!ෝપෝ ෝપ ෝ પપ ෝ પ પપ પ ෝપપ ෝ પપ ! પપ પ ෝપ ! ෝ ෝ !ෝ ෝ પ Þo પ
!NNNIIIVVVEEERRRSSSIIIDDDAAADDDEEE FFFEEEDDDEEERRRAAALLL DDDOOO MMMAAARRRAAANNNHHHÃÃÃOOOCCCEEENNNTTTRRROOO DDDEEE CCCIIIEEENNNCCCIIIAAASSS EEEXXXAAATTTAAASSS EEE TTTEEECCCNNNOOOLLLOOOGGGIIIAAADDDEEEPPPAAARRRTTTAAAMMMEEENNNTTTOOO DDDEEE EEENNNGGGEEENNNHHHAAARRRIIIAAACCC!RRRSSSOOO::: EEENNNGGGEEENNNHHHAAARRRIIIAAA EEELLLÉÉÉTTTRRRIIICCCAAADDDIIISSSCCCIIIPPPLLLIIINNNAAA::: CCCOOONNNTTTRRROOOLLLEEE III
!RRROOOJJJEEETTTOOO DDDEEE CCCOOONNNTTTRRROOOLLLEEE ––– CCCOOONNNTTTRRROOOLLLEEE DDDEEE !ÊÊÊNNNDDDUUULLLOOO IIINNNVVVEEERRRTTTIIIDDDOOO CCCOOOMMM BBBAAASSSEEE MMMÓÓÓVVVEEELLL
JJJaaannneeeiiirrrooo dddeee 222000111000SSSãããooo LLLuuuííísss ––– MMMAAA
222
!NNNIIIVVVEEERRRSSSIIIDDDAAADDDEEE FFFEEEDDDEEERRRAAALLL DDDOOO MMMAAARRRAAANNNHHHÃÃÃOOOCCCEEENNNTTTRRROOO DDDEEE CCCIIIEEENNNCCCIIIAAASSS EEEXXXAAATTTAAASSS EEE TTTEEECCCNNNOOOLLLOOOGGGIIIAAADDDEEEPPPAAARRRTTTAAAMMMEEENNNTTTOOO DDDEEE EEENNNGGGEEENNNHHHAAARRRIIIAAACCC!RRRSSSOOO::: EEENNNGGGEEENNNHHHAAARRRIIIAAA EEELLLÉÉÉTTTRRRIIICCCAAADDDIIISSSCCCIIIPPPLLLIIINNNAAA::: CCCOOONNNTTTRRROOOLLLEEE III
!RRROOOJJJEEETTTOOO DDDEEE CCCOOONNNTTTRRROOOLLLEEE ––– CCCOOONNNTTTRRROOOLLLEEE DDDEEE !ÊÊÊNNNDDDUUULLLOOO IIINNNVVVEEERRRTTTIIIDDDOOO CCCOOOMMM BBBAAASSSEEE MMMÓÓÓVVVEEELLL
AAAllluuunnnooo::: FFFrrraaannnccciiissscccooo dddaaasss CCChhhaaagggaaasss LLLiiinnnhhhaaarrreeesss SSSiiilllvvvaaa JJJúúúnnniiiooorrr CCCóóódddiiigggooo::: EEEEEE000777111111111-‐-‐-666777
JJJaaannneeeiiirrrooo dddeee 222000111000SSSãããooo LLLuuuííísss ––– MMMAAA
333
SSSuuummmááárrriiioooiii... LLLiiissstttaaa dddeee SSSiiiggglllaaasss qqquuueee ssseeerrrãããooo uuutttiiillliiizzzaaadddaaasss nnnooo dddeeecccooorrrrrreeerrr dddooo ppprrrooojjjeeetttooo ............................................................................................................................................. 444
iiiiii... LLLiiissstttaaa dddeee fffiiiggguuurrraaasss .................................................................................................................................................................................................................................................................................................................................................................. 444
111... TTTííítttuuulllooo ............................................................................................................................................................................................................................................................................................................................................................................................................... 555
222... OOObbbjjjeeetttiiivvvooosss dddooo PPPrrrooojjjeeetttooo....................................................................................................................................................................................................................................................................................................................................... 555
333... IIInnntttrrroooddduuuçççãããooo....................................................................................................................................................................................................................................................................................................................................................................................... 666
444... MMMooodddeeelllaaagggeeemmm dddooo PPPrrrooojjjeeetttooo........................................................................................................................................................................................................................................................................................................................... 777
444...111... MMMooodddeeelllaaagggeeemmm dddooo PPPêêênnnddduuulllooo ............................................................................................................................................................................................................................................................................................. 777
444...222... MMMooodddeeelllaaagggeeemmm dddooo CCCaaarrrrrrooo............................................................................................................................................................................................................................................................................................................ 999
444...333... MMMooodddeeelllaaagggeeemmm dddooo SSSiiisssttteeemmmaaa................................................................................................................................................................................................................................................................................................ 999
555... SSSiiimmmuuulllaaaçççãããooo.................................................................................................................................................................................................................................................................................................................................................................................... 111000
555...111... PPPaaarrrâââmmmeeetttrrrooosss dddooo MMMooodddeeelllooo............................................................................................................................................................................................................................................................................................. 111000
555...222... SSSiiisssttteeemmmaaa eeemmm MMMaaalllhhhaaa AAAbbbeeerrrtttaaa.............................................................................................................................................................................................................................................................................. 111000
555...222...111... RRReeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo................................................................................................................................................................................................................................................................................. 111111
555...222...222... RRReeessspppooossstttaaa aaaooo DDDeeegggrrraaauuu .................................................................................................................................................................................................................................................................................... 111222
555...222...333... RRReeessspppooossstttaaa eeemmm fffrrreeeqqqüüüêêênnnccciiiaaa ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee......................................................................................................................................................... 111333
555...333... SSSiiisssttteeemmmaaa eeemmm MMMaaalllhhhaaa FFFeeeccchhhaaadddaaa cccooommm CCCooonnntttrrrooolllaaadddooorrr PPPIIIDDD ............................................................................................................................................................ 111444
555...333...111... RRReeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo................................................................................................................................................................................................................................................................................. 111555
555...333...222... RRReeessspppooossstttaaa aaaooo DDDeeegggrrraaauuu .................................................................................................................................................................................................................................................................................... 111666
555...333...333... RRReeessspppooossstttaaa eeemmm fffrrreeeqqqüüüêêênnnccciiiaaa ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee......................................................................................................................................................... 111777
555...333...444... GGGrrráááfffiiicccooo dddooosss PPPóóólllooosss eee ZZZeeerrrooosss ......................................................................................................................................................................................................................................................... 111888
555...444... SSSiiisssttteeemmmaaa cccooommm CCCooommmpppeeennnsssaaadddooorrr LLLeeeaaaddd-‐-‐-lllaaaggg (((MMMééétttooodddooo dddooo LLLuuugggaaarrr dddaaasss RRRaaaííízzzeeesss))) ..................................................................... 111999
555...444...111... RRReeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo................................................................................................................................................................................................................................................................................. 222111
555...444...222... RRReeessspppooossstttaaa aaaooo DDDeeegggrrraaauuu .................................................................................................................................................................................................................................................................................... 222222
555...444...333... RRReeessspppooossstttaaa eeemmm FFFrrreeeqqquuuêêênnnccciiiaaa ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee ...................................................................................................................................................... 222333
555...555... SSSiiisssttteeemmmaaa CCCooonnntttrrrooolllaaadddooo aaatttrrraaavvvééésss dddeee AAAlllooocccaaaçççãããooo dddeee PPPóóólllooosss ......................................................................................................................................................... 222444
555...555...111... RRReeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo................................................................................................................................................................................................................................................................................. 222666
555...555...222... RRReeessspppooossstttaaa aaaooo DDDeeegggrrraaauuu .................................................................................................................................................................................................................................................................................... 222666
