ECE-320: Linear Control Systems Homework 1. 1) For the following transfer functions, determine both the impulse response and the unit step response.

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1 Due: Mnday Marh 4, 6 at the beginning f la ECE-: Linear Cntrl Sytem Hmewrk ) Fr the fllwing tranfer funtin, determine bth the imule rene and the unit te rene. Srambled Anwer: H ( ) H ( ) ( )( ) ( )( ) ) 8 5 ( ) H ( H 4 4t t t t t / t/ h( t) e in( t) u( t), h( t) e u( t) e u( t) te u( t), h( t) e u( t) e u( t), t t t/ t / h( t) e ( t) u( t) e in( t) u( t), y( t) u( t) e u( t) e u( t), t t t t t y( t) u( t) e in( t) u( t) e ( t) u( t), y( t) e u( t) e u( t) te u( t), 8 4t 4t y( t) u( t) e in(t) ut ( ) e ( t) u( t) ) Fr the fllwing tranfer funtin 5 H ( ) H ( ) H() H ( ) H ( ) By muting the invere Lalae tranfrm hw that the te rene are given by t t t t y( t) e ( t) e in( t) u( t) y( t) e in( t) e ( t) ( ) u t t t 4 8 t 4 t y( t) e in( t) e ( t) u( t) y( t) e in( t) e ( t) u() t 7 7 y() t ( t ) u ( t ) 4 4

2 ) (Man Rule) Fr the blk diagram hwn belw, determine a rrending ignal flw diagram and hw that the led l tranfer funtin i GG G G4 ( GG H GH GG H) Hytem G G H G H G G H 4) (Mdel Mathing) Cnider the fllwing led l ytem, with lant G () and ntrller G (). One way t he the ntrller i t try and make yur led l ytem math a tranfer funtin that yu he (hene the name mdel mathing). Let aume that ur deired led l tranfer funtin, G (), ur lant an be written in term f numeratr and denminatr a N () () N G( ) G( ) D ( ) D ( ) N( ) D( ) Shw that ur ntrller i then G () N ( ) D ( ) N ( ) Nte that there are me retritin here, in that fr imlementatin ure the ntrller mut be table, and it mut be rer. 5) Fr the fllwing ytem, with lant G (), and ntrller G () a) Uing the reult frm rblem, determine the ntrller that the led l ytem mathe a end rder ITAE (Integral f Time and Ablute Errr) timal ytem, i.e., that the led l tranfer funtin i G ().4 ( ) Anwe. G (), nte that there i a le/zer anellatin between the ntrller and the lant and (.4 ) there i a le at zer in the ntrller.

3 b) Shw that the daming rati fr thi ytem i.7, the led l le f thi ytem are at.7 j.74. Fr fater rene huld be large r mall? ) Determine the ntrller that the led l ytem mathe a third rder deadbeat ytem, i.e., that the led l tranfer funtin i G ().9. ( ) An. G (), nte that there i a le/zer anellatin between the ntrller and the (.9. ) lant and there i a le at zer in the ntrller. 6) Cnider the fllwing imle feedbak ntrl blk diagram. The lant i G(). The inut i a unit 4 te. a) Determine the ettling time and teady tate errr f the lant alne (auming there i n feedbak) b) Auming a rrtinal ntrller, G () k, determine the led l tranfer funtin, G () ) Auming a rrtinal ntrller, G () k, determine the value f k the teady tate errr fr a unit te i., and the rrending ettling time fr the ytem. d) Auming a rrtinal ntrller, G () k,determine the value f k the ettling time i.5 end, and the rrending teady tate errr. e) Auming an integral ntrller, G ( ) k /, determine led l tranfer funtin, G () i f) Auming an integral ntrller, G( ) ki /, determine the value f ki the teady tate errr fr a unit te i le than. and the ytem i table. Partial Anwer: T, e.5, k 8, k, T., e.5, k i 4 7) Cnider the fllwing imle feedbak ntrl blk diagram. The lant i G(). 7 a) What i the bandwidth f the lant alne (auming there i n feedbak) b) Auming a rrtinal ntrller, G () k, determine the led l tranfer funtin, G () ) Auming a rrtinal ntrller, G () k, determine the value f k the bandwidth f the led l ytem i 7 rad/e.

4 d) Auming the rrtinal ntrller frm rblem, determine the ettling time and the teady tate errr fr a unit te. Partial Anwer: 7, 5, 7/7, 4/7 8) (Matlab/Simulink) Dwnlad and unmre the file Mdel_Mathing.rar frm the la webite. The file ledl.lx i a Simulink file the fr a imle led l ytem, a hwn belw in in Figure. Figure. Simle feedbak ntrl ytem In thi ytem the lant i the thing we want t ntrl, the ntrller mdifie the behavir f the lant, and we have inluded a limit n the ntrl effrt (the Saturatin blk). Many real ytem have limit n the ntrl effrt, a yu will ee. The file ledl_driver.m i the Matlab driver file that lad arameter and tranfer funtin int the Matlab wrkae fr the Simulink file t ue. Uing thee tw rgram we an lk at me mdel mathing ytem and me examle f what haen when the mdel de nt math the real ytem. Al thi give u a review f Matlab and Simulink. If yu run the rgram a they are yu huld get a rene like that hwn in Figure 4 n the next age. Thi figure hw the rene t the lant (the thing we want t ntrl) in the t grah. The middle grah hw a ntrl ytem uing mdel mathing fr a m inut. In thi grah the red line hw the rene f the led l tranfer funtin we exet t get if the mathematial mdel f the lant i exat, and the red line hw the reult uing the mdel mathing ntrl ytem. Yu huld nte that the ntrl ytem ha a fater rene, maller verht, and a teady tate errr f zer (the utut i equal t the inut in teady tate). The lat grah hw the ntrl effrt f the ytem. Mt ratial ytem have a limit n the allwed ntrl effrt ine real ytem (like am) tend t aturate. a) Mdify the arameter t get the fatet rene yu an withut aturating the ntrl effrt (the ntrl effrt huld remain belw ). Nte that the ntrl effrt i maximum at the beginning and then die dwn, thi i mmn fr ntrller. At thi int ur mdel f the lant and the true lant have the tranfer funtin G ().Turn in yur grah. In the next few art we will lk at examle when the mdel i nt mathed exatly. Fr eah f thee art ue the value f yu determined in art a.

5 b) Aume the true lant ha the tranfer funtin G (). Thi invlve hanging num_g and den_g. Then adjut Tf the ytem reahe teady tate. Rerun the imulatin and turn in yur reult. Yu huld ntie that the mdel mathing ntrller de nt fllw the ideal (the red mdel in the grah), but i de eventually nverge t a ytem with zer teady tate errr. Turn in yur grah. 5 5 ) Aume the true lant ha the tranfer funtin G (). Rerun the imulatin and turn in yur reult. d) Aume the true lant ha the tranfer funtin G (). Rerun the imulatin and turn in. yur reult. e) Aume the true lant ha the tranfer funtin G (). Set the final time t.5 end (Tf =.5) and rerun the imulatin and turn in yur reult. What haen in thi ae? 4 Oen L Plant y (m) Time (e).5 Cledl Mdel Mathing, = 5rad/e y (m).5 Mdel Mathing Mdel Time (e) Cntrl Effrt u (m) Time (e) Figure 4. Rene f a lant and an ITAE led l mdel mathing ytem.