken o fully characerize he overall sysem, no jus he plan. To make he closed-loop response measuremen, no special echniques beyond hose previously discussed in his noe are required. Simply excie he conrol sysem a is command inpu and measure he response. The measuremen configuraion is shown in Figure 1. The remaining wo measuremens (open-loop response and he plan dynamics) will be discussed in he following secions. The physical sysem The DUT used in he previous measuremen examples is a real physical sysem. Unil his poin, i has simply been reaed as a mildly non-linear single-inpu single-oupu sysem wih measuremen noise. In fac, he sysem ha has been measured, is he conrol sysem shown in Figure 15 (beneah he broken line). Several measuremen configuraions were seup wih a roary swich (shown as A, B, C, D below SigLab's inpus). All he previous measuremen examples were made wih he selecor in he B posiion. This conrol sysem hardware will be explored in he remaining measuremen examples. The complee conrol sysem consiss of he conroller and plan. The plan is a single inpu single oupu sysem wih impulse response h (). The conroller is a wo inpu single oupu sysem. The oupu of he conroller is a linear combinaion of he command inpu and he posiion inpu from he plan. The command and posiion inpus are convolved wih he conroller dynamics represened by c () and g (), and he difference is aken o produce he conroller oupu ygh( ). The equaion for his operaion is shown inside he conroller block. The noise due o he normal operaion of he conrol sysem loop is represened by nr(). The noise source, nr(), is acually anoher SigLab generaing a combinaion of a periodic sawooh and bandlimied random disurbance. In he disk drive scenario his noise signal would be due o imperfecions in he servo rack, plaer or rack eccenriciy, spindle bearing imperfecions, or oher mechanical errors. Therefore, in he real world, he user has lile or no conrol over his error source. For he purposes of his noe, he error source can be urned off when desired. This allows a comparison o be made beween measuremens wih and wihou noise. I is a common pracice o add a summing circui in he feedback loop of he conrol sysem. This allows he measuremen insrumenaion o injec a signal ino he sysem. In his example he summing circui is beween he conroller oupu ygh( )and he plan command inpu u (). The exac posiion of his circui in he conrol loop may vary from design o design bu he principles o be discussed are no changed significanly. Measuremen configuraions Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 11
Figure 15 shows a swich which can selec one of four ypical conrol sysem measuremen configuraions. This swich is no usually presen in an acual es seup, bu i is shown here o allow a comparison of several possible measuremen configuraions. I should be noed ha posiion A of he selecor swich is he only configuraion of he hree ha does no saisfy he requiremen ha he exciaion be measured direcly by SigLab per Figure 1. The oher configuraions (B, C, D) mee his requiremen will herefore provide unbiased ransfer funcion measuremens. Noaion The following discussion will use noaion where upper case leers refer o he coninuous ime Fourier ransform of he signal as in (3a) and (3b). X( ω ) = F( x( )) (3a) G ( ω ) = F( g( )) (3b) For all he examples, he equaions were wrien in erms of ω, bu he display resuls are in erms of f where f = ω 2π. Configuraion A: direc esimae of he open-loop ransfer funcion using he broad-band FFT analyzer Measuring he open-loop response of a conrol sysem is a common requiremen. This measuremen informaion is imporan in assessing he sabiliy of he loop. The open-loop ransfer funcion G( ω) H( ω) can be measured direcly by selecing swich SigLab OK Model 20-22 Power On xm() ym() Noe: swiches are ganged x() A B D C n () R Command y () gh = c() n () R g() y () p Posiion yp() Posiion Conroller Noe: *= ime-domain convoluion Plan Command h() u () ygh() k 1 + Summing circui added for es. k 0 Noe: c() and g() are conroller impulse responses Figure 15 12 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