555...555...333... RRReeessspppooossstttaaa eeemmm fffrrreeeqqqüüüêêênnnccciiiaaa ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee......................................................................................................................................................... 222777
666... SSScccrrriiipppttt dddooo MMMAAATTTLLLAAABBB ............................................................................................................................................................................................................................................................................................................................................. 222888
777... CCCooonnncccllluuusssãããooo.................................................................................................................................................................................................................................................................................................................................................................................... 333111
888... RRReeefffeeerrrêêênnnccciiiaaasss ........................................................................................................................................................................................................................................................................................................................................................................... 333222
444
iii... LLLiiissstttaaa dddeee SSSiiiggglllaaasss qqquuueee ssseeerrrãããooo uuutttiiillliiizzzaaadddaaasss nnnooo dddeeecccooorrrrrreeerrr dddooo ppprrrooojjjeeetttooo
XXXcccggg === dddiiissstttââânnnccciiiaaa dddooo ccceeennntttrrrooo dddaaa ooorrriiigggeeemmm aaatttééé ooo ccceeennntttrrrooo dddeee mmmaaassssssaaa dddooo pppêêênnnddduuulllooo eeemmm rrreeelllaaaçççãããooo aaaooo eeeiiixxxooo xxxYYYcccggg === dddiiissstttââânnnccciiiaaa dddooo ccceeennntttrrrooo dddaaa ooorrriiigggeeemmm aaatttééé ooo ccceeennntttrrrooo dddeee mmmaaassssssaaa dddooo pppêêênnnddduuulllooo eeemmm rrreeelllaaaçççãããooo aaaooo eeeiiixxxooo yyyHHH === fffooorrrçççaaa nnnaaa dddiiirrreeeçççãããooo hhhooorrriiizzzooonnntttaaalllVVV === fffooorrrçççaaa nnnaaa dddiiirrreeeçççãããooo vvveeerrrtttiiicccaaalllmmm === mmmaaassssssaaa dddooo pppêêênnnddduuulllooo222lll === cccooommmppprrriiimmmeeennntttooo dddooo pppêêênnnddduuulllooo
ââânnnggguuulllooo dddooo pppêêênnnddduuulllooo eeemmm rrreeelllaaaçççãããooo ààà llliiinnnhhhaaa vvveeerrrtttiiicccaaalllMMM===mmmaaassssssaaa dddooo cccaaarrrrrrooouuu === fffooorrrçççaaa aaapppllliiicccaaadddaaa aaaooo cccaaarrrrrroooiii === mmmooommmeeennntttooo dddeee iiinnnééérrrccciiiaaa dddooo pppêêênnnddduuulllooo (((mmmlll²²///333 pppaaarrraaa uuummmaaa hhhaaasssttteee uuunnniiifffooorrrmmmeee)))
iiiiii... LLLiiissstttaaa dddeee fffiiiggguuurrraaasss
FFFiiiggguuurrraaa 111 ––– MMMooodddeeelllooo dddooo SSSiiisssttteeemmmaaa CCCaaarrrrrrooo-‐-‐- pppêêênnnddduuulllooo iiinnnvvveeerrrtttiiidddoooFFFiiiggguuurrraaa 222 ––– SSSiiisssttteeemmmaaa dddeee cccoooooorrrdddeeennnaaadddaaasss dddooo cccooonnnjjjuuunnntttooo dddooo pppêêênnnddduuulllooo iiinnnvvveeerrrtttiiidddoooFFFiiiggguuurrraaa 333 ––– DDDiiiaaagggrrraaammmaaa dddeee cccooorrrpppooo llliiivvvrrreee dddooo sssiiisssttteeemmmaaaFFFiiiggguuurrraaa 444 ––– DDDiiiaaagggrrraaammmaaa dddeee bbblllooocccooosss dddooo sssiiisssttteeemmmaaa eeemmm mmmaaalllhhhaaa aaabbbeeerrrtttaaa FFFiiiggguuurrraaa 555 ––– RRReeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo nnnooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa aaabbbeeerrrtttaaaFFFiiiggguuurrraaa 666 ––– RRReeessspppooossstttaaa aaaooo dddeeegggrrraaauuu nnnooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa aaabbbeeerrrtttaaaFFFiiiggguuurrraaa 777 ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee nnnooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa aaabbbeeerrrtttaaaFFFiiiggguuurrraaa 888 ––– DDDiiiaaagggrrraaammmaaa dddeee bbblllooocccooosss dddooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa fffeeeccchhhaaadddaaa cccooommm cccooonnntttrrrooolllaaadddooorrr !IIIDDDFFFiiiggguuurrraaa 999 ––– RRReeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo nnnooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa fffeeeccchhhaaadddaaa cccooommm cccooonnntttrrrooolllaaadddooorrr !IIIDDDFFFiiiggguuurrraaa 111000 ––– RRReeessspppooossstttaaa aaaooo dddeeegggrrraaauuu nnnooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa fffeeeccchhhaaadddaaa cccooommm cccooonnntttrrrooolllaaadddooorrr !IIIDDD FFFiiiggguuurrraaa 111111 ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee nnnooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa fffeeeccchhhaaadddaaa cccooommm cccooonnntttrrrooolllaaadddooorrr !IIIDDDFFFiiiggguuurrraaa 111222 ––– GGGrrráááfffiiicccooo dddooosss pppóóólllooosss eee zzzeeerrrooosss nnnooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa fffeeeccchhhaaadddaaa cccooommm cccooonnntttrrrooolllaaadddooorrr !IIIDDDFFFiiiggguuurrraaa 111333 ––– GGGrrráááfffiiicccooo dddaaasss rrraaaííízzzeeesss dddooo sssiiisssttteeemmmaaa cccooommm cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaagggFFFiiiggguuurrraaa 111444 ––– ZZZoooooommm dddooo gggrrráááfffiiicccooo dddaaasss rrraaaííízzzeeesss dddooo sssiiisssttteeemmmaaa cccooommm cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaagggFFFiiiggguuurrraaa 111555 ––– RRReeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo dddooo sssiiisssttteeemmmaaa cccooommm cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaagggFFFiiiggguuurrraaa 111666 ––– RRReeessspppooossstttaaa aaaooo dddeeegggrrraaauuu dddooo sssiiisssttteeemmmaaa cccooommm cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaagggFFFiiiggguuurrraaa 111777 ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee dddooo sssiiisssttteeemmmaaa cccooommm cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaagggFFFiiiggguuurrraaa 111888 ––– GGGrrráááfffiiicccooo dddooosss pppóóólllooosss eee zzzeeerrrooosss nnnooo sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa aaabbbeeerrrtttaaaFFFiiiggguuurrraaa 111999 ––– GGGrrráááfffiiicccooo dddooosss pppóóólllooosss eee zzzeeerrrooosss nnnooo sssiiisssttteeemmmaaa cccooommm aaalllooocccaaaçççãããooo dddeee pppóóólllooosssFFFiiiggguuurrraaa 222000 ––– RRReeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo nnnooo sssiiisssttteeemmmaaa cccooommm aaalllooocccaaaçççãããooo dddeee pppóóólllooosssFFFiiiggguuurrraaa 222111 ––– RRReeessspppooossstttaaa aaaooo dddeeegggrrraaauuu nnnooo sssiiisssttteeemmmaaa cccooommm aaalllooocccaaaçççãããooo dddeee pppóóólllooosssFFFiiiggguuurrraaa 222222 ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee nnnooo sssiiisssttteeemmmaaa cccooommm aaalllooocccaaaçççãããooo dddeee pppóóólllooosss
555
111... TTTííítttuuulllooo
CCCooonnntttrrrooollleee dddeee uuummm pppêêênnnddduuulllooo iiinnnvvveeerrrtttiiidddooo aaatttrrraaavvvééésss dddeee uuummmaaa bbbaaassseee mmmóóóvvveeelll cccooonnntttrrrooolllaaadddaaa...
222... OOObbbjjjeeetttiiivvvooosss dddooo PPPrrrooojjjeeetttooo
OOO ppprrrooojjjeeetttooo cccooonnnsssiiisssttteee eeemmm uuummm cccaaarrrrrrooo,,, qqquuueee ééé iiimmmpppuuulllsssiiiooonnnaaadddooo pppooorrr uuummmaaa fffooorrrçççaaa FFF,,, cccooommm uuummm pppêêênnnddduuulllooo iiinnnvvveeerrrtttiiidddooo fffiiixxxaaadddooo pppeeelllaaa bbbaaassseee,,, cccooommmooo pppooodddeee ssseeerrr ooobbbssseeerrrvvvaaadddooo nnnaaa fffiiiggguuurrraaa aaabbbaaaiiixxxooo:::
FFFiiiggguuurrraaa 111
OOO ppprrriiimmmeeeiiirrrooo ooobbbjjjeeetttiiivvvooo ééé aaatttrrraaavvvééésss dddaaa fffooorrrçççaaa aaapppllliiicccaaadddaaa nnnooo cccaaarrrrrrooo,,, eeevvviiitttaaarrr qqquuueee ooo pppêêênnnddduuulllooo cccaaaiiiaaa,,, eeennntttãããooo ssseee dddeeevvveee cccrrriiiaaarrr uuummmaaa ppplllaaannntttaaa qqquuueee mmmooodddeeellleee eeesssssseee sssiiisssttteeemmmaaa ttteeennndddooo cccooommm eeennntttrrraaadddaaa dddooo sssiiisssttteeemmmaaa aaa fffooorrrçççaaa aaapppllliiicccaaadddaaa sssooobbbrrreee ooo cccaaarrrrrrooo,,, uuu,,, eee sssaaaííídddaaa ooo ââânnnggguuulllooo ...
DDDeeepppoooiiisss dddeee cccrrriiiaaadddooo aaa ppplllaaannntttaaa,,, fffaaazzzeeerrr uuummmaaa aaannnááállliiissseee dddeeesssssseee sssiiisssttteeemmmaaa cccooommm rrreeelllaaaçççãããooo àààsss rrreeessspppooossstttaaasss aaaooo iiimmmpppuuulllsssooo,,, aaaooo dddeeegggrrraaauuu,,, fffaaazzzeeerrr ooo dddiiiaaagggrrraaammmaaa dddeee BBBooodddeee cccooommm eee ssseeemmm ooo cccooonnntttrrrooolllaaadddooorrr !IIIDDD,,, pppaaarrraaa aaa pppaaarrrtttiiirrr dddaaaííí vvveeerrriiifffiiicccaaarrr ssseee hhhááá ooouuu nnnãããooo aaa nnneeeccceeessssssiiidddaaadddeee dddooo mmmeeesssmmmooo...
TTTaaammmbbbééémmm aaannnaaallliiisssaaarrr ooo ttteeemmmpppooo dddeee aaacccooommmooodddaaaçççãããooo (((SSSeeettttttllliiinnnggg TTTiiimmmeee))),,, ttteeemmmpppooo dddeee pppiiicccooo (((!eeeaaakkk TTTiiimmmeee))),,, MMMáááxxxiiimmmooo sssooobbbrrreee-‐-‐-sssiiinnnaaalll,,, ooo gggaaannnhhhooo,,, ooosss lllooocccaaaiiisss dddaaa rrraaaííízzzeeesss,,, dddeeennntttrrreee ooouuutttrrraaasss iiinnnfffooorrrmmmaaaçççõõõeeesss qqquuueee ssseeerrrãããooo dddiiissspppooossstttaaasss aaabbbaaaiiixxxooo...
666
333... IIInnntttrrroooddduuuçççãããooo
OOO pppêêênnnddduuulllooo iiinnnvvveeerrrtttiiidddooo ééé uuummm mmmeeecccaaannniiisssmmmooo cccooommm cccaaarrraaacccttteeerrríííssstttiiicccaaa dddiiinnnâââmmmiiicccaaa iiinnntttrrriiinnnssseeecccaaammmeeennnttteee iiinnnssstttááávvveeelll eee rrreeeppprrreeessseeennntttaaa uuummmaaa ppplllaaatttaaafffooorrrmmmaaa úúútttiiilll pppaaarrraaa ooo eeessstttuuudddooo dddeee mmmuuuiiitttooosss ooouuutttrrrooosss mmmeeecccaaannniiisssmmmooosss cccooommmpppllleeexxxooosss cccooommmooo ooo mmmooodddeeelllooo dddeee cccooonnntttrrrooollleee dddeee pppooosssiiiçççãããooo dddeee uuummm fffoooggguuueeettteee nnnaaa fffaaassseee dddeee lllaaannnçççaaammmeeennntttooo (((aaannnaaallliiisssaaannndddooo sssooommmeeennnttteee 222DDD))),,, cccooonnntttrrrooollleee dddeee mmmííísssssseeeiiisss eee eeetttccc... UUUmmmaaa aaannnaaalllooogggiiiaaa sssiiimmmpppllleeesss ééé aaa bbbrrriiinnncccaaadddeeeiiirrraaa dddeee eeeqqquuuiiillliiibbbrrraaarrr uuummm lllááápppiiisss ooouuu uuummm cccaaabbbooo dddeee vvvaaassssssooouuurrraaa nnnaaa pppooonnntttaaa dddooosss dddeeedddooosss... !aaarrraaa cccooonnnssseeeggguuuiiirrr uuummmaaa cccooonnndddiiiçççãããooo rrreeelllaaatttiiivvvaaammmeeennnttteee eeessstttááávvveeelll ééé nnneeeccceeessssssááárrriiiooo fffiiicccaaarrr cccooonnnssstttaaannnttteeemmmeeennnttteee mmmooovvveeennndddooo aaa mmmãããooo dddeee fffooorrrmmmaaa aaa mmmaaannnttteeerrr ooo eeeiiixxxooo dddooo cccaaabbbooo dddeee vvvaaassssssooouuurrraaa nnnaaasss ppprrroooxxxiiimmmiiidddaaadddeeesss dddaaa sssuuuaaa pppooosssiiiçççãããooo vvveeerrrtttiiicccaaalll...
