ENGR 7181 LECTURE NOTES WEEK 5 Dr. Amir G. Aghdam Concordia University

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1 ENGR 78 LETURE NOTES WEEK 5 r. mir G. dam onordia Univrity ilinar Tranformation - W will now introdu anotr mtod of tranformation from -plan to t - plan and vi vra. - Ti tranformation i bad on t trapoidal approximation for intration in T domain. onidr a T tranfr funtion. W av: u t yt Fiur 5.: n intrator a a T LT ytm y y u τ dτ y u u - n t T domain ti approximation an b writtn a t followin quation: y y u u - Tain t -tranform of bot id w an find t tranfr funtion of t T quivalnt modl for an intrator a follow: d - Ti impli tat in ti tranformation w av t followin mappin from t -plan to t -plan and vi vra: 5.a Ltur Not Prpard by mir G. dam

2 - n otr word: 5.b d - Ti tranformation i alld bilinar tranformation and i omtim rfrrd to a Tutin bilinar tranformation. - Not tat unli tp-invariant tranformation bilinar tranformation i a on-toon mappin from t -plan to t -plan and vi vra. Howvr imilar to t tp-invarian mtod ti tranformation alo prrv tability. n otr word any point in t lft alf -plan i mappd into a point inid t unit irl in t -plan and any point in t rit alf -plan i mappd into a point outid t unit irl in t -plan. ny point on t imainary axi of t - plan i mappd to a point on t unit irl in t -plan. - Fiur 5. iv a lar omparion btwn t pol mappin from t -plan to t -plan tp-invariant and bilinar tranformation. - m{} b a d -plan R{}. m{} m{} b Unit irl -plan -plan b d a R{} a -/ R{} abd 0 a b d π 0 π d abd 0 Ltur Not Prpard by mir G. dam

3 - T tat-pa matri of t T quivalnt modl uin bilinar tranformation an b oaind a follow. onidr a T LT ytm wit t followin tatpa quation in t -domain: Fiur 5.: Mappin of t important point Xundr Xtp-invariant U and bilinar tranformation 5. Y X U - Uin 5.a w will av t followin quation in t -domain: X X U Y X U - T tat quation an b rwrittn a follow: X X U - Tu t T quivalnt modl in t tim domain will b: x x x x u u x x u x x u 5. x u - fin w a follow: w : x x x u - Suitutin 5.4 in 5. w will av: w u Tu: w w u u Ltur Not Prpard by mir G. dam

4 Ltur Not Prpard by mir G. dam 4 u w u w u w w - On t otr and t output quation an b writtn a follow: u w u x y - Ti an b ummarid a follow: wr: - T abov formula an b ud only if t matrix i invrtibl i.. if i not an invalu of. - Not tat t tat-pa raliation for t T quivalnt modl i not uniqu. For xampl all of t followin raliation iv t T quivalnt modl for 5.: 5.5

5 Ltur Not Prpard by mir G. dam 5 - Howvr t T quivalnt modl ivn by 5.5 a a baland ain btwn t input and output i.. bot and ar multiplid by t am ain matrix in t T quivalnt modl. MTL ommand y_ddy tutin omput t T quivalnt modl for t T ytm dfind by y_ uin t bilinar tranformation. - T frquny rpon of t ytm undr bilinar tranformation an b oaind a follow: tan - Ti quation man tat on an find t frquny rpon of t T quivalnt modl undr bilinar tranformation from t frquny rpon of t T ytm uin t followin tranformation: tan wr rprnt frquny in t T domain. - Similarly on an writ: tan

6 6 - Ti man tat t low frquny 0 in t T domain i quivalnt to 0 in t T domain and t i frquny ± in t T domain i quivalnt to ± in t T domain. - Sin 0 i quivalnt to 0 ti impli tat t ain of t T ytm and it T quivalnt undr bilinar tranformation ar qual. - t an b aily vrifid tat undr bilinar tranformation a T ytm wit a band-limitd frquny rpon will alway rult in a T quivalnt modl wit band-limitd frquny rpon i..: ' ' ' b - 0 for > b 0 for b < < b b : tan - Exampl 5.: Plot t manitud frquny rpon of t T ytm and it T quivalnt modl for t ytm of Exampl 4.: 0. undr bilinar tranformation for two amplin frquni 0 rad/ and rad/ and ompar t rult wit to oaind uin tp-invariant mtod. - Solution: Uin MTL w will av t followin plot: Ltur Not Prpard by mir G. dam

7 7 Manitud Frquny rad/ Fiur 5.: Frquny rpon of t T ytm of Exampl 5. 0 rad/ Manitud Frquny rad/ 0 rad/ Manitud Frquny rad/ Fiur 5.4: Frquny rpon of t T quivalnt modl for t ytm of Exampl 5. undr bilinar tranformation for 0 rad/ rn urv and rad/ rd urv Ltur Not Prpard by mir G. dam

8 8 - n ontrol ytm on an din a T ontrollr uin any din tniqu and apply any of t dirtiation mtod diud in t la to find t T quivalnt modl for t ontrollr and u it wit t ytm. T anwr to wi diitiation tniqu to u dpnd on your appliation. Non of t tniqu ar prft but in nral t ilinar tranformation map t ntir frquny rpon of t dird filtr t mpul nvariant tranformation prrv t impul-rpon and t Stp nvariant prrv t tprpon. So if you nd frquny prforman u a filtrin a inal your bt bt i to u t ilinar tranformation. f you nd tim prforman your bt bt i to u t mpul nvariant or Stp nvariant tranformation. T matd Z tranform i vry bai and a no ditinuiin fatur. - To av an ida about ow pri diffrnt dirtiation mtod ar w will introdu dirtiation rror nxt w. Rfrn: T. n and.. Frani Optimal Sampld-ata ontrol Sytm availabl at ttp:// G.F. Franlin J.. Powll and M.L. Worman iital ontrol of ynami Sytm rd Edition ddion-wly 998. Ltur Not Prpard by mir G. dam