posiion A. This connecs plan inpu u () o SigLab channel 1 and he conroller response ygh( )o SigLab channel 2. Exciaion o he overall sysem is provided by feeding x () ino he summing amplifier. The advanage of his configuraion is he measuremen of he open-loop ransfer funcion is made direcly. No deailed undersanding or inclusion of he parameers of he es summing circui is required. The approach is inuiively appealing due o is simpliciy. I is imporan o noe ha he exciaion signal x (), is no being measured by SigLab, and herefore, he esimaor given by (1) will yield a biased ransfer funcion esimae due o he noise source nr(). If, however, nr()is negligible, he ransfer funcion can be esimaed wih (1) wihou significan error. Oher connecion variaions are possible such as measuring beween he plan inpu u ()and oupu yp()bu he resuls will be similar o resuls obained using swich posiion A. Figure 16 shows he resul of his direc measuremen in a noise free siuaion (where nr() is se o zero). The resuling ransfer funcion daa will be viewed as being he rue open-loop ransfer funcion of he sysem. Figure 16 - The open-loop response measured direcly under noise free condiions. Now, he measuremen will be repeaed, bu he noise source will be reacivaed and se o he level used in he measuremens shown in Figure 5. Figure 17 - Direc open-loop measuremen wih noise. A visual comparison beween Figure 16 and Figure 17 indicaes ha a nearly 15 db measuremen error exiss a low frequencies. Using a few simple MATLAB commands he magniude difference in db beween he wo measuremens may be ploed. This error is shown in Figure 18. 20 10 0 db -10-20 Error in db using direc open loop measuremen 10 2 Herz 10 3 10 4 Figure 18 - Esimaion error using he direc open-loop measuremen when noise is presen. One housand measuremen averages were used o compue he ransfer funcion esimae in Figure 17. The coherence looks decepively good. The fac is, his measuremen canno be improved by any increase in averaging because i is a biased measuremen. Due o he sysem's non-linear behavior, he exciaion level canno be increased. The underlying problem is he esimaor defined in (1) canno be successfully used since here is significan noise in boh he Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 13
exciaion measuremen xm()and he response measuremen ym(). To make maers worse, he measuremen noise on hese wo inpu channels is correlaed since his noise is a resul of he single noise source nr(). Here is a case where he usual measuremen qualiy indicaors (good coherence and he smooh, credible ransfer funcion magniude) all poin o a good measuremen, bu, since he assumpions of Figure 1 were no observed, he measuremen is seriously flawed. Configuraion A: direc esimae of he open-loop ransfer funcion using swep-sine analysis There are ways, however, o comba his correlaed noise. Of course, nohing is free: i coss measuremen ime and here is risk of error. As previously discussed, he swepsine echnique uses racking band-pass filers on he measuremen channels. As he bandwidh of hese racking filers is reduced, he effec of noise nr()is minimized on boh measuremen channels. If nr() is a broad-band signal, i is easy o see how his filering can improve he measuremens. Ofen, however, nr() conains large periodic componens e.g. a muliples of he plaer roaional speed in he case of disk drives. In his case, he user mus eiher carefully srucure he measuremen so ha hese noise componens do no lie wihin he racking filer bandwidh a he desired measuremen frequencies, or accep reduced accuracy a frequencies where hey do. The resuls of using he swep-sine echnique are shown in Figure 19. This measuremen is in good agreemen wih ha of Figure 16 excep for he spike a around 9000 Hz due o a harmonic of he periodic componen of he noise. I should be recognized ha he swep-sine echnique also relies on he esimaor given in (1). The racking filers and single frequency sine exciaion ofen allow good measuremens o be made even when he assumpions of lineariy and noiseless inpu measuremens are invalid. Since he coherence esimae (2) is based on he same measuremen assumpions, i is also suspec. However, he larges single disadvanage of he swep-sine echnique is ha of measuremen ime. As he racking filer bandwidh is reduced o improve he measuremen, he measuremen ime naurally increases. For example, his paricular measuremen ook over 5 minues o complee. Figure 19 - Swep-sine's digial racking filers can be used o reduce he effec of he noise on he inpu channels. Configuraion B: esimaing open-loop dynamics from a closed-loop measuremen. A popular alernaive mehod o direcly measuring he open-loop ransfer funcion involves making an unbiased ransfer funcion measuremen of he closed-loop response which relaes ygh( )and x (). This esimae is hen mapped (or ransformed) o he open-loop ransfer funcion. Wih he selecor swich in posiion B, he following equaion relaing ygh( ) and x ()in he frequency domain may be wrien: 14 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