UUUmmm pppêêênnnddduuulllooo iiinnnvvveeerrrtttiiidddooo tttííípppiiicccooo ééé uuummm dddiiissspppooosssiiitttiiivvvooo fffííísssiiicccooo qqquuueee ssseee cccooonnnsssiiisssttteee dddeee uuummmaaa bbbaaarrrrrraaa ccciiilllííínnndddrrriiicccaaa,,, uuusssuuuaaalllmmmeeennnttteee mmmeeetttááállliiicccaaa,,, aaa qqquuuaaalll ééé llliiivvvrrreee pppaaarrraaa mmmooovvviiimmmeeennntttaaarrr eeemmm tttooorrrnnnooo dddeee uuummm pppooonnntttooo fffiiixxxooo... EEEsssssseee pppooonnntttooo ééé mmmooonnntttaaadddooo eeemmm uuummm cccaaarrrrrrooo qqquuueee pppooorrr sssuuuaaa vvveeezzz ééé llliiivvvrrreee pppaaarrraaa mmmooovvveeerrr nnnaaa dddiiirrreeeçççãããooo hhhooorrriiizzzooonnntttaaalll... OOO cccaaarrrrrrooo ééé aaaccciiiooonnnaaadddooo pppooorrr uuummm mmmoootttooorrr qqquuueee pppooodddeee eeexxxeeerrrccceeerrr uuummmaaa fffooorrrçççaaa vvvaaarrriiiááávvveeelll nnnooo dddeeessslllooocccaaammmeeennntttooo dddooo mmmeeesssmmmooo... AAA hhhaaasssttteee nnnaaatttuuurrraaalllmmmeeennnttteee ttteeennndddeee aaa cccaaaiiirrr,,, pppoooiiisss sssuuuaaa pppooosssiiiçççãããooo vvveeerrrtttiiicccaaalll ééé uuummmaaa cccooonnndddiiiçççãããooo dddeee eeeqqquuuiiilllíííbbbrrriiiooo iiinnnssstttááávvveeelll... UUUsssaaa-‐-‐-ssseee uuummmaaa mmmaaalllhhhaaa dddeee cccooonnntttrrrooollleee cccooommm ooo ooobbbjjjeeetttiiivvvooo dddeee eeessstttaaabbbiiillliiizzzaaarrr aaa hhhaaasssttteee dddooo pppêêênnnddduuulllooo nnnaaa pppooosssiiiçççãããooo vvveeerrrtttiiicccaaalll... IIIssssssooo ééé pppooossssssííívvveeelll eeexxxeeerrrccceeennndddooo-‐-‐-ssseee uuummmaaa fffooorrrçççaaa aaatttrrraaavvvééésss dddooo mmmooovvviiimmmeeennntttooo dddooo cccaaarrrrrrooo qqquuueee ttteeennndddeee aaa cccooonnntttrrraaabbbaaalllaaannnçççaaarrr aaa dddiiinnnâââmmmiiicccaaa nnnaaatttuuurrraaalll dddooo pppêêênnnddduuulllooo... AAA iiinnnttteeennnsssiiidddaaadddeee dddaaa fffooorrrçççaaa pppooodddeee ssseeerrr cccooonnntttrrrooolllaaadddaaa aaa pppaaarrrtttiiirrr dddaaa iiinnnfffooorrrmmmaaaçççãããooo dddaaa pppooosssiiiçççãããooo aaannnggguuulllaaarrr dddaaa hhhaaasssttteee...
777
444... MMMooodddeeelllaaagggeeemmm dddooo PPPrrrooojjjeeetttooo
CCCooommmooo pppooodddeee ssseeerrr vvviiissstttooo nnnaaa FFFiiiggguuurrraaa 111,,, ooo mmmooodddeeelllooo eeemmm qqquuueeessstttãããooo pppooossssssuuuiii uuummm cccaaarrrrrrooo eee uuummm pppêêênnnddduuulllooo,,, eeennntttãããooo aaa mmmooodddeeelllaaagggeeemmm pppooodddeee ssseeerrr fffeeeiiitttooo eeemmm ddduuuaaasss eeetttaaapppaaasss pppaaarrraaa fffaaaccciiillliiitttaaarrr ooo ppprrroooccceeessssssooo... UUUmmmaaa pppaaarrraaa ooo cccaaarrrrrrooo,,, eee ooouuutttrrraaa pppaaarrraaa ooo pppêêênnnddduuulllooo,,, cccooonnnsssiiidddeeerrraaannndddooo qqquuueee aaa sssaaaííídddaaa dddaaa fffuuunnnçççãããooo dddooo cccaaarrrrrrooo,,, ooo dddeeessslllooocccaaammmeeennntttooo xxx,,, ssseeerrrááá aaa eeennntttrrraaadddaaa dddaaa fffuuunnnçççãããooo dddooo pppêêênnnddduuulllooo...
444...111...MMMooodddeeelllaaagggeeemmm dddooo PPPêêênnnddduuulllooo
FFFiiiggguuurrraaa 222
!eeelllaaa fffiiiggguuurrraaa 222,,, ttteeemmmooosss qqquuueee:::
cg pX X l sen
coscg pY Y l2
2 cgdH m Xdt2
2 cgdV m Y mgdt
888
FFFiiiggguuurrraaa 333
EEE aaagggooorrraaa,,, cccooonnnsssiiidddeeerrraaannndddooo ooo dddiiiaaagggrrraaammmaaa dddeee cccooorrrpppooo llliiivvvrrreee aaaccciiimmmaaa,,, fffiiiggguuurrraaa 333,,, ooo mmmooovvviiimmmeeennntttooo rrroootttaaaccciiiooonnnaaalll dddaaa hhhaaasssttteee dddooo pppêêênnnddduuulllooo eeemmm tttooorrrnnnooo dddeee ssseeeuuu ccceeennntttrrrooo dddeee gggrrraaavvviiidddaaadddeee pppooodddeee ssseeerrr dddaaadddooo cccooommmooo:::
2
2 cosdi V l sen H ldt
***CCCooonnnsssiiidddeeerrraaannndddooo qqquuueee ooo ooobbbjjjeeetttiiivvvooo ééé mmmaaannnttteeerrr ooo pppêêênnnddduuulllooo nnnaaa pppooosssiiiçççãããooo vvveeerrrtttiiicccaaalll,,, ééé dddeee ssseee iiimmmaaagggiiinnnaaarrr qqquuueee ooo ââânnnggguuulllooo nnnãããooo ssseeejjjaaa mmmuuuiiitttooo gggrrraaannndddeee,,, eee pppaaarrraaa ââânnnggguuulllooosss pppeeeqqquuueeennnooosss pppooodddeeemmmooosss aaadddmmmiiitttiiirrr qqquuueee::: ssseeennn === çççãããooo ssseeerrrááá uuutttiiillliiizzzaaadddaaa pppaaarrraaa fffaaaccciiillliiitttaaarrr ooosss cccááálllcccuuulllooosss... AAA eeeqqquuuaaaçççãããooo dddooo mmmooovvviiimmmeeennntttooo rrroootttaaaccciiiooonnnaaalll fffiiicccaaarrrááá aaassssssiiimmm:::
2
2
di Vl Hldt
SSSuuubbbssstttiiitttuuuiiinnndddooo VVV eee HHH nnnaaa eeeqqquuuaaaçççãããooo:::.. .. ..
.. .. ..2
.. ..2
( )
( )
cg
cg
cg
i mgl m X ml l
i mgl m X l ml
i ml ml X mgl
AAA pppaaarrrtttiiirrr dddaaa eeeqqquuuaaaçççãããooo aaaccciiimmmaaa,,, pppooodddeeemmmooosss ooobbbttteeerrr aaa fffuuunnnçççãããooo tttrrraaannnsssfffeeerrrêêênnnccciiiaaa dddooo pppêêênnnddduuulllooo,,, qqquuueee ééé fffeeeiiitttaaa aaassssssiiimmm:::
2 2 2
2 2 2
( ) ( ) ( ) ( )
[( ) ] ( ) ( )cg
cg
i ml s s mlX s s mgl s
i ml s mgl s mlX s s
EEEnnntttãããooo,,, ooobbbttteeemmmooosss aaa FFFuuunnnçççãããooo TTTrrraaannnsssfffeeerrrêêênnnccciiiaaa dddooo !êêênnnddduuulllooo:::
2 2
2
[( ) ] ( )( )cgi ml s mgl sX s
mlseeeqqquuuaaaçççãããooo III
999
444...222...MMMooodddeeelllaaagggeeemmm dddooo CCCaaarrrrrrooo
AAA mmmooodddeeelllaaagggeeemmm dddooo cccaaarrrrrrooo ééé mmmaaaiiisss sssiiimmmpppllleeesss,,, pppoooiiisss cccooommmooo pppooodddeeemmmooosss ooobbbssseeerrrvvvaaarrr pppeeelllaaa FFFiiiggguuurrraaa 333,,, aaasss fffooorrrçççaaasss aaapppllliiicccaaadddaaasss nnneeellleee sssãããooo aaa uuu eee HHH,,, eee dddeee aaacccooorrrdddooo cccooommm aaa ssseeeggguuunnndddaaa llleeeiii dddeee NNNeeewwwtttooonnn ttteeemmmooosss:::
2
2 cgdM X u Hdt
SSSuuubbbssstttiiitttuuuiiinnndddooo HHH eee fffaaazzzeeennndddooo uuummm rrreeeaaarrrrrraaannnjjjooo dddaaa eeeqqquuuaaaçççãããooo:::..
..( )
cg cgMx u mx ml
u M m x mlUUUtttiiillliiizzzaaannndddooo aaa TTTrrraaannnsssfffooorrrmmmaaadddaaa dddeee LLLaaappplllaaaccceee:::
2 2( ) ( ) ( ) ( )cgU s M m s X s mls s eeeqqquuuaaaçççãããooo IIIIII
444...333...MMMooodddeeelllaaagggeeemmm dddooo SSSiiisssttteeemmmaaa
CCCooommmooo fffoooiii dddiiitttooo aaannnttteeerrriiiooorrrmmmeeennnttteee ooo sssiiisssttteeemmmaaa cccooommmpppllleeetttooo cccooonnnsssiiisssttteee nnnaaa jjjuuunnnçççãããooo dddooo cccaaarrrrrrooo eee dddooo pppêêênnnddduuulllooo,,, ooonnndddeee aaa eeennntttrrraaadddaaa dddooo sssiiisssttteeemmmaaa ééé aaa fffooorrrçççaaa uuu eee aaa sssaaaííídddaaa ééé ooo ââânnnggguuulllooo ãããooo pppaaarrraaa ooobbbttteeerrrmmmooosss aaa FFFuuunnnçççãããooo TTTrrraaannnsssfffeeerrrêêênnnccciiiaaa dddooo sssiiisssttteeemmmaaa,,, ttteeemmmooosss qqquuueee nnnooosss uuutttiiillliiizzzaaarrr dddaaa eeeqqquuuaaaçççãããooo III eee sssuuubbbssstttiiitttuuuiiirrr nnnaaa eeeqqquuuaaaçççãããooo IIIIII:::
2 22 2
2
2 2 2 2
2 2 2
2 2 2
[( ) ] ( )( ) ( ) ( )
( )[( ) ] ( ) ( ) ( )( )
( )( ) ( ) ( )( ) ( )
( )( ) ( ) ( )( ) ( )
i ml s mgl sU s M m s ml s smls
M m i ml s mgl s ml s sU sml
M m i ml s M m mgl mlsU s sml
M m i ml ml s M m mglU s s
ml
EEE eeennnfffiiimmm,,, ooobbbttteeemmmooosss aaa FFFuuunnnçççãããooo TTTrrraaannnsssfffeeerrrêêênnnccciiiaaa dddaaa !lllaaannntttaaa:::
2 2 2
( )( ) [( )( ) ( ) ] ( )s ml
U s s M m i ml ml M m mgleeeqqquuuaaaçççãããooo IIIIIIIII
111000
555... SSSiiimmmuuulllaaaçççãããooo
AAAsss eeetttaaapppaaasss dddeee sssiiimmmuuulllaaaçççãããooo cccooonnnsssiiisssttteeemmm nnnaaasss ssseeeggguuuiiinnnttteeesss::: ppprrriiimmmeeeiiirrraaammmeeennnttteee cccooommm aaa fffuuunnnçççãããooo dddaaa ppplllaaannntttaaa (((eeeqqquuuaaaçççãããooo IIIIIIIII))) eeemmm mmmaaalllhhhaaa aaabbbeeerrrtttaaa,,, iiirrreeemmmooosss aaannnaaallliiisssaaarrr aaa rrreeessspppooossstttaaa dddeeesssssseee sssiiisssttteeemmmaaa aaa eeennntttrrraaadddaaasss iiimmmpppuuulllsssooo,,, dddeeegggrrraaauuu eee aaannnaaallliiissseee dddaaa rrreeessspppooossstttaaa eeemmm fffrrreeeqqqüüüêêênnnccciiiaaa... EEEmmm ssseeeggguuuiiidddaaa ssseeerrrááá fffeeeiiitttooo ooo cccooonnntttrrrooollleee !IIIDDD pppaaarrraaa aaa ppplllaaannntttaaa eee eeennntttãããooo rrreeepppeeetttiiirrreeemmmooosss aaasss mmmeeesssmmmaaasss aaannnaaallliiissseeesss pppaaarrraaa qqquuueee ssseee pppooossssssaaa fffaaazzzeeerrr uuummm eeessstttuuudddooo cccooommmpppaaarrraaatttiiivvvooo eeennntttrrreee eeelllaaasss eee vvveeerrriiifffiiicccaaarrr aaa eeefffiiiccciiiêêênnnccciiiaaa dddooo cccooonnntttrrrooolllaaadddooorrr...
