Lecture 17 Errors in Matlab s Turbulence PSD and Shaping Filter Expressions

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1 Lectre 7 Errors in Matlab s Trblence PSD and Shaping Filter Expressions b Peter J Sherman /7/7 [prepared for AERE 355 class] In this brief note we will show that the trblence power spectral densities (psds) and corresponding shaping filters for the Drden and von Karman wind trblence models in Matlab s Aerospace toolbox are incorrect The Relation Between the PSD and the Atocorrelation Fnction- The psd is defined as ) R ( ) i ( e d where R ( ) E[ ( t) ( t )] is the atocorrelation fnction This definition reslts in the Forier Transform pair relation R this relation gives, in particlar: R () ( ) i ( e d For a process ) ( ) (t) with E [ ( t)], d () A Note on Units- In () it is not the area associated with () that eqals the total power that area scaled b / This scale factor converts rad/sec to Hz However, R the nits of (t) Rather, the total power is () has nits that are the sqare of (eg velocit, volts, etc) It does not have anthing to do with nits of Hz Hence, to better nderstand the role that nits pla, it wold be better define the freqenc variable f / that has nits of Hz Then df d/ and f Then () becomes: R () ( ) f f df ( ) This makes it clear that the nits of ( ) ( ) are power/hz B retaining onl ( ), as in (), one is more likel to presme that its nits are power/(rad/sec) And this is reasonable, since it clearl is a fnction of Unfortnatel, this presmption is incorrect Now, one might reasonabl sggest that () be written as: R () ( ) f df ( ) In fact, some books do se ( ) The problem with this becomes clear when working with shaping filters Sch filters are specified as, sa, Hs (); that is, the are fnctions of the Laplace variable s i, where the nits of are rad/sec Hence, it is trivial to get H ( s i) from a table of Laplace transform pairs However, sppose that H ( s) V ( s) / F( s) with nits of velocit/force Then H( i) V ( i) / F( i) also has these nits The qantit Hi ( ) is called the sstem Freqenc Response Fnction (FRF) It is a fnction of Nowhere are there nits of Hz in Hi ( ) Even so, the inverse Laplace transform of Hs () can be compted as: it h( t) H ( i) e d In the context of this integral, the nits of Hi ( ) are (velocit/force)/hz If or head is spinning at this point, then o are in good compan

2 Modeling Trblence as the Otpt of a Shaping Filter with White Noise Inpt- This section relies on the following it where Y( i ) lim T ( t) e dt T FACT : ( ) EY ( i) Note : The above definition of the Forier transform Yi ( ) it Y ( i) ( t) e dt T is not consistent with the standard definition: In this athor s opinion, this definition shold be reserved for onl deterministic t (), sch as fnctions given in tables of Laplace transform pairs However, almost all books on both linear sstems and random processes se it While it is correct for a deterministic t (), it is not correct for a reglar random process t () Sppose that we want to model a process Then clearl Y( i) H( i) U( i) Hence, (t) as the otpt of a transfer fnction H(s) having the arbitrar inpt (t) ( ) E Y ( i) EH ( i) U ( i) EH ( i) U ( i) H ( i) EU ( i) H ( i) ( ) () In () the qantit E (X ) is the expected vale of the random variable X It is eas to show that E( cx ) ce( X ) It was this relation that was sed in (), wherein c H ( i) and X U ( i) Definition A random process (t) is said to be a standard white noise process if it has ( ) for all FACT : A standard white noise process fnction (t) has atocorrelation fnction R ( ) ( ), where ( ) is the Dirac delta To prove FACT, simpl take the Forier transform of R ( ) : Hence, the white noise variance (ie total power) is exist! Even so, it is a valable mathematical concept i i ( ) R ( ) e d ( ) e d (3) () In other words, continos time white noise does not Now, sppose that we want to model a process (t) is the otpt of a transfer fnction H (s) having the standard white noise inpt (t) Then from () and (3): Sbstitting (4) into () gives:, scaled b / ( ) H ( i) (4) H ( i) d In words, the variance is the total area associated with H ( i) for

