ENGR 78 LECTURE NOTES WEEK Dr. ir G. ga Concoria Univrity DT Equivalnt Tranfr Function for SSO Syt - So far w av tui DT quivalnt tat pac ol uing tp-invariant tranforation. n t ca of SSO yt on can u t following tp to fin t tranfr function of t DT quivalnt ol irctly. Conir a CT LT yt wit t tranfr function g ˆ. Fin t invr Laplac tranfor of y t an rprnt t tp rpon of t CT yt. g ˆ. Ti ti ignal i not by Fin t z-tranfor of y k wr rprnt t apling prio. Ti function i not by Y z an rprnt t tp rpon of t DT quivalnt ol. Divi Y z by t z-tranfor of t unit tp ignal z t tranfr function of t DT quivalnt ol. - T abov tp can b iply own a follow: to fin gˆ z gˆ gˆ z z Zˆ - Eapl.: Fin t DT quivalnt ol for t yt of Eapl. irctly fro t tranfr function of t CT yt. - Solution: W will av: On t otr an: ˆ g z z z ˆ gˆ Z ˆ Z Lctur Not Prpar by ir G. ga
ˆ Z z z z Tu: ˆ g z z z wic i t a rult obtain fro t DT quivalnt tat pac ol. Controllability of Syt - Controllability: control yt i fin to b controllabl if for any arbitrary initial tat tr it an uncontrain control ignal u tat can bring t tat to t origin of t tat pac in a finit ti. - Racability: control yt i fin to b racabl if tr it an uncontrain control ignal u tat can tranfr t tat of t yt fro t origin to any arbitrary point in t tat pac in a finit ti. - T abov finition of controllability an racability ar vry gnral an can b u for any CT an DT yt. - Conir now a CT LT yt wit t following tat-pac rprntation: & t t u t y t C t Du t. wr n t R i t tat vctor u t R i t input vctor an y t R i t output vctor. atric. n n R n R C R n an D R ar contant - For CT LT yt t concpt of controllability an racability ar quivalnt. n otr wor tranfrring t tat fro any arbitrary point to t origin an tranfrring t tat fro t origin to any arbitrary point ar quivalnt. n fact if t abov yt i controllabl or racabl on can u an uncontrain control ignal to tranfr t tat of t yt fro any arbitrary point to any otr arbitrary point in t tat pac in finit ti. Lctur Not Prpar by ir G. ga
- Controllability Tt: T CT LT yt. i controllabl iff any of t following two conition ol: i T following controllability atri i full-rank: n [ L ] ii T following atri i full-rank for any copl calar : [ ] - T con tt i rfrr to a Popov-lvitc-Hautu PH tt. - Sinc t atri i alway full-rank if i not an ignvalu of t rank conition of ii uffic to b cck only for qual to t ignvalu of. - T concpt of controllability an racability for DT LT yt ar not quivalnt. Ti i own in t following apl. - Eapl.: Conir t following DT LT yt: [ k ] [ u[ [ k ] [ Cck t controllability an racability of ti yt. - Solution: To cck controllability w au tat t final tat i qual to zro t origin of t tat pac. W will av: [ k ] [ k ] [ [ u[ Ti ipli tat t tat of t yt can b tranfrr fro any arbitrary [] point [] in t tat pac to t origin in on tp by cooing t [] following control ignal u [ ] [] [] Ti an tat t yt i controllabl. u now tat t initial tat i t origin of t tat pac i.. W will av: []. Lctur Not Prpar by ir G. ga
