Homework 6 AERE331 Spring 2019 Due 4/24(W) Name Sec. 1 / 2

Transcription

1 Homework 6 AERE33 Spring 9 Due 4/4(W) Name Sec / PROBLEM (5p In PROBLEM 4 of HW4 we used he frequency domain o design a yaw/rudder feedback conrol sysem for a plan wih ransfer funcion 46 Gp () s The conroller we obained was s 84 Gc ( 356 s 76s455 s 757 Noice ha he upper break frequency of he conroller is 757 For his reason, one migh anicipae ha we will need a Nyquis frequency well above his value o arrive a a digial conrol sysem Since en imes his frequency is where he conroller phase is close o ero, choose r/ s (a)(p (i) Use he cd command wih he imp flag o obain he expression for he digial conroller (ii) Overlay Bode plos of he analog and digial conrollers (iii) Discuss he accuracy of he digial approximaion of he analog conroller [See (a)] (i): (ii): See plo a righ (iii): N Figure (a) Analog & digial conroller Bode plos (b)(5p Regardless of your answer o (iii) in (a), in he following pars you will design a beer digial conroller To his end, noe firs ha here is no s/ ransform pair for Gc () s in Table 8 on p6 Hence, express i as: s 84 s Gc ( 356 K K H( H( s 757 s s Give, and hen use he appropriae enry in Table 8 on o arrive a H () However, be sure o include a facor of T Do no include any numbers here (c)(5p The erm s Y() s H() s is he real issue In fac, were you o ake a course in digial conrol sysems, you s U() s would address more sophisicaed ways o arrive a H () (eg [], []) Show ha y( ) y( ) d u( ) [] Book Secion 83 (Tusin s Mehod) including Example 8 [] hp://wwwengrusaskca/classes/ee/48/noes/myee48-p9-freqresppidpdf T (d)(5p From (c) we have y( T ) u( T ) y( ) d, which gives y( ) d u( T ) y( T ) From his, we T hen have y( ) d y( ) d y( ) d y( ) d u( T ) y( T ) T T Using he approximaion y( ) d T y( ) gives: T y( ) d T y( ) u( T ) y( T ) (*) Subsiue (*) ino (c) o arrive a he difference equaion: yk ( T) yk uk u, where k yk y( kt ) T

2 (e)(5p From (d) we have Y( ) H(), where T U() From (b) you should have found ( ) T H where Hence he digial conroller ransfer funcion is: G ( c ) K H ( ) H ( ) (i) Use he f(h,t) command o obain G () c e T (ii) Overlay he Bode plo of he analog conroller and his digial conroller Then discuss how well he digial one approximaes he analog one [See (e)] (i): (ii: Figure (e) Analog & digial conroller Bode plos (f)(p Use he cd command wih he oh flag o arrive a overlaid Bode plos of Gp() s nd Gp() Then discuss how well he discree model approximaes he rue plan ransfer funcion [See (f)] Figure (f) Analog & digial plan Bode plos (g)(p Overlay (i) CL Bode plos and (ii) sep responses of he analog and digial sysems Then commen Figure (g) Analog & digial CL Bode plos (LEFT) and sep responses (RIGHT)

3 3 PROBLEM (35p The orsional dynamics of a saellie solar panel ( are G ( For he closed loop command sysem, he p T( s s 5 T( PD conroller G ( K( s ) will be used o place CL c E( conjugae poles wih opimal damping, 77 and sec (a)(5p Use he rlocus command o verify ha he needed gain [See (a)] K 8 Figure (a) Roo locus plo (b)(p For ( 87( s ), here is no enry in Table 8 for obaining G c () G c (i) For T 5sec use he backward difference approximaion of a derivaive o arrive a G () c s and G () c o validae your design G c () (ii) Overlay Bode plos of Figure (b) Bode plos for G c ( and G c () (c)(5p Overlay he CL sep responses for he analog and digial closed loop sysems Discuss how well ( kt ) approximaes [See (c)] () Figure (c) CL sep response for () and ( kt ) (d)(p You should have found ha he responses in (c) compare very well You should have also found ha he digial sysem open loop ransfer funcion is G( ) [This includes K 87 ] (i) Use a rlocus plo daa cursor re: G ( ) o recover he CL complex poles for W () (ii) Use he relaion st e o recover he associaed s-domain poles (iii) compare hese o he poles of W ( per ζ and τ [NOTE: You will need o really oom ino he region near he origin] Figure (d) Roo locus plo

4 4 [See (d)] (ii): (iii): (e)(5p Use he command pole(w) o recover he associaed s-domain poles, and compare hem o hose of hey differ from hose in (d), offer a possible reason for he difference [See (e)] W ( If

5 5 PROBLEM 3(5p [See Example 8 on p67] Le ( ) 5s Gc s and T 5sec The Tusin-based digial s conroller is Gc () 7778 (a)(4p (i) Verify his using he cd command (ii) Overlay he analog and digial conroller sep responses [Include your code HERE] (b)(3p Arrive a he conroller difference equaion Figure 3(a) Conroller sep responses (c)(4p For Gp () s ss ( ) he analog CL TF is 5s W() s 3 s s 6s (i): Compue and give W( ) [Do no use Simulink] (ii) Overlay he CL sep responses [See 3(c)] (i): Figure 3(c) CL sep responses for W() s and W( ) (d)(4p I should be clear ha he sep response for W( ) is in discree ime This is because he command inpu is in discree ime If he command inpu is viewed as a sep-ype inpu in coninuous ime, hen he oupu will be a coninuous ime approximaion of he analog sysem sep response (i) Use he command W_ha = dc(wtus, oh ) o compue and give Ws ˆ () (ii) Overlay he sep responses for W() s and Ws ˆ () Figure 3(d) Sep responses for W() s and Ws ˆ ()

6 6 Malab Code %PROGRAM NAME: hw6m (4//9) %PROBLEM %================================== %PROBLEM %====================================== %PROBLEM 3 % %Example 8- p67 %