3 3 Fact 3: For an chosen n n k k / Let i( k) U( ) k e Then U ( ) n k and n let { } ~ iid N, E [The reader shold notice that in this Forier transform expression we have, indeed, inclded the T n, per Note ] Proof: EU ( ) n n k, l E( ) e k l i[( k l) ] where n / for k l E( kl ) So: n EU( ) for k l n k n Matlab Validation: %PROGRAM NAME: whitenoise_psdm del=; %Var this to validate that psd= for ANY del n=; nsim=^5; std=/sqrt(del); =normrnd(,std,n,nsim); U=sqrt(del/n)*fft(); S=mean(abs(U)^,); S=S(:n/); SdB=*log(S); wn=pi/del; dw=*wn/n; w=:dw:wn-dw; figre() plot(w,sdb) xlabel('freqenc (rad/sec)') label('db') title('simlation-based Standard White Noise PSD') grid axis([,wn,-,]) Example To illstrate the importance of this scale factor, consider the following Drden model MIL-F-8785C for the forward speed trblence psd [See Matlab (ML) docmentation]: ( ML) L ( ) ; (5) [( L / ) ] The limits reflect the fact that (5) is the -sided psd This is times the -sided psd Moreover, even thogh it is expressed as a fnction of ( rad / s), it is has incorporated a factor /, so that it has nits ( fps ) / Hz The combination of the factors of and /, reslts in a net factor of ( ML) L ( ) d dv v / d Let v ( L / ) Then d ( / L) dv, and so: [( L / ) ] dv v its nits are ( fps ) / Hz tan The corresponding -sided psd associated with (5) is, therefore, v in (5) To validate this claim write This confirms the fact that (5) is, indeed, a -sided psd, and that L ) ; [( L / ) ] ( (6) It is the -sided psd (6) that pertains to (4) Even so, the Matlab shaping filter is given as:

4 4 H ( ML) L ( s) (7) ( L / ) s In (7) the variable s is the standard Laplace transform representation sed in Matlab elsewhere, as well as in practicall all textbooks on this topic Specificall, s i, where the nits for are The inpt to (7) is white trblence, rad /sec and the otpt is the Drden trblence Hence, the nits for (7) are ; even thogh it is portraed as having nits /( rad / sec) for s i No table of Laplace transform pairs ses nits / Hz Frthermore, for s i, (7) is defined over, as was (5) For s i Hence, the correct shaping filter is not (7) Rather, it is: / Hz, all tables (inclding Matlab s tf e-tables) assme the defined region L H ( s) (8) ( L / ) s Conseqentl, were one to se (7) to simlate trblence sing standard white noise, the reslting trblence standard deviation / 56 In words, the simlated standard deviation wold be approximatel half of what one shold expect The following code simlates trblence with desired according to (7): dt=ta/; tmax=5; t=:dt:tmax; t=t'; n=length(t); stdu=/sqrt(dt); =normrnd(,stdu,n,); L=75; =8; stdg=; gs=stdg*sqrt(*l/(pi*)); ta=l/; H=tf(gs,[ta ]); g=lsim(h,,t); varg=var(g) stdg=sqrt(varg) plot(t,g) histogram(g, Normalization, pdf ) stdg = 557 The shaping filter (6) is clearl not the correct filter! It shold NOT inclde the factor of /

5 5 CONCLUSIONS: (C): The Matlab trblence psd is a -sided psd and has nits ( fps ) /( rad / s), that psd shold be mltiplied b ( fps ) / Hz To obtain the -sided psd in nits (C): The Matlab shaping filter is incorrect, per the accepted definition of standard white noise To obtain the correct filter it shold be mltiplied b (C3): The Matlab shaping filter will be correct if the driving white noise has standard deviation definition of standard white noise is at odds with the se of the standard deviation psd for the white noise sing noise in this fashion / dt / dt / dt However, While it is tre that the -sided is ( fps ) / Hz, there is no textbook that I am aware of that addresses white While I am sre that the simlation of trblence sing Simlink ields correct reslts, man researchers prefer to se more direct methods sch as the one sed in the above example If, in fact, Simlink incldes the standard deviation / dt for the white noise standard deviation, and ses the -sided shaping filter, then it wold be beneficial to researchers if it wold clearl state the same in the docmentation Sch docmentation wold also be valable for edcators who se Matlab in relation to topics sch as random processes, time series, and linear sstems/controls It is extremel frstrating for stdents to realize that alternative methods sch as -sided psds scaled b /, and transfer fnctions that are similarl scaled

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