VVVaaallleee llleeemmmbbbrrraaarrr qqquuueee aaa sssiiimmmuuulllaaaçççãããooo rrreeeaaallliiizzzaaadddaaa nnneeesssssseee ppprrrooojjjeeetttooo fffoooiii tttooodddaaa fffeeeiiitttaaa nnnooo MMMAAATTTLLLAAABBB...
555...111...PPPaaarrrâââmmmeeetttrrrooosss dddooo MMMooodddeeelllooo
!aaarrraaa aaa sssiiimmmuuulllaaaçççãããooo ooocccooorrrrrreeerrr cccooommm sssuuuccceeessssssooo,,, ééé ppprrreeeccciiisssooo dddeeefffiiinnniiirrr aaalllggguuunnnsss pppaaarrrâââmmmeeetttrrrooosss qqquuueee nnnãããooo fffooorrraaammm dddeeefffiiinnniiidddooosss qqquuuaaannndddooo fffeeeiiitttaaa aaa fffuuunnnçççãããooo tttrrraaannnsssfffeeerrrêêênnnccciiiaaa...
!aaarrrâââmmmeeetttrrrooo SSSííímmmbbbooolllooo VVVaaalllooorrr UUUnnniiidddaaadddeeeMMMaaassssssaaa dddooo cccaaarrrrrrooo MMM 111,,,000000 KKKgggMMMaaassssssaaa dddooo pppêêênnnddduuulllooo mmm 000,,,666000 KKKgggCCCooommmppprrriiimmmeeennntttooo dddooo pppêêênnnddduuulllooo lll 000,,,444000 mmmMMMooommmeeennntttooo dddeee IIInnnééérrrccciiiaaa iii 000,,,000333222 NNN...mmm²²gggrrraaavvviiidddaaadddeee ggg 999,,,888 mmm///sss²²
555...222...SSSiiisssttteeemmmaaa eeemmm MMMaaalllhhhaaa AAAbbbeeerrrtttaaa
OOO sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa aaabbbeeerrrtttaaa pppooodddeee ssseeerrr rrreeeppprrreeessseeennntttaaadddooo pppeeelllooo ssseeeggguuuiiinnnttteee dddiiiaaagggrrraaammmaaa dddeee bbblllooocccooosss:::
FFFiiiggguuurrraaa 444
OOOnnndddeee aaa ppplllaaannntttaaa ééé dddaaadddaaa pppeeelllaaa eeeqqquuuaaaçççãããooo IIIIIIIII,,, nnnaaa sssiiimmmuuulllaaaçççãããooo,,, ooo rrreeesssuuullltttaaadddooo ooobbbtttiiidddooo dddaaa fffuuunnnçççãããooo tttrrraaannnsssfffeeerrrêêênnnccciiiaaa cccooommm ooosss pppaaarrrâââmmmeeetttrrrooosss iiinnnssseeerrriiidddooosss fffoooiii:::
TTTrrraaannnsssfffeeerrr fffuuunnnccctttiiiooonnn::: 000...222444
-‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐-000...111444777222 sss^^^222 -‐-‐- 333...777666666
111111
555...222...111... RRReeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo
EEEmmm ppprrroooccceeessssssaaammmeeennntttooo dddeee sssiiinnnaaaiiisss,,, aaa rrreeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo dddeee uuummm sssiiisssttteeemmmaaa ééé aaa sssaaaííídddaaa dddeeellleee qqquuuaaannndddooo eeemmm sssuuuaaa eeennntttrrraaadddaaa ééé cccooolllooocccaaadddooo uuummm iiimmmpppuuulllsssooo...
EEEmmm sssiiisssttteeemmmaaasss llliiinnneeeaaarrreeesss iiinnnvvvaaarrriiiaaannnttteeesss nnnooo ttteeemmmpppooo (((LLLIIITTT))) aaa rrreeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo ééé uuummmaaa cccaaarrraaacccttteeerrríííssstttiiicccaaa iiimmmpppooorrrtttaaannntttíííssssssiiimmmaaa... EEElllaaa pppeeerrrmmmiiittteee cccaaalllcccuuulllaaarrr aaa sssaaaííídddaaa dddooo sssiiisssttteeemmmaaa pppaaarrraaa qqquuuaaalllqqquuueeerrr sssiiinnnaaalll dddeee eeennntttrrraaadddaaa,,, aaatttrrraaavvvééésss dddaaa cccooonnnvvvooollluuuçççãããooo dddeeesssttteee cccooommm aaa rrreeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo dddooo sssiiisssttteeemmmaaa...
FFFiiiggguuurrraaa 555
CCCooommmooo pppôôôdddeee ssseeerrr ooobbbssseeerrrvvvaaadddooo nnnaaa fffiiiggguuurrraaa,,, ooo rrreeesssuuullltttaaadddooo nnnãããooo fffoooiii ooo eeessspppeeerrraaadddooo,,, ooo qqquuueee mmmooossstttrrraaa qqquuueee eeesssssseee sssiiisssttteeemmmaaa ééé aaallltttaaammmeeennnttteee iiinnnssstttááávvveeelll... IIIssssssooo dddeeemmmooonnnssstttrrraaa qqquuueee aaaooo aaapppllliiicccaaarrr ooo iiimmmpppuuulllsssooo ssseeemmm cccooonnntttrrrooollleee ooo ââânnnggguuulllooo cccrrreeesssccceee ssseeemmm pppaaarrraaarrr...
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
7x 10
5 Impulse Response
Time (sec)
Ampli
tude
111222
555...222...222... RRReeessspppooossstttaaa aaaooo DDDeeegggrrraaauuu
AAA rrreeessspppooossstttaaa dddeeegggrrraaauuu ééé aaa rrreeessspppooossstttaaa dddooo sssiiisssttteeemmmaaa aaa uuummmaaa eeennntttrrraaadddaaa cccooonnnssstttaaannnttteee,,, eee cccooommmooo rrreeesssuuullltttaaadddooo eeessspppeeerrraaadddooo,,, ooo sssiiisssttteeemmmaaa dddeeevvveee ssseeeggguuuiiirrr eeessssssaaa eeennntttrrraaadddaaa...
FFFiiiggguuurrraaa 666
MMMaaaiiisss uuummmaaa vvveeezzz pppuuudddeeemmmooosss ooobbbssseeerrrvvvaaarrr qqquuueee ooo gggrrráááfffiiicccooo dddooo sssiiisssttteeemmmaaa eeemmm mmmaaalllhhhaaa aaabbbeeerrrtttaaa nnnãããooo cccooorrrrrreeessspppooonnndddeee cccooommm ooo eeessspppeeerrraaadddooo,,, mmmooossstttrrraaannndddooo nnnooovvvaaammmeeennnttteee aaa sssuuuaaa iiinnnssstttaaabbbiiillliiidddaaadddeee... MMMooossstttrrraaa qqquuueee tttaaammmbbbééémmm ooo ââânnnggguuulllooo cccrrreeesssccceee ssseeemmm pppaaarrraaarrr cccooommm rrreeessspppooossstttaaa aaa eeennntttrrraaadddaaa dddeeegggrrraaauuu... IIIssssssooo pppooodddeee ssseeerrr ooobbbssseeerrrvvvaaadddooo aaatttrrraaavvvééésss dddaaa fffuuunnnçççãããooo !ttteeepppiiinnnfffooo dddooo MMMAAATTTLLLAAABBB,,, qqquuueee rrreeetttooorrrnnnaaa ooosss pppaaarrrâââmmmeeetttrrrooosss dddaaa rrreeessspppooossstttaaa aaaooo dddeeegggrrraaauuu... !oooiiisss ooo sssiiisssttteeemmmaaa nnnãããooo cccooonnnssseeeggguuueee eeessstttaaabbbiiillliiizzzaaarrr,,, nnnãããooo ooobbbttteeennndddooo aaassssssiiimmm,,, nnneeennnhhhuuummm pppaaarrrâââmmmeeetttrrrooo...
pppaaarrraaammmeeetttrrrooosss___dddeeegggrrraaauuu___mmmaaalllhhhaaa___aaabbbeeerrrtttaaa ===
RRRiiissseeeTTTiiimmmeee::: NNNaaaNNNSSSeeettttttllliiinnngggTTTiiimmmeee::: NNNaaaNNNSSSeeettttttllliiinnngggMMMiiinnn::: NNNaaaNNNSSSeeettttttllliiinnngggMMMaaaxxx::: NNNaaaNNNOOOvvveeerrrssshhhoooooottt::: NNNaaaNNNUUUnnndddeeerrrssshhhoooooottt::: NNNaaaNNN!eeeaaakkk::: IIInnnfff!eeeaaakkkTTTiiimmmeee::: IIInnnfff
0 0.5 1 1.5 2 2.5 30
2
4
6
8
10
12
14x 10
4 Step Response
Time (sec)
Am
plitu
de
111333
555...222...333... RRReeessspppooossstttaaa eeemmm fffrrreeeqqqüüüêêênnnccciiiaaa ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee
RRReeessspppooossstttaaa eeemmm fffrrreeeqqquuuêêênnnccciiiaaa ééé aaa aaannnááállliiissseee dddooo cccooommmpppooorrrtttaaammmeeennntttooo dddeee uuummm sssiiisssttteeemmmaaaqqquuuaaannntttooo aaaooo ssseeeuuu gggaaannnhhhooo nnnuuummmaaa ccceeerrrtttaaa fffaaaiiixxxaaa dddeee fffrrreeeqqquuuêêênnnccciiiaaa (((ooouuu eeemmm aaalllggguuunnnsss cccaaasssooosss,,, vvveeellloooccciiidddaaadddeee aaannnggguuulllaaarrr)))...
