Well-Posedness of Feedback Loop:

Transcription

1 ntena Stabiity We-oedne of Feedback Loop: onide the foowing feedback ytem - u u p d i d y Let be both pope tanfe function. Howeve u n d di 3 3 ote that the tanfe function fom the extena igna n d d to u i ae not pope. Hence the feedback ytem i not phyicay eaizabe. n Sping E854 - GGZ ntena Stabiity age

2 ntena Stabiity We-oedne of Feedback Loop: efinition 5. feedback ytem i aid to be we ae we - defined pope. - poed if a coed - oop tanfe matice Let : egoup the extena input igna into the feedback oop a. w w egoup input igna a e e w e w e Lemma 5. Figue 5. The feedback ytem above i we - poed if i invetibe. ony if Sping E854 - GGZ ntena Stabiity age

3 Sping E854 - GGZ age 3 ntena Stabiity ntena Stabiity ntena Stabiity We-oedne of Feedback Loop: The ytem can be epeented by e w e e w e oof: ote: invetibe invetibe i invetibe. the tanfe function to that contant tem of thi i equivaent ote exit i pope. to that poedne i equivaent - we Thu Then equation can be ewiten a. w w e invetibe invetibe eaization Fo tate pace

4 Sping E854 - GGZ age 4 ntena Stabiity ntena Stabiity ntena Stabiity ntena Stabiity: RH e e w w to beong to fom the tanfe matix The ytem defined in Figue 5. i aid to be intenay tabe if efinition 5. Then tabiity. Fo exampe: intena fou tanfe matice fo to tet each of neceay ufficient ote that it i w w e e

5 ntena Stabiity ntena Stabiity: Remak 5. ntenatabiity i a that a igna in a ytem bounded povided that the injected igna at any ocation ae bounded. ooay 5. Suppoe RH baic equiement fo a if it i we- poed oof: RH pactica feedback ytem.t guaantee.then the ytem in Figue 5. i intenaytabe if RH. we - poed ony RH. we - poed RH ince invetibe RH Sping E854 - GGZ ntena Stabiity age 5

6 ntena Stabiity ntena Stabiity: ooay 5.3 Suppoe RH if it i we- poed.then the ytem in Figue 5. i intenaytabe if RH. ony ooay 5.4 Suppoe if RH ight haf pane. RH. Then the ytem in Figue 5. i intenay tabe if ony o equivaenty det ha no zeo in the coed Theoem 5.5: The ytem i intenay tabe if ony if it i we- poed i the numbe of open hp poe of n poe of epectivey; ii i tabe. p n k n p n k ae open hp Sping E854 - GGZ ntena Stabiity age 6

7 Sping E854 - GGZ age 7 ntena Stabiity ntena Stabiity ntena Stabiity ntena Stabiity: [ ] [ ] i tabe See obem 5. the ytem i intenay tabe if Hence whee oof: then ume that

8 ntena Stabiity ntena Stabiity: ume that the ytem i intena tabe then we have ii note that ince i tabiizabe i detectabe iff [ ] ae. O theei no untabe unobevabe contoabe mode o in othe wod thee i no untabe poe/zeo canceation of. RH ; Sping E854 - GGZ ntena Stabiity age 8

9 Sping E854 - GGZ age 9 ntena Stabiity ntena Stabiity ntena Stabiity ntena Stabiity: det whee det then tanfe matice : be Let n exampe: no poe/zeo canceation fo thee i n thi cae tabe intena not have zeo at coed RH doe ote :det

10 ntena Stabiity opime Factoization ove RH : onide two poynomia aid to be copime if no common zeo. m n with fo exampe ea coefficient ae thei geatet common divio io equivaenty thee i Two poynomia n m ae copime if ony if thee exit x y uch that x n y m ezout identity. poynomia Simiaiy two tanfe function m n in RH if thee exit x y RH uch that x n y m ae aid to be copime ove RH Sping E854 - GGZ ntena Stabiity age

11 ntena Stabiity opime Factoization ove RH : efinition 5.3 Two matice RH coumn if numbe of thee exit matice [ X Y ] Simiaiy two matice RH ow if ae ight copime ove thee exit matice X [ ] Y ae eft copime ove Y uch that X X Y. Y ote that thee definition ae equivaent to ay that matice ight invetibe epectivey. X X in RH RH Y in if. RH they have the ame numbe of RH if uch that they have the ame [ ] ae eft Sping E854 - GGZ ntena Stabiity age

12 ntena Stabiity opime Factoization ove RH : Right eft copime factoization Let be a pope ea matix. ight copime factoization cf of whee ae ight copime ove RH. i a factoization Simiaiy eft copime factoization cf of ae eft copime ove RH. i a factoization whee matix R copime factoization X Y RH i aid to have doube copime factoization if uch that a eft copime factoization X Y Y X. thee exit a ight X Y Sping E854 - GGZ ntena Stabiity age

