1it is said to be overdamped. When 1, the roots of

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1 Homework 3 AERE573 Fall 8 Due /8(M) SOLUTIO PROBLEM (4pt) Coider a D order uderdamped tem trafer fuctio H( ) ratio The deomiator i the tem characteritic polomial P( ) (a)(5pt) Ue the quadratic formula, to how that the root of thi polomial are: id where d for ( ) 4, ( i ( g with dampig () i Remark The frequec d i called the tem damped atural frequec, ad the frequec i called the tem udamped atural frequec The parameter i called the tem dampig ratio Whe the tem i aid to be criticall damped, ad whe it i aid to be overdamped Whe, the root of P( ) will be egative real umber I thi cae, the tem will exhibit o atural ocillatio Moreover, the value of become meaigle It i ol for that the umerical value of ha meaig I thi cae, the d atural ocillatio of the tem will occur at The will ol occur at ( ) o phical tem ha, for, if it did, it would be a perpetual motio tem Hece, i the phical world oe will ever oberve Clearl though, for, oe ca get a prett good idea of what it i d whe the tem ha zero dampig (b)(pt) Shade the regio i the left upper quadrat of complex plae (ie the -plae) i which the root id will lie uder each of the followig cotrait: [OTE: Be ure to iclude the mbolic value o the appropriate axi ad jutif our plot with math] (i) d (ii) (iii) (iv) i i i i / / Figure b(i) d Figure b(ii) Figure (b)(iii) Figure (b)(iv) Jutificatio: (i) cotat imagiar part ; (ii) cotat real part ; (iii) otig that ( ) (, it follow that co, or co (iv) From (iii) we foud that

2 (c)(5pt) Recall that H ( i) g 4 H( ir) The ue thi to how that r r ( ) H( ir) g g 4 Settig ( r ) i( r) r r ( ) d( f / g) / dx [ f ' g fg ']/ g f g i the tem Frequec Repoe Fuctio (FRF) Let H ( ir) r / Show that i a maximum whe d H ( ir) / dr ad uig the calculu reult, if we et thi equal to zero, the we have ' ' ad 4 f g fg Applig thi to re H ( ir) 4 3 g r r ( ) give: g d[ r r ( )]/ dr 4r 4 r( ) r ( ) Hece, r /, or with Remark The frequec re i called the tem reoat frequec It i ot the damped atural frequec Hece, while i the time domai we ee atural ocillatio occurrig at peak magitude at re d d, i the frequec domai we ee ol the (d)(5pt) Recall that for a iput u( t) u i( t), the tead tate output will be ( t) um ( )i[ t ( )], where M ( ) H( i) ad ( ) arg[ Hi ( )] Ue thi to explai wh the parameter i called the tem tatic gai Explaatio: For it i clear that M() g I thi cae, the iput i the cotat u() t u (ie it i tatic) Hece, the tead tate repoe ( t) um () i alo cotat, or tatic The ratio of cotat output to the cotat iput i therefore Hece, at frequecie cloe to zero the gai of the FRF i g Whe the frequec of the iput i exactl zero, the iput i M() g Thi i wh i called the tem tatic gai g g (e)(5pt) ow coider that cae where r / H ( i r) H ( ir) 4 (ie at high frequecie Defie H ( ir) log H ( ir) H ( ir) ha a lope of -4/decade) g H( ir) log H( ir) log 4 Hece, for r, r r ( ) g 4 H( ir) log 4 log g log r log g 4log r Similarl, r 4 Show that H ( i r) log g log ( r) log g 4log r 4log H( i r) 4