OOO gggrrráááfffiiicccooo ooonnndddeee ééé aaannnaaallliiisssaaadddaaa aaa rrreeessspppooossstttaaa eeemmm fffrrreeeqqquuuêêênnnccciiiaaa dddeee uuummmaaa rrreeedddeee ééé gggeeerrraaalllmmmeeennnttteee uuummmaaa cccuuurrrvvvaaa dddeee BBBooodddeee... AAA cccuuurrrvvvaaa dddeee BBBooodddeee cccooonnnsssiiisssttteee dddeee uuummm dddiiiaaagggrrraaammmaaa cccooommm uuummmaaa eeessscccaaalllaaa llliiinnneeeaaarrr dddeee gggaaannnhhhooo nnnaaa ooorrrdddeeennnaaadddaaa (((eeemmm dddeeeccciiibbbéééiii! (((dddBBB))) ooouuu eeemmm VVVooolllttt pppooorrr VVVooolllttt(((VVV///VVV)))... AAA uuunnniiidddaaadddeee mmmaaaiii! uuutttiiillliiizzzaaadddaaa ééé dddBBB))) eee uuummmaaa eeessscccaaalllaaa lllooogggaaarrrííítttiiimmmiiicccaaa nnnaaa aaabbbccciiissssssaaa dddeee fffrrreeeqqquuuêêênnnccciiiaaa (((eeemmm HHHeeerrrtttzzz (((HHHzzz))) ooouuu eeemmm vvveeellloooccciiidddaaadddeee aaannnggguuulllaaarrr (((rrraaaddd///sss))))))...
NNNooo dddiiiaaagggrrraaammmaaa dddeee BBBooodddeee sssãããooo aaappprrreeessseeennntttaaadddaaasss ddduuuaaasss cccuuurrrvvvaaasss,,, aaa CCCuuurrrvvvaaa dddeee GGGaaannnhhhooo,,, qqquuueee ééé uuummmaaa rrreeeppprrreeessseeennntttaaaçççãããooo dddaaa eeevvvooollluuuçççãããooo dddeee GGG(((jjjwww))) eeemmm eeessscccaaalllaaa lllooogggaaarrrííítttmmmiiicccaaa dddooo sssiiisssttteeemmmaaa,,, eee aaa CCCuuurrrvvvaaa dddeee FFFaaassseeesss,,, rrreeeppprrreeessseeennntttaaaçççãããooo dddaaa eeevvvooollluuuçççãããooo eeemmm eeessscccaaalllaaa lllooogggaaarrrííítttmmmiiicccaaa dddaaa fffrrreeeqqqüüüêêênnnccciiiaaa...
FFFiiiggguuurrraaa 777
AAAnnnaaallliiisssaaannndddooo ooo gggrrráááfffiiicccooo dddeee mmmaaagggnnniiitttuuudddeee ssseee pppeeerrrccceeebbbeee ooo qqquuuãããooo iiinnnssstttááávvveeelll ééé ooo sssiiisssttteeemmmaaa,,, pppoooiiisss aaa cccuuurrrvvvaaa eeessstttááá tttooodddaaa aaabbbaaaiiixxxooo dddaaa fffrrreeeqqqüüüêêênnnccciiiaaa dddeee cccooorrrttteee (((-‐-‐-333dddBBB)))... EEE aaa cccuuurrrvvvaaa dddeee fffaaassseee mmmooossstttrrraaa qqquuueee nnnãããooo hhhooouuuvvveee dddeeefffaaasssaaammmeeennntttooo...
-80
-70
-60
-50
-40
-30
-20
Mag
nitu
de (d
B)
100
101
102
-181
-180.5
-180
-179.5
-179
Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
111444
555...333...SSSiiisssttteeemmmaaa eeemmm MMMaaalllhhhaaa FFFeeeccchhhaaadddaaa cccooommm CCCooonnntttrrrooolllaaadddooorrr PPPIIIDDD
OOO sssiiisssttteeemmmaaa dddeee mmmaaalllhhhaaa aaabbbeeerrrtttaaa pppooodddeee ssseeerrr rrreeeppprrreeessseeennntttaaadddooo pppeeelllooo ssseeeggguuuiiinnnttteee dddiiiaaagggrrraaammmaaa dddeee bbblllooocccooosss:::
FFFiiiggguuurrraaa 888
AAA fffuuunnnçççãããooo tttrrraaannnsssfffeeerrrêêênnnccciiiaaa dddeeesssssseee sssiiisssttteeemmmaaa ééé ooobbbtttiiidddooo aaatttrrraaavvvééésss dddaaa fffuuunnnçççãããooo fffeeeeeedddbbbaaaccckkk dddooo MMMAAATTTLLLAAABBB... EEEssssssaaa fffuuunnnçççãããooo fffaaazzz eeexxxaaatttaaammmeeennnttteee ooo qqquuueee aaa fffiiiggguuurrraaa 888 mmmooossstttrrraaa,,, aaa cccooonnnvvvooollluuuçççãããooo dddooo CCCooonnntttrrrooolllaaadddooorrr !IIIDDD cccooommm aaa !lllaaannntttaaa dddooo sssiiisssttteeemmmaaa cccooommm rrreeeaaallliiimmmeeennntttaaaçççãããooo uuunnniiitttááárrriiiaaa... FFFooorrraaammm eeexxxeeecccuuutttaaadddooosss vvvááárrriiiooosss ttteeesssttteeesss cccooommm aaasss cccooonnntttaaannnttteeesss dddooo cccooonnntttrrrooolllaaadddooorrr pppaaarrraaa qqquuueee hhhooouuuvvveeesssssseee uuummmaaa rrreeessspppooossstttaaa sssaaatttiiisssfffaaatttóóórrriiiaaa tttaaannntttooo pppaaarrraaa aaasss aaalllooocccaaaçççõõõeeesss dddaaasss rrraaaííízzzeeesss cccooommmooo pppaaarrraaa ooosss pppaaarrrâââmmmeeetttrrrooosss dddooo dddeeegggrrraaauuu (((ttteeemmmpppooo dddeee pppiiicccooo,,, ttteeemmmpppooo dddeee aaacccooommmooodddaaaçççãããooo,,, eeetttccc...)))...
AAAssssssiiimmm,,, ooosss vvvaaalllooorrreeesss dddeee KKK dddooo cccooonnntttrrrooolllaaadddooorrr sssãããooo:::kp = 140;kd = 20;ki = 180;
TTTeeemmmooosss qqquuueee aaa FFFuuunnnçççãããooo TTTrrraaannnsssfffeeerrrêêênnnccciiiaaa dddooo CCCooonnntttrrrooolllaaadddooorrr ééé:::
_ ip dKFT controlador K K ss
FFFaaazzz-‐-‐-ssseee aaa cccooonnnvvvooollluuuçççãããooo cccooommm aaa FFFuuunnnçççãããooo TTTrrraaannnsssfffeeerrrêêênnnccciiiaaa dddaaa !lllaaannntttaaa aaatttrrraaavvvééésss dddaaa fffuuunnnçççãããooo fffeeeeeedddbbbaaaccckkk:::
FT_malha_fechada = feedback(FT_malha_aberta*FT_controlador,1)
EEE aaassssssiiimmm,,, ooobbbtttêêêmmm-‐-‐-ssseee aaa FFFuuunnnçççãããooo TTTrrraaannnsssfffeeerrrêêênnnccciiiaaa dddeee MMMaaalllhhhaaa FFFeeeccchhhaaadddaaa dddooo SSSiiisssttteeemmmaaa:::
TTTrrraaannnsssfffeeerrr fffuuunnnccctttiiiooonnn::: 444...888 sss^^^222 +++ 333333...666 sss +++ 444333...222-‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐-000...111444777222 sss^^^333 +++ 444...888 sss^^^222 +++ 222999...888333 sss +++ 444333...222
111555
555...333...111... RRReeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo
AAAgggooorrraaa cccooommm aaa eeennntttrrraaadddaaa dddooo CCCooonnntttrrrooolllaaadddooorrr !IIIDDD,,, pppooodddeeemmmooosss aaannnaaallliiisssaaarrr aaa rrreeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo nnnooo SSSiiisssttteeemmmaaa dddeee MMMaaalllhhhaaa FFFeeeccchhhaaadddaaa eee cccooommmpppaaarrraaarrr cccooommm ooo SSSiiisssttteeemmmaaa dddeee MMMaaalllhhhaaa AAAbbbeeerrrtttaaa...
FFFiiiggguuurrraaa 999
CCCooommmpppaaarrraaannndddooo aaa fffiiiggguuurrraaa 999 cccooommm aaa fffiiiggguuurrraaa 555,,, vvveeemmmooosss aaa mmmeeelllhhhooorrraaa qqquuueee hhhooouuuvvveee nnnooo sssiiisssttteeemmmaaa,,, rrreeessspppooonnndddeeennndddooo dddeee fffooorrrmmmaaa eeessstttááávvveeelll ààà eeennntttrrraaadddaaa iiimmmpppuuulllsssooo...
0 0.5 1 1.5 2 2.5 3-5
0
5
10
15
20
25
30
35Impulse Response
Time (sec)
Ampl
itude
111666
555...333...222... RRReeessspppooossstttaaa aaaooo DDDeeegggrrraaauuu
CCCooommmooo dddiiitttooo aaannnttteeerrriiiooorrrmmmeeennnttteee,,, aaagggooorrraaa ssseeerrrááá fffeeeiiitttaaa aaa cccooommmpppaaarrraaaçççãããooo eee aaannnttteeesss eee dddeeepppoooiiisss dddaaa eeennntttrrraaadddaaa dddooo cccooonnntttrrrooolllaaadddooorrr !IIIDDD...
FFFiiiggguuurrraaa 111000
CCCooonnnsssiiidddeeerrraaannndddooo aaa iiinnnssstttaaabbbiiillliiidddaaadddeee qqquuueee eeerrraaa ooo sssiiisssttteeemmmaaa aaannnttteeesss,,, fffiiiggguuurrraaa 666,,, pppooodddeee-‐-‐-ssseee pppeeerrrccceeebbbeeerrr qqquuueee hhhooouuuvvveee uuummm cccooonnnsssiiidddeeerrrááávvveeelll aaajjjuuusssttteee,,, qqquuueee aaapppeeesssaaarrr dddeee aaappprrreeessseeennntttaaarrr uuummm MMMppp
(((mmmáááxxxiiimmmooo sssooobbbrrreee-‐-‐-sssiiinnnaaalll))) uuummm pppooouuucccooo gggrrraaannndddeee,,, cccooommmpppaaarrraaannndddooo cccooommm aaa sssiiisssttteeemmmaaa eeemmm mmmaaalllhhhaaa aaabbbeeerrrtttaaa,,, pppooodddeee-‐-‐-ssseee dddiiizzzeeerrr qqquuueee ooo sssiiisssttteeemmmaaa ssseee tttooorrrnnnooouuu eeessstttááávvveeelll...