Ygh( ω) = NR( ω) C( ω) X( ω) G( ω) H( ω) 1+ G( ω) H( ω) 1+ G( ω) H( ω) This equaion assumes he es summer has gains of k = k =. 0 1 10 (4) Examining (4) shows ha he response Y gh ( ω ) conains a noise erm due o he sysem operaion N R ( ω ) and a erm due o he exciaion X( ω ). This is he same siuaion as shown in Figure 1 bu in frequency domain erms: e.g. he sysem response is corruped by addiive noise ha is no necessarily whie or random. Again, for he resuling ransfer funcion o be unbiased, he only underlying measuremen assumpion is ha he noise is uncorrelaed wih he exciaion. If he ransfer funcion relaing Y gh ( ω) and X( ω ) is measured, a simple mapping will provide he open-loop ransfer funcion G( ω) H( ω ). consans. The denominaor erm k 0 is k1 paricularly imporan, since ofen here are one or more inegraors in he loop. This forces lim k T( ω) = ω 0 k 0 1 Therefore, a low frequencies, he esimaed open-loop ransfer funcion is a sensiive funcion of he above raio. Since he ransfer funcion esimae of T( ω) is unbiased, his simple mapping (closed o open-loop) provides an unbiased mehod of esimaing he combined conroller and plan dynamics. Figure 20 may look familiar. I is he same measuremen ha was made in he iniial par of he his noe. This measuremen is now referred o as he closed-loop ransfer funcion T( ω ). Firs, le he measured ransfer funcion be defined as: Ygh( ω) T( ω) = (5) X( ω) Then: G( ω) H( ω) = 1 T( ω) k1 k0 + T( ω) k 1 In he mapping given by (6), he summing circui gain consans ( k0, k1) are now included as variables. (6) For he bes mapping resuls, i is imporan o accuraely know he values of hese Figure 20 - Closed-loop ransfer funcion measuremen T( ). ω Conrol sysems engineers ofen need o display boh he magniude and phase of he ransfer funcion in a Bode plo forma shown in figure 21. Wih simple poin and click operaions, he SigLab sofware performs he mapping in (6), displays he resuls in he Bode forma, and displays gain and phase margins. Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 15
ransfer funcion esimae is made, i will be unbiased since his exciaion is being measured. Define his ransfer funcion and coherence measuremen as: H ) YX ( ω ) and ) C YX ( ω ) respecively. Figure 21 - Bode plo of closed o open-loop mapping wih gain and phase margins. The open-loop ransfer funcion compued by mapping he closed-loop measuremen is in excellen agreemen wih he direc (noisefree) measuremen made in Figure 16. The mapping echnique is popular because i provides a rusworhy esimae of he openloop ransfer funcion wih minimal effor. Configuraion C and D: esimaion of he plan ransfer funcion from wo measuremens. Ofen, a measuremen of a single secion of he conrol sysem is desired. For insance, he plan ransfer funcion H( ω ) is commonly required as inpu daa for frequency domain sysem idenificaion. If he conroller dynamics, G( ω ), are known, a division will provide H( ω ) from he openloop funcion G( ω) H( ω ). However, a direc measuremen of H( ω ) is sill ofen preferable. As previously shown, he swepsine echnique can someimes be used wih success o make measuremens ha violae he requiremens in Figure 1, bu his is risky business. A wo-sep measuremen procedure is a good is approach for measuring he plan dynamics. Wih he swich in Figure 15 in he C posiion, he oupu of he plan yp()is being measured by SigLab's channel 2 and he exciaion ino he es summing circui x () by SigLab's channel 1. If a Wih he swich in posiion D anoher unbiased ransfer funcion esimae can be made relaing he exciaion x () and plan inpu u (). This ransfer funcion and coherence measuremen is defined as: H ) UX ( ω ) and C ) UX ( ω ) respecively. These independen unbiased esimaes can be combined o provide an unbiased esimae of he plan ransfer funcion (7) ) ) HYX( ω) H( ω) = ) (7) H ( ω) UX as well as an composie coherence (8). ) ) ) C( ω) = C ( ω) C ( ω) (8) YX UX The advanage of his echnique over aemping o measure he plan direcly (e.g. relaing u ()and yp() while ignoring x ()), is ha he esimaes will be unbiased and herefore wih sufficien averaging converge o he correc resuls. If hree measuremen channels are available, he wo ransfer funcion esimaes can be made simulaneously. In fac, by wriing (7) and (8) in erms of (1) and (2), he resul is ye anoher ransfer funcion esimaor: P xy ( ω ) H ( ω ) = (9) P ( ω ) xu and composie coherence indicaor: 2 ( ) 2 P ( ) xy ω Pxu ω C ( ω ) = (10) P 2 ( ) P ( ) P xx ω yy ω uu ( ω ) 16 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