4 [] [] [] [] u[] Ti ipli tat tarting fro t origin t con tat variabl [ ] can b tranfrr to any arbitrary point in t tat pac in on tp but [ ] will tay k k in t origin an cannot b ift to any arbitrary point in a finit nubr of tp. Ti an tat t yt i not racabl. - t can b conclu fro Eapl. tat controllability an racability of DT LT yt ar iffrnt. - Conir a DT LT yt wit t following tat pac rprntation: [ k ] y[ C [ [ D u[ u[. wr n [ R i t tat vctor u [ R i t input vctor an y[ R i t output vctor. R n n R n C R n an D R ar contant atric. For a DT LT yt wit a non-ingular tat atri controllability an racability ar quivalnt. t i to b not tat if a DT yt i obtain by icrtizing a CT yt t rultant tat atri will b noningular. - n ti cour w will rfr to a DT LT yt a controllabl if tr it an uncontrain control ignal fin ovr a finit nubr of apl uc tat tarting fro any arbitrary initial tat t tat can b tranfrr to any ir tat in finit nubr of apl ti i in fact quivalnt to racability. - Controllability tt for t DT LT yt. i iilar to tat of t CT countrpart.. T DT LT yt. i controllabl iff any of t following two conition ol: i T following controllability atri i full-rank: n [ L ] ii T following atri i full-rank for any copl calar z : [ z ] Lctur Not Prpar by ir G. ga
5 Sinc t atri z i alway full-rank if z i not an ignvalu of t rank conition of ii uffic to b cck only for z qual to t ignvalu of. Obrvability of Syt - yt i fin to b obrvabl if vry initial tat can b trin fro t obrvation itory of output in a finit ti givn t control ignal. - Obrvability Tt: T CT LT yt. i obrvabl iff any of t following two conition ol: i T following obrvability atri i full-rank: C C M n C ii T following atri i full-rank for any copl calar : - T con tt i rfrr to a Popov-lvitc-Hautu PH tt. - Obrvability tt for t DT LT yt. i iilar to tat of t CT countrpart.. T DT LT yt. i obrvabl iff any of t following two conition ol: i T following obrvability atri i full-rank: C C C M C ii T following atri i full-rank for any copl calar z : C n z Lctur Not Prpar by ir G. ga
6 - n gnral controllability tll u about t faibility to control a plant wil obrvability tll u about wtr it i poibl to know wat i appning in a givn yt by obrving it output. - yt wo uncontrollabl o ar tabl i rfrr to a a tabilizabl yt. Likwi a yt wo unobrvabl o ar tabl i rfrr to a a tctabl yt. Effct of Sapling - W will now invtigat t ffct of apling in icrtization unr tpinvariant tranforation. Can w av a controllabl/obrvabl CT yt wic a an uncontrollabl/unobrvabl DT quivalnt an vic vra? - Conir t following t-up []: u [ H ut G yt S y [ Figur.: apl-ata yt u for invtigating t apling ffct - u tat t apling frquncy in ra/c i not by : wr i t apling prio. Lt t CT yt G av t following tranfr function: gˆ - u now tat t unit tp ignal i appli at u [. W will av: u[ u[ u t u t Y y t in t t - y [ i t apl vrion of t tp-rpon y t wit t apling frquncy a follow: y[ y t in k in k ink k t kt Lctur Not Prpar by ir G. ga
Lctur Not Prpar by ir G. ga 7 - Ti an tat SGH G i a zro yt an ˆ z g. On t otr an by icrtizing a zro yt w will alo gt G. n otr wor uing t abov apling prio icrtization of t givn CT yt will not b uniqu. - n tr of tat ol w will av: ˆ D C g - t i to b not tat t ignvalu of t atri wic ar in fact t pol of t CT tranfr function too ar locat at j ±. - T DT quivalnt ol uing t tp-invariant tranforation can b obtain for ti yt a follow: co in in co } { t t t t L L t co in in co τ τ - Ti an tat altoug t pair i controllabl in t DT quivalnt ol i not. n otr wor t particular coic of apling prio for ti apl i not goo in t n tat it o not prrv controllability. Ti apling prio i call patological. - Dfinition. []: Conir a CT yt wit t following tat atric: D C