!ooodddeeemmmooosss ooobbbttteeerrr ooosss vvvaaalllooorrreeesss dddooosss pppaaarrrâââmmmeeetttrrrooosss dddaaa rrreeessspppooossstttaaa aaaooo dddeeegggrrraaauuu mmmaaaiiisss uuummm vvveeezzz uuutttiiillliiizzzaaannndddooo aaa fffuuunnnçççãããooo !ttteeepppiiinnnfffooo dddooo MMMAAATTTLLLAAABBB...
pppaaarrraaammmeeetttrrrooosss___dddeeegggrrraaauuu___mmmaaalllhhhaaa___fffeeeccchhhaaadddaaa ===
RRRiiissseeeTTTiiimmmeee::: 000...000444666777SSSeeettttttllliiinnngggTTTiiimmmeee::: 111...222222777333SSSeeettttttllliiinnngggMMMiiinnn::: 111...000000000000SSSeeettttttllliiinnngggMMMaaaxxx::: 111...111999999000OOOvvveeerrrssshhhoooooottt::: 111999...999000333111UUUnnndddeeerrrssshhhoooooottt::: 000!eeeaaakkk::: 111...111999999000!eeeaaakkkTTTiiimmmeee::: 000...111666999888
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Ampl
itude
111777
555...333...333... RRReeessspppooossstttaaa eeemmm fffrrreeeqqqüüüêêênnnccciiiaaa ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee
FFFiiiggguuurrraaa 111111
OOO dddiiiaaagggrrraaammmaaa mmmooossstttrrraaa qqquuueee hhhááá uuummm dddeeefffaaasssaaammmeeennntttooo nnnaaa fffaaassseee dddooo sssiiisssttteeemmmaaa eee mmmooossstttrrraaa tttaaammmbbbééémmm qqquuueee ooo sssiiisssttteeemmmaaa aaagggeee cccooommmooo uuummm pppaaassssssaaa-‐-‐-bbbaaaiiixxxaaa tttrrraaabbbaaalllhhhaaannndddooo cccooommm gggaaannnhhhooo uuunnniiitttááárrriiiooo ooouuu ppprrróóóxxxiiimmmooo dddeeellleee aaatttééé aaatttiiinnngggiiirrr sssuuuaaa fffrrreeeqqqüüüêêênnnccciiiaaa dddeee cccooorrrttteee eeemmm 333888...555 rrraaaddd///ssseeeggg...
-30
-20
-10
0
10
Mag
nitu
de (d
B)
System: FT_malha_fechadaFrequency (rad/sec): 38.5Magnitude (dB): -3
10-1
100
101
102
103
-90
-45
0
45
Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
111888
555...333...444... GGGrrráááfffiiicccooo dddooosss PPPóóólllooosss eee ZZZeeerrrooosss
FFFiiiggguuurrraaa 111222
AAAtttrrraaavvvééésss dddaaa fffuuunnnçççãããooo [[[zzz ppp]]] fffoooiii pppooossssssííívvveeelll ooobbbttteeerrr ooosss vvvaaalllooorrreeesss dddooosss pppóóólllooosss eee zzzeeerrrooosss:::UUUtttiiillliiizzzaaammmooosss ooosss vvvaaalllooorrreeesss dddaaa FFFTTT dddeee mmmaaalllhhhaaa fffeeeccchhhaaadddaaa eee aaapppllliiicccaaammmooosss aaa fffuuunnnçççãããooo:::num2 = [4.8 33.6 43.2];den2 = [0.1472 4.8 29.83 43.2];[z p]=tf2zp(num2,den2)
CCCuuujjjooosss vvvaaalllooorrreeesss ooobbbtttiiidddooosss sssãããooo:::zzz ===
-‐-‐-555...333000222888 -‐-‐-111...666999777222
ppp ===
-‐-‐-222444...999666111111 -‐-‐-555...555111666111 -‐-‐-222...111333111555
-25 -20 -15 -10 -5 0-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
10.90.9760.990.9960.9980.999
1
1
0.90.9760.990.9960.9980.999
1
1
5101520
Pole-Zero Map
Real Axis
Imag
inar
y Ax
is
111999
555...444...SSSiiisssttteeemmmaaa cccooommm CCCooommmpppeeennnsssaaadddooorrr LLLeeeaaaddd-‐-‐-lllaaaggg (((MMMééétttooodddooo dddooo LLLuuugggaaarrr dddaaasss RRRaaaííízzzeeesss)))
OOO mmmééétttooodddooo rrreeeppprrreeessseeennntttaaa gggrrraaafffiiicccaaammmeeennnttteee ooosss pppóóólllooosss dddaaa fffuuunnnçççãããooo dddeee tttrrraaannnsssfffeeerrrêêênnnccciiiaaa eeemmm mmmaaalllhhhaaa fffeeeccchhhaaadddaaa eee aaasss sssuuuaaasss vvvááárrriiiaaasss lllooocccaaallliiizzzaaaçççõõõeeesss,,, eeemmm fffuuunnnçççãããooo dddaaa vvvaaarrriiiaaaçççãããooo dddeee aaalllggguuummm pppaaarrrâââmmmeeetttrrrooo ppprrreeessseeennnttteee nnnaaa fffuuunnnçççãããooo dddeee tttrrraaannnsssfffeeerrrêêênnnccciiiaaa (((ooo gggaaannnhhhooo KKK)))...
OOO llluuugggaaarrr dddaaasss rrraaaííízzzeeesss (((rrroooooottt lllooocccuuu!))) nnnooosss dddááá aaa tttrrraaajjjeeetttóóórrriiiaaa qqquuueee aaasss rrraaaííízzzeeesss vvvãããooo ttteeerrr qqquuuaaannndddooo nnnóóósss fffeeeccchhhaaarrrmmmooosss aaa mmmaaalllhhhaaa dddeee cccooonnntttrrrooollleee,,, ooouuu ssseeejjjaaa,,, aaa mmmiiigggrrraaaçççãããooo dddaaasss rrraaaííízzzeeesss dddaaa mmmaaalllhhhaaa aaabbbeeerrrtttaaa pppaaarrraaa aaa mmmaaalllhhhaaa fffeeeccchhhaaadddaaa cccooommm ooo cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaaggg...
TTTeeemmmooosss qqquuueee aaa FFFuuunnnçççãããooo TTTrrraaannnsssfffeeerrrêêênnnccciiiaaa dddooo CCCooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaaggg ééé dddaaadddaaa pppooorrr:::
( )( )_( )( )k s a s cFT comps b s c
!ooodddeeemmmooosss ooobbbssseeerrrvvvaaarrr qqquuueee kkk ssseeerrrááá aaa vvvaaarrriiiááávvveeelll nnnooo gggrrráááfffiiicccooo eee dddeeefffiiinnniiimmmooosss ooosss ooouuutttrrrooosss pppaaarrrâââmmmeeetttrrrooosss eeemmm:::
a = 7; b = 1; c = 6; d = 300;
UUUtttiiillliiizzzaaammmooosss aaa fffuuunnnçççãããooo rrrlllooocccuuu! dddooo MMMAAATTTLLLAAABBB dddaaa cccooonnnvvvooollluuuçççãããooo FFFTTT (((FFFuuunnnçççãããooo TTTrrraaannnsssfffeeerrrêêênnnccciiiaaa))) dddooo sssiiisssttteeemmmaaa eeemmm mmmaaalllhhhaaa aaabbbeeerrrtttaaa cccooommm aaa FFFTTT dddooo cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaaggg pppaaarrraaa ooobbbssseeerrrvvvaaarrr ooo gggrrráááfffiiicccooo dddaaasss rrraaaííízzzeeesss:::
rlocus(FT_comp*FT_malha_aberta)
eee ooobbbttteeemmmooosss ooo gggrrráááfffiiicccooo:::
FFFiiiggguuurrraaa 111333
-1200 -1000 -800 -600 -400 -200 0 200-300
-200
-100
0
100
200
300Root Locus
Real Axis
Imag
inar
y Ax
is
222000
EEE aaappprrroooxxxiiimmmaaannndddooo aaa fffiiiggguuurrraaa pppaaarrraaa pppeeerrrtttooo dddaaa ooorrriiigggeeemmm,,, ttteeemmmooosss:::
FFFiiiggguuurrraaa 111444
NNNaaa fffiiiggguuurrraaa 111333,,, uuutttiiillliiizzzaaammmooosss aaa fffuuunnnçççãããooo rrrlllooocccfffiiinnnddd pppaaarrraaa ooobbbttteeerrr ooo vvvaaalllooorrr dddooo gggaaannnhhhooo eeemmm uuummmaaa dddeeettteeerrrmmmiiinnnaaadddaaa pppooosssiiiçççãããooo,,, eee ooo vvvaaalllooorrr ooobbbtttiiidddooo fffoooiii:::
[k]=rlocfind(FT_comp*FT_malha_aberta)k =
3.98e+003
OOObbbtttiiidddooo ooo gggaaannnhhhooo,,, pppooodddeee-‐-‐-ssseee fffaaazzzeeerrr aaa MMMaaalllhhhaaa fffeeeccchhhaaadddaaa cccooommm ooo cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaaggg eee rrreeeaaallliiimmmeeennntttaaaçççãããooo uuunnniiitttááárrriiiaaa,,, dddaaa mmmeeesssmmmaaa fffooorrrmmmaaa qqquuueee fffiiizzzeeemmmooosss pppaaarrraaa ooo cccooonnntttrrrooolllaaadddooorrr !IIIDDD:::
FT_MF_comp=feedback(FT_malha_aberta,k*FT_comp)
EEE ooobbbttteeemmmooosss aaa FFFTTT:::TTTrrraaannnsssfffeeerrr fffuuunnnccctttiiiooonnn:::
000...222444 sss^^^222 +++ 222666444 sss +++ 222444000000000-‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐--‐-‐-000...111444777222 sss^^^444 +++ 111666111...999 sss^^^333 +++ 111...555666777eee000000444 sss^^^222 +++ 666...888444555eee000000444 sss +++ 222...444666111eee000000444
CCCooommm FFFTTT dddooo cccooommmpppeeennnsssaaadddooorrr llleeeaaaddd-‐-‐-lllaaaggg eeemmm mmmaaalllhhhaaa fffeeeccchhhaaadddaaa,,, jjjááá ééé pppooossssssííívvveeelll ooobbbttteeerrr aaa rrreeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo eee aaaooo dddeeegggrrraaauuu dddooo sssiiisssttteeemmmaaa...
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10Root Locus
Real Axis
Imag
inar
y Ax
is
222111
555...444...111... RRReeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo
FFFiiiggguuurrraaa 111555
CCCooommmpppaaarrraaadddaaa ààà rrreeessspppooossstttaaa aaaooo iiimmmpppuuulllsssooo dddooo cccooonnntttrrrooolllaaadddooorrr !IIIDDD (((fffiiiggguuurrraaa 999))),,, ooobbbssseeerrrvvvaaa-‐-‐-ssseee qqquuueee eeelllaaa pppooossssssuuuiii uuummm ttteeemmmpppooo dddeee aaacccooommmooodddaaaçççãããooo mmmuuuiiitttooo mmmaaaiiiooorrr...