13 ntena Stabiity opime Factoization ove RH : Theoem 5.6 Suppoe i a pope ea ationa matix i a tabiizabe detectabe eaization. Let L ae both tabe define Then i atified. Y X F F F F L be uch that F L L L X Y F ae cf cf epectivey fiuthemoe equation 5.7 Sping E854 - GGZ ntena Stabiity age 3 L

14 Sping E854 - GGZ age 4 ntena Stabiity ntena Stabiity ntena Stabiity oof: [ ] [ ] L F F F L F F L F F L F L L F L G G X Y Y X F F L F X Y G F L L L Y X G G G G o ote that ote that Then et ytem connection Reca eia opime Factoization ove RH :

15 ntena Stabiity opime Factoization ove RH : oof: Since L L F L F F [ ] [ ] Theefoe { } L F L F F { } L F F L L F X Y Y F [ ] [ L] X ɶ ɶ Remak 5. ote that if i tabe then we can take X X Y Y. Sping E854 - GGZ ntena Stabiity age 5

16 ntena Stabiity opime Factoization ove RH : Remak 5.3 The copime factoization of conto intepetation. Fo intance natuay fom changing conto a tanfe ight copime factoization come out vaiabe function can be given a feedback by a tate feedback. onide the tate - pace equation fo a pant xɺ x u : y x u uch that F i tabe. Then Then u v uch that y xɺ u y with tate feedback F x v Fx v F F F x v y v u ;that i u : F F Fx v. Sping E854 - GGZ ntena Stabiity age 6

17 ntena Stabiity opime Factoization ove RH : Reca the cf' cf' of U U Lemma 5.7 onide the ytem decibed in Fig 5.. The foowing condition ae equivaent i The feedback ytem i intenay tabe. U ii i invetibe in RH U iii i invetibe in RH iv U i invetibe in RH v U i invetibe in RH. : Sping E854 - GGZ ntena Stabiity age 7

18 Sping E854 - GGZ age 8 ntena Stabiity ntena Stabiity ntena Stabiity oof: opime Factoization ove RH : ae copime ince ote that the matice o that ow o equivaenty ote that the ytem i intenay tabe if U U U U RH RH

19 ntena Stabiity opime Factoization ove RH : ae copime thee exit matice.t. U X Y X Y RH X Y X Y U U Y X Y X X Y Y X Y RH. Thi pove the equivaence of. ote X X Y that 3 can be poved in a imia way. Sping E854 - GGZ ntena Stabiity age 9

20 ntena Stabiity opime Factoization ove RH : ondition 4 5 ae impied by condition 3 by ɶ Uɶ U ɶ U ɶ ɶ ɶ ɶ U ɶ Since eft-h ide i invetibe in RH o in ight-h ide. Theefoe condition 4 5 ae atified. We need to how eithe condition 4 o 5 impie condition. onide ɶ Uɶ ɶ ɶ ɶ U ɶ Uɶ if RH i.e. ɶ U ɶ RH o condition 5 i atified. RH Sping E854 - GGZ ntena Stabiity age

21 Sping E854 - GGZ age ntena Stabiity ntena Stabiity ntena Stabiity ooay 5.8 opime Factoization ove RH : F L L L U F F L F U L F L F U U RH U U U U RH tate pace eaization fo thee matice can be given by et of paticua Then a ae tabe. be uch that et Futhemoe uch that with Then thee exit a contoe. ove cf coeponding cf be the matix ationa be a pope ea Let

22 ntena Stabiity opime Factoization ove RH : oce of finding a copime factoization fo a caa tanfe function i uppoe num den ii et α be any tabe poynomia with the ame ode a den num den iii Then n / m whee n m i a factoization α α copime of. Sping E854 - GGZ ntena Stabiity age

23 ntena Stabiity opime Factoization ove RH : Scaa Exampe: Let eect α 3. Then 3 n whee n m m 3 i a copime factoization of. To find x y uch that x n y m. onide a tabiization contoe fo : Then u / v whee u v i a copime factoization Then m v n u x u / β y v / β : β Sping E854 - GGZ ntena Stabiity age 3

24 ntena Stabiity opime Factoization ove RH : Sytematic poce of finding a copime factoization i Let be a tabiizabe detectabe eaization of ii Find a tabiization tate feedback gain F F tabe uing eithe poe aignment o LQR deign iii Find a tabe obeve gain L L tabe uing eithe poe aignment o aman fite deign iv Let F L L L L U U F ɶ ɶ & F ɶ ɶ F whee ɶ ɶ ɶ U ɶ ɶ U ɶ Sping E854 - GGZ ntena Stabiity age 4

25 ntena Stabiity opime Factoization ove RH : O exampe: 4 3 Let be a tabiizabe 3 4 detectabe eaization. The tabiization gain F can be found by..333 >> F -pace[- -3] F.67 tabe etimation gain can be found by >> L -q''eyeeye L eig L { ±.3454 j} Sping E854 - GGZ ntena Stabiity age 5

26 ntena Stabiity opime Factoization ove RH : Then U F F F L & U L F whee U ɶ Uɶ ɶ L ɶ L Sping E854 - GGZ ntena Stabiity age 6