8 T apling frquncy i patological wit rpct to if a two ignvalu wit qual ral part but tir iaginary part iffr by an intgr ultipl of.g. α ± j k k K. Otrwi t apling frquncy i non-patological. - Eapl. []: Conir t CT LT yt. an au tat t ignvalu of t atri ar: ± j ± j Fin t patological apling frqunci. - Solution: W av two t of ignvalu wit qual ral part. an 4 ± j all av a ral part qual to an 5 6 ± j bot av a ral part qual to. T iffrnc btwn t iaginary part of t firt t { 4 } ar an an t iffrnc btwn t iaginary part of t con t } i 4. Tu all apling frqunci in t following t ar patological: { 5 6 4 : k K U : k K U : k K k k k Sinc 4 i iviibl by an t t of all patological apling frqunci can b iplifi a follow: 4 : k K k t i to b not tat t uppr boun for t lnt of t abov t i qual to 4. Tu t apling frquncy will b non-patological if ufficint conition it i gratr tan 4. - Patological apling can alo b icu uing pctral apping tor. t i known tat if a function f i analytic at t ignvalu of tn t ignvalu of f ar qual to f wr rprnt t ignvalu of. Ti an tat t ignvalu of ar qual to. Sinc t Lctur Not Prpar by ir G. ga
9 function i prioic wit prio j two CT o wit a itanc qual to k along t iaginary ai will b app to on point in t icrtiz ol. Ti an tat t ap i not on-to-on an t inforation about t o will b lot if t apling prio i patological. - W will now icu t ffct of apling on controllability an obrvability. - Tor. []: u tat t CT yt. i icrtiz wit a nonpatological apling frquncy to obtain t DT ol.. Tn: controllabl controllabl C obrvabl C obrvabl - Proof []: W will prov it for obrvability firt iplication. T proof for controllability i iilar. u tat t apling frquncy i non-patological an C i obrvabl. To prov tat C i obrvabl w will ow tat all t ignvalu of if tn: ar obrvabl. Uing t PH tt w ut ow tat rank n C rank n C wr an rprnt t ignvalu of an rpctivly. Dfin t following function: g Ti function i analytic for all not tat vn for t function will b analytic u to t cancllation of t pol wit a zro at tat point. Morovr zro of a follow: g will b locat at { } :. Ti can alo b writtn Lctur Not Prpar by ir G. ga
or: { : jk k ± ± ± K} { : jk k ± ± ± K} Ti ipli tat for a non-patological apling prio tr i no intrction btwn t t of zro of g an t t of ignvalu of. Sinc t ignvalu of t atri g i qual to g i i K n wr i rprnt t ignvalu of trfor i not an ignvalu of g. Ti an tat t atri g i invrtibl. On t otr an: g g Sinc tu: Sinc t atri C g C g i invrtibl an inc t invrtibility of g an g ar quivalnt on can conclu tat: rank rank C C Ti coplt t proof. - t i to b not tat non-patological apling i only a ufficint conition for controllability an obrvability of DT quivalnt ol. - T following tor ow tat w cannot gain controllability or obrvability by apling. - Tor. []: f t CT yt. i icrtiz to obtain t DT ol. tn: not controllabl not controllabl C not obrvabl C not obrvabl Lctur Not Prpar by ir G. ga
Lctur Not Prpar by ir G. ga - Proof []: W will only proof t firt iplication. T proof for unobrvability i iilar. f t pair i not controllabl w will av: [ ] n < rank for an ignvalu of. Ti an tat tr i a nonzro vctor uc tat: [ ] n otr wor: Tu: an: tc. Now fro: L!! t can b conclu tat:!!! L L Trfor: an:.
Ti ipli tat: wic i quivalnt to: t t t t fro quation. rank [ ] [ ] n < Ti an tat i an uncontrollabl ignvalu of. - T rult for tabilizability an tctability ar vry iilar to controllability an obrvability. n otr wor on can ay: - Tor.: u tat t CT yt. i icrtiz to obtain t DT ol.. Tn: not tabilizabl not tabilizabl C not tctabl C not tctabl Furtror if t apling frquncy i non-patological tn: tabilizabl tabilizabl C tctabl C tctabl Rfrnc: [] T. Cn an.. Franci Optial Sapl-Data Control Syt availabl at ttp://www.control.utoronto.ca/popl/prof/franci/_book.pf. [] G.F. Franklin J.D. Powll an M.L. Workan Digital Control of Dynaic Syt r Eition ion-wly 998. Lctur Not Prpar by ir G. ga