0 5 10 15 20 25 30 35 40 45 500
0.05
0.1
0.15
0.2
0.25
0.3
0.35Impulse Response
Time (sec)
Ampl
itude
222222
555...444...222... RRReeessspppooossstttaaa aaaooo DDDeeegggrrraaauuu
FFFiiiggguuurrraaa 111666
OOOsss pppaaarrrâââmmmeeetttrrrooosss ooobbbtttiiidddooosss aaatttrrraaavvvééésss dddaaa fffuuunnnçççãããooo !ttteeepppiiinnnfffooo sssãããooo:::
aaannnsss ===
RRRiiissseeeTTTiiimmmeee::: 555...666000444000SSSeeettttttllliiinnngggTTTiiimmmeee::: 111000...111555333888SSSeeettttttllliiinnngggMMMiiinnn::: 000...888777888444SSSeeettttttllliiinnngggMMMaaaxxx::: 000...999777555222OOOvvveeerrrssshhhoooooottt::: 000UUUnnndddeeerrrssshhhoooooottt::: 000!eeeaaakkk::: 000...999777555222!eeeaaakkkTTTiiimmmeee::: 222555...111555444444
EEE mmmaaaiiisss uuummmaaa vvveeezzz,,, aaassssssiiimmm cccooommmooo nnnooo iiimmmpppuuulllsssooo,,, ooobbbssseeerrrvvvaaammmooosss qqquuueee ooo ttteeemmmpppooo dddeee aaacccooommmooodddaaaçççãããooo ééé mmmuuuiiitttooo mmmaaaiiiooorrr qqquuueee eeemmm rrreeelllaaaçççãããooo aaaooo dddeeegggrrraaauuu dddooo cccooonnntttrrrooolllaaadddooorrr !IIIDDD,,, eee aaallleeemmm dddeeellleee,,, ooo ttteeemmmpppooo dddeee pppiiicccooo eee ooo ttteeemmmpppooo dddeee sssuuubbbiiidddaaa tttaaammmbbbééémmm...
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Step Response
Time (sec)
Ampl
itude
222333
555...444...333... RRReeessspppooossstttaaa eeemmm FFFrrreeeqqquuuêêênnnccciiiaaa ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee
FFFiiiggguuurrraaa 111777
OOO sssiiisssttteeemmmaaa cccooonnntttiiinnnuuuaaa aaagggiiinnndddooo cccooommm uuummm pppaaassssssaaa-‐-‐-bbbaaaiiixxxaaa aaatttééé aaatttiiinnngggiiirrr sssuuuaaa fffrrreeeqqqüüüêêênnnccciiiaaa dddeee cccooorrrttteee dddeee 000...333777111 rrraaaddd///ssseeeggg...
-200
-150
-100
-50
0
Mag
nitu
de (d
B)
System: FT_MF_compFrequency (rad/sec): 0.371Magnitude (dB): -3.04
10-2
10-1
100
101
102
103
104
-180
-135
-90
-45
0
Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
222444
555...555...SSSiiisssttteeemmmaaa CCCooonnntttrrrooolllaaadddooo aaatttrrraaavvvééésss dddeee AAAlllooocccaaaçççãããooo dddeee PPPóóólllooosss
OOO ppprrrooojjjeeetttooo dddeee uuummm cccooonnntttrrrooolllaaadddooorrr pppooorrr aaalllooocccaaaçççãããooo dddeee pppóóólllooosss vvviiisssaaa uuutttiiillliiizzzaaarrr uuummmaaa lllóóógggiiicccaaa dddeee cccooonnntttrrrooollleee qqquuueee eeessstttaaabbbiiillliiizzzeee uuummm sssiiisssttteeemmmaaa,,, aaa ppprrriiinnncccííípppiiiooo iiinnnssstttááávvveeelll... AAAssssssiiimmm,,, ééé pppooossssssííívvveeelll dddeeessslllooocccaaarrr ooosss pppóóólllooosss dddaaa mmmaaatttrrriiizzz dddeee tttrrraaannnsssfffeeerrrêêênnnccciiiaaa dddaaa pppaaarrrttteee pppooosssiiitttiiivvvaaa dddooo ppplllaaannnooo RRReeeaaalll pppaaarrraaa aaa pppaaarrrttteee nnneeegggaaatttiiivvvaaa,,, eeessstttaaabbbiiillliiizzzaaannndddooo ooo sssiiisssttteeemmmaaa...
!rrriiimmmeeeiiirrraaammmeeennnttteee uuutttiiillliiizzzaaammmooosss aaa fffuuunnnçççãããooo [[[zzz ppp]]] nnnaaa FFFTTT dddaaa ppplllaaannntttaaa pppaaarrraaa ooobbbttteeerrr ooosss vvvaaalllooorrreeesss dddooosss pppóóólllooosss qqquuuaaannndddooo ooo sssiiisssttteeemmmaaa aaaiiinnndddaaa ééé eeessstttááávvveeelll...
zzz ===
EEEmmmppptttyyy mmmaaatttrrriiixxx::: 000-‐-‐-bbbyyy-‐-‐-111
ppp ===
555...000555777999 -‐-‐-555...000555777999
DDDaaaííí,,, pppeeerrrccceeebbbeeemmmooosss qqquuueee aaa ppplllaaannntttaaa nnnãããooo pppooossssssuuuiii zzzeeerrrooosss eee qqquuueee tttaaammmbbbééémmm pppooossssssuuuiii uuummm pppóóólllooo pppooosssiiitttiiivvvooo,,, ooo qqquuueee fffaaazzz ooo sssiiisssttteeemmmaaa ssseee tttooorrrnnnaaarrr iiinnnssstttááávvveeelll... EEEllleeesss tttaaammmbbbééémmm pppooodddeeemmm ssseeerrr ooobbbssseeerrrvvvaaadddooosss pppeeelllaaa fffiiiggguuurrraaa aaabbbaaaiiixxxooo:::
FFFiiiggguuurrraaa 111888
-6 -4 -2 0 2 4 6-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
10.70.910.960.98
0.99
0.996
0.998
1
0.70.910.960.980.99
0.996
0.998
1
123456
Pole-Zero Map
Real Axis
Imag
inar
y Ax
is
222555
AAA pppaaarrrtttiiirrr dddaaaííí,,, dddeeevvveee-‐-‐-ssseee cccrrriiiaaarrr uuummm cccooonnntttrrrooolllaaadddooorrr qqquuueee fffaaaçççaaa cccooommm qqquuueee ooo pppóóólllooo pppooosssiiitttiiivvvooo vvvááá pppaaarrraaa ooo eeeiiixxxooo nnneeegggaaatttiiivvvooo,,, eeennntttãããooo,,, pppooorrr ttteeennntttaaatttiiivvvaaasss fffoooiii ooobbbtttiiidddooo aaa FFFTTT dddooo cccooonnntttrrrooolllaaadddooorrr:::
TTTrrraaannnsssfffeeerrr fffuuunnnccctttiiiooonnn:::555000 sss +++ 111000000000
NNNooosss uuutttiiillliiizzzaaannndddooo mmmaaaiiisss uuummmaaa vvveeezzz dddaaa fffuuunnnçççãããooo fffeeeeeedddbbbaaaccckkk,,, cccrrriiiaaammmooosss uuummmaaa FFFTTT dddeee mmmaaalllhhhaaa fffeeeccchhhaaadddaaa pppaaarrraaa eeesssssseee sssiiisssttteeemmmaaa:::
FT_MF_aloc = feedback(FT_malha_aberta*FT_aloc,1)
Transfer function: 12 s + 240-------------------------0.1472 s^2 + 12 s + 236.2
DDDeeepppoooiiisss dddeee cccrrriiiaaadddaaa aaa nnnooovvvaaa FFFTTT,,, uuutttiiillliiizzzaaammmooosss aaa fffuuunnnçççãããooo pppzzzmmmaaappp,,, qqquuueee ooorrriiigggiiinnnaaa uuummm gggrrráááfffiiicccooo,,, pppaaarrraaa lllooocccaaallliiizzzaaarrr ooosss pppóóólllooosss eee zzzeeerrrooosss dddaaa nnnooovvvaaa fffuuunnnçççãããooo:::
FFFiiiggguuurrraaa 111999
AAAgggooorrraaa cccooommm eeesssssseee sssiiisssttteeemmmaaa mmmaaaiiisss eeessstttááávvveeelll,,, vvvaaammmooosss ppprrrooocccuuurrraaarrr aaasss rrreeessspppooossstttaaasss aaaooo iiimmmpppuuulllsssooo,,, dddeeegggrrraaauuu eee eeemmm fffrrreeeqqqüüüêêênnnccciiiaaa...
Pole-Zero Map
Real Axis
Imag
inar
y Ax
is
-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
10.9980.9990.9991
1
1
0.970.9940.9980.9990.9991
1
1
10203040
System: FT_MF_alocPole : -48.3Damping: 1Overshoot (%): 0Frequency (rad/sec): 48.3
System: FT_MF_alocPole : -33.2Damping: 1Overshoot (%): 0Frequency (rad/sec): 33.2
System: FT_MF_alocZero : -20Damping: 1Overshoot (%): 0Frequency (rad/sec): 20
0.970.994
222666
555...555...111... RRReeessspppooossstttaaa aaaooo IIImmmpppuuulllsssooo
fffiiiggguuurrraaa 222000
OOObbbssseeerrrvvvaaa-‐-‐-ssseee qqquuueee eeessssssaaa rrreeessspppooossstttaaa ééé tttãããooo rrrááápppiiidddaaa qqquuuaaannntttooo aaaooo dddooo cccooonnntttrrrooolllaaadddooorrr !IIIDDD...
555...555...222... RRReeessspppooossstttaaa aaaooo DDDeeegggrrraaauuu
FFFiiiggguuurrraaa 222111
DDDooosss cccooonnntttrrrooolllaaadddooorrreeesss,,, ééé ooo qqquuueee aaappprrreeessseeennntttaaa aaa rrreeessspppooossstttaaa mmmaaaiiisss rrrááápppiiidddaaa,,, ooobbbssseeerrrvvveee ooosss pppaaarrrâââmmmeeetttrrrooosss:::
0 0.05 0.1 0.15-10
0
10
20
30
40
50
60
70
80
90Impulse Response
Time (sec)
Ampl
itude
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Ampl
itude
222777
aaannnsss ===
RRRiiissseeeTTTiiimmmeee::: 000...000111888555SSSeeettttttllliiinnngggTTTiiimmmeee::: 000...111333333888SSSeeettttttllliiinnngggMMMiiinnn::: 000...999666333222SSSeeettttttllliiinnngggMMMaaaxxx::: 111...111444111666OOOvvveeerrrssshhhoooooottt::: 111222...333666888111UUUnnndddeeerrrssshhhoooooottt::: 000!eeeaaakkk::: 111...111444111666!eeeaaakkkTTTiiimmmeee::: 000...000555111000
555...555...333... RRReeessspppooossstttaaa eeemmm fffrrreeeqqqüüüêêênnnccciiiaaa ––– DDDiiiaaagggrrraaammmaaa dddeee BBBooodddeee
FFFiiiggguuurrraaa 222222
AAAtttuuuaaa mmmaaaiiisss uuummmaaa vvveeezzz cccooommmooo fffiiillltttrrrooo pppaaassssssaaa-‐-‐-bbbaaaiiixxxaaa aaatttééé sssuuuaaa fffrrreeeqqqüüüêêênnnccciiiaaa dddeee cccooorrrttteee dddeee 111000000 rrraaaddd///ssseeeggg ssseee tttooorrrnnnaaannndddooo ooo qqquuueee tttrrraaabbbaaalllhhhaaa eeemmm mmmaaaiiisss fffrrreeeqqqüüüêêênnnccciiiaaasss dddooo qqquuueee ooosss ooouuutttrrrooosss...
