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

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1 Homework 3 AERE573 Fall 08 Due 0/8(M) ame PROBLEM (40pts) Cosider a D order uderdamped system trasfer fuctio H( s) s ratio 0 The deomiator is the system characteristic polyomial P( s) s s (a)(5pts) Use the quadratic formula, to show that the roots of this polyomial are: s id where d for 0 gs with dampig s () Remark The frequecy d is called the system damped atural frequecy, ad the frequecy called the system udamped atural frequecy The parameter is called the system dampig ratio Whe system is said to be critically damped, ad whe the it is said to be overdamped Whe, the roots of P( s) s s will be egative real umbers I this case, the system will ehibit o atural oscillatios Moreover, the value of becomes meaigless It is oly for that the umerical value of has meaig I this case, the d 0 atural oscillatios of the system will occur at They will oly occur at ( 0) o physical system has 0, for, if it did, it would be a perpetual motio system Hece, i the physical world oe will ever observe Clearly though, for, oe ca get a pretty good idea of what it is whe the system has zero dampig is (b)(0pts) Shade the regio i the left upper quadrat of comple plae (ie the s-plae) i which the root s id will lie uder each of the followig costraits: [OTE: Be sure to iclude the symbolic values o the appropriate ais ad justify your 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) Justificatio:

2 (c)(5pts) Recall that H ( s i) g s 4 H( ir) The use this to show that r r ( ) is the system Frequecy Respose Fuctio (FRF) Let H ( ir) r / Show that is a maimum whe res Remark The frequecy res is called the system resoat frequecy It is ot the damped atural frequecy Hece, while i the time domai we see atural oscillatios occurrig at peak magitude at res d d, i the frequecy domai we see oly the (d)(5pts) Recall that for a iput u( t) u0 si( t), the steady state output will be y( t) u0m ( )si[ t ( )], where M ( ) H( i) ad ( ) arg[ Hi ( )] Use this to eplai why the parameter is called the system static gai Eplaatio: g s (e)(5pts) ow cosider that case where r / H ( i0 r) H ( ir) 40 (ie at high frequecies Defie H ( ir) 0log 0 H ( ir) H ( ir) has a slope of -40/decade) Show that

3 3 PROBLEM (40pts) Cosider the followig secod order system: (a)(5pts) Use the Matlab commad impulse to obtai a plot of the system impulse respose, Also, obtai the samplig iterval, T, that was used [See (a)] The samplig time used was h(t) s H ( s) () s s 5 Figure (a): (a) blue, (b) red, ad (g) correlatio black (b)(5pts) (i) Use the table of Laplace trasforms that is posted i the Lecture Summary folder to obtai the epressio for (ii) Overlay a plot of it o your plot i (a) ad (iii) commet h(t) (c)(5pts) (i)use the Matlab commad bode to obtai a plot of the system frequecy respose fuctio (FRF) (ii) Take the maimum frequecy to be the yquist frequecy, ad fid the correspodig samplig iterval, T (iii)compare this value to the samplig period used to obtai your plot i (a) (i) [See (c)] (ii) (iii) (d)(5pts) To simulate a radom process, y(t) Figure (c) FRF plot that has a psd with the shape of the FRF magitude i (c), we will simulate a white oise process u (t) Use the geeral Wieer-Kichie relatio S ( ) E[ X ( i) ] to show that S y ( ) H( i) c, where c is the (badlimited) white oise psd value c (e)(0pts) Recall Ry (0) y S y ( ) d So, from (d): y H( ) d (i) Use the Matlab commad itegral to obtai the umerical value of H ( ) From this value ad the badwidth iformatio fid the value for [See (e)] d (ii)for this value ad with 0, fid the umerical value for c (iii) u y

4 4 (f)(5pts) (i)use the Matlab commad lsim to simulate the system respose to a white oise iput with the variace you foud i (e) Your simulatio should start with poits Discard the first 500 poits, ad plot the remaiig data (ii)commet o whether the rage of values seems to be correct based o the rage [See (f)] 4 y Figure (f) Partial realizatio (g)(5pts)(i)use the Matlab commad corr to obtai a ubiased estimate of { R ( ) ; 0 7sec} ad overlay it o Figure (a) (ii) Commet [See (g)] y

5 5 PROBLEM 3(0pts) Cosider a trasfer fuctio (a)(5pts) For a uit step impulse, f ( t) ( t) 0 0t H ( s) From etry 4 i the posted table: h( t) 0e s 0, recall that from the covolutio itegral, h ( t) h( ) ( t ) d For a itegratio time step, approimate this itegral usig your epressio for h(t) 0 i (a) Approimate the uit 0 impulse by ( k) / for k 0 Show that the epressio for the summatio leads to h( ) 0( e ) C (b)(5pts) Deote h( k) h k The the z-trasform of the sequece I class, the followig power series relatio was proved: relatio to show that the z-trasform of { h k } k k0 is defied as Thus, for k H ( z) h k z k 0, k 0 C h( ) C is H ( z) for z-values that satisfy z k Use this last z (c)(5pts) Recall that the fuctio s z z ) e z e, ad so for s i, we have i ( is periodic, with frequecy / z e i It should be clear that, as a fuctio of frequecy, rad/sec It follows that we eed oly cosider frequecies i the iterval [ /, / ] The frequecy / rad/sec is called the yquist frequecy ow, for a sufficietly 0 low frequecy,, we have H( i ) 0, which is the static gai of H (s) For the same we have i 0 i C z ( ) e Hece, for this, we have H( e i ), ad so the static gai of the sampled system H (z) is ot (or 0) Ad so, cosider the scaled discrete system FRF H( z) H( z) For 00 sec write your ow C Matlab code (do ot use ay of Matlab s codes) to compute ad plot the magitude (i ) ad the phase (i degrees) of i the discrete Frequecy Respose Fuctio (FRF) H ( e ) for radial frequecies i the iterval [ 00 /, / ] Also, use the semilog commad so that your frequecy ais is spaced logarithmically, ad use a frequecy step size of 00 [Place your code i a Appedi at the ed of this homework] [See 3(e)] i Figure 3(e) FRF for H ( e ) (blue) ad for H ( i) (red)

6 (d)(5pts) I would hope that it bothers you that we simply applied a fudge factor to make thigs work out i (c) It should! Rather tha just applyig a fudge factor, i this part you will try to discover that somethig was missig i (d) To this ed, begi by recallig the defiitio: st H( s) h( ) e d Approimate this itegral as a Riema sum, call it The epress it i terms of the variable H ( z) k 0 h k z k You should discover that compare it to the value of the fudge factor z s e 0 Call this H (z) H (z) differs from ( ) / C H (s) Fially, compare this to the defiitio of the z-trasform H (z) by a costat Compute the value of this costat ad 6

7 7 Appedi %PROGRAM AME: hw3m 9/3/8 %PROBLEM : %(a): %(b): %(c): %(e): Fid total area uder Hw) ^: %(f): %(g) %========================================================== % PROBLEM 3: %(e) %(g)