3 3 PROBLEM (4pt) Coider the followig ecod order tem: (a)(5pt) Ue the Matlab commad impule to obtai a plot of the tem impule repoe, Alo, obtai the amplig iterval, T, that wa ued [See (a)] The amplig time ued wa h(t) T 46ec Amplitude 5 5 H ( ) () 5 Impule Reoe from (a) (blue),from (b) (red) & Correlatio Fc (black) Time (ecod) Figure (a): (a) blue, (b) red, ad (g) correlatio black (b)(5pt) (i) Ue the table of Laplace traform that i poted i the Lecture Summar folder to obtai the expreio for h (t) (ii) Overla a plot of it o our plot i (a) ad (iii) commet (i) H( ) From 5 ( ) d ( ) 4 we obtai: t h( t) e co( d t) i(t ) (ii) The plot (red) i i Figure (a) (iii)the are exactl the ame Bode Diagram (c)(5pt) (i)ue the Matlab commad bode to obtai a plot of the tem frequec repoe fuctio (FRF) (ii) Take the maximum frequec to be the quit frequec, ad fid the correpodig amplig iterval, T (iii)compare thi value to the amplig period ued to obtai our plot i (a) (i) [See (c)] (ii) r / f 595Hz f 383Hz T 34ec (iii) Thi ample iterval i about /3 of that ued i (a) (d)(5pt) To imulate a radom proce, a white oie proce S ( ) H( i) u(t) (t) Magitude () Phae (deg) Frequec (rad/) Figure (c) FRF plot that ha a pd with the hape of the FRF magitude i (c), we will imulate Ue the geeral Wieer-Kichie relatio S x ( ) E[ X ( i) ] to how that c, where c i the (badlimited) white oie pd value S ( ) E[ Y ( i) ] E[ H( i) U( i) ] H( i) E[ U( i) ] H( i) S ( ) H( i) c (e)(pt) Recall R () S ( d ) c So, from (d): H( ) d (i) Ue the Matlab commad itegral to obtai the umerical value of H ( ) From thi value ad the badwidth iformatio fid the value for u d (ii)for thi value ad with, fid the umerical value for c (iii) [See (e)] (i) H( ) d 5 (ii) 5c c 95 (iii) c( f ) cf 95(383) 66 u amp u

4 (f)(5pt) (i)ue the Matlab commad lim to imulate the tem repoe to a white oie iput with the variace ou foud i (e) Your imulatio hould tart with 5 +5 poit Dicard the firt 5 poit, ad plot the remaiig data (ii)commet o whether the rage of value eem to be correct baed o the rage 4 [See (f)] (i) See plot at right Sice, Thi matche the rage how Partial Realizatio of w (t) Figure (f) Partial realizatio x 4 4 (g)(5pt)(i)ue the Matlab commad xcorr to obtai a ubiaed etimate of { R ( ) ; 7ec} ad overla it o Figure (a) (ii) Commet [See (g)(i)it i the plot i black (ii)the correlatio fuctio i ver imilar to the impule repoe

5 5 PROBLEM 3(pt) Coider a trafer fuctio (a)(5pt) For a uit tep impule, f ( t) ( t) t H ( ) From etr 4 i the poted table: h( t) e, recall that from the covolutio itegral, h ( t) h( ) ( t ) d For a itegratio time tep, approximate thi itegral uig our expreio for h(t) i (a) Approximate the uit impule b ( k) / for k Show that the expreio for the ummatio lead to h( ) ( e ) C ( k) ( ) h( ) e [( k) / ] e ( e ) k (b)(5pt) Deote h( k) h The the z-traform of the equece k { h k } k x I cla, the followig power erie relatio wa proved: x x relatio to how that the z-traform of k k k k k C H( z) hk z C z C ( z ) z (c)(5pt) Recall that the fuctio k k k z z ) e i defied a Thu, for k H ( z) h k z k x, k C h( ) C i H ( z) for z-value that atif z z e, ad o for i, we have i ( i periodic, with frequec / x k Ue thi lat x z z e i It hould be clear that, a a fuctio of frequec, rad/ec It follow that we eed ol coider frequecie i the iterval [ /, / ] The frequec / rad/ec i called the quit frequec ow, for a ufficietl low frequec,, we have H( i ), which i the tatic gai of H () For the ame we have i i C z ( ) e Hece, for thi, we have H( e i ), ad o the tatic gai of the ampled tem H (z) i ot (or ) Ad o, coider the caled dicrete tem FRF H( z) H( z) For ec write our ow C Matlab code (do ot ue a of Matlab code) to compute ad plot the magitude (i ) ad the phae (i degree) of i the dicrete Frequec Repoe Fuctio (FRF) H ( e ) for radial frequecie i the iterval [ /, / ] Alo, ue the emilogx commad o that our frequec axi i paced logarithmicall, ad ue a frequec tep ize of [Place our code i a Appedix at the ed of thi homework] [See 3(e)] Scaled H(z) & H()FRF Magitude Scaled H(z)& H()FRF Phae Degree -5-3 i Figure 3(e) FRF for H ( e ) (blue) ad for H ( i) (red)

6 (d)(5pt) I would hope that it bother ou that we impl applied a fudge factor to make thig work out i (c) It hould! Rather tha jut applig a fudge factor, i thi part ou will tr to dicover that omethig wa miig i (d) To thi ed, begi b recallig the defiitio: t H( ) h( ) e d Approximate thi itegral a a Riema um, call it The expre it i term of the variable H ( z) k h k z k You hould dicover that compare it to the value of the fudge factor H( ) z e Call thi H (z) H (z) differ from ( ) / C H () Fiall, compare thi to the defiitio of the z-traform H (z) b a cotat Compute the value of thi cotat ad t ( k) k h( ) e d h( k) e h z H ( z) Hece, H (z) i (c) i miig 34 [It wa k k k caceled out b the approximate delta fuctio] The fudge factor we ued i (e) wa 97 3 Thee C are almot idetical 6

7 7 Appedix %PROGRAM AME: hw3m 9/3/8 %PROBLEM : %(a): =[ ]; d=[ 5]; H=tf(,d); [h,t]=impule(h); figure() impule(h) T=t()-t() %(b): r=root(d); wd=ab(imag(r())); tau=-real(r()); hh=exp(-t)*(*co(wd*t)-(/wd)*i(wd*t)); hold o plot(t,hh,'r','liewidth',) title('impule Reoe from (a) (blue) & from (b) (red') %(c): figure() bode(h); wmax=;%thi wa obtaied from the Bode plot famp=wmax/pi; T=/famp %(e): Fid total area uder Hw) ^: H =@(w) (ab(((()*w*i+())/(d(3)*(i*w)^+d()*i*w+d()))))^; RH=*itegral(H,,wmax)/(*pi) vary=; c=vary/rh %white oie pd value varu=c*famp % PART (f): = ^5; u=ormrd(,qrt(varu),+5,); t=:t:(+5-)*t; t=t'; =lim(u,h,t); t=t(:); =(5:+5); figure() plot(t,) title('partial Realizatio of w (t)') %(g) maxlag = fix(7/t); rhat=xcorr(,maxlag,'ubiaed'); rhat=rhat(maxlag+:*maxlag+); tau=t*(:maxlag); tau=tau'; figure() plot(tau,rhat,'k--','liewidth',) title('impule Reoe from (a) (blue),from (b) (red) & Correlatio Fc (black)') %========================================================== % PROBLEM 3: %(e) C = ; alpha = 97; del = *pi; dw = ; w = *pi/del:dw:pi/del; z = exp(w*del*i); H = C*(-alpha*z^-)^-; HH = ((-alpha)/c)*h; M = *log(ab(hh)); Ph = (8/pi)*agle(HH); figure(3) ubplot(,,), emilogx(w,m)

8 label('') title('scaled H(z)FRF Magitude') ubplot(,,), emilogx(w,ph) label('degree') title('scaled H(z)FRF Phae') %(g) % The cotiuou-time tem FRF =tf(,[ ]); [Mag,Phae]=bode(,w); lw = legth(w); Mag = rehape(mag,,lw); Phae = rehape(phae,,lw); Mag = *log(mag); figure(3) ubplot(,,), emilogx(w,m) hold o ubplot(,,), emilogx(w,mag,'r') label('') title('scaled H(z) & H()FRF Magitude') ubplot(,,), emilogx(w,ph) hold o ubplot(,,), emilogx(w,phae,'r') label('degree') title('scaled H(z)& H()FRF Phae') 8