-25
-20
-15
-10
-5
0
5
Mag
nitu
de (d
B) System: FT_MF_alocFrequency (rad/sec): 100Magnitude (dB): -3
100
101
102
103
-90
-45
0
Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
222888
666... SSScccrrriiipppttt dddooo MMMAAATTTLLLAAABBB
% UNIVERSIDADE FEDERAL DO MARANHÃO% CENTRO DE CIÊNCIAS EXATAS E TECNOLOGIA% FRANCISCO DAS CHAGAS LINHARES SILVA JÚNIOR% CÓDIGO: EE07111-67% CONTROLE I% % CONTROLE DE UM SISTEMA CARRO-PÊNDULO INVERTIDO
clear allclc
% Parâmetros do Sistema
M = 1;m = 0.6;l = 0.4;I = 0.032;g = 9.80665;
x = [(M+m)*(I+m*l^2)-(m*l)^2]; % utilizado para simplificação do cáculo
num = [m*l];den = [x 0 -(M+m)*m*g*l];
% Função Trasferência de Malha AbertaFT_malha_aberta = tf(num,den)
% Resposta do Sistema em Malha aberta ao Impulsofigure (1)impulse (FT_malha_aberta)grid on
% Resposta do Sistema em Malha aberta ao Degraufigure(2)step(FT_malha_aberta)grid onparametros_degrau_malha_aberta = stepinfo(FT_malha_aberta)
% Resposta em Frequência do Sistema em Malha abertafigure(3)bode(FT_malha_aberta)grid on
% Aplicando o controlador PID
kp = 140;kd = 20;ki = 180;FT_controlador=tf([kd kp ki],[1 0]);
% Malha fechada do controle PIDFT_malha_fechada = feedback(FT_malha_aberta*FT_controlador,1)
t=0:0.01:3;
222999
% Resposta ao impulso do Sistema em Malha fechada do controle PIDfigure(4)impulse(FT_malha_fechada,t)grid on
% Resposta ao degrau do Sistema em Malha fechada do controle PIDfigure(5)step(FT_malha_fechada,t)grid onparametros_degrau_malha_fechada = stepinfo(FT_malha_fechada)
% Resposta em frequência do Sistema em Malha fechada do controle PIDfigure(6)bode(FT_malha_fechada)grid on
% Gráfico dos pólos e zerosfigure(7)pzmap(FT_malha_fechada)grid onnum2 = [4.8 33.6 43.2];den2 = [0.1472 4.8 29.83 43.2];[z p]=tf2zp(num2,den2)
% Controle Do Sistema com compensador lead-lag
a = 70; b = 1000; c = 6; d = 100;numc = [1 (a+c) a*c];denc = [1 (b+d) b*d];% Função transferência do compensadorFT_comp=tf(numc,denc)
% Grafico das raizes da convolucao da planta com o compensadorfigure(8)rlocus(FT_comp*FT_malha_aberta)
% Determinação do ganho Kfigure(9)rlocus(FT_comp*FT_malha_aberta)axis([-10 10 -10 10])%[k]=rlocfind(FT_comp*FT_malha_aberta)k=3.98e+003FT_MF_comp=feedback(FT_malha_aberta,k*FT_comp)
t2=0:0.01:50;
% Resposta ao impulso do Sistema em Malha fechada do compensador lead-lagfigure(10)impulse(FT_MF_comp,t2)grid on
% Resposta ao degrau do Sistema em Malha fechada do compensador lead-lagfigure(11)step(FT_MF_comp,t2)grid onstepinfo(FT_MF_comp)
333000
% Resposta em frequencia do Sistema em Malha fechada do compensador lead-lagfigure (12)bode(FT_MF_comp)grid on
% Controle por Alocação de Pólos
% determinacao dos polos em malha aberta[z p]=tf2zp(num,den)
% grafico dos polos em malha abertafigure(13)pzmap(FT_malha_aberta)grid on
% determinacao do ganho para deslocar os polosnumal = [50 1000];denal = [1];FT_aloc = tf(numal,denal)FT_MF_aloc = feedback(FT_malha_aberta*FT_aloc,1)
% grafico dos polos ja com eles deslocadosfigure(14)pzmap(FT_MF_aloc)grid on
% Resposta ao impulso do Sistema em Malha fechada com alocação de polosfigure(15)impulse(FT_MF_aloc)grid on
% Resposta ao degrau do Sistema em Malha fechada com alocação de polosfigure (16)step(FT_MF_aloc)grid onstepinfo(FT_MF_aloc)
% Resposta em frequencia do Sistema em Malha fechada com alocação de polosfigure(17)bode(FT_MF_aloc)grid on
333111
777... CCCooonnncccllluuusssãããooo
AAAtttrrraaavvvééésss dddeeesssssseee ppprrrooojjjeeetttooo dddeee cccooonnntttrrrooollleee,,, fffoooiii pppooossssssííívvveeelll pppooorrr eeemmm ppprrrááátttiiicccaaa,,, pppeeelllooo mmmeeennnooosss aaatttrrraaavvvééésss dddaaa sssiiimmmuuulllaaaçççãããooo ooo cccooonnnttteeeúúúdddooo eeessstttuuudddaaadddooo nnnaaa cccaaadddeeeiiirrraaa dddeee CCCooonnntttrrrooollleee III... EEEllleee tttaaammmbbbééémmm nnnooosss “““fffooorrrçççooouuu””” aaa eeessstttuuudddaaarrr mmmaaaiiisss aaa fffuuunnndddooo sssiiisssttteeemmmaaasss iiinnnssstttááávvveeeiiisss pppaaarrraaa pppooodddeeerrr eeennnfffiiimmm eeessstttaaabbbiiillliiizzzááá-‐-‐-lllooosss aaatttrrraaavvvééésss dddeee cccooonnntttrrrooolllaaadddooorrreeesss eee cccooommmpppeeennnsssaaadddooorrreeesss...
ÉÉÉ ccclllaaarrrooo qqquuueee ooo eeessstttuuudddooo dddeee cccooonnntttrrrooolllaaadddooorrreeesss !IIIDDD,,, cccooommmpppeeennnsssaaadddooorrreeesss llleeeaaaddd-‐-‐-lllaaaggg,,, eee aaalllooocccaaaçççãããooo dddeee pppóóólllooosss tttaaammmbbbééémmm fffooorrraaammm bbbaaassstttaaannnttteee aaappprrrooofffuuunnndddaaadddooosss,,, dddaaannndddooo pppaaarrraaa ooo eeessstttuuudddaaannnttteee uuummm bbbooommm eeemmmbbbaaasssaaammmeeennntttooo ppprrraaatttiiicccooo,,, pppoooiiisss aaatttrrraaavvvééésss dddaaasss sssiiimmmuuulllaaaçççõõõeeesss fffeeeiiitttaaasss eeemmm MMMAAATTTLLLAAABBB,,, fffoooiii pppooossssssííívvveeelll aaannnaaallliiisssaaarrr aaasss rrreeessspppooossstttaaasss dddeeesssssseee sssiiisssttteeemmmaaasss qqquuueee sssãããooo uuummmaaa aaappprrroooxxxiiimmmaaaçççãããooo dddooo qqquuueee aaacccooonnnttteeeccceee nnnaaa rrreeeaaallliiidddaaadddeee,,, dddeeeiiixxxaaannndddooo ooo aaallluuunnnooo mmmaaaiiisss mmmoootttiiivvvaaadddooo pppaaarrraaa aaappprrreeennndddeeerrr...
!ooorrrtttaaannntttooo,,, aaacccrrreeedddiiitttooo qqquuueee eeesssttteee ppprrrooojjjeeetttooo fffoooiii dddeee fffuuunnndddaaammmeeennntttaaalll iiimmmpppooorrrtttââânnnccciiiaaa pppaaarrraaa ooo aaappprrreeennndddiiizzzaaadddooo iiinnnttteeellleeeccctttuuuaaalll dddooosss aaallluuunnnooosss... EEE tttaaammmbbbééémmm aaacccrrreeedddiiitttooo qqquuueee ssseeeuuusss rrreeesssuuullltttaaadddooosss fffooorrraaammm bbbaaassstttaaannnttteee sssaaatttiiisssfffaaatttóóórrriiiooosss...
333222
888... RRReeefffeeerrrêêênnnccciiiaaasss
hhhttttttppp::://////wwwwwwwww...dddeeelll...uuufffmmmsss...bbbrrr///DDDEEELLL///pppeeettt///aaapppooossstttiiilllaaasss///MMMAAATTTLLLAAABBB///CCCooonnntttrrrooollleee%%%222000cccooommm%%%222000MMMAAATTTLLLAAABBB...dddoooccchhhttttttppp::://////wwwwwwwww...uuufffsssmmm...bbbrrr///gggeeepppoooccc///rrreeennneeesss///TTTeeemmmppplllaaattteeesss///aaarrrqqquuuiiivvvooosss///eeelllccc111000333111///EEELLLCCC111000333111...LLL777...333...pppdddfffhhhttttttppp::://////wwwwwwwww...mmmeee...cccmmmuuu...eeeddduuu///ccctttmmmsss///cccooonnntttrrrooolllsss///ccctttmmmsss///eeexxxaaammmpppllleeesss///pppeeennnddd///iiinnnvvvrrrlll...hhhtttmmmhhhttttttppp::://////wwwwwwwww...mmmeee...cccmmmuuu...eeeddduuu///ccctttmmmsss///cccooonnntttrrrooolllsss///ccctttmmmsss///eeexxxaaammmpppllleeesss///pppeeennnddd///iiinnnvvvpppiiiddd...hhhtttmmmhhhttttttppp::://////mmmeeemmmbbbeeerrrsss...fffooorrrtttuuunnneeeccciiitttyyy...cccooommm///cccaaammmuuusssppp///rrreeelllaaatttooorrriiiooosss///cccooonnntttrrrooollleee222///TTTrrraaabbbaaalllhhhooo%%%222000dddeee%%%222000CCCooonnntttrrrooollleee...dddooocccaaadddmmm-‐-‐-nnneeettt-‐-‐-aaa...uuunnniiifffeeeiii...eeeddduuu...bbbrrr///ppphhhlll///pppdddfff///000000333000777111444...pppdddfffBBBOOOLLLTTTOOONNN,,, WWW... EEEnnngggeeennnhhhaaarrriiiaaa dddeee CCCooonnntttrrrooollleee,,, MMMaaakkkrrrooonnn BBBooooookkksss dddooo BBBrrraaasssiiilll,,, SSSãããooo !aaauuulllooo,,, 111999999333...FFFRRRAAANNNKKKLLLIIINNN,,, GGG... FFF...;;; !OOOWWWEEELLLLLL,,, JJJ... DDD...;;; EEEMMMAAAMMMIII-‐-‐-NNNAAAEEEIIINNNIII,,, AAA... FFFeeeeeedddbbbaaaccckkk CCCooonnntttrrrooolll ooofff DDDyyynnnaaammmiiiccc SSSyyysssttteeemmmsss,,, AAAddddddiiisssooonnn-‐-‐-WWWeeesssllleeeyyy !uuubbbllliiissshhhiiinnnggg CCCooommmpppaaannnyyy,,,222ªªª EEEdddiiitttiiiooonnn,,, MMMaaassssssaaaccchhhuuussseeettttttsss,,, 111999999111...OOOGGGAAATTTAAA,,, KKK... EEEnnngggeeennnhhhaaarrriiiaaa dddeee CCCooonnntttrrrooollleee MMMooodddeeerrrnnnooo,,, !eeeaaarrrsssooonnn EEEddduuucccaaatttiiiooonnn dddooo BBBrrraaasssiiilll,,, SSSãããooo !aaauuulllooo,,, 222000000333...