Using a hree channel simulaneous measuremen, he measuremen speed will double and he end resul will be a bi more accurae, bu hree channels are no mandaory. Figure 22 - Plan ransfer funcion H( ω). In order o verify he previous dual ransfer echnique, a noise free measuremen of he plan is made. Since measuremen noise can be eliminaed (for his example), he plan ransfer funcion can be accuraely measured by he direc means. The ransfer funcion is shown in Figure 22. Now, he ask is o esimae he plan ransfer funcion under he same condiions ha were presen (noise and non-lineariy) for he open-loop measuremen using (7) and (8). Since SigLabs can be combined o creae muli-channel sysems, wo SigLabs were linked for he following measuremens. The wo ransfer funcion configuraions correspond o swich posiions C and D. Figure 23 shows he H ) YX ( ω ) ransfer funcion esimae made wih he swich in he C posiion. Noe ha he coherence is close o zero a he low frequency end of he measuremen. However, since he measuremen is unbiased, i will converge o he acual ransfer funcion given sufficien averaging. Even wih 1000 averages, SigLab ook less han one minue o complee his measuremen. Figure 24 shows he second ransfer funcion measuremen now made wih he swich in posiion D. The coherence in his measuremen is also very low a low frequencies. Figure 24 - Transfer funcion H ) UX ( ω ), corresponding o swich posiion D. The simple MATLAB scrip file in Lising 1, was used o compue and plo he plan ransfer funcion esimae H( ) ω ). Figure 23 - Transfer funcion H ) YX ( ω ), corresponding o swich posiion C. Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 17
The resuling plan ransfer funcion plo is shown in Figure 25. A comparison beween Figure 22 and 25 shows he excellen agreemen beween he plan ransfer funcion esimae using (9) and he plan ransfer funcion which was measured direcly in Figure 22. To ge a closer look a he difference beween H( ) ω ) and he H( ω ) scrip M-file was exended o compue and plo he magniude (in db) difference beween he measuremens. Figure 25 - Plan ransfer funcion esimaed on a non-linear and noisy sysem by he mehod in Lising 1. The error is ploed in Figure 26. There is excellen agreemen beween his esimae and he acual plan, even a he low frequencies where he coherence of he esimaor is almos zero. The errors increased a he high frequency end of he measuremen where he plan response is rapidly rolling off. This of lile concern since i is well beyond he ineresing dynamics of he plan. 2 error beween acual plan and dual measuremen 1 % M-file dual_x.m load dblx1.vna -ma % his is a 3 channel meas H=XferDa(:,2)./XferDa(:,1); % he plan ransfer funcion Coh = CohDa(:,2).*CohDa(:,1); % composie coherence semilogx(fvec,20*log10(abs(h)),... 'color','whie'); axis([25,10000,-50,10]); ile('plan ransfer funcion'); xlabel('herz'); ylabel('db'); % Lising 1 - M-file scrip compuing H( ) ω ). 0 db -1-2 10 2 10 3 10 4 Herz Figure 26 - Magniude difference beween ) H( ω ) and H( ω ). The composie coherence is also easy o compue and i is ploed in Figure 27. Noe he dips due o he periodic componens in he noise nr(). These coherence and ransfer funcion esimaes can serve as he inpu o frequency domain idenificaion algorihms. 10 Plan ransfer Composie Coherence 0 1-10 0.8-20 db 0.6-30 0.4-40 -50 10 2 10 3 10 4 Herz 0.2 0 10 2 10 3 10 4 Herz 18 Applicaion Noe 5.1 Esimaing Transfer Funcion wih DSPT SigLab
Figure 27 - Composie coherence calculaion from Lising 1. Conclusion Making ransfer funcion esimaes on noisy non-linear sysems is far more difficul han in he noise-free, linear case. A high qualiy measuremen can be obained even under adverse condiions, by using eiher he closed o open-loop mapping echniques, or by making wo unbiased ransfer funcion esimaes and combining hem. The measuremen seup and assumpions oulined in Figure 1 should be observed for opimal resuls. Alhough he bulk of he examples presened used he broad-band FFT based esimaion echnique, he swep-sine analysis will do as well or beer. If he measuremen resuls are suspec using he broad-band FFT echnique, i is pruden o repea he measuremen wih swep-sine o ge a differen measuremen viewpoin. If a meaningful ransfer funcion exiss, swep-sine will do he job when all else fails. Applicaion Noe 5.1 Esimaing Transfer Funcions wih DSPT SigLab. 19
For more informaion conac: DSP Technology Inc., Signal Analysis Group 48500 Kao Road, Fremon, CA 94538 Phone: (510) 657-7555 Fax: (510) 657-7576 1 Welch, The use of Fas Fourier Transform for he Esimaion of Power Specra: A Mehod Based on Time Averaging over Shor Modified Periodograms, IEEE Transacions on Audio and Elecroacousics, vol AU-15, June 1967, pp. 70-73. see also T. P. Krauss, L. Shure, J. N Lile, Signal Processing Toolbox Users Guide, The MahWorks, pp 1-72 - 1-73, June 1994. 1996 DSP Technology Inc. DSPT, DSP Technology Inc., and DSPT SigLab are rademarks of DSP Technology, Inc. MATLAB is a regisered rademark and Handle Graphics is a rademark of The MahWorks, Incorporaed. Oher produc and rade names are rademarks or regisered rademarks of heir respecive holders. Prined